Inverse-strongly Monotone Mappings Research Articles

Analysis of Subgradient Extragradient Method for Variational Inequality Problems and Null Point Problems

In this paper, we introduce a new numerical method for finding a common solution to variational inequality problems involving monotone mappings and null point problems involving a finite family of inverse-strongly monotone mappings. The method is inspired by the subgradient extragradient method and the regularization method. Strong convergence results of the proposed algorithms have been obtained under some suitable conditions.

A Simple proof for the algorithms of relaxed $(u, v)$-cocoercive mappings and $alpha$-inverse strongly monotone mappings

In this paper, a simple proof is presented for the Convergence of the algorithms for the class of relaxed $(u, v)$-cocoercive mappings and $alpha$-inverse strongly monotone mappings. Based on $alpha$-expansive maps, for example, a simple proof of the convergence of the recent iterative algorithms by relaxed $(u, v)$-cocoercive mappings due to Kumam-Jaiboon is provided. Also a simple proof for the convergence of the iterative algorithms by inverse-strongly monotone mappings due to Iiduka-Takahashi in a special case is provided. These results are an improvement as well as a refinement of previously known results.

Dual Variable Inertial Accelerated Algorithm for Split System of Null Point Equality Problems

In this paper, we introduce a novel dual variable algorithm accelerated via inertial extrapolation for solving the generalized split inverse problem given as a task of finding the common null point equality of finite family of inverse-strongly monotone mappings in Hilbert spaces. The proposed method uses the dual variable and self-adaptive technique, along with an inertial strategy which aims at getting accelerated convergence. We establish the convergence Theorem of the proposed method under some suitable assumptions. We also apply our result to solve several types of problems. Our results improve and generalize the corresponding several known results announced by many others. Numerical experiments are presented to show the efficiency and comparative performance of the proposed algorithm.

Extension and Application of the Yamada Iteration Algorithm in Hilbert Spaces

In this paper, based on the Yamada iteration, we propose an iteration algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping. We obtain a weak convergence theorem in Hilbert space. In particular, the set of zero points of an inverse strongly-monotone mapping can be transformed into the solution set of the variational inequality problem. Further, based on this result, we also obtain some new weak convergence theorems which are used to solve the equilibrium problem and the split feasibility problem.

Δ-convergence theorems for inverse-strongly monotone mappings in CAT(0) spaces

Δ-convergence theorems for inverse-strongly monotone mappings in CAT(0) spaces

Strong convergence of Halpern iteration for products of finitely many resolvents of maximal monotone operators in Banach spaces

In this paper, using Bregman functions, we introduce a new Halpern-type iterative algorithm for finding common zeros of finitely many maximal monotone operators and obtain a strongly convergent iterative sequence to the common zeros of these operators in a reflexive Banach space. Furthermore, we study Halpern-type iterative schemes for finding common solutions of a finite system of equilibrium problems and null spaces of a -inverse strongly monotone mapping in a 2-uniformly convex Banach space. Some applications of our results to the solution of equations of Hammerstein-type are presented. Our scheme has an advantage that we do not use any projection of a point on the intersection of closed and convex sets which creates some difficulties in a practical calculation of the iterative sequence. So the simple construction of Halpern iteration provides more flexibility in defining the algorithm parameters which is important from the numerical implementation perspective. Presented results improve and generalize many known results in the current literature.

Strong convergence theorem for a common fixed point of a finite family of strictly pseudo-contractive mappings and a strictly pseudononspreading mapping

AbstractIn this paper, we introduce a new mapping in a real Hilbert space to prove a strong convergence theorem for finding a common fixed point of a finite family of strictly pseudo-contractive mappings and a strictly pseudononspreading mapping. Moreover, we also obtain a strong convergence theorem for a finite family of inverse-strongly monotone mappings and a strictly pseudononspreading mapping.

Strong Convergence of a New Composite Iterative Method for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

In this paper, first we introduce a new mapping for finding a common fixed point of an infinite family of nonexpansive mappings then we consider iterative method for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for $\alpha$-inverse-strongly monotone mapping in a Hilbert space. We show that under suitable conditions, the sequence converges strongly to a common element of the above three sets. Our results presented in this paper improve and extend other results.

On a finite family of variational inclusions with the constraints of generalized mixed equilibrium and fixed point problems

Abstract In this paper, we introduce two iterative algorithms for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions. MSC:49J30, 47H09, 47J20, 49M05.

A New Iterative Method for the Set of Solutions of Equilibrium Problems and of Operator Equations with Inverse-Strongly Monotone Mappings

The purpose of the paper is to present a new iteration method for finding a common element for the set of solutions of equilibrium problems and of operator equations with a finite family ofλi-inverse-strongly monotone mappings in Hilbert spaces.

Strong convergence of a general iterative algorithm in Hilbert spaces

In this paper, the problem of approximating a common element in the common fixed point set of an infinite family of nonexpansive mappings, in the solution set of a variational inequality involving an inverse-strongly monotone mapping and in the solution set of an equilibrium problem is investigated based on a general iterative algorithm. Strong convergence of the iterative algorithm is obtained in the framework of Hilbert spaces. The results obtained in this paper improve the corresponding results announced by many authors.

Strong Convergence Theorems for Maximal and Inverse-Strongly Monotone Mappings in Hilbert Spaces and Applications

In this paper, we prove two strong convergence theorems for finding a common point of the set of zero points of the addition of an inverse-strongly monotone mapping and a maximal monotone operator and the set of zero points of a maximal monotone operator, which is related to an equilibrium problem in a Hilbert space. Such theorems improve and extend the results announced by Y. Liu (Nonlinear Anal. 71:4852–4861, 2009). As applications of the results, we present well-known and new strong convergence theorems which are connected with the variational inequality, the equilibrium problem and the fixed point problem in a Hilbert space.

VISCOSITY APPROXIMATION METHODS FOR NONEXPANSIVE SEMINGROUPS AND MONOTONE MAPPPINGS

Abstract. Let C be a nonempty closed convex subset of real Hilbertspace Hand F = fS(t) : t0ga nonexpansive self-mapping semigroupof C, and f: C!Cis a xed contractive mapping. Consider the processfx n g:(x n+1 = n x n + (1 n )z n z n = n f(x n ) + (1 n )S(t n )P C (x n r n Ax n ):It is shown that fx n gconverges strongly to a common element of the setof xed points of nonexpansive semigroups and the set of solutions of thevariational inequality for an inverse strongly-monotone mapping whichsolves some variational inequality. 1. IntroductionLet H be a real Hilbert space with inner product h;iand induced normkk, Cbe a nonempty closed convex subset of H. S: C!Cis nonexpansiveif kS(x) S(y)kkx yk;for all x;y2C. The set of xed points of SisF(S) = fx2C: x= Sxg. We assume that F(S) 6= ˚, it is well known thatF(S) is closed convex.A nonexpansive semigroup is a family F = fS(t) : t0gof self-mapping ofCif the following conditions are satis ed:(a) S(0)x= xfor all x2C;(b) S(s+ t) = S(s)S(t) for all s;t0;(c) For each t>0, kS(t)x S(t)ykkx yk;x;y2C:(d) For each x2C, the mapping S()xis continuous.In this paper, we use Fto denote the set of common xed points of F; that is,F= fx2C: S(t)x= x;t0g=

Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems

In this article, zero points of the sum of a maximal monotone operator and an inverse-strongly monotone mapping, solutions of a monotone variational inequality, and fixed points of a strict pseudocontraction are investigated. A hybrid projection iterative algorithm is considered for analyzing the convergence of the iterative sequences. Strong convergence theorems are established in the framework of real Hilbert spaces without any compact assumptions. Some applications of the main results are also provided. AMS Classification: 47H05; 47H09; 47J25; 90C33.

CONVERGENCE THEOREMS FOR VARIATIONAL INEQUALITIES EQUILIBRIUM PROBLEMS AND NONEXPANSIVE MAPPINGS BY HYBRID METHOD

In this paper, we introduce iterative schemes for finding a common element of the set of common fixed points for a left amenable semigroup of nonexpansive mappings, the set of solutions of the variational inequalities for a family of $\alpha$-inverse-strongly monotone mappings and the set of solutions of a system of equilibrium problems in a Hilbert space. We establish weak and strong convergence theorems for the sequences generated by our proposed schemes. Moreover, we present various applications to the additive semigroup of nonnegative real numbers and families of strictly pseudocontractive mappings.

TWO GENERALIZED STRONG CONVERGENCE THEOREMS OF HALPERN’S TYPE IN HILBERT SPACES AND APPLICATIONS

Let $C$ be a closed convex subset of a real Hilbert space $H$. Let $A$ be an inverse-strongly monotone mapping of $C$ into $H$ and let $B$ be a maximal monotone operator on $H$ such that the domain of $B$ is included in $C$. We introduce two iteration schemes of finding a point of $(A+B)^{-1}0$, where $(A+B)^{-1}0$ is the set of zero points of $A+B$. Then, we prove two strong convergence theorems of Halpern's type in a Hilbert space. Using these results, we get new and well-known strong convergence theorems in a Hilbert space.

A new iterative method for finding common solutions of generalized equilibrium problem, fixed point problem of infinite k-strict pseudo-contractive mappings, and quasi-variational inclusion problem

In this article, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems, the set of common fixed point for a family of infinite k-strict pseudo-contractive mappings, and the set of solutions of the variational inclusion problem with multi-valued maximal monotone mappings and inverse-strongly monotone mappings in Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extends the recent results in G.L.Acedo and H.K.Xu [2], Zhang, Lee and Chan [8], Takahashi and Toyoda [9], Takahashi and Takahashi [10] and S. S. Chang, H. W. Joseph Lee and C. K. Chan [11], S.Takahashi and W.Takahashi [12]. Moreover, the method of proof adopted in this article is different from those of [4] and [12].

CONVERGENCE THEOREMS FOR VARIATIONAL INEQUALITIES EQUILIBRIUM PROBLEMS AND NONEXPANSIVE MAPPINGS BY HYBRID METHOD

In this paper, we introduce iterative schemes for finding a common element of the set of common fixed points for a left amenable semigroup of nonexpansive mappings, the set of solutions of the variational inequalities for a family of $\alpha$-inverse-strongly monotone mappings and the set of solutions of a system of equilibrium problems in a Hilbert space. We establish weak and strong convergence theorems for the sequences generated by our proposed schemes. Moreover, we present various applications to the additive semigroup of nonnegative real numbers and families of strictly pseudocontractive mappings.

TWO GENERALIZED STRONG CONVERGENCE THEOREMS OF HALPERN’S TYPE IN HILBERT SPACES AND APPLICATIONS

Let $C$ be a closed convex subset of a real Hilbert space $H$. Let $A$ be an inverse-strongly monotone mapping of $C$ into $H$ and let $B$ be a maximal monotone operator on $H$ such that the domain of $B$ is included in $C$. We introduce two iteration schemes of finding a point of $(A+B)^{-1}0$, where $(A+B)^{-1}0$ is the set of zero points of $A+B$. Then, we prove two strong convergence theorems of Halpern's type in a Hilbert space. Using these results, we get new and well-known strong convergence theorems in a Hilbert space.

A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

We introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse‐strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.