Geraghty-type theorems in modular metric spaces with an application to partial differential equation

Parin Chaipunya,Yeol Je Cho,Poom Kumam

doi:10.1186/1687-1847-2012-83

Abstract

Abstract In this article, we prove some fixed point theorems of Geraghty-type concerning the existence and uniqueness of fixed points under the setting of modular metric spaces. Also, we give an application of our main results to establish the existence and uniqueness of a solution to a nonhomogeneous linear parabolic partial differential equation in the last section. Mathematics Subject Classification (2010): 47H10, 54H25, 35K15.

Highlights

Introduction and preliminariesThroughout this article, let R+ denote the set of all positive real numbers and let R+ denote the set of all nonnegative real numbers.Since the year 1922, Banach’s contraction principle, due to its simplicity and applicability, has became a very popular tool in modern analysis, especially in nonlinear analysis including its applications to differential and integral equations, variational inequality theory, complementarity problems, equilibrium problems, minimization problems and many others
We prove a generalization of Geraghty’s theorem which improves the result of Eshagi Gordji et al [13] under the influence of a modular metric space
Let Xω be a complete modular metric space with a partial ordering ⊑ and f be a self-mapping on Xω such that, for each l >0, there exists h(l) Î (0, l) such that ψ(ωλ(fx, fy)) ≤α(ψ(ωλ(x, y)))ψ(ωλ+η(λ)(x, y)) + β(ψ(ωλ(x, y)))ψ(ωλ(x, fx)) (2:1) + γ (ψ(ωλ(x, y)))ψ(ωλ(y, fy)), where ψ ∈ ̄ and (α, β, γ ) ∈ S3 with a(t) + 2 max{supt≥0 b(t), supt≥0 g(t)}

Summary

Introduction

Introduction and preliminariesThroughout this article, let R+ denote the set of all positive real numbers and let R+ denote the set of all nonnegative real numbers.Since the year 1922, Banach’s contraction principle, due to its simplicity and applicability, has became a very popular tool in modern analysis, especially in nonlinear analysis including its applications to differential and integral equations, variational inequality theory, complementarity problems, equilibrium problems, minimization problems and many others. [11]Let (X, d) be a complete metric space and f be a self-mapping on X such that there exists β ∈ S satisfying d(fx, fy) ≤ β(d(x, y))d(x, y) for all x, y Î X.

Results

Conclusion