Positive Solution for the Integral and Infinite Point Boundary Value Problem for Fractional-Order Differential Equation Involving a Generalized -Laplacian Operator
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.
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.
(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  [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.
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 ).
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
3. Related Lemmas
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
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
, 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
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 .
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
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
Since is normal with the constant of normality , then for and so then which contradicts (10).
Consequently, for ,
Remark 12. If , then and for ,
Lemma 13.  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
In addition, from equation we deduce that with . The Fubini’s theorem gives
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
For , as and , we have and gives
For , we have and with leads then
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 ,
Moreover, the linear operator is compact, and is completely continuous, increasing, positively 1- homogeneous and verifying where with
4. Main Results