Abstract and Applied Analysis

Abstract and Applied Analysis / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 2127071 | https://doi.org/10.1155/2020/2127071

Nadir Benkaci-Ali, "Positive Solution for the Integral and Infinite Point Boundary Value Problem for Fractional-Order Differential Equation Involving a Generalized -Laplacian Operator", Abstract and Applied Analysis, vol. 2020, Article ID 2127071, 11 pages, 2020. https://doi.org/10.1155/2020/2127071

Positive Solution for the Integral and Infinite Point Boundary Value Problem for Fractional-Order Differential Equation Involving a Generalized -Laplacian Operator

Academic Editor: Changbum Chun
Received09 Jun 2020
Accepted31 Jul 2020
Published28 Aug 2020

Abstract

In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving -Laplacian operator where is a continuous function and is the Riemann-Liouville derivative for . By using some properties of fixed point index, we obtain the existence results and give an example at last.

1. Introduction

Our aim in this article is to study the existence of a nontrivial positive solution to the following integral and infinite point boundary-value problem involving a two-dimensional -Laplacian operator where is the Riemann-Liouville derivative for , and and for such that

Throughout this paper, we assume that the following conditions are satisfied;

(A1) for with

(A2) is continuous and is an integrable function.

(A3)

(A4) is continuous and for , the function is odd and increasing, is the inverse function of denoted by where is continuous.

(A5) There exist with such that with and

Boundary value problems involving a -Laplacian operator have attracted a great deal of attention in the last ten years (see [1] [29]). At the same time, boundary value problems with fractional-order differential equations involving -Laplacian are of great importance and are an interesting class of problems. Such kind of BVPs in Banach space has been studied by many authors, see, for example [1013] and the references therein. Noting that the generalized -Laplacian operator can turn into the well-known -Laplacian operator when we replace by , so our results extend and enrich some existing papers.

By using the homotopy deformation property of the fixed point index, our paper aims at investigating the existence of at least one positive solution for bvp 1.

The paper is organized as follows. In the first section, we recall some lemmas giving fixed point index calculations. In the second section, we present a fixed point formulation for bvp (1), and we close this section by some lemmas making use of homotopical arguments. After that, we give our main results and their proofs and we end by giving as an example, a problem involving a sum of many -Laplacian operators.

2. Preliminaries

For the sake of completeness, let us recall some basic facts needed in this paper. Let be a real Banach space equipped with its norm noted , is the set of all linear continuous mapping from into . For , denotes the spectral radius of . A nonempty closed convex subset of is said to be a cone if and for all .

Let be a cone in . A cone induces a partial ordering “”, defined so that if and only if .

is said to be normal if there exists a positive constant such that for all ,

is said to be positive in if , it is said to be strongly positive in if and , and it is said to be -normal if for all ,

Let be a real Banach space and let be a cone.

Let , be the ball of radius in and a completely continuous mapping, where . We will use the following lemmas concerning computations of the fixed point index, , for a compact map (see [14]).

Lemma 1. If for all , then

Lemma 2. If for all , then

Lemma 3. If for all , then

Lemma 4. If for all , then

Lemma 5. If for all and , then

Let be an operator and be a cone of a real Banach space E, and consider the partial ordering “” in , defined so that if and only if .

Let , and consider the following cone.

and the positive value where

Remark 6. It is clear that If is completely continuous, then from Lemmas 4 and 5, there exist such that .

Lemma 7. Assume that is increasing, positively 1-homogeneous, and completely continuous, such that .
If there exist such that , then

Proof. In first, we claim .

Let such that . Since , is strictly increasing and positively 1-homogenuous, we have then with we deduce and so

Now, we show that

In the contrary, we assume that there exist and with Such that

For , and so

Then,

, and we obtain which is a contradiction.

Remark 8. If is a normal cone in a Banach space , with the constant of normality (i.e, if ), then

Since for

In the following lemma, we assume that is a positively 1-homogeneous and completely continuous operator, and is a completely continuous, increasing and positively 1-homogeneous operator, such that where , , and is a normal cone in a Banach space , with the constant of normality .

Lemma 9. Let be continuous mappings with such that

Suppose that there exist and with such that

If then there exist such that for all

Moreover, if then there exist such that for all

Proof. In first, we show that there exist such that for all

We consider the homotopy

We show that there exist such that for all the equation has not solutions in . In the contrary, we assume that for all , there exist and such that

By dividing the above equation by , we obtain

From (5), then the sequences are bounded, and we deduce from the compactness of and , that admits a convergent subsequence also denoted by .

Let and

By using the conditions (31) and (33), it follows from 12 that for all

With the fact that we have and so where

Then which contradicts (10).

Then, there exist such that for all the equation has not the solutions in and by invariance property of fixed point index, we deduce that for all

By the fact that , we have from the excision property of the fixed point index that

Then

Now, we assume that the condition (36) holds.

By using Lemma 4, we prove that there exists such that, for all

In the contrary, we assume that for all there exist and such that

By the condition (33), we have with

As we have

Set

We have

Since is normal with the constant of normality , then for and so then which contradicts (10).

Consequently, for ,

Definition 10. [10, 11] The Riemann-Liouville fractional integral of order of is defined by where is the gamma function.

Definition 11. [10, 11] The Riemann-Liouville fractional derivative of order of a function is defined by where is the integer part of .

Remark 12. If , then and for ,

Lemma 13. [15] Let , and let be an integrable function in . where , , and is the integer part of .

Lemma 14. Let , and

Then the unique solution of is given by with and with and such that where with

Proof. By Lemma 13, equation gives

Since , we have that and with

And also from Lemma 13, we have

If , then the condition leads , and if , the equation leads with

Then

In addition, from equation we deduce that with . The Fubini’s theorem gives

Then where

Consequently, the solution of (19) is with with and such that where

This finishes the proof.

Lemma 15. For , we have where with

Proof. It is clear that the right hand inequality is obvious.

Now, we show that where with

Let .

For , as and , we have and gives

For , we have and with leads then

Remark 16. The function defined by (20) is continuous, and from Lemma 15, we have .

According to Lemma 14, is solution of 1 if and only if is a fixed point of the operator <