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This paper explores fixed points for both contractive and non-contractive mappings in traditional b-metric spaces, preordered b-metric spaces, and random b-metric spaces. Our findings provide insights into the behavior of mappings under various constraints and extend our approach to include coincidence and common fixed-point theorems in these spaces. We present new examples and graphical representations for the first time, offering novel results and enhancing several related findings in the literature, while broadening the scope of earlier works of Ran and Reurings, Nieto and Rodríguez-López, Górnicki, and others. [ABSTRACT FROM AUTHOR]

This paper explores fixed points for both contractive and non-contractive mappings in traditional b-metric spaces, preordered b-metric spaces, and random b-metric spaces. Our findings provide insights into the behavior of mappings under various constraints and extend our approach to include coincidence and common fixed-point theorems in these spaces. We present new examples and graphical representations for the first time, offering novel results and enhancing several related findings in the literature, while broadening the scope of earlier works of Ran and Reurings, Nieto and Rodríguez-López, Górnicki, and others.

In this paper, we introduce a new inertial type algorithm with self -- adaptive step -- size technique for approximating a common element of the set of solutions of pseudomonotone variational inequality problem and the set of common fixed point of a finite family of generic generalized nonspreading mappings in uniformly smooth and 2 -- uniformly convex real Banach space. Furthermore, we prove a strong convergence theorem of our algorithm without prior knowledge of the Lipschitz constant of the operator under some mild assumptions. We also give numerical examples to illustrate the performance of our algorithm. Our result generalize and improve many existing results in the literature. [ABSTRACT FROM AUTHOR]

In this work, we give necessary and sufficient conditions for the existence of solutions to the variational inequality problem: find x 0 ∈ K such that ⟨ F (x 0) , y - x 0 ⟩ ≥ 0 , for every y ∈ K , where K is a nonempty closed convex subset of a real Hilbert space H and F : K → H is a monotone and continuous operator. These characterizations are given in terms of approximate fixed points sequences, as well as by Leray–Schauder condition. We apply our obtained results in a constrained convex minimization problem. [ABSTRACT FROM AUTHOR]

We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1n∑i=0n−1Ti(x)$ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard’s iteration {Tn(x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

In this article, we introduced an iterative scheme for finding a common element of the set of fixed points of a multi-valued hemicontractive-type mapping, the set of common solutions of a finite family of split equilibrium problems and the set of common solutions of a finite family of variational inequality problems in real Hilbert spaces. Moreover, the sequence generated by the proposed algorithm is proved to be strongly convergent to a common solution of these three problems under mild conditions on parameters. Our results improve and generalize many well-known recent results existing in the literature in this field of research.

In this paper, we first introduce a monotone mapping and its resolvent in general metric spaces.Then, we give two new iterative methods by combining the resolvent method with Halpern's iterative method and viscosity approximation method for finding a fixed point of monotone mappings and a solution of variational inequalities. We prove convergence theorems of the proposed iterations in CAT(0) metric spaces.

Ran and Reurings (Proc Am Math Soc 132(5):1435–1443, 2003) extended the Banach contraction principle to the setting of partially ordered metric spaces and recently Proinov (J Fixed Point Theory Appl 22:21, 2020) extended and unified many earlier fixed point theorems. In this paper we will present analogous results for the significantly wider class of mappings on preordered metric spaces. We give non-trivial examples of Kannan-type mappings. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized explicit algorithm for approximating the common solution of generalized split feasibility problem and the fixed point of demigeneralized mapping in uniformly smooth and 2-uniformly convex real Banach spaces. The generalized split feasibility problem is a general mathematical problem in the sense that it unifies several mathematical models arising in (symmetry and non-symmetry) optimization theory and also finds many applications in applied science. We designed the algorithm in such a way that the convergence analysis does not need a prior estimate of the operator norm. More so, we establish the strong convergence of our algorithm and present some computational examples to illustrate the performance of the proposed method. In addition, we give an application of our result for solving the image restoration problem and compare with other algorithms in the literature. This result improves and generalizes many important related results in the contemporary literature.

In this manuscript we introduce a new class of monotone generalized nonexpansive mappings and establish some weak and strong convergence theorems for Krasnoselskii iteration in the setting of a Banach space with partial order. We consider also an application to the space L1([0,1]). Our results generalize and unify the several related results in the literature.

The purpose of this note is to discuss the recent paper of Espínola and Wiśnicki about the fixed point theory of monotone nonexpansive mappings. In their work, it is claimed that most of the fixed point results of this class of mappings boil down to the classical Knaster-Tarski fixed point theorem. We will show that their approach is very restrictive and fail to have any meaningful usefulness in applications.

Recently, researchers are showing more interest on both modular vector spaces and modular function spaces. Looking at the number of results it is pertinent to say that, exploration in this direction especially in the area of fixed point theory and applications is still ongoing, many good results can still be unveiled. As a contribution from our part, we study some fixed point results in modular vector spaces associated with order relation. As an application, we were able to study the existence of fixed point(s) of both depolarizing quantum operation and Markov operators through modular functions/modular spaces. The awareness on the importance of quantum theory and Economics globally were the sole motivations of the application choices in our work. Our work complement the existing results. In fact, it adds to the number of application areas that modular vector/function spaces covered. [ABSTRACT FROM AUTHOR]

We present some new coincidence fixed point theorems for generalized multi-valued weak L-contraction mappings. Our outcomes extend several recent results in the framework of complete metric spaces endowed with a graph. Two illustrative examples are included and some consequences are derived. [ABSTRACT FROM AUTHOR]

A coincidence point of a pair of mappings is an element, at which these mappings take on the same value. Coincidence points of mappings of partially ordered spaces were studied by A.V. Arutyunov, E.S. Zhukovskiy, and S.E. Zhukovskiy (see Topology and its Applications, 2015, V. 179, No. 1, p. 13–33); they proved, in particular, that an orderly covering mapping and a monotone one, both acting from a partially ordered space to a partially ordered one, possess a coincidence point. In this paper, we study the existence of a coincidence point of a pair of mappings that act from a partially ordered space to a set, where no binary relation is defined, and, consequently, it is impossible to define covering and monotonicity properties of mappings. In order to study the mentioned problem, we define the notion of a "quasi-coincidence" point. We understand it as an element, for which there exists another element, which does not exceed the initial one and is such that the value of the first mapping at it equals the value of the second mapping at the initial element. It turns out that the following condition is sufficient for the existence of a coincidence point: any chain of "quasi-coincidence" points is bounded and has a lower bound, which also is a "quasi-coincidence" point. We give an example of mappings that satisfy the above requirements and do not allow the application of results obtained for coincidence points of orderly covering and monotone mappings. In addition, we give an interpretation of the stability notion for coincidence points of mappings that act in partially ordered spaces with respect to their small perturbations and establish the corresponding stability conditions. [ABSTRACT FROM AUTHOR]

Let (X , ∥ ⋅ ∥) {(X,\lVert\,\cdot\,\rVert)} be a Banach space. Let C be a nonempty, bounded, closed and convex subset of X and let T : C → C {T:C\rightarrow C} be a G-monotone nonexpansive mapping. In this work, it is shown that the Mann iteration sequence defined by x n + 1 = t n ⁢ T ⁢ (x n) + (1 - t n) ⁢ x n , n = 1 , 2 , ... , x_{n+1}=t_{n}T(x_{n})+(1-t_{n})x_{n},\quad n=1,2,\dots, proves the existence of a fixed point of G-monotone nonexpansive mappings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the monotone Caristi inward mappings. As an example, we show that monotone inward contraction mappings are monotone Caristi inward mappings. A general fixed point theorem for such mappings is given. A mutlivalued version of these results is also introduced. [ABSTRACT FROM AUTHOR]

In this manuscript we introduce a new class of monotone generalized nonexpansive mappings and establish some weak and strong convergence theorems for Krasnoselskii iteration in the setting of a Banach space with partial order. We consider also an application to the space L1([0,1]). Our results generalize and unify the several related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce dualistic contractive mappings and use such mappings to prove some fixed point theorems. The results extend various comparable results existing in the literature. Moreover, we give examples that show the superiority and effectiveness of our results among corresponding fixed point theorems in partial metric spaces. [ABSTRACT FROM AUTHOR]

Let C be a nonempty, bounded, closed, and convex subset of a Banach space X and T : C → C be a monotone asymptotically nonexpansive mapping. In this paper, we investigate the existence of fixed points of T. In particular, we establish an analogue to the original Goebel and Kirk’s fixed point theorem for asymptotically nonexpansive mappings. [ABSTRACT FROM AUTHOR]

In this manuscript, we discuss the latest fixed point results of monotone mappings. The fixed point theory of such mappings has seen a tremendous interest in the last decade since the publication of Ran and Reurings paper in 2004. Fixed point theory for monotone mappings is useful and has many applications. For example when one is looking for a positive or negative solution, the use of the classical fixed point results is not adapted in this situation. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and investigate a more generalized hybrid shrinking projection algorithm for finding a common solution for a system of generalized mixed equilibrium problems. A accelerated strong convergence theorem of common solutions is established in the framework of a non-uniformly convex Banach space. These new results improve and extend the previously known ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of generalized nonexpansive mappings. Some new fixed point theorems for these mappings are obtained. [ABSTRACT FROM AUTHOR]

We prove the existence of fixed points of monotone ρ-nonexpansive mappings in ρ-uniformly convex modular function spaces. This is the modular version of Browder and Gohde fixed point theorems for monotone mappings. We also discuss the validity of this result in modular function spaces where the modular is uniformly convex in every direction. This property has never been considered in the context of modular spaces. [ABSTRACT FROM AUTHOR]

The purpose of this note is to give a natural approach to the extensions of the Banach contraction principle in metric spaces endowed with a partial order, a directed graph or a binary relation in terms of extended quasi-metric. This novel approach is new and may open the door to other new fixed point theorems. The case of multivalued mappings is also discussed and an analogue result to Nadler's fixed point theorem in extended quasi-metric spaces is given. [ABSTRACT FROM AUTHOR]

Let X be a Banach space or a complete hyperbolic metric space. Let C be a nonempty, bounded, closed, and convex subset of X and $T: C \rightarrow C$ be a monotone nonexpansive mapping. In this paper, we show that if X is a Banach space which is uniformly convex in every direction or a uniformly convex hyperbolic metric space, then T has a fixed point. This is the analog to Browder and Göhde's fixed point theorem for monotone nonexpansive mappings. [ABSTRACT FROM AUTHOR]

A system of variational inequality and fixed point problems is considered. Two algorithms have been constructed. Our algorithms can find the minimum norm solution of this system of variational inequality and fixed point problems. [ABSTRACT FROM AUTHOR]

In this work, we discuss the existence of fixed points of monotone nonexpansive mappings defined on partially ordered Banach spaces. This work is a continuity of the previous works of Ran and Reurings, Nieto et al., and Jachimsky done for contraction mappings. As an application, we discuss the existence of solutions to an integral equations. [ABSTRACT FROM AUTHOR]

In this work we define the new concept of monotone pointwise contraction mappings in Banach and metric spaces. Then we prove the existence of fixed points of such mappings. [ABSTRACT FROM AUTHOR]

In this work, we give a new definition of G-monotone pointwise contraction mappings in metric spaces endowed with a graph. Then we obtain sufficient conditions for the existence of a fixed point for such mappings. The proofs are based on the crucial inequality (GK). [ABSTRACT FROM AUTHOR]

Let the set $C \subset L_{1}([0,1])$ be nonempty, convex and compact for the convergence almost everywhere and $T: C \rightarrow C$ be a monotone nonexpansive mapping. In this paper, we study the behavior of the Krasnoselskii-Ishikawa iteration sequence $\{f_{n}\}$ defined by $f_{n+1} = \lambda f_{n} + (1-\lambda)T(f_{n})$, $n = 1, 2, \ldots$ , $\lambda\in(0,1)$. Then we prove a fixed point theorem for these mappings. Our result is new and was never investigated. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to address a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. Indeed, if A and B are non-void subsets of a partially ordered set that is equipped with a metric, and S is a non-self mapping from A to B, this paper scrutinizes the existence of an optimal approximate solution, called a best proximity point of the mapping S, to the operator equation Sx = x where S is a continuous, proximally monotone, ordered proximal contraction. Further, this paper manifests an iterative algorithm for discovering such an optimal approximate solution. As a special case of the result obtained in this article, an interesting fixed point theorem on partially ordered sets is deduced. [ABSTRACT FROM AUTHOR]

In this paper, two iterative algorithms are considered for a generalized Ky Fan inequality and a fixed point problem of asymptotically strict pseudocontractions in the intermediate sense. Strong and weak convergence theorems are established in real Hilbert spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping. We introduce a modified hybrid Mann iterative scheme with perturbed mapping which is based on well-known CQ method, Mann iteration method and hybrid (or outer approximation) method. We establish a strong convergence theorem for three sequences generated by this modified hybrid Mann iterative scheme with perturbed mapping. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings. [ABSTRACT FROM AUTHOR]

In this paper, we present a new iterative method for finding a common element of the set of solutions of generalized equilibrium problems and the set of fixed points of a strict pseudocontractive mapping. Furthermore, we prove that the proposed iterative method has strong convergence under some mild conditions imposed on algorithm parameters. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the following minimization problem where is a bounded linear operator, is some constant, is a potential function for , is the set of fixed points of nonexpansive mapping and is the solution set of an equilibrium problem. This paper introduces two new algorithms (one implicit and one explicit) that can be used to find the solution of the above minimization problem. [Copyright &y& Elsevier]

In this article, we present a new iterative method based on the relaxed hybrid method and the extragradient method for finding a common element of the set of solutions of generalized equilibrium problems and the set of fixed points of a strictly pseudocontractive mapping and the set of a general system of variational inequality with inverse strongly monotone mappings in a real Hilbert space. Furthermore, we prove that the proposed iterative method has strong convergence under some mild conditions imposed on algorithm parameters. [ABSTRACT FROM AUTHOR]

We study the convergence of an iterative algorithm for finding common fixed points of finitely many Bregman firmly nonexpansive operators in reflexive Banach spaces. Our algorithm is based on the concept of the so-called shrinking projection method and it takes into account possible computational errors. We establish a strong convergence theorem and then apply it to the solution of convex feasibility and equilibrium problems, and to finding zeroes of two different classes of nonlinear mappings. [ABSTRACT FROM AUTHOR]

In this paper, we devote to find the solution of the following quadratic minimization problem where Ω is the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping and the solution set of some variational inequality. In order to solve the above minimization problem, we first construct an implicit algorithm by using the projection method. Further, we suggest an explicit algorithm by discretizing this implicit algorithm. Finally, we prove that the proposed implicit and explicit algorithms converge strongly to a solution of the above minimization problem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a weak relatively nonexpansive mapping, the set of solutions of the variational inequality for the monotone mapping and the set of solutions of an equilibrium problem in a 2-uniformly convex and uniformly smooth Banach space. Then we show that the iterative sequence converges strongly to a common element of the three sets. In this paper, we also give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping in Banach space . [Copyright &y& Elsevier]

In this paper, we introduce two iterative schemes by extragradient-like methods for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in a Hilbert space. We obtain a strong convergence theorem and a weak convergence theorem for the sequences generated by these processes. Based on these two results, we also get some new and interesting results. The results in this paper generalize and extend some well-known strong convergence theorems and weak convergence theorems in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate the problem of finding a common element of the set of fixed points of a non-expansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping. We introduce a hybrid Mann iterative scheme with perturbed mapping which is based on the well-known Mann iteration method and hybrid (or outer approximation) method. We establish a strong convergence theorem for three sequences generated by this hybrid Mann iterative scheme with perturbed mapping. Utilizing this theorem, we also construct an iterative process for finding a common fixed point of two mappings, one of which is non-expansive and the other taken from the more general class of Lipschitz pseudocontractive mappings. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive. [Copyright &y& Elsevier]

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points F( S) of a nonexpansive mapping S and the set of solutions Ω A of the variational inequality for a monotone, Lipschitz continuous mapping A. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of $${F(S)\cap\Omega_{A}}$$. As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of an infinite (a finite) family of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. We obtain some weak convergence theorems for the sequences generated by these processes in Hilbert spaces. The results in this paper generalize, extend and unify some well-known weak convergence theorems in the literature. [Copyright &y& Elsevier]

In this paper, we introduce an iterative scheme using an extragradient method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a nonexpansive mapping and the set of the variational inequality for a monotone, Lipschitz-continuous mapping. We obtain a weak convergence theorem for three sequences generated by this process. Based on this result, we also obtain several interesting results. The results in this paper generalize and extend some well-known weak convergence theorems in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new iterative scheme based on the extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz continuous mapping is proposed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Applications to optimization problems are given. [Copyright &y& Elsevier]

In this paper, we propose a generalized explicit algorithm for approximating the common solution of generalized split feasibility problem and the fixed point of demigeneralized mapping in uniformly smooth and 2-uniformly convex real Banach spaces. The generalized split feasibility problem is a general mathematical problem in the sense that it unifies several mathematical models arising in (symmetry and non-symmetry) optimization theory and also finds many applications in applied science. We designed the algorithm in such a way that the convergence analysis does not need a prior estimate of the operator norm. More so, we establish the strong convergence of our algorithm and present some computational examples to illustrate the performance of the proposed method. In addition, we give an application of our result for solving the image restoration problem and compare with other algorithms in the literature. This result improves and generalizes many important related results in the contemporary literature. [ABSTRACT FROM AUTHOR]