Open Access

Dirichlet problem in Banach spaces: the bound sets approach

Boundary Value Problems20132013:25

DOI: 10.1186/1687-2770-2013-25

Received: 19 October 2012

Accepted: 16 January 2013

Published: 11 February 2013

Abstract

The existence and localization result is obtained for a multivalued Dirichlet problem in a Banach space. The upper-Carathéodory and Marchaud right-hand sides are treated separately because in the latter case, the transversality conditions derived by means of bounding functions can be strictly localized on the boundaries of bound sets.

MSC:34A60, 34B15, 47H04.

Keywords

Dirichlet problem bounding functions solutions in a given set condensing multivalued operators

1 Introduction

The Dirichlet problem and its special case with homogeneous boundary conditions, usually called the Picard problem, belong to the most frequently studied boundary value problems. A lot of results concerning the standard problem for scalar second-order ordinary differential equations were generalized in various directions.

In Euclidean spaces, besides many extensions to vector equations, vector inclusions were under consideration, e.g., in [14]. In abstract spaces, usually in Banach and Hilbert spaces, equations, e.g., in [511] and inclusions, e.g., in [9, 12, 13] were treated.

Sadovskii’s or Darbo’s fixed point theorems, jointly with the usage of a measure of noncompactness, were applied in [5, 8, 9, 11]. Kakutani’s or Ky Fan’s fixed point theorems were applied with the upper and lower solutions technique in [9] and with a measure of noncompactness in [13]. On the other hand, continuation principles were employed in [2, 4, 7].

The main aim of our present paper is an extension of the finite-dimensional results in [2, 4] into infinite-dimensional ones. We were also stimulated by the work of Jean Mawhin in [7], where degree arguments were applied to the Dirichlet problem in a Hilbert space probably for the first time, and in [14], where a bound sets approach was systematically developed. Hence, besides these two approaches, our extension consists in the consideration of differential inclusions in rather general Banach spaces and the usage of a measure of noncompactness. Similar results were already obtained in an analogous way by ourselves for Floquet problems in [1518].

Besides the existence, the localization of solutions will be obtained in our main theorems (see Theorem 5.1 and Theorem 5.2). Unlike in [10], where the solutions belong to a positively invariant set, in our paper, some trajectories can escape from the prescribed set of candidate solutions. Moreover, the associated bound set need not be compact as in [10]. Similarly, the main difference between our results and those in [9, 13] consists in the application of a continuation principle jointly with a bound sets approach, which allows us to check fixed point free boundaries of given bound sets. This, in particular, means that, unlike in [9, 13], some trajectories can again escape from the prescribed set of candidate solutions in a transversal way.

Let E be a Banach space (with the norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq1_HTML.gif) satisfying the Radon-Nikodym property (e.g., reflexivity) and let us consider the Dirichlet boundary value problem (b.v.p.)
x ¨ ( t ) F ( t , x ( t ) , x ˙ ( t ) ) for a.a.  t [ 0 , T ] , x ( 0 ) = x ( T ) = 0 , } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ1_HTML.gif
(1)

where F : [ 0 , T ] × E × E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq2_HTML.gif is an upper-Carathéodory mapping or a globally upper semicontinuous mapping with compact, convex values (for the related definitions, see Section 2).

The main purpose of the present paper is to prove the existence of a Carathéodory solution x A C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq3_HTML.gif to problem (1) in a given set Q. This will be achieved by means of a suitable continuation principle. The crucial condition of the continuation principle described in Section 3 consists in guaranteeing the fixed point free boundary of Q w.r.t. an admissible homotopical bridge starting from (1) (see condition (v) in Proposition 3.1 below). This requirement will be verified by means of Lyapunov-like bounding functions, i.e., via a bound sets technique. That is also why the whole Section 4 is devoted to this technique applied to Dirichlet problem (1). We will distinguish two cases, namely when F is an upper-Carathéodory mapping and when F is globally upper semicontinuous (i.e., a Marchaud mapping). Unlike in the first case, the second one allows us to apply bounding functions which can be strictly localized at the boundaries of given bound sets.

2 Preliminaries

Let E be a Banach space having the Radon-Nikodym property (see, e.g., [[19], pp.694-695]), i.e., if for every finite measure space ( M , Σ , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq4_HTML.gif and every vector measure m : Σ E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq5_HTML.gif of bounded variation, which is absolutely continuous w.r.t. μ, we can find a Bochner integrable function f : M E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq6_HTML.gif such that
m ( C ) = C f ( ν ) d μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equa_HTML.gif

for each C Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq7_HTML.gif. Let [ 0 , T ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq8_HTML.gif be a closed interval. By the symbol L 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq9_HTML.gif, we will mean the set of all Bochner integrable functions x : [ 0 , T ] E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq10_HTML.gif. For the definition and properties, see, e.g., [[19], pp.693-701].

The symbol A C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq11_HTML.gif will denote the set of functions x : [ 0 , T ] E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq12_HTML.gif whose first derivative x ˙ ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq13_HTML.gif is absolutely continuous. Then x ¨ L 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq14_HTML.gif and the fundamental theorem of calculus (the Newton-Leibniz formula) holds (see, e.g., [[15], pp.243-244], [[19], pp.695-696]). In the sequel, we will always consider A C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq15_HTML.gif as a subspace of the Banach space C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq16_HTML.gif.

Given C E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq17_HTML.gif and ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq18_HTML.gif, the symbol B ( C , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq19_HTML.gif will denote, as usually, the set C + ε B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq20_HTML.gif, where B is the open unit ball in E, i.e., B = { x E x < 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq21_HTML.gif.

We will also need the following definitions and notions from multivalued analysis. Let X, Y be two metric spaces. We say that F is a multivalued mapping from X to Y (written F : X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq22_HTML.gif) if for every x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq23_HTML.gif, a nonempty subset F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq24_HTML.gif of Y is given. We associate with F its graph Γ F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq25_HTML.gif, the subset of X × Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq26_HTML.gif, defined by
Γ F : = { ( x , y ) X × Y y F ( x ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equb_HTML.gif

A multivalued mapping F : X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq27_HTML.gif is called upper semicontinuous (shortly, u.s.c.) if for each open subset U Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq28_HTML.gif, the set { x X F ( x ) U } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq29_HTML.gif is open in X.

A multivalued mapping F : X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq27_HTML.gif is called compact if the set F ( X ) = x X F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq30_HTML.gif is contained in a compact subset of Y; it is called quasi-compact if it maps compact sets onto relatively compact sets; and completely continuous if it maps bounded sets onto relatively compact sets.

We say that a multivalued mapping F : [ 0 , T ] Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq31_HTML.gif with closed values is a step multivalued mapping if there exists a finite family of disjoint measurable subsets I k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq32_HTML.gif, k = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq33_HTML.gif such that [ 0 , T ] = I k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq34_HTML.gif and F is constant on every I k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq32_HTML.gif. A multivalued mapping F : [ 0 , T ] Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq31_HTML.gif with closed values is called strongly measurable if there exists a sequence of step multivalued mappings { F n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq35_HTML.gif such that d H ( F n ( t ) , F ( t ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq36_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq37_HTML.gif for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif, where d H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq39_HTML.gif stands for the Hausdorff distance.

It is well known that if Y is a Banach space, then a strongly measurable mapping F : [ 0 , T ] Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq31_HTML.gif with compact values possesses a single-valued strongly measurable selection (see, e.g., [12, 20]).

A multivalued mapping F : [ 0 , T ] × X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq40_HTML.gif is called an upper-Carathéodory mapping if the map F ( , x ) : [ 0 , T ] Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq41_HTML.gif is strongly measurable for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq23_HTML.gif, the map F ( t , ) : X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq42_HTML.gif is u.s.c. for almost all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq43_HTML.gif and the set F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq44_HTML.gif is compact and convex for all ( t , x ) [ 0 , T ] × X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq45_HTML.gif.

Let us note that if X , Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq46_HTML.gif are Banach spaces, then an upper-Carathéodory mapping F : [ 0 , T ] × X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq40_HTML.gif is weakly superpositionally measurable, i.e., that for each continuous g : [ 0 , T ] X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq47_HTML.gif, the composition F ( , g ( ) ) : [ 0 , T ] Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq48_HTML.gif possesses a single-valued measurable selection (see, e.g., [12, 20]).

A multivalued mapping F : [ 0 , T ] × X × X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq49_HTML.gif is called Lipschitzian in ( x , y ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq50_HTML.gif if there exists a constant L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq51_HTML.gif such that
d H ( F ( t , x 1 , y 1 ) , F ( t , x 2 , y 2 ) ) L ( x 1 x 2 + y 1 y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equc_HTML.gif

for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif and for all x 1 , x 2 , y 1 , y 2 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq52_HTML.gif.

For more details concerning multivalued analysis, see, e.g., [12, 15, 20, 21].

In the sequel, the measure of noncompactness will also be employed.

Definition 2.1 Let N be a partially ordered set, E be a Banach space and let P ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq53_HTML.gif denote the family of all nonempty subsets of E. A function β : P ( E ) N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq54_HTML.gif is called a measure of noncompactness (m.n.c.) in E if β ( co Ω ¯ ) = β ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq55_HTML.gif for all Ω P ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq56_HTML.gif, where co Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq57_HTML.gif denotes the closed convex hull of Ω.

An m.n.c. β is called:
  1. (i)

    monotone if β ( Ω 1 ) β ( Ω 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq58_HTML.gif for all Ω 1 Ω 2 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq59_HTML.gif,

     
  2. (ii)

    nonsingular if β ( { x } Ω ) = β ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq60_HTML.gif for all x E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq61_HTML.gif and Ω E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq62_HTML.gif,

     
  3. (iii)

    invariant with respect to the union with compact sets if β ( K Ω ) = β ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq63_HTML.gif for every relatively compact K E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq64_HTML.gif and every Ω E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq62_HTML.gif,

     
  4. (iv)

    regular when β ( Ω ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq65_HTML.gif if and only if Ω is relatively compact,

     
  5. (v)

    algebraically semi-additive if β ( Ω 1 + Ω 2 ) β ( Ω 1 ) + β ( Ω 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq66_HTML.gif for all Ω 1 , Ω 2 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq67_HTML.gif.

     

Definition 2.2 An m.n.c. β with values in a cone of a Banach space has the semi-homogeneity property if β ( t Ω ) = | t | β ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq68_HTML.gif for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq69_HTML.gif and all Ω E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq62_HTML.gif.

It is obvious that an m.n.c. which is invariant with respect to the union with compact sets is also nonsingular.

The typical example of an m.n.c. is the Hausdorff measure of noncompactness γ defined, for all Ω E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq62_HTML.gif, by
γ ( Ω ) : = inf { ε > 0 | x 1 , , x n E : Ω i = 1 n B ( { x i } , ε ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equd_HTML.gif

The Hausdorff measure of noncompactness is monotone, nonsingular, algebraically semi-additive and has the semi-homogeneity property.

Let { f n } L 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq70_HTML.gif be such that f n ( t ) α ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq71_HTML.gif, γ ( { f n ( t ) } ) c ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq72_HTML.gif for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif, all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq73_HTML.gif and suitable α , c L 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq74_HTML.gif, then (cf. [20])
γ ( { 0 T f n ( t ) d t } ) 2 0 T c ( t ) d t for a.a.  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ2_HTML.gif
(2)
Moreover, for all subsets Ω of E (see, e.g., [18]),
γ ( λ [ 0 , 1 ] λ Ω ) γ ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ3_HTML.gif
(3)
Let us now introduce the function
μ ( Ω ) : = max { w n } n Ω ( sup t [ 0 , T ] [ γ ( { w n ( t ) } n ) + γ ( { w ˙ n ( t ) } n ) ] , mod C ( { w n } n ) + mod C ( { w ˙ n } n ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ4_HTML.gif
(4)

defined on the bounded set Ω C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq75_HTML.gif, where the ordering is induced by the positive cone in R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq76_HTML.gif and where mod C ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq77_HTML.gif denotes the modulus of continuity of a subset Ω C ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq78_HTML.gif.a Such a μ is an m.n.c. in C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq16_HTML.gif, as shown in the following lemma (proven in [16]), where the properties of μ will be also discussed.

Lemma 2.1 The function μ given by (4) defines an m.n.c. in C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq16_HTML.gif; such an m.n.c. μ is monotone, invariant with respect to the union with compact sets and regular.

The m.n.c. μ defined by (4) will be used in order to solve problem (1) (cf. Theorem 5.1).

Definition 2.3 Let E be a Banach space and X E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq79_HTML.gif. A multivalued mapping F : X E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq80_HTML.gif with compact values is called condensing with respect to an m.n.c. β (shortly, β-condensing) if for every Ω X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq81_HTML.gif such that β ( F ( Ω ) ) β ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq82_HTML.gif, it holds that Ω is relatively compact.

A family of mappings G : X × [ 0 , 1 ] E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq83_HTML.gif with compact values is called β-condensing if for every Ω X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq84_HTML.gif such that β ( G ( Ω × [ 0 , 1 ] ) ) β ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq85_HTML.gif, it holds that Ω is relatively compact.

The following convergence result will be also employed.

Lemma 2.2 (cf. [[15], Lemma III.1.30])

Let E be a Banach space and assume that the sequence of absolutely continuous functions x k : [ 0 , T ] E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq86_HTML.gif satisfies the following conditions:
  1. (i)

    the set { x k ( t ) | k N } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq87_HTML.gif is relatively compact for every t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif,

     
  2. (ii)

    there exists α L 1 ( [ 0 , T ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq89_HTML.gif such that x ˙ k ( t ) α ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq90_HTML.gif for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif and for all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq91_HTML.gif,

     
  3. (iii)

    the set { x ˙ k ( t ) | k N } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq92_HTML.gif is weakly relatively compact for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif.

     
Then there exists a subsequence of { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq93_HTML.gif (for the sake of simplicity denoted in the same way as the sequence) converging to an absolutely continuous function x : [ 0 , T ] E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq94_HTML.gif in the following way:
  1. 1.

    { x k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq93_HTML.gif converges uniformly to x in C ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq95_HTML.gif,

     
  2. 2.

    { x ˙ k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq96_HTML.gif converges weakly in L 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq97_HTML.gif to x ˙ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq98_HTML.gif.

     

The following lemma is well known when the Banach spaces E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq99_HTML.gif and E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq100_HTML.gif coincide (see, e.g., [[22], p.88]). The present slight modification for E 1 E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq101_HTML.gif was proved in [23].

Lemma 2.3 Let [ 0 , T ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq102_HTML.gif be a compact interval, let E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq99_HTML.gif, E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq103_HTML.gif be Banach spaces and let F : [ 0 , T ] × E 1 E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq104_HTML.gif be a multivalued mapping satisfying the following conditions:
  1. (i)

    F ( , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq105_HTML.gif has a strongly measurable selection for every x E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq106_HTML.gif,

     
  2. (ii)

    F ( t , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq107_HTML.gif is u.s.c. for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif,

     
  3. (iii)

    the set F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq44_HTML.gif is compact and convex for all ( t , x ) [ 0 , T ] × E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq108_HTML.gif.

     
Assume in addition that for every nonempty, bounded set Ω E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq109_HTML.gif, there exists ν = ν ( Ω ) L 1 ( [ 0 , T ] , ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq110_HTML.gif such that
F ( t , x ) ν ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Eque_HTML.gif

for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq111_HTML.gif and every x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq112_HTML.gif. Let us define the Nemytskiǐ operator N F : C ( [ 0 , T ] , E 1 ) L 1 ( [ 0 , T ] , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq113_HTML.gif in the following way: N F ( x ) : = { f L 1 ( [ 0 , T ] , E 2 ) f ( t ) F ( t , x ( t ) ) , a.e. on [ 0 , T ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq114_HTML.gif for every x C ( [ 0 , T ] , E 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq115_HTML.gif. Then, if sequences { x k } C ( [ 0 , T ] , E 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq116_HTML.gif and { f k } L 1 ( [ 0 , T ] , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq117_HTML.gif, f k N F ( x k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq118_HTML.gif, k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq119_HTML.gif, are such that x k x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq120_HTML.gif in C ( [ 0 , T ] , E 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq121_HTML.gif and f k f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq122_HTML.gif weakly in L 1 ( [ 0 , T ] , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq123_HTML.gif, then f N F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq124_HTML.gif.

3 Continuation principle

The proof of the main result (cf. Theorem 5.1 below) will be based on the combination of a bound sets technique together with the following continuation principle developed in [16].

Proposition 3.1 Let us consider the general multivalued b.v.p.
x ¨ ( t ) φ ( t , x ( t ) , x ˙ ( t ) ) for a.a. t [ 0 , T ] , x S , } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ5_HTML.gif
(5)
where φ : [ 0 , T ] × E × E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq125_HTML.gif is an upper-Carathéodory mapping and S A C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq126_HTML.gif. Let H : [ 0 , T ] × E × E × E × E × [ 0 , 1 ] E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq127_HTML.gif be an upper-Carathéodory mapping such that
H ( t , c , d , c , d , 1 ) φ ( t , c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ6_HTML.gif
(6)
for all ( t , c , d ) [ 0 , T ] × E × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq128_HTML.gif. Moreover, assume that the following conditions hold:
  1. (i)
    There exist a closed set S 1 S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq129_HTML.gif and a closed, convex set Q C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq130_HTML.gif with a nonempty interior IntQ such that each associated problem
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equf_HTML.gif
     
where q Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq131_HTML.gif and λ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq132_HTML.gif, has a nonempty, convex set of solutions (denoted by T ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq133_HTML.gif).
  1. (ii)
    For every nonempty, bounded set Ω E × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq134_HTML.gif, there exists ν Ω L 1 ( [ 0 , T ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq135_HTML.gif such that
    H ( t , x , y , q ( t ) , q ˙ ( t ) , λ ) ν Ω ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equg_HTML.gif
     
for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif and all ( x , y ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq136_HTML.gif, q Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq131_HTML.gif and λ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq137_HTML.gif.
  1. (iii)

    The solution mapping T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq138_HTML.gif is quasi-compact and μ-condensing with respect to a monotone and nonsingular m.n.c. μ defined on C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq16_HTML.gif.

     
  2. (iv)

    For each q Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq131_HTML.gif, the set of solutions of the problem P ( q , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq139_HTML.gif is a subset of IntQ, i.e., T ( q , 0 ) Int Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq140_HTML.gif for all q Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq131_HTML.gif.

     
  3. (v)

    For each λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq141_HTML.gif, the solution mapping T ( , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq142_HTML.gif has no fixed points on the boundary ∂Q of Q.

     

Then the b.v.p. (5) has a solution in Q.

The proof of the continuation principle is based on the fact that the family P ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq143_HTML.gif of problems depending on two parameters q Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq131_HTML.gif and λ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq132_HTML.gif is associated to the original b.v.p. (5). This family is defined in such a way that if T : Q × [ 0 , 1 ] A C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq144_HTML.gif is its corresponding solution mapping, then all fixed points of the map T ( , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq145_HTML.gif are solutions of (5) (see condition (6)).

4 Bound sets technique

The continuation principle formulated in Proposition 3.1 requires, in particular, the existence of a suitable set Q A C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq146_HTML.gif of candidate solutions. The set Q should satisfy the transversality condition (v), i.e., it should have a fixed-point free boundary with respect to the solution mapping T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq138_HTML.gif. Since the direct verification of the transversality condition is usually a difficult task, we will devote this section to a bound sets technique which can be used for guaranteeing such a condition. For this purpose, we will define the set Q as Q : = C 1 ( [ 0 , T ] , K ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq147_HTML.gif, where K is nonempty and open in E and K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq148_HTML.gif denotes its closure.

Hence, let us consider the Dirichlet boundary value problem (1) and let V : E R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq149_HTML.gif be a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq150_HTML.gif-function satisfying

(H1) V | K = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq151_HTML.gif,

(H2) V ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq152_HTML.gif for all x K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq153_HTML.gif.

Definition 4.1 A nonempty open set K E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq154_HTML.gif is called a bound set for the b.v.p. (1) if every solution x of (1) such that x ( t ) K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq155_HTML.gif for each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif does not satisfy x ( t ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq156_HTML.gif for any t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq157_HTML.gif.

Let E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq158_HTML.gif be the Banach space dual to E and let us denote by , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq159_HTML.gif the pairing (the duality relation) between E and E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq158_HTML.gif, i.e., for all Φ E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq160_HTML.gif and x E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq161_HTML.gif, we put Φ ( x ) : = Φ , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq162_HTML.gif. The proof of the following proposition is quite analogous to the finite-dimensional case considered in [4]. Nevertheless, for the sake of completeness, we present it here, too.

Proposition 4.1 Let K E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq163_HTML.gif be an open set such that 0 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq164_HTML.gif and F : [ 0 , T ] × E × E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq165_HTML.gif be an upper-Carathéodory mapping. Assume that the function V C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq166_HTML.gif has a locally Lipschitzian Fréchet derivative V ˙ x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq167_HTML.gif and satisfies conditions (H1) and (H2). Suppose, moreover, that there exists ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq168_HTML.gif such that, for all x K ¯ B ( K , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq169_HTML.gif, t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq170_HTML.gif and y E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq171_HTML.gif, at least one of the following conditions:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ7_HTML.gif
(7)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ8_HTML.gif
(8)

holds for all w F ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq172_HTML.gif. Then K is a bound set for the Dirichlet problem (1).

Proof Let x : [ 0 , T ] K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq173_HTML.gif be a solution of problem (1). We assume, by a contradiction, that there exists t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq174_HTML.gif such that x ( t ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq175_HTML.gif. The point t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq176_HTML.gif must lie in ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq177_HTML.gif according to the Dirichlet boundary conditions and the fact that 0 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq164_HTML.gif.

Since V ˙ x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq178_HTML.gif is locally Lipschitzian, there exist a neighborhood U of x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq179_HTML.gif and a constant L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq180_HTML.gif such that V ˙ | U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq181_HTML.gif is Lipschitzian with a constant L. Let δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq182_HTML.gif be such that x ( t ) U B ( K , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq183_HTML.gif for each t [ t δ , t + δ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq184_HTML.gif.

In order to get the desired contradiction, let us define the function g : [ 0 , T ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq185_HTML.gif as the composition g ( t ) : = ( V x ) ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq186_HTML.gif. According to the regularity properties of x and V, g C 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq187_HTML.gif. Since g ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq188_HTML.gif and g ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq189_HTML.gif for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif, t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq190_HTML.gif is a local maximum point for g. Therefore, g ˙ ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq191_HTML.gif. Moreover, there exist points t ( t δ , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq192_HTML.gif, t ( t , t + δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq193_HTML.gif such that g ˙ ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq194_HTML.gif and g ˙ ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq195_HTML.gif.

Since g ˙ ( t ) = V ˙ x ( t ) , x ˙ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq196_HTML.gif, where V ˙ x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq197_HTML.gif is locally Lipschitzian and x ˙ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq198_HTML.gif is absolutely continuous on [ t δ , t ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq199_HTML.gif, g ¨ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq200_HTML.gif exists for a.a. t [ t δ , t + δ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq201_HTML.gif. Consequently,
0 g ˙ ( t ) = g ˙ ( t ) g ˙ ( t ) = t t g ¨ ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ9_HTML.gif
(9)
and
0 g ˙ ( t ) = g ˙ ( t ) g ˙ ( t ) = t t g ¨ ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ10_HTML.gif
(10)
At first, let us assume that condition (7) holds and let t ( t , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq202_HTML.gif be such that g ¨ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq200_HTML.gif and x ¨ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq203_HTML.gif exist. Then
lim h 0 x ˙ ( t + h ) x ˙ ( t ) h = x ¨ ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equh_HTML.gif
and so there exists a function a ( h ) , a ( h ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq204_HTML.gif as h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq205_HTML.gif, such that for each h,
x ˙ ( t + h ) = x ˙ ( t ) + h [ x ¨ ( t ) + a ( h ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equi_HTML.gif
Moreover, since x C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq206_HTML.gif, there exists a function b ( h ) , b ( h ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq207_HTML.gif as h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq205_HTML.gif, such that for each h,
x ( t + h ) = x ( t ) + h [ x ˙ ( t ) + b ( h ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equj_HTML.gif
Consequently, we obtain
g ¨ ( t ) = lim h 0 g ˙ ( t + h ) g ˙ ( t ) h = lim sup h 0 g ˙ ( t + h ) g ˙ ( t ) h = lim sup h 0 V ˙ x ( t + h ) , x ˙ ( t + h ) V ˙ x ( t ) , x ˙ ( t ) h = lim sup h 0 V ˙ x ( t ) + h [ x ˙ ( t ) + b ( h ) ] , x ˙ ( t ) + h [ x ¨ ( t ) + a ( h ) ] V ˙ x ( t ) , x ˙ ( t ) h lim sup h 0 V ˙ x ( t ) + h x ˙ ( t ) , x ˙ ( t ) + h [ x ¨ ( t ) + a ( h ) ] V ˙ x ( t ) , x ˙ ( t ) h L | b ( h ) | x ˙ ( t ) + h [ x ¨ ( t ) + a ( h ) ] = lim sup h 0 V ˙ x ( t ) + h x ˙ ( t ) , x ˙ ( t ) + h x ¨ ( t ) V ˙ x ( t ) , x ˙ ( t ) h L | b ( h ) | x ˙ ( t ) + h [ x ¨ ( t ) + a ( h ) ] + V ˙ x ( t ) + h x ˙ ( t ) , a ( h ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equk_HTML.gif
Since V ˙ x ( t ) + h x ˙ ( t ) , a ( h ) L | b ( h ) | x ˙ ( t ) + h [ x ¨ ( t ) + a ( h ) ] 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq208_HTML.gif as h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq209_HTML.gif,
g ¨ ( t ) lim sup h 0 V ˙ x ( t ) + h x ˙ ( t ) , x ˙ ( t ) + h x ¨ ( t ) V ˙ x ( t ) , x ˙ ( t ) h = lim sup h 0 V ˙ x ( t ) + h x ˙ ( t ) V ˙ x ( t ) , x ˙ ( t ) h + V ˙ x ( t ) + h x ˙ ( t ) , x ¨ ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equl_HTML.gif
Moreover, for every x , w E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq210_HTML.gif and h R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq211_HTML.gif, we have that
V ˙ x + h y , w = V ˙ x , w + [ V ˙ x + h y , w V ˙ x , w ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equm_HTML.gif
According to the Lipschitzianity of V ˙ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq212_HTML.gif, when | h | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq213_HTML.gif is sufficiently small, we have that
| V ˙ x + h y , w V ˙ x , w | = | V ˙ x + h y V ˙ x , w | V x + h y V ˙ x w L | h | y w , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equn_HTML.gif
where L denotes the local Lipschitz constant of V ˙ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq212_HTML.gif in a neighborhood of x. It implies that
lim h 0 V ˙ x + h y , w V ˙ x , w = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equo_HTML.gif
and then
lim h 0 V ˙ x + h y , w = V ˙ x , w . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equp_HTML.gif
Therefore,
g ¨ ( t ) lim sup h 0 V ˙ x ( t ) + h x ˙ ( t ) V ˙ x ( t ) , x ˙ ( t ) h + V ˙ x ( t ) , x ¨ ( t ) > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equq_HTML.gif

according to assumption (7), it leads to a contradiction with inequality (9).

Secondly, let us assume that condition (8) holds and let s ( t , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq214_HTML.gif be such that g ¨ ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq215_HTML.gif and x ¨ ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq216_HTML.gif exist. Then it is possible to show, using the same procedure as before, that according to assumption (8),
g ¨ ( s ) lim sup h 0 + V ˙ x ( s ) + h x ˙ ( s ) V ˙ x ( s ) , x ˙ ( s ) h + V ˙ x ( s ) , x ¨ ( s ) > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equr_HTML.gif

which leads to a contradiction with inequality (10).

Therefore, we get the contradiction in case that at least one of conditions (7), (8) holds which completes the proof. □

If the mapping F ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq217_HTML.gif is globally u.s.c. in ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq218_HTML.gif, then the transversality conditions can be localized directly on the boundary of K, as will be shown in the following proposition, whose proof is again quite analogous to the finite-dimensional case considered in [2].

Proposition 4.2 Let K E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq163_HTML.gif be a nonempty open set such that 0 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq219_HTML.gif and F : [ 0 , T ] × E × E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq2_HTML.gif be an upper semicontinuous multivalued mapping with compact, convex values. Assume that there exists a function V C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq220_HTML.gif with a locally Lipschitzian Fréchet derivative V ˙ x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq178_HTML.gif which satisfies conditions (H1) and (H2). Suppose, moreover, that for all x K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq221_HTML.gif, t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq170_HTML.gif and y E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq222_HTML.gif with
V ˙ x , y = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ11_HTML.gif
(11)
the following condition holds:
lim inf h 0 V ˙ x + h y , y h + V ˙ x , w > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ12_HTML.gif
(12)

for all w F ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq223_HTML.gif. Then K is a bound set for problem (1).

Proof Let x : [ 0 , T ] K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq173_HTML.gif be a solution of problem (1). We assume, by a contradiction, that there exists t 0 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq224_HTML.gif such that x ( t 0 ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq225_HTML.gif. Since 0 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq219_HTML.gif and x satisfies Dirichlet boundary conditions, t 0 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq226_HTML.gif.

Let us define the function g : [ t 0 , T t 0 ] ( , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq227_HTML.gif as the composition g ( h ) : = ( V x ) ( t 0 + h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq228_HTML.gif. Then g ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq229_HTML.gif and g ( h ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq230_HTML.gif for all h [ t 0 , T t 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq231_HTML.gif, i.e., there is a local maximum for g at the point 0, and so g ˙ ( 0 ) = V ˙ x ( t 0 ) , x ˙ ( t 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq232_HTML.gif. Consequently, v : = x ˙ ( t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq233_HTML.gif satisfies condition (11).

Since V ˙ x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq178_HTML.gif is locally Lipschitzian, there exist a neighborhood U of x ( t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq234_HTML.gif and a constant L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq180_HTML.gif such that V ˙ | U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq181_HTML.gif is Lipschitzian with a constant L.

Let { h k } k = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq235_HTML.gif be an arbitrary decreasing sequence of positive numbers such that h k 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq236_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq237_HTML.gif, x ( t 0 + h ) U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq238_HTML.gif for all h ( 0 , h 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq239_HTML.gif.

Since g ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq229_HTML.gif and g ( h ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq240_HTML.gif for all h ( 0 , h k ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq241_HTML.gif, there exists, for each k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq242_HTML.gif, h k ( 0 , h k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq243_HTML.gif such that g ˙ ( h k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq244_HTML.gif.

Since x C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq206_HTML.gif, for each k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq245_HTML.gif,
x ( t 0 + h k ) = x ( t 0 ) + h k [ x ˙ ( t 0 ) + b k ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ13_HTML.gif
(13)

where b k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq246_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq247_HTML.gif.

Let
ζ : = { x ˙ ( t 0 + h k ) x ˙ ( t 0 ) h k , k N } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equs_HTML.gif
and let ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq168_HTML.gif be given. As a consequence of the regularity assumptions imposed on F and of the continuity of both x and x ˙ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq248_HTML.gif, there exists δ ¯ = δ ¯ ( ε ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq249_HTML.gif such that for each t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq250_HTML.gif, | t t 0 | δ ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq251_HTML.gif, it follows that
F ( t , x ( t ) , x ˙ ( t ) ) F ( t 0 , x ( t 0 ) , x ˙ ( t 0 ) ) + ε B ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equt_HTML.gif
Subsequently, according to the mean-value theorem (see, e.g., [[24], Theorem 0.5.3]), there exists k ε N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq252_HTML.gif such that for each k > k ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq253_HTML.gif,
x ˙ ( t 0 + h k ) x ˙ ( t 0 ) h k = 1 h k t 0 t 0 + h k x ¨ ( s ) d s F ( t 0 , x ( t 0 ) , x ˙ ( t 0 ) ) + ε B ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equu_HTML.gif
Therefore,
ζ { x ˙ ( t 0 + h k ) x ˙ ( t 0 ) h k , k = 1 , 2 , , k ( ε ) } F ( t 0 , x ( t 0 ) , x ˙ ( t 0 ) ) + ε B ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equv_HTML.gif
Since F has compact values and ε is arbitrary, we obtain that ζ is a relatively compact set. Thus, there exist a subsequence, for the sake of simplicity denoted as the sequence, of { x ˙ ( t 0 + h k ) x ˙ ( t 0 ) h k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq254_HTML.gif and w E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq255_HTML.gif such that
x ˙ ( t 0 + h k ) x ˙ ( t 0 ) h k w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ14_HTML.gif
(14)
as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq256_HTML.gif implying, for the arbitrariness of ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq168_HTML.gif,
w F ( t 0 , x ( t 0 ) , x ˙ ( t 0 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equw_HTML.gif
As a consequence of the property (14), there exists a sequence { a k } k = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq257_HTML.gif, a k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq258_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq237_HTML.gif, such that
x ˙ ( t 0 + h k ) = x ˙ ( t 0 ) + h k [ w + a k ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ15_HTML.gif
(15)
for each k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq245_HTML.gif. Since h k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq259_HTML.gif and g ˙ ( h k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq244_HTML.gif, in view of (13) and (15),
0 g ˙ ( h k ) h k = V ˙ x ( t 0 + h k ) , x ˙ ( t 0 + h k ) h k = V ˙ x ( t 0 ) + h k [ x ˙ ( t 0 ) + b k ] , x ˙ ( t 0 ) + h k [ w + a k ] h k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equx_HTML.gif
Since h k ( 0 , h k ) ( 0 , h 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq260_HTML.gif for all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq261_HTML.gif, we have, according to (13), that x ( t 0 ) + h k [ x ˙ ( t 0 ) + b k ] U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq262_HTML.gif for each k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq261_HTML.gif. Since b k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq263_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq256_HTML.gif, it is possible to find k 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq264_HTML.gif such that for all k k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq265_HTML.gif, it holds that x ( t 0 ) + x ˙ ( t 0 ) h k U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq266_HTML.gif. By means of the local Lipschitzianity of V ˙ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq212_HTML.gif, for all k k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq265_HTML.gif,
0 g ˙ ( h k ) h k = V ˙ x ( t 0 ) + h k [ x ˙ ( t 0 ) + b k ] V ˙ x ( t 0 ) + h k x ˙ ( t 0 ) + V ˙ x ( t 0 ) + h k x ˙ ( t 0 ) , x ˙ ( t 0 ) + h k [ w + a k ] h k V ˙ x ( t 0 ) + h k x ˙ ( t 0 ) , x ˙ ( t 0 ) + h k [ w + a k ] h k L | b k | x ˙ ( t 0 ) + h k [ w + a k ] = V ˙ x ( t 0 ) + h k x ˙ ( t 0 ) , x ˙ ( t 0 ) + h k w h k L | b k | x ˙ ( t 0 ) + h k [ w + a k ] + V ˙ x ( t 0 ) + h k x ˙ ( t 0 ) , a k = V ˙ x ( t 0 ) + h k x ˙ ( t 0 ) , x ˙ ( t 0 ) h k + V ˙ x ( t 0 ) , w L | b k | x ˙ ( t 0 ) + h k [ w + a k ] + V ˙ x ( t 0 ) + h k x ˙ ( t 0 ) , a k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equy_HTML.gif
Since V ˙ x ( t 0 ) + h k x ˙ ( t 0 ) , a k L | b k | x ˙ ( t 0 ) + h k [ w + a k ] 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq267_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq268_HTML.gif,
lim inf h 0 + V ˙ x ( t 0 ) + h k x ˙ ( t 0 ) , x ˙ ( t 0 ) h k + V ˙ x ( t 0 ) , w 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ16_HTML.gif
(16)
If we consider, instead of the sequence { h k } k = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq269_HTML.gif, an increasing sequence { h ¯ k } k = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq270_HTML.gif of negative numbers such that h ¯ k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq271_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq272_HTML.gif, x ( t 0 + h ) U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq273_HTML.gif for all h ( h ¯ 1 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq274_HTML.gif, we are able to find, for each k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq245_HTML.gif, h ¯ k ( h ¯ k , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq275_HTML.gif such that g ˙ ( h ¯ k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq276_HTML.gif. Therefore, using the same procedure as in the first part of the proof, we obtain, for k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq119_HTML.gif sufficiently large, that
0 g ˙ ( h ¯ k ) h ¯ k V ˙ x ( t 0 ) + h ¯ k x ˙ ( t 0 ) , x ˙ ( t 0 ) h ¯ k + V ˙ x ( t 0 ) , w ¯ L | b ¯ k | x ˙ ( t 0 ) + h ¯ k [ w ¯ + a ¯ k ] + V ˙ x ( t 0 ) + h ¯ k x ˙ ( t 0 ) , a ¯ k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equz_HTML.gif

where a ¯ k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq277_HTML.gif, b ¯ k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq278_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq237_HTML.gif and w ¯ F ( t 0 , x ( t 0 ) , x ˙ ( t 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq279_HTML.gif.

This means that V ˙ x ( t 0 ) + h ¯ k x ˙ ( t 0 ) , a ¯ k L | b ¯ k | x ˙ ( t 0 ) + h ¯ k [ w ¯ + a ¯ k ] 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq280_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq268_HTML.gif, which implies
lim inf h 0 V ˙ x ( t 0 ) + h x ˙ ( t 0 ) , x ˙ ( t 0 ) h + V ˙ x ( t 0 ) , w ¯ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ17_HTML.gif
(17)

Inequalities (16) and (17) are in a contradiction with condition (12), because x ( t 0 ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq281_HTML.gif, x ˙ ( t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq282_HTML.gif satisfies condition (11) and w , w ¯ F ( t 0 , x ( t 0 ) , x ˙ ( t 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq283_HTML.gif. □

Remark 4.1 One can readily check that for V C 2 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq284_HTML.gif, inequalities (7) and (8), as well as (12), become
V ¨ x ( y ) , y + V ˙ x , w > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equaa_HTML.gif

with t, x, y, w as in Proposition 4.1 or in Proposition 4.2.

The typical case occurs when E = H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq285_HTML.gif is a Hilbert space, , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq286_HTML.gif denotes the scalar product and
V ( x ) : = 1 2 ( x 2 R 2 ) = 1 2 ( x , x R 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equab_HTML.gif
for some R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq287_HTML.gif. In this case, V C 2 ( H , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq288_HTML.gif and it is not difficult to see that conditions (7) and (8), as well as (12), become
y , y + x , w > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equac_HTML.gif

with t, x, y and w as in Proposition 4.1 or in Proposition 4.2, where K : = { x H x < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq289_HTML.gif.

Definition 4.2 A C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq150_HTML.gif-function V : E R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq290_HTML.gif with a locally Lipschitzian Fréchet derivative V ˙ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq212_HTML.gif which satisfies conditions (H1), (H2) and all assumptions in Proposition 4.1 or Proposition 4.2 is called a bounding function for problem (1).

5 Existence and localization results

Combining the continuation principle with the bound sets technique, we are ready to state the main result of the paper concerning the solvability and localization of a solution of the multivalued Dirichlet problem (1).

Theorem 5.1 Consider the Dirichlet b.v.p. (1), where F : [ 0 , T ] × E × E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq2_HTML.gif is an upper-Carathéodory multivalued mapping. Assume that K E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq291_HTML.gif is an open, convex set containing  0. Furthermore, let the following conditions be satisfied:

( 5 i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq292_HTML.gif) γ ( F ( t , Ω 1 × Ω 2 ) ) g ( t ) ( γ ( Ω 1 ) + γ ( Ω 2 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq293_HTML.gif for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq294_HTML.gif and each bounded Ω 1 , Ω 2 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq295_HTML.gif, where g L 1 ( [ 0 , T ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq296_HTML.gif and γ is the Hausdorff measure of noncompactness in E.

( 5 i i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq297_HTML.gif) For every nonempty, bounded set Ω E × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq134_HTML.gif, there exists ν Ω L 1 ( [ 0 , T ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq135_HTML.gif such that
F ( t , x , y ) ν Ω ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ18_HTML.gif
(18)

for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif and all ( x , y ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq136_HTML.gif,

( 5 i i i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq298_HTML.gif) ( T + 4 ) g L 1 ( [ 0 , T ] , [ 0 , ) ) < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq299_HTML.gif.

Finally, let there exist a function V C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq166_HTML.gif with a locally Lipschitzian Fréchet derivative V ˙ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq300_HTML.gif satisfying conditions (H1), (H2), and at least one of conditions (7), (8) for a suitable ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq301_HTML.gif, all x K ¯ B ( K , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq302_HTML.gif, t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq170_HTML.gif, y E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq171_HTML.gif, λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq141_HTML.gif and w λ F ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq303_HTML.gif. Then the Dirichlet b.v.p. (1) admits a solution whose values are located in K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq148_HTML.gif.

Proof Let us define the closed set S = S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq304_HTML.gif by
S : = { x A C 1 ( [ 0 , T ] , E ) : x ( T ) = x ( 0 ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equad_HTML.gif

and let the set Q of candidate solutions be defined as Q : = C 1 ( [ 0 , T ] , K ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq305_HTML.gif. Because of the convexity of K, the set Q is closed and convex.

For all q Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq131_HTML.gif and λ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq132_HTML.gif, consider still the associated fully linearized problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equae_HTML.gif

and denote by T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq138_HTML.gif a solution mapping which assigns to each ( q , λ ) Q × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq306_HTML.gif the set of solutions of P ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq143_HTML.gif. We will show that the family of the above b.v.p.s P ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq143_HTML.gif satisfies all assumptions of Proposition 3.1.

In this case, φ ( t , x , x ˙ ) = F ( t , x , x ˙ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq307_HTML.gif which, together with the definition of P ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq143_HTML.gif, ensures the validity of (6).

ad (i) In order to verify condition (i) in Proposition 3.1, we need to show that for each ( q , λ ) Q × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq308_HTML.gif, the problem P ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq143_HTML.gif is solvable with a convex set of solutions. So, let ( q , λ ) Q × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq306_HTML.gif be arbitrary and let f q ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq309_HTML.gif be a strongly measurable selection of F ( , q ( ) , q ˙ ( ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq310_HTML.gif. The homogeneous problem corresponding to b.v.p. P ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq143_HTML.gif,
x ¨ ( t ) = 0 for a.a.  t [ 0 , T ] , x ( T ) = x ( 0 ) = 0 , } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ19_HTML.gif
(19)
has only the trivial solution, and therefore the single-valued Dirichlet problem
x ¨ ( t ) = λ f q ( t ) for a.a.  t [ 0 , T ] , x ( T ) = x ( 0 ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equaf_HTML.gif
admits a unique solution x q , λ ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq311_HTML.gif which is one of solutions of P ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq143_HTML.gif. This is given, for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif, by x q , λ ( t ) = 0 T G ( t , s ) λ f q ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq312_HTML.gif, where G is the Green function associated to the homogeneous problem (19). The Green function G and its partial derivative t G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq313_HTML.gif are defined by (cf., e.g., [[12], pp.170-171])
G ( t , s ) = { ( s T ) t T for all  0 t s T , ( t T ) s T for all  0 s t T , t G ( t , s ) = { ( s T ) T for all  0 t s T , s T for all  0 s t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equag_HTML.gif

Thus, the set of solutions of P ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq143_HTML.gif is nonempty. The convexity of the solution sets follows immediately from the properties of a mapping F and the fact that problems P ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq143_HTML.gif are fully linearized.

ad (ii) Assuming that H : [ 0 , T ] × E × E × E × E × [ 0 , 1 ] E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq314_HTML.gif is defined by H ( t , x , y , q , r , λ ) : = λ F ( t , q , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq315_HTML.gif, condition (ii) in Proposition 3.1 is ensured directly by assumption (5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq316_HTML.gif ).

ad (iii) Since the verification of condition (iii) in Proposition 3.1 is technically the most complicated, it will be subdivided into two parts: (iii1) the quasi-compactness of the solution operator T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq138_HTML.gif, (iii2) the condensity of T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq138_HTML.gif w.r.t. the monotone and nonsingular (cf. Lemma 2.1) m.n.c. μ defined by (4).

ad (iii1) Let us firstly prove that the solution mapping T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq138_HTML.gif is quasi-compact. Since C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq317_HTML.gif is a metric space, it is sufficient to prove the sequential quasi-compactness of  T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq138_HTML.gif. Hence, let us consider the sequences { q n } , { λ n } , q n Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq318_HTML.gif, λ n [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq319_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq320_HTML.gif such that q n q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq321_HTML.gif in C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq317_HTML.gif and λ n λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq322_HTML.gif. Moreover, let x n T ( q n , λ n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq323_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq73_HTML.gif. Then there exists, for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq73_HTML.gif, f n ( ) F ( , q n ( ) , q ˙ n ( ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq324_HTML.gif such that
x ¨ n ( t ) = λ n f n ( t ) for a.a.  t [ 0 , T ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ20_HTML.gif
(20)

and that x n ( T ) = x n ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq325_HTML.gif.

Since q n q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq326_HTML.gif and q ˙ n q ˙ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq327_HTML.gif in C ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq328_HTML.gif, there exists a bounded Ω E × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq329_HTML.gif such that ( q n ( t ) , q ˙ n ( t ) ) Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq330_HTML.gif for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif and n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq331_HTML.gif. Therefore, there exists, according to condition ( 5 i i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq297_HTML.gif), ν Ω L 1 ( [ 0 , T ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq135_HTML.gif such that f n ( t ) ν Ω ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq332_HTML.gif for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq333_HTML.gif and a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq334_HTML.gif.

Moreover, for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq333_HTML.gif and a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif,
x n ( t ) = λ n 0 T G ( t , s ) f n ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equah_HTML.gif
and
x ˙ n ( t ) = λ n 0 T t G ( t , s ) f n ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equai_HTML.gif
Thus, x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq335_HTML.gif satisfies, for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq333_HTML.gif and a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq43_HTML.gif, x n ( t ) a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq336_HTML.gif and x ˙ n ( t ) b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq337_HTML.gif, where
a : = T 4 0 T ν Ω ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equaj_HTML.gif
and
b : = 0 T ν Ω ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equak_HTML.gif
Furthermore, for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq333_HTML.gif and a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif, we have
x ¨ n ( t ) ν Ω ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equal_HTML.gif

Hence, the sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq338_HTML.gif and { x ˙ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq339_HTML.gif are bounded and { x ¨ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq340_HTML.gif is uniformly integrable.

Since the sequences { q n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq341_HTML.gif, { q ˙ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq342_HTML.gif are converging, we obtain, in view of ( 5 i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq292_HTML.gif),
γ ( { f n ( t ) } ) g ( t ) ( γ ( { q n ( t ) } ) + γ ( { q ˙ n ( t ) } ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equam_HTML.gif

for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif, which implies that { f n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq343_HTML.gif is relatively compact.

For all ( t , s ) [ 0 , T ] × [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq344_HTML.gif, the sequence { G ( t , s ) f n ( s ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq345_HTML.gif is relatively compact as well since, according to the semi-homogeneity of the Hausdorff m.n.c.,
γ ( { G ( t , s ) f n ( s ) } ) | G ( t , s ) | γ ( { f n ( s ) } ) = 0 for all  ( t , s ) [ 0 , T ] × [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ21_HTML.gif
(21)
Moreover, by means of (2), (3), (21) and the semi-homogeneity of the Hausdorff m.n.c.,
γ ( { x n ( t ) } ) γ ( λ [ 0 , 1 ] λ { 0 T G ( t , s ) f n ( s ) d s } ) γ ( { 0 T G ( t , s ) f n ( s ) d s } ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equan_HTML.gif
By similar reasonings, we can also get
γ ( { x ˙ n ( t ) } ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equao_HTML.gif

by which { x n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq346_HTML.gif, { x ˙ n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq347_HTML.gif are relatively compact for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif. Moreover, since x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq335_HTML.gif satisfies for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq348_HTML.gif equation (20), { x ¨ n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq349_HTML.gif is relatively compact for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif. Thus, according to Lemma 2.2, there exist a subsequence of { x ˙ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq339_HTML.gif, for the sake of simplicity denoted in the same way as the sequence, and x C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq350_HTML.gif such that { x ˙ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq339_HTML.gif converges to x ˙ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq248_HTML.gif in C ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq95_HTML.gif and { x ¨ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq340_HTML.gif converges weakly to x ¨ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq351_HTML.gif in L 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq9_HTML.gif. Therefore, the mapping T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq138_HTML.gif is quasi-compact.

ad (iii2) In order to show that T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq138_HTML.gif is μ-condensing, where μ is defined by (4), we will prove that any bounded subset Θ Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq352_HTML.gif such that μ ( T ( Θ × [ 0 , 1 ] ) ) μ ( Θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq353_HTML.gif is relatively compact. Let { x n } n T ( Θ × [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq354_HTML.gif be a sequence such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equap_HTML.gif
Then we can find { q n } n Θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq355_HTML.gif, { f n } n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq356_HTML.gif satisfying f n ( t ) F ( t , q n ( t ) , q ˙ n ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq357_HTML.gif for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif and { λ n } n [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq358_HTML.gif such that for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif,
x n ( t ) = λ n 0 T G ( t , s ) f n ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ22_HTML.gif
(22)
and
x ˙ n ( t ) = λ n 0 T t G ( t , s ) f n ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ23_HTML.gif
(23)
In view of ( 5 i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq292_HTML.gif), we have, for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equaq_HTML.gif

Since { q n } n Θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq359_HTML.gif and Θ is bounded in C 1 ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq16_HTML.gif, by means of ( 5 i i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq297_HTML.gif), we get the existence of ν Θ L 1 ( [ 0 , T ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq360_HTML.gif such that f n ( t ) ν Θ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq361_HTML.gif for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif and all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq331_HTML.gif. This implies G ( t , s ) f n ( t ) | G ( t , s ) | ν Θ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq362_HTML.gif for a.a. t , s [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq363_HTML.gif and all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq331_HTML.gif.

Moreover, by virtue of the semi-homogeneity of the Hausdorff m.n.c., for all ( t , s ) [ 0 , T ] × [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq364_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equar_HTML.gif
According to (2), (3) and (22), we so obtain for each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif,
γ ( { x n ( t ) , n N } ) γ ( { 0 T G ( t , s ) f n ( s ) d s , n N } ) 2 T 4 g L 1 sup t [ 0 , T ] ( γ ( { q n ( t ) , n N } ) + γ ( { q ˙ n ( t ) , n N } ) ) = T 2 g L 1 S , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equas_HTML.gif
where
S : = sup t [ 0 , T ] ( γ ( { q n ( t ) , n N } ) + γ ( { q ˙ n ( t ) , n N } ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equat_HTML.gif
By the similar reasonings, we can obtain that for each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif,
γ ( { x ˙ n ( t ) , n N } ) 2 g L 1 S , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equau_HTML.gif
when starting from condition (23). Subsequently,
γ ( { x n ( t ) , n N } ) + γ ( { x ˙ n ( t ) , n N } ) T + 4 2 g L 1 S , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equav_HTML.gif
yielding
sup t [ 0 , T ] ( γ ( { x n ( t ) , n N } ) + γ ( { x ˙ n ( t ) , n N } ) ) T + 4 2 g L 1 S . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ24_HTML.gif
(24)
Since μ ( T ( Θ × [ 0 , 1 ] ) ) μ ( Θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq365_HTML.gif and { q n } n Θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq359_HTML.gif, we so get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equaw_HTML.gif
and, in view of (24) and ( 5 i i i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq298_HTML.gif), we have that
sup t [ 0 , T ] ( γ ( { q n ( t ) , n N } ) + γ ( { q ˙ n ( t ) , n N } ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equax_HTML.gif
Inequality (24) implies that
sup t [ 0 , T ] ( γ ( { x n ( t ) , n N } ) + γ ( { x ˙ n ( t ) , n N } ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ25_HTML.gif
(25)
Now, we show that both the sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq338_HTML.gif and { x ˙ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq339_HTML.gif are equi-continuous. Let Θ ˜ E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq366_HTML.gif be such that q n ( t ) Θ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq367_HTML.gif and q ˙ n ( t ) Θ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq368_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq333_HTML.gif and t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq88_HTML.gif. Thus, we get that x ¨ n ( t ) = λ n f n ( t ) ν Θ ˜ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq369_HTML.gif, where ν Θ ˜ L 1 ( [ 0 , T ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq370_HTML.gif comes from ( 5 i i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq297_HTML.gif), and so { x ¨ n } n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq371_HTML.gif is uniformly integrable. This implies that { x ˙ n } n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq372_HTML.gif is equi-continuous. Moreover, according to (23), we obtain that
x ˙ n ( t ) 0 T ν Θ ˜ ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equay_HTML.gif
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq333_HTML.gif and t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif, implying that { x ˙ n } n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq373_HTML.gif is bounded; consequently, also { x n } n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq374_HTML.gif is equi-continuous. Therefore,
mod C ( { x n } ) = mod C ( { x ˙ n } ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equaz_HTML.gif
In view of (25), we have so obtained that
μ ( T ( Θ × [ 0 , 1 ] ) ) = ( 0 , 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equba_HTML.gif

Hence, also μ ( Θ ) = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq375_HTML.gif and since μ is regular, we have that Θ is relatively compact. Therefore, condition (iii) in Proposition 3.1 holds.

ad (iv) For all q Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq131_HTML.gif, the problem P ( q , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq139_HTML.gif has only the trivial solution. Since 0 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq164_HTML.gif, condition (iv) in Proposition 3.1 is satisfied.

ad (v) Let q Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq376_HTML.gif be a solution of the b.v.p. P ( q , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq377_HTML.gif for some λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq141_HTML.gif, i.e., a fixed point of the solution mapping T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq138_HTML.gif. In view of conditions (7), (8) (see Proposition 4.1), K is, for all λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq141_HTML.gif, a bound set for the problem
q ¨ ( t ) λ F ( t , q ( t ) , q ˙ ( t ) ) , for a.a.  t [ 0 , T ] , x ( T ) = x ( 0 ) = 0 . } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbb_HTML.gif

This implies that q Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq378_HTML.gif, which ensures condition (v) in Proposition 3.1. □

If the mapping F ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq217_HTML.gif is globally u.s.c. in ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq218_HTML.gif (i.e., a Marchaud map), then we are able to improve Theorem 5.1 in the following way.

Theorem 5.2 Consider the Dirichlet b.v.p. (1), where F : [ 0 , T ] × E × E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq2_HTML.gif is an upper semicontinuous mapping with compact, convex values. Assume that K E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq291_HTML.gif is an open, convex set containing 0. Moreover, let conditions ( 5 i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq292_HTML.gif), ( 5 i i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq297_HTML.gif), ( 5 i i i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq298_HTML.gif) from Theorem  5.1 be satisfied.

Furthermore, let there exist a function V C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq166_HTML.gif with a locally Lipschitz Frechét derivative V ˙ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq300_HTML.gif satisfying (H1) and (H2). Moreover, let, for all x K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq379_HTML.gif, t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq380_HTML.gif, λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq141_HTML.gif and y E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq222_HTML.gif satisfying (11), condition (12) hold for all w λ F ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq381_HTML.gif. Then the Dirichlet b.v.p. (1) admits a solution whose values are located in K ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq148_HTML.gif.

Proof The verification is quite analogous as in Theorem 5.1 when just replacing the usage of Proposition 4.1 by Proposition 4.2. □

6 Illustrative example

Example 6.1 Let E = H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq285_HTML.gif be a Hilbert space and let us consider the Dirichlet b.v.p.
x ¨ ( t ) F 1 ( t , x ( t ) , x ˙ ( t ) ) + F 2 ( t , x ( t ) , x ˙ ( t ) ) , for a.a.  t [ 0 , T ] , x ( 0 ) = x ( T ) = 0 , } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ26_HTML.gif
(26)
where
  1. (i)
    F 1 : [ 0 , T ] × H × H H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq382_HTML.gif is an upper-Carathéodory multivalued mapping and F 1 ( t , , ) : H × H H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq383_HTML.gif is completely continuous for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif such that
    F 1 ( t , x , y ) ν 1 ( t , D 0 , D 1 ) L 1 ( [ 0 , T ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbc_HTML.gif
     
for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif and all x , y H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq384_HTML.gif with x D 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq385_HTML.gif, y D 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq386_HTML.gif,
  1. (ii)
    F 2 : [ 0 , T ] × H × H H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq387_HTML.gif is a Carathéodory multivalued mapping such that
    F 2 ( t , 0 , 0 ) ν 2 ( t ) L 1 ( [ 0 , T ] , [ 0 , ) )  for a.a.  t [ 0 , T ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbd_HTML.gif
     
and F 2 ( t , , ) : H × H H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq388_HTML.gif is Lipschitzian for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif with the Lipschitz constant
L < 2 T ( T + 4 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Eqube_HTML.gif
Moreover, suppose that
  1. (iii)
    there exist R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq287_HTML.gif and ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq168_HTML.gif such that, for all x H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq389_HTML.gif with R ε < x R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq390_HTML.gif, t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq170_HTML.gif, y H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq391_HTML.gif, λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq141_HTML.gif and w λ ( F 1 ( t , x , y ) + F 2 ( t , x , y ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq392_HTML.gif, we have
    y , y + x , w > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbf_HTML.gif
     

Then the Dirichlet problem (26) admits, according to Theorem 5.1, a solution x ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq393_HTML.gif such that x ( t ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq394_HTML.gif for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif.

Indeed. The properties of F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq395_HTML.gif guarantee that F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq395_HTML.gif satisfies the inequality (cf., e.g., [20])
γ ( F 2 ( t , Ω 1 × Ω 2 ) ) L ( γ ( Ω 1 ) + γ ( Ω 2 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equ27_HTML.gif
(27)

for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq294_HTML.gif and every bounded Ω 1 , Ω 2 H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq396_HTML.gif, where γ stands for the Hausdorff measure of noncompactness in H.

Since F 1 ( t , , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq397_HTML.gif is completely continuous and thanks to the algebraic semi-additivity of γ, inequality (27) can be rewritten into
γ ( F 1 ( t , Ω 1 × Ω 2 ) + F 2 ( t , Ω 1 × Ω 2 ) ) L ( γ ( Ω 1 ) + γ ( Ω 2 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbg_HTML.gif

for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq398_HTML.gif and every bounded Ω 1 , Ω 2 H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq396_HTML.gif, i.e., ( 5 i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq399_HTML.gif, for g : = L < 2 T ( T + 4 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq400_HTML.gif (cf. ( 5 i i i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq298_HTML.gif)).

Moreover, according to the Lipschitzianity of F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq395_HTML.gif, the following inequalities take place:
d H ( F 2 ( t , x , y ) , 0 ) d H ( F 2 ( t , x , y ) , F 2 ( t , 0 , 0 ) ) + d H ( F 2 ( t , 0 , 0 ) , 0 ) L ( x + y ) + ν 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbh_HTML.gif

for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif and all x , y H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq384_HTML.gif.

Thus, for x D 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq385_HTML.gif, y D 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq386_HTML.gif, we arrive at
F 1 ( t , x , y ) + F 2 ( t , x , y ) L ( D 0 + D 1 ) + ν 1 ( t , D 0 , D 1 ) + ν 2 ( t ) : = ν Ω ( t ) L 1 ( [ 0 , T ] , [ 0 , ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbi_HTML.gif

i.e., (18) in ( 5 i i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq297_HTML.gif).

Finally, in view of Remark 4.1, we can define the bounding function V C 2 ( H , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq401_HTML.gif by the formula
V ( x ) : = 1 2 ( x , x R 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbj_HTML.gif

and the bound set K as K : = { x H x < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq402_HTML.gif in order to get a claim.

Remark 6.1 Consider again (26) in a Hilbert space H, but let this time F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq403_HTML.gif, F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq395_HTML.gif be globally u.s.c. mappings with compact, convex values ( F 2 ( [ 0 , T ] , 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq404_HTML.gif is compact (cf., e.g., [[15], Proposition I.3.20]) and, in particular, bounded) such that
  1. (i)
    F 1 ( t , , ) : H × H H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq383_HTML.gif is a completely continuous mapping for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif such that
    F 1 ( t , x , y ) ν 1 ( t , D 0 , D 1 ) L 1 ( [ 0 , T ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbk_HTML.gif
     
for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif and all x , y H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq384_HTML.gif with x D 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq405_HTML.gif, y D 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq406_HTML.gif.
  1. (ii)
    F 2 ( t , , ) : H × H H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq407_HTML.gif is a Lipschitzian mapping for a.a. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif with the Lipschitz constant
    L < 2 T ( T + 4 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbl_HTML.gif
     
(iii usc ) There exists R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq287_HTML.gif such that, for all x H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq389_HTML.gif with x = R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq408_HTML.gif, t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq170_HTML.gif, y H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq409_HTML.gif satisfying x , y = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq410_HTML.gif, λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq141_HTML.gif and w λ ( F 1 ( t , x , y ) + F 2 ( t , x , y ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq392_HTML.gif, we have
y , y + x , w > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbm_HTML.gif

Applying now Theorem 5.2, by the analogous arguments as in Example 6.1, the Dirichlet problem (26) admits a solution x ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq393_HTML.gif such that x ( t ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq411_HTML.gif for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq38_HTML.gif.

Remark 6.2 Since the solution derivative x ˙ ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq13_HTML.gif takes the form
x ˙ ( t ) 0 T t G ( t , s ) [ F 1 ( s , x ( s ) , x ˙ ( s ) ) + F 2 ( s , x ( s ) , x ˙ ( s ) ) ] d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbn_HTML.gif
where
t G ( t , s ) = { ( s T ) T for all  0 t s T , s T for all  0 s t T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbo_HTML.gif
and so | t G ( t , s ) | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq412_HTML.gif for all t , s [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq413_HTML.gif, we obtain (under the above assumptions) the implicit inequality
D 1 1 1 L T [ 0 T ν 1 ( t , R , D 1 ) d t + 0 T ν 2 ( t ) d t + L R T ] for all  t [ 0 , T ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbp_HTML.gif

for D 1 : = max t [ 0 , T ] x ˙ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq414_HTML.gif.

Thus, for F 1 ( t , x , y ) F 1 ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq415_HTML.gif, we have ν 1 ( t , R , D 1 ) ν 1 ( t , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq416_HTML.gif, and subsequently
x ˙ ( t ) 1 1 L T [ 0 T ν 1 ( t , R ) d t + 0 T ν 2 ( t ) d t + L R T ] for all  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbq_HTML.gif
Similarly, if F 1 : [ 0 , T ] × H × H H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq417_HTML.gif is compact, then
0 T ν 1 ( t , R , D 1 ) d t C 1 T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbr_HTML.gif
holds with a suitable constant C 1 F 1 ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq418_HTML.gif, and the following estimate holds:
x ˙ ( t ) 1 1 L T [ C 1 T + L R T + 0 T ν 2 ( t ) d t ] for all  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_Equbs_HTML.gif

Because of the Dirichlet boundary conditions x ( 0 ) = x ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq419_HTML.gif for H = R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq420_HTML.gif, there exists a zero point t 0 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq421_HTML.gif of x ˙ ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq13_HTML.gif, i.e., x ˙ ( t 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq422_HTML.gif, by which the same estimates can be also obtained without an explicit usage of the Green function above. Otherwise, it is not so easy to obtain such estimates, because Rolle’s theorem fails in general.

For obtaining the estimation of the solution derivative x ˙ ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq13_HTML.gif in a Hilbert space H, one can also apply, under natural assumptions, the p-Nagumo condition derived in [7].

Endnote

a The m.n.c. mod C ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq77_HTML.gif is monotone, nonsingular and algebraically subadditive on C ( [ 0 , T ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq423_HTML.gif (cf., e.g., [20]) and it is equal to zero if and only if all the elements x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-25/MediaObjects/13661_2012_Article_279_IEq112_HTML.gif are equi-continuous.

Declarations

Acknowledgements

The first and third authors were supported by the grant PrF_2012_017. The second author was supported by the national research project PRIN “Ordinary Differential Equations and Applications”.

Authors’ Affiliations

(1)
Department of Mathematical Analysis, Faculty of Science, Palacký University
(2)
Department of Engineering Sciences and Methods, University of Modena and Reggio Emilia

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