Some fixed point theorems for (psi -phi )-almost weak contractions in S-metric spaces solving conformable differential equations

Hemant Kumar Nashine,Rabha W Ibrahim,G S Saluja

doi:10.1186/s13660-020-02386-w

Hemant Kumar Nashine, Rabha W Ibrahim + Show 1 more

Open Access

https://doi.org/10.1186/s13660-020-02386-w

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Abstract

The aim of this paper is to establish some fixed point and common fixed point theorems for (psi -phi )-almost weak contractions in complete S-metric spaces, followed by some supportive examples. Our results extend and generalize several results existing in the literature. We employ the outcomes of the fixed point theorems to establish the existence and uniqueness of a solution for a class of conformable differential equations which is a new branch of fractional calculus.

Highlights

Then forBy Corollary 2.5, T has a unique fixed point 0 in X
Introduction and preliminariesFixed point theory is one of the noteworthy and stimulating themes of nonlinear functional analysis that blends topology, analysis, and applied mathematics
Through the Banach fixed point theorem, one can get a unique solution of any nonlinear equation if we convert it into operator form, and it is a contraction operator in a complete metric space

Summary

Then for

By Corollary 2.5, T has a unique fixed point 0 in X. ⎨0 if x = y = z, S(x, y, z) = ⎩max{x, y, z} if otherwise, for all x, y, z ∈ X. Let T : X → X be a mapping defined. T satisfies inequality (2.23) so that T is a weakly contractive map. By Corollary 2.3, T has a unique fixed point and clearly it is 0 in X. (X, S) is a complete S-metric space.

Then choosing

Note that CC satisfies the integral equation

Conclusion