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In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition 0 < κ < 1 2 , as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is E = l p , 2 ≤ p < ∞ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition 0 < κ ≤ 1 2 , then E is necessarily l 2 , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented. [ABSTRACT FROM AUTHOR]

In this note we correct an error in the paper (Ariza-Ruiz et al. in J Optim Theory Appl 167:409–429, 2015). [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]

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]

We explore the intermixed method for finding a common element of the intersection of the solution set of a mixed variational inequality and the fixed point set of a nonexpansive mapping. We point out that Khuangsatung and Kangtunyakarn's statement [J. Inequal. Appl. 2023:1, 2023] regarding the resolvent utilized in their paper is not correct. To resolve this gap, we employ the epigraphical projection and the product space approach. In particular, we obtain a strong convergence theorem with a weaker assumption. [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 study the generalized multiple-set split feasibility problem including the common fixed-point problem for a finite family of generalized demimetric mappings and the monotone inclusion problem in 2-uniformly convex and uniformly smooth Banach spaces. We propose an inertial Halpern-type iterative algorithm for obtaining a solution of the problem and derive a strong convergence theorem for the algorithm. Then, we apply our convergence results to the convex minimization problem, the variational inequality problem, the multiple-set split feasibility problem and the split common null-point problem in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove strong convergence theorem of the general inertial Mann–Halpern algorithm for nonexpansive mappings in the setting of Hilbert spaces. We also prove weak convergence theorem of the general inertial Mann algorithm for k-strict pseudo-contractive mappings in the setting of Hilbert spaces. These convergence results extend and generalize some existing results in the literature. Finally, we provide examples to verify our main results. [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]

In this paper, we introduce an intermixed algorithm with viscosity technique for finding a common solution of the combination of mixed variational inequality problems and the fixed-point problem of a nonexpansive mapping in a real Hilbert space. Moreover, we propose the mathematical tools related to the combination of mixed variational inequality problems in the second section of this paper. Utilizing our mathematical tools, a strong convergence theorem is established for the proposed algorithm. Furthermore, we establish additional conclusions concerning the split-feasibility problem and the constrained convex-minimization problem utilizing our main result. Finally, we provide numerical experiments to illustrate the convergence behavior of our proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solution of the generalized equilibrium problem (GEP) in the framework of an Hadamard manifold. Using the KKM lemma, we prove the existence of solution of the GEP and give the properties of the resolvent function associated with the problem under consideration. Furthermore, we introduce an iterative algorithm for approximating a common solution of the GEP and a fixed point problem. Using the proposed method, we obtain and prove a strong convergence theorem for approximating a solution of the GEP, which is also a fixed point of a nonexpansive mapping under some mild conditions. We give an application of our convergence result to a solution of the convex minimization problem. To illustrate the convergence of the method, we report some numerical experiments. The result in this paper extends the study of the GEP from the linear settings to the Hadamard manifolds. [ABSTRACT FROM AUTHOR]

The study of the zero existence problem for a nonnegative set-valued functional on a metric space is continued. The zero existence problem for a functional related by a certain -continuity condition to a parametric family of -search functionals on an open subset of a metric space is examined. A theorem containing several sufficient conditions for this functional to have zeros is proved. As corollaries of this result, theorems on the existence of coincidence and fixed points are also proved for set-valued mappings related by the -continuity condition to families of set-valued mappings with the property that the existence of coincidence and fixed points in an open subset of a metric space is preserved under parameter variation. For uniformly convex metric spaces, analogs of M. Edelstein's 1972 asymptotic center theorem and M. Frigon's 1996 fixed point theorem for nonexpansive mappings of Banach spaces are obtained and compared with the main results of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iteration process, called multi-valued F-iteration process, for the approximation of fixed points. We introduce a new class of multi-valued generalized nonexpansive mappings satisfying a B γ , μ property. Moreover, we establish certain weak and strong convergence theorems in uniformly convex Banach spaces. We also discuss the stability of the modified F-iteration process. Furthermore, a numerical example is presented to illustrate the superiority of our results. [ABSTRACT FROM AUTHOR]

The paper is devoted to the introduction of a new class of generalized set-valued nonlinear variational-like inequality problems in the setting of Banach spaces. By means of the notion of P- η -proximal mapping, we prove its equivalence with a class of generalized implicit Wiener–Hopf equations and employ the obtained equivalence relationship and Nadler's technique to suggest a new iterative algorithm for finding an approximate solution of the considered problem. The existence of solution and the strong convergence of the sequences generated by our proposed iterative algorithm to the solution of our considered problem are verified. The problem of finding a common element of the set of solutions of a generalized nonlinear variational-like inequality problem and the set of fixed points of a total asymptotically nonexpansive mapping is also investigated. The final section deals with the investigation and analysis of the main results appeared in Kazmi and Bhat (Appl Math Comput 166:164–180, 2005) and some comments relating to them are given. The results presented in this article extend and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, for one thing, we introduce positively expansive system and positively expansive measure for nonautonomous dynamical systems. And we study the relation between positively expansive measure and the maximal cardinalities of separated set. We obtain the following main results. Firstly, we prove that every nonautonomous dynamical system with positively expansive measure implies there are e > 0 and a sequence of (n, e)-separated sets whose cardinalities go to infinite as n → ∞ . Secondly, we point out that every positively expansive system also implies the above properties. Thirdly, for each nonautonomous dynamical system that satisfies the above properties, there is a Borel probability measure such that the measure of the set of stable points of nonautonomous dynamical systems is zero. For another, we introduce positively countably expansive and positively measure expansive for nonautonomous dynamical systems. In addition, we prove that a nonautonomous dynamical system is positively measure expansive if and only if it is positively countably expansive. Finally, we introduce stable points and equicontinuous for a nonautonomous dynamical system and study some relevant properties. This paper primarily generalizes the main results obtained by Morales to nonautonomous dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, we first study G-κ-strictly pseudocontractive mappings and we establish a strong convergence theorem for finding the fixed points of two G-κ-strictly pseudocontractive mappings, two G-nonexpansive mappings, and two G-variational inequality problems in a Hilbert space endowed with a directed graph without the Property G. Moreover, we prove an interesting result involving the set of fixed points of a G-κ-strictly pseudocontractive and G-variational inequality problem and if Λ is a G-κ-strictly pseudocontractive mapping, then I − Λ is a G − (1 − κ) 2 -inverse strongly monotone mapping, shown in Lemma 3.3. In support of our main result, some examples are also presented. [ABSTRACT FROM AUTHOR]

Primal-dual hybrid gradient (PDHG) method is a canonical and popular prototype for solving saddle point problem (SPP). However, the nonlinear coupling term in SPP excludes the application of PDHG on far-reaching real-world problems. In this paper, following the seminal work by Valkonen (Inverse Problems 30, 2014), we devise a variant iterative scheme for solving SPP with nonlinear function by exerting an alternative extrapolation procedure. The novel iterative scheme falls exactly into the proximal point algorithmic framework without any residuals, which indicates that the associated inclusion problem is "nearer" to the KKT mapping induced by SPP. Under the metrically regular assumption on KKT mapping, we simplify the local convergence of the proposed method on contractive perspective. Numerical simulations on a PDE-constrained nonlinear inverse problem demonstrate the compelling performance of the proposed method. [ABSTRACT FROM AUTHOR]

This paper investigates optimal error bounds and convergence rates for general Mann iterations for computing fixed-points of non-expansive maps. We look for iterations that achieve the smallest fixed-point residual after n steps, by minimizing a worst-case bound ‖ x n - T x n ‖ ≤ R n derived from a nested family of optimal transport problems. We prove that this bound is tight so that minimizing R n yields optimal iterations. Inspired from numerical results we identify iterations that attain the rate R n = O (1 / n) , which we also show to be the best possible. In particular, we prove that the classical Halpern iteration achieves this optimal rate for several alternative stepsizes, and we determine analytically the optimal stepsizes that attain the smallest worst-case residuals at every step n, with a tight bound R n ≈ 4 n + 4 . We also determine the optimal Halpern stepsizes for affine non-expansive maps, for which we get exactly R n = 1 n + 1 . Finally, we show that the best rate for the classical Krasnosel'skiĭ–Mann iteration is Ω (1 / n) , and present numerical evidence suggesting that even extended variants cannot reach a faster rate. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative algorithm for approximating a common element of the set of solutions of an equilibrium problem, a common zero of a finite family of monotone operators and the set of fixed points of nonexpansive mappings in Hadamard spaces. We also give numerical examples to solve a nonconvex optimization problem in a Hadamard space to support our main result. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a variant of the proximal point method for obtaining weakly efficient solutions of convex vector optimization problems in real Hilbert spaces, with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. The proposed method is a hybrid scheme that combines proximal point type iterations and projections onto some special halfspaces in order to achieve the strong convergence to a weakly efficient solution. To the best of our knowledge, this is the first time that proximal point type method with strong convergence has been considered in the literature for solving vector/multiobjective optimization problems in infinite dimensional Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper, we propose a viscosity-type and self-adaptive iterative algorithm resulting in strong convergence theorem for finding the common solution of generalized split system of variational inclusion problem. The self-adaptive technique presented in this paper is a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. As an application of our result, we study the strong convergence theorem to approximate the solution of generalized forms of multiple-set split feasibility problem, split system of equilibrium problem and split system of minimization problem. Our result generalize, improve and unify various results in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a simple and easily computable algorithm for finding a common solution to split-equilibrium problems and fixed-point problems in the framework of real Hilbert spaces. New self-adaptive step sizes are adopted for the avoidance of Lipschitz constants that are not practically implemented. Furthermore, an inertial term is incorporated to speed up the rate of convergence, a condition that is very desirable in applications. A strong convergence is obtained under some mild assumptions, which is not limited to the fact that the bifunctions are pseudomonotone operators. This condition is better, weaker, and more general than being strongly pseudomonotone or monotone. Our result improves and extends already announced results in this direction of research. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the convergence theorems in CAT (κ) spaces with k > 0 for total asymptotically nonexpansive mappings which are essentially wider than nonexpansive mappings, asymptotically nonexpansive mapping, and asymptotically nonexpansive mappings in the intermediate sense. Our results generalize, unify, and improve several comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

This paper introduces a triple-adaptive subgradient extragradient process with extrapolation to solve a bilevel split pseudomonotone variational inequality problem (BSPVIP) with the common fixed point problem constraint of finitely many nonexpansive mappings. The problem under consideration is in real Hilbert spaces, where the BSPVIP involves a fixed point problem of demimetric mapping. The proposed rule exploits the strong monotonicity of one operator at the upper level and the pseudomonotonicity of another mapping at the lower level. The strong convergence result for the proposed algorithm is established under some suitable assumptions. In addition, a numerical example is given to demonstrate the viability of the proposed rule. Our results improve and extend some recent developments to a great extent. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of asymptotic measure expansivity for diffeomorphisms to flows on a compact metric space, and prove that there exists an asymptotic measure expansive flow in that space. Then we consider the hyperbolicity of the asymptotic measure expansive vector fields on a closed smooth Riemannian manifold M. More precisely, we prove that any C1 stably asymptotic measure expansive vector field on M satisfy Axiom A without cycles, and it is quasi-Anosov. On the other hand, we show that C1 generically, every asymptotic measure expansive vector field on M satisfies Axiom A without cycles. Moreover, we study the asymptotic measure expansivity of the homoclinic classes which are the delegate of the invariant subsets for given systems, prove that C1 stably and C1 generically, the asymptotic measure expansive homoclinic class is hyperbolic. Furthermore, we consider the hyperbolicity of asymptotic measure expansive divergence-free vector fields for C1 stably and C1 generic point of view. We also apply the our results to the divergence-free vector fields. [ABSTRACT FROM AUTHOR]

The goal of this paper is to develop some fundamental and important nonlinear analysis for single-valued mappings under the framework of p-vector spaces, in particular, for locally p-convex spaces for 0 < p ≤ 1 . More precisely, based on the fixed point theorem of single-valued continuous condensing mappings in locally p-convex spaces as the starting point, we first establish best approximation results for (single-valued) continuous condensing mappings, which are then used to develop new results for three classes of nonlinear mappings consisting of 1) condensing; 2) 1-set contractive; and 3) semiclosed 1-set contractive mappings in locally p-convex spaces. Next they are used to establish the general principle for nonlinear alternative, Leray–Schauder alternative, fixed points for nonself mappings with different boundary conditions for nonlinear mappings from locally p-convex spaces, to nonexpansive mappings in uniformly convex Banach spaces, or locally convex spaces with the Opial condition. The results given by this paper not only include the corresponding ones in the existing literature as special cases, but are also expected to be useful tools for the development of new theory in nonlinear functional analysis and applications to the study of related nonlinear problems arising from practice under the general framework of p-vector spaces for 0 < p ≤ 1 . Finally, the work presented by this paper focuses on the development of nonlinear analysis for single-valued (instead of set-valued) mappings for locally p-convex spaces. Essentially, it is indeed the continuation of the associated work given recently by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022); therein, the attention is given to the study of nonlinear analysis for set-valued mappings in locally p-convex spaces for 0 < p ≤ 1 . [ABSTRACT FROM AUTHOR]

In this paper, an inertial S-iteration iterative process for approximating a common fixed point of a finite family of quasi-Bregman nonexpansive mappings is introduced and studied in a reflexive Banach space. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, extend, and generalize some recent results in the literature. [ABSTRACT FROM AUTHOR]

This paper reports the hyperbolic space version of the iterative scheme F. First, under mild conditions, we obtain the Δ and strong convergence of this scheme in the class of mappings enriched with condition (E). To support these results, and to show the high accuracy of the F iterative scheme in the class of mappings enriched with condition (E), we suggest an appropriate numerical example and show that its F iterative scheme is more effective than some other iterative schemes. After this, we establish strong convergence, of this scheme for the general class of contractive-like operators. An example to support this result is also presented and the efficiency of the F iterative scheme is tested with the other iterative schemes. Finally, the weak w2–stability and data dependency results are established. Our findings are new and improve some recent results of the current literature. [ABSTRACT FROM AUTHOR]

In this paper, using the Bregman distance, we introduce a new projection-type algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points. Then the strong convergence of the sequence generated by the algorithm will be established under suitable conditions. Finally, using MATLAB software, we present a numerical example to illustrate the convergence performance of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction. [ABSTRACT FROM AUTHOR]

This paper proposes, in the framework of Hadamard manifolds, two iterative schemes for approximating a solution of a variational inequality problem involving a nonexpansive mapping with a fixed point set of another nonexpansive mapping as constraint. The first scheme is a modified Halpern iteration and the second is a viscosity-type iteration with a weakly contraction mapping. We also discuss some special cases of the mentioned problem. Numerical examples are provided to illustrate the algorithm's numerical behavior. [ABSTRACT FROM AUTHOR]

This paper first introduces a new iterative method for weak and strong convergence theorem to demonstrate the estimation potential for a fixed point of the cutter and the finite general split feasibility problem. Consequently, the set of fixed points of a quasi-nonexpansive mapping and the finite general split feasibility problem, the constrained minimization problem, and the general constrained minimization problems are proved using our main results. Finally, we give two numerical examples to advocate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study the method of cyclic projections for inconsistent convex feasibility problems in a Hilbert space under the presence of computational errors. We show that our algorithm generates a good approximate solution, if computational errors are bounded from above by a small positive constant. Our main goal is, for a known computational error, to find out what approximate solution can be obtained and how many iterates one needs for this. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. Our first objective is to establish a new equivalence relationship between the graph convergence of a sequence of A ^ -maximal m-relaxed η -accretive mappings and their associated resolvent operators, respectively, to a given A ^ -maximal m-relaxed η -accretive mapping and its associated resolvent operator. We construct an iterative scheme for approximating a common element of the set of solutions of a system of generalized variational-like inclusions and the set of fixed points of a generalized nearly asymptotically nonexpansive mapping in the setting of real Banach spaces. As an application, the obtained equivalence conclusion is employed and under some appropriate conditions, the strong convergence of the sequence generated by our suggested iterative algorithm to a common element of the above-mentioned two sets is verified. The second objective of this paper is to investigate and analyze the concept of H(.,.)-mixed mapping and the relevant results appeared in the literature. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a class of generalized nonexpansive mappings called generalized nearly asymptotically nonexpansive mappings and to show that it properly includes the class of nearly asymptotically nonexpansive mappings. We investigate the problem of approximating a common element of the set of solutions of a system of generalized nonlinear variational-like inclusions involving P-η-accretive mappings and of the set of fixed points of a generalized nearly asymptotically nonexpansive mapping. To this end, we suggest a new iterative algorithm with mixed errors. As an application of the obtained equivalence, we prove the strong convergence and stability of the sequence generated by the proposed iterative algorithm to a common point of the two sets mentioned above. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to prove the mean ergodic theorem for nonexpansive mappings and semigroups in locally compact Hadamard spaces, including finite dimensional Hadamard manifolds. The main tool for proving ergodic convergence is the almost periodicity of orbits of a nonexpansive mapping. Therefore, in the first part of the paper, we study almost periodicity (and as a special case, periodicity) in Hadamard spaces. Then, we prove a mean ergodic theorem for nonexpansive mappings and continuous semigroups of contractions in locally compact Hadamard spaces. Finally, an application to the asymptotic behavior of the first order evolution equation associated to the monotone vector field on Hadamard manifolds is presented. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iterative algorithm to find the common element of set of solutions of a split equilibrium problem and set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. The new method is used to prove the strong convergence for the result of this paper. The result extends the corresponding one in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of positive solutions for nonlinear fourth-order equation with clamped beam boundary conditions u ′ ′ ′ ′ (x) = λ a (x) f (u (x)) for x ∈ (0 , 1) , u (0) = u (1) = u ′ (0) = u ′ (1) = 0 , where λ > 0 is a parameter, a ∈ L 1 (0 , 1) may change sign and f ∈ C ([ 0 , ∞) , [ 0 , ∞)) . We establish some new results of existence of positive solutions to this problem if the nonlinearity f is monotone on [ 0 , ∞) . The proofs of our main results are based upon a monotone iteration technique and the Schauder's fixed point theorem. Finally, an example is presented to illustrate the application of our main results. [ABSTRACT FROM AUTHOR]

The motive of this paper is to propose a new iterative algorithm for evaluating a solution of constrained minimization problem, variational inequality and split feasibility problem and common fixed point of two generalized α -nonexpansive mappings. We obtain few convergence results in the setting of uniformly convex Banach space. We also present some numerical examples for supporting our main results and to demonstrate the convergence behaviour of the obtained process. [ABSTRACT FROM AUTHOR]

Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and nonexpansive fixed point iterations with a new tool called the scaled relative graph. The SRG provides a correspondence between nonlinear operators and subsets of the 2D plane. Under this framework, a geometric argument in the 2D plane becomes a rigorous proof of convergence. [ABSTRACT FROM AUTHOR]

In this paper, we consider the split equality fixed point problem for quasi-pseudo-contractive mappings without a priori knowledge of operator norms in Hilbert spaces, which includes split feasibility problem, split equality problem, split fixed point problem, etc., as special cases. A unified framework for the study of this kind of problems and operators is provided. The results presented in the paper extend and improve many recent results. [ABSTRACT FROM AUTHOR]

Let C be a closed affine subset of a real Hilbert space H and T : C → C be a nonexpansive mapping. In this paper, for any fixed u ∈ C, a general Halpern iteration process: x 0 ∈ C , x n + 1 = t n u + (1 − t n) T x n , n ≥ 0 , is considered for finding a fixed point of T nearest to u, where the parameter sequence {tn} is selected in the real number field, ℝ . The core problem to be addressed in this paper is to find the optimal parameter sequence so that this iteration process has the optimal convergence rate and to give some numerical results showing advantages of our algorithms. Also, we study the problem of selecting the optimal parameters for a general viscosity approximation method and apply the results obtained from this study to solve a class of variational inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we propose two improved fixed point iterative methods for tensor complementarity problem (TCP), which decrease the number of fixed point iterations. First, based on the tensor splitting, we develop two-step fixed point iterative method and prove that this method converges to a solution of TCP. Then, we present subspace fixed point iterative method for TCP with L -tensor and this method still holds the monotone convergence property. Numerical experiments illustrate the effectiveness of our proposed methods. [ABSTRACT FROM AUTHOR]

We use techniques originating from the subdiscipline of mathematical logic called 'proof mining' to provide rates of metastability and—under a metric regularity assumption—rates of convergence for a subgradient-type algorithm solving the equilibrium problem in convex optimization over fixed-point sets of firmly nonexpansive mappings. The algorithm is due to H. Iiduka and I. Yamada who in 2009 gave a noneffective proof of its convergence. This case study illustrates the applicability of the logic-based abstract quantitative analysis of general forms of Fejér monotonicity as given by the second author in previous papers. [ABSTRACT FROM AUTHOR]

A survey of the existing literature reveals that results on quasimonotone variational inequality problems are scanty in the literature. Moreover, the few existing results are either obtained in finite dimensional Hilbert spaces or the authors were only able to obtain weak convergence results in infinite dimensional Hilbert spaces. In this paper, we study the quasimonotone variational inequality problem and variational inequality problem without monotonicity. We introduce two new inertial iterative schemes with self-adaptive step sizes for approximating a solution of the variational inequality problem. Our proposed methods combine the inertial Tseng extragradient method with viscosity approximation method. We prove some strong convergence results for the proposed algorithms without the knowledge of the Lipschitz constant of the cost operator in infinite dimensional Hilbert spaces. Finally, we provide some numerical experiments to demonstrate the efficiency of our proposed methods in comparison with some recently announced results in the literature in this direction. [ABSTRACT FROM AUTHOR]

It is known that several optimization problems can be converted to a fixed point problem for which the underline fixed point operator is an averaged quasi-nonexpansive mapping and thus the corresponding fixed point method utilizes to solve the considered optimization problem. In this paper, we consider a fixed point method involving inertial extrapolation step with relaxation parameter to obtain a common fixed point of a countable family of averaged quasi-nonexpansive mappings in real Hilbert spaces. Our results bring a unification of several versions of fixed point methods for averaged quasi-nonexpansive mappings considered in the literature and give several implications of our results. We also give some applications to monotone inclusion problem with three-operator splitting method and composite convex and non-convex relaxed inertial proximal methods to solve both convex and nonconvex reweighted l Q regularization for recovering a sparse signal. Finally, some numerical experiments are drawn from sparse signal recovery to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Several efficient methods have been developed in the literature for approximating solutions of fixed point and optimization problems. However, the S-iteration process has been shown to outperform many of these existing methods. In this paper, we study the problem of finding the common solution of split variational inclusion problem, equilibrium problem and common fixed point of nonexpansive mappings. We introduce an improved S-iteration method, which combines inertial and viscosity techniques with self-adaptive step size for approximating the solution of the problem in the framework of Hilbert spaces. Moreover, under some mild conditions we prove strong convergence theorem for the proposed algorithm without the knowledge of the operator norm and we apply our result to study split minimization problem, split feasibility problem and relaxed split feasibility problem. Finally, we present some numerical experiments with graphical illustrations to demonstrate the implementability and efficiency of our proposed method in comparison with some existing state of the art methods in the literature. [ABSTRACT FROM AUTHOR]