Existence of the Solutions of Nonlinear Fractional Differential Equations Using the Fixed Point Technique in Extended b-Metric Spaces

Liliana Guran,Monica-Felicia Bota

doi:10.3390/sym13020158

Liliana Guran, Monica-Felicia Bota

Open Access

https://doi.org/10.3390/sym13020158

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Abstract

The purpose of this paper is to prove fixed point theorems for cyclic-type operators in extended b-metric spaces. The well-posedness of the fixed point problem and limit shadowing property are also discussed. Some examples are given in order to support our results, and the last part of the paper considers some applications of the main results. The first part of this section is devoted to the study of the existence of a solution to the boundary value problem. In the second part of this section, we study the existence of solutions to fractional boundary value problems with integral-type boundary conditions in the frame of some Caputo-type fractional operators.

Highlights

Introduction and PreliminariesExistence of the Solutions ofIn 1922, Banach proved an interesting fixed point theorem for metric spaces, known as the famous “Banach contraction principle”
One important paper that deals with fixed point theory for cyclic contractions is [2], where some fixed point results for cyclic mappings are proved
The purpose of this paper is to extend the previous results to the class of extended b-metric spaces and to discuss the well-posedness and the limit shadowing property of the fixed point problem

Summary

Introduction

In 1922, Banach proved an interesting fixed point theorem for metric spaces (see [1]), known as the famous “Banach contraction principle”. Generalising the Banach contraction principle has been considered in a variety of ways. One of these is the consideration of different types of operators that satisfy some contraction conditions. Different authors proved fixed point theorems for operators that satisfy a cyclic-type contraction condition. The results are extended in the paper [3], where the authors considered generalisation of the contraction condition. R. George et al in [4] considered various types of cyclic contractions, such as Kannan, Chatterjee, and Ćirić, and proved the existence and uniqueness theorems for these classes of operators. Other results that involve the notion of cyclic contraction, including applications to integral equations, can be found in [5,6,7,8]

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