 Research
 Open access
 Published:
ScorzaDragoni approach to Dirichlet problem in Banach spaces
Boundary Value Problems volume 2014, Article number: 23 (2014)
Abstract
Hartmantype conditions are presented for the solvability of a multivalued Dirichlet problem in a Banach space by means of topological degree arguments, bounding functions, and a ScorzaDragoni approximation technique. The required transversality conditions are strictly localized on the boundaries of given bound sets. The main existence and localization result is applied to a partial integrodifferential equation involving possible discontinuities in state variables. Two illustrative examples are supplied. The comparison with classical singlevalued results in this field is also made.
MSC:34A60, 34B15, 47H04.
1 Introduction
In this paper, we will establish sufficient conditions for the existence and localization of strong solutions to a multivalued Dirichlet problem in a Banach space via degree arguments combined with a bound sets technique. More precisely, Hartmantype conditions (cf. [1]), i.e. sign conditions w.r.t. the first state variable and growth conditions w.r.t. the second state variable, will be presented, provided the righthand side is a multivalued upperCarathéodory mapping which is γregular w.r.t. the Hausdorff measure of noncompactness γ.
The main aim will be twofold: (i) strict localization of sign conditions on the boundaries of bound sets by means of a technique originated by ScorzaDragoni [2], and (ii) the application of the obtained abstract result (see Theorem 3.1 below) to an integrodifferential equation involving possible discontinuities in a state variable. The first aim allows us, under some additional restrictions, to extend our earlier results obtained for globally upper semicontinuous righthand sides and partly improve those for upperCarathéodory righthand sides (see [3]). As we shall see, the latter aim justifies such an abstract setting, because the problem can be transformed into the form of a differential inclusion in a Hilbert {L}^{2}space. Roughly speaking, problems of this type naturally require such an abstract setting. In order to understand in a deeper way what we did and why, let us briefly recall classical results in this field and some of their extensions.
Hence, consider firstly the Dirichlet problem in the simplest vector form:
where f:[0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n} is, for the sake of simplicity allowing the comparison of the related results, a continuous function.
The first existence results, for a bounded f in (1), are due to ScorzaDragoni [4, 5]. Let us note that his name in the title is nevertheless related to the technique developed in [2] rather than to the existence results in [4, 5].
It is well known (see e.g. [3, 6–13]) that the problem (1) is solvable on various levels of generality provided:
(i_{sign}) \mathrm{\exists}R>0 such that \u3008f(t,x,y),x\u3009>0, for (t,x,y)\in [0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n} with \parallel x\parallel =R,
(ii_{growth}) \mathrm{\exists}{C}_{1}\ge 0,{C}_{2}\ge 0 such that {C}_{1}R<1 and \parallel f(t,x,y)\parallel \le {C}_{1}{\parallel y\parallel}^{2}+{C}_{2}, for (t,x,y)\in [0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n} with \parallel x\parallel \le R.
Let us note that the existence of the same constant R>0 in (i_{sign}) and (ii_{growth}) can be assumed either explicitly as in [6, 7, 9, 11, 13] or it follows from the assumptions as those in [8, 10, 12].
(1) Hartmann [9] (cf. also [1]) generalized both conditions as follows:
(i_{H}) \mathrm{\exists}R>0 such that \u3008f(t,x,y),x\u3009+{\parallel y\parallel}^{2}>0, for t\in [0,1] and (x,y)\in {\mathbb{R}}^{n}\times {\mathbb{R}}^{n} such that \parallel x\parallel =R and \u3008x,y\u3009=0,
(i_{H}) the wellknown BernsteinNagumoHartman condition (for its definition and more details, see e.g. [1, 14]).
Let us note that the strict inequality in (i_{H}) can be replaced by a nonstrict one (see e.g. [[1], Chapter XII,II,5], [[11], Corollary 6.2]).

(2)
Lasota and Yorke [10] improved condition (i_{sign}) with suitable constants {K}_{1}\ge 0 and {K}_{2}>0 in the following way:
(i_{LY}) \u3008f(t,x,y),x\u3009+{\parallel y\parallel}^{2}\ge {K}_{1}(1+\parallel x\parallel +\u3008x,y\u3009)+{K}_{2}\parallel y\parallel,
but for t\in [0,1], (x,y)\in {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}, and replaced (ii_{growth}) by the BernsteinNagumoHartman condition.
Since (i_{LY}) implies (cf. [10]) the existence of a constant K\ge 0 such that
for (t,x,y)\in [0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}, the sign condition (i_{LY}) is obviously more liberal than (i_{sign}) as well as than (i_{H}), on the intersection of their domains.
If {K}_{1}>0 in (i_{LY}), then constant {K}_{2} can be even equal to zero, i.e. {K}_{2}=0, in (i_{LY}) (see e.g. [[7], Corollary V.26 on p.74]). Moreover, the related BernsteinNagumoHartman condition can only hold for x in a suitable convex, closed, bounded subset of {\mathbb{R}}^{n} (see again e.g. [7]).
(3) Following the ideas of Mawhin in [7, 11, 12], Amster and Haddad [6] demonstrated that an open, bounded subset of {\mathbb{R}}^{n}, say D\subset {\mathbb{R}}^{n}, need not be convex, provided it has a {C}^{2}boundary ∂D such that condition (i_{H}) can be generalized as follows:
(i_{AH}) \u3008f(t,x,y),{n}_{x}\u3009\ge {I}_{x}(y), (t,x,y)\in [0,1]\times T\partial D\times {\mathbb{R}}^{n}, with \u3008{n}_{x},y\u3009=0,
where {n}_{x} is the outerpointing normal unit vector field, T\partial D denotes the tangent vector bundle and {I}_{x}(y) stands for the second fundamental form of the hypersurface.
Since for the ball D:=B(0,R), R>0, we can have
condition (i_{AH}) is obviously more general than the original Hartman condition (i_{H}).
Nevertheless, the growth condition takes there only the form (ii_{growth}), namely with \parallel x\parallel \le R replaced by x\in \overline{D}, where R denotes, this time, the radius of D.
For a convex, open, bounded subset D\subset {\mathbb{R}}^{n}, the particular case of (i_{AH}) can read as follows:
(i_{conv}) \u3008f(t,x,y),{n}_{x}\u3009>0, for (t,x,y)\in [0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n} with x\in \partial D and \u3008{n}_{x},y\u3009=0,
which is another wellknown generalization of (i_{sign}).
(4) In a Hilbert space H, for a completely continuous mapping f, Mawhin [12] has shown that, for real constants a, b, c such that a+b<1, condition (i_{sign}) can be replaced in particular by
(i_{M}) \u3008f(t,x,y),x\u3009\ge (a{\parallel x\parallel}^{2}+b\parallel x\parallel \parallel y\parallel +c\parallel x\parallel ), (t,x,y)\in [0,1]\times H\times H,
and (ii_{growth}) by an appropriate version of the BernsteinNagumoHartman condition.
(5) In a Banach space E, Schmitt and Thompson [13] improved, for a completely continuous mapping f, condition (i_{conv}) in the sense that the strict inequality in (i_{conv}) can be replaced by a nonstrict one. More concretely, if there exists a convex, open, bounded subset D\subset E of E with 0\in D such that
(i_{ST}) \u3008f(t,x,y),{n}_{x}\u3009\ge 0, for (t,x,y)\in [0,1]\times E\times E, with x\in \partial D and \u3008{n}_{x},y\u3009=0,
where \u3008\cdot ,\cdot \u3009 denotes this time the pairing between E and its dual {E}^{\prime}, jointly with the appropriate BernsteinNagumoHartman condition, then the problem (1) admits a solution whose values are located in \overline{D} (see [[13], Theorem 4.1]).
In the Carathéodory case of f:[0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n} in (1), for instance, the strict inequality in condition (i_{sign}) can be replaced, according to [[8], Theorem 6.1], by a nonstrict one and the constants {C}_{1}, {C}_{2} can be replaced without the requirement {C}_{1}R<1, but globally in [0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}, by functions {c}_{1}(t,x), {c}_{2}(t,x) which are bounded on bounded sets. Moreover, system (1) can be additively perturbed, for the same goal, by another Carathéodory function which is sublinear in both states variables x and y.
On the other hand, the Carathéodory case brings about some obstructions in a strict localization of sign conditions on the boundaries of bound sets (see e.g. [3, 15]). The same is also true for other boundary value problems (for Floquet problems, see e.g. [16–18]). Therefore, there naturally exist some extensions of classical results in this way. Further extensions concern problems in abstract spaces, functional problems, multivalued problems, etc. For the panorama of results in abstract spaces, see e.g. [19], where multivalued problems are also considered.
Nevertheless, let us note that in abstract spaces, it is extremely difficult (if not impossible) to avoid the convexity of given bound sets, provided the degree arguments are applied for noncompact maps (for more details, see [20]).
In this light, we would like to modify in the present paper the Hartmantype conditions (i_{sign}), (ii_{growth}) at least in the following way:

the given space E to be Banach (or, more practically, Hilbert),

the righthand side to be a multivalued upperCarathéodory mapping F which is γregular w.r.t. (x,y)\in E\times E and either globally measurable or globally quasicompact,

the inequality in (i_{sign}) to hold w.r.t. x strictly on the boundary ∂D of a convex, bounded subset D\subset E (or, more practically, of the ball B(0,R)\subset E),

condition (ii_{growth}) to be replaced by a suitable growth condition which would allow us reasonable applications (the usage of the BernsteinNagumoHartmantype condition will be employed in this context by ourselves elsewhere).
Hence, let E be a separable Banach space (with the norm \parallel \cdot \parallel) satisfying the RadonNikodym property (e.g. reflexivity, see e.g. [[21], pp.694695]) and let us consider the Dirichlet boundary value problem (b.v.p.)
where F:[0,T]\times E\times E\u22b8E is an upperCarathéodory multivalued mapping.
Let us note that in the entire paper all derivatives will be always understood in the sense of Fréchet and, by the measurability, we mean the one with respect to the Lebesgue σalgebra in [0,T] and the Borel σalgebra in E.
The notion of a solution will be understood in a strong (i.e. Carathéodory) sense. Namely, by a solution of problem (2) we mean a function x:[0,T]\to E whose first derivative \dot{x}(\cdot ) is absolutely continuous and satisfies (2), for almost all t\in [0,T].
The solution of the b.v.p. (2) will be obtained as the limit of a sequence of solutions of approximating problems that we construct by means of a ScorzaDragonitype result developed in [22]. The approximating problems will be treated by means of the continuation principle developed in [19].
2 Preliminaries
Let E be as above and [0,T]\subset \mathbb{R} be a closed interval. By the symbol {L}^{1}([0,T],E), we shall mean the set of all Bochner integrable functions x:[0,T]\to E. For the definition and properties of Bochner integrals, see e.g. [[21], pp.693701]. The symbol A{C}^{1}([0,T],E) will be reserved for the set of functions x:[0,T]\to E whose first derivative \dot{x}(\cdot ) is absolutely continuous. Then \ddot{x}\in {L}^{1}([0,T],E) and the fundamental theorem of calculus (the NewtonLeibniz formula) holds (see e.g. [[21], pp.695696], [[23], pp.243244]). In the sequel, we shall always consider A{C}^{1}([0,T],E) as a subspace of the Banach space {C}^{1}([0,T],E) and by the symbol \mathcal{L}(E) we shall mean the Banach space of all linear, bounded transformations L:E\to E endowed with the supnorm.
Given C\subset E and \epsilon >0, the symbol B(C,\epsilon ) will denote, as usually, the set C+\epsilon B, where B is the open unit ball in E centered at 0, i.e. B=\{x\in E\mid \parallel x\parallel <1\}. In what follows, the symbol μ will denote the Lebesgue measure on ℝ.
Let {E}^{\prime} be the Banach space dual to E and let us denote by \u3008\cdot ,\cdot \u3009 the pairing (the duality relation) between E and {E}^{\prime}, i.e., for all \mathrm{\Phi}\in {E}^{\prime} and x\in E, we put \mathrm{\Phi}(x)=:\u3008\mathrm{\Phi},x\u3009.
We recall also the Pettis measurability theorem which will be used in Section 4 and which we state here in the form of proposition.
Proposition 2.1 [[24], p.278]
Let (X,\mathrm{\Sigma}) be a measure space, E be a separable Banach space. Then f:X\to E is measurable if and only if for every e\in {E}^{\prime} the function e\circ f:X\to \mathbb{R} is measurable with respect to Σ and the Borel σalgebra in ℝ.
We shall 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\u22b8Y) if, for every x\in X, a nonempty subset F(x) of Y is given. We associate with F its graph {\mathrm{\Gamma}}_{F}, the subset of X\times Y, defined by {\mathrm{\Gamma}}_{F}:=\{(x,y)\in X\times Y\mid y\in F(x)\}.
A multivalued mapping F:X\u22b8Y is called upper semicontinuous (shortly, u.s.c.) if, for each open subset U\subset Y, the set \{x\in X\mid F(x)\subset U\} is open in X.
Let J\subset \mathbb{R} be a compact interval. A mapping F:J\u22b8Y, where Y is a separable metric space, is called measurable if, for each open subset U\subset Y, the set \{t\in J\mid F(t)\subset U\} belongs to a σalgebra of subsets of J.
A multivalued mapping F:X\u22b8Y is called compact if the set F(X)={\bigcup}_{x\in X}F(x) is contained in a compact subset of Y and it is called quasicompact if it maps compact sets onto relatively compact sets.
Let J\subset \mathbb{R} be a given compact interval. A multivalued mapping F:J\times X\u22b8Y, where Y is a separable Banach space, is called an upperCarathéodory mapping if the map F(\cdot ,x):J\u22b8Y is measurable, for all x\in X, the map F(t,\cdot ):X\u22b8Y is u.s.c., for almost all t\in J, and the set F(t,x) is compact and convex, for all (t,x)\in J\times X.
The technique that will be used for proving the existence and localization result consists in constructing a sequence of approximating problems. This construction will be made on the basis of the ScorzaDragonitype result developed in [22] (cf. also [25]).
For more details concerning multivalued analysis, see e.g. [23, 26, 27].
Definition 2.1 An upperCarathéodory mapping F:[0,T]\times X\times X\u22b8X is said to have the ScorzaDragoni property if there exists a multivalued mapping {F}_{0}:[0,T]\times X\times X\u22b8X\cup \{\mathrm{\varnothing}\} with compact, convex values having the following properties:

(i)
{F}_{0}(t,x,y)\subset F(t,x,y), for all (t,x,y)\in [0,T]\times X\times X,

(ii)
if u,v:[0,T]\to X are measurable functions with v(t)\in F(t,u(t),\dot{u}(t)), for a.a. t\in [0,T], then also v(t)\in {F}_{0}(t,u(t),\dot{u}(t)), for a.a. t\in [0,T],

(iii)
for every \epsilon >0, there exists a closed {I}_{\epsilon}\subset [0,T] such that \mu ([0,T]\setminus {I}_{\epsilon})<\epsilon, {F}_{0}(t,x,y)\ne \mathrm{\varnothing}, for all (t,x,y)\in {I}_{\epsilon}\times X\times X, and {F}_{0} is u.s.c. on {I}_{\epsilon}\times X\times X.
The following two propositions are crucial in our investigation. The first one is almost a direct consequence of the main result in [22] (cf. [25] and [[16], Proposition 2]). The second one allows us to construct a sequence of approximating problems of (2).
Proposition 2.2 Let E be a separable Banach space and F:[0,T]\times E\times E\u22b8E be an upperCarathéodory mapping. If F is globally measurable or quasicompact, then F has the ScorzaDragoni property.
Proposition 2.3 (cf. [[18], Theorem 2.2])
Let E be a Banach space and K\subset E a nonempty, open, convex, bounded set such that 0\in K. Moreover, let \epsilon >0 and V:E\to \mathbb{R} be a Fréchet differentiable function with \dot{V} Lipschitzian in \overline{B(\partial K,\epsilon )} satisfying

(H1) V{}_{\partial K}=0,

(H2) V(x)\le 0, for all x\in \overline{K},

(H3) \parallel \dot{V}(x)\parallel \ge \delta, for all x\in \partial K, where \delta >0 is given.
Then there exist k\in (0,\epsilon ] and a bounded Lipschitzian function \varphi :\overline{B(\partial K,k)}\to E such that \u3008{\dot{V}}_{x},\varphi (x)\u3009=1, for every x\in \overline{B(\partial K,k)}.
Remark 2.1 Let us note that the function x\to \varphi (x)\parallel {\dot{V}}_{x}\parallel, where ϕ and {\dot{V}}_{x} are the same as in Proposition 2.3, is Lipschitzian and bounded in \overline{B(\partial K,k)}. The symbol {\dot{V}}_{x} denotes as usually the first Fréchet derivative of V at x.
Example 2.1 If V satisfies all the assumptions of Proposition 2.3, then it is easy to prove the existence of \sigma \in (0,\epsilon ] such that \parallel {\dot{V}}_{x}\parallel \ge \frac{\delta}{2}, for all x\in \overline{B(\partial K,\sigma )}. Consequently, when E is an arbitrary Hilbert space, we can define \varphi :\overline{B(\partial K,\sigma )}\to E by the formula
which satisfies all the properties mentioned in Proposition 2.3.
Definition 2.2 Let N be a partially ordered set, E be a Banach space and let P(E) denote the family of all nonempty bounded subsets of E. A function \beta :P(E)\to N is called a measure of noncompactness (m.n.c.) in E if \beta (\overline{co\mathrm{\Omega}})=\beta (\mathrm{\Omega}), for all \mathrm{\Omega}\in P(E), where \overline{co\mathrm{\Omega}} denotes the closed convex hull of Ω.
A m.n.c. β is called:

(i)
monotone if \beta ({\mathrm{\Omega}}_{1})\le \beta ({\mathrm{\Omega}}_{2}), for all {\mathrm{\Omega}}_{1}\subset {\mathrm{\Omega}}_{2}\subset E,

(ii)
nonsingular if \beta (\{x\}\cup \mathrm{\Omega})=\beta (\mathrm{\Omega}), for all x\in E and \mathrm{\Omega}\subset E.
If N is a cone in a Banach space, then a m.n.c. β is called:

(iii)
semihomogeneous if \beta (t\mathrm{\Omega})=t\beta (\mathrm{\Omega}), for every t\in \mathbb{R} and every \mathrm{\Omega}\subset E,

(iv)
regular when \beta (\mathrm{\Omega})=0 if and only if Ω is relatively compact,

(v)
algebraically subadditive if \gamma ({\mathrm{\Omega}}_{1}+{\mathrm{\Omega}}_{2})\le \gamma ({\mathrm{\Omega}}_{1})+\gamma ({\mathrm{\Omega}}_{2}), for all {\mathrm{\Omega}}_{1},{\mathrm{\Omega}}_{2}\subset E.
The typical example of an m.n.c. is the Hausdorff measure of noncompactness γ defined, for all \mathrm{\Omega}\subset E by
The Hausdorff m.n.c. is monotone, nonsingular, semihomogeneous and regular. Moreover, if M\in \mathcal{L}(E) and \mathrm{\Omega}\subset E, then (see, e.g., [27])
Let E be a separable Banach space and {\{{f}_{n}\}}_{n}\subset {L}^{1}([0,T],E) be such that \parallel {f}_{n}(t)\parallel \le \alpha (t), \gamma ({\{{f}_{n}(t)\}}_{n})\le c(t), for a.a. t\in [0,T], all n\in \mathbb{N} and suitable \alpha ,c\in {L}^{1}([0,T],\mathbb{R}), then (cf. [27])
Moreover, if h:E\u22b8E is LLipschitzian, then
for all bounded \mathrm{\Omega}\subset E.
Furthermore, for all subsets Ω of E (see e.g. [17]),
Let us now introduce the function
defined on the bounded \mathrm{\Omega}\subset {C}^{1}([0,T],E), where the ordering is induced by the positive cone in {\mathbb{R}}^{2} and where {mod}_{C}(\mathrm{\Omega}) denotes the modulus of continuity of a subset \mathrm{\Omega}\subset C([0,T],E).^{a} It was proved in [19] that the function α given by (7) is an m.n.c. in {C}^{1}([0,T],E) that is monotone, nonsingular and regular.
Definition 2.3 Let E be a Banach space and X\subset E. A multivalued mapping F:X\u22b8E with compact values is called condensing with respect to an m.n.c. β (shortly, βcondensing) if, for every bounded \mathrm{\Omega}\subset X such that \beta (F(\mathrm{\Omega}))\ge \beta (\mathrm{\Omega}), we see that Ω is relatively compact.
A family of mappings G:X\times [0,1]\u22b8E with compact values is called βcondensing if, for every bounded \mathrm{\Omega}\subset X such that \beta (G(\mathrm{\Omega}\times [0,1]))\ge \beta (\mathrm{\Omega}), we see that Ω is relatively compact.
The proof of the main result (cf. Theorem 3.1 below) will be based on the following slight modification of the continuation principle developed in [19]. Since the proof of this modified version differs from the one in [19] only slightly in technical details, we omit it here.
Proposition 2.4 Let us consider the b.v.p.
where \phi :[0,T]\times E\times E\u22b8E is an upperCarathéodory mapping and S\subset A{C}^{1}([0,T],E). Let H:[0,T]\times E\times E\times E\times E\times [0,1]\u22b8E be an upperCarathéodory mapping such that
Moreover, assume that the following conditions hold:

(i)
There exist a closed set {S}_{1}\subset S and a closed, convex set Q\subset {C}^{1}([0,T],E) with a nonempty interior IntQ such that each associated problem
P(q,\lambda )\phantom{\rule{1em}{0ex}}\begin{array}{l}\ddot{x}(t)\in H(t,x(t),\dot{x}(t),q(t),\dot{q}(t),\lambda ),\phantom{\rule{1em}{0ex}}\mathit{\text{for a.a.}}t\in [0,T],\\ x\in {S}_{1},\end{array}\}
where q\in Q and \lambda \in [0,1], has a nonempty, convex set of solutions (denoted by \mathfrak{T}(q,\lambda )).

(ii)
For every nonempty, bounded set \mathrm{\Omega}\subset E\times E\times E\times E, there exists {\nu}_{\mathrm{\Omega}}\in {L}^{1}([0,T],[0,\mathrm{\infty})) such that
\parallel H(t,x,y,u,v,\lambda )\parallel \le {\nu}_{\mathrm{\Omega}}(t),
for a.a. t\in [0,T] and all (x,y,u,v)\in \mathrm{\Omega} and \lambda \in [0,1].

(iii)
The solution mapping \mathfrak{T} is quasicompact and μcondensing with respect to a monotone and nonsingular m.n.c. μ defined on {C}^{1}([0,T],E).

(iv)
For each q\in Q, the set of solutions of problem P(q,0) is a subset of IntQ, i.e. \mathfrak{T}(q,0)\subset IntQ, for all q\in Q.

(v)
For each \lambda \in (0,1), the solution mapping \mathfrak{T}(\cdot ,\lambda ) has no fixed points on the boundary ∂Q of Q.
Then the b.v.p. (8) has a solution in Q.
3 Main result
Combining the foregoing continuation principle with the ScorzaDragonitype technique (cf. Proposition 2.2), we are ready to state the main result of the paper concerning the solvability and localization of a solution of the multivalued Dirichlet problem (2).
Theorem 3.1 Consider the Dirichlet b.v.p. (2). Suppose that F:[0,T]\times E\times E\u22b8E is an upperCarathéodory mapping which is either globally measurable or quasicompact. Furthermore, let K\subset E be a nonempty, open, convex, bounded subset containing 0 of a separable Banach space E satisfying the RadonNikodym property. Let the following conditions ({2}_{\mathrm{i}})({2}_{\mathrm{iii}}) be satisfied:
({2}_{\mathrm{i}}) \gamma (F(t,{\mathrm{\Omega}}_{1}\times {\mathrm{\Omega}}_{2}))\le g(t)(\gamma ({\mathrm{\Omega}}_{1})+\gamma ({\mathrm{\Omega}}_{2})), for a.a. t\in [0,T] and each {\mathrm{\Omega}}_{1}\subset \overline{K}, and each bounded {\mathrm{\Omega}}_{2}\subset E, where g\in {L}^{1}([0,T],[0,\mathrm{\infty})) and γ is the Hausdorff m.n.c. in E.
({2}_{\mathrm{ii}}) For every nonempty, bounded \mathrm{\Omega}\subset E, there exists {\nu}_{\mathrm{\Omega}}\in {L}^{1}([0,T],[0,\mathrm{\infty})) such that
for a.a. t\in [0,T] and all (x,y)\in \mathrm{\Omega}\times E.
({2}_{\mathrm{iii}})
Furthermore, let there exist \epsilon >0 and a function V\in {C}^{2}(E,\mathbb{R}), i.e. a twice continuously differentiable function in the sense of Fréchet, satisfying (H1)(H3) (cf. Proposition 2.3) with Fréchet derivative \dot{V} Lipschitzian in \overline{B(\partial K,\epsilon )}.^{b} Let there still exist h>0 such that
where {\ddot{V}}_{x}(v) denotes the second Fréchet derivative of V at x in the direction (v,v)\in E\times E. Finally, let
for a.a. t\in (0,T) and all x\in \partial K, v\in E, and w\in F(t,x,v).
Then the Dirichlet b.v.p. (2) admits a solution whose values are located in \overline{K}. If, moreover, 0\notin F(t,0,0), for a.a. t\in [0,T], then the obtained solution is nontrivial.
Proof Since the proof of this result is rather technical, it will be divided into several steps. At first, let us define the sequence of approximating problems. For this purpose, let k be as in Proposition 2.3 and consider a continuous function \tau :E\to [0,1] such that \tau (x)=0, for all x\in E\setminus B(\partial K,k), and \tau (x)=1, for all x\in \overline{B(\partial K,\frac{k}{2})}. According to Proposition 2.3 (see also Remark 2.1), the function \stackrel{\u02c6}{\varphi}:E\to E, where
is well defined, continuous and bounded.
Since the mapping (t,x,y)\u22b8F(t,x,y) has, according to Proposition 2.2, the ScorzaDragoni property, we are able to find a decreasing sequence {\{{J}_{m}\}}_{m} of subsets of [0,T] and a mapping {F}_{0}:[0,T]\times E\times E\u22b8E\cup \{\mathrm{\varnothing}\} with compact, convex values such that, for all m\in \mathbb{N},

\mu ({J}_{m})<\frac{1}{m},

[0,T]\setminus {J}_{m} is closed,

(t,x,y)\u22b8{F}_{0}(t,x,y) is u.s.c. on [0,T]\setminus {J}_{m}\times E\times E,

{\nu}_{\overline{K}} is continuous in [0,T]\setminus {J}_{m} (cf. e.g.[2]).
If we put J={\bigcap}_{m=1}^{\mathrm{\infty}}{J}_{m}, then \mu (J)=0, {F}_{0}(t,x,y)\ne \mathrm{\varnothing}, for all t\in [0,T]\setminus J, the mapping (t,x,y)\u22b8{F}_{0}(t,x,y) is u.s.c. on [0,T]\setminus J\times E\times E and {\nu}_{\overline{K}} is continuous in [0,T]\setminus J.
For each m\in \mathbb{N}, let us define the mapping {F}_{m}:[0,T]\times E\times E\u22b8E with compact, convex values by the formula
Let us consider the b.v.p.
Now, let us verify the solvability of problems ({P}_{m}). Let m\in \mathbb{N} be fixed. Since {F}_{0} is globally u.s.c. on [0,T]\setminus J\times E\times E, {F}_{m}(\cdot ,x,y) is measurable, for each (x,y)\in E\times E, and, due to the continuity of \stackrel{\u02c6}{\varphi}, {F}_{m}(t,\cdot ,\cdot ) is u.s.c., for all t\in [0,T]\setminus J. Therefore, {F}_{m} is an upperCarathéodory mapping. Moreover, let us define the upperCarathéodory mapping {H}_{m}:[0,T]\times E\times E\times E\times E\times [0,1]\u22b8E by the formula
Let us show that, when m\in \mathbb{N} is sufficiently large, all assumptions of Proposition 2.4 (for \phi (t,x,\dot{x}):={F}_{m}(t,x,\dot{x})) are satisfied.
For this purpose, let us define the closed set S={S}_{1} by
and let the set Q of candidate solutions be defined as Q:={C}^{1}([0,T],\overline{K}). Because of the convexity of K, the set Q is closed and convex.
For all q\in Q and \lambda \in [0,1], consider still the associated fully linearized problem
and denote by {\mathfrak{T}}_{m} the solution mapping which assigns to each (q,\lambda )\in Q\times [0,1] the set of solutions of {P}_{m}(q,\lambda ).
ad (i) In order to verify condition (i) in Proposition 2.4, we need to show that, for each (q,\lambda )\in Q\times [0,1], the problem {P}_{m}(q,\lambda ) is solvable with a convex set of solutions. So, let (q,\lambda )\in Q\times [0,1] be arbitrary and let {f}_{q,\lambda}(\cdot ) be a measurable selection of {H}_{m}(\cdot ,q(\cdot ),\dot{q}(\cdot ),\lambda ), which surely exists (see, e.g., [[27], Theorem 1.3.5]). According to ({2}_{\mathrm{ii}}) and the definition of {H}_{m}, it is also easy to see that {f}_{q,\lambda}\in {L}^{1}([0,T],E). The homogeneous problem corresponding to b.v.p. {P}_{m}(q,\lambda ),
has only the trivial solution, and therefore the singlevalued Dirichlet problem
admits a unique solution {x}_{q,\lambda}(\cdot ) which is one of solutions of {P}_{m}(q,\lambda ). This is given, for a.a. t\in [0,T], by {x}_{q,\lambda}(t)={\int}_{0}^{T}G(t,s){f}_{q,\lambda}(s)\phantom{\rule{0.2em}{0ex}}ds, where G is the Green function associated to the homogeneous problem (12). The Green function G and its partial derivative \frac{\partial}{\partial t}G are defined by (cf. e.g. [[28], pp.170171])
Thus, the set of solutions of {P}_{m}(q,\lambda ) is nonempty. The convexity of the solution sets follows immediately from the definition of {H}_{m} and the fact that problems {P}_{m}(q,\lambda ) are fully linearized.
ad (ii) Let \mathrm{\Omega}\subset E\times E\times E\times E be bounded. Then, there exists a bounded {\mathrm{\Omega}}_{1}\subset E such that \mathrm{\Omega}\subset {\mathrm{\Omega}}_{1}\times {\mathrm{\Omega}}_{1}\times {\mathrm{\Omega}}_{1}\times {\mathrm{\Omega}}_{1} and, according to ({2}_{\mathrm{ii}}) and the definition of {H}_{m}, there exists \stackrel{\u02c6}{J}\subset [0,T] with \mu (\stackrel{\u02c6}{J})=0 such that, for all t\in [0,T]\setminus (J\cup \stackrel{\u02c6}{J}), (x,y,u,v)\in \mathrm{\Omega} and \lambda \in [0,1],
Therefore, the mapping {H}_{m}(t,q(t),\dot{q}(t),\lambda ) satisfies condition (ii) from Proposition 2.4.
ad (iii) Since the verification of condition (iii) in Proposition 2.4 is technically the most complicated, it will be split into two parts: (iii_{1}) the quasicompactness of the solution operator {\mathfrak{T}}_{m}, (iii_{2}) the condensity of {\mathfrak{T}}_{m} w.r.t. the monotone and nonsingular m.n.c. α defined by (7).
ad (iii_{1}) Let us firstly prove that the solution mapping {\mathfrak{T}}_{m} is quasicompact. Since {C}^{1}([0,T],E) is a complete metric space, it is sufficient to prove the sequential quasicompactness of {\mathfrak{T}}_{m}. Hence, let us consider the sequences \{{q}_{n}\}, \{{\lambda}_{n}\}, {q}_{n}\in Q, {\lambda}_{n}\in [0,1], for all n\in \mathbb{N}, such that {q}_{n}\to q in {C}^{1}([0,T],E) and {\lambda}_{n}\to \lambda. Moreover, let {x}_{n}\in {\mathfrak{T}}_{m}({q}_{n},{\lambda}_{n}), for all n\in \mathbb{N}. Then there exists, for all n\in \mathbb{N}, {k}_{n}(\cdot )\in {F}_{0}(\cdot ,{q}_{n}(\cdot ),{\dot{q}}_{n}(\cdot )) such that
where
and that
Since {q}_{n}\to q and {\dot{q}}_{n}\to \dot{q} in C([0,T],E), there exists a bounded \mathrm{\Omega}\times \mathrm{\Omega}\subset E\times E such that ({q}_{n}(t),{\dot{q}}_{n}(t))\in \mathrm{\Omega}\times \mathrm{\Omega}, for all t\in [0,T] and n\in \mathbb{N}. Therefore, there exists, according to condition ({2}_{\mathrm{ii}}), {\nu}_{\mathrm{\Omega}}\in {L}^{1}([0,T],[0,\mathrm{\infty})) such that \parallel {f}_{n}(t)\parallel \le \varpi (t), for every n\in \mathbb{N} and a.a. t\in [0,T], where \varpi (t):={\nu}_{\mathrm{\Omega}}(t)+2{\nu}_{\overline{K}}(t)\cdot {max}_{x\in \overline{B(\partial K,\epsilon )}}\parallel \stackrel{\u02c6}{\varphi}(x)\parallel.
Moreover, for every n\in \mathbb{N} and a.a. t\in [0,T],
and
Thus, {x}_{n} satisfies, for every n\in \mathbb{N} and a.a. t\in [0,T], \parallel {x}_{n}(t)\parallel \le a and \parallel {\dot{x}}_{n}(t)\parallel \le b, where
Furthermore, for every n\in \mathbb{N} and a.a. t\in [0,T], we have
Hence, the sequences \{{x}_{n}\} and \{{\dot{x}}_{n}\} are bounded and \{{\ddot{x}}_{n}\} is uniformly integrable.
For each t\in [0,T], the properties of the Hausdorff m.n.c. yield
Since {q}_{n}(t)\in \overline{K}, for all t\in [0,T] and all n\in \mathbb{N}, it follows from condition ({2}_{\mathrm{i}}) that, for a.a. t\in [0,T],
Since the function x\to \varphi (x)\parallel {\dot{V}}_{x}\parallel is Lipschitzian on \overline{B(\partial K,\epsilon )} with some Lipschitz constant \stackrel{\u02c6}{L}>0 (see Remark 2.1), we get
Since {q}_{n}\to q and {\dot{q}}_{n}\to \dot{q} in C([0,T],E), we get, for all t\in [0,T], \gamma ({\{{q}_{n}(t)\}}_{n})=\gamma ({\{{\dot{q}}_{n}(t)\}}_{n})=0, which implies that \gamma ({\{{f}_{n}(t)\}}_{n})=0, for all t\in [0,T].
For all (t,s)\in [0,T]\times [0,T], the sequence \{G(t,s){f}_{n}(s)\} is relatively compact as well since, according to the semihomogeneity of the Hausdorff m.n.c.,
Moreover, by means of (4) and (18),
By similar reasoning, we also get
by which \{{x}_{n}(t)\}, \{{\dot{x}}_{n}(t)\} are relatively compact, for all t\in [0,T].
Moreover, since {x}_{n} satisfies for all n\in \mathbb{N} (13), \{{\ddot{x}}_{n}(t)\} is relatively compact, for a.a. t\in [0,T]. Thus, according to [[23], Lemma III.1.30], there exist a subsequence of \{{\dot{x}}_{n}\}, for the sake of simplicity denoted in the same way as the sequence, and x\in {C}^{1}([0,T],E) such that \{{\dot{x}}_{n}\} converges to \dot{x} in C([0,T],E) and \{{\ddot{x}}_{n}\} converges weakly to \ddot{x} in {L}^{1}([0,T],E). According to the classical closure results (cf. e.g. [[27], Lemma 5.1.1]), x\in {\mathfrak{T}}_{m}(q,\lambda ), which implies the quasicompactness of {\mathfrak{T}}_{m}.
ad (iii_{2}) In order to show that, for m\in \mathbb{N} sufficiently large, {\mathfrak{T}}_{m} is αcondensing with respect to the m.n.c. α defined by (7), let us consider a bounded subset \mathrm{\Theta}\subset Q such that \alpha ({\mathfrak{T}}_{m}(\mathrm{\Theta}\times [0,1]))\ge \alpha (\mathrm{\Theta}). Let \{{x}_{n}\}\subset {\mathfrak{T}}_{m}(\mathrm{\Theta}\times [0,1]) be a sequence such that
At first, let us show that the set {\mathfrak{T}}_{m}(\mathrm{\Theta}\times [0,1]) is bounded. If x\in {\mathfrak{T}}_{m}(\mathrm{\Theta}\times [0,1]), then there exist q\in \mathrm{\Theta}, \lambda \in [0,1] and k(\cdot )\in {F}_{0}(\cdot ,q(\cdot ),\dot{q}(\cdot )) such that
with f(t)=\lambda k(t)+{\nu}_{\overline{K}}(t)({\chi}_{{J}_{m}}(t)+\frac{1}{m})\stackrel{\u02c6}{\varphi}(q(t)), for a.a. t\in [0,T].
Since Θ is bounded, there exists \mathrm{\Omega}\subset E such that q(t)\in \mathrm{\Omega}, for all q\in \mathrm{\Theta} and all t\in [0,T]. Hence, according to ({2}_{\mathrm{ii}}), there exists {\nu}_{\mathrm{\Omega}}\in {L}^{1}([0,T]) such that \parallel k(t)\parallel \le {\nu}_{\mathrm{\Omega}}(t), for a.a. t\in [0,T]. Consequently
Similarly,
Thus, the set {\mathfrak{T}}_{m}(\mathrm{\Theta}\times [0,1]) is bounded.
Moreover, we can find \{{q}_{n}\}\subset \mathrm{\Theta}, \{{\lambda}_{n}\}\subset [0,1] and \{{k}_{n}\} satisfying, for a.a. t\in [0,T], {k}_{n}(t)\in {F}_{0}(t,{q}_{n}(t),{\dot{q}}_{n}(t)), such that, for all t\in [0,T], {x}_{n}(t) and {\dot{x}}_{n}(t) are defined by (15) and (16), respectively, where {f}_{n}(t) is defined by (14).
By similar reasoning as in the part ad (iii_{1}), we obtain
for a.a. t\in [0,T], and that
Since {k}_{n}(t)\in {F}_{0}(t,{q}_{n}(t),{\dot{q}}_{n}(t)), for a.a. t\in [0,T], and {q}_{n}\in \mathrm{\Theta}, for all n\in \mathbb{N}, where Θ is a bounded subset of {C}^{1}([0,T],E), there exists \mathrm{\Omega}\subset \overline{K} such that {q}_{n}(t)\in \mathrm{\Omega}, for all n\in \mathbb{N} and t\in [0,T]. Hence, it follows from condition ({2}_{\mathrm{ii}}) that
This implies \parallel G(t,s){f}_{n}(t)\parallel \le G(t,s)({\nu}_{\mathrm{\Omega}}(t)+2\cdot {\nu}_{\overline{K}}(t)\cdot {max}_{x\in \overline{B(\partial K,\epsilon )}}\parallel \stackrel{\u02c6}{\varphi}(x)\parallel ), for a.a. t,s\in [0,T] and all n\in \mathbb{N}.
Moreover, by virtue of the semihomogeneity of the Hausdorff m.n.c., for all (t,s)\in [0,T]\times [0,T], we have
Let us denote
and
According to (4) and (15) we thus obtain for each t\in [0,T],
By similar reasonings, we can see that, for each t\in [0,T],
when starting from condition (16). Subsequently,
Since we assume that \alpha ({\mathfrak{T}}_{m}(\mathrm{\Theta}\times [0,1]))\ge \alpha (\mathrm{\Theta}) and {\{{q}_{n}\}}_{n}\subset \mathrm{\Theta}, we get
Since we have, according to ({2}_{\mathrm{iii}}), \frac{T+4}{4}{\parallel g\parallel}_{{L}^{1}}<1, we can choose {m}_{0}\in \mathbb{N} such that, for all m\in \mathbb{N}, m\ge {m}_{0}, we have
Therefore, we get, for sufficiently large m\in \mathbb{N}, the contradiction \mathcal{S}<\mathcal{S} which ensures the validity of condition (iii) in Proposition 2.4.
ad (iv) For all q\in Q, the set {\mathfrak{T}}_{m}(q,0) coincides with the unique solution {x}_{m} of the linear system
According to (15) and (16), for all t\in [0,T],
and
where {\phi}_{m}(t):={\nu}_{\overline{K}}(t)({\chi}_{{J}_{m}}(t)+\frac{1}{m})\stackrel{\u02c6}{\varphi}(q(t)).
Since
we have, for all t\in [0,T],
Let us now consider r>0 such that rB\subset K. Then it follows from (21) that we are able to find {m}_{0}\in \mathbb{N} such that, for all m\in \mathbb{N}, m\ge {m}_{0}, and t\in [0,T], \parallel {x}_{m}\parallel \le r. Therefore, for all m\in \mathbb{N}, m\ge {m}_{0}, {\mathfrak{T}}_{m}(q,0)\subset IntQ, for all q\in Q, which ensures the validity of condition (iv) in Proposition 2.4.
ad (v) The validity of the transversality condition (v) in Proposition 2.4 can be proven quite analogously as in [16] (see pp.4043 in [16]) with the following differences:

due to the Dirichlet boundary conditions, {t}_{0} belongs to the open interval (0,T),

since A(t)=B(t)=0, we have p(t)={\nu}_{\overline{K}}(t).
In this way, we can prove that there exists {m}_{0}\in \mathbb{N} such that every problem ({P}_{m}), where m\ge {m}_{0}, satisfies all the assumptions of Proposition 2.4. This implies that every such ({P}_{m}) admits a solution, denoted by {x}_{m}, with {x}_{m}(t)\in \overline{K}, for all t\in [0,T]. By similar arguments as in [16], but with the expression Z(4Zk+1) replaced by \frac{T}{4}, according to condition ({2}_{\mathrm{ii}}), we can obtain the result that there exists a subsequence, denoted as the sequence, and a function x\in A{C}^{1}([0,T],E) such that {x}_{m}\to x and {\dot{x}}_{m}\to \dot{x} in C([0,T],E) and also {\ddot{x}}_{m}\rightharpoonup x in {L}^{1}([0,T],E), when m\to \mathrm{\infty}. Thus, a classical closure result (see e.g. [[27], Lemma 5.1.1]) guarantees that x is a solution of (2) satisfying x(t)\in \overline{K}, for all t\in [0,T], and the sketch of proof is so complete. □
The case when F={F}_{1}+{F}_{2}, with {F}_{1}(t,\cdot ,\cdot ) to be completely continuous and {F}_{2}(t,\cdot ,\cdot ) to be Lipschitzian, for a.a. t\in [0,T], represents the most classical example of a map which is γregular w.r.t. the Hausdorff measure of noncompactness γ. The following corollary of Theorem 3.1 can be proved quite analogously as in [[3], Example 6.1 and Remark 6.1].
Corollary 3.1 Let E=H be a separable Hilbert space and let us consider the Dirichlet b.v.p.:
where

(i)
{F}_{1}:[0,T]\times H\times H\u22b8H is an upperCarathéodory, globally measurable, multivalued mapping and {F}_{1}(t,\cdot ,\cdot ):H\times H\u22b8H is completely continuous, for a.a. t\in [0,T], such that
\parallel {F}_{1}(t,x,y)\parallel \le {\nu}_{1}(t,{D}_{0}),
for a.a. t\in [0,T], all x\in H with \parallel x\parallel \le {D}_{0}, where {D}_{0}>0 is an arbitrary constant, {\nu}_{1}\in {L}^{1}([0,T],[0,\mathrm{\infty})), and all y\in H,

(ii)
{F}_{2}:[0,T]\times H\times H\u22b8H is a Carathéodory multivalued mapping such that
\parallel {F}_{2}(t,0,0)\parallel \le {\nu}_{2}(t),\phantom{\rule{1em}{0ex}}\mathit{\text{for a.a.}}t\in [0,T],
where {\nu}_{2}\in {L}^{1}([0,T],[0,\mathrm{\infty})), and {F}_{2}(t,\cdot ,\cdot ):H\times H\u22b8H is Lipschitzian, for a.a. t\in [0,T], with the Lipschitz constant
Moreover, suppose that

(iii)
there exists R>0 such that, for all x\in H with \parallel x\parallel =R, t\in (0,T), y\in H and w\in {F}_{1}(t,x,y)+{F}_{2}(t,x,y), we have
\u3008x,w\u3009>0.
Then the Dirichlet problem (22) admits, according to Theorem 3.1, a solution x(\cdot ) such that \parallel x(t)\parallel \le R, for all t\in [0,T].
Remark 3.1 For {F}_{2}(t,x,y)\equiv 0, the completely continuous mapping {F}_{1}(t,x,y) allows us to make a comparison with classical singlevalued results recalled in the Introduction. Unfortunately, our {F}_{1} in (i) (see also ({2}_{\mathrm{ii}}) in Theorem 3.1) is the only mapping which is (unlike in [[3], Example 6.1 and Remark 6.1], where under some additional restrictions quite liberal growth restrictions were permitted) globally bounded w.r.t. y\in H. Furthermore, our sign condition in (iii) is also (unlike again in [[3], Example 6.1 and Remark 6.1], where under some additional restrictions the Hartmantype condition like (i_{H}) in the Introduction was employed) the most restrictive among their analogies in [6–13]. On the other hand, because of multivalued upperCarathéodory maps {F}_{1}+{F}_{2} in a Hilbert space which are γregular, our result has still, as far as we know, no analogy at all.
4 Illustrative examples
The first illustrative example of the application of Theorem 3.1 concerns the integrodifferential equation
involving discontinuities in a state variable. In this equation, the nonlocal diffusion term {\int}_{\mathbb{R}}k(x,y)u(t,y)\phantom{\rule{0.2em}{0ex}}dy replaces the classical diffusion behavior given by {u}_{xx}(t,x). In dispersal models such an integral term takes into account the longdistance interactions between individuals (see e.g. [29]). Moreover, when φ is linear in {u}_{t}, (23) can be considered as an alternative version of the classical telegraph equation (see e.g. [30] and the references therein), where the classical diffusivity is replaced by the present nonlocal diffusivity.
Telegraph equations appear in many fields such as modeling of an anomalous diffusion, a wave propagation phenomenon, subdiffusive systems or modeling of a pulsate blood flow in arteries (see e.g. [31, 32]).
For the sake of simplicity, we will discuss here only the case when φ is globally bounded w.r.t. {u}_{t}. On the other hand, for nonstrictly localized transversality conditions as in [3], for instance, a suitable linear growth estimate w.r.t. {u}_{t} can be permitted.
Example 4.1 Let us consider the integrodifferential equation (23) with \phi :[0,T]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}, b:[0,T]\to \mathbb{R}, k:\mathbb{R}\times \mathbb{R}\to \mathbb{R}, \psi :\mathbb{R}\to \mathbb{R} and p:\mathbb{R}\to \mathbb{R}. We assume that

(a)
φ is Carathéodory, i.e. \phi (\cdot ,x,y) is measurable, for all x,y\in \mathbb{R}, and \phi (t,\cdot ,\cdot ) is continuous, for a.a. t\in [0,T]; \phi (t,x,\cdot ) is L(t)Lipschitzian with L\in {L}^{1}([0,T]); \phi (t,x,y)\le {\phi}_{1}(t){\phi}_{2}(x), for a.a. t\in [0,T] and all x,y\in \mathbb{R}, where {\phi}_{1}\in {L}^{1}([0,T]) and {\phi}_{2}\in {L}^{2}(\mathbb{R}); \phi (t,x,0)\ne 0, for all a.a. t\in [0,T] and all x\in \mathbb{R},

(b)
b\in {L}^{1}([0,T]) and satisfies b(t)\ge {b}_{0}>1, for a.a. t\in [0,T],

(c)
k\in {L}^{2}(\mathbb{R}\times \mathbb{R}) with {\parallel k\parallel}_{{L}^{2}(\mathbb{R}\times \mathbb{R})}=1,

(d)
p(r)\ge 0, for all r\in \mathbb{R}; and there can exist {r}_{1}<{r}_{2}<\cdots <{r}_{k} such that p(\cdot ) is continuous, for r\ne {r}_{i}, and p(\cdot ) has discontinuities at {r}_{i}, for i=1,\dots ,k, with p({r}_{i}^{\mp}):={lim}_{r\to {r}_{i}^{\mp}}p(r)\in \mathbb{R},

(e)
f is LLipschitzian; L>0; f(0)=0; and xf(x)>0, for all x\ne 0,

(f)
\psi \in {L}^{2}(\mathbb{R}) with {\parallel \psi \parallel}_{{L}^{2}(\mathbb{R})}=1.
Since the function p can have some discontinuities, a solution of (23) satisfying the Dirichlet conditions
will be appropriately interpreted in the sense of Filippov. More precisely, let us define P:\mathbb{R}\u22b8\mathbb{R} by the formula
A function u(t,x) is said to be a solution of (23), (24) if u(t,\cdot )\in {L}^{2}(R), for all t\in [0,T], the map [0,T]\to {L}^{2}(R) defined by t\to u(t,\cdot ) is {C}^{1} if it is a solution of the inclusion
and if it satisfies (24).
If we further assume the existence of R>0 such that
and that
where
then the problem (23), (24) has a solution, in the sense of Filippov, satisfying {\parallel u(t,\cdot )\parallel}_{{L}^{2}(\mathbb{R})}\le R, for a.a. t\in [0,T].
In fact, problem (25), (24) can be transformed into the abstract setting
where y(t):=u(t,\cdot )\in {L}^{2}(\mathbb{R}), for all t\in [0,T], and F:[0,T]\times {L}^{2}(\mathbb{R})\times {L}^{2}(\mathbb{R})\u22b8{L}^{2}(\mathbb{R}) is defined by
where \stackrel{\u02c6}{\phi}:[0,T]\times {L}^{2}(\mathbb{R})\to {L}^{2}(\mathbb{R}), (t,y)\mapsto (x\mapsto \phi (t,x,y(x))), K:{L}^{2}(\mathbb{R})\to {L}^{2}(\mathbb{R}), w\mapsto (x\mapsto {\int}_{\mathbb{R}}k(x,y)w(y)\phantom{\rule{0.2em}{0ex}}dy), \stackrel{\u02c6}{f}:{L}^{2}(\mathbb{R})\to {L}^{2}(\mathbb{R}), y\mapsto (x\mapsto f(y(x))) and \stackrel{\u02c6}{F}:{L}^{2}(\mathbb{R})\u22b8{L}^{2}(\mathbb{R}), y\u22b8\{p\stackrel{\u02c6}{f}(y):p\in P({\int}_{\mathbb{R}}\psi (x)y(x)\phantom{\rule{0.2em}{0ex}}dx)\}.
Let us now examine the properties of F. According to (a), \stackrel{\u02c6}{\phi} is well defined. Given y\in {L}^{2}(\mathbb{R}), let us show that \stackrel{\u02c6}{\phi}(\cdot ,y) is measurable. For this purpose, let Ψ be an arbitrary element in the dual space {({L}^{2}(\mathbb{R}))}^{\prime} of {L}^{2}(\mathbb{R}). Hence, there exists \psi \in {L}^{2}(\mathbb{R}) such that \mathrm{\Psi}(z)={\int}_{\mathbb{R}}\psi (x)z(x)\phantom{\rule{0.2em}{0ex}}dx, for all z\in {L}^{2}(\mathbb{R}), and consequently the composition \mathrm{\Psi}\circ \stackrel{\u02c6}{\phi}(\cdot ,y):[0,T]\to \mathbb{R} is such that t\to {\int}_{\mathbb{R}}\psi (x)\phi (t,x,y(x))\phantom{\rule{0.2em}{0ex}}dx. Since φ is Carathéodory, it is globally measurable, and so the mapping (t,x)\to \psi (x)\phi (t,x,y(x)) is globally measurable as well. This implies that, according to the Fubini Theorem, the mapping \mathrm{\Psi}\circ \stackrel{\u02c6}{\phi}(\cdot ,y) is measurable, too. Finally, since Ψ was arbitrary, according to the Pettis Theorem (see Proposition 2.1), \stackrel{\u02c6}{\phi}(\cdot ,y) is measurable.
Furthermore, let us show that \stackrel{\u02c6}{F} is u.s.c. For this purpose, let {y}_{0}\in {L}^{2}(\mathbb{R}) be fixed.

(i)
If {r}_{0}:={\int}_{\mathbb{R}}\psi (x){y}_{0}(x)\phantom{\rule{0.2em}{0ex}}dx\ne {r}_{i}, i=1,2,\dots ,k, then it is possible to find \delta >0 such that \stackrel{\u02c6}{F}:B({y}_{0},\delta )\to {L}^{2}(\mathbb{R}) is singlevalued, i.e. \stackrel{\u02c6}{F}(y)=p(r)\stackrel{\u02c6}{f}(y), r:={\int}_{\mathbb{R}}\psi (x)y(x)\phantom{\rule{0.2em}{0ex}}dx\in [{r}_{0}\delta ,{r}_{0}+\delta ], for all y\in B({y}_{0},\delta ) and {r}_{i}\notin [{r}_{0}\delta ,{r}_{0}+\delta ], for i=1,2,\dots ,k. Since p is continuous in [{r}_{0}\delta ,{r}_{0}+\delta ] and \stackrel{\u02c6}{f} is Lipschitzian, \stackrel{\u02c6}{F} is continuous in B({y}_{0},\delta ).

(ii)
Let {r}_{0}={r}_{j}, for some j\in \{i=1,2,\dots ,k\} and let U\subset {L}^{2}(\mathbb{R}) be open and such that \stackrel{\u02c6}{F}({y}_{0})\subset U. Moreover, let \sigma >0 be such that r:={\int}_{\mathbb{R}}\psi (x)y(x)\phantom{\rule{0.2em}{0ex}}dx\ne {r}_{i}, i\ne j, for any y\in B({y}_{0},\sigma ). This implies that \stackrel{\u02c6}{F}(y) is equal either to p(r)\stackrel{\u02c6}{f}(y) or to P({r}_{j})\stackrel{\u02c6}{f}(y), for all y\in B({y}_{0},\sigma ). If r<{r}_{j} is such that \stackrel{\u02c6}{F}(y)=p(r)\stackrel{\u02c6}{f}(y), then
\begin{array}{rcl}{\parallel \stackrel{\u02c6}{F}(y)p\left({r}_{j}^{}\right)\stackrel{\u02c6}{f}({y}_{0})\parallel}_{{L}^{2}(\mathbb{R})}& =& {\parallel p(r)\stackrel{\u02c6}{f}(y)p\left({r}_{j}^{}\right)\stackrel{\u02c6}{f}({y}_{0})\parallel}_{{L}^{2}(\mathbb{R})}\\ \le & p(r)p\left({r}_{j}^{}\right)\cdot \parallel \stackrel{\u02c6}{f}(y)\parallel +p\left({r}_{j}^{}\right)\cdot \parallel \stackrel{\u02c6}{f}(y)\stackrel{\u02c6}{f}({y}_{0})\parallel ,\end{array}
which implies that it is possible to find {\sigma}_{1}>0 such that F(y)\subset U, for all y\in B({y}_{0},{\sigma}_{1}). Similarly, we would obtain the same when assuming r>{r}_{j}.
If \stackrel{\u02c6}{F}(y)=P({r}_{j})\stackrel{\u02c6}{f}(y) then, for every p\in P({r}_{j}),
which implies that also in this case it is possible to find {\sigma}_{2}>0 such that F(y)\subset U, for all y\in B({y}_{0},{\sigma}_{2}).
Moreover, according to (a) and (c), \stackrel{\u02c6}{\phi} is a Carathéodory mapping such that \stackrel{\u02c6}{\phi}(t,\cdot ) is L(t)Lipschitzian, for all t\in [0,T], and K is well defined and 1Lipschitzian. It can also be shown that, according to (d) and (e), \stackrel{\u02c6}{F} has compact and convex values. Therefore, the mapping F is globally measurable, and so has the ScorzaDragoni property (cf. Proposition 2.2).
Let us now verify particular assumptions of Theorem 3.1.
Let {\mathrm{\Omega}}_{1}\subset \{y\in {L}^{2}(\mathbb{R})\mid {\parallel y\parallel}_{{L}^{2}(\mathbb{R})}\le R\}. Then, according to (f),
for all y\in {\mathrm{\Omega}}_{1}. Hence,
where m is defined by (28).
Thus,
according to the Lipschitzianity of \stackrel{\u02c6}{f} and property (6). For a.a. t\in [0,T] and all {\mathrm{\Omega}}_{2}\subset {L}^{2}(\mathbb{R}), we have
and so condition ({2}_{\mathrm{i}}) is satisfied with g(t)=L(t)+b(t)+1+m\cdot L. The obtained form of g(t) together with assumption (27) directly guarantee the condition ({2}_{\mathrm{iii}}). It can also be easily shown that properties of F ensure the validity of condition ({2}_{\mathrm{ii}}).
In order to verify conditions imposed on a bounding function, let us define V:{L}^{2}(\mathbb{R})\to \mathbb{R}, \alpha \to \frac{1}{2}({\parallel \alpha \parallel}_{{L}^{2}(\mathbb{R})}^{2}{R}^{2}). The function V\in {C}^{2}({L}^{2}(\mathbb{R}),\mathbb{R}) with {\dot{V}}_{x}:h\to \u3008x,h\u3009 obviously satisfies (10), so it is only necessary to check condition (11). Thus, let \alpha \in {L}^{2}(\mathbb{R}), {\parallel \alpha \parallel}_{{L}^{2}(\mathbb{R})}=R, t\in (0,T), v\in {L}^{2}(\mathbb{R}) and z\in F(t,\alpha ,v). Then there exists {p}^{\ast}\in P({\int}_{\mathbb{R}}\psi (x)\alpha (x)\phantom{\rule{0.2em}{0ex}}dx) such that
Moreover, since {p}^{\ast}\ge 0 and {\int}_{\mathbb{R}}\alpha (x)f(\alpha (x))\phantom{\rule{0.2em}{0ex}}dx\ge 0, we see that
and since
we see that
The properties (a)(f) together with the wellknown Hölder inequality then yield
in view of condition (26), (30), and (31).
Hence, the Dirichlet problem (29) admits, according to Theorem 3.1, a solution y satisfying {\parallel y(t)\parallel}_{{L}^{2}(\mathbb{R})}\le R, for a.a. t\in (0,T). If u(t,x):=y(t)(x), then u is a solution of (24), (25) which is the Filippov solution of the original problem (23), (24).
Finally, we can sum up the above result in the form of the following theorem.
Theorem 4.1 Let the assumptions (a)(f) be satisfied. If still conditions (26), (27) hold, then the problem (23), (24) admits a nontrivial solution u in the sense of Fillippov such that {\parallel u(t,\cdot )\parallel}_{{L}^{2}(\mathbb{R})}\le R.
Remark 4.1 In [[13], Example 5.2], the following formally simpler integrodifferential equation in ℝ:
with nonhomogeneous Dirichlet conditions
was solved provided \tilde{k}:[0,1]\times [0,1]\to (0,\mathrm{\infty}) is a positive kernel of the HilbertSchmidttype and the norms {\parallel {u}_{0}\parallel}_{{L}^{2}([0,1])} and {\parallel {u}_{1}\parallel}_{{L}^{2}([0,1])} are finite.
After the homogenization of boundary conditions, the Dirichlet problem takes the form
where \tilde{\phi}(t,x):={\int}_{0}^{1}\tilde{k}(x,y)\{[{u}_{1}(y){u}_{0}(y)]t+{u}_{0}(y)\}\phantom{\rule{0.2em}{0ex}}dy.
Thus, it can be naturally extended onto the infinite strip [0,1]\times \mathbb{R}, into the form (23), (24), where
and b(t)\equiv 0, p(r)\equiv 0 or f(s)\equiv 0.
The result in [[13], Example 5.2] cannot be, however, deduced from Theorem 3.1, because condition (b) in Example 4.1 cannot be satisfied in this way.
On the other hand, the linear term with coefficient b could not be implemented in their equation, because it is not completely continuous in (36) below, as required in [13].
In view of the arguments in Remark 4.1, we can conclude by the second illustrative example.
Example 4.2 Consider the following nonhomogeneous Dirichlet problem in ℝ:
where \tilde{k}:[0,1]\times [0,1]\to (0,\mathrm{\infty}) is a positive kernel of the HilbertSchmidttype such that
and b\in {L}^{1}((0,1)) is such that b(t)\ge {b}_{0}>0, for a.a. t\in (0,1).
Furthermore, let there exist a constant L<\frac{4}{5} such that
The properties of {u}_{0} and {u}_{1} guarantee that there exists B\ge 0 such that
We will show that, under (33) and (34), problem (32) is solvable, in the abstract setting, by means of Corollary 3.1.
Problem (32) can be homogenized as follows:
where
with w(t,x):={\int}_{0}^{1}\tilde{k}(x,y)\{[{u}_{1}(y){u}_{0}(y)]t+{u}_{0}(y)\}\phantom{\rule{0.2em}{0ex}}dy.
Since the HilbertSchmidt operator
where \overline{k}(y)(\cdot ):=\tilde{k}(x,y) is well known to be completely continuous (cf. [[13], Example 5.2]) and b(t)(\cdot ):{L}^{2}([0,1])\to {L}^{2}([0,1]) is, according to (33), LLipschitzian with L<\frac{4}{5}, conditions (i), (ii) in Corollary 3.1 can be easily satisfied, for \overline{u}(t):=u(t,x), \overline{u}\in {L}^{2}([0,1]),
where \overline{\phi}(t):=b(t)\{[{\overline{u}}_{1}{\overline{u}}_{0}]t+{\overline{u}}_{0}\}+\overline{w}(t), \overline{w}(t):=w(t,x),
and
In this setting, problem (35) takes the abstract form as (22), namely
Since \u3008f(\overline{u}),\overline{u}\u3009\ge 0 holds, for all \overline{u}\in {L}^{2}([0,1]) (see [[13], Example 5.2]) one can check that the strict inequality in (iii) in Corollary 3.1 can be easily satisfied, for (32), whenever
Hence, applying Corollary 3.1, problem (36) admits a solution, say \stackrel{\u02c6}{u}(\cdot ), such that
where R satisfies (37), and subsequently the same is true for (32), i.e.
as claimed.
After all, we can sum up the sufficient conditions for the existence of a solution \stackrel{\u02c6}{u} of (32) satisfying (38) as follows:

\tilde{k} is a positive kernel of the HilbertSchmidt operator with the finite norm
{k}_{0}:={\parallel \tilde{k}\parallel}_{{L}^{2}([0,1]\times [0,1])}<\mathrm{\infty}, 
there exists {b}_{0}>0, L<\frac{4}{5}: {b}_{0}\le b(t)\le L, for a.a. t\in (0,1),

condition (37) holds.
Endnotes
^{a}The m.n.c. {mod}_{C}(\mathrm{\Omega}) is a monotone, nonsingular and algebraically subadditive on C([0,T],E) (cf. e.g. [27]) and it is equal to zero if and only if all the elements x\in \mathrm{\Omega} are equicontinuous.
^{b}Since a {C}^{2}function V has only a locally Lipschitzian Fréchet derivative \dot{V} (cf. e.g. [21]), we had to assume explicitly the global Lipschitzianity of \dot{V} in a noncompact set \overline{B(\partial K,\epsilon )}.
References
Hartman P: Ordinary Differential Equations. Willey, New York; 1964.
ScorzaDragoni G: Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un’altra variabile. Rend. Semin. Mat. Univ. Padova 1948, 17: 102106.
Andres J, Malaguti L, Pavlačková M: Dirichlet problem in Banach spaces: the bound sets approach. Bound. Value Probl. 2013., 2013: Article ID 25
ScorzaDragoni G: Sui sistemi di equazioni integrali non lineari. Rend. Semin. Mat. Univ. Padova 1936, 7: 135.
ScorzaDragoni G: Sul problema dei valori ai limiti per i systemi di equazioni differenziali del secondo ordine. Boll. Unione Mat. Ital. 1935, 14: 225230.
Amster P, Haddad J: A HartmanNagumo type conditions for a class of contractible domains. Topol. Methods Nonlinear Anal. 2013, 41(2):287304.
Gaines RE, Mawhin J LNM 568. In Coincidence Degree, and Nonlinear Differential Equations. Springer, Berlin; 1977.
Granas A, Guenther RB, Lee JW: Some existence principles in the Carathéodory theory of nonlinear differential system. J. Math. Pures Appl. 1991, 70: 153196.
Hartman P: On boundary value problems for systems of ordinary differential equations. Trans. Am. Math. Soc. 1960, 96: 493509. 10.1090/S00029947196001245535
Lasota A, Yorke JA: Existence of solutions of twopoint boundary value problems for nonlinear systems. J. Differ. Equ. 1972, 11(3):509518. 10.1016/00220396(72)900630
Mawhin J: Boundary value problems for nonlinear second order vector differential equations. J. Differ. Equ. 1974, 16: 257269. 10.1016/00220396(74)900138
Mawhin J: Two point boundary value problems for nonlinear second order differential equations in Hilbert spaces. Tohoku Math. J. 1980, 32: 225233. 10.2748/tmj/1178229639
Schmitt K, Thompson RC: Boundary value problems for infinite systems of secondorder differential equations. J. Differ. Equ. 1975, 18(2):277295. 10.1016/00220396(75)900637
Mawhin J: The BernsteinNagumo problem and twopoint boundary value problems for ordinary differential equations. In Qualitative Theory of Differential Equations Edited by: Farkas M. 1981, 709740. Budapest
Pavlačková, M: A ScorzaDragoni approach to Dirichlet problem with an upperCarathéodory righthand side. Topol. Methods Nonlinear Anal. (to appear)
Andres J, Malaguti L, Pavlačková M: A ScorzaDragoni approach to secondorder boundary value problems in abstract spaces. Appl. Math. Inform. Sci. 2012, 6(2):177192.
Andres J, Malaguti L, Taddei V: On boundary value problems in Banach spaces. Dyn. Syst. Appl. 2009, 18: 275302.
Cecchini S, Malaguti L, Taddei V: Strictly localized bounding functions and Floquet boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2011., 2011: Article ID 47
Andres J, Malaguti L, Pavlačková M: On secondorder boundary value problems in Banach spaces: a bound sets approach. Topol. Methods Nonlinear Anal. 2011, 37(2):303341.
Andres J, Väth M: Coincidence index for noncompact mappings on nonconvex sets. Nonlinear Funct. Anal. Appl. 2002, 7(4):619658.
Papageorgiou NS, KyritsiYiallourou ST: Handbook of Applied Analysis. Springer, Berlin; 2009.
Rzezuchowski T: ScorzaDragoni type theorem for upper semicontinuous multivalued functions. Bull. Acad. Pol. Sci., Sér. Sci. Math. 1980, 28(12):6166.
Andres J, Górniewicz L Topological Fixed Point Theory and Its Applications 1. In Topological Fixed Point Principles for Boundary Value Problems. Kluwer Academic, Dordrecht; 2003.
Pettis BJ: On the integration in vector spaces. Trans. Am. Math. Soc. 1938, 44(2):277304. 10.1090/S00029947193815019708
Bader R, Kryszewski W: On the solution set of differential inclusions and the periodic problem in Banach spaces. Nonlinear Anal. 2003, 54(4):707754. 10.1016/S0362546X(03)000981
Hu S, Papageorgiou NS: Handbook of Multivalued Analysis. Volume I: Theory. Kluwer Academic, Dordrecht; 1997.
Kamenskii MI, Obukhovskii VV, Zecca P: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter, Berlin; 2001.