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In this study, we introduce a method for finding common fixed points of a finite family of (η i , k i) -enriched strictly pseudocontractive (ESPC) maps and (η i , β i) -enriched strictly pseudononspreading (ESPN) maps in the setting of real Hilbert spaces. Further, we prove the strong convergence theorem of the proposed method under mild conditions on the control parameters. Our main results are also applied in proving strong convergence theorems for η i -enriched nonexpansive, strongly inverse monotone, and strictly pseudononspreading maps. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature. [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]

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]

The main purpose of this research article is to generalize Kannan-type fixed-point (FP) theorems for single-valued mappings and Chatterjea-type FP result for fuzzy mappings (FMs) in the context of complete strong b-metric spaces (MSs). Moreover, fuzzy FPs are established for Suzuki-type fuzzy contraction in the setting of complete strong b-MSs. The conclusions are supported by nontrivial examples to enhance the validity of the results obtained in this study. In addition, previous findings have been made as corollaries from the relevant literature. The numerous implications that this technique has across the literature improve and integrate our findings. Applications of some of the results obtained are also incorporated. [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 some user-friendly versions of integral-type fixed-point results and give some modifications of the classical Banach contraction principle by constructing a special type of contractive restrictions of integral forms for weak contraction mappings defined on 1 E -complete multiplicative partial cone metric spaces over Banach algebras and formulate some existence and uniqueness results regarding the fixed-point theorems using some integrative conditions. Moreover, we validate the significance our results and exploit them to find the unique solution of a fractional nonlinear differential equation of Caputo type, which complements some previously well-known generalizations found in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose two inertial algorithms with a new self-adaptive step size for approximating a solution of the split common null-point problem in the framework of Banach spaces. The step sizes are adaptively updated over each iteration by a simple process without the prior knowledge of the operator norm of the bounded linear operator. Under suitable conditions, we prove the weak-convergence results for the proposed algorithms in p-uniformly convex and uniformly smooth Banach spaces. Finally, we give several numerical results in both finite- and infinite-dimensional spaces to illustrate the efficiency and advantage of the proposed methods over some existing methods. Also, data classifications of heart diseases and diabetes mellitus are presented as the applications of our methods. [ABSTRACT FROM AUTHOR]

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space. [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]

A family of inertial extragradient-type algorithms is proposed for solving convex pseudomonotone variational inequality with fixed-point problems, where the involved mapping for the fixed point is a ρ-demicontractive mapping. Under standard hypotheses, the generated iterative sequence achieves strong convergence to the common solution of the variational inequality and fixed-point problem. Some special cases and sufficient conditions that guarantee the validity of the hypotheses of the convergence statements are also discussed. Numerical applications in detail illustrate the theoretical results and comparison with existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we establish the stability and strong convergence theorems, for the three-step iteration with perturbations for a T-Ciric quasicontraction, in the environment of the CAT(0) space. Finally, an application to the integral-type contraction and an example are shown. [ABSTRACT FROM AUTHOR]

In this paper, we propose new methods for finding a common solution to pseudomonotone and Lipschitz-type equilibrium problems, as well as a fixed-point problem for demicontractive mapping in real Hilbert spaces. A novel hybrid technique is used to solve this problem. The method shown here is a hybrid of the extragradient method (a two-step proximal method) and a modified Mann-type iteration. Our methods use a simple step-size rule that is generated by specific computations at each iteration. A strong convergence theorem is established without knowing the operator's Lipschitz constants. The numerical behaviors of the suggested algorithms are described and compared to previously known ones in many numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a simple proof of the convergence of the recent iterative algorithm by relaxed (u , v) -cocoercive mappings due to Imnang (J. Inequal. Appl. 2013:249, 2013) is presented. [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 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]

With the help of CN-inequality, we study fixed point of multi-valued mappings with closed bounded images and establish some strong convergence theorems involving a countable family of demicontractive mappings in Hadamard spaces. Furthermore, we use the established theorems to deduce some theorems involving a family of minimization problems, variational inequality problems, and monotone inclusion problems. We finally give examples to illustrate the results. The results obtained herein generalise some recent results 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 objective of this paper is to introduce a new motif of α ∗ -ψ-Λ-contraction multivalued mappings, some novel fixed-point and coincidence-point results for this contraction will be investigated in the scope of F-metric spaces, and some examples are given to illustrate our main results and we derive the existence and uniqueness of a solution of a functional equation to support our main result. [ABSTRACT FROM AUTHOR]

This paper aims at proposing an iterative algorithm for finding an element in the intersection of the solutions set of a system of variational inclusions and the fixed-points set of a total uniformly L-Lipschitzian mapping. Applying the concepts of graph convergence and the resolvent operator associated with an Ĥ-accretive mapping, a new equivalence relationship between graph convergence and resolvent-operator convergence of a sequence of Ĥ-accretive mappings is established. As an application of the obtained equivalence relationship, the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the above two sets is proved under some suitable hypotheses imposed on the parameters and mappings. At the same time, the notion of H (⋅ , ⋅) -accretive mapping that appeared in the literature, where H (⋅ , ⋅) is an α, β-generalized accretive mapping, is also investigated and analyzed. We show that the notions H (⋅ , ⋅) -accretive and Ĥ-accretive operators are actually the same, and point out some comments on the results concerning them that are available in the literature. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new type of non-expansive mapping, namely weakly K-nonexpansive mapping, which is weaker than non-expansiveness and stronger than quasi-nonexpansiveness. We prove some weak and strong convergence results using weakly K-nonexpansive mappings. Also, we define weakly (α , K) -nonexpansive mapping and using it prove one stability result for JF-iterative process. Some prominent examples are presented illustrating the facts. A numerical example is given to compare the convergence behavior of some known iterative algorithms for weakly K-nonexpansive mappings. Moreover, we show by example that the class of α-nonexpansive mappings due to Aoyama and Kohsaka and the class of generalized α-nonexpansive mappings due to Pant and Shukla are independent. Finally, our fixed point theorem is applied to obtain a solution of a nonlinear fractional differential equation. [ABSTRACT FROM AUTHOR]

In a real Hilbert space, let the VIP, GSVI, HVI, and CFPP denote a variational inequality problem, a general system of variational inequalities, a hierarchical variational inequality, and a common fixed-point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping, respectively. We design two Mann implicit composite subgradient extragradient algorithms with line-search process for finding a common solution of the CFPP, GSVI, and VIP. The suggested algorithms are based on the Mann implicit iteration method, subgradient extragradient method with line-search process, and viscosity approximation method. Under mild assumptions, we prove the strong convergence of the suggested algorithms to a common solution of the CFPP, GSVI, and VIP, which solves a certain HVI defined on their common solutions set. [ABSTRACT FROM AUTHOR]

In this paper, we present new iterative techniques for approximating the solution of an equilibrium problem involving a pseudomonotone and a Lipschitz-type bifunction in Hilbert spaces. These techniques consist of two computing steps of a proximal-type mapping with an inertial term. Improved simplified stepsize rules that do not involve line search are investigated, allowing the method to be implemented more quickly without knowing the Lipschitz-type constants of a bifunction. The iterative sequences converge weakly on a specific solution to the problem when the control parameter conditions are properly specified. The numerical tests were carried out, and the results demonstrated the applicability and quick convergence of innovative approaches over earlier ones. [ABSTRACT FROM AUTHOR]

We introduce a new subgradient extragradient algorithm utilizing the concept of the set of solutions of the split modified system of variational inequality problems (SMSVIP). Our main theorem is weak convergence theorem for such an algorithm for approximating the fixed point problem in a real Hilbert space. We also apply these results to approximate the split minimization problem. In the last section, we provide an example to illustrate the potential of our main theorem. [ABSTRACT FROM AUTHOR]

In this article, we propose a faster iterative scheme, called the AH iterative scheme, for approximating fixed points of contractive-like mappings and Reich–Suzuki-type nonexpansive mappings. We show that the AH iterative scheme converges faster than a number of existing iterative schemes for contractive-like mappings. The w 2 -stability result of the new iterative scheme is established and a supportive example is provided to illustrate the notion of w 2 -stability. Then, we prove weak and several strong convergence results of our new iterative scheme for fixed points of Reich–Suzuki-type nonexpansive mappings. Further, we carry out a numerical experiment to illustrate the efficiency of our new iterative scheme. As an application, we use our main result to prove the existence of a solution of a mixed-type nonlinear integral equation. An illustrative example to validate the result in our application is also given. Our results extend and generalize several related results in the existing literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study an inertial hybrid iterative method for solving generalized equilibrium problems involving Bregman relatively nonexpansive mappings in Banach spaces. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational example to emphasize the efficiency and relevancy of main result. [ABSTRACT FROM AUTHOR]

In this paper, we consider a distributed optimization problem of minimizing sum of convex functions over the intersection of fixed-point constraints. We propose a distributed method for solving the problem. We prove the convergence of the generated sequence to the solution of the problem under certain assumption. We further discuss the convergence rate with an appropriate positive stepsize. A numerical experiment is given to show the effectiveness of the obtained theoretical result. [ABSTRACT FROM AUTHOR]

We use the notion of Halpern-type sequence recently introduced by the present authors to conclude two strong convergence theorems for solving the bilevel equilibrium problems proposed by Yuying et al. and some authors. Our result excludes some assumptions as were the cases in their results. [ABSTRACT FROM AUTHOR]

In a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a three-step iteration process is stable for contractive-like mappings. It is also proved analytically and numerically that the considered process converges faster than some remarkable iterative processes for contractive-like mappings. Furthermore, some convergence results are proved for the mappings satisfying Suzuki's condition (C) in uniformly convex Banach spaces. A couple of nontrivial numerical examples are presented to support the main results and the visualization is showed by Matlab. Finally, by utilizing our main result the solution of a nonlinear fractional differential equation is approximated. [ABSTRACT FROM AUTHOR]

He (J. Inequal. Appl. 2012:Article ID 162 2012) introduced the proximal point CQ algorithm (PPCQ) for solving the split equilibrium problem (SEP). However, the PPCQ converges weakly to a solution of the SEP and is restricted to monotone bifunctions. In addition, the step-size used in the PPCQ is a fixed constant μ in the interval (0 , 1 ∥ A ∥ 2) . This often leads to excessive numerical computation in each iteration, which may affect the applicability of the PPCQ. In order to overcome these intrinsic drawbacks, we propose a robust step-size { μ n } n = 1 ∞ which does not require computation of ∥ A ∥ and apply the adaptive step-size rule on { μ n } n = 1 ∞ in such a way that it adjusts itself in accordance with the movement of associated components of the algorithm in each iteration. Then, we introduce a self-adaptive extragradient-CQ algorithm (SECQ) for solving the SEP and prove that our proposed SECQ converges strongly to a solution of the SEP with more general pseudomonotone equilibrium bifunctions. Finally, we present a preliminary numerical test to demonstrate that our SECQ outperforms the PPCQ. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a regularization method for solving the variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters, which allow us to obtain a short proof of another strong convergence theorem for this problem. We also apply our main result to the fixed point problem of the nonexpansive variational inequality problem, the common fixed point problem of nonexpansive strict pseudocontractions, the convex minimization problem, and the split feasibility problem. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique. [ABSTRACT FROM AUTHOR]

Every non-convex pair (C , D) may not have proximal normal structure even in a Hilbert space. In this article, we use cyclic relatively nonexpansive maps with respect to orbits to show the presence of best proximity points in C ∪ D , where C ∪ D is a cyclic T-regular set and (C , D) is a non-empty, non-convex proximal pair in a real Hilbert space. Moreover, we show the presence of best proximity points and fixed points for non-cyclic relatively nonexpansive maps with respect to orbits defined on C ∪ D , where C and D are T-regular sets in a uniformly convex Banach space satisfying T (C) ⊆ C , T (D) ⊆ D wherein the convergence of Kranoselskii's iteration process is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze an iterative algorithm to approximate a common solution of a hierarchical fixed point problem for nonexpansive mappings, a system of variational inequalities, and a split equilibrium problem in Hilbert spaces. Under some suitable conditions imposed on the sequences of parameters, we prove that the sequence generated by the proposed iterative method converges strongly to a common element of the solution set of these three kinds of problems. The results obtained here extend and improve the corresponding results of the relevant literature. [ABSTRACT FROM AUTHOR]

In this article, we investigate the notion of the pre-quasi norm on a generalized Cesàro backward difference sequence space of non-absolute type (Ξ (Δ , r)) ψ under definite function ψ. We introduce the sufficient set-up on it to form a pre-quasi Banach and a closed special space of sequences (sss), the actuality of a fixed point of a Kannan pre-quasi norm contraction mapping on (Ξ (Δ , r)) ψ , it supports the property (R) and has the pre-quasi normal structure property. The existence of a fixed point of the Kannan pre-quasi norm nonexpansive mapping on (Ξ (Δ , r)) ψ and the Kannan pre-quasi norm contraction mapping in the pre-quasi Banach operator ideal constructed by (Ξ (Δ , r)) ψ and s-numbers has been determined. Finally, we support our results by some applications to the existence of solutions of summable equations and illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modified proximal point algorithm based on the Thakur iteration process to approximate the common element of the set of solutions of convex minimization problems and the fixed points of two nearly asymptotically quasi-nonexpansive mappings in the framework of CAT (0) spaces. We also prove the Δ-convergence of the proposed algorithm. We also provide an application and numerical result based on our proposed algorithm as well as the computational result by comparing our modified iteration with previously known Sahu's modified iteration. [ABSTRACT FROM AUTHOR]

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 paper, we propose two iterative schemes for approximating solutions of split common fixed point problems in multiple linear operators case. The first algorithm implements the Krasnosel'skiĭ–Mann iteration with an inertial effect for which the weak convergence is established under mild assumptions. With the tool of nearly contractive mappings, we introduce a viscosity-type iteration which ensures strong convergence. We apply our results to solve a multiple split monotone variational inclusion problem. A numerical example is given to demonstrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The proximal gradient method is a highly powerful tool for solving the composite convex optimization problem. In this paper, firstly, we propose inexact inertial acceleration methods based on the viscosity approximation and proximal scaled gradient algorithm to accelerate the convergence of the algorithm. Under reasonable parameters, we prove that our algorithms strongly converge to some solution of the problem, which is the unique solution of a variational inequality problem. Secondly, we propose an inexact alternated inertial proximal point algorithm. Under suitable conditions, the weak convergence theorem is proved. Finally, numerical results illustrate the performances of our algorithms and present a comparison with related algorithms. Our results improve and extend the corresponding results reported by many authors recently. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and study the mixed set-valued vector inverse quasi-variational inequality problems (MSVIQVIPs) and to obtain error bounds for this kind of MSVIQVIP in terms of the residual gap function, the regularized gap function, and the D-gap function. These bounds provide effective estimated distances between an arbitrary feasible point and the solution set of mixed set-valued vector inverse quasi-variational inequality problem. The results presented in the paper improve and generalize some recent results. [ABSTRACT FROM AUTHOR]

We propose and analyze an inertial iterative algorithm to approximate a common solution of generalized equilibrium problem, variational inequality problem, and fixed point problem in the framework of a 2-uniformly convex and uniformly smooth real Banach space. Further, we study the convergence analysis of our proposed iterative method. Finally, we give application and a numerical example to illustrate the applicability of the main algorithm. [ABSTRACT FROM AUTHOR]

Let X be a uniformly convex and q-uniformly smooth Banach space with 1 < q ≤ 2 . In the framework of this space, we are concerned with a composite gradient-like implicit rule for solving a hierarchical monotone variational inequality with the constraints of a system of monotone variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonlinear operators { S n } n = 0 ∞ . Our rule is based on the Korpelevich extragradient method, the perturbation mapping, and the W-mappings constructed by { S n } n = 0 ∞ . [ABSTRACT FROM AUTHOR]

For a variational inequality problem, the inertial projection and contraction method have been studied. It has a weak convergence result. In this paper, we propose a strong convergence iterative method for finding a solution of a variational inequality problem with a monotone mapping by projection and contraction method and inertial hybrid algorithm. Our result can be used to solve other related problems in Hilbert spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to solve the hierarchical variational inequality with the constraint of a general system of variational inequalities in a uniformly convex and 2-uniformly smooth Banach space. We introduce implicit and explicit iterative algorithms which converge strongly to a unique solution of the hierarchical variational inequality problem. Our results improve and extend the corresponding results announced by some authors. [ABSTRACT FROM AUTHOR]

In this paper, a new form of the symmetric vector equilibrium problem is introduced and, by mixing properties of the nonlinear scalarization mapping and the maximal element lemma, an existence theorem for it is established. We show that Ky Fan's lemma, as a usual technique for proving the existence results for equilibrium problems, implies the maximal element lemma, while it is useless for proving the main theorem of this paper. Our results can be viewed as an extension and improvement of the main results obtained by Farajzadeh (Filomat 29(9):2097-2105, 2015) and some corresponding results that appeared in this area by relaxing the lower semicontinuity, quasiconvexity on the mappings and being nontrivial of the dual cones. Finally, some examples are given to support the main results. [ABSTRACT FROM AUTHOR]

The split feasibility problem (SFP) is finding a point $x\in C$ such that $Ax\in Q$ , where C and Q are nonempty closed convex subsets of Hilbert spaces $H_{1}$ and $H_{2}$ , and $A:H_{1}\rightarrow H_{2}$ is a bounded linear operator. Byrne's CQ algorithm is an effective algorithm to solve the SFP, but it needs to compute $\|A\|$ , and sometimes $\|A\| $ is difficult to work out. López introduced a choice of stepsize $\lambda_{n}$ , $\lambda_{n}=\frac{\rho_{n}f(x_{n})}{\|\nabla f(x_{n})\| ^{2}}$ , $0<\rho_{n}<4$ . However, he only obtained weak convergence theorems. In order to overcome the drawbacks, in this paper, we first provide a regularized CQ algorithm without computing $\|A\|$ to find the minimum-norm solution of the SFP and then obtain a strong convergence theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain refined stability results and alternative stability results for additive and quadratic functional equations using direct method in modular spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the algorithm proposed in recent years by Censor, Gibali and Reich, which solves split variational inequality problem, and Korpelevich's extragradient method, which solves variational inequality problems. As our main result, we propose an iterative method for finding an element to solve a class of split variational inequality problems under weaker conditions and get a weak convergence theorem. As applications, we obtain some new weak convergence theorems by using our weak convergence result to solve related problems in nonlinear analysis and optimization. [ABSTRACT FROM AUTHOR]

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Assume that g is a real-valued convex function and the gradient ∇ g is $\frac{1}{L}$ -ism with $L>0$ . Let $0<\lambda <\frac{2}{L+2}$ , $0<\beta_{n}<1$ . We prove that the sequence $\{x_{n}\} $ generated by the iterative algorithm $x_{n+1}=P_{C}(I-\lambda(\nabla g+\beta_{n}I))x_{n}$ , $\forall n\geq0$ converges strongly to $q\in U$ , where $q=P_{U}(0)$ is the minimum-norm solution of the constrained convex minimization problem, which also solves the variational inequality $\langle-q, p-q\rangle\leq0$ , $\forall p\in U$ . Under suitable conditions, we obtain some strong convergence theorems. As an application, we apply our algorithm to solving the split feasibility problem in Hilbert spaces. [ABSTRACT FROM AUTHOR]

We know that variational inequality problem is very important in the nonlinear analysis. The main purpose of this paper is to propose an iterative method for finding an element of the set of solutions of a variational inequality problem with a monotone and Lipschitz continuous mapping in Hilbert space. This iterative method is based on the extragradient method. We get a weak convergence theorem. Using this result, we obtain three weak convergence theorems for the equilibrium problem, the constrained convex minimization problem, and the split feasibility problem. [ABSTRACT FROM AUTHOR]

Two new ergodic convergence theorems for approximating the common element of the set of zero points of an m-accretive mapping and the set of fixed points of an infinite family of non-expansive mappings in a real smooth and uniformly convex Banach space are obtained, which improves some of the previous work. The computational experiments to demonstrate the effectiveness of the proposed iterative algorithms in this paper are conducted. [ABSTRACT FROM AUTHOR]