Abstract
The main aim of this work is to introduce the new concept of λ − Υ , χ -contraction self-mappings and prove the existence of χ -fixed points for such mappings in metric spaces. Our results generalize and improve some results in existing literature. Moreover, some fixed point results in partial metric spaces can be derived from our χ -fixed points results. Finally, the existence of solutions of nonlinear integral equations is investigated via the theoretical results in this work.
Highlights
Introduction and PreliminariesOne of the most famous metrical fixed point theorem is the Banach contraction principle (BCP) which is the classical tool for solving several nonlinear problems
In [1], on the basis of the probabilistic metric space and the S-metric space, Hu and Gu introduced the concept of the probabilistic metric space, which is called the Menger probabilistic S-metric space. ey proved some fixed point theorems in the framework of Menger probabilistic S-metric spaces
In [2], using the notion of the cyclic representation of a nonempty set with respect to a pair of mappings, Mohanta and Biswas obtained coincidence points and common fixed points of a pair of self-mappings satisfying a type of contraction condition involving comparison functions and (w)-comparison functions in partial metric spaces
Summary
One of the most famous metrical fixed point theorem is the Banach contraction principle (BCP) which is the classical tool for solving several nonlinear problems. According to the published work of Matthews [3], fixed point results in partial metric spaces have been investigated widely by many mathematicians. Several χ-fixed point results for mappings satisfying the generalized Banach contractive condition based on the idea of new control function are proved in [4]. They claimed that some fixed point results in partial metric spaces can be derived from these χ-fixed point results in metric spaces. Introduced the ideas of (Υ, χ)-contractions and (Υ, χ)-weak contractions and proved existence of χ-fixed point for such mappings as follows. E existence results of χ-fixed points for such contraction mappings in metric spaces are provided. We apply the theoretical results in this work to prove the existence of solutions of nonlinear integral equations
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