Positive Solution of Fractional Differential Equations with Integral Boundary Condition

https://doi-004.org/6812/17647859605357

Submission: 13/06/2025

Acceptance: 10/11/2025

Published: 03/12/2025

Lilia ZENKOUFI

 Department of Mathematics. Faculty of Sciences

University 8 may 1945 Guelma, Algeria

Laboratory of Applied Mathematics and Modeling “LMAM”

e-mail:  zenkoufi@yahoo.fr

Abstract: This present work concerns the study of a class of nonlinear fractional differential equations with an integral condition, by the help of some fixed point theorems. We establish the uniqueness result by the Banach contraction principle and to prove the existence of positive solution we use a cone fixed point theorem due to Guo-Krasnoselskii by introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions. Two examples are also included to illustrate our results.

Keywords: Cone, fractional differential equations, fixed-point theorem, Integral condition, Riemann-Liouville fractional derivative.

Mathematics Subject Classifications: 34B10, 34B15.

Introduction

  Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. Fractional boundary value problems have been widely studied in the last decades and many monographs and books are devoted to this subject we refer to [1,4,5,7-13,15-19,21,23] and their references.

Fractional and ordinary boundary value problems with integral conditions have been investigated by many authors see . The history of fractional calculus can be traced back to the 17th century, when the German mathematician Gottfried Leibniz first mentioned the concept of fractional differentiation in a letter to his colleague Johann Bernoulli. However, the development of fractional calculus as a field of study actually began in the 19th century, with the work of several mathematicians, including Augustin-Louis Cauchy, Liouville, and Riemann. In the early 20th century, the French mathematician Paul Lévy used fractional calculus to model random processes, and it was subsequently used in the study of fractals and other areas of mathematics. We can cite the paper where Wenxia Wang studied the following fractional integral boundary value problem (BVP) with a parameter

 Finally,

 The proof is complete.

To use the fixed point theorem, according to  we define the operator  as

 Then, we have the following  

 Lemma 8:  The operator  is completely continuous.

 Proof:    is continuous.

From the continuity of   and  , we conclude that  is continous operator

  Let    a bounded subset. we will prove that  is relatively compact,  

   is uniformly bounded.

Then for any  there exists a constant   such that   for some   we have:

 Then,

 then,    uniformly bounded.

    is equicontinuous.

Because  is continuous on   is uniformly continuous on    ( likewise for   )  Thus for any    there existe  such that    and    let       and , one has:

 Consequently,  is equicontinuous. From Arzela-Ascoli theorem, we deduce that  completely continuous operator.

Uniqueness solution

In this section, we prove the uniqueness result by the Banach contraction principle.

 Theorem 9:  Assume that there are  such that

 and if

 Then the boundary value broblem  has a unique solution in  

 Proof: We shall 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.

Existence of positive solution

In this section we investigate the positivity of solutions for the fractional boundary value problem   , for this we make the following hypotheses.

  For any positive numbers  there exists a continuous function    and    such that

Let  so that  is a Banach space endowed with the norm   

The main result of this section is the following well-known Guo-Krasnosel’skii fixed point theorem on cone.

 Theorem 10:    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   

 Definition 11: A function  is called positive solution for the boundary value problem    if    .

 Lemma 12:  Let   , then the solution  of the fractional boundary value problem    is nonnegative and satisfies

 Proof:  Let  it is obvious that  is nonnegative,  From  and we have

 Hence

 On the other hand, for all    we obtain

 Therefore, we have

 The proof is complete.

Let  be the cone of nonnegative function in  with the following form

   is a nonempty closed and convex subset of  hence it is a cone.

Denote    and    for 

 Lemma 13:  Soppose that   hold and  Then  is completely continuous.

 Proof:   For any  it follows from  that

 On the other hand, for all    we obtain

 Therefore, we have

 Noticing the continuity of  and  it i seasy to see that  is continous in   Next, we show  is compact. For any   we have

 Then,

 where, there exists a constant , such that   Then,   is uniformly bounded.

Because  is continuous on   is uniformly continuous on   ( likewise for ) Thus for any   there existe  such that  and  if  and     Then, for any  and    such that   , we have

 We can see that the functions in  are equicontinuous. So,   is relatively compact in . Therefore,  is compact in , and thus   is completely continuous.

We introduce the following height functions to control the growth of the nonlinear term  :

 Theorem 14:  Suppose that  hold and there exist two positive numbers   such that one of the following conditions is satisfied:

and,    

and,   

where,   is nondecreasing on  for any  

Then the boundary value problem  has at least one strictly increasing positive solution  such that  

 Proof:  Without loss of generality, we only prove  

If  then  and   By the definition of    we know that

 By  we have that

 If  then  and   By the definition of  we know that

 By  we have that

 By  (Guo-Krasnosel’skii fixed point theorem),  has a fixed point   From  we know that  is a solution of  and    Because    we get that  is a positive solution for . And, we have

 which implies that  is a strictly increasing positive solution. The proof is completed.

Examples

In order to illustrate our results, we give the following examples.

 Example 15:  Consider the following fractional boundary value problem

   (P₁)

 Let,

 and,

 Then,

 and,

 Hence, by  , the boundary value problem  has a unique solution in  

 Example 16:  Consider the following boundary value problem

       (P₂)

 Let,

 And,

 Obviously,    For any positive numbers , it is easy to see that  hold for    The height functions    and    satisfy the following inequality:

 and    is nondecreasing on    for any   

It follows that

 And

 By , we get that  has at least one strictly increasing positive solution    such that   

Conclusion

In this paper, we considered a fractional differential equation involving Riemann-Liouville fractional derivative of order  . We studied the uniqueness of solution by using the Banach contraction principle and, we established the existence of positive solution by Guo-Krasnosel’skii fixed point theorem by introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions. As application, examples are presented to illustrate the main results.

Author:

Lilia ZENKOUFI

Department of Mathematics. Faculty of Sciences

University 8 may 1945 Guelma, Algeria

Laboratory of Applied Mathematics and Modeling “LMAM”

E-mail:  zenkoufi@yahoo.fr

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