Study boundary problem with Integral condition for Fractional Differential Equations

Many years ago there was a great interest in studying the existence of positive solutions for fractional differential equations. Many researchers have considered the existence of positive solutions of non-linear differential equations of non-integer order with integral boundary value conditions using fixed point theorems. G.wang et al.(2012) investigated the following fractional differential equations DW(t) + f(t,W(t)) = 0, 0 < t < 1 with integral boundary value condition W(0) = W(0) = 0 , W (1) = λ∫ W(s)ds 1 0 were 2 < α ≤ 3 λ is a positive number (0 < λ < 2), Dis the standard fractional derivative equation of Caputo who obtained his results by means of Guo-krosnosel'skii theorem in a cone also A.Cabada et at. (2013) established the following non-linear fractional differential equation with integral boundary value conditions DW(t) + f(t,W(t)) = 0 , 0 < t < 1 W(0) = W(0) = 0 ,W(1) = λ ∫ W(s)ds , were 2 < α ≤ 3 , λ > 0 , λ ≠ α , 1 0 Dis Riemann –Liovuville standard fractional derivative. and f is a continuous function. The results were based on Guo-krasnosel'skii fixed point theorem in a cone . In this paper we investigate the existent results of a positive solution for the integral boundary value conditions of the following system of equations: D h(t) + k(t, h(t)) = 0 , t ∈ (0,1) h(0) = h(0) = h(0) = 0 , h(1) = δ∫ h(n)dn 1 0 where 3< β ≤ 4 , δ is a positive number , δ ≠ 3 , D denotes Caputo standard derivative and k is a continuous function.Our work based on Banach's and Schauder's theorem. Keyword: Fractional Differential Equation, Integral boundary value Conditions, Schauder's theorem, Green function. 1.Introduction In the last few decades, fractional order calculus has been of the most rapidly developing areas of mathematical analysis. In fact, a natural phenomenon may depend not only on the time instant but also on the previous time history, which can be successfully modeled by fractional calculus. Also fractional differential equations Study boundary problem with Integral condition for Fractional Differential Equations 239 play an important role because of their application in various fields of science such as mathematics, physics, mechanism, economics, engineering and biological sciences etc.(see[1-4]). Integral boundary conditions have several applications in real-life problem such as population dynamics, underground water flow, and blood flow problems. For a detailed description of the integral boundary conditions, we refer in this study to some recent papers [5,6,7]. Our establishment is the existence of positive solutions of the integral boundary conditions of the following fractional differential equation:D h(t) + k(t, h(t)) = 0 , 0 < t < 1 (1) h(0) = h(0) = h(0) = 0 , h(1) = δ ∫ h(n)dn 1 0 (2) where 3 < β ≤ 4 , 0 < δ < 3 , D denotes Caputo standard derivative and k: [0,1] × [0,∞) → [0,∞) is a continuous function. Firstly we obtain the exact expression of the Green's function related to the liner equation Dh(t) + g(t) = 0 , 0 < t < 1 (3) corresponded to the integral condition h(0) = h(0) = h(0) = 0 , h(1) = δ ∫ h(ρ)dρ 1 0 Our work is based on Banach's and Schauder's fixed point theorems. 2.Preliminaries In this article , we present some definitions, notations, lemmas and theorems from fractional calculus theorem. Definition 2.1.[1] Capote derivative fractional order α of the function J: [0,∞) → R is defined as : DJ(t) = 1 Γ(ρ − α) ∫ Jρ(c) (t − c)1+α−n t

is Riemann -Liovuville standard fractional derivative. and is a continuous function. The results were based on Guo-krasnosel'skii fixed point theorem in a cone .
In this paper we investigate the existent results of a positive solution for the integral boundary value conditions of the following system of equations: where 3< ≤ 4 , is a positive number , ≠ 3 , denotes Caputo standard derivative and is a continuous function.Our work based on Banach's and Schauder's theorem.

1.Introduction
In the last few decades, fractional order calculus has been of the most rapidly developing areas of mathematical analysis. In fact, a natural phenomenon may depend not only on the time instant but also on the previous time history, which can be successfully modeled by fractional calculus. Also fractional differential equations

Study boundary problem with Integral condition for Fractional Differential Equations
play an important role because of their application in various fields of science such as mathematics, physics, mechanism, economics, engineering and biological sciences etc.(see [1][2][3][4]).
Integral boundary conditions have several applications in real-life problem such as population dynamics, underground water flow, and blood flow problems. For a detailed description of the integral boundary conditions, we refer in this study to some recent papers [5,6,7].
Our establishment is the existence of positive solutions of the integral boundary conditions of the following fractional differential equation:- denotes Caputo standard derivative and Firstly we obtain the exact expression of the Green's function related to the liner equation corresponded to the integral condition Our work is based on Banach's and Schauder's fixed point theorems.

2.Preliminaries
In this article , we present some definitions, notations, lemmas and theorems from fractional calculus theorem.

Study boundary problem with Integral condition for Fractional Differential Equations
The proof is complete. □

3.The Main Result
Note Then the boundary value problem (1)-(2) has a unique solution.

Proof:
Let the operator : → be defined as : is the Green's function given by (4) points we will show that is a contraction and from lemma 2.1; the fixed points of are solutions to (1)- (2).

Study boundary problem with Integral condition for Fractional Differential Equations
Hence, we have The theorem is proved. □ Our second result is based on the Schauder's theorem. (A3) There exists a constant > 0 such that where 1 = Sup{ ( ), ∈ [0,1]}. If the above conditions hold, then the problem(1)-(2) has at least one solution.
The proof will be in four steps.
Step1: First we show that is continuous.
So the operator is completely continuous according to Arzela-Ascoli theorem.

Conclusion
In this paper, we study boundary value problem with integral conditions for fractional differential equation involving Caputo derivative of order ∈ (3,4). We establish the existence and uniqueness results of the solution. Our first result is based on Banach contraction principle and our second result is based on Schauder's fixed point theorem.