# Boundary Value Problems forq-Difference Inclusions

Authors: Bashir Ahmad; Sotiris K. Ntouyas
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We investigate the existence of solutions for a class of second-orderq-difference inclusions with nonseparated boundary conditions. By using suitable fixed-point theorems, we study the cases when the right-hand side of the inclusions has convex as well as nonconvex values.


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## Body

## 1. Introduction

The discretization of the ordinary differential equations is an important and necessary step towards finding their numerical solutions. Instead of the standard discretization based on the arithmetic progression, one can use an equally efficientq-discretization related to geometric progression. This alternative method leads to q-difference equations, which in the limit q→1 correspond to the classical differential equations. q-difference equations are found to be quite useful in the theory of quantum groups [1]. For historical notes and development of the subject, we refer the reader to [2, 3] while some recent results on q-difference equations can be found in [4–6]. However, the theory of boundary value problems for nonlinear q-difference equations is still in the initial stages, and many aspects of this theory need to be explored.Differential inclusions arise in the mathematical modelling of certain problems in economics, optimal control, stochastic analysis, and so forth and are widely studied by many authors; see [7–13] and the references therein. For some works concerning difference inclusions and dynamic inclusions on time scales, we refer the reader to the papers [14–17].In this paper, we study the existence of solutions for second-orderq-difference inclusions with nonseparated boundary conditions given by
(1.1)Dq2u(t)∈F(t,u(t)),0≤t≤T,(1.2)u(0)=ηu(T),Dqu(0)=ηDqu(T),
where F:[0,T]×ℝ→𝒫(ℝ) is a compact valued multivalued map, 𝒫(ℝ) is the family of all subsets of ℝ, T is a fixed constant, and η≠1 is a fixed real number.The aim of our paper is to establish some existence results for the Problems (1.1)-(1.2), when the right-hand side is convex as well as nonconvex valued. First of all, an integral operator is found by applying the tools of q-difference calculus, which plays a pivotal role to convert the given boundary value problem to a fixed-point problem. Our approach is simpler as it does not involve the typical series solution form of q-difference equations. The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we will combine 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, while in the third result, we will use the fixed-point theorem for generalized contraction multivalued maps due to Wegrzyk. The methods used are standard; however, their exposition in the framework of Problems (1.1)-(1.2) is new.The paper is organized as follows: in Section2, we recall some preliminary facts that we need in the sequel, and we prove our main results in Section 3.


## 2. Preliminaries

In this section, we introduce notation, definitions, and preliminary facts which we need for the forthcoming analysis.


### 2.1.q-Calculus

Let us recall some basic concepts ofq-calculus [1–3].For0<q<1, we define the q-derivative of a real-valued function f as
(2.1)Dqf(t)=f(t)-f(qt)(1-q)t,Dqf(0)=lim⁡t→0Dqf(t).
The higher-order q-derivatives are given by
(2.2)Dq0f(t)=f(t),Dqnf(t)=DqDqn-1f(t),n∈N.
The q-integral of a function f defined in the interval [a,b] is given by
(2.3)∫axf(t)dqt:=∑n=0∞x(1-q)qnf(xqn)-af(qna),x∈[a,b],
and for a=0, we denote
(2.4)Iqf(x)=∫0xf(t)dqt=∑n=0∞x(1-q)qnf(xqn),
provided the series converges. If a∈[0,b] and f is defined in the interval [0,b], then
(2.5)∫abf(t)dqt=∫0bf(t)dqt-∫0af(t)dqt.
Similarly, we have
(2.6)Iq0f(t)=f(t),Iqnf(t)=IqIqn-1f(t),n∈N.
Observe that
(2.7)DqIqf(x)=f(x),
and if f is continuous at x=0, then
(2.8)IqDqf(x)=f(x)-f(0).
In q-calculus, the integration by parts formula is
(2.9)∫0xf(t)Dqg(t)dqt=[f(t)g(t)]0x-∫0xDqf(t)g(qt)dqt.


### 2.2. Multivalued Analysis

Let us recall some basic definitions on multivalued maps [18, 19].LetX denote a normed space with the norm |·|. A multivalued map G:X→𝒫(X) is convex (closed) valued if G(x) is convex (closed) for all x∈X. G is bounded on bounded sets if G(B)=∪x∈BG(x) is bounded in X for all bounded sets B in X (i.e., supx∈B{sup{|y|:y∈G(x)}}<∞). G is called upper semicontinuous (u.s.c.) on X if for each x0∈X, the set G(x0) is a nonempty closed subset of X, and if for each open set N of X containing G(x0), there exists an open neighborhood N0 of x0 such that G(N0)⊆N. G is said to be completely continuous if G(ℬ) is relatively compact for every bounded set B in X. If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e., xn→x*,  yn→y*, yn∈G(xn) imply y*∈G(x*)). G has a fixed-point if there is x∈X such that x∈G(x). The fixed-point set of the multivalued operator G will be denoted by FixG.For more details on multivalued maps, see the books of Aubin and Cellina [20], Aubin and Frankowska [21], Deimling [18], and Hu and Papageorgiou [19].LetC([0,T],ℝ) denote the Banach space of all continuous functions from [0,T] into ℝ with the norm
(2.10)‖u‖∞=sup⁡{|u(t)|:t∈[0,T]}.
Let L1([0,T],ℝ) be the Banach space of measurable functions u:[0,T]→ℝ which are Lebesgue integrable and normed by
(2.11)‖u‖L1=∫0T|u(t)|dt,∀u∈L1([0,T],R).Definition 2.1.
A multivalued mapG:[0,T]→𝒫(ℝ) with nonempty compact convex values is said to be measurable if for any x∈ℝ, the function
(2.12)t⟼d(x,F(t))=inf⁡{|x-z|:z∈F(t)}
is measurable.Definition 2.2.
A multivalued mapF:[0,T]×ℝ→𝒫(ℝ) is said to be Carathéodory if(i)
t↦F(t,x) is measurable for each x∈ℝ,(ii)
x↦F(t,x) is upper semicontinuous for almost all t∈[0,T].Further a Carathéodory function F is called L1-Carathéodory if(iii)
for eachα>0, there exists φα∈L1([0,T],ℝ+) such that
(2.13)‖F(t,x)‖=sup⁡{|v|:v∈F(t,x)}≤φα(t)
for all ∥x∥∞≤α and for a.e. t∈[0,T].LetE be a Banach space, let X be a nonempty closed subset of E, and let G:X→𝒫(E) be a multivalued operator with nonempty closed values. G is lower semicontinuous (l.s.c.) if the set {x∈X:G(x)∩B≠∅} is open for any open set  B in  E. Let A be a subset of [0,T]×ℝ. A is ℒ⊗ℬ measurable if A belongs to the σ-algebra generated by all sets of the form 𝒥×D, where 𝒥 is Lebesgue measurable in [0,T] and D is Borel measurable in ℝ. A subset A of L1([0,T],ℝ) is decomposable if for all u,v∈A and 𝒥⊂[0,T] measurable, the function uχ𝒥+vχJ-𝒥∈A, where χ𝒥 stands for the characteristic function of 𝒥.Definition 2.3.
IfF:[0,T]×ℝ→𝒫(ℝ) is a multivalued map with compact values and u(·)∈C([0,T],ℝ), then F(·,·) is of lower semicontinuous type if
(2.14)SF(u)={w∈L1([0,T],R):w(t)∈F(t,u(t))fora.e.t∈[0,T]}
is lower semicontinuous with closed and decomposable values.Let(X,d) be a metric space associated with the norm |·|. The Pompeiu-Hausdorff distance of the closed subsets A,B⊂X is defined by
(2.15)dH(A,B)=max⁡{d*(A,B),d*(B,A)},d*(A,B)=sup⁡{d(a,B):a∈A},
where d(x,B)=infy∈Bd(x,y).Definition 2.4.
A functionl:ℝ+→ℝ+ is said to be a strict comparison function (see [25]) if it is continuous strictly increasing and ∑n=1∞ln(t)<∞, for each t>0.Definition 2.5.
A multivalued operatorN on X with nonempty values in X is called(a)
γ-Lipschitz if and only if there exists γ>0 such that
(2.16)dH(N(x),N(y))≤γd(x,y),foreachx,y∈X,(b)
a contraction if and only if it isγ-Lipschitz with γ<1,(c)
a generalized contraction if and only if there is a strict comparison functionl:ℝ+→ℝ+ such that
(2.17)dH(N(x),N(y))≤l(d(x,y)),foreachx,y∈X.The following lemmas will be used in the sequel.Lemma 2.6 (see [22]).
LetX be a Banach space. Let F:[0,T]×X→𝒫(X) be an L1-Carathéodory multivalued map with SF≠∅, and let Γ be a linear continuous mapping from L1([0,T],X) to C([0,T],X), then the operator
(2.18)Γ∘SF:C([0,T],X)⟶P(C([0,T],X))
defined by (Γ∘SF)(x)=Γ(SF(x)) has compact convex values and has a closed graph operator in C([0,T],X)×C([0,T],X).In passing, we remark that ifdimX<∞, then SF(x)≠∅ for any x(·)∈C([0,T],X) with F(·,·) as in Lemma 2.6.Lemma 2.7 (nonlinear alternative for Kakutani maps [23]).
LetE be a Banach space, C, a closed convex subset of E, U an open subset of C and 0∈U. Suppose that F:U¯→𝒫c,cv(C) is an upper semicontinuous compact map; here, 𝒫c,cv(C) denotes the family of nonempty, compact convex subsets of C, then either(i)
F has a fixed-point in U¯,(ii)
or there is au∈∂U and λ∈(0,1) with u∈λF(u).Lemma 2.8 (see [24]).
LetY be a separable metric space, and let N:Y→𝒫(L1([0,T],ℝ)) be a lower semicontinuous multivalued map with closed decomposable values, then N(·) has a continuous selection; that is, there exists a continuous mapping (single-valued) g:Y→L1([0,T],ℝ) such that g(y)∈N(y) for every y∈Y.Lemma 2.9 (Wegrzyk's fixed-point theorem [25, 26]).
Let(X,d) be a complete metric space. If N:X→𝒫(X) is a generalized contraction with nonempty closed values, then FixN≠∅.Lemma 2.10 (Covitz and Nadler's fixed-point theorem [27]).
Let(X,d) be a complete metric space. If N:X→𝒫(X) is a multivalued contraction with nonempty closed values, then N has a fixed-point z∈X such that z∈N(z), that is, FixN≠∅.


## 2.1.q-Calculus

Let us recall some basic concepts ofq-calculus [1–3].For0<q<1, we define the q-derivative of a real-valued function f as
(2.1)Dqf(t)=f(t)-f(qt)(1-q)t,Dqf(0)=lim⁡t→0Dqf(t).
The higher-order q-derivatives are given by
(2.2)Dq0f(t)=f(t),Dqnf(t)=DqDqn-1f(t),n∈N.
The q-integral of a function f defined in the interval [a,b] is given by
(2.3)∫axf(t)dqt:=∑n=0∞x(1-q)qnf(xqn)-af(qna),x∈[a,b],
and for a=0, we denote
(2.4)Iqf(x)=∫0xf(t)dqt=∑n=0∞x(1-q)qnf(xqn),
provided the series converges. If a∈[0,b] and f is defined in the interval [0,b], then
(2.5)∫abf(t)dqt=∫0bf(t)dqt-∫0af(t)dqt.
Similarly, we have
(2.6)Iq0f(t)=f(t),Iqnf(t)=IqIqn-1f(t),n∈N.
Observe that
(2.7)DqIqf(x)=f(x),
and if f is continuous at x=0, then
(2.8)IqDqf(x)=f(x)-f(0).
In q-calculus, the integration by parts formula is
(2.9)∫0xf(t)Dqg(t)dqt=[f(t)g(t)]0x-∫0xDqf(t)g(qt)dqt.


## 2.2. Multivalued Analysis

Let us recall some basic definitions on multivalued maps [18, 19].LetX denote a normed space with the norm |·|. A multivalued map G:X→𝒫(X) is convex (closed) valued if G(x) is convex (closed) for all x∈X. G is bounded on bounded sets if G(B)=∪x∈BG(x) is bounded in X for all bounded sets B in X (i.e., supx∈B{sup{|y|:y∈G(x)}}<∞). G is called upper semicontinuous (u.s.c.) on X if for each x0∈X, the set G(x0) is a nonempty closed subset of X, and if for each open set N of X containing G(x0), there exists an open neighborhood N0 of x0 such that G(N0)⊆N. G is said to be completely continuous if G(ℬ) is relatively compact for every bounded set B in X. If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e., xn→x*,  yn→y*, yn∈G(xn) imply y*∈G(x*)). G has a fixed-point if there is x∈X such that x∈G(x). The fixed-point set of the multivalued operator G will be denoted by FixG.For more details on multivalued maps, see the books of Aubin and Cellina [20], Aubin and Frankowska [21], Deimling [18], and Hu and Papageorgiou [19].LetC([0,T],ℝ) denote the Banach space of all continuous functions from [0,T] into ℝ with the norm
(2.10)‖u‖∞=sup⁡{|u(t)|:t∈[0,T]}.
Let L1([0,T],ℝ) be the Banach space of measurable functions u:[0,T]→ℝ which are Lebesgue integrable and normed by
(2.11)‖u‖L1=∫0T|u(t)|dt,∀u∈L1([0,T],R).Definition 2.1.
A multivalued mapG:[0,T]→𝒫(ℝ) with nonempty compact convex values is said to be measurable if for any x∈ℝ, the function
(2.12)t⟼d(x,F(t))=inf⁡{|x-z|:z∈F(t)}
is measurable.Definition 2.2.
A multivalued mapF:[0,T]×ℝ→𝒫(ℝ) is said to be Carathéodory if(i)
t↦F(t,x) is measurable for each x∈ℝ,(ii)
x↦F(t,x) is upper semicontinuous for almost all t∈[0,T].Further a Carathéodory function F is called L1-Carathéodory if(iii)
for eachα>0, there exists φα∈L1([0,T],ℝ+) such that
(2.13)‖F(t,x)‖=sup⁡{|v|:v∈F(t,x)}≤φα(t)
for all ∥x∥∞≤α and for a.e. t∈[0,T].LetE be a Banach space, let X be a nonempty closed subset of E, and let G:X→𝒫(E) be a multivalued operator with nonempty closed values. G is lower semicontinuous (l.s.c.) if the set {x∈X:G(x)∩B≠∅} is open for any open set  B in  E. Let A be a subset of [0,T]×ℝ. A is ℒ⊗ℬ measurable if A belongs to the σ-algebra generated by all sets of the form 𝒥×D, where 𝒥 is Lebesgue measurable in [0,T] and D is Borel measurable in ℝ. A subset A of L1([0,T],ℝ) is decomposable if for all u,v∈A and 𝒥⊂[0,T] measurable, the function uχ𝒥+vχJ-𝒥∈A, where χ𝒥 stands for the characteristic function of 𝒥.Definition 2.3.
IfF:[0,T]×ℝ→𝒫(ℝ) is a multivalued map with compact values and u(·)∈C([0,T],ℝ), then F(·,·) is of lower semicontinuous type if
(2.14)SF(u)={w∈L1([0,T],R):w(t)∈F(t,u(t))fora.e.t∈[0,T]}
is lower semicontinuous with closed and decomposable values.Let(X,d) be a metric space associated with the norm |·|. The Pompeiu-Hausdorff distance of the closed subsets A,B⊂X is defined by
(2.15)dH(A,B)=max⁡{d*(A,B),d*(B,A)},d*(A,B)=sup⁡{d(a,B):a∈A},
where d(x,B)=infy∈Bd(x,y).Definition 2.4.
A functionl:ℝ+→ℝ+ is said to be a strict comparison function (see [25]) if it is continuous strictly increasing and ∑n=1∞ln(t)<∞, for each t>0.Definition 2.5.
A multivalued operatorN on X with nonempty values in X is called(a)
γ-Lipschitz if and only if there exists γ>0 such that
(2.16)dH(N(x),N(y))≤γd(x,y),foreachx,y∈X,(b)
a contraction if and only if it isγ-Lipschitz with γ<1,(c)
a generalized contraction if and only if there is a strict comparison functionl:ℝ+→ℝ+ such that
(2.17)dH(N(x),N(y))≤l(d(x,y)),foreachx,y∈X.The following lemmas will be used in the sequel.Lemma 2.6 (see [22]).
LetX be a Banach space. Let F:[0,T]×X→𝒫(X) be an L1-Carathéodory multivalued map with SF≠∅, and let Γ be a linear continuous mapping from L1([0,T],X) to C([0,T],X), then the operator
(2.18)Γ∘SF:C([0,T],X)⟶P(C([0,T],X))
defined by (Γ∘SF)(x)=Γ(SF(x)) has compact convex values and has a closed graph operator in C([0,T],X)×C([0,T],X).In passing, we remark that ifdimX<∞, then SF(x)≠∅ for any x(·)∈C([0,T],X) with F(·,·) as in Lemma 2.6.Lemma 2.7 (nonlinear alternative for Kakutani maps [23]).
LetE be a Banach space, C, a closed convex subset of E, U an open subset of C and 0∈U. Suppose that F:U¯→𝒫c,cv(C) is an upper semicontinuous compact map; here, 𝒫c,cv(C) denotes the family of nonempty, compact convex subsets of C, then either(i)
F has a fixed-point in U¯,(ii)
or there is au∈∂U and λ∈(0,1) with u∈λF(u).Lemma 2.8 (see [24]).
LetY be a separable metric space, and let N:Y→𝒫(L1([0,T],ℝ)) be a lower semicontinuous multivalued map with closed decomposable values, then N(·) has a continuous selection; that is, there exists a continuous mapping (single-valued) g:Y→L1([0,T],ℝ) such that g(y)∈N(y) for every y∈Y.Lemma 2.9 (Wegrzyk's fixed-point theorem [25, 26]).
Let(X,d) be a complete metric space. If N:X→𝒫(X) is a generalized contraction with nonempty closed values, then FixN≠∅.Lemma 2.10 (Covitz and Nadler's fixed-point theorem [27]).
Let(X,d) be a complete metric space. If N:X→𝒫(X) is a multivalued contraction with nonempty closed values, then N has a fixed-point z∈X such that z∈N(z), that is, FixN≠∅.


## 3. Main Results

In this section, we are concerned with the existence of solutions for the Problems (1.1)-(1.2) when the right-hand side has convex as well as nonconvex values. Initially, we assume that F is a compact and convex valued multivalued map.To define the solution for the Problems (1.1)-(1.2), we need the following result.Lemma 3.1.
Suppose thatσ:[0,T]→ℝ is continuous, then the following problem
(3.1)Dq2u(t)=σ(t),a.e.t∈[0,T],u(0)=ηu(T),Dqu(0)=ηDqu(T)
has a unique solution
(3.2)u(t)=∫0TG(t,qs)σ(s)dqs,
where G(t,qs) is the Green's function given by
(3.3)G(t,qs)=1(η-1)2{η(η-1)(qs-t)+ηT,if0≤t<s≤T,(η-1)(qs-t)+ηT,if0≤s≤t≤T.Proof.
In view of (2.7) and (2.9), the solution of Dq2u=σ(t) can be written as
(3.4)u(t)=∫0t(t-qs)σ(s)dqs+a1t+a2,
where a1, a2 are arbitrary constants. Using the boundary conditions (1.2) and (3.4), we find that
(3.5)a1=-η(η-1)∫0Tσ(s)dqs,a2=η2T(η-1)2∫0Tσ(s)dqs-η(η-1)∫0T(T-qs)σ(s)dqs.
Substituting the values of a1 and a2 in (3.4), we obtain (3.2).Let us denote(3.6)G1=max⁡t,s∈[0,T]|G(t,qs)|.Definition 3.2.
A functionu∈C([0,T],ℝ) is said to be a solution of (1.1)-(1.2) if there exists a function v∈L1([0,T],ℝ) with v(t)∈F(t,x(t)) a.e. t∈[0,T] and
(3.7)u(t)=∫0TG(t,qs)v(s)dqs,
where G(t,qs) is given by (3.3).Theorem 3.3.
Suppose that(H1)
the mapF:[0,T]×ℝ→𝒫(ℝ) has nonempty compact convex values and is Carathéodory,(H2)
there exist a continuous nondecreasing functionψ:[0,∞)→(0,∞) and a function p∈L1([0,T],ℝ+) such that
(3.8)‖F(t,u)‖P:=sup⁡{|v|:v∈F(t,u)}≤p(t)ψ(‖u‖∞)
for each (t,u)∈[0,T]×ℝ,(H3)
there exists a numberM>0 such that
(3.9)MG1ψ(M)‖p‖L1>1,then the BVP (1.1)-(1.2) has at least one solution.Proof.
In view of Definition3.2, the existence of solutions to (1.1)-(1.2) is equivalent to the existence of solutions to the integral inclusion
(3.10)u(t)∈∫0TG(t,qs)F(s,u(s))dqs,t∈[0,T].
Let us introduce the operator
(3.11)N(u):={h∈C([0,T],R):h(t)=∫0TG(t,qs)v(s)dqs,v∈SF,u}.
We will show thatN satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps.
Step 1 (N(u) is convex for each u∈C([0,T],ℝ)).
Indeed, ifh1, h2 belong to N(u), then there exist v1,v2∈SF,u such that for each t∈[0,T], we have
(3.12)hi(t)=∫0TG(t,qs)vi(s)dqs,(i=1,2).
Let 0≤d≤1, then, for each t∈[0,T], we have
(3.13)(dh1+(1-d)h2)(t)=∫0TG(t,qs)[dv1(s)+(1-d)v2(s)]dqs.
Since SF,u is convex (because F has convex values); therefore,
(3.14)dh1+(1-d)h2∈N(u).Step 2 (N maps bounded sets into bounded sets in C([0,T],ℝ)).
LetBm={u∈C([0,T],ℝ):∥u∥∞≤m,m>0} be a bounded set in C([0,T],ℝ) and u∈Bm, then for each h∈N(u), there exists v∈SF,u such that
(3.15)h(t)=∫0TG(t,qs)v(s)dqs.
Then, in view of (H2), we have
(3.16)|h(t)|≤∫0T|G(t,qs)||v(s)|dqs≤G1∫0Tp(s)ψ(‖u‖∞)dqs≤G1ψ(m)∫0Tp(s)dqs.
Thus,
(3.17)‖h‖∞≤G1ψ(m)‖p‖L1.Step 3 (N maps bounded sets into equicontinuous sets of C([0,T],ℝ)).
Letr1,r2∈[0,T],  r1<r2 and Bm be a bounded set of C([0,T],ℝ) as in Step 2 and x∈Bm. For each h∈N(u)
(3.18)|h(r2)-h(r1)|≤∫0T|G(r2,s)-G(r1,s)||v(s)|dqs≤ψ(‖u‖∞)∫0T|G(r2,s)-G(r1,s)|p(s)dqs≤ψ(m)∫0T|G(r2,s)-G(r1,s)|p(s)dqs.
The right-hand side tends to zero as r2-r1→0. As a consequence of Steps 1 to 3 together with the Arzelá-Ascoli Theorem, we can conclude that  N:C([0,T],ℝ)→𝒫(C([0,T],ℝ)) is completely continuous.Step 4 (N has a closed graph).
Letun→u*,hn∈N(un), and  hn→h*. We need to show that h*∈N(u*). hn∈N(un) means that there exists vn∈SF,un such that, for each t∈[0,T],
(3.19)hn(t)=∫0TG(t,qs)vn(s)dqs.
We must show that there exists h*∈SF,u* such that, for each t∈[0,T],
(3.20)h*(t)=∫0TG(t,qs)v*(s)dqs.
Clearly, we have
(3.21)‖hn-h*‖∞⟶0asn⟶∞.
Consider the continuous linear operator
(3.22)Γ:L1([0,T],R)⟶C([0,T],R),
defined by
(3.23)v⟼(Γv)(t)=∫0TG(t,qs)v(s)dqs.
From Lemma 2.6, it follows that Γ∘SF is a closed graph operator. Moreover, we have
(3.24)hn(t)∈Γ(SF,un).
Since un→u*, it follows from Lemma 2.6 that
(3.25)h*(t)=∫0TG(t,qs)v*(s)dqs
for some v*∈SF,u*.Step 5 (a priori bounds on solutions).
Letu be a possible solution of the Problems (1.1)-(1.2), then there exists v∈L1([0,T],ℝ) with v∈SF,u such that, for each t∈[0,T],
(3.26)u(t)=∫0TG(t,qs)v(s)dqs.
For each t∈[0,T], it follows by (H2) and (H3) that
(3.27)|u(t)|≤G1∫0Tp(s)ψ(‖u‖∞)dqs≤G1ψ(‖u‖∞)∫0Tp(s)dqs.
Consequently,
(3.28)‖u‖∞G1ψ(‖u‖∞)‖p‖L1≤1.
Then by (H3), there exists M such that ∥u∥∞≠M.
Let(3.29)U={u∈C([0,T],R):‖u‖∞<M+1}.
The operator N:U¯→𝒫(C([0,T],ℝ)) is upper semicontinuous and completely continuous. From the choice of U, there is no u∈∂U such that u∈λN(u) for some λ∈(0,1). Consequently, by Lemma 2.7, it follows that N has a fixed-point u in U¯ which is a solution of the Problems (1.1)-(1.2). This completes the proof.Next, we study the case whereF is not necessarily convex valued. Our approach here is based on the nonlinear alternative of Leray-Schauder type combined with the selection theorem of Bressan and Colombo for lower semicontinuous maps with decomposable values.Theorem 3.4.
Suppose that the conditions (H2) and (H3) hold. Furthermore, it is assumed that(H4)
F:[0,T]×ℝ→𝒫(ℝ) has nonempty compact values and
(a)
(t,u)↦F(t,u) is ℒ⊗ℬ measurable,(b)
u↦F(t,u) is lower semicontinuous for a.e. t∈[0,T],(H5)
for eachρ>0, there exists φρ∈L1([0,T],ℝ+) such that
(3.30)‖F(t,u)‖=sup⁡{|v|:v∈F(t,u)}≤φρ(t)∀‖u‖∞≤ρand fora.e.t∈[0,T].
then, the BVP (1.1)-(1.2) has at least one solution.Proof.
Note that (H4) and (H5) imply thatF is of lower semicontinuous type. Thus, by Lemma 2.8, there exists a continuous function f:C([0,T],ℝ)→L1([0,T],ℝ) such that f(u)∈ℱ(u) for all u∈C([0,T],ℝ). So we consider the problem
(3.31)Dq2u(t)=f(u(t)),0≤t≤T,u(0)=ηu(T),Dqu(0)=ηDqu(T).
Clearly, if u∈C([0,T],ℝ) is a solution of (3.31), then u is a solution to the Problems  (1.1)-(1.2). Transform the Problem (3.31) into a fixed-point theorem
(3.32)u(t)=(N¯u)(t),t∈[0,T],
where
(3.33)(N¯u)(t)=∫0TG(t,qs)f(u(s))dqs,t∈[0,T].
We can easily show that N¯ is continuous and completely continuous. The remainder of the proof is similar to that of Theorem 3.3.Now, we prove the existence of solutions for the Problems  (1.1)-(1.2) with a nonconvex valued right-hand side by applying Lemma 2.9 due to Wegrzyk.Theorem 3.5.
Suppose that(H6)
F:[0,T]×ℝ→𝒫(ℝ) has nonempty compact values and F(·,u) is measurable for each u∈ℝ,(H7)
dH(F(t,u),F(t,u¯))≤k(t)l(|u-u¯|) for almost all t∈[0,1] and u,u¯∈ℝ with k∈L1([0,1],ℝ+) and d(0,F(t,0))≤k(t) for almost all t∈[0,1], where l:ℝ+→ℝ+ is strictly increasing,then the BVP (1.1)-(1.2) has at least one solution on [0,T] if γl:ℝ+→ℝ+ is a strict comparison function, where γ=G1∥k∥L1.Proof.
Suppose thatγl:ℝ+→ℝ+ is a strict comparison function. Observe that by the assumptions (H6) and (H7), F(·,u(·)) is measurable and has a measurable selection v(·) (see Theorem  3.6 [28]). Also k∈L1([0,1],ℝ) and
(3.34)|v(t)|≤d(0,F(t,0))+Hd(F(t,0),F(t,u(t)))≤k(t)+k(t)l(|u(t)|)≤(1+l(‖u‖∞))k(t).
Thus, the set SF,u is nonempty for each u∈C([0,T],ℝ).
As before, we transform the Problems (1.1)-(1.2) into a fixed-point problem by using the multivalued map N given by (3.11) and show that the map N satisfies the assumptions of Lemma 2.9. To show that the map N(u) is closed for each u∈C([0,T],ℝ), let (un)n≥0∈N(u) such that un→ũ in C([0,T],ℝ), then ũ∈C([0,T],ℝ) and there exists vn∈SF,u such that, for each t∈[0,T],
(3.35)un(t)=∫0TG(t,qs)vn(s)dqs.
As F has compact values, we pass onto a subsequence to obtain that vn converges to v in L1([0,T],ℝ). Thus, v∈SF,u and for each t∈[0,T],
(3.36)un(t)⟶ũ(t)=∫0TG(t,qs)v(s)dqs.
So, ũ∈N(u) and hence N(u) is closed.
Next, we show that(3.37)dH(N(u),N(u¯))≤γl(‖u-u¯‖∞)for eachu,u¯∈C([0,T],R).
Let u,u¯∈C([0,T],ℝ) and h1∈N(u). Then, there exists v1(t)∈SF,u such that for each t∈[0,T],
(3.38)h1(t)=∫0TG(t,qs)v1(s)dqs.
From (H7), it follows that
(3.39)dH(F(t,u(t)),F(t,u¯(t)))≤k(t)l(|u(t)-u¯(t)|).
So, there exists w∈F(t,u¯(t)) such that
(3.40)|v1(t)-w|≤k(t)l(|u(t)-u¯(t)|),t∈[0,T].
Define U:[0,T]→𝒫(ℝ) as
(3.41)U(t)={w∈R:|v1(t)-w|≤k(t)l(|u(t)-u¯(t)|)}.
Since the multivalued operator U(t)∩F(t,u¯(t)) is measurable (see Proposition  3.4 in [28]), there exists a function v2(t) which is a measurable selection for U(t)∩F(t,u¯(t)). So, v2(t)∈F(t,u¯(t)), and for each t∈[0,T],
(3.42)|v1(t)-v2(t)|≤k(t)l(|u(t)-u¯(t)|).
For each t∈[0,T], let us define
(3.43)h2(t)=∫0TG(t,qs)v2(s)dqs,
then
(3.44)|h1(t)-h2(t)|≤∫0T|G(t,qs)||v1(s)-v2(s)|dqs≤G1∫0Tk(s)l(‖u-u¯‖)dqs.
Thus,
(3.45)‖h1-h2‖∞≤G1‖k‖L1l(‖u-u¯‖∞)=γl(‖u-u¯‖∞).
By an analogous argument, interchanging the roles of u and u¯, we obtain
(3.46)dH(N(u),N(u¯))≤G1‖k‖L1l(‖u-u¯‖∞)=γl(‖u-u¯‖∞)
for each u,u¯∈C([0,T],ℝ). So, N is a generalized contraction, and thus, by Lemma 2.9, N has a fixed-point u which is a solution to (1.1)-(1.2). This completes the proof.Remark 3.6.
We notice that Theorem3.5 holds for several values of the function l, for example, l(t)=ln(1+t)/χ, where χ∈(0,1), l(t)=t, and so forth. Here, we emphasize that the condition (H7) reduces to dH(F(t,u),F(t,u¯))≤k(t)|u-u¯| for l(t)=t, where a contraction principle for multivalued map due to Covitz and Nadler [27] (Lemma 2.10) is applicable under the condition G1∥k∥L1<1. Thus, our result dealing with a nonconvex valued right-hand side of (1.1) is more general, and the previous results for nonconvex valued right-hand side of the inclusions based on Covitz and Nadler's fixed-point result (e.g., see [8]) can be extended to generalized contraction case.Remark 3.7.
Our results correspond to the ones for second-orderq-difference inclusions with antiperiodic boundary conditions (u(0)=-u(T), Dqu(0)=-Dqu(T)) for η=-1. The results for an initial value problem of second-order q-difference inclusions follow for η=0. These results are new in the present configuration.Remark 3.8.
In the limitq→1, the obtained results take the form of their “continuous” (i.e., differential) counterparts presented in Sections  4 (ii) for λ1=λ2=η,μ1=0=μ2 of [29].Example 3.9.
Consider a boundary value problem of second-orderq-difference inclusions given by
(3.47)Dq2u(t)∈F(t,u(t)),0≤t≤1u(0)=-u(1),Dqu(0)=-Dqu(1),
where η=-1 and F:[0,1]×ℝ→𝒫(ℝ) is a multivalued map given by
(3.48)(t,u)⟶F(t,u)=[u3u3+3+t3+3,uu+1+t+1].
For f∈F, we have
(3.49)|f|≤max⁡(u3u3+3+t3+3,uu+1+t+1)≤5,u∈R.
Thus,
(3.50)‖F(t,u)‖P:=sup⁡{|y|:y∈F(t,u)}≤5=p(t)ψ(‖u‖∞),u∈R,
with p(t)=1, ψ(∥u∥∞)=5. Further, using the condition
(3.51)MG1ψ(M)‖p‖L1>1,
we find that M>5G2, where G2=G1∣η=-1,T=1. Clearly, all the conditions of Theorem 3.3 are satisfied. So, the conclusion of Theorem 3.3 applies to the Problem (3.47).



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*Source: 292860-2011-04-18.xml*