Fixed Point Theorems on Some New Types of Cyclic Contractions and Their Generalization

There are different types of contraction in the existing literature for the generalization of Banach’s contraction principle. Our aim in this paper is to generalize cyclic contraction so that it can explain all types of cyclic contraction as a particular case. Besides all contractions in the existing literature we introduce some new types of cyclic contraction before defining the generalized cyclic contraction.

Ulam–Hyers Stability and Uniqueness for Nonlinear Sequential Fractional Differential Equations Involving Integral Boundary Conditions

Fractional-order boundary value problems are used to model certain phenomena in chemistry, physics, biology, and engineering. However, some of these models do not meet the existence and uniqueness required in the mainstream of mathematical processes. Therefore, in this paper, the existence, stability, and uniqueness for the solution of the coupled system of the Caputo-type sequential fractional differential equation, involving integral boundary conditions, was discussed, and investigated. Leray–Schauder’s alternative was applied to derive the existence of the solution, while Banach’s contraction principle was used to examine the uniqueness of the solution. Moreover, Ulam–Hyers stability of the presented system was investigated. It was found that the theoretical-related aspects (existence, uniqueness, and stability) that were examined for the governing system were satisfactory. Finally, an example was given to illustrate and examine certain related aspects.

Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces

This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some examples for the illustration of our main theorems.

Nonlocal and multiple-point fractional boundary value problem in the frame of a generalized Hilfer derivative

The aim of this manuscript is to handle the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equations involving a ξ-Hilfer derivative. The used fractional operator is generated by the kernel of the kind $k(\vartheta,s)=\xi (\vartheta )-\xi (s)$ k ( ϑ , s ) = ξ ( ϑ ) − ξ ( s ) and the operator of differentiation ${ D}_{\xi } = ( \frac{1}{\xi ^{\prime }(\vartheta )}\frac{d}{d\vartheta } ) $ D ξ = ( 1 ξ ′ ( ϑ ) d d ϑ ) . The existence and uniqueness of solutions are established for the considered system. Our perspective relies on the properties of the generalized Hilfer derivative and the implementation of Krasnoselskii’s fixed point approach and Banach’s contraction principle with respect to the Bielecki norm to obtain the uniqueness of solution on a bounded domain in a Banach space. Besides, we discuss the Ulam–Hyers stability criteria for the main fractional system. Finally, some examples are given to illustrate the viability of the main theories.

Existence and Uniqueness of Solutions for Coupled Impulsive Fractional Pantograph Differential Equations with Antiperiodic Boundary Conditions

In this paper, we investigate the solutions of coupled fractional pantograph differential equations with instantaneous impulses. The work improves some existing results and contributes toward the development of the fractional differential equation theory. We first provide some definitions that will be used throughout the paper; after that, we give the existence and uniqueness results that are based on Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Two examples are given in the last part to support our study.

Multiple Solutions for a Sixth Order Boundary Value Problem

This work presents conditions for the existence of multiple solutions for a sixth order equation with homogeneous boundary conditions using Avery Peterson's theorem. In addition, non-trivial examples are presented and a new numerical method based on the Banach's Contraction Principle is introduced.

On the nonlinear Hadamard-type integro-differential equation

This paper studies uniqueness of solutions for a nonlinear Hadamard-type integro-differential equation in the Banach space of absolutely continuous functions based on Babenko’s approach and Banach’s contraction principle. We also include two illustrative examples to demonstrate the use of main theorems.

Stability analysis of initial value problem of pantograph-type implicit fractional differential equations with impulsive conditions

In this paper, we study an initial value problem for a class of impulsive implicit-type fractional differential equations (FDEs) with proportional delay terms. Schaefer’s fixed point theorem and Banach’s contraction principle are the key tools in obtaining the required results. We apply our results to a numerical problem for demonstration purpose.

Uniqueness of the Hadamard-type integral equations

The goal of this paper is to study the uniqueness of solutions of several Hadamard-type integral equations and a related coupled system in Banach spaces. The results obtained are new and based on Babenko’s approach and Banach’s contraction principle. We also present several examples for illustration of the main theorems.

Some significant remarks on multivalued Perov type contractions on cone metric spaces with a directed graph

<abstract><p>Using the approach of so-called c-sequences introduced by the fifth author in his recent work, we give much simpler and shorter proofs of multivalued Perov's type results with respect to the ones presented in the recently published paper by M. Abbas et al. Our proofs improve, complement, unify and enrich the ones from the recent papers. Further, in the last section of this paper, we correct and generalize the well-known Perov's fixed point result. We show that this result is in fact equivalent to Banach's contraction principle.</p></abstract>