Abstract
In this article, we introduce and establish various approaches related to the F-contraction using new sorts of contractions, namely the extended F B e -contraction, the extended F B e -expanding contraction, and the extended generalized F B e -contraction. Thereafter, we propose a simple and efficient solution for non-linear integral equations using the fixed point technique in the setting of a B e -metric space. Moreover, to address conceptual depth within this approach, we supply illustrative examples where necessary.
Highlights
Renowned for the “Fredholm Integral Equation” Erik Ivar Fredholm [1], a mathematician and researcher par excellence, has provided research contributions on various aspects of integral equation theory. Inspired by his great work, many fixed point researchers have focused their work on solving the Fredholm integral equation [2,3,4,5]
A mapping H : X → X is said to be an F-contraction if there exists τ > 0 such that for all x, y ∈ X, d(H x, H y) > 0 ⇒ τ + F (d(H x, H y)) ≤ F (d( x, y))
Motivated by the above facts, we establish fixed point theorems by using F-contractions in the context of an extended b-metric space since it was very hard to obtain fixed points via the Warkowski [15] approach, which gives a solutions for non-linear integral equations by using the fixed point technique
Summary
Department of Mathematics, Basic Sciences and Humanities, GMR Institute of Technology, Rajam 532127, Andhra Pradesh, India College of Computer and Information Sciences, Majmaah University, Majmaah 11952, Saudi Arabia Distinguished University Professor of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA Received: 8 January 2019; Accepted: 28 January 2019; Published: 12 February 2019
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