Multiple Positive Solutions for a Fractional Boundary Value Problem with Fractional Integral Deviating Argument

Abstract

This work is devoted to the existence of positive solutions for a fractional boundary value problem with fractional integral deviating argument. The proofs of the main results are based on Guo-Krasnoselskii fixed point theorem and Avery and Peterson fixed point theorem. Two examples are given to illustrate the obtained results, ending the paper.

1. Introduction

The study of differential equations with deviating arguments has known a rapid development and still attracts most attention; it is a vast domain constantly enriched and fruitful. Differential equations with deviating arguments appear in many areas of sciences and technology such as in the study of problems related with combustion in rocket motion, in the theory of automatic control, in economics, and biological systems.

There exists a vast amount of literature devoted to the investigation of boundary value problems with deviating arguments (see [1–4]); one can cite the paper of Haloi et al. [5], where the authors studied an abstract initial value problem with deviating arguments by the help of Sobolevskii and Tanabe theory and fixed point theorems. However, few papers exist concerning fractional boundary value problems with deviating arguments, see [3, 4, 6, 7]. We can cite the work of Jankowski [8], where the author studied the positivity of solutions for fourth-order differential equations with deviating arguments and integral boundary conditions, by Avery and Peterson theorem.

Fractional differential equations have been proved to be valuable tools in the modelling of many phenomena in various fields of engineering, such as rheology, fluid flows, electrical networks, viscoelasticity, chemical physics, biosciences, signal processing, systems control theory, electrochemistry, mechanics and diffusion processes; for more details on this subject we refer the reader to [9–12].

In this work, we study a fractional boundary value problem with fractional deviating argument : where , , denotes the Caputo fractional derivative, is a given function, , . We assume that the deviating argument satisfies the delay property for any . Let us remark that when we apply the fractional integral to the deviating argument , then the problem becomes more delayed in time. The existence of a deviation delay in time is necessary in some situations, to avoid the unstable combustion in liquid rocket engine and to contribute in the conversion of the fuel mixture into the product of combustion. No contributions exist, as far as we know, concerning the positivity of solutions for the fractional differential equation with deviating argument .

The literature on fractional boundary value problems with deviating argument is very reduced; we can cite Ntouyas et al. [6] work, where the authors investigated the existence of positive solutions to fractional differential equations with advanced arguments by Guo-Krasnoselskii fixed point theorem. Other interesting results on fractional boundary value problems can be found in [13–21].

The organization of this paper is as follows. In Section 2, we provide some necessary background materials and definitions. In Section 3, we present some lemmas which are useful to obtain our main results. In Section 4, we discuss the existence of at least one positive solution of problem by using Guo-Krasnoselskii fixed point theorem on cone; then, under some sufficient conditions on the nonlinear source term, we apply Avery-Peterson theorem to prove the existence of at least three positive solutions. At the end of this section, we give two examples to verify the obtained results.

2. Background Materials and Definitions

In this section, we introduce definitions from fractional calculus theory and preliminary facts which are used throughout this paper.

Definition 2.1 (see [10]). The integral is called Riemann-Liouville fractional integral of of order when the right side exists. Here is the gamma function.

Definition 2.2 (see [10]). For a function given on the interval , the Caputo fractional derivative of of order is defined by where and denotes the integer part of .

Lemma 2.3 (see [10]). Letting and letting , then the following relations hold: , and , .

Lemma 2.4 (see [10]). For , , the homogenous fractional differential equation has a solution where , , and .

Denote by the Banach space of Lebesgue integrable functions from into with the norm . Define , equipped with the norm .

The following lemmas give some properties of Riemann-Liouville fractional integrals and Caputo fractional derivative.

Lemma 2.5 (see [10]). Let , . Then and , for all .

Lemma 2.6 (see [10]). Let . Then the formula holds almost everywhere on , for , and it is valid at any point if .

Now we present the necessary definitions from the theory of cone in Banach spaces.

Definition 2.7. A nonempty subset of a Banach space is called a cone if is convex, closed and satisfies the conditions(i) for all and any ,(ii) imply .

Definition 2.8. A map is said to be a nonnegative continuous convex functional on a cone of a real Banach space if is continuous and Similarly is a nonnegative continuous concave functional on a cone of a real Banach space if

Definition 2.9. A mapping is called completely continuous if it is continuous and maps bounded sets into relatively compact sets.
In the following, we state some fixed point theorems.

Theorem 2.10 (Guo-Krasnoselskii fixed point theorem on cone [22]). Let be a Banach space, and let be a cone. Assume and are open subsets of with , and let be a completely continuous operator such that(i), , and , ,(ii), , and , .
Then has a fixed point in .

Theorem 2.11 (Avery and Peterson fixed point theorem [23]). Let be a cone in a real Banach space . Let and be continuous, nonnegative, and convex functionals on , let be a continuous nonnegative, and concave functional on , and let be continuous and nonnegative functional on satisfying for . Define the sets, , , , . For and positive numbers we have and for any . Assume is completely continuous and there exists positive numbers , , and with such that(S1) and for ,(S2) for with ,(S3) and for with .
Then has at least three positive fixed points such that , for . , , with and .

3. Some Lemmas

We start by solving an auxiliary problem which allows us to get the expression of the solution. Let us consider the following problem:

Lemma 3.1. Assuming that and , then the problem has a unique solution given by where

Proof. Using Lemmas 2.3 and 2.4, we get The boundary conditions give , and the condition implies . Substituting , and by their values in (3.3) we get the desired result.

Some useful estimates of the Green function are given hereafter.

Lemma 3.2. If , then for all , the Green function is nonnegative, continuous and satisfies(i),(ii) where .

Proof. The proof is easy; then we omit it.

Lemma 3.3. Assuming that and , then the unique solution of problem satisfies on . Moreover we have

Proof. From and hypothesis on the function we deduce that consequently the function is concave. Moreover from the condition we can say that the values and have the same sign. Since , then on . Let us analyze the boundary conditions; from and the condition we conclude that is also negative; consequently the function is decreasing. From the above discussion we conclude that and . The concavity of allows to write Using the condition it yields then Since , we have so, Using the assumption , we obtain consequently This achieves the proof.

4. Existence of Positive Solutions

We first state the assumptions that will be used to prove our main results:(H1) and satisfies the delay property , for all ,(H2) and ,(H3) and .

Define the operator by

Definition 4.1. A function is called positive solution of problem if , for all and it satisfies the boundary conditions in .
Let us introduce the following notations , . The case and is called superlinear case, and the case and is called sublinear case.

Theorem 4.2. Assuming that (H1)–(H3) hold, then the problem has at least one positive solution in both cases, superlinear as well as sublinear.

Proof. We apply Guo-Krasnoselskii fixed point theorem on cone. Define the cone . It is easy to check that is a nonempty closed and convex subset of , so it is a cone. One can check that . It is obvious that is continuous since and are continuous. Let us prove that is completely continuous mapping on .
Claim 1. is uniformly bounded, where . Let us remark that if , then In fact from the delay property, the properties of Riemann fractional integrals and , we have In view of the concavity of it yields Therefore Since the functions and are continuous, then there exists a constant such that for any . By virtue of Lemma 3.2 we obtain Hence is uniformly bounded.
Claim 2. is equicontinuous. We have for any Therefore Thus is equicontinuous; from Ascoli-Arzela Theorem, we deduce that is completely continuous.
Now, we prove the sublinear case. Since , then for any there exists , such that for any , then . Set . Letting , we can see that ; then from (4.5) we have With the help of (4.4) we get In view of assumption (H2) we can choose ; therefore (4.10) becomes .
On the other hand since , we deduce that for any there exists , such that for any , then . Setting and , then and for we have ; consequently . Taking Lemma 3.2 and (4.5) into account it yields Then if we choose , we get for all . The second part of Theorem 2.10 implies that has a fixed point in ; that means that has at least a positive solution in . Arguing as above, we prove the superlinear case. The proof is complete.

Let us introduce the following functionals. Defining on , the nonnegative, continuous, and concave functional by , then . Defining the nonnegative, continuous, and convex functionals and on by and the nonnegative continuous functional on by , then for .

Theorem 4.3. Let assumptions (H1)–(H3) hold, and assume that there exist positive constants such that , , and(i) for ,(ii) for ,(iii) for .
Then problem has at least three positive solutions such that , for . , , with and .

Proof. To prove the existence of three positive solutions, we apply Theorem 2.11. Proceeding analogously as in the proof of Theorem 4.2, we prove that the mapping is completely continuous on .
Claim 1. .
Letting , then ; since , for all , then in view of (4.5) we have . Thus with the help of assumption (i) it yields and hence .
Claim 2. (S1) holds; that is, and for . Let ; since and , then
Moreover we have Thus , so .
Letting , then and ; thus by virtue of Lemma 3.2 and assumption (ii), we obtain
So condition (S1) is satisfied.
Claim 3. (S2) holds. Letting such , then This implies that claim (S2) holds true.
Claim 4. (S3) holds. Letting , then , and then . Let with ; using Lemma 3.2 and assumption (iii) it yields Then (S3) is satisfied.
Finally we conclude by Theorem 2.11 that there exist at least three positive solutions such that , for . , , with and . The proof of Theorem 4.2 is complete.