Positive solutions of higher order fractional integral boundary value problem with a parameter ∗

In this paper, we study a higher-order fractional differential equation with integral boundary conditions and a parameter. Under different conditions of nonlinearity, existence and nonexistence results for positive solutions are derived in terms of different intervals of parameter. Our approach relies on the Guo–Krasnoselskii fixed point theorem on cones.

Boundary value problems (BVPs for short) with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems and arise in the study of various biological, physical and chemical processes [5,6,31,44] such as heat conduction, thermo-elasticity, chemical engineering, underground water flow and plasma physics.The existence of solutions or positive solutions for such class of problems has attracted much attention; see, for example, [8, 12-15, 20-24, 26, 27, 29, 30, 35, 37, 41-43, 47, 48, 50-52] and the references therein.
Recently, Gunendi and Yaslan [17] considered the multi-point BVP for higher order fractional differential equation The existence results of at least one, two and three positive solutions are obtained by the four functionals fixed point theorem, the Avery-Henderson fixed point theorem and the Legget-Williams fixed point theorem, respectively.
In the present paper, we consider the more general fractional differential equation integral BVP (1).Under different conditions of the function f , existence and nonexistence results for positive solutions are derived in terms of different intervals of parameter λ.Our approach relies on the Guo-Krasnoselskii fixed point theorem on cones.
We express the fixed point operator with a Green's function, which is a convolution.The idea constructing Green's functions as convolutions of Green's functions for lower order BVPs is from the work of Eloe and Neugebauer [10].The paper [10] contains some interesting ideas and develops the convolution method to several families of BVPs.
This paper is arranged as follows.In Section 2, we present some definitions and preliminary lemmas.In Section 3, we establish the existence and nonexistence of positive solutions for BVP (1) by using the fixed point theorem on cones.An example is also given to illustrate the main results in Section 4.
X. Hao et al.

Preliminaries
We present the definitions of fractional calculus and some auxiliary results that are useful to the proof of our main results.Definition 1. (See [32,34,36,55].)The Riemann-Liouville fractional integral of order α > 0 of a function h : (0, +∞) → R is given by provided the right-hand side is pointwise defined on (0, +∞).Definition 2. (See [32,34,36,55].)The Riemann-Liouville fractional derivative of order α > 0 of a continuous function h : (0, +∞) → R is given by where n is the smallest integer not less than α, provided that the right-hand side is pointwise defined on (0, +∞).Lemma 1. (See [32,34,36].) where Using Lemma 1, by arguments similar to Lemma 2.4 in [17], we have the following result. where By direction computations we obtain the properties of H(κ; t, s).
Now we consider the following integral BVP: then G 0 (t, s) is the Green's function of the following homogeneous differential equation BVP: −u (t) = 0, t ∈ (0, 1), Define then a(t) and b(t) are the solutions of −a (t) = 0, t ∈ (0, 1), respectively.Denote We will use the following assumption: if and only if u can be expressed by where Lemma 5. (See [33].) where Lemma 6. Assume that (H) holds, then where Proof.By using Lemma 5, for any t, s ∈ [0, 1], we obtain On the other hand, by Lemma 5, we deduce By using Lemmas 2 and 4 a solution of integral equation is a solution for BVP (1).As in [10], the integral equation can be rewritten in terms of a Green's function, which is a convolution of G and H.In fact, where Proof.By Lemma 3 and expression of G(κ; t, s) it is easy to see that (i) holds.In the following, we will prove (ii).By using Lemma 6, for any t, s ∈ [0, 1], we deduce G(κ; t, s) .
It is easy to see that P is a cone in E. We define the operator T : It is clear that if u ∈ P is a fixed point of T , then u is a positive solution of BVP (1).By using standard arguments we obtain the following lemma with respect to completely continuous operator.
Lemma 8. Assume that (H) holds, then T : P → P is a completely continuous operator.
The main tool in the paper is the following Guo-Krasnoselskii fixed point theorem on cones.
Otherwise, we suppose that BVP (1) has a positive solution u, then u(t) = T u(t) λ This contradiction shows that BVP (1) has no positive solution.
Proof.By definitions of f i 0 and f i ∞ there exists m > 0 such that f (t, x) mx, t ∈ [0, 1], x 0.