On the Correlation between Banach Contraction Principle and Caristi’s Fixed Point Theorem in b-Metric Spaces

We solve a question posed by E. Karapinar, F. Khojasteh and Z.D. Mitrović in their paper “A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces”. We also characterize the completeness of b-metric spaces with the help of a variant of the contractivity condition introduced by the authors in the aforementioned article.

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.

A generalization of Matkowski’s and Suzuki’s fixed point theorems

In this paper, we present a generalization of Suzuki’s fixed point theorem and the Matkowski contraction principle for a system of transformations on the finite product of metric spaces.

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.

Uniqueness and stability for generalized Caputo fractional differential quasi-variational inequalities

In this paper, we study a new kind of generalized Caputo fractional differential quasi-variational inequalities in Hilbert spaces. We prove the uniqueness and the stability of the abstract inequality by using generalized singular Gronwall’s lemma, projection operators, and contraction principle. Finally, an example is given to illustrate the abstract results.

Functional differential equations with maxima, via step by step contraction principle

T. A. Burton presented in some examples of integral equations a notion of progressive contractions on C([a,\infty \lbrack). In 2019, I. A. Rus formalized this notion (I. A. Rus, Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle, Advances in the Theory of Nonlinear Analysis and its Applications, 3 (2019) No. 3, 111–120), put “step by step” instead of “progressive” in this notion, and give some variant of step by step contraction principle in the case of operators with Volterra property on C([a,b],\mathbb{B)} and C([a,\infty \lbrack,\mathbb{B}) where \mathbb{B} is a Banach space. In this paper we use the abstract result given by I. A. Rus, to study some classes of functional differential equations with maxima.