Q-INTEGRAL EQUATIONS of FRACTIONAL ORDERS 1. Introduction in This Paper, We Are Concerned with the Following Functional Equation

Home , Quadratic integral

Electronic Journal of Diﬀerential Equations, Vol. 2016 (2016), No. 17, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

Q-INTEGRAL EQUATIONS OF FRACTIONAL ORDERS

MOHAMED JLELI, MOHAMMAD MURSALEEN, BESSEM SAMET

Abstract. The aim of this paper is to study the existence of solutions for a class of q-integral equations of fractional orders. The techniques in this work are based on the measure of non-compactness argument and a generalized version of Darbo’s theorem. An example is presented to illustrate the obtained result.

1. Introduction In this paper, we are concerned with the following functional equation Z t f(t, x(b(t))) (α−1) x(t) = F t, x(a(t)), (t − qs) u(s, x(s)) dqs , t ∈ I, (1.1) Γq(α) 0 where α > 1, q ∈ (0, 1), I = [0, 1], f, u : [0, 1] × R → R, a, b : I → I and F : I × R × R → R. Equation (1.1) can be written as α x(t) = F t, x(a(t)), f(t, x(b(t)))Iq u(·, x(·))(t) , t ∈ I, α where Iq is the q-fractional integral of order α defined by (see [1]) Z t α 1 (α−1) Iq h(t) = (t − qs) h(s) dqs, t ∈ [0, 1]. Γq(α) 0 Using a measure of non-compactness argument combined with a generalized version of Darbo’s theorem, we provide sufficient conditions for the existence of at least one solution to (1.1). We present also some examples to illustrate our obtained result. The measure of non-compactness argument was used by several authors to study the existence of solutions for various classes of integral equations. As examples, we refer the reader to Aghajani et al [2, 4, 5], Bana´set al [10, 14, 15], Bana´sand Goebel [11], Bana´sand Rzepka [16], Bana´sand Martinon [12], Caballero et al [20, 21, 22], Darwish [25], Çakan and Ozdemir [23], Dhage and Bellale [28], Mursaleen and Mohiuddine [39], Mursaleen and Alotaibi [37], and the references therein. For other applications of the measure of non-compactness concept, we refer to [13, 38].

2010 Mathematics Subject Classiﬁcation. 31A10, 26A33, 47H08. Key words and phrases. q-fractional integral; existence; measure of non-compactness. c 2016 Texas State University. Submitted October 2, 2015. Published January 8, 2016. 1 2 M. JLELI, M. MURSALEEN, B. SAMET EJDE-2016/17

In [24], via a measure of non-compactness concept, Darwish obtained an existence result for the singular integral equation y(t) Z t u(s, y(s)) y(t) = f(t) + 1−α ds, t ∈ [0, 1], α > 0. Γ(α) 0 (t − s) The above equation can be written in the form y(t) = f(t) + y(t)Iαu(·, y(·))(t), t ∈ [0, 1], where Iα is the Riemann-Liouville fractional integral of order α defined by (see [41]) Z t α 1 h(s) I h(t) = 1−α ds, t ∈ [0, 1]. Γ(α) 0 (t − s) Following the above work, there has been a great interest in the study of functional equations involving fractional integrals. In this direction, we refer the reader to [3, 19, 26, 27, 17, 18, 29] and the references therein. The concept of q-calculus (quantum calculus) was introduced by Jackson (see [33, 34]). This subject is rich in history and has several applications (see [30, 35]). Fractional q-difference concept was initiated by Agarwal and by Al-Salam (see [1, 7]). Because of the considerable progress in the study of fractional differential equations, a great interest appeared from many authors in studying fractional q- difference equations (see for examples [6, 7, 31, 32, 36] and the references therein). The paper is organized as follows. In Section 2, we fix some notations that will be used through this paper, we recall some basic tools on q-calculus and we collect some basic definitions and facts on the measure of non-compactness concept. In Section 3, we state and prove our main result. Next, we present an illustrative example.

2. Preliminaries At first, we recall some concepts on fractional q-calculus and present additional properties that will be used later. For more details, we refer to [1, 8, 40]. Let q be a positive real number such that q 6= 1. For x ∈ R, the q-real number [x]q is defined by 1 − qx [x] = . q 1 − q The q-shifted factorial of real number x is defined by k−1 Y i (x, q)0 = 1, (x, q)k = (1 − xq ), k = 1, 2,..., ∞. i=0 For (x, y) ∈ R2, the q-analog of (x − y)k is defined by k−1 Y (x − y)(0) = 1, (x − y)(k) = (x − qiy), k = 1, 2,... i=0 For β ∈ R,(x, y) ∈ R2 and x ≥ 0, ∞ Y x − yqi (x − y)(β) = xβ . x − yqβ+i i=0 Note that if y = 0, then x(β) = xβ. The following inequality (see [31]) will be used later. EJDE-2016/17 Q-INTEGRAL EQUATIONS OF FRACTIONAL ORDERS 3

Lemma 2.1. If β > 0 and 0 ≤ a ≤ b ≤ t, then (t − b)(β) ≤ (t − a)(β). The q-gamma function is given by (1 − q)(x−1) Γ (x) = , x 6∈ {0, −1, −2,... }. q (1 − q)x−1 We have the following property

Γq(x + 1) = [x]qΓq(x). Let f : [0, b] → R (b > 0) be a given function. The q-integral of the function f is given by Z t ∞ X n n Iqf(t) = f(s) dqs = t(1 − q) f(tq )q , t ∈ [0, b]. 0 n=0 If c ∈ [0, b], we have Z b Z b Z c f(s) dqs = f(s) dqs − f(s) dqs. c 0 0 Lemma 2.2. If f : [0, 1] → R is a continuous function, then Z t Z t

f(s) dqs ≤ |f(s)| dqs, t ∈ [0, 1]. 0 0

Note that in general, if 0 ≤ t1 ≤ t2 ≤ 1 and f : [0, 1] → R is a continuous function, the inequality

Z t2 Z t2

f(s) dqs ≤ |f(s)| dqs t1 t1 is not satisfied. We remark that in many papers dealing with q-difference boundary value problems, the use of such inequality yields wrong results. As a counter- example, we refer the reader to [8, Page.12]. Let f : [0, 1] → R be a given function. The fractional q-integral of order α ≥ 0 0 of the function f is given by Iq f(t) = f(t) and Z t α 1 (α−1) Iq f(t) = (t − qs) f(s) dqs, t ∈ [0, 1], α > 0. Γq(α) 0 Note that for α = 1, we have 1 Iq f(t) = Iqf(t), t ∈ [0, 1]. If f ≡ 1, then α 1 α Iq 1(t) = t , t ∈ [0, 1]. Γq(α + 1) Now, we recall the notion of measure of non-compactness, which is the main tool used in this paper. Let E be a Banach space over R with respect to a certain norm k · k. We denote by BE the set of all nonempty and bounded subsets of E. A mapping σ : BE → [0, ∞) is a measure of non-compactness in E (see [13]) if the following conditions are satisfied:

(i) for all M ∈ BE, we have σ(M) = 0 implies M is precompact; 4 M. JLELI, M. MURSALEEN, B. SAMET EJDE-2016/17

(ii) for every pair (M1,M2) ∈ BE × BE, we have

M1 ⊆ M2 =⇒ σ(M1) ≤ σ(M2);

(iii) for every M ∈ BE, σ(M) = σ(M) = σ(coM), where coM denotes the closed convex hull of M;

(iv) for every pair (M1,M2) ∈ BE × BE and λ ∈ (0, 1), we have

σ(λM1 + (1 − λ)M2) ≤ λσ(M1) + (1 − λ)σ(M2);

(v) if {Mn} is a sequence of closed and decreasing (w.r.t ⊆) sets in BE such ∞ that σ(Mn) → 0 as n → ∞, then M∞ := ∩n=1Mn is nonempty. Let Λ be the set of functions η : [0, ∞) → [0, ∞) such that (1) η is a non-decreasing function; (2) η is an upper semi-continuous function; (3) η(s) < s, for all s > 0. For our purpose, we need the following generalized version of Darbo’s theorem (see [2]). Lemma 2.3. Let D be a nonempty, bounded, closed and convex subset of a Banach space E. Let T : D → D be a continuous mapping such that σ(TM) ≤ η(σ(M)),M ⊆ D, where η ∈ Λ and σ is a measure of non-compactness in E. Then T has at least one ﬁxed point.

Lemma 2.4. Let η1, η2 ∈ Λ and τ ∈ (0, 1). Then the function γ : [0, ∞) → [0, ∞) deﬁned by

γ(t) = max{η1(t), η2(t), τ t}, t ≥ 0 belongs to the set Λ.

2 Proof. Let (t, s) ∈ R be such that 0 ≤ t ≤ s. Since η1, η2 are non-decreasing and τ ∈ (0, 1), we have

ηi(t) ≤ ηi(s) ≤ γ(s), i = 1, 2 , τt ≤ τs ≤ γ(s), which yield γ(t) ≤ γ(s). Therefore, γ is a non-decreasing function. Now, for all s > 0, we have ηi(s) < s (for i = 1, 2) and τ s < s. Since γ(s) ∈ {η1(s), η2(s), τ s}, we obtain γ(s) < s, s > 0. On the other hand, it is well-known that the maximum of ﬁnitely many upper semi-continuous functions is upper semi-continuous. As consequence, the function γ belongs to the set Λ.

In what follows let E = C(I; R) be the set of real and continuous functions in the compact set I. It is well-known that E is a Banach space with respect to the norm kxk = max{|x(t)| : t ∈ I}, x ∈ E. EJDE-2016/17 Q-INTEGRAL EQUATIONS OF FRACTIONAL ORDERS 5

Now, let M ∈ BE be ﬁxed. For (x, ρ) ∈ M × (0, ∞), we denote by ω(x, ρ) the modulus of continuity of x; that is,

ω(x, ρ) = sup{|x(t1) − x(t2)| :(t1, t2) ∈ I × I, |t1 − t2| ≤ ρ}. Further on let us deﬁne ω(M, ρ) = sup{ω(x, ρ): x ∈ M}.

Deﬁne the mapping σ : BE → [0, ∞) by σ(M) = lim ω(M, ρ),M ∈ B . ρ→0+ E Then σ is a measure of non-compactness in E (see [11]).

3. Main result Deﬁne the operator T on E = C(I; R) by Z t f(t, x(b(t))) (α−1) (T x)(t) = F t, x(a(t)), (t − qs) u(s, x(s)) dqs , (x, t) ∈ E × I. Γq(α) 0 We consider the assumption (A1) The functions f, u : [0, 1] × R → R, a, b : I → I and F : [0, 1] × R × R → R are continuous.

Proposition 3.1. Under assumption (A1), the operator T maps E into itself. Proof. From assumption (A1), we have just to show that the operator H deﬁned on E by Z t (α−1) (Hx)(t) = (t − qs) u(s, x(s)) dqs, (x, t) ∈ E × I (3.1) 0 maps E into itself. To do this, let us ﬁx x ∈ E. For all t ∈ I, we have Z t (α−1) (Hx)(t) = (t − qs) u(s, x(s)) dqs 0 ∞ X = t(1 − q) qn(t − qn+1t)(α−1)u(tqn, x(tqn)) n=0 ∞ X = tα(1 − q) qn(1 − qn+1)(α−1)u(tqn, x(tqn)). n=0 On the other hand, since 0 < qn+1 < 1, using Lemma 2.1, we have (1 − qn+1)(α−1) ≤ (1 − 0)(α−1) = 1. Then by the continuity of u and using the Weierstrass convergence theorem, we obtain the desired result. We consider also the following assumptions.

(A2) There exist a constant CF > 0 and a non-decreasing function ϕF : [0, ∞) → [0, ∞) such that

|F (t, x, y)−F (t, z, w)| ≤ ϕF (|x−z|)+CF |y−w|, (t, x, y, z, w) ∈ I ×R×R×R×R.

(A3) There exists a constant Cf > 0 such that

|f(t, x) − f(t, y)| ≤ Cf |x − y|, (t, x, y) ∈ I × R × R. 6 M. JLELI, M. MURSALEEN, B. SAMET EJDE-2016/17

(A4) There exists a non-decreasing and continuous function ϕu : [0, ∞) → [0, ∞) such that

|u(t, x) − u(t, y)| ≤ ϕu(|x − y|), (t, x, y) ∈ I × R × R, ϕu(t) < t, t > 0, u(t, 0) = 0, t ∈ I.

(A5) There exists r0 > 0 such that ∗ ∗ CF (Cf r0 + f )ϕu(r0) ϕF (r0) + F + ≤ r0, Γq(α + 1) where F ∗ = max{|F (t, 0, 0)| : t ∈ I} and f ∗ = max{|f(t, 0)| : t ∈ I}.

Let the closed ball of center 0 and radius r0 be denote by

B(0, r0) = {x ∈ E : kxk ≤ r0}.

Proposition 3.2. Under assumptions (A1)–(A5), the operator T maps B(0, r0) into itself.

Proof. Let x ∈ B(0, r0). Using the considered assumptions, for all t ∈ I, we have |(T x)(t)| f(t, x(b(t))) Z t (α−1) ≤ F t, x(a(t)), (t − qs) u(s, x(s)) dqs − F (t, 0, 0) Γq(α) 0 + |F (t, 0, 0)| Z t |f(t, x(b(t)))| (α−1) ∗ ≤ ϕF (|x(a(t))|) + CF (t − qs) |u(s, x(s))| dqs + F Γq(α) 0 |f(t, x(b(t))) − f(t, 0)| + |f(t, 0)| ≤ ϕF (kxk) + CF Γq(α) Z t (α−1) ∗ × (t − qs) |u(s, x(s))| dqs + F 0 ∗ Z t (Cf |x(b(t))| + f ) ϕu(kxk) (α−1) ∗ ≤ ϕF (kxk) + CF (t − qs) dqs + F Γq(α) 0 ∗ (Cf kxk + f ) ϕu(kxk) α ∗ ≤ ϕF (kxk) + CF t + F Γq(α + 1) ∗ (Cf r0 + f ) ϕu(r0) ∗ ≤ ϕF (r0) + CF + F . Γq(α + 1) Therefore, ∗ (Cf r0 + f ) ϕu(r0) ∗ kT xk ≤ ϕF (r0) + CF + F , x ∈ B(0, r0). Γq(α + 1) Using the above inequality and assumption (A5), we obtain the desired result. Proposition 3.3. Under assumptions (A1)–(A5), the operator T maps continu- ously B(0, r0) into itself.

Proof. Deﬁne the operators γ1, γ2 and γ3 on E by (γ1x)(t) = t, (x, t) ∈ E × I, (γ2x)(t) = x(a(t)), (x, t) ∈ E × I, EJDE-2016/17 Q-INTEGRAL EQUATIONS OF FRACTIONAL ORDERS 7

(γ3x)(t) = f(t, x(b(t))), (x, t) ∈ E × I. Obviously, γ1 : E → E is continuous. Moreover, for all x, y ∈ E, we have

|(γ2x)(t) − (γ2y)(t)| = |x(a(t)) − y(a(t))| ≤ kx − yk, t ∈ I, which implies that

kγ2x − γ2yk ≤ kx − yk, (x, y) ∈ E × E. Therefore, γ2 is uniformly continuous on E. Similarly, for all x, y ∈ E, for all t ∈ I, we have

|(γ3x)(t) − (γ3y)(t)| = |f(t, x(b(t))) − f(t, y(b(t)))|

≤ Cf |x(b(t)) − y(b(t))k ≤ Cf kx − yk, which implies kγ3x − γ3yk ≤ Cf kx − yk, (x, y) ∈ E × E. Then γ3 is also uniformly continuous on E. So, in order to prove that T is continuous on B(0, r0), we only need to show that the operator H deﬁned by (3.1) is continuous on B(0, r0). To do this, let us consider ε > 0 and (x, y) ∈ B(0, r0) × B(0, r0) such that kx − yk ≤ ε. For all t ∈ I, we have Z t Z t (α−1) (α−1) (Hx)(t) − (Hy)(t) = (t − qs) u(s, x(s)) dqs − (t − qs) u(s, y(s)) dqs 0 0 Z t (α−1) = (t − qs) (u(s, x(s)) − u(s, y(s))) dqs. 0 Set

ur0 (ε) = sup{|u(t, x) − u(t, y)| : t ∈ I, (x, y) ∈ [−r0, r0] × [−r0, r0], |x − y| ≤ ε}, we obtain α t ur0 (ε) (Hx)(t) − (Hy)(t) ≤ ur0 (ε) ≤ , [α]q [α]q for all t ∈ I. Therefore, u (ε) kHx − Hyk ≤ r0 . [α]q Passing to the limit as ε → 0+ and using the uniform continuity of u on the compact set I × [−r0, r0], we obtain u (ε) lim r0 = 0, ε→0+ [α]q which completes the proof. To prove our main result, the following additional assumptions are needed.

(A6) The function ϕF : [0, ∞) → [0, ∞) is continuous and it satisﬁes ϕF (s) < s for s > 0. (A7) The function a : I → I satisﬁes

|a(t) − a(s)| ≤ ϕa(|t − s|), (t, s) ∈ I × I,

where ϕa : [0, ∞) → [0, ∞) is non-decreasing and limt→0+ ϕa(t) = 0. (A8) The function b : I → I satisﬁes

|b(t) − b(s)| ≤ ϕb(|t − s|), (t, s) ∈ I × I,

where ϕb : [0, ∞) → [0, ∞) is non-decreasing and limt→0+ ϕb(t) = 0. 8 M. JLELI, M. MURSALEEN, B. SAMET EJDE-2016/17

(A9) We suppose that

Γq(α + 1) CF ∗ 0 < ϕu(r0) < and (Cf r0 + f ) < 1. CF Cf Γq(α) Our main result is the following. Theorem 3.4. Under assumptions (A1)–(A9), Equation (1.1) has at least one ∗ ∗ solution x ∈ C(I; R) satisfying kx k ≤ r0.

Proof. From Proposition 3.3, we know that T : B(0, r0) → B(0, r0) is a continuous operator. Now, let M be a nonempty subset of B(0, r0). Let ρ > 0, x ∈ M and (t1, t2) ∈ I × I be such that |t1 − t2| ≤ ρ. Without restriction of the generality, we may assume that t1 ≥ t2. We have

|(T x)(t1) − (T x)(t2)| f(t , x(b(t ))) Z t1 1 1 (α−1) = F t1, x(a(t1)), (t1 − qs) u(s, x(s)) dqs Γq(α) 0 f(t , x(b(t ))) Z t2 2 2 (α−1) − F t2, x(a(t2)), (t2 − qs) u(s, x(s)) dqs Γq(α) 0 f(t , x(b(t ))) Z t1 ≤ F t , x(a(t )), 1 1 (t − qs)(α−1)u(s, x(s)) d s 1 1 Γ (α) 1 q q 0 (3.2) f(t , x(b(t ))) Z t1 1 1 (α−1) − F t2, x(a(t1)), (t1 − qs) u(s, x(s)) dqs Γq(α) 0 f(t , x(b(t ))) Z t1 1 1 (α−1) + F t2, x(a(t1)), (t1 − qs) u(s, x(s)) dqs Γq(α) 0 f(t , x(b(t ))) Z t2 2 2 (α−1) − F t2, x(a(t2)), (t2 − qs) u(s, x(s)) dqs Γq(α) 0 = (I) + (II). • Estimate for (I). We have f(t , x(b(t ))) Z t1 1 1 (α−1) (t1 − qs) u(s, x(s)) dqs Γq(α) 0 Z t1 |f(t1, x(b(t1)))| (α−1) ≤ (t1 − qs) |u(s, x(s))| dqs Γq(α) 0 Z t1 |f(t1, x(b(t1))) − f(t1, 0)| + |f(t1, 0)| (α−1) ≤ (t1 − qs) ϕu(|x(s)|) dqs Γq(α) 0 ∗ (Cf |x(b(t1))| + f )ϕu(kxk) α ≤ t1 Γq(α + 1) (C kxk + f ∗)ϕ (kxk) ≤ f u Γq(α + 1) (C r + f ∗)ϕ (r ) ≤ f 0 u 0 = D. Γq(α + 1) Set C(F, δ) = sup |F (t, x, y) − F (s, x, y)| :(t, s) ∈ I × I, |t − s| ≤ ρ,

x ∈ [−r0, r0], y ∈ [−D,D] , EJDE-2016/17 Q-INTEGRAL EQUATIONS OF FRACTIONAL ORDERS 9 we obtain (I) ≤ C(F, δ). (3.3) • Estimate for (II). We have

(II) ≤ ϕF (|x(a(t1)) − x(a(t2))|) C Z t1 F (α−1) + f(t1, x(b(t1))) (t1 − qs) u(s, x(s)) dqs Γq(α) 0 Z t2 (α−1) − f(t2, x(b(t2))) (t2 − qs) u(s, x(s)) dqs . 0 On the other hand, |x(a(t1)) − x(a(t2))| ≤ ω(x ◦ a, ρ), which yields ϕF (|x(a(t1)) − x(a(t2))|) ≤ ϕF (ω(x ◦ a, ρ)). Now, we have Z t1 (α−1) f(t1, x(b(t1))) (t1 − qs) u(s, x(s)) dqs 0 Z t2 (α−1) − f(t2, x(b(t2))) (t2 − qs) u(s, x(s)) dqs 0 Z t1 (α−1) ≤ f(t1, x(b(t1))) (t1 − qs) u(s, x(s)) dqs 0 Z t1 (α−1) − f(t2, x(b(t2))) (t1 − qs) u(s, x(s)) dqs 0 Z t1 (α−1) + f(t2, x(b(t2))) (t1 − qs) u(s, x(s)) dqs 0 Z t2 (α−1) − f(t2, x(b(t2))) (t2 − qs) u(s, x(s)) dqs 0 |f(t , x(b(t ))) − f(t , x(b(t )))|ϕ (kxk) ≤ 1 1 2 2 u [α]q Z t1 (α−1) + |f(t2, x(b(t2)))| (t1 − qs) u(s, x(s)) dqs 0 Z t2 (α−1) − (t2 − qs) u(s, x(s)) dqs 0 = (III) + (IV ). Let us deﬁne

ωf (r0, ρ) = sup{|f(t, x) − f(s, x)| :(t, s) ∈ I × I, |t − s| ≤ ρ, x ∈ [−r0, r0]}. Then ϕu(kxk) (III) ≤ |f(t1, x(b(t1))) − f(t1, x(b(t2)))| [α]q

ϕu(kxk) + |f(t1, x(b(t2))) − f(t2, x(b(t2)))| [α]q [C |x(b(t )) − x(b(t ))| + ω (r , ρ)] ϕ (r ) ≤ f 1 2 f 0 u 0 [α]q 10 M. JLELI, M. MURSALEEN, B. SAMET EJDE-2016/17

[C ω(x ◦ b, ρ) + ω (r , ρ)] ϕ (r ) ≤ f f 0 u 0 . [α]q Now, let us estimate (IV ). At ﬁrst, we have

≤ ϕu(ω(x, ρ)) + Aρ, where 0 0 0 0 4 0 Aρ = sup |N (τ, s, x) − N (τ , s , x)| :(τ, s, τ , s ) ∈ I , |τ − τ | ≤ ρ, 0 |s − s | ≤ ρ, x ∈ [−r0, r0] and α N (τ, s, x) = τ u(s, x), (τ, s, x) ∈ I × I × R. Then, we obtain Z t1 Z t2 (α−1) (α−1) (t1 − qs) u(s, x(s)) dqs − (t2 − qs) u(s, x(s)) dqs 0 0 ≤ ϕu(ω(x, ρ)) + Aρ. As consequence, we have ∗ (IV ) ≤ (Cf r0 + f )(ϕu(ω(x, ρ)) + Aρ). Using the above inequalities, we obtain

CF [Cf ω(x ◦ b, ρ) + ωf (r0, ρ)]ϕu(r0) (II) ≤ ϕF (ω(x ◦ a, ρ)) + Γq(α) [α]q ∗ + (Cf r0 + f )(ϕu(ω(x, ρ)) + Aρ) . Now, observe that from assumption (A7), we have ω(x ◦ a, ρ) = sup{|x(a(t)) − x(a(s))| :(t, s) ∈ I × I, |t − s| ≤ ρ}

≤ sup{|x(µ) − x(ν)| :(µ, ν) ∈ I × I, |µ − ν| ≤ ϕa(ρ)}

= ω(x, ϕa(ρ)). Similarly, from assumption (A8), we have

ω(x ◦ b, ρ) ≤ ω(x, ϕb(ρ)). EJDE-2016/17 Q-INTEGRAL EQUATIONS OF FRACTIONAL ORDERS 11

Then

CF [Cf ω(x, ϕb(ρ)) + ωf (r0, ρ)]ϕu(r0) (II) ≤ ϕF (ω(x, ϕa(ρ))) + Γq(α) [α]q (3.4) ∗ + (Cf r0 + f )(ϕu(ω(x, ρ)) + Aρ) . Next, using (3.2), (3.3) and (3.4), we obtain

CF [Cf ω(x, ϕb(ρ)) + ωf (r0, ρ)]ϕu(r0) ω(T x, ρ) ≤ C(F, δ) + ϕF (ω(x, ϕa(ρ))) + Γq(α) [α]q ∗ + (Cf r0 + f )(ϕu(ω(x, ρ)) + Aρ) , which yields

ω(T M, ρ) ≤ C(F, δ) + ϕF (ω(M, ϕa(ρ))) C [C ω(M, ϕ (ρ)) + ω (r , ρ)]ϕ (r ) + F f b f 0 u 0 Γq(α) [α]q ∗ + (Cf r0 + f )(ϕu(ω(M, ρ)) + Aρ) . Recall that from assumptions (A7)–(A8), we have

lim ϕa(t) = lim ϕb(t) = 0. ρ→0+ ρ→0+ Then passing to the limit as ρ → 0+ in the above inequality, we obtain

CF Cf σ(M)ϕu(r0) ∗ σ(TM) ≤ ϕF (σ(M)) + + (Cf r0 + f )ϕu(σ(M)) . Γq(α) [α]q Therefore, σ(TM) ≤ η(σ(M)), where η(t) = max{ϕF (t), Lϕu(t), Nt}, t ≥ 0, with CF ∗ CF Cf L = (Cf r0 + f ),N = ϕu(r0). Γq(α) Γq(α + 1) From assumption (A9) and Lemma 2.4, the function η belongs also to the set Λ. Finally, applying Lemma 2.3, we obtain the existence of at least one fixed point of the operator T in B(0, r0), which is a solution to (1.1). We end the paper with the following illustrative example. Consider the integral equation Z t t x(t) t x(t) (α−1) x(s) x(t) = + + [α]q + (t − qs) 2 dqs, (3.5) 32 4 2 4 0 (2 + s ) for t ∈ I = [0, 1], where α > 1 and q ∈ (0, 1). Observe that (3.5) can be written in the form (1.1), where a(t) = t, t ∈ I, b(t) = t, t ∈ I, t x F (t, x, y) = + + Γ (α + 1)y, (t, x, y) ∈ I × × , 32 4 q R R t x f(t, x) = + , (t, x) ∈ I × , 2 4 R 12 M. JLELI, M. MURSALEEN, B. SAMET EJDE-2016/17 x u(t, x) = , (t, x) ∈ I × . (2 + t2) R Now, let us check that the required assumptions by Theorem 3.4 are satisfied. • Assumption (A1). It is trivial. • Assumption (A2). For all (t, x, y, z, w) ∈ I × R × R × R × R, we have x z |F (t, x, y) − F (t, z, w)| = + Γ (α + 1)y − − Γ (α + 1)w 4 q 4 q |x − z| ≤ + Γ (α + 1)|y − w|. 4 q Then assumption (A2) is satisfied with t ϕ (t) = , t ≥ 0, F 4 CF = Γq(α + 1). • Assumption (A3). For all (t, x, y) ∈ I × R × R, we have |x − y| |f(t, x) − f(t, y)| = . 4 1 Then assumption (A3) is satisfied with Cf = 4 . • Assumption (A4). For all (t, x, y) ∈ I × R × R, we have |x − y| |x − y| |u(t, x) − u(t, y)| = ≤ . 2 + t2 2 t Take ϕu(t) = 2 , t ≥ 0, assumption (A4) holds. ∗ 1 ∗ 1 • Assumption (A5). At first, in our case, we have F = 32 and f = 2 . Now, the inequality ∗ ∗ CF (Cf r0 + f )ϕu(r0) ϕF (r0) + F + ≤ r0 Γq(α + 1) is equivalent to 1 r2 − 4r + ≤ 0. 0 0 4 √ √ 4− 15 4+ 15 The above inequality is satisfied for any r0 ∈ [ 2 , 2 ]. • Assumptions (A6)–(A8) are trivial. • Assumption (A9). The inequality

Γq(α + 1) 0 < ϕu(r0) < CF Cf is equivalent to 0 < r0 < 8. The inequality

CF ∗ (Cf r0 + f ) < 1 Γq(α) is equivalent to 4 r0 < − 2. [α]q A simple computation gives us that √ √ 4 − 15 4 + 15 4 , ∩ 0, − 2 6= ∅ 2 2 [α]q for α = 3/2 and q = 1/2. Therefore, all the assumptions (A1)–(A9) are satisﬁed for α = 3/2 and q = 1/2. By Theorem 3.4, we have the following result. EJDE-2016/17 Q-INTEGRAL EQUATIONS OF FRACTIONAL ORDERS 13

Theorem 3.5. For (α, q) = (3/2, 1/2), Equation (3.5) has at least one solution x∗ ∈ C(I; R). Acknowledgements. The ﬁrst and third authors would like to extend their sin- cere appreciation to the Deanship of Scientiﬁc Research at King Saud University for its funding of this research through the International Research Group Project no. IRG14-04.

References [1] R. P. Agarwal; Certain fractional q-integrals and q-derivatives, Proc. Cambridge Philos. Soc., 66 (1969), 365–370. [2] A. Aghajani, R. Allahyari, M. Mursaleen; A generalization of Darbo’s theorem with application to the solvability of systems of integral equations, J. Comput. Appl. Math., 260 (2014), 68–77. [3] A. Aghajani, J. Bana´s,Y. Jalilian; Existence of solutions for a class of nonlinear Volterra singular integral equations, Comput. Math. Appl., 62 (2011), 1215–1227. [4] A. Aghajani, M. Mursaleen, A. Shole Haghighi; Fixed point theorems for Meir-Keeler con- densing operators via measure of noncompactness, Acta Math. Sci., 35 B (3) (2015), 552–566. [5] A. Aghajani, E. Pourhadi, J.J. Trujillo; Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 16 (4) (2013), 962–977. [6] B. Ahmad, J. Nieto; Basic theory of nonlinear third-order q-difference equations and inclu- sions, Math. Model. Anal., 18 (1) (2013), 122–135. [7] W. Al-Salam; Some fractional q-integrals and q-derivatives, Proc. Edinb. Math. Soc., 15 (2) (1966/1967), 135–140 . [8] M. H. Annaby, Z. S. Mansour; q-fractional calculus and equations, Lecture Notes in Mathe- matics 2056, Springer-Verlag, Berlin, 2012. [9] F. Atici, P. Eloe; Fractional q-calculus on a time scale, J. Nonlinear Math. Phys., 14 (3) (2007), 341–352. [10] J. Bana´s,J. Caballero, J. Rocha, K. Sadarangani; Monotonic solutions of a class of quadratic integral equations of Volterra type, Comput. Math. Appl., 49 (2005), 943–952. [11] J. Bana´s,K. Goebel; Measures of Noncompactness in Banach Spaces, in: Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcel Dekker, New York and Basel, 1980. [12] J. Bana´s,A. Martinon; Monotonic solutions of a quadratic integral equation of Volterra type, Comput. Math. Appl., 47 (2004), 271–279. [13] J. Bana´s,M. Mursaleen; Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New Dehli, 2014. [14] J. Bana´s,M. Lecko, W.G. El-Sayed; Existence theorems of some quadratic integral equations, J. Math. Anal. Appl., 222 (1998), 276–285. [15] J. Bana´s,J. Rocha, K. Sadarangani; Solvability of a nonlinear integral equation of Volterra type, J. Comput. Appl. Math., 157 (2003), 31–48. [16] J. Bana´s,B. Rzepka; On existence and asymptotic stability of solutions of a nonlinear integral equation, J. Math. Anal. Appl., 284 (2003), 165–173. [17] J. Bana´s,B. Rzepka; Monotonic solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl., 332 (2007), 1370–1378. [18] J. Bana´s,D. O’Regan; On existence and local attractivity of solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl., 345 (2008), 573-582. [19] M. Benchohra, D. Seba; Integral equations of fractional order with multiple time delays in banach spaces, Electron. J. Differential Equations, 56 (2012), 1–8. [20] J. Caballero, B. López, K. Sadarangani; On monotonic solutions of an integral equation of Volterra type with supremum, J. Math. Anal. Appl., 305 (2005), 304–315. [21] J. Caballero, B. López, K. Sadarangani; Existence of nondecreasing and continuous solutions for a nonlinear integral equation with supremum in the kernel, Z. Anal. Anwend., 26 (2007), 195–205. [22] J. Caballero, J. Rocha, K. Sadarangani; On monotonic solutions of an integral equation of Volterra type, J. Comput. Appl. Math., 174 (2005), 119–133. 14 M. JLELI, M. MURSALEEN, B. SAMET EJDE-2016/17

[23] U. Çakan, I. Ozdemir; An application of Darbo fixed-point theorem to a class of functional integral equations, Numer. Funct. Anal. Optim. 36 (2015), 29–40. [24] M. A. Darwish; On quadratic integral equation of fractional orders, J. Math. Anal. Appl., 311 (2005), 112–119. [25] M. A. Darwish; On solvability of some quadratic functional-integral equation in Banach algebra, Commun. Appl. Anal. 11 (2007), 441–450. [26] M. A. Darwish; On monotonic solutions of a singular quadratic integral equation with supremum, Dynam. Systems Appl., 17 (2008) 539–550. [27] M. A. Darwish, K. Sadarangani; On a quadratic integral equation with supremum involving Erdélyi-Kober fractional order, Mathematische Nachrichten, 288 (5-6) (2015) 566–576. [28] B. C. Dhage, S. S. Bellale; Local asymptotic stability for nonlinear quadratic functional integral equations, Electron. J. Qual. Theory Differ. Equ., 10 (2008), 1–13. [29] A. M. A. El-Sayed, H. H. G. Hashem; Existence results for coupled systems of quadratic integral equations of fractional orders, Optimization Letters, 7 (2013), 1251—260. [30] T. Ernst; The History of q-Calculus and a New Method, U. U. D. M. Report 2000:16, ISSN 1101–3591, Department of Mathematics, Uppsala University, 2000. [31] R. Ferreira,; Nontrivial solutions for fractional q-difference boundary value problems, Elec- tron. J. Qual. Theory Differ. Equ., 70 (2010), 1–10. [32] R. Ferreira; Positive solutions for a class of boundary value problems with fractional q- differences, Comput. Math. Appl., 61 (2011), 367–373. [33] F. H. Jackson; On q-functions and a certain difference operator, Trans. Roy Soc. Edin., 46 (1908), 253–281. [34] F. H. Jackson; On q-definite integrals, Quart. J. Pure and Appl. Math., 41 (1910), 193–203. [35] V. Kac, P. Cheung; Quantum calculus, Springer Science & Business Media, 2002. [36] S. Liang, J. Zhang; Existence and uniqueness of positive solutions for three-point boundary value problem with fractional q-differences, J. Appl. Math. Comput., 40 (2012), 277–288. [37] M. Mursaleen, A. Alotaibi; Infinite system of differential equations in some BK spaces, Abstract Appl. Anal., Volume 2012, Article ID 863483, 20 pages. [38] M. Mursaleen, A. K. Noman; Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal., 73 (2010), 2541–2557. [39] M. Mursaleen, S. A. Mohiuddine; Applications of measures of noncompactness to the infinite system of differential equations in lp spaces, Nonlinear Anal. 75 (2012), 2111–2115. [40] P. M. Rajković,S. D. Marinković,M. S. Stanković; On q-analogues of Caputo derivative and Mittag-Leffler function, Fract. Calc. Appl. Anal., 10 (2007), 359–373. [41] S.G. Samko, A. A. Kilbas, O. I. Marichev; Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon et alibi, 1993.

Mohamed Jleli Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia E-mail address: [email protected]

Mohammad Mursaleen Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India E-mail address: [email protected]

Bessem Samet Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia E-mail address: [email protected]