Abstract
Existence and uniqueness of fixed points of a mapping defined on partially ordered G-metric spaces is discussed. The mapping satisfies contractive conditions based on certain classes of functions. The results are applied to the problems involving contractive conditions of integral type and to a particular type of initial value problems for the nonhomogeneous heat equation in one dimension. This work is a generalization of the results published recently in (Gordji et al. in Fixed Point Theory Appl. 2012:74, 2012, doi:10.1186/1687-1812-2012-74) to G-metric space. MSC:47H10, 54H25.
Highlights
Introduction and preliminariesOne of the most common applications of the fixed point theory is the problem of existence and uniqueness of solutions of initial and boundary value problems for differential and integral equations
In, Mustafa and Sims [ ] introduced the concept of a G-metric and G-metric space, which is a generalization of metric space
We present some fixed point theorems on G-metric spaces and investigate the existence of solutions of an initial value problem for a partial differential equation, more precisely, a nonlinear one dimensional heat equation
Summary
Introduction and preliminariesOne of the most common applications of the fixed point theory is the problem of existence and uniqueness of solutions of initial and boundary value problems for differential and integral equations. A function f : (X, G) → (X , G ) is said to be G-continuous at a point a ∈ X if and only if for every ε > , there exists δ > such that x, y ∈ X and G(a, x, y) < δ implies G (f (a), f (x), f (y)) < ε. Let denote the class of the functions ψ : [ , +∞) → [ , +∞) satisfying the following conditions:
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