Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 11, Number 2, pp. 93–104 (2016) http://campus.mst.edu/adsa Solvability of a Maximum Quadratic Integral Equation of Arbitrary Orders Mohamed Abdalla Darwish King Abdulaziz University Sciences Faculty for Girls Department of Mathematics Jeddah, Saudi Arabia [email protected] Johnny Henderson Baylor University Department of Mathematics Waco, Texas 76798-7328, USA Johnny [email protected] Kishin Sadarangani Universidad de Las Palmas de Gran Canaria Departamento de Matematicas´ Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain [email protected] Abstract We investigate a new quadratic integral equation of arbitrary orders with maxi- mum and prove an existence result for it. We will use a fixed point theorem due to Darbo as well as the monotonicity measure of noncompactness due to Banas´ and Olszowy to prove that our equation has at least one solution in C[0; 1] which is monotonic on [0; 1]. AMS Subject Classifications: 45G10, 47H09, 45M99. Keywords: Fractional, monotonic solutions, monotonicity measure of noncompact- ness, quadratic integral equation, Darbo theorem. Received August 24, 2016; Accepted November 27, 2016 Communicated by Martin Bohner 94 M. A. Darwish, J. Henderson and K. Sadarangani 1 Introduction In several papers, among them [1,11], the authors studied differential and integral equa- tions with maximum. In [6–9] Darwish et al. studied fractional integral equations with supremum. Also, in [4, 5], Caballero et al. studied the Volterra quadratic integral equa- tions with supremum. They showed that these equations have monotonic solutions in the space C[0; 1]. Darwish [7] generalized and extended the Caballero et al. [4] results to the case of quadratic fractional integral equations with supremum. In this paper we will study the fractional quadratic integral equation with maximum Z t 0 (T y)(t) ' (s)κ(t; s) max[0,σ(s)] jy(τ)j y(t) = f(t) + 1−β ds; t 2 J = [0; 1]; 0 < β < 1; Γ(β) 0 ('(t) − '(s)) (1.1) where f; ' : J ! R, T : C(J) ! C(J), σ : J ! J and κ : J × J ! R+. By using the monotonicity measure of noncompactness due to Banas´ and Olszowy [3] as well as the Darbo fixed point theorem, we prove the existence of monotonic solutions to (1.1) in C[0; 1]. Now, we assume that (E; k · k) is a real Banach space. We denote by B(x; r) the closed ball centred at x with radius r and Br ≡ B(θ; r), where θ is a zero element of E. We let X ⊂ E. The closure and convex closure of X are denoted by X and ConvX, respectively. The symbols X + Y and λY are using for the usual algebraic operators on sets and ME and NE stand for the families defined by ME = fA ⊂ E : A 6= ;;A is boundedg and NE = fB ⊂ ME : B is relatively compactg, respectively. Definition 1.1 (See [2]). A function µ : ME ! [0; +1) is called a measure of non- compactness in E if the following conditions: ◦ 1 ; 6= fX 2 ME : µ(X) = 0g = kerµ ⊂ NE, 2◦ if X ⊂ Y , then µ(X) ≤ µ(Y ), 3◦ µ(X) = µ(X) = µ(ConvX), 4◦ µ(λX + (1 − λ)Y ) ≤ λµ(X) + (1 − λ)µ(Y ), 0 ≤ λ ≤ 1 and ◦ 5 if (Xn) is a sequence of closed subsets of ME with Xn ⊃ Xn+1(n = 1; 2; 3; :::) 1 and lim µ(Xn) = 0 then X1 = \n=1Xn 6= ;, n!1 hold. We will establish our result in the Banach space C(J) of all defined, real and con- tinuous functions on J ≡ [0; 1] with standard norm kyk = maxfjy(τ)j : τ 2 Jg. Next, we define the measure of noncompactness related to monotonicity in C(J); see [2, 3]. Let ; 6= Y ⊂ C(J) be a bounded set. For y 2 Y and " ≥ 0, the modulus of continuity of the function y , denoted by !(y; "), is defined by !(y; ") = supfjy(t) − y(s)j : t; s 2 J; jt − sj ≤ "g: A Maximum Quadratic Integral Equation 95 Moreover, we let !(Y; ") = supf!(y; "): y 2 Y g and !0(Y ) = lim !(Y; "): "!0 Define d(y) = sup (jy(t) − y(s)j − [y(t) − y(s)]) t;s2J; s≤t and d(Y ) = sup d(y): y2Y Notice that all functions in Y are nondecreasing on J if and only if d(Y ) = 0. Now, we define the map µ on MC(J) as µ(Y ) = d(Y ) + !0(Y ): Clearly, µ satisfies all conditions in Definition 3, and therefore, it is a measure of non- compactness in C(J) [3]. Definition 1.2. Let P : M ! E be a continuous mapping, where ; 6= M ⊂ E. Suppose that P maps bounded sets onto bounded sets. Let Y be any bounded subset of M with µ(PY ) ≤ αµ(Y ), α ≥ 0, then P is called verify the Darbo condition with respect to a measure of noncompactness µ. In the case α < 1, the operator P is said to be a contraction with respect to µ. Theorem 1.3 (See [10]). Let ; 6= Ω ⊂ E be a closed, bounded and convex set. If P :Ω ! Ω is a continuous contraction mapping with respect to µ, then P has a fixed point in Ω. We will need the following two lemmas in order to prove our results [4]. Lemma 1.4. Let r : J ! J be a continuous function and y 2 C(J). If, for t 2 J, (F y)(t) = max jy(τ)j; [0,σ(t)] then F y 2 C(J). Lemma 1.5. Let (yn) be a sequence in C(J) and y 2 C(J). If (yn) converges to y 2 C(J), then (F yn) converges uniformly to F y uniformly on J. 96 M. A. Darwish, J. Henderson and K. Sadarangani 2 Main Theorem Let us consider the following assumptions: (a1) f 2 C(J). Moreover, f is nondecreasing and nonnegative on J. (a2) The operator T : C(J) ! C(J) is continuous and satisfies the Darbo condition with a constant c for the measure of noncompactness µ . Moreover, T y ≥ 0 if y ≥ 0. (a3) There exist constants a; b ≥ 0 such that j(T y)(t)j ≤ a + bkyk 8y 2 C(J); t 2 J. 1 (a4) The function ' : J ! R is C (J) and nondecreasing. (a5) The function κ : J ×J ! R+ is continuous on J ×J and nondecreasing 8t and s separately. Moreover, κ∗ = sup κ(t; s). (t;s)2J×J (a6) The function σ : J ! J is nondecreasing and continuous on J. (a7) 9 r0 > 0 such that κ∗r (a + br ) kfk + 0 0 ('(1) − '(0))β ≤ r (2.1) Γ(β + 1) 0 ck∗r and 0 < ('(1) − '(0))−β. Γ(β + 1) Now, we define two operators K and F on C(J) as follows Z t 0 1 ' (s)κ(t; s) max[0,σ(s)] jy(τ)j (Ky)(t) = 1−β ds (2.2) Γ(β) 0 ('(t) − '(s)) and (Fy)(t) = f(t) + (T y)(t) · (Ky)(t); (2.3) respectively. Solving (1.1) is equivalent to find a fixed point of the operator F. Under the above assumptions, we will prove the following theorem. Theorem 2.1. Assume the assumptions (a1) − (a7) are satisfied. Then (1.1) has at least one solution y 2 C(J) which is nondecreasing on J. Proof. First, we claim that the operator F transforms C(J) into itself. For this, it is sufficient to show that if y 2 C(J), then Ky 2 C(J). Let y 2 C(J) and t1; t2 2 J (t1 ≤ t2) such that jt2 − t1j ≤ " for fixed " > 0, then we have j(Ky)(t2) − (Ky)(t1)j A Maximum Quadratic Integral Equation 97 Z t2 0 1 ' (s)κ(t2; s) max[0,σ(s)] jy(τ)j = 1−β ds Γ(β) 0 ('(t2) − '(s)) Z t1 0 ' (s)κ(t1; s) max[0,σ(s)] jy(τ)j − 1−β ds 0 ('(t1) − '(s)) Z t2 0 1 ' (s)κ(t2; s) max[0,σ(s)] jy(τ)j ≤ 1−β ds Γ(β) 0 ('(t2) − '(s)) Z t2 0 ' (s)κ(t1; s) max[0,σ(s)] jy(τ)j − 1−β ds 0 ('(t2) − '(s)) Z t2 0 1 ' (s)κ(t1; s) max[0,σ(s)] jy(τ)j + 1−β ds Γ(β) 0 ('(t2) − '(s)) Z t1 0 ' (s)κ(t1; s) max[0,σ(s)] jy(τ)j − 1−β ds 0 ('(t2) − '(s)) Z t1 0 1 ' (s)κ(t1; s) max[0,σ(s)] jy(τ)j + 1−β ds Γ(β) 0 ('(t2) − '(s)) Z t1 0 ' (s)κ(t1; s) max[0,σ(s)] jy(τ)j − 1−β ds 0 ('(t1) − '(s)) Z t2 0 1 ' (s)jκ(t2; s) − κ(t1; s)j max[0,σ(s)] jy(τ)j ≤ 1−β ds Γ(β) 0 ('(t2) − '(s)) Z t2 0 Z t1 1 ' (s)jκ(t1; s)j max[0,σ(s)] jy(τ)j 1 + 1−β ds + jκ(t1; s)j Γ(β) t1 ('(t2) − '(s)) Γ(β) 0 0 β−1 β−1 ×' (s) j('(t2) − '(s)) − ('(t1) − '(s)) j max jy(τ)j ds [0,σ(s)] kyk Z t2 '0(s) κ∗kyk ≤ !κ("; :) 1−β ds + Γ(β) 0 ('(t2) − '(s)) Γ(β) Z t1 0 β−1 β−1 × ' (s)[('(t1) − '(s)) − ('(t2) − '(s)) ]ds 0 Z t2 '0(s) + 1−β ds t1 ('(t2) − '(s)) kyk κ∗kyk = ! ("; :)('(t ) − '(0))β + Γ(β + 1) κ 2 Γ(β + 1) β β β × ('(t1) − '(0)) − ('(t2) − '(0)) + 2('(t2) − '(t1)) kyk 2κ∗kyk ≤ ! ("; :)('(t ) − '(0))β + ('(t ) − '(t ))β Γ(β + 1) κ 2 Γ(β + 1) 2 1 kyk 2κ∗kyk ≤ ! ("; :)('(1) − '(0))β + [!('; ")]β; (2.4) Γ(β + 1) κ Γ(β + 1) where we used !κ("; :) = sup jκ(t; s) − κ(τ; s)j t; τ2J; jt−τ|≤" 98 M.