Extension of Darbo’s fixed point theorem via shifting distance functions and its application

Hemant Kumar Nashine,Anupam Das

doi:10.15388/namc.2022.27.25203

Hemant Kumar Nashine, Anupam Das

Open Access

https://doi.org/10.15388/namc.2022.27.25203

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Abstract

In this paper, we discuss solvability of infinite system of fractional integral equations (FIE) of mixed type. To achieve this goal, we first use shifting distance function to establish a new generalization of Darbo’s fixed point theorem, and then apply it to the FIEs to establish the existence of solution on tempered sequence space. Finally, we verify our results by considering a suitable example.

Highlights

Integral equations have multiple practical applications in modelling specific real world problems and different types of real-life situations, e.g., in laws of physics, in the theory of radioactive transmission, in the theory of statistical mechanics, and in the cytotoxic activity
There are many real-life problems, which can be modelled by infinite systems of integral equations with fractional order in a very effective manner
In [16], Harjani et al established sufficient condition about the length of the interval for the existence and uniqueness of mild solutions to a fractional boundary value problem with Sturm–Liouville boundary conditions when the data function is of Lipschitzian type

Summary

Introduction

Integral equations have multiple practical applications in modelling specific real world problems and different types of real-life situations, e.g., in laws of physics, in the theory of radioactive transmission, in the theory of statistical mechanics, and in the cytotoxic activity. In [16], Harjani et al established sufficient condition about the length of the interval for the existence and uniqueness of mild solutions to a fractional boundary value problem with Sturm–Liouville boundary conditions when the data function is of Lipschitzian type. They have presented an application of our result to the eigenvalues problem and its connection with a Lyapunov-type inequality. In [18], Kataria et al have established the existence of mild solution for noninstantaneous impulsive fractional-order integro–differential equations with local and nonlocal conditions in Banach space.

New results

Measure of noncompactness

Infinite systems of mixed type fractional integral equations