Volume 2025, Issue 1 9932318

Research Article

Open Access

A Higher Order Nonresonant p-Laplacian Boundary Value Problem on an Unbounded Domain

S. A. Iyase,

S. A. Iyase

orcid.org/0000-0001-6028-0559

Department of Mathematics , Covenant University , Ota , Ogun State , Nigeria , covenantuniversity.edu.ng

Search for more papers by this author

O. F. Imaga,

Corresponding Author

O. F. Imaga

[email protected]

orcid.org/0000-0002-3252-3530

Department of Mathematics , Covenant University , Ota , Ogun State , Nigeria , covenantuniversity.edu.ng

Search for more papers by this author

S. A. Iyase,

S. A. Iyase

orcid.org/0000-0001-6028-0559

Department of Mathematics , Covenant University , Ota , Ogun State , Nigeria , covenantuniversity.edu.ng

Search for more papers by this author

O. F. Imaga,

Corresponding Author

O. F. Imaga

[email protected]

orcid.org/0000-0002-3252-3530

Department of Mathematics , Covenant University , Ota , Ogun State , Nigeria , covenantuniversity.edu.ng

Search for more papers by this author

First published: 13 March 2025

https://doi.org/10.1155/aaa/9932318

Academic Editor: Douglas R. Anderson

Share a link

Email
Wechat
Bluesky

Abstract

In this article, we prove the existence of at least one solution to the nonresonant higher-order p-Laplacian boundary value problem of the form:

with the nonresonant condition

. We employ the Leray–Schauder continuation principle and some apriori estimates to obtain our result.

MSC2010 Classification: 34B10, 34B15

1. Introduction

The purpose of this study is to derive existence result for the p-Laplacian boundary value problem:

()

where φ_p(s) = |s|^p−2s, p > 1,

satisfies Caratheodory’s condition with respect to L¹[0, ∞), 0 ≤ η_j,

, q > 1,

, j = 1, 2, …, n, σ(r) ∈ C[0, ∞), σ(r) > 0 and

. This problem is nonresonant if the associated problem:

()

subjected to Equation (2) has only the trivial solution.

In [1], Gupta studied the nonresonant p-Laplacian boundary value problem of the form:

for which existence result was obtained under nonresonant condition using topological degree theory.

In recent years, boundary value problems with p-Laplacian have received a lot of attention in the literature. Existence results have been obtained by utilising degree theory, fixed point theorems, upper and lower solution techniques and monotone iteration method. For some results in these directions, see [1–11] and the references therein. Boundary value problems with a p-Laplacian occur in diverse areas such as in non-linear elasticity, blood flow models, non-Newtonian mechanics, glaciology, et cetera. To the best of our knowledge, higher-order p-Laplacian boundary value problems at nonresonance on an unbounded domain have not received much attention in the literature. This paper, therefore, attempts to bridge this gap.

This paper is organised as follows: in Section 2, we provide background definitions and some key theorems and lemmas including the apriori estimates that will be used in the subsequent sections. Section 3 is devoted to proving the main result, and in Section 4, we provide an example to validate our result.

2. Preliminaries

We recall some theorems and definitions. Let X and Y be Banach spaces with the norms ‖⋅‖_X and ‖⋅‖_Y, respectively. Let AC[0, ∞) be the space of absolutely continuous function on [0, ∞) and let:

()

with the norm:

()

Then, X is a Banach space.

Let Y = L¹[0, ∞) with the norm:

()

We define the mapping N : X → Y by:

()

Definition 1. The map is L¹[0, ∞) Caratheodory if the following conditions are fulfilled:

(i)
for each , g(r, u) is Lebesgue measurable;
(ii)
for a.e. r ∈ [0, ∞), g(r, u) is continuous on ;
(iii)
for each b > 0, there exists α_b ∈ L¹[0, ∞) such that for a.e. r ∈ [0, ∞) and every u such that ‖u‖ < b, we have:
()

We now state the Leray–Schauder continuation principle.

Theorem 1. [12] Let L : X → X be a compact mapping and X a Banach space. Suppose that there exists b > 0 such that if u = λLu for λ ∈ [0, 1] and ‖u‖ ≤ b, then, L has a fixed point in X.

In what follows, we shall apply the above theorem which has been used by researchers to obtain existence results. However, since we are dealing with an unbounded domain [0, ∞), the compactness principle on the bounded interval [0, 1] is not applicable here. We shall, therefore, apply the following result:

Theorem 2. [13] Let X be the space of all bounded continuous vector valued functions on [0, ∞) and W ⊂ X. Then, W is relatively compact on X if the following conditions hold:

(i)
W is bounded in X;
(ii)
the functions from W are equicontinuous on any compact interval of [0, ∞);
(iii)
the functions from W are equiconvergent at infinity.

Lemma 1. The boundary value problem (2) and (3) has only the trivial solution if and only if .

Proof. From Equation (3), we obtain:

()

So that,

()

and hence,

()

where

For non-resonance, we must have B = C = 0. From u(0) = 0, we obtain C = 0, while from Equation (3) we derive:

which implies,

that is,

()

From Equation (2), we have:

()

Therefore, from Equations (12) and (13), we obtain:

Since

, we have B = 0.

In what follows, we shall assume the following constants:

()

Lemma 2. Let k ∈ Y, then the unique solution of the equation:

()

subjected to Equation (2) is given by:

()

Proof. From Equation (16), we obtain:

so that,

()

From Equation (16), we have:

so that,

Thus,

()

Therefore, from Equation (18), we derive:

Lemma 3. Let k ∈ Y, then the solution Equation (17) satisfies:

()

and thus,

()

where A₁ and A₂ are as in Equations (14) and (15).

Proof. From Equation (17), we obtain:

Also,

Hence,

()

where A = max(A₁, A₂).

Let T : Y → X be defined by:

then, T is well defined for all k ∈ Y and is the unique solution of:

subjected to Equation (2).

Let L = TN : X → X be defined by:

then, L is well defined. We prove the compactness of L by applying Theorem 2.

Lemma 4. The mapping L : X → X is compact.

Proof. Let W ⊂ X be bounded with:

()

Since

is L¹[0, ∞) Caratheodory, there exists α_b ∈ Y such that |Nu(r)| ≤ α_b(r) for b ∈ [0, ∞) and u ∈ W. Therefore, for u ∈ W and 0 ≤ i ≤ n − 2:

()

For i = n − 1,

()

Equations (24) and (25) implies that:

()

This shows that L(W) is bounded in X. Next, we prove that L(W) is equicontinuous. Let r₁, r₂ ∈ [0, D], D ∈ (0, ∞) with r₁ < r₂ and u ∈ W. Then, for 0 ≤ i ≤ n − 2, we have:

For i = n − 1, we have:

and hence,

Therefore, L(W) is equicontinuous on every compact subset of [0, ∞). Next, we show that L ∈ W is equiconvergent at infinity. For u ∈ W and for 0 ≤ i ≤ n − 2, we have:

For i = n − 1, we derive:

This implies that L(W) is equiconvergent at infinity. Thus, all the conditions of Theorem 2 are satisfied, and hence, L(W) is, therefore, relatively compact. The continuity of L(W) follows from the Lebesgue dominated convergence theorem.

3. Main Result

Theorem 3. Let be a L¹[0, ∞) Caratheodory function. Suppose there exist positive functions b_i, a : [0, ∞) → [0, ∞) such that:

()

with,

()

where A is defined in Equation (27). Then, the boundary value problem (1) and (2) has at least one solution for every a ∈ L¹[0, ∞).

Proof. Let u be any solution of the equation:

subjected to Equation (2), λ ∈ [0, 1]. Then,

which yields,

Then, from Equation (27), we derive:

()

This implies that the set of solutions of Equations (1) and (2) are a priori bounded by a constant independent of λ and of solutions. This proves the theorem.

4. Example

We consider the boundary value problem:

()

where σ(r) = e^r, p = 3,

and a ∈ Y, a > 0.

All the conditions of Theorem 3 are verified, and thus, Equations (31) and (32) has at least one solution for every a ∈ Y.

Disclosure

An earlier version of this work has been presented as a conference paper titled ‘A third-order p-Laplacian boundary value problem on an unbounded domain’.

Conflicts of Interest

The authors declare no conflicts of interest.

Author Contributions

The first author conceived the idea and developed the theory. The second author did the literature review. The results where discussed by all authors who also contributed to the final manuscript.

Funding

The authors received no specific funding for this research.

Acknowledgments

The authors would like to thank the Covenant University for their support during the study period. The authors received no specific funding for this work.

Open Research

Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

1 Gupta C. P., A Non-Resonant Generalised Multipoint Boundary Value Problem of Dirichlet Type Involving a p-Laplacian Type Operator, Electronic Journal of Differential Equations. (2007) 15, no. 27, 127–139.
Google Scholar
2 Gupta C. P., A Non-Resonant Multipoint Boundary Value Problem of Neumann-Dirichlet Type for a p-Laplacian Type Operator, Dynamical Systems and Application. (2004) 4, 439–442.
Google Scholar
3 Iyase S. and Opanuga A., Higher Order Nonlocal Boundary Value Problems at Resonance on the Half-line, European Journal of Pure and Applied Mathematics. (2020) 13, no. 1, 33–47, https://doi.org/10.29020/nybg.ejpam.v13i1.3539.
10.29020/nybg.ejpam.v13i1.3539
Web of Science® Google Scholar
4 Iyase S. A. and Imaga O. F., A Third-Order p-Laplacian Boundary Value Problem on an Unbounded Domain, Turkish Journal of Mathematics. (2021) 45, 2382–2392.
10.3906/mat-2002-89
Web of Science® Google Scholar
5 Iyase S. A. and Imaga O. F., Higher Order p-Laplacian Boundary Value Problems With Integral Boundary Conditions on the Half-Line, Bulletin of the Malaysian Mathematical Sciences Society. (2021) 6, no. 44, 4127–4141.
10.1007/s40840-021-01159-y
Google Scholar
6 Kim C. G., Solvability of Multipoint Boundary Value Problem on the Half-Line, Journal of Nonlocal Science and Applications. (2012) 5, 27–33.
Google Scholar
7 Kosmatov N., Second Order Boundary Value Problems on an Unbounded Domain, Nonlinear Analysis: Theory, Methods & Applications. (2008) 68, no. 4, 875–882, https://doi.org/10.1016/j.na.2006.11.043, 2-s2.0-36749007705.
10.1016/j.na.2006.11.043
Web of Science® Google Scholar
8 Liu Y., Non-Homogeneous Boundary Value Problems of Higher Order Differential Equations With p-Laplacian, Electronic Journal of Differential Equations. (2008) 20, 1–43.
Google Scholar
9 Wang W. and Yang X., Nonlocal Boundary Value Problems With Resonant or Non-Resonant Conditions, Boundary Value Problems. (2013) 2013, 238.
10.1186/1687-2770-2013-238
Google Scholar
10 Yang A., Henderson J., and Wang D., The Study of Higher-Order Resonance and Non-Resonant Boundary Value Problems, Electronic Journal of Qualitative Theory of Differential Equations. (2016) 3, 1–10.
Google Scholar
11 Liu M. and Pei M., Solvability of a Nonlinear Second Order m-Point Boundary Value Problem With p-Laplacian at Resonance, Boundary Value Problems. (2024) 46, 425–431.
CAS Google Scholar
12 Zeidler E., Nonlinear Functional Analysis and Its Applications, 1986, Springer Verlag, New York.
10.1007/978-1-4612-4838-5
Google Scholar
13 Agarwal R. P. and Regan D. O.’, Infinte Interval Problems for Differential, Difference and Integral Equations, 2011, Kluwer Academic Publishers, London, U.K.
Google Scholar

All articles

A Higher Order Nonresonant p-Laplacian Boundary Value Problem on an Unbounded Domain

Abstract

1. Introduction

2. Preliminaries

3. Main Result

4. Example

Disclosure

Conflicts of Interest

Author Contributions

Funding

Acknowledgments

Open Research

Data Availability Statement

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

A Higher Order Nonresonant p-Laplacian Boundary Value Problem on an Unbounded Domain

Abstract

1. Introduction

2. Preliminaries

3. Main Result

4. Example

Disclosure

Conflicts of Interest

Author Contributions

Funding

Acknowledgments

Open Research

Data Availability Statement

References

References

Related

Information