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We obtain results on the existence and approximation of fixed points of enriched contractions in quasi-Banach spaces and thus extend the previous results for enriched contractions defined on Banach spaces [Berinde, V.; Păcurar, M. Approximating fixed points of enriched contractions in Banach spaces. J. Fixed Point Theory Appl. 22 (2020), no. 2, Paper No. 38, 10 pp.]. The theoretical results are illustrated by means of an appropriate example of enriched contraction on a quasi-Banach space which is not a Banach space and thus show that our new results are effective generalizations of the previous ones in literature. [ABSTRACT FROM AUTHOR]

A very impressive research work has been devoted in the last two decades to obtaining fixed point theorems in quasimetric spaces (also called b-metric spaces). Some incorrect and incomplete references with respect to the early developments on fixed point theory in b-metric spaces are though perpetually taking over from the existing publications to the new ones. Starting from this fact, our main aim in this note is threefold: (1) to briefly survey the early developments in the fixed point theory on quasimetric spaces (b-metric spaces); (2) to collect some relevant bibliography related to this topic; (3) to discuss some other aspects of current interest in the fixed point theory on quasimetric spaces (b-metric spaces). [ABSTRACT FROM AUTHOR]

In this paper, we propose a strongly convergent simultaneous cutter method for finding the minimal norm solution over the intersection of fixed point sets of cutters. The proposed method is the combination of the simultaneous cutter method and the strongly variance of Krasnosel'skiĭ-Mann method. We show a strong convergence result of the sequence generated by the proposed method to the unique minimal norm solution. We finally present the numerical experiments on the minimal norm solution over a finite number of linear feasibility problem. [ABSTRACT FROM AUTHOR]

In this paper we provide a new approach in the study of a variational-hemivariational inequality in Hilbert space, based on the theory of maximal monotone operators and the Banach fixed point theorem. First, we introduce the inequality problem we are interested in, list the assumptions on the data and show that it is governed by a multivalued maximal monotone operator. Then, we prove that solving the variational-hemivariational inequality is equivalent to finding a fixed point for the resolvent of this operator. Based on this equivalence result, we use the Banach contraction principle to prove the unique solvability of the problem. Moreover, we construct the corresponding Picard, Krasnoselski and Mann iterations and deduce their convergence to the unique solution of the variational-hemivariational inequality. [ABSTRACT FROM AUTHOR]

In this paper some existence and stability results for cyclic graphical contractions in complete metric spaces are given. An application to a coupled fixed point problem is also derived. [ABSTRACT FROM AUTHOR]

We prove that the convergence theorems for Mann iteration used for approximation of the fixed points of demicontractive mappings in Hilbert spaces can be derived from the corresponding convergence theorems in the class of quasi-nonexpansive mappings. Our derivation is based on an important auxiliary lemma (Lemma 3.2), which shows that if T is k-demicontractive, then for any λ ∈ (0, 1 − k), Tλ is quasi-nonexpansive. In this way we obtain a unifying technique of proof for various well known results in the fixed point theory of demicontractive mappings. We illustrate this reduction technique for the case of two classical convergence results in the class of demicontractive mappings: [Măruşter, Şt. The solution by iteration of nonlinear equations in Hilbert spaces. Proc. [ABSTRACT FROM AUTHOR]

In this paper we give some fixed point theorems for nonself generalized contractions on a large Kasahara space, which generalize some results given by I.A. Rus and M.-A. S¸erban (I.A. Rus, M.-A. S¸erban, Some fixed point theorems for nonself generalized contractions, Miskolc Math. Notes, 17(2016), no.2, 1021-1031) and by S. Reich and A.J. Zaslavski (S. Reich, A.J. Zaslavski, A note on Rakotch contractions, Fixed Point Theory, 9(2008), no.1, 267-273) in complete metric spaces. We prove our results without using the completeness of the metric structure. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a common fixed point theory for a pair of multivalued operators in complete metric spaces. Existence and stability results for the common fixed point problem will be discussed. The approach is based on some fixed point results for multivalued graphical contractions and for multivalued operators of Feng-Liu type. Our results extend to the multivalued case some recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we present a Tseng-type self-adaptive algorithm for solving a variational inequality and a fixed point problem involving pseudomonotone and pseudocontractive operators in Hilbert spaces. A weak convergent result for such algorithm is proved under a weaker assumption than sequentially weakly continuous imposed on the pseudomonotone operator. Some corollaries are also included. [ABSTRACT FROM AUTHOR]

In this work, we analyse the class of strictly pseudocontractive mappings in general metric spaces by providing a comprehensive and appropriate definition of a strictly pseudocontractive mapping, which serves as a natural extension of the existing notion. Moreover, we establish its various characterizations and explore several significant properties of these mappings in relation to fixed point theory in CAT(0) spaces. Specifically, we establish that these mappings are Lipschitz continuous, satisfying the demiclosedness-type property, and possessing a closed convex fixed point set. Furthermore, we show that the fixed points of the mappings can be effectively approximated using an iterative scheme for fixed points of nonexpansive mappings. The results in this work contribute to a deeper understanding of strictly pseudocontractive mappings and their applicability in the context of fixed point theory in metric spaces. [ABSTRACT FROM AUTHOR]

In this work, two inertial algorithms for approximating common elements of the sets of solutions of three important problems are constructed. The first problem is a generalized mixed equilibrium one involving relaxed monotone mapping, the second is a zero problem of inverse strongly monotone mappings, while the third one is a fixed point problem of a family of relatively nonexpansive mappings. The first algorithm is a shrinking projection type for a common solution of all the three problems. The second is a generalized Alber projection free method for the second and the third problems. Each of the devised algorithms uses the conjugate gradient-like direction, which allows it to accelerate its iterates toward a solution of the problems. The strong convergence theorem for each of the algorithms is formulated and proved in a real 2 - uniformly convex and uniformly smooth Banach space. Additionally, the applications of our algorithms to convex optimization problems and image recovery problems are studied. The advantages and computational efficiency of our methods are analyzed based on their numerical performance in comparison to some of the existing and recently proposed methods using numerical example. [ABSTRACT FROM AUTHOR]

Let (X, //⋅//) be a Banach space and T : A → X a contraction mapping, where A ⊂ X is a closed set. Consider a sequence {xn} ⊂ A and define the sequence {yn} ⊂ X, by yn = xn +T xσ(n), where {σ(n)} is a sequence of natural numbers. We highlight some general conditions so that the two sequences {xn} and {yn} are simultaneously convergent. Both cases: 1) σ(n) < n, for all n, and 2) σ(n) ≥ n, for all n, are discussed. In the first case, a general Picard iteration procedure is inferred. The results are then extended to sequences of mappings and some appropriate applications are also proposed. [ABSTRACT FROM AUTHOR]

We show that the choice of stepsize in Korpelevich’s extragradient algorithm is sharp, while the choice of stepsize in Popov’s extragradient algorithm can be relaxed. We also extend Korpelevich’s extragradient algorithm and Popov’s extragradient algorithm (with larger stepsize) to the infinite-dimensional Hilbert space framework, with weak convergence. [ABSTRACT FROM AUTHOR]

We apply the concept of quasilinearization to introduce some enriched classes of Banach contraction mappings and analyse the fixed points of such mappings in the setting of Hadamard spaces. We establish existence and uniqueness of the fixed point of such mappings. To approximate the fixed points, we use an appropriate Krasnoselskij-type scheme for which we establish ∆ and strong convergence theorems. Furthermore, we discuss the fixed points of local enriched contractions and Maia-type enriched contractions in Hadamard spaces setting. In addition, we establish demiclosedness-type property of enriched nonexpansive mappings. Finally, we present some special cases and corresponding fixed point theorems. [ABSTRACT FROM AUTHOR]

We study the space of all continuous and increasing self-mappings of the real interval [0, 1] equipped with the topology of uniform convergence. In particular, we show that most such functions have at least two different fixed points. [ABSTRACT FROM AUTHOR]

We give some extensions of the beautiful 1968 fixed point theorem of Maia [Maia, M. G. Un'osservazione sulle contrazioni metriche. (Italian) Rend. Sem. Mat. Univ. Padova 40 (1968), 139-143] to three classes of enriched contractive mappings in Banach spaces: enriched contractions, Kannan enriched contractions and Ć irić-Reich-Rus contractions. [ABSTRACT FROM AUTHOR]

We prove Δ-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. [ABSTRACT FROM AUTHOR]