Ulam stability for a quasi-monotone elliptic system of Laplacian equations

Subject Areas : Analyze

Mehdi Choubin ¹ , Mohammad Bagher Ghaemi ^{2 *}

1 - Department of Mathematics, Velayat University, Iranshahr, Iran,
2 - Department of Mathematics, Iran University of Science and Technology, Tehran, Iran

Received: 2022-12-11 Accepted : 2024-04-29 Published : 2025-07-29

Keywords: Ulam stability, quasi-monotone elliptic system, upper and lower solutions.,

Abstract :

The method of upper and lower solutions is a well-known tool that has been used to prove results of the existence of solutions for many classes of boundary value problems involving ordinary and partial differential equations. It is a well-known fact that the existence of a lower and a upper solutions α and β satisfying α≤β, implies the existence of a solution u with α≤u≤β. In this paper, we study the Hyers-Ulam-Rassias stability of the quasi-monotone(cooperative) elliptic system of Laplacian equations (*) ■(-Δu=F(x,u,v), x∈Ω,@-Δv=G(x,u,v), x∈Ω,) with homogeneous Dirichlet boundary conditions which arise in different applications such as population dynamics and population genetics, where Ω is a bounded domain in R^n with a smooth boundary ∂Ω, Δz is the n-dimensional Laplacian operator defined by Δz:=(∂^2 z)/(∂x_1^2 )+⋯+(∂^2 z)/(∂x_n^2 ) and F,G:¯Ω×R^2→R are functions. The aim of this paper is to investigate the Hyers-Ulam-Rassias stability of (*) by using the new method of upper and lower solutions in two sense of clasical and weak.

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