On the Solvability of a Nonlinear Hadamard Fractional Differential Equation with an Integral Boundary Condition
https://doi-001.org/1025/17620672786522
Lilia ZENKOUFI 1 and Hamid BOULARES 2
1Department of Mathematics. Faculty of MISM, University 8 may 1945 Guelma, Algeria.
Laboratory of Applied Mathematics and Modeling “LAMM”. E-mail: zenkoufi@yahoo.fr
2Laboratory of Analysis and Control of Differential Equations “ACED”, Faculty MISM. Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria. E-mail: boulareshamid@gmail.com
Submitted 01.01.2025 ; Accepted: 02.06.2025
Abstract
The aim of this work is to investigate the existence, uniqueness, and positivity of a solution to a nonlinear Hadamard fractional differential equation supplemented with an integral boundary condition. Our approach leverages several key theorems from nonlinear functional analysis: the Leray-Schauder nonlinear alternative, the Banach contraction mapping principle, and the Guo-Krasnosel’skii fixed point theorem on cone expansion and compression. Finally, we provide illustrative examples to demonstrate the applicability of our theoretical findings.
Keywords: Cone, fixed point theorem, Hadamard fractional differential equations, Integral condition.
Mathematics Subject Classifications: 34B10,34B15,26A33.
- Introduction
The ability of fractional calculus to capture non-local effects, such as memory and hereditary properties, makes it exceptionally well-suited for describing complex phenomena in materials science, viscoelasticity, and anomalous transport. This has driven intense interest in the analysis of fractional-order boundary value problems (BVPs). A central challenge in this field is proving the existence and uniqueness of solutions, especially for nonlinear equations where analytical solutions are often unattainable. To address this, researchers frequently employ methods from nonlinear functional analysis, with fixed point theorems (e.g., Banach, Schaefer, Krasnoselskii) serving as a primary technique for proving existence results.
In authors studied the existence of at least three positive solutions to the following singular boundary value problem:
where and is the standard Caputo derivative.
Wengui Yang [18] applied the Leray-Schauder nonlinear alternative and Krasnosel’skii’s fixed point theorem to prove the existence of positive solutions for a class of coupled semipositone Hadamard fractional differential equations with integral boundary conditions.
In authors investigated the existence criteria for the following problem:
where denotes the Hadamard fractional derivative of order is a continuous function, are given points with and are appropriate real numbers.
Let be the Banach space of continuous functions endowed with the norm
Motivated by the work discussed above and others we investigate to the following nonlinear Hadamard fractional differential equation with integral boundary condition:
where is the Hadamard fractional derivative of fractional order is a real number, and
The organization of this paper is as follows. We begin in Section 2 with preliminary material, including key definitions, lemmas, and a study of the Green’s function properties. The section also outlines the fixed point theorems employed in subsequent sections. Sections 3 and 4 are devoted to stating and proving the main results on the existence, uniqueness, and positivity of solutions, achieved via the Leray-Schauder nonlinear alternative, the Banach contraction principle, and the Guo-Krasnosel’skii fixed point theorem. Concluding examples that illustrate the applicability of our theorems are given in Section 5.
- Preliminaries
We introduce some necessary definitions, lemmas and theorems which will be used in this paper.
Definition 1. The fractional integral
where , is called Riemann-Liouville fractional integral of order of a function and is the gamma function.
Definition 2. The Riemann-Liouville fractional derivative of order , of a continuous function is given by
Where is the gamma function, and with denoting the greatest integer less than or equal to . It is assumed that the right-hand side is pointwise defined on .
Definition 3. The Hadamard fractional integral of order , for a continuous function is given by
Definition 4. Let and its integer part. The Hadamard fractional derivative of order of the function is defined as
Where
Lemma 5. Assume that with a frational derivative of order that belongs to Then
for some
Definition 6. Let be a Banach space and The operator is a contraction operator if there is an such that imply
Theorem 7. Let be a nonempty closed convex subset of a Banach space and be a contraction operator. Then there is a unique with
Theorem 8. Let be a Banach space, and let be a cone. Assume are open subsets of with and let
be a completely continuous operator. In addition suppose either
and or
and
holds. Then has a fixed point in
Lemma 9. let Then the Hadamard fractional boundary value problem
has a unique solution, given by
Where,
and,
Proof . The solution of the Hadamard differential equation in can be written as the equivalent integral equation
From the boundary condition we get And,
From the boundary condition we deduce that
Then
So,
Integrating this result with respect to from to we obtain
Therefore,
Where is defined by The proof is complete.
Now we give some properties of the Green function.
Lemma 10. The function defined by satisfies the following properties
and .
Where, and for
Combining and we obtain
Lemma 11. The function defined by satisfies the following properties
and .
Proof. The continuity of is easily checked.
If it is easy to see that and
If , we have
Then
And,
which implies that is the monotone nondecreasing function, so
On the other hand,
which implies
Finally
The proof is complete.
We now note that is the solution of problem
if and only if
is a fixed point of the operator
The operator is continuous in view of continuity of
and
And by means of the Arzelà-Ascoli theorem,
is completely continuous
.
- Existence and Uniqueness results
In this section, we prove the uniqueness result via Banach contraction principle.
Theorem 12. Assume that there are such that
and if
Then, the problem has a unique solution in
Proof. We will use the Banach contraction principle to prove that the operator defined by
has a fixed point. Now we will prove that
is a contraction. Let
we get
So, we can obtain
By using
Obviously, we have
so, the contraction principle ensures the uniqueness of a solution for the fractional boundary value problem This finishes the proof.
The existence results are based on the following Leray-Schauder nonlinear alternative.
Lemma 13. Let be Banach space and be a bounded open subset of , . be a completely continuous operator. Then, either there exists , such that , or there exists a fixed point
Theorem 14. Assume that there exist nonnegative functions such that
and
Then the BVP has at least one solution .
Proof. To prove this we apply
First, we need to prove that
is completely continuous:
is continues and
is continuous nonnegative function, we get that
is continuous.
Let
a bounded subset. we will prove that
is relatively compact:
For some
we have:
From the above inequalities we have
This shows that
then, uniformly bounded.
The continuity of
implies that, for any
there exists a constant
such that
if
then
We have:
So,
As ,the right-hand side of the above inequality tends to zero,
consequently
is equicontinuous. From Arzela-Ascoli theorem, we deduce that
is a completely continuous operator.
Now, we prove that there exists a point which satisfies
.
Consider with
We assume that
sush that
then
We also have,
This shows that
From this we get
consequently this contradicts
By applying
has a fixed point
which is a solution of the Hadamard fractional boundary value broblem
The proof is complete.
- Positivity results
In this section, we discuss the existence of positive solution for the Hadamard fractional boundary value problem . We make the following additional assumptions.
(Q1) where
and
(Q2)
Definition 15. A function is called positive solution for the fractional boundary value problem if and satisfies, the
Lemma 16. Let , the unique solution of the fractional boundary value problem is nonnegative and satisfies
Proof. Let it is obvious that
is nonnegative,
From
we have
and,
Hence, for all we obtain
The proof is complete.
Definition 17. We define the cone by
is a non-empty closed and convex subset of
We define an operator as follows:
The operator is continuous in view of continuity of
and
And by means of the Arzelà-Ascoli theorem,
is completely continuous.
By we have
and,
Therefore,
Lemma 18. The operator defined in is completely continuous and satisfies
To establish the existence of positive solutions for problem we will employ the Guo-Krasnosel’skii fixed point theorem.
The main result of this section is the following:
Theorem 19. Let and hold, and assume that
Then the problem has at least one positive solution in the case
and or
and
Proof. We will prove that the problem has at least one positive solution in both cases, superlinear and sublinear. For this we use . We prove the superlinear case.
Since then for any such that for . Let be an open set in defined by
then, for any it yields
If we choose then it yields
Now from then such that for . Let
Denote by the open set
For any we have
let then
and choosing we get
By the first part of ,
has at least one fixed point in
such that;
This completes the superlinear case of
. Case II Now, we assume that
and
(sublinear case)
Proceding as above and by the second part of
we prove the sublinear case. This achieves the proof of
- Examples
In order to illustrate our result, we give the following examples:
Example 20. Consider the following fractional boundary value problem
set
and
One can choose
are nonnegative functions, where
and,
Hence, by , the Hadamard fractional boundary value problem has a unique solution in
Example 21. Consider the following fractional boundary value problem
set
Where,
are nonnegative functions, where
and
Hence, by , the Hadamard fractional boundary value problem has at least one solution in
Example 22. Consider the following fractional boundary value problem
where,
and
Then
By the fractional boundary value problem has at least one positive solution.
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