Abstract
The aim of this paper is to introduce a notion of ϕ , F -contraction defined on a metric space with w -distance. Moreover, fixed-point theorems are given in this framework. As an application, we prove the existence and uniqueness of a solution for the nonlinear Fredholm integral equations. Some illustrative examples are provided to advocate the usability of our results.
Highlights
IntroductionBy a contraction on a metric space ðX, dÞ, we understand a mapping T : X ⟶ X satisfying for all x, y ∈ X : dðTx, TyÞ ≤ kdðx, yÞ, where k is a real in 1⁄20, 1Þ
By a contraction on a metric space ðX, dÞ, we understand a mapping T : X ⟶ X satisfying for all x, y ∈ X : dðTx, TyÞ ≤ kdðx, yÞ, where k is a real in 1⁄20, 1Þ.In 1922, Banach proved the following theorem.Theorem 1
In 2012, Wardowski [3] introduced the concept of F-contraction, using this concept, he proved the existence and uniqueness of a fixed point in complete metric spaces
Summary
By a contraction on a metric space ðX, dÞ, we understand a mapping T : X ⟶ X satisfying for all x, y ∈ X : dðTx, TyÞ ≤ kdðx, yÞ, where k is a real in 1⁄20, 1Þ. In 2012, Wardowski [3] introduced the concept of F-contraction, using this concept, he proved the existence and uniqueness of a fixed point in complete metric spaces. This direction has been studied and generalized in different spaces, and various fixed-point theorems are developed [4, 5]. In 2016, Piri and Kumam [7] introduced the modified generalized F-contractions, by combining the ideas of Dung and Hang [8], Piri and Kumam [9], Wardowski [3], and Wardowski and Van Dung [10], and gave some fixedpoint result for these type mappings on complete metric space.
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