We investigate the existence of solutions for a class of second-order q-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.
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 efficient q-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-order q-difference inclusions with nonseparated boundary conditions given by
Dq2u(t)∈F(t,u(t)),0≤t≤T,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 Section 2, 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 of q-calculus [1–3].
For 0<q<1, we define the q-derivative of a real-valued function f as
Dqf(t)=f(t)-f(qt)(1-q)t,Dqf(0)=limt→0Dqf(t).
The higher-order q-derivatives are given by
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
∫axf(t)dqt:=∑n=0∞x(1-q)qnf(xqn)-af(qna),x∈[a,b],
and for a=0, we denote
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
∫abf(t)dqt=∫0bf(t)dqt-∫0af(t)dqt.
Similarly, we have
Iq0f(t)=f(t),Iqnf(t)=IqIqn-1f(t),n∈N.
Observe that
DqIqf(x)=f(x),
and if f is continuous at x=0, then
IqDqf(x)=f(x)-f(0).
In q-calculus, the integration by parts formula is
∫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].
Let X 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].
Let C([0,T],ℝ) denote the Banach space of all continuous functions from [0,T] into ℝ with the norm
‖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
‖u‖L1=∫0T|u(t)|dt,∀u∈L1([0,T],R).
Definition 2.1.
A multivalued map G:[0,T]→𝒫(ℝ) with nonempty compact convex values is said to be measurable if for any x∈ℝ, the function
t⟼d(x,F(t))=inf{|x-z|:z∈F(t)}
is measurable.
Definition 2.2.
A multivalued map F:[0,T]×ℝ→𝒫(ℝ) is said to be Carathéodory if
t↦F(t,x) is measurable for each x∈ℝ,
x↦F(t,x) is upper semicontinuous for almost all t∈[0,T].
Further a Carathéodory function F is called L1-Carathéodory if
for each α>0, there exists φα∈L1([0,T],ℝ+) such that
‖F(t,x)‖=sup{|v|:v∈F(t,x)}≤φα(t)
for all ∥x∥∞≤α and for a.e. t∈[0,T].
Let E 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.
If F:[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
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
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 function l: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 operator N on X with nonempty values in X is called
γ-Lipschitz if and only if there exists γ>0 such that
dH(N(x),N(y))≤γd(x,y),foreachx,y∈X,
a contraction if and only if it is γ-Lipschitz with γ<1,
a generalized contraction if and only if there is a strict comparison function l:ℝ+→ℝ+ such that
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]).
Let X 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
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 if dimX<∞, 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]).
Let E 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
F has a fixed-point in U¯,
or there is a u∈∂U and λ∈(0,1) with u∈λF(u).
Lemma 2.8 (see [24]).
Let Y 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.
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 that σ:[0,T]→ℝ is continuous, then the following problem
Dq2u(t)=σ(t),a.e.t∈[0,T],u(0)=ηu(T),Dqu(0)=ηDqu(T)
has a unique solution
u(t)=∫0TG(t,qs)σ(s)dqs,
where G(t,qs) is the Green's function given by
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
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
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
G1=maxt,s∈[0,T]|G(t,qs)|.
Definition 3.2.
A function u∈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
u(t)=∫0TG(t,qs)v(s)dqs,
where G(t,qs) is given by (3.3).
Theorem 3.3.
Suppose that
the map F:[0,T]×ℝ→𝒫(ℝ) has nonempty compact convex values and is Carathéodory,
there exist a continuous nondecreasing function ψ:[0,∞)→(0,∞) and a function p∈L1([0,T],ℝ+) such that
‖F(t,u)‖P:=sup{|v|:v∈F(t,u)}≤p(t)ψ(‖u‖∞)
for each (t,u)∈[0,T]×ℝ,
there exists a number M>0 such that
MG1ψ(M)‖p‖L1>1,
then the BVP (1.1)-(1.2) has at least one solution.
Proof.
In view of Definition 3.2, the existence of solutions to (1.1)-(1.2) is equivalent to the existence of solutions to the integral inclusion
u(t)∈∫0TG(t,qs)F(s,u(s))dqs,t∈[0,T].
Let us introduce the operator
N(u):={h∈C([0,T],R):h(t)=∫0TG(t,qs)v(s)dqs,v∈SF,u}.
We will show that N 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, if h1, h2 belong to N(u), then there exist v1,v2∈SF,u such that for each t∈[0,T], we have
hi(t)=∫0TG(t,qs)vi(s)dqs,(i=1,2).
Let 0≤d≤1, then, for each t∈[0,T], we have
(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,
dh1+(1-d)h2∈N(u).
Step 2 (N maps bounded sets into bounded sets in C([0,T],ℝ)).
Let Bm={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
h(t)=∫0TG(t,qs)v(s)dqs.
Then, in view of (H2), we have
|h(t)|≤∫0T|G(t,qs)||v(s)|dqs≤G1∫0Tp(s)ψ(‖u‖∞)dqs≤G1ψ(m)∫0Tp(s)dqs.
Thus,
‖h‖∞≤G1ψ(m)‖p‖L1.
Step 3 (N maps bounded sets into equicontinuous sets of C([0,T],ℝ)).
Let r1,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)|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).
Let un→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],
hn(t)=∫0TG(t,qs)vn(s)dqs.
We must show that there exists h*∈SF,u* such that, for each t∈[0,T],
h*(t)=∫0TG(t,qs)v*(s)dqs.
Clearly, we have
‖hn-h*‖∞⟶0asn⟶∞.
Consider the continuous linear operator
Γ:L1([0,T],R)⟶C([0,T],R),
defined by
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
hn(t)∈Γ(SF,un).
Since un→u*, it follows from Lemma 2.6 that
h*(t)=∫0TG(t,qs)v*(s)dqs
for some v*∈SF,u*.
Step 5 (a priori bounds on solutions).
Let u 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],
u(t)=∫0TG(t,qs)v(s)dqs.
For each t∈[0,T], it follows by (H2) and (H3) that
|u(t)|≤G1∫0Tp(s)ψ(‖u‖∞)dqs≤G1ψ(‖u‖∞)∫0Tp(s)dqs.
Consequently,
‖u‖∞G1ψ(‖u‖∞)‖p‖L1≤1.
Then by (H3), there exists M such that ∥u∥∞≠M.
Let
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 where F 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
F:[0,T]×ℝ→𝒫(ℝ) has nonempty compact values and
(t,u)↦F(t,u) is ℒ⊗ℬ measurable,
u↦F(t,u) is lower semicontinuous for a.e. t∈[0,T],
for each ρ>0, there exists φρ∈L1([0,T],ℝ+) such that
‖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 that F 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
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
u(t)=(N¯u)(t),t∈[0,T],
where
(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
F:[0,T]×ℝ→𝒫(ℝ) has nonempty compact values and F(·,u) is measurable for each u∈ℝ,
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 that γl: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
|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],
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],
un(t)⟶ũ(t)=∫0TG(t,qs)v(s)dqs.
So, ũ∈N(u) and hence N(u) is closed.
Next, we show that
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],
h1(t)=∫0TG(t,qs)v1(s)dqs.
From (H7), it follows that
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
|v1(t)-w|≤k(t)l(|u(t)-u¯(t)|),t∈[0,T].
Define U:[0,T]→𝒫(ℝ) as
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],
|v1(t)-v2(t)|≤k(t)l(|u(t)-u¯(t)|).
For each t∈[0,T], let us define
h2(t)=∫0TG(t,qs)v2(s)dqs,
then
|h1(t)-h2(t)|≤∫0T|G(t,qs)||v1(s)-v2(s)|dqs≤G1∫0Tk(s)l(‖u-u¯‖)dqs.
Thus,
‖h1-h2‖∞≤G1‖k‖L1l(‖u-u¯‖∞)=γl(‖u-u¯‖∞).
By an analogous argument, interchanging the roles of u and u¯, we obtain
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 Theorem 3.5 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-order q-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 limit q→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-order q-difference inclusions given by
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
(t,u)⟶F(t,u)=[u3u3+3+t3+3,uu+1+t+1].
For f∈F, we have
|f|≤max(u3u3+3+t3+3,uu+1+t+1)≤5,u∈R.
Thus,
‖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
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).
Acknowledgments
The authors thank the referees for their comments. The research of B. Ahmad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
KacV.CheungP.2002New York, NY, USASpringerJacksonF. H.On q-definite integrals191041193203GasperG.RahmanM.1990Cambridge, UKCambridge University PressAnnabyM. H.MansourZ. S.q-Taylor and interpolation series for Jackson q-difference operators200834414724832-s2.0-4264908501010.1016/j.jmaa.2008.02.033ZBL1149.40001DobrogowskaA.OdzijewiczA.Second order q-difference equations solvable by factorization method200619313193462-s2.0-3364619968110.1016/j.cam.2005.06.009ZBL1119.39017El-ShahedM.HassanH. A.Positive solutions of q-Difference equation20101385173317382-s2.0-7795115427210.1090/S0002-9939-09-10185-5ZBL1201.39003AyoolaE. O.Quantum stochastic differential inclusions satisfying a general lipschitz condition2008173-44875022-s2.0-76149087480ZBL1202.81130BelarbiA.BenchohraM.Existence results for nonlinear boundary-value problems with integral boundary conditions20052005061102-s2.0-13444302404ZBL1075.34015BenaïmM.HofbauerJ.SorinS.Stochastic approximations and differential inclusions, Part II: applications20063146736952-s2.0-3384723019010.1287/moor.1060.0213ChangY. K.LiW. T.NietoJ. J.Controllability of evolution differential inclusions in Banach spaces20076726236322-s2.0-3404727009010.1016/j.na.2006.06.018ZBL1128.93005NtouyasS. K.Neumann boundary value problems for impulsive differential inclusions200922113ZBL1195.34041SimsenJ.GentileC. B.Systems of p-Laplacian differential inclusions with large diffusion201036825255372-s2.0-7795288765310.1016/j.jmaa.2010.02.006ZBL1191.35163SmirnovG. V.2002Providence, RI, USAAmerican Mathematical SocietyApreuteseiN.ApreuteseiG.A Trotter-Kato type result for a second order difference inclusion in a Hilbert space201036111952042-s2.0-7034939901610.1016/j.jmaa.2009.08.065ZBL1181.47066AticiF. M.BilesD. C.First order dynamic inclusions on time scales200429212222372-s2.0-184255497810.1016/j.jmaa.2003.11.053ZBL1064.34009CerneaA.GeorgescuC.Necessary optimality conditions for differential-difference inclusions with state constraints2007334143532-s2.0-3424928198410.1016/j.jmaa.2006.12.020ZBL1124.49018ChangY. K.LiW. T.Existence results for second-order dynamic inclusion with m-point boundary value conditions on time scales20072088858912-s2.0-3404716525110.1016/j.aml.2006.09.004ZBL1154.34310DeimlingK.1992Berlin, GermanyWalter De GruyterHuS.PapageorgiouN.1997Dodrecht, The NetherlandsKluwer AcademicAubinJ. P.CellinaA.1984Heidelberg, GermanySpringerAubinJ. P.FrankowskaH.1990Boston, Mass, USABirkhäauserLasotaA.OpialZ.An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations196513781786ZBL0151.10703GranasA.DugundjiJ.2003New York, NY, USASpringerBressanA.ColomboG.Extensions and selections of maps with decomposable values1988906986ZBL0677.54013LazǎrT. A.PetruşelA.ShahzadN.Fixed points for non-self operators and domain invariance theorems20097011171252-s2.0-5574911140210.1016/j.na.2007.11.037WegrzykR.Fixed point theorems for multifunctions and their applications to functional equations1982201128CovitzH.NadlerS. B.Jr.Multi-valued contraction mappings in generalized metric spaces1970815112-s2.0-5124919181410.1007/BF02771543ZBL0192.59802CastaingC.ValadierM.1977Heidelberg, GermanySpringerLecture Notes in Mathematics 580AhmadB.basher_qau@yahoo.comNtouyasS. K.sntouyas@uoi.grSome existence results for boundary value problems of fractional differential inclusions with non-separated boundary conditions201071117ZBL1206.26005