https://doi.org/10.65281/641590
Submission date : 02.12.2023 ,
Acceptanc Date: 19.03.2024,
Publication Date : 15 .01.2025
Lilia ZENKOUFI
Department of Mathematics. Faculty of Sciences
University 8 may 1945 Guelma, Algeria
Laboratoire de Mathématiques Appliquées et de Modélisation “LMAM”
e-mail : zenkoufi@yahoo.fr
Abstract: In this paper, we apply Krasnoselskii’s fixed point theorem and the well-known Guo-Krasnoselskii fixed point theorem in cone to prove the existence and the positivity of nontrivial solutions for fifth-order boundary value problems. Two examples are given to illustrate our results.
Keywords: Boundary value problem, differential equation, existence of solution, positive solution, fixed point theorem.
Mathematics Subject Classifications: 34B10, 34B15.
Introduction
Multipoint boundary value problem (BVP) plays an important part in various fields, such as fluid mechanics, physics, engineering, and many other branches of applied mathematics. Recently certain three point boundary value problems for nonlinear ordinary differential equations have been studied by many authors. The literature concerning these problems is extensive and application of theorems of functional analysis has attracted more interest see .
In 2003, Chen Yang considered a nonlinear third-order three-point boundary value problem and gave the existence and uniqueness of solutions by constructing Green’s function and using its properties
In 2018, N. Bouteraa and S. Benaicha by imposing certain conditions on the nonlinear term f, they constructed a lower and a higher solution to prove that a solution exists for a type of nonlinear third-order nonlocal boundary value problem.
The aim of this work is to study the existence and the positivity of solutions for the following nonlinear fifth-order boundary value problem.
Motivated by several results, we consider the three-point boundary value problem for fifth-order differential equations (1.1). By applying Krasnosel’skii’s fixed-point theorem, we discuss the existence solutions, to demonstrate the existence results, we transformed the posed problem into a sum of a contraction and a compact operator. To prove the positivity results : we expressed the Green function associated to the posed problem, then we apply the well-known Guo-Krasnoselskii fixed point theorem in cone. We ended this work with two examples illustrating the obtain results.
and
then,
which implies the Lemma 01.
Existence results
We state a known result due to Krasnoselskii.
Theorem 02 : Let be a closed convex and nonempty subset of a Banach space . Let and be two operators such that
whenever
is compact and continuous.
is a contraction.
Then there exists such that
Theorem03 : Assume that and there exists a nonnegative function such that
(3.1)
Then problem has at least one solution on
Hence, we get
Positive results
To establish the existence of positive solutions to the posed problem, we employ the following Guo-Krasnosel’skii fixed point theorem.
Theorem 04 : 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
holds. Then has a fixed point in
Lemma 05: The solution of boundary value problem can be expressed as
(4.1)
(4.2)
Proof : We have
So,
And, that is equivalent to
The proof is complete.
Lemma 06: is strictly increasing in the first variable, and satisfies the following properties
and .
Definition 07: We define an operator by
(4.3)
Lemma 08: An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.
Lemma 09: Let the unique solution of the boundary value problem is nonnegative and satisfies
Definition 10: We define the cone by
Lemma 11: The operator defined in is completely continuous and satisfies
The main result of this section is the following
Theorem 12: Let and hold, and assume that
Then problem has at least one positive solution in the case
Proof : We shall prove that problem has at least one positive solution in both cases, superlinear and sublinear, for this we use 04. 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
Therefore
If we choose then it yields
Now from then such that for . Let
Denote by the open set
For any have
alors, Let then
And choosing we get
By the first part of 04, has at least one fixed point in such that, This completes the superlinear case of 12.
Case II Now, we assume that and (sublinear case). Proceding as above and by the second part of 04, we prove the sublinear case. This achieves the proof of 12.
Example
In order to illustrate our results, we give the following examples
Example 01: Consider the following boundary value problem
We have .
And,
Therefore, by 03, problem has at least one solution in with
Example 02: Consider the following boundary value problem
By 12 the boundary value problem has at least one positive solution.
Conclusion
The objective of this work is to study the existence and the positivity of solutions for the boundary value problems generated by a fifth-order differential equations by applying Krasnosel’skii fixed point theorem and the well-known Guo-Krasnoselskii fixed point theorem in cone, where the boundary conditions are imposed in three-points of the domain. We ended this work with two examples illustrating the obtain results.
Author:
Lilia ZENKOUFI
Department of Mathematics. Faculty of Sciences
University 8 may 1945 Guelma, Algeria
Laboratory of Applied Mathematics and Modeling “LAMM”
E-mail: zenkoufi@yahoo.fr
Declarations:
Competing interests
The author declares no conflicts of interest.
References
- Burton, T.A., Furumochi, T.: Krasnoselskii’s fixed point theorem and stability. Nonlinear Anal., Theory Methods Appl. 49(4), 445-454, 2002.
- Chen Yang: Existence and Uniqueness of Solutions for a Third-Order Three-Point Boundary Value Problem via Measure of Noncompactness. HindawiJournal of Mathematics. Volume 2022, Article ID 1157154, 8 pages, 2022.
- K. Deimling : Nonlinear functional analysis, Springer, Berlin, 1985.
- D.Guo,V.Lakshmikantham : Nonlinear problems in abstract cones, Academic Press, San Diego, 1988.
- C. P. Gupta : Solvability of a three-point nonlinear boundary value problem for a second order differential equation, J. Math. Anal. Appl. 168, (1992), 540- 551, 1992.
- G. Infante, J. R. L. Webb : Three point boundary value problems with solutions that change sign, J. Integ. Eqns Appl., 15 (2003), 37-57, 2003.
- Le X. Phan D. : Existence of positive solutions for a multi-point four-order boundary value problem. Electronic journal of Differential Equations, Vol. 2011, pp. 1-10, 2011.
- Lilia Zenkouf : Existence and uniqueness solution for integral boundary value problem of fractional differential equation. New Trends in Mathematical Sciences, BISKA. NTMSCI 10 Special Issue, No. 1, 90-94, 2022.
- Li, X, Song, L, Wei, J : Positive solutions for boundary value problem of nonlinear fractional functional differential equations. Appl. Math. Comput. 217 (22), 9278-9285 (2011).
- R. Ma, A Survey On nonlocal boundary value problems. Applied Mathematics E-Notes, 7 257-279, 2007.
- R. Ma and H. Wang : On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal., 59 (1995), 225-231, 1995.
- Noureddine Bouteraa and Slimane Benaicha: Existence of solution for third-order three-point boundary value problem. MATHEMATICA, 60 (83), No 1, 2018, pp. 21–31, 2018.
- L. Shuhong, Y-P. Sun : Nontrivial solution of a nonlinear second order three point boundary value problem, Appl. math. j(2007), 22(1), 37-47.
- Y-P. Sun; Nontrivial solution for a three-point boundary-value problem, E.J.D.E, Vol. 2004(2004), No. 111, 1-10.
- S. Zhang : Existence results of positive solutions to boundary value problem for fractional dfferential equation, Positivity, 13: 583–599, 2009.