Open Access

On fractional differential inclusions with anti-periodic type integral boundary conditions

Boundary Value Problems20132013:82

DOI: 10.1186/1687-2770-2013-82

Received: 4 October 2012

Accepted: 12 March 2013

Published: 10 April 2013

Abstract

This paper investigates the existence of solutions for fractional differential inclusions of order q ( 2 , 3 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq1_HTML.gif with anti-periodic type integral boundary conditions by means of some standard fixed point theorems for inclusions. Our results include the cases when the multivalued map involved in the problem has convex as well as non-convex values. The paper concludes with an illustrative example.

MSC:34A60, 34A08.

Keywords

fractional differential inclusions anti-periodic integral boundary conditions fixed point theorems

1 Introduction

The topic of fractional differential equations and inclusions has recently emerged as a popular field of research due to its extensive development and applications in several disciplines such as physics, mechanics, chemistry, engineering, etc. [15]. An important characteristic of a fractional-order differential operator, in contrast to its integer-order counterpart, is its nonlocal nature. This feature of fractional-order operators (equations) is regarded as one of the key factors for the popularity of the subject. As a matter of fact, the use of fractional-order operators in the mathematical modeling of several real world processes gives rise to more realistic models as these operators are capable of describing memory and hereditary properties. For some recent results on fractional differential equations, see [622] and the references cited therein, whereas some recent work dealing with fractional differential inclusions can be found in [2328].

In this paper, we study a boundary value problem of fractional differential inclusions with anti-periodic type integral boundary conditions given by
{ D q c x ( t ) x ( t ) F ( t , x ( t ) ) , 0 < t < T , 2 < q 3 , x ( j ) ( 0 ) λ j x ( j ) ( T ) = μ j 0 T g j ( s , x ( s ) ) d s , j = 0 , 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equ1_HTML.gif
(1.1)

where D q c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq2_HTML.gif denotes the Caputo derivative of fractional order q, x ( j ) ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq3_HTML.gif denotes j th derivative of x ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq4_HTML.gif with x ( 0 ) ( ) = x ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq5_HTML.gif, F : [ 0 , T ] × R P ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq6_HTML.gif is a multivalued map, P ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq7_HTML.gif is the family of all subsets of , g j : [ 0 , T ] × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq8_HTML.gif are given continuous functions and λ j , μ j R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq9_HTML.gif ( λ j 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq10_HTML.gif).

The present work is motivated by a recent paper [22], where the authors considered (1.1) with F as a single-valued map. The existence of solutions for problem (1.1) has been discussed for the cases when the right-hand side is convex as well as non-convex valued. The first result is based on the nonlinear alternative of Leray-Schauder type, whereas the second result is established by combining the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values. In the third result, we use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler. Though the methods used are well known, their exposition in the framework of problem (1.1) is new. We recall some preliminary facts about fractional calculus and multivalued maps in Section 2, while the main results are presented in Section 3.

2 Preliminaries

2.1 Fractional calculus

Let us recall some basic definitions of fractional calculus [13].

Definition 2.1 Let h : [ 0 , ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq11_HTML.gif be an ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq12_HTML.gif-times absolutely continuous function. Then the Caputo derivative of fractional order ν for h is defined as
D ν c h ( t ) = 1 Γ ( n ν ) 0 t ( t s ) n ν 1 h ( n ) ( s ) d s , n 1 < ν < n , n = [ ν ] + 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equa_HTML.gif

where [ ν ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq13_HTML.gif denotes the integer part of the real number ν.

Definition 2.2 The Riemann-Liouville fractional integral of order ν is defined as
I ν h ( t ) = 1 Γ ( ν ) 0 t g ( s ) ( t s ) 1 ν d s , ν > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equb_HTML.gif

provided the integral exists.

Definition 2.3 A function x A C 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq14_HTML.gif is called a solution of problem (1.1) if there exists a function v L 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq15_HTML.gif with v ( t ) F ( t , x ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq16_HTML.gif, a.e. [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq17_HTML.gif such that D α x ( t ) = v ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq18_HTML.gif, a.e. [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq19_HTML.gif and x ( j ) ( 0 ) λ j x ( j ) ( T ) = μ j 0 T g j ( s , x ( s ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq20_HTML.gif, j = 0 , 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq21_HTML.gif.

In the sequel, the following lemma plays a pivotal role.

Lemma 2.4 ([22])

For a given y C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq22_HTML.gif and 2 < q 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq23_HTML.gif, the unique solution of the equation D q c x ( t ) = y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq24_HTML.gif, t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq25_HTML.gif subject to the boundary conditions of (1.1) is given by
x ( t ) = 0 t ( t s ) q 1 Γ ( q ) y ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) y ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) y ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) y ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equ2_HTML.gif
(2.1)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equc_HTML.gif

2.2 Basic concepts of multivalued analysis

Let us begin this section with some basic concepts of multi-valued maps [29, 30].

Let X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq26_HTML.gif denote a normed space equipped with the norm | | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq27_HTML.gif. A multivalued map G : X P ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq28_HTML.gif is

  • convex (closed) valued if G ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq29_HTML.gif is convex (closed) for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq30_HTML.gif;

  • bounded on bounded sets if G ( B ) = x B G ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq31_HTML.gif is bounded in X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq26_HTML.gif for all bounded sets B in X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq26_HTML.gif, that is, sup x B { sup { | y | : y G ( x ) } } < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq32_HTML.gif;

  • upper semi-continuous (u.s.c.) on X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq26_HTML.gif if for each x 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq33_HTML.gif, the set G ( x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq34_HTML.gif is a nonempty closed subset of X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq26_HTML.gif, and if for each open set N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq35_HTML.gif of X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq26_HTML.gif containing G ( x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq34_HTML.gif, there exists an open neighborhood N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq36_HTML.gif of x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq37_HTML.gif such that G ( N 0 ) N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq38_HTML.gif;

  • completely continuous if G ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq39_HTML.gif is relatively compact for every bounded set B in X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq26_HTML.gif.

Remark 2.5 If the multivalued map G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq40_HTML.gif is completely continuous with nonempty compact values, then G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq40_HTML.gif is u.s.c. if and only if G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq40_HTML.gif has a closed graph, that is, x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq41_HTML.gif, y n y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq42_HTML.gif, y n G ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq43_HTML.gif imply that y G ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq44_HTML.gif.

Definition 2.6 The multivalued map G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq40_HTML.gif has a fixed point if there is x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq30_HTML.gif such that x G ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq45_HTML.gif. The fixed point set of the map G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq40_HTML.gif is denoted by FixG.

Definition 2.7 A multivalued map G : [ 0 , T ] P ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq46_HTML.gif with nonempty compact convex values is said to be measurable if for any x R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq47_HTML.gif, the function
t d ( x , F ( t ) ) = inf { | x y | : y F ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equd_HTML.gif

is measurable.

Let C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq48_HTML.gif denote the Banach space of all continuous functions from [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq17_HTML.gif into with the norm
x = sup { | x ( t ) | : t [ 0 , T ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Eque_HTML.gif
Let L 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq49_HTML.gif be the Banach space of measurable functions x : [ 0 , T ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq50_HTML.gif which are Lebesgue integrable and normed by
x L 1 = 0 T | x ( t ) | d t for all  x L 1 ( [ 0 , T ] , R ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equf_HTML.gif

Definition 2.8 A multivalued map G : [ 0 , T ] × R P ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq51_HTML.gif is called Carathéodory if t G ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq52_HTML.gif is measurable for each x R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq53_HTML.gif and x G ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq54_HTML.gif is upper semicontinuous for almost all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq55_HTML.gif. A Carathéodory function G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq40_HTML.gif is said to be L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq56_HTML.gif-Carathéodory if, for each δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq57_HTML.gif, there exists φ δ L 1 ( [ 0 , T ] , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq58_HTML.gif such that G ( t , x ) = sup { | v | : v G ( t , x ) } φ δ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq59_HTML.gif for all x δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq60_HTML.gif and for a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq61_HTML.gif.

For each y C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq62_HTML.gif, we define the set of selections of F by
S F , y : = { v L 1 ( [ 0 , T ] , R ) : v ( t ) F ( t , y ( t ) )  for a.e.  t [ 0 , T ] } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equg_HTML.gif

Let W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq63_HTML.gif denote a nonempty closed subset of a Banach space E, and let G : W P ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq64_HTML.gif be a multivalued operator with nonempty closed values. The map G is lower semi-continuous (l.s.c.) if the set { y W : G ( y ) B } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq65_HTML.gif is open for any open set B in E. Let A be a subset of [ 0 , T ] × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq66_HTML.gif. A is L B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq67_HTML.gif measurable if A belongs to the σ-algebra generated by all sets of the form J × D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq68_HTML.gif, where J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq69_HTML.gif is Lebesgue measurable in [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq17_HTML.gif and D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq70_HTML.gif is Borel measurable in . A subset A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq71_HTML.gif of L 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq72_HTML.gif is decomposable if for all u , v A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq73_HTML.gif and measurable J [ 0 , T ] = J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq74_HTML.gif, the function u χ J + v χ J J A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq75_HTML.gif, where χ J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq76_HTML.gif stands for the characteristic function of J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq69_HTML.gif.

Definition 2.9 Let Y be a separable metric space. A multivalued operator N : Y P ( L 1 ( [ 0 , T ] , R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq77_HTML.gif has the property (BC) if N is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values.

Let F : [ 0 , T ] × R P ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq78_HTML.gif be a multivalued map with nonempty compact values. Define a multivalued operator F : C ( [ 0 , T ] × R ) P ( L 1 ( [ 0 , T ] , R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq79_HTML.gif associated with F as
F ( x ) = { w L 1 ( [ 0 , T ] , R ) : w ( t ) F ( t , x ( t ) )  for a.e.  t [ 0 , 1 ] } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equh_HTML.gif

which is called the Nemytskii operator associated with F.

Definition 2.10 Let F : [ 0 , T ] × R P ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq80_HTML.gif be a multivalued function with nonempty compact values. We say F is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator is lower semi-continuous and has nonempty closed and decomposable values.

Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq81_HTML.gif be a metric space induced from the normed space ( X ; ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq82_HTML.gif. Consider H d : P ( X ) × P ( X ) R { } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq83_HTML.gif given by
H d ( A , B ) = max { sup a A d ( a , B ) , sup b B d ( A , b ) } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equi_HTML.gif

where d ( A , b ) = inf a A d ( a ; b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq84_HTML.gif and d ( a , B ) = inf b B d ( a ; b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq85_HTML.gif. Then ( P b , c l ( X ) , H d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq86_HTML.gif is a metric space and ( P c l ( X ) , H d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq87_HTML.gif is a generalized metric space (see [31]), where P c l ( X ) = { Y P ( X ) : Y  is closed } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq88_HTML.gif, and P b , c l ( X ) = { Y P ( X ) : Y  is bounded and closed } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq89_HTML.gif.

Definition 2.11 A multivalued operator N : X P c l ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq90_HTML.gif is called γ-Lipschitz if and only if there exists γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq91_HTML.gif such that H d ( N ( x ) , N ( y ) ) γ d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq92_HTML.gif for each x , y ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq93_HTML.gif and is a contraction if and only if it is γ-Lipschitz with γ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq94_HTML.gif.

3 Existence results

3.1 The Carathéodory case

We recall the following lemmas to prove the existence of solutions for problem (1.1) when the multivalued map F in (1.1) is of Carathéodory type.

Lemma 3.1 (Nonlinear alternative for Kakutani maps) [32]

Let E be a Banach space, let C be a closed convex subset of E, let U be an open subset of C, and 0 U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq95_HTML.gif. Suppose that F : U ¯ P c p , c ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq96_HTML.gif is an upper semicontinuous compact map; here P c p , c ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq97_HTML.gif denotes the family of nonempty, compact convex subsets of C. Then either
  1. (i)

    F has a fixed point in U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq98_HTML.gif, or

     
  2. (ii)

    there is an u U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq99_HTML.gif and λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq100_HTML.gif with u λ F ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq101_HTML.gif.

     

Lemma 3.2 ([33])

Let X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq26_HTML.gif be a Banach space, and let P c p , c ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq102_HTML.gif denote a family of nonempty, compact and convex subsets of X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq26_HTML.gif. Let F : [ 0 , T ] × R P c p , c ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq103_HTML.gif be an L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq56_HTML.gif-Carathéodory multivalued map, and let Θ be a linear continuous mapping from L 1 ( [ 0 , T ] , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq104_HTML.gif to C ( [ 0 , T ] , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq105_HTML.gif. Then the operator
Θ S F : C ( [ 0 , T ] , X ) P c p , c ( C ( [ 0 , T ] , X ) ) , x ( Θ S F ) ( x ) = Θ ( S F , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equj_HTML.gif

is a closed graph operator in C ( [ 0 , T ] , X ) × C ( [ 0 , T ] , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq106_HTML.gif.

Theorem 3.3 Suppose that

(H1) F : [ 0 , T ] × R P ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq107_HTML.gif is Carathéodory and has nonempty compact and convex values;

(H2) there exists a continuous nondecreasing function ψ : [ 0 , ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq108_HTML.gif and a function p L 1 ( [ 0 , T ] , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq109_HTML.gif such that
F ( t , x ) P : = sup { | y | : y F ( t , x ) } p ( t ) ψ ( x ) for each ( t , x ) [ 0 , T ] × R ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equk_HTML.gif
(H3) there exist continuous nondecreasing functions ψ j : [ 0 , ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq110_HTML.gif and functions p j L 1 ( [ 0 , T ] , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq111_HTML.gif such that
| g j ( t , x ) | p j ( t ) ψ j ( x ) , j = 0 , 1 , 2 , for each ( t , x ) [ 0 , T ] × R ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equl_HTML.gif
(H4) there exists a constant M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq112_HTML.gif such that
M ψ ( M ) Ω 1 p L 1 + ψ 0 ( M ) | μ 0 ξ 1 | p 0 L 1 + ψ 1 ( M ) | μ 1 η 2 | p 1 L 1 + ψ 2 ( M ) | μ 2 η 1 | p 2 L 1 > 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equm_HTML.gif
where
Ω 1 = T q 1 Γ ( q ) { 1 + | λ 0 ξ 1 | + | λ 1 η 2 | ( q 1 ) T 1 + | λ 2 η 1 | q ( q 1 ) ( q 2 ) T 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equn_HTML.gif

Then the boundary value problem (1.1) has at least one solution on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq17_HTML.gif.

Proof Define the operator Ω F : C ( [ 0 , T ] , R ) P ( C ( [ 0 , T ] , R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq113_HTML.gif by
Ω F ( x ) = { h C ( [ 0 , T ] , R ) : h ( t ) = { 0 t ( t s ) q 1 Γ ( q ) v ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s , 0 t 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equo_HTML.gif

for v S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq114_HTML.gif. We will show that Ω F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq115_HTML.gif satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As the first step, we show that Ω F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq115_HTML.gif is convex for each x C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq116_HTML.gif. This step is obvious since S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq117_HTML.gif is convex (F has convex values), and therefore we omit the proof.

In the second step, we show that Ω F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq115_HTML.gif maps bounded sets (balls) into bounded sets in C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq118_HTML.gif. For a positive number ρ, let B ρ = { x C ( [ 0 , T ] , R ) : x ρ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq119_HTML.gif be a bounded ball in C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq120_HTML.gif. Then, for each h Ω F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq121_HTML.gif, x B ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq122_HTML.gif, there exists v S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq123_HTML.gif such that
h ( t ) = 0 t ( t s ) q 1 Γ ( q ) v ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equp_HTML.gif
Then for t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq25_HTML.gif we have
| h ( t ) | 0 t ( t s ) q 1 Γ ( q ) | v ( s ) | d s + | λ 0 ξ 1 | 0 T ( T s ) q 1 Γ ( q ) | v ( s ) | d s + | λ 1 η 2 | 0 T ( T s ) q 2 Γ ( q 1 ) | v ( s ) | d s + | λ 2 η 1 | 0 T ( T s ) q 3 Γ ( q 2 ) | v ( s ) | d s + | μ 0 ξ 1 | 0 T | g 0 ( s , x ( s ) ) | d s + | μ 1 η 2 | 0 T | g 1 ( s , x ( s ) ) | d s + | μ 2 η 1 | 0 T | g 2 ( s , x ( s ) ) | d s ψ ( x ) { T q 1 Γ ( q ) + | λ 0 ξ 1 | T q 1 Γ ( q ) + | λ 1 η 2 | T q 2 Γ ( q 1 ) + | λ 2 η 1 | T q 3 Γ ( q 2 ) } 0 T p ( s ) d s + ψ 0 ( x ) | μ 0 ξ 1 | 0 T p 0 ( s ) d s + ψ 1 ( x ) | μ 1 η 2 | 0 T p 1 ( s ) d s + ψ 2 ( x ) | μ 2 η 1 | 0 T p 2 ( s ) d s ψ ( x ) Ω 1 p L 1 + ψ 0 ( x ) | μ 0 ξ 1 | p 0 L 1 + ψ 1 ( x ) | μ 1 η 2 | p 1 L 1 + ψ 2 ( x ) | μ 2 η 1 | p 2 L 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equq_HTML.gif
Thus,
h ψ ( ρ ) Ω 1 p L 1 + ψ 0 ( ρ ) | μ 0 ξ 1 | p 0 L 1 + ψ 1 ( ρ ) | μ 1 η 2 | p 1 L 1 + ψ 2 ( ρ ) | μ 2 η 1 | p 2 L 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equr_HTML.gif
Now we show that Ω F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq115_HTML.gif maps bounded sets into equicontinuous sets of C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq118_HTML.gif. Let t , t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq124_HTML.gif with t < t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq125_HTML.gif and x B ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq122_HTML.gif. For each h Ω F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq126_HTML.gif, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equs_HTML.gif

Obviously, the right-hand side of the above inequality tends to zero independently of x B ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq122_HTML.gif as t t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq127_HTML.gif. As Ω F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq115_HTML.gif satisfies the above three assumptions, it follows by the Ascoli-Arzelá theorem that Ω F : C ( [ 0 , T ] , R ) P ( C ( [ 0 , T ] , R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq128_HTML.gif is completely continuous.

In our next step, we show that Ω F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq115_HTML.gif has a closed graph. Let x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq129_HTML.gif, h n Ω F ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq130_HTML.gif and h n h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq131_HTML.gif. Then we need to show that h Ω F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq132_HTML.gif. Associated with h n Ω F ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq130_HTML.gif, there exists v n S F , x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq133_HTML.gif such that for each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq61_HTML.gif,
h n ( t ) = 0 t ( t s ) q 1 Γ ( q ) v n ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v n ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v n ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v n ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x n ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x n ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x n ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equt_HTML.gif
Thus it suffices to show that there exists v S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq134_HTML.gif such that for each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq61_HTML.gif,
h ( t ) = 0 t ( t s ) q 1 Γ ( q ) v ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equu_HTML.gif
Let us consider the continuous linear operator Θ : L 1 ( [ 0 , T ] , R ) C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq135_HTML.gif given by
f Θ ( f ) ( t ) = 0 t ( t s ) q 1 Γ ( q ) v ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equv_HTML.gif
Observe that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equw_HTML.gif
Thus, it follows by Lemma 3.2 that Θ S F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq136_HTML.gif is a closed graph operator. Further, we have h n ( t ) Θ ( S F , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq137_HTML.gif. Since x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq129_HTML.gif, therefore, we have
h ( t ) = 0 t ( t s ) q 1 Γ ( q ) v ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equx_HTML.gif

for some v S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq134_HTML.gif.

Finally, we show there exists an open set U C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq138_HTML.gif with x Ω F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq139_HTML.gif for any λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq100_HTML.gif and all x U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq140_HTML.gif. Let λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq141_HTML.gif and x λ Ω F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq142_HTML.gif. Then there exists v L 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq143_HTML.gif with v S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq123_HTML.gif such that, for t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq61_HTML.gif, we have
h ( t ) = 0 t ( t s ) q 1 Γ ( q ) v ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equy_HTML.gif
and using the computations of the second step above, we have
h ψ ( x ) { T q 1 Γ ( q ) + | λ 0 ξ 1 | T q 1 Γ ( q ) + | λ 1 η 2 | T q 2 Γ ( q 1 ) + | λ 2 η 1 | T q 3 Γ ( q 2 ) } 0 T p ( s ) d s + ψ 0 ( x ) | μ 0 ξ 1 | 0 T p 0 ( s ) d s + ψ 1 ( x ) | μ 1 η 2 | 0 T p 1 ( s ) d s + ψ 2 ( x ) | μ 2 η 1 | 0 T p 2 ( s ) d s ψ ( x ) Ω 1 p L 1 + ψ 0 ( x ) | μ 0 ξ 1 | p 0 L 1 + ψ 1 ( x ) | μ 1 η 2 | p 1 L 1 + ψ 2 ( x ) | μ 2 η 1 | p 2 L 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equz_HTML.gif
Consequently, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equaa_HTML.gif
In view of (H4), there exists M such that x M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq144_HTML.gif. Let us set
U = { x C ( [ 0 , T ] , R ) : x < M } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equab_HTML.gif

Note that the operator Ω F : U ¯ P ( C ( [ 0 , T ] , R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq145_HTML.gif is upper semicontinuous and completely continuous. From the choice of U, there is no x U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq146_HTML.gif such that x λ Ω F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq147_HTML.gif for some λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq100_HTML.gif. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we deduce that Ω F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq115_HTML.gif has a fixed point x U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq148_HTML.gif which is a solution of problem (1.1). This completes the proof. □

3.2 The lower semicontinuous case

This section deals with the case when F is not necessarily convex valued. Our strategy to deal with this problem is based on the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo for lower semi-continuous maps with decomposable values.

Lemma 3.4 (Bressan and Colombo [34])

Let Y be a separable metric space, and let N : Y P ( L 1 ( [ 0 , T ] , R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq149_HTML.gif be a multivalued operator satisfying the property (BC). Then N has a continuous selection, that is, there exists a continuous function (single-valued) g : Y L 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq150_HTML.gif such that g ( x ) N ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq151_HTML.gif for every x Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq152_HTML.gif.

Theorem 3.5 Assume that (H2), (H3), (H4) and the following condition hold:

(H4) F : [ 0 , T ] × R P ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq107_HTML.gif is a nonempty compact-valued multivalued map such that
  1. (a)

    ( t , x ) F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq153_HTML.gif is L B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq154_HTML.gif measurable;

     
  2. (b)

    x F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq155_HTML.gif is lower semicontinuous for each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq61_HTML.gif;

     

then the boundary value problem (1.1) has at least one solution on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq17_HTML.gif.

Proof It follows from (H2) and (H4) that F is of l.s.c. type. Then from Lemma 3.4, there exists a continuous function f : A C 1 ( [ 0 , T ] , R ) L 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq156_HTML.gif such that f ( x ) F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq157_HTML.gif for all x C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq158_HTML.gif.

Consider the problem
{ D q c x ( t ) x ( t ) = f ( x ( t ) ) , t [ 0 , T ] , x ( j ) ( 0 ) λ j x ( j ) ( T ) = μ j 0 T g j ( s , x ( s ) ) d s , j = 0 , 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equ3_HTML.gif
(3.1)
Observe that if x A C 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq159_HTML.gif is a solution of (3.1), then x is a solution to problem (1.1). In order to transform problem (3.1) into a fixed point problem, we define the operator Ω F ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq160_HTML.gif as
Ω F ¯ x ( t ) = 0 t ( t s ) q 1 Γ ( q ) f ( x ( s ) ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) f ( x ( s ) ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) f ( x ( s ) ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) f ( x ( s ) ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equac_HTML.gif

It can easily be shown that Ω F ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq160_HTML.gif is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 3.3. So, we omit it. This completes the proof. □

3.3 The Lipschitz case

Here we show the existence of solutions for problem (1.1) with a nonconvex valued right-hand side by applying a fixed point theorem for multivalued maps due to Covitz and Nadler [35].

Lemma 3.6 ([35])

Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq161_HTML.gif be a complete metric space. If N : X P c l ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq162_HTML.gif is a contraction, then Fix N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq163_HTML.gif.

Theorem 3.7 Assume that the following conditions hold:

(A1) F : [ 0 , T ] × R P c p ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq164_HTML.gif is such that F ( , x ) : [ 0 , T ] P c p ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq165_HTML.gif is measurable for each x R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq53_HTML.gif;

(A2) H d ( F ( t , x ) , F ( t , x ¯ ) ) m ( t ) | x x ¯ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq166_HTML.gif for almost all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq167_HTML.gif and x , x ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq168_HTML.gif with m L 1 ( [ 0 , T ] , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq169_HTML.gif and d ( 0 , F ( t , 0 ) ) m ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq170_HTML.gif for almost all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq61_HTML.gif;

(A3) There exist constants c j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq171_HTML.gif, j = 0 , 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq21_HTML.gif, such that
| g j ( t , x ) g j ( t , y ) | c j | x y | , t [ 0 , T ] , j = 0 , 1 , 2 , x , y R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equad_HTML.gif
Then the boundary value problem (1.1) has at least one solution on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq17_HTML.gif if
Ω 1 m L 1 + { c 0 | μ 1 ξ 1 | + c 1 | μ 2 η 2 | + c 2 | μ 3 η 1 | } T < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equae_HTML.gif
Proof Observe that the set S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq117_HTML.gif is nonempty for each x C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq158_HTML.gif by the assumption (A1), so F has a measurable selection (see Theorem III.6 [36]). Now we show that the operator Ω F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq115_HTML.gif, defined in the beginning of the proof of Theorem 3.3, satisfies the assumptions of Lemma 3.6. To show that Ω F ( x ) P c l ( ( C [ 0 , T ] , R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq172_HTML.gif for each x C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq116_HTML.gif, let { u n } n 0 Ω F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq173_HTML.gif be such that u n u ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq174_HTML.gif in C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq175_HTML.gif. Then u C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq176_HTML.gif and there exists v n S F , x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq133_HTML.gif such that, for each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq61_HTML.gif,
u n ( t ) = 0 t ( t s ) q 1 Γ ( q ) v n ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v n ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v n ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v n ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equaf_HTML.gif
As F has compact values, we pass onto a subsequence (if necessary) to obtain that v n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq177_HTML.gif converges to v in L 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq178_HTML.gif. Thus, v S F , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq123_HTML.gif and for each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq167_HTML.gif, we have
u n ( t ) u ( t ) = 0 t ( t s ) q 1 Γ ( q ) v ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equag_HTML.gif

Hence, u Ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq179_HTML.gif.

Next we show that there exists δ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq180_HTML.gif such that
H d ( Ω F ( x ) , Ω F ( x ¯ ) ) δ x x ¯ for each  x , x ¯ A C 1 ( [ 0 , T ] , R ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equah_HTML.gif
Let x , x ¯ A C 1 ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq181_HTML.gif and h 1 Ω F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq182_HTML.gif. Then there exists v 1 ( t ) F ( t , x ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq183_HTML.gif such that, for each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq61_HTML.gif,
h 1 ( t ) = 0 t ( t s ) q 1 Γ ( q ) v 1 ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v 1 ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v 1 ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v 1 ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equai_HTML.gif
By (H3), we have
H d ( F ( t , x ) , F ( t , x ¯ ) ) m ( t ) | x ( t ) x ¯ ( t ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equaj_HTML.gif
So, there exists w F ( t , x ¯ ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq184_HTML.gif such that
| v 1 ( t ) w | m ( t ) | x ( t ) x ¯ ( t ) | , t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equak_HTML.gif
Define U : [ 0 , T ] P ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq185_HTML.gif by
U ( t ) = { w R : | v 1 ( t ) w | m ( t ) | x ( t ) x ¯ ( t ) | } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equal_HTML.gif

Since the multivalued operator U ( t ) F ( t , x ¯ ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq186_HTML.gif is measurable (Proposition III.4 [36]), there exists a function v 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq187_HTML.gif which is a measurable selection for U. So, v 2 ( t ) F ( t , x ¯ ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq188_HTML.gif and for each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq61_HTML.gif, we have | v 1 ( t ) v 2 ( t ) | m ( t ) | x ( t ) x ¯ ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq189_HTML.gif.

For each t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq61_HTML.gif, let us define
h 2 ( t ) = 0 t ( t s ) q 1 Γ ( q ) v 2 ( s ) d s λ 0 ξ 1 0 T ( T s ) q 1 Γ ( q ) v 2 ( s ) d s + λ 1 η 2 0 T ( T s ) q 2 Γ ( q 1 ) v 2 ( s ) d s + λ 2 η 1 0 T ( T s ) q 3 Γ ( q 2 ) v 2 ( s ) d s μ 0 ξ 1 0 T g 0 ( s , x ( s ) ) d s + μ 1 η 2 0 T g 1 ( s , x ( s ) ) d s + μ 2 η 1 0 T g 2 ( s , x ( s ) ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equam_HTML.gif
Thus,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equan_HTML.gif
Hence,
h 1 h 2 [ Ω 1 m L 1 + { c 0 | μ 1 ξ 1 | + c 1 | μ 2 η 2 | + c 2 | μ 3 η 1 | } T ] x x ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equao_HTML.gif
Analogously, interchanging the roles of x and x ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq190_HTML.gif, we obtain
H d ( Ω F ( x ) , Ω F ( x ¯ ) ) δ x x ¯ [ Ω 1 m L 1 + { c 0 | μ 1 ξ 1 | + c 1 | μ 2 η 2 | + c 3 | μ 2 η 1 | } T ] x x ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equap_HTML.gif

Since Ω F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq115_HTML.gif is a contraction, it follows by Lemma 3.6 that Ω F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq115_HTML.gif has a fixed point x which is a solution of (1.1). This completes the proof. □

Example 3.8

Consider the following boundary value problem of fractional differential inclusions:
{ D 5 / 2 c x ( t ) F ( t , x ( t ) ) , t [ 0 , 1 ] , x ( 0 ) + x ( 1 ) = 0 1 x ( s ) 3 ( 1 + s ) 2 d s , x ( 0 ) + x ( 1 ) = 1 2 0 1 e s x ( s ) 3 ( 1 + e s ) d s , x ( 0 ) + x ( 1 ) = 1 3 0 1 x ( s ) 3 ( 1 + e s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equ4_HTML.gif
(3.2)
where F : [ 0 , 1 ] × R P ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq191_HTML.gif is a multivalued map given by
x F ( t , x ) = [ | x | 3 10 ( | x | 3 + 3 ) , | sin x | 9 ( | sin x | + 1 ) + 1 12 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equaq_HTML.gif
For f F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq192_HTML.gif, we have
| f | max ( | x | 3 10 ( | x | 3 + 3 ) , | sin x | 9 ( | sin x | + 1 ) + 1 12 ) 7 36 , x R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equar_HTML.gif
Thus,
F ( t , x ) P : = sup { | y | : y F ( t , x ) } 7 36 = p ( t ) ψ ( x ) , x R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equas_HTML.gif
with p ( t ) = 1 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq193_HTML.gif, ψ ( x ) = 7 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq194_HTML.gif. Here
g 0 ( t , x ) = x ( t ) 3 ( 1 + t ) 2 , g 1 ( t , x ) = e t x ( t ) 3 ( 1 + e t ) , g 2 ( t , x ) = x ( t ) 3 ( 1 + e t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equat_HTML.gif

and λ 0 = λ 1 = λ 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq195_HTML.gif, μ 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq196_HTML.gif, μ 1 = 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq197_HTML.gif, μ 2 = 1 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq198_HTML.gif.

Clearly, ξ 1 = 1 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq199_HTML.gif, ξ 2 = 1 / 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq200_HTML.gif, ξ 3 = 1 / 16 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq201_HTML.gif, η 1 = 1 / 16 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq202_HTML.gif, η 2 = 1 / 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq203_HTML.gif, | g 0 ( t , x ) | 1 3 x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq204_HTML.gif, | g 1 ( t , x ) | 1 3 x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq205_HTML.gif, | g 2 ( t , x ) | 1 3 x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq206_HTML.gif with ψ ( M ) = 7 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq207_HTML.gif, ψ 0 ( M ) = ψ 1 ( M ) = ψ 2 ( M ) = M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq208_HTML.gif, p L 1 = p 0 L 1 = p 1 L 1 = p 2 L 1 = 1 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq209_HTML.gif, and Ω 1 = 85 π 32 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq210_HTML.gif. In view of the condition
M ψ ( M ) Ω 1 p L 1 + ψ 0 ( M ) | μ 0 ξ 1 | p 0 L 1 + ψ 1 ( M ) | μ 1 η 2 | p 1 L 1 + ψ 2 ( M ) | μ 2 η 1 | p 2 L 1 > 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_Equau_HTML.gif

we find that M > 595 π 904 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq211_HTML.gif. Thus, all the conditions of Theorem 3.3 are satisfied. So, there exists at least one solution of problem (3.2) on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-82/MediaObjects/13661_2012_Article_337_IEq19_HTML.gif.

Declarations

Acknowledgements

The authors are grateful to the anonymous referees for their useful comments. The research of B. Ahmad and A. Alsaedi was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University
(2)
Department of Mathematics, University of Ioannina

References

  1. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon; 1993.
  2. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
  3. Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
  4. Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.
  5. Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus: Models and Numerical Methods. World Scientific, Boston; 2012.
  6. Agarwal RP, Benchohra M, Hamani S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 2010, 109: 973-1033. 10.1007/s10440-008-9356-6MathSciNetView Article
  7. Bai ZB: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 2010, 72: 916-924. 10.1016/j.na.2009.07.033MathSciNetView Article
  8. Balachandran K, Trujillo JJ: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Anal. 2010, 72: 4587-4593. 10.1016/j.na.2010.02.035MathSciNetView Article
  9. Baleanu D, Mustafa OG: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 2010, 59: 1835-1841. 10.1016/j.camwa.2009.08.028MathSciNetView Article
  10. Hernandez E, O’Regan D, Balachandran K: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 2010, 73(10):3462-3471. 10.1016/j.na.2010.07.035MathSciNetView Article
  11. Wang Y, Liu L, Wu Y: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 2011, 74: 3599-3605. 10.1016/j.na.2011.02.043MathSciNetView Article
  12. Ford NJ, Morgado ML: Fractional boundary value problems: analysis and numerical methods. Fract. Calc. Appl. Anal. 2011, 14: 554-567.MathSciNet
  13. Ahmad B, Nieto JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 36
  14. Ahmad B, Nieto JJ, Alsaedi A: Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions. Acta Math. Sci. 2011, 31: 2122-2130.MathSciNetView Article
  15. Ahmad B, Ntouyas SK: Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions. Electron. J. Differ. Equ. 2012., 2012: Article ID 98
  16. Wang G, Agarwal RP, Cabada A: Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Lett. 2012, 25: 1019-1024. 10.1016/j.aml.2011.09.078MathSciNetView Article
  17. Bai ZB, Sun W: Existence and multiplicity of positive solutions for singular fractional boundary value problems. Comput. Math. Appl. 2012, 63: 1369-1381. 10.1016/j.camwa.2011.12.078MathSciNetView Article
  18. Sakthivel R, Mahmudov NI, Nieto JJ: Controllability for a class of fractional-order neutral evolution control systems. Appl. Math. Comput. 2012, 218: 10334-10340. 10.1016/j.amc.2012.03.093MathSciNetView Article
  19. Ahmad B, Nieto JJ, Alsaedi A, El-Shahed M: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl. 2012, 13: 599-606. 10.1016/j.nonrwa.2011.07.052MathSciNetView Article
  20. Ahmad B, Ntouyas SK: Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with strip conditions. Bound. Value Probl. 2012., 2012: Article ID 55
  21. Agarwal RP, O’Regan D, Stanek S: Positive solutions for mixed problems of singular fractional differential equations. Math. Nachr. 2012, 285(1):27-41. 10.1002/mana.201000043MathSciNetView Article
  22. Ahmad B, Ntouyas SK: A boundary value problem of fractional differential equations with anti-periodic type integral boundary conditions. J. Comput. Anal. Appl. 2013, 15: 1372-1380.MathSciNet
  23. Henderson J, Ouahab A: Fractional functional differential inclusions with finite delay. Nonlinear Anal. 2009, 70: 2091-2105. 10.1016/j.na.2008.02.111MathSciNetView Article
  24. Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 2009, 49: 605-609. 10.1016/j.mcm.2008.03.014MathSciNetView Article
  25. Cernea A: On the existence of solutions for nonconvex fractional hyperbolic differential inclusions. Commun. Math. Anal. 2010, 9(1):109-120.MathSciNet
  26. Hamani S, Benchohra M, Graef JR: Existence results for boundary-value problems with nonlinear fractional differential inclusions and integral conditions. Electron. J. Differ. Equ. 2010., 2010: Article ID 20
  27. Agarwal RP, Ahmad B, Alsaedi A, Shahzad N: On the dimension of the solution set for semilinear fractional differential inclusions. Abstr. Appl. Anal. 2012., 2012: Article ID 305924
  28. Ahmad B, Ntouyas SK: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 2012, 15: 362-382.MathSciNet
  29. Deimling K: Multivalued Differential Equations. de Gruyter, Berlin; 1992.View Article
  30. Hu S, Papageorgiou N: Handbook of Multivalued Analysis, Theory I. Kluwer Academic, Dordrecht; 1997.View Article
  31. Kisielewicz M: Differential Inclusions and Optimal Control. Kluwer Academic, Dordrecht; 1991.
  32. Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2005.
  33. Lasota A, Opial Z: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1965, 13: 781-786.MathSciNet
  34. Bressan A, Colombo G: Extensions and selections of maps with decomposable values. Stud. Math. 1988, 90: 69-86.MathSciNet
  35. Covitz H, Nadler SB Jr.: Multivalued contraction mappings in generalized metric spaces. Isr. J. Math. 1970, 8: 5-11. 10.1007/BF02771543MathSciNetView Article
  36. Castaing C, Valadier M Lecture Notes in Mathematics 580. In Convex Analysis and Measurable Multifunctions. Springer, Berlin; 1977.View Article

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