On convergence theorems for single-valued and multi-valued mappings in p-uniformly convex metric spaces

We prove ∆-convergence and strong convergence theorems of an iterative sequence generated by the Ishikawa’s method to a fixed point of a single-valued quasi-nonexpansive mappings in p-uniformly convex metric spaces without assuming the metric convexity assumption. As a consequence of our single-valued version, we obtain a result for multi-valued mappings by showing that every multi-valued quasi-nonexpansive mapping taking compact values admits a quasi-nonexpansive selection whose fixed-point set of the selection is equal to the strict fixed-point set of the multi-valued mapping. In particular, we immediately obtain all of the convergence theorems of Laokul and Panyanak [Laokul, T.; Panyanak, B. A generalization of the (CN) inequality and its applications. Carpathian J. Math. 36 (2020), no. 1, 81–90] and we show that some of their assumptions are superfluous.

Data Dependence, Strict Fixed Point Results, and Well-Posedness of Multivalued Weakly Picard Operators

In this paper, we introduce the notion of s , r -contractive multivalued weakly Picard operators via simulation functions, named as Z s , r -contractions. We present some related fixed point theorems. We investigate data dependence and strict fixed point results. The well-posedness for such operators is also considered. Moreover, we generalize the results of Moţ and Petruşel. To show the usability of our results, we give some examples and an application to resolve a functional equation arising in dynamical systems.

A generalization of the (CN) inequality and its applications

We extend the (CN) inequality of Bruhat and Tits in CAT(0) spaces to the general setting of uniformly convex hyperbolic spaces. We also show that, under some appropriate conditions, the sequence of Ishikawa iteration defined by Panyanak converges to a strict fixed point of a multi-valued Suzuki mapping.

AN AFFIRMATIVE ANSWER TO QUASI-CONTRACTION'S OPEN PROBLEM UNDER SOME LOCAL CONSTRAINTS IN JS-METRIC SPACES

In this work, we first present JS-Pompeiu-Hausdorff metric in JS metric spaces and then introduce well-behaved quasi-contraction in order to find an affirmative answer to quasi-contractions’ open problem under some local constraints in JS-metric spaces. In the literature, this problem solved when the constant modules α ∈ [0,1/2] and when α ∈ (1/2,1], finding conditions by which the set of all fixed points be non-empty, has remained open yet. Moreover, we support our result by a notable example. Finally, by taking into account the approximate strict fixed point property we present some worthwhile open problems in these spaces.

Some variants of Ćirić’s multi-valued contraction principle

In this article, a study of the fixed point problem for Ćirić type multi-valued operators is presented. More precisely, some variants ofĆirić’s contraction principle for multi-valued operators, as well as a strict fixed point principle forĆirić type multi-valued will be given.

Approximation of a Common Element of the Fixed Point Sets of Multivalued Strictly Pseudocontractive-Type Mappings and the Set of Solutions of an Equilibrium Problem in Hilbert Spaces

The strong convergence of a hybrid algorithm to a common element of the fixed point sets of multivalued strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces is obtained using a strict fixed point set condition. The obtained results improve, complement, and extend the results on multivalued and single-valued mappings in the contemporary literature.