Fixed Point Theorems for Multi-Valued Contractions in $b$ -Metric Spaces With Applications to Fractional Differential and Integral Equations

Abstract

The aim of this manuscript is to establish common fixed points results for multi-valued mappings via generalized rational type contractions in complete b-metric spaces. Using the derived results, existence of solutions to certain integral equations and fractional differential equations in the frame of Caputo fractional derivative are studied. Examples are provided for the authenticity of the presented work.INDEX TERMS Common fixed points, b-metric space, set valued mappings, generalized rational type contraction, system of integral equation.

Orthogonal m-metric spaces and an application to solve integral equations

“…And in this context, we have improved and generalized some well-known results in the literature and developed some fixed point theorems for the proposed contraction. Inspired by the work of many scientist [14][15][16][17][18] regarding fixed point theorems to prove the existence and uniqueness of solutions for certain types of integral equations related to fractional type differential equations, we have also developed an illustrative example and a possible application for the system of Fredholm integral equations to claim the validity of the proposed results.…”

Applications of some fixed point theorems for fractional differential equations with Mittag-Leffler kernel

“…In the last decades, two topics have been densely studied: "fixed point theory" and "fractional differential/integral equations". Recently, several significant results have been recorded [7,31,32].…”

On existence–uniqueness results for proportional fractional differential equations and incomplete gamma functions

“…Fixed point techniques are always used to prove existence and uniqueness of ordinary and fractional dynamic equations [21][22][23][24][25][26][27]. It turns out that the structure of the kernel of fractional operator affects the applied analysis technique in proving the existence and uniqueness of solution or its stability criteria due to the natural appearance of the exponential function in the kernel of proportional fractional point technique in proving the existence and uniqueness of solutions for fractional differential equations in the setting of GPF derivatives.…”