Existence of positive solutions for boundary value problems of fractional functional differential equations

This paper deals with the existence of positive solutions for a boundary value problem involving a nonlinear functional differential equation of fractional ordergiven by Du(t) + f(t,ut) = 0, t 2 (0,1), 2 < � � 3, u ' (0) = 0, u ' (1) = bu ' (�), u0 = �. Our results are based on the nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed point theorem.

For τ > 0, we denote by C τ the Banach space of all continuous functions ψ : [−τ, 0] → R endowed with the sup-norm If u : [−τ, 1] → R, then for any t ∈ [0, 1], we denote by u t the element of C τ defined by In this paper we investigate a fractional order functional differential equation of the form D α u(t) + f (t, u t ) = 0, t ∈ (0, 1), 2 < α ≤ 3, (1.1)where D α is the standard Riemann-Liouville fractional order derivative, f (t, u t ) : [0, 1] × C τ → R is a continuous function, associated with the boundary condition and the initial condition where 0 < η < 1, 1 < b < 1 η α−2 , and φ is an element of the space To the best of the authors knowledge, no one has studied the existence of positive solutions for problem (1.1)- (1.3).The aim of this paper is to fill the gap in the relevant literatures.The key tools in finding our main results are the nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed point theorem.
Definition 2.1.The Riemann-Liouville fractional derivative of order α > 0 of a continuous function where n = [α] + 1 and [α] denotes the integer part of number α, provided that the right side is pointwise defined on (0, ∞).Definition 2.2.The Riemann-Liouville fractional integral of order α of a function f : (0, ∞) → R is defined as provided that the integral exists.
The following lemma is crucial in finding an integral representation of the boundary value problem Lemma 2.1 [11].Suppose that u ∈ C(0, 1) ∩ L(0, 1) with a fractional derivative of order α > 0. Then , then the boundary value problem has a unique solution where and Proof.By Lemma 2.1, the solution of (2.1) can be written as Using the boundary conditions (2.2), we find that c 2 = c 3 = 0, and Hence, the unique solution of BVP (2.1), (2.2) is The proof is complete.
To establish the existence of solutions for (1.1)-(1.3),we need the following known results.Theorem 2.4 (Krasnosel'skii [23]).Let E be a Banach space and let K be a cone in E. Assume that Lemma 2.5.G(t, s) has the following properties.
Proof.It is easy to check that (i) holds.Next, we prove (ii) holds.If t ≥ s, then The proof is complete.

Main results
In the sequel we shall denote by C 0 [0, 1] the space of all continuous functions x : [0, 1] → R with x(0) = 0.This is a Banach space when it is furnished with the usual sup-norm and observe that x t (s; φ) ∈ C τ .
For u ∈ P , we define the operator T φ as follows: It is easy to know that fixed points of T φ are solutions of the BVP (1.1)-(1.3).
In this paper, we assume that 0 < τ < 1, φ ∈ C + τ (0), and we make use of the following assumption: In view of Lemma 2.5, we have which shows that T φ P ⊂ P .Moreover, similar to the proof of Lemma 3.2 in [10], it is easy to check that T φ : P → P is completely continuous.Lemma 3.2.If 0 < τ < 1 and u ∈ P , then we have Proof.From the definition of u t (s; φ), for t ≥ τ , we have Thus, we get for u ∈ P that We are now in a position to present and prove our main results.Theorem 3.3.Let (H 1 ) holds.Suppose that the following conditions are satisfied: (H 2 ) there exist a continuous function a : [0, 1] → [0, +∞) and a continuous, nondecreasing function 3) has at least one positive solution.
Proof.We shall apply Theorem 2.3 (the nonlinear alternative of Leray-Schauder type) to prove that T φ has at least one positive solution.
Let U = {u ∈ P : u < r}, where r is as in (H 3 ).Assume that there exist u ∈ P and λ ∈ (0, 1) such that u = λT φ u, we claim that u = r.In fact, if u = r, we have By the definition of u s (•; φ), we easily obtain that Thus by (3.4), (3.5) and the nondecreasing of F , we get that And there exists a constant r 1 > 0 (r 1 < r 2 ) satisfying Then BVP (1.1)-( 1.3) has a positive solution.
Proof.In (H 4 ), let p(t) ≡ M and L(x) ≡ x, then by (3.12) and (3.14), we have that (3.8) holds.In fact, we have Moreover, let c(t) ≡ m and J(x) ≡ x in (H 5 ), then by (3.13) and (3.15), we have that (3.9) holds.In fact, we have So all conditions of Theorem 3. Having in mind the proof of Theorem 3.4, one can easily conclude the following results.

Examples
To illustrate our results we present the following examples.
Example 4.1.Consider the boundary value problem of fractional order functional differential equations where With the aid of computation we have that

Theorem 2 . 3 (
Nonlinear alternative of Leray-Schauder [22]).Let E be a Banach space with C ⊂ E EJQTDE, 2010 No. 30, p. 3 closed and convex.Assume that U is a relatively open subset of C with 0 ∈ U and T : U → C is completely continuous.Then either (i) T has a fixed point in U, or (ii) there exists u ∈ ∂U and γ ∈ (0, 1) with u = γT u.
By a solution of the boundary value problem (1.1)-(1.3)we mean a function u ∈ C 0 [0, 1] such that D α u exists on [0, 1] and u satisfies boundary condition (1.2) and for a certain φ the relation EJQTDE, 2010 No. 30, p. 5