Abstract
In this paper, we study a fixed point problem for certain rational contractions on γ-complete metric spaces. Uniqueness of the fixed point is obtained under additional conditions. The Ulam–Hyers–Rassias stability of the problem is investigated. Well-posedness of the problem and the data dependence property are also explored. There are several corollaries of the main result. Finally, our fixed point theorem is applied to solve a problem of integral equation. There is no continuity assumption on the mapping.
Highlights
Introduction and mathematical backgroundIn this paper, we consider a rational contraction on metric spaces and investigate the fixed point problem associated with it
We consider a rational contraction on metric spaces and investigate the fixed point problem associated with it
We investigate Ulam–Hyers–Rassias stability of the fixed point problem
Summary
We consider a rational contraction on metric spaces and investigate the fixed point problem associated with it. Our investigation of the different aspects of the fixed point problem is performed in a metric space without completeness property. Many metric fixed point results were proved and applied to different problem arising in mathematics. (See [23].) A metric space (Z, d) is said to have regular property with respect to a mapping γ : Z × Z → [0, ∞) if for any sequence {an} in Z with limit a ∈ Z, γ(an, an+1) 1 implies γ(an, a) 1 for all n. 2. We prove our main result with a generalized notion of completeness assumption of the underlying space and without the continuity assumption on the mapping. We apply our theorem to a problem of an integral equation
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