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The purpose of this short note is to present a correction of the proof of the main result given in the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings,” Demonstr. Math. 53 (2020), 38–43.

Let H be a Hilbert space and let C be a closed convex nonempty subset of H and $$T : C\rightarrow H$$ a non-self nonexpansive mapping. A map $$h : C\rightarrow R$$ defined by $$h(x) := \inf \{\lambda \ge 0 : \lambda x+(1-\lambda )Tx \in C \}$$ . Then, for a fixed $$x_0 \in C$$ and for $$\begin{aligned}{\alpha _0} = \max \left\{ {\frac{1}{2},h({x_0})} \right\}\end{aligned}$$ , Krasnoselskii-Mann algorithm is defined by $$x_{n+1}=\alpha _n+(1-\alpha _n)Tx_n,$$ where $$\alpha _{n+1}=\max \{\alpha _n, h(x_{x_{n+1}})\}$$ . Recently, Colao and Marino (Fixed Point Theory Appl 2015:39, 2015) have proved both weak and strong convergence theorems when C is a strictly convex set and T is an inward mapping. Meanwhile, they proposed a open question for a countable family of non-self nonexpansive mappings. In this article, authors will give an answer and will prove the further generalized results with the examples to support them. [ABSTRACT FROM AUTHOR]

The aim of this paper is to exhibit a novel two-step iterative algorithm named PV algorithm to determine the fixed points of weak contractions in Banach spaces. Data dependence result is also obtained. It is proved that this PV iterative algorithm converges strongly to the fixed point of weak contractions. This iteration is almost stable for weak contraction. Furthermore, it is proved that rate of convergence of the PV iterative algorithm is faster than Picard, Ishikawa, Mann, S,normal-S, Varat, and F* algorithms. Examples are also given to support the main result. The results of this paper are original and will further enrich the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified Halpern inertial method for approximating solutions of split feasibility problem and fixed point problem of Bregman strongly nonexpansive mappings in the framework of p-uniformly convex and uniformly smooth real Banach spaces. We establish a strong convergence result for the sequence generated by our iterative scheme under some mild conditions without the computation of the operator norm. We state some consequences and present some examples to show the efficiency and implementation of our proposed method. The result discussed in this paper extends and generalizes many recent results in this direction. Our result extends and complements some related results in literature. [ABSTRACT FROM AUTHOR]

In this paper, we study a variational inequality problem which is defined over the the solution of multiple set split common fixed point problem for a finite family of generalized demimetric mappings. We present an accelerated steepest descent algorithm for solving this problem and then establish strong convergence of the proposed method under standard and mild conditions. We apply our proposed algorithm to solve some problems, including variational inequalities over multiple set split feasibility problem, variational inequalities over multiple set split zero point problem and the convex optimization problem. [ABSTRACT FROM AUTHOR]

We devise an iterative algorithm incorporating inertial techniques to approximate the shared solution of a generalized equilibrium problem, a fixed point problem for a finite family of relatively nonexpansive multivalued mappings, and a variational inequality problem. Our discussion encompasses the strong convergence of the proposed algorithm and highlights specific outcomes derived from our theorem. Additionally, we provide a computational analysis to underscore the significance of our findings and draw comparisons. The results presented in this paper serve to extend and unify numerous previously established outcomes in this particular research domain. [ABSTRACT FROM AUTHOR]

In this paper, we study a modified projection algorithm combined with an inertial effect whose sequence approximates a common solution of variational inequality problems and split fixed point problems (VSFPP) involving a nonexpansive map, Lipschitz pseudocontractive map and a Lipschitz pseudo-monotone map in real Hilbert spaces. Strong convergence of the sequence generated by the algorithm is obtained without prior knowledge of the Lipschitz constant of the Lipschitz pseudo-monotone map used in this paper. Also, the theorem proved is applied to approximate solutions of some nonlinear analysis and optimization problems. Numerical examples are also given to study the effect of the inertial terms in our algorithm in comparison with some related recent algorithms without inertial term. Finally, the theorem proved improves, extends and unifies some related recent results in the literature. [ABSTRACT FROM AUTHOR]

Let A and B be two nonempty subsets of a normed space X, and let T : A ∪ B → A ∪ B be a cyclic (resp., noncyclic) mapping. The objective of this paper is to establish weak conditions on T that ensure its relative nonexpansiveness. The idea is to recover the results mentioned in two papers by Matkowski (Banach J. Math. Anal. 2:237–244, 2007; J. Fixed Point Theory Appl. 24:70, 2022), by replacing the nonexpansive mapping f : C → C with a cyclic (resp., noncyclic) relatively nonexpansive mapping to obtain the best proximity pair. Additionally, we provide an application to a functional equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss some results concerning the demiclosedness principle of generalized, nonexpansive mappings in uniformly convex spaces. Further, we present some new fixed-point theorems for generalized nonexpansive mappings in different settings of Banach spaces. [ABSTRACT FROM AUTHOR]

In this research paper, we propose a new iterative algorithm for finding a common solution to fixed point problems of demicontractive mapping and variational inequality problems which involves monotone and Lipschitz continuous operators in the framework of real Hilbert spaces. We incorporate a viscosity iterative technique, using subgradient extragradient method, we prove under standard assumptions that the iterative sequence generated from our algorithm strongly converges to the solution set, assuming the solution set is consistent. Furthermore, we adopt a self-adaptive stepsize that is being generated at each iteration, which is independent of the Lipschitz constant of the singled-valued operator. Our result is an improvement and an extension of many results in this direction. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze four inertial algorithms for solving fixed point problems of Bregman quasi-nonexpansive mappings in the framework of reflexive Banach spaces. Our first two algorithms, we propose inertial-like methods based on Mann-type and Halpern-type iterations, and in the others, we propose relaxed inertial-like methods based on Mann-type and Halpern-type iterations. The weak and strong convergence of the algorithms are established under some appropriate conditions on the parameters. As an application, we utilize our main results to find a zero of the sum of Bregman inverse strongly monotone mappings and maximal monotone operators in real reflexive Banach spaces. Also, we provide several numerical experiments to show the convergence behaviour of our algorithms in both finite-dimensional and infinite-dimensional spaces. Finally, we further utilize our algorithms to numerically solve the data classification problems of lung cancer. [ABSTRACT FROM AUTHOR]

This paper attempts to prove the Lipschitz continuity of the resolvent operator associated with a (P , η) -accretive mapping and compute an estimate of its Lipschitz constant. This is done under some new appropriate conditions that are imposed on the parameter and mappings involved in it; with the goal of approximating a common element of the solution set of a system of generalized variational-like inclusions and the fixed point set of a total asymptotically nonexpansive mapping in the framework of real Banach spaces. A new iterative algorithm based on the resolvent operator technique is proposed. Under suitable conditions, we prove the strong convergence of the sequence generated by our proposed iterative algorithm to a common element of the two sets mentioned above. The final section is dedicated to investigating and analyzing the notion of a generalized H(.,.)-accretive mapping introduced and studied by Kazmi et al. (Appl Math Comput 217:9679–9688, 2011). In this section, we provide some comments based on the relevant results presented in their work. [ABSTRACT FROM AUTHOR]

In this paper, we propose two new self-adaptive relaxed CQ algorithms to solve the split feasibility problem with multiple output sets, which involve the computation of projections onto half-spaces instead of the computation onto the closed convex sets. Our proposed algorithms with selection technique reduce the computation of projections. And then, as a generalization, we construct two new algorithms to solve the variational inequalities over the solution set of split feasibility problem with multiple output sets. More importantly, strong convergence of all proposed algorithms is proved under suitable conditions. Finally, we conduct numerical experiments to show the efficiency and accuracy of our algorithms compared to some existing results. [ABSTRACT FROM AUTHOR]

This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem. [ABSTRACT FROM AUTHOR]

The notion of approximate fixed point sequence, emphasized in Chidume (Geometric properties of Banach spaces and nonlinear iterations. Lecture Notes in Mathematics, 1965. Springer-Verlag London, Ltd., London, 2009), is a very useful tool in proving convergence theorems for fixed point iterative schemes in the class of nonexpansive-type mappings. In the present paper, our aim is to present simple and unified alternative proofs of some classical fixed point theorems emerging from Banach contraction principle, by using a technique based on the concepts of approximate fixed point sequence and graphic contraction. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an interlacing condition on the elements of a family of operators that allows us to gather together a number of results on fixed points and common fixed points for single and families of mappings defined on metric spaces. The innovative concept studied here deals with nonexpansivity conditions with respect to orbits and under assumptions that only depend on the features of the closed balls of the metric space. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two types of weak enriched contractions, namely weak enriched F -contraction, weak enriched F ′ -contraction, and k-fold averaged mapping based on Kirk's iterative algorithm of order k. The types of contractions introduced herein unify, extend, and generalize several existing classes of enriched and weak enriched contraction mappings. Moreover, K-fold averaged mappings can be viewed as a generalization of the averaged mappings and double averaged mappings. We then prove the existence of a unique fixed point of the k-fold averaged mapping associated with weak enriched contractions introduced herein. We study necessary conditions that guarantee the equality of the sets of fixed points of the k-fold averaged mapping and weak enriched contractions. We show that an appropriate Kirk's iterative algorithm can be used to approximate a fixed point of a k-fold averaged mapping and of the two weak enriched contractions. We also study the well-posedness, limit shadowing property, and Ulam–Hyers stability of the k-fold averaged mapping. We provide necessary conditions that ensure the periodic point property of each illustrated weak enriched contraction. Some examples are presented to show that our results are a potential generalization of the comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new algorithm of multi-step inertial form for approximating a common solution of Split Generalized Mixed Equilibrium Problem (SGMEP) and Minimization Problem (MP) in real Hilbert spaces. Motivated by the multi-step inertial technique, we incorporate the multi-step inertial term to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity and lower semicontinuity of the SGMEP associated mappings, we establish the strong convergence of the scheme. Some numerical examples are presented to illustrate the performance of our method as well as comparing it with the non-inertial version. [ABSTRACT FROM AUTHOR]

In this article, we introduce the multiple-sets split equality variational inequality problem which includes the split feasibility problem, split variational inequality problem, split equality problem and multiple-sets split variational inequality problem to mention a few. Also, we prove a strong convergence theorem for approximation the solution of multiple-sets split equality variational inequality problem and fixed point problem of multi-valued quasi-nonepansive mappings in real Hilbert spaces using a modified Halpern iterative algorithm. The iterative algorithm employed in this paper is designed in such a way that its implementation does not require the estimation of the operator norms. Lastly, we present some consequences and a numerical example to illustrate the performance of our main result. Our result extends and complements many related results in the literature. [ABSTRACT FROM AUTHOR]

Cyclic mappings describe fixed paths for which each point is sequentially transmitted from one set to another. Cyclic mappings satisfying certain cyclic contraction conditions have been used to obtain the best proximity points, which constitute a suitable framework for the mirror reflection model. Alternative contraction mappings introduced by Chen (Symmetry 11:750, 2019) built a new model containing several mirrors in which the light reflected from a mirror does not go to the next mirror sequentially, and its path may diverge to any other mirror. The aim of this paper is to present a new variant of alternative contraction called alternative p-contraction and study its properties. The best proximity point result for such contractions under the alternative UC property is proved. An example to support the result proved herein is provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative process for approximating common fixed points of two non-self mappings in the setting of CAT(0) spaces. Then we establish Δ -convergence and strong convergence results for two nonexpansive non-self mappings under appropriate conditions. Moreover, we establish strong convergence theorems for approximating common fixed points of two Lipschitz quasi-nonexpansive non-self mappings under some additional conditions. Our results improve, complement and unify most of the results in the literature. [ABSTRACT FROM AUTHOR]

A modular string averaging procedure (MSA, for short) for a finite number of operators was first introduced by Reich and Zalas in 2016. The MSA concept provides a flexible algorithmic framework for solving various feasibility problems such as common fixed point and convex feasibility problems. In 2001 Bauschke and Combettes introduced the notion of coherence and applied it to proving weak and strong convergence of many iterative methods. In 2019 Barshad, Reich and Zalas proposed a stronger variant of coherence which provides a more convenient sufficient convergence condition for such methods. In this paper we combine the ideas of both modular string averaging and coherence. Focusing on extending the above MSA procedure to an infinite sequence of operators with admissible controls, we establish strong coherence of its output operators. Various applications of these concepts are presented with respect to weak and strong convergence. They also provide important generalizations of known results, where the weak convergence of sequences of operators generated by the MSA procedure with intermittent controls was considered. [ABSTRACT FROM AUTHOR]

This paper studies modified extragradient methods with inertial extrapolation step and self-adaptive step-sizes for solving equilibrium problems in real Hilbert spaces. Strong convergence results are obtained under the assumption that the bifunction is pseudomonotone and satisfies the Lipchitz-type condition. Our method of proof is of independent interest and different from the recent arguments used in related papers on strong convergence methods with inertial steps for equilibrium problems. Numerical implementations and comparisons are given to support the theoretical findings. [ABSTRACT FROM AUTHOR]