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] [2–9]). 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 [10–13] 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

can be written as where

Remark 17. Let be the Banach space equipped with the sup-norm and the cone

We have from 2 of the condition (A5) that where are the inverse functions of , respectively, defined by and where is the inverse function of defined in the condition (A5).

Remark 18. There exist such that for all ,

Remark 19. By Lemma 15 and Remark 18, we have for where and with

Moreover, the linear operator is compact, and is completely continuous, increasing, positively 1- homogeneous and verifying where with

4. Main Results

Set

Theorem 20. Assume that there exist and such that and with then problem 1 has at least one nontrivial positive solution.

Proof. In first, we show that where

From (25), we have

For , and from the definition of the constant , we have that then

Then,

By Lemma 1,

Now, by using Lemma 9, we show that there exists such that

In first, we have from Remark 19 that where

As , then the condition (31) of Lemma 9 is satisfied.

Now, we have from (26), for with

Then, there exists such that and set

We have for with

Moreover, from Remark 18, for , and Then, from (27), we have

By Lemma 9, there exist such that

Consequently, has at least one fixed point in , which is a nontrivial positive solution for problem 1.

Set and

Theorem 21. Assume that there exist and such that and then problem 1 has at least one nontrivial positive solution.

Proof. In first, by using Lemma 3, we show that there exists such that In the contrary, we assume that there exists a sequence in with such that

From (28), there exist and such that

Then, for where

Then, with it follows the following contradiction

Then, there exists such that

Now, we prove that

Let with

We have and from (29)

Then,

From Lemma 1, we have

Consequently, has at least one fixed point in , which is a nontrivial positive solution for problem 1.

Example 22. We consider the following -Laplacian boundary value problem where is the -Laplacian operator defined in as with

We consider the problem 1 with and We assume that the conditions (A1), (A2), and (A3) are satisfied, and verifies (A4) and (A5) with and

Set and

We deduce from Theorems 20 and 21 that, if there exist and such that verifies one of the following conditions;

(H1) and or

(H2) and then problem (30) has at least one nontrivial positive solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.