A FOUR-POINT NONLOCAL INTEGRAL BOUNDARY VALUE PROBLEM FOR FRACTIONAL DIFFERENTIAL EQUATIONS OF ARBITRARY ORDER

This paper studies a nonlinear fractional differential equation of an arbitrary order with four-point nonlocal integral boundary conditions. Some existence results are obtained by applying standard fixed point theorems and Leray-Schauder degree theory. The involvement of nonlocal parameters in four-point integral boundary conditions of the problem makes the present work distinguished from the available literature on four-point integral boundary value problems which mainly deals with the four-point boundary conditions restrictions on the solution or gradient of the solution of the problem. These integral conditions may be regarded as strip conditions involving segments of arbitrary length of the given interval. Some illustrative examples are presented.


Introduction
Boundary value problems for nonlinear fractional differential equations have recently been studied by several researchers.Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes.These characteristics of the fractional derivatives make the fractional-order models more realistic and practical than the classical integer-order models.As a matter of fact, fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc. [17,18,19,20].Some recent work on boundary value problems of fractional order can be found in [1,2,3,6,7,8,9,11,12,13,14,15,16,22]  In this paper, we consider a boundary value problem of nonlinear fractional differential equations of an arbitrary order with four-point integral boundary conditions given by x(s)ds, x ′ (0) = 0, x ′′ (0) = 0, . . ., x (m−2) (0) = 0, where c D q denotes the Caputo fractional derivative of order q, f : [0, 1] × X → X is continuous and α, β ∈ R. Here, (X, • ) is a Banach space and C = C([0, 1], X) denotes the Banach space of all continuous functions from [0, 1] → X endowed with a topology of uniform convergence with the norm denoted by • .
Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, etc.For a detailed description of the integral boundary conditions, we refer the reader to the papers [4,5] and references therein.It has been observed that the limits of integration in the integral part of the boundary conditions are usually taken to be fixed, for instance, from 0 to 1 in case the independent variable belongs to the interval [0, 1].In the present study, we have introduced a nonlocal type of integral boundary conditions with limits of integration involving the parameters 0 < ξ, η < 1.It is imperative to note that the available literature on nonlocal boundary conditions is confined to the nonlocal parameters involvement in the solution or gradient of the solution of the problem.The present work is motivated by a recent article [10], in which some existence results were obtained for nonlinear fractional differential equations with three-point nonlocal integral boundary conditions.

Preliminaries
Let us recall some basic definitions of fractional calculus [17,18,20].Definition 2.1 For a function g : [0, ∞) → R, the Caputo derivative of fractional order q is defined as where [q] denotes the integer part of the real number q. EJQTDE, 2011 No. 22, p. 2 Definition 2.2 The Riemann-Liouville fractional integral of order q is defined as provided the integral exists.
For the forthcoming analysis, we need the following assumption: (H) Assume that f : [0, 1] × X → X is a jointly continuous function and maps bounded subsets of [0, 1] × X into relatively compact subsets of X.
Furthermore, we need the following fixed point theorem to prove the existence of solutions for the problem at hand.Proof.Clearly, continuity of the operator F follows from the continuity of f.Let Ω ⊂ C be bounded.Then, ∀x ∈ Ω, by the assumption (H), there exists L 1 > 0 such that |f (t, x)| ≤ L 1 .Thus, we have which implies that (Fx) ≤ L 2 .Furthermore, Hence, for t 1 , t 2 ∈ [0, 1], we have This implies that F is equicontinuous on [0, 1].Thus, by the Arzela-Ascoli theorem, we have that F(Ω)(t) is relatively compact in X for every t, and so the operator F : C → C is completely continuous.
which, in view of (3.3), yields Fx ≤ x , x ∈ ∂Ω 1 .Therefore, by Theorem 2.1, the operator F has at least one fixed point, which in turn implies that the problem (1.1) has at least one solution.2 Theorem 3.2 Assume that f : [0, 1] × X → X is a jointly continuous function and satisfies the condition with L < 1/ϑ, where ϑ is given by (2.6).Then the boundary value problem (1.1) has a unique solution.
Proof.Setting sup t∈[0,1] |f (t, 0)| = M and choosing r ≥ ϑM 1 − Lϑ , we show that FB r ⊂ B r , where B r = {x ∈ C : x ≤ r}.For x ∈ B r , we have: Now, for x, y ∈ C and for each t ∈ [0, 1], we obtain EJQTDE, 2011 No. 22, p. 9 where ϑ is given by (2.6).Observe that L depends only on the parameters involved in the problem.As L < 1/ϑ, therefore F is a contraction.Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem).2 Our next existence result is based on Leray-Schauder degree theory.
Theorem 3.3 Suppose that (H) holds.Furthermore, it is assumed that there exist , where θ is given by (2.6) and M > 0 such that |f (t, x)| ≤ κ|x|+M for all t ∈ [0, 1], x ∈ X.Then the boundary value problem (1.1) has at least one solution.
Proof.Consider a fixed point problem where F is defined by (2.5).In view of the fixed point problem where I denotes the identity operator.By the nonzero property of Leray-Schauder degree, h 1 (t) = x − λFx = 0 for at least one x ∈ B R .In order to prove (3.6), we assume that x = λFx for some λ ∈ [0, 1] and for all t ∈ [0, 1] so that where m − 1 < q ≤ m, m ≥ 2.
It can easily be verified that all the assumptions of Theorem 3.1 hold.Consequently, the conclusion of Theorem 3.1 implies that the problem (4.1) has at least one solution.So f (t, x) ≤ 1 2 x + 1. Clearly M = 1, κ = 1/2, ϑ = 1.081553 × 10 −03 , and κ < 1/ϑ.Thus, all the conditions of Theorem 3.3 are satisfied and consequently the problem (4.3) has at least one solution.

Theorem 2 . 1 [ 21 ] 5 Lemma 3 . 1
Let X be a Banach space.Assume that Ω is an open bounded subset of X with θ ∈ Ω and let T : Ω → X be a completely continuous operator such thatT u ≤ u , ∀u ∈ ∂Ω.Then T has a fixed point in Ω.EJQTDE, 2011 No. 22, p.The operator F : C → C is completely continuous.