Abstract

We investigate the existence of multiple positive solutions to the nonlinear -fractional boundary value problem , , by using a fixed point theorem in a cone.

1. Introduction

Fractional differential calculus is a discipline to which many researchers are dedicating their time, perhaps because of its demonstrated applications in various fields of science and engineering [1]. Many researchers studied the existence of solutions to fractional boundary value problems, for example, [2–5].

The -difference calculus or quantum calculus is an old subject that was initially developed by Jackson [6, 7], basic definitions and properties of -difference calculus can be found in the book mentioned in [8].

The fractional -difference calculus had its origin in the works by Al-Salam [9] and Agarwal [10]. More recently, maybe due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional -difference calculus were made, for example, -analogues of the integral and differential fractional operators properties such as MittageLeffler function [11], just to mention some.

El-Shahed and Hassan [12] studied the existence of positive solutions of the -difference boundary value problem:

Ferreira [13] considered the existence of positive solutions to nonlinear -difference boundary value problem: In other paper, Ferreira [14] studied the existence of positive solutions to nonlinear -difference boundary value problem:

In this paper, we consider the existence of positive solutions to nonlinear -difference equation: with the boundary conditions where and is the fractional -derivative of the Caputo type.

2. Preliminaries of Fractional -Calculus

Let and define [8] The -analogue of the power is If is not a positive integer, then Note that if , then . The -gamma function is defined by and satisfies .

The -derivative of a function is here defined by and -derivatives of higher order by The -integral of a function defined in the interval is given by If and is defined in the interval , its integral from to is defined by Similarly as done for derivatives, an operator can be defined, namely, The fundamental theorem of calculus applies to these operators and , that is, and if is continuous at , then Basic properties of the two operators can be found in the book mentioned in [8]. We now point out three formulas that will be used later ( denotes the derivative with respect to variable ) [13]

Remark 2.1. We note that if and , then [13].
The following definition was considered first in [10].

Definition 2.2. Let and be a function defined on . The fractional -integral of the Riemann-Liouville type is and

Definition 2.3 (see [15]). The fractional -derivative of the Riemann-Liouville type of order is defined by and where is the smallest integer greater than or equal to .

Definition 2.4 (see [15]). The fractional -derivative of the Caputo type of order is defined by where is the smallest integer greater than or equal to .

Lemma 2.5. Let and let be a function defined on . Then, the next formulas hold: (1), (2).

The next result is important in the sequel. It was proved in a recent work by the author of [13].

Theorem 2.6. Let and . Then, the following equality holds:

Theorem 2.7 (see [15]). Let and . Then, the following equality holds:

3. Fractional Boundary Value Problem

We will consider now the question of existence of positive solutions to the following problem: where is the fractional -derivative of the Caputo type, and such that . To that end, we need the following theorem.

Theorem 3.1 (see [16, 17]). Let be a Banach space, and is a cone in . Assume that and are open subsets of with and .
Let be completely continuous operator. In addition suppose either: : and or : and ,holds. Then, has a fixed point in .

Lemma 3.2. Let ; then the boundary value problem has a unique solution where

Proof. We may apply Lemma 2.5 and Theorem 2.7; we see that By using the boundary conditions , we get Differentiating both sides of (3.6), one obtains, with the help of (2.13), and (2.14) Then, by the condition , we have The proof is complete.

Lemma 3.3. Function defined above satisfies the following conditions:

Proof. We start by defining two functions It is clear that . Now, and, in view of Remark 2.1, for Therefore, . Moreover, for fixed , that is, is an increasing function of . Obviously, is increasing in , therefore; is an increasing function of for fixed . This concludes the proof of (3.9).
Suppose now that . Then, If , Then and this finishes the proof of (3.10).

Remark 3.4. If we let , then Let be the Banach space endowed with norm . Let [18] for a given and define the cone by and the operator is defined by

Remark 3.5. It follows from the nonnegativeness and continuity of , and that the operator is completely continuous [3]. Moreover, for on and that is, .
Throughout this section, we will use the following notations: It is obvious that . Also, we define

Theorem 3.6. Let with . And assume that the following assumptions are satisfied: (H1); (H2)There exist constant and such that Then, the BVP (3.1) has at least two positive solutions , such that

Proof. First, since , for any there exists such that Letting , for any , we get which yields Thus, Second, in view of , then for any , there exists such that and we consider two cases.Case 1 (2.12). Suppose that is unbounded; then from , there is such that then, from (3.28) and (3.29), one has For , we get Case 2 (2.13). Suppose that is bounded and . Taking , for , we get Hence, in either case, we always may set such that Thus, Finally, set , for , since for , and hence, for any , from (H2), we can get which implies Thus, Hence, since and from (3.27), (3.34), and (3.37), it follows from the additivity of the fixed point index that Thus, has two positive solutions such that for . So, the proof is complete.