Abstract
In this paper, inspired by Jleli and Samet (Journal of Inequalities and Applications 38 (2014) 1–8), we introduce two new classes of auxiliary functions and utilize the same to define ( θ , ψ ) R -weak contractions. Utilizing ( θ , ψ ) R -weak contractions, we prove some fixed point theorems in the setting of relational metric spaces. We employ some examples to substantiate the utility of our newly proven results. Finally, we apply one of our newly proven results to ensure the existence and uniqueness of the solution of a Volterra-type integral equation.
Highlights
Fixed point theory remains a very important and popular tool in pure, as well as applied mathematics, especially in the existence and uniqueness theories
The Banach contraction principle [1] is one of the pivotal results of fixed point theory, which asserts that every contraction mapping defined on a complete metric space ( E, d) to itself always admits a unique fixed point
As the Banach contraction principle and its extensions are existence and uniqueness results, they are very effectively utilized in several kinds of applications in the entire domain of mathematical and physical sciences, which includes economics
Summary
Fixed point theory remains a very important and popular tool in pure, as well as applied mathematics, especially in the existence and uniqueness theories. The Banach contraction principle [1] is one of the pivotal results of fixed point theory, which asserts that every contraction mapping defined on a complete metric space ( E, d) to itself always admits a unique fixed point. This principle is a very effective and popular tool for guaranteeing the existence and uniqueness of the solution of certain problems arising within and beyond mathematics. To introduce the notion of (θ, ψ)R -weak contraction; to prove our results in the setting of relational metric spaces; to adopt some examples substantiating the utility of our proven results; to utilize our newly proven results and establish an existence and uniqueness result for the solution of a Volterra-type integral equation
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