Monotone Mapping In Hilbert Space Research Articles

A System of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Pseudocontractive Mappings in Hilbert Spaces

AbstractWe introduce and analyze a new iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of a system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Furthermore, we prove new strong convergence theorems for a new iterative algorithm under some mild conditions. Finally, we also apply our results for solving convex feasibility problems in Hilbert spaces. The results obtained in this paper improve and extend the corresponding results announced by Qin and Kang (2010) and the previously known results in this area.

A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings

The purpose of this paper is to consider a new hybrid relaxed extragradient method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a finite family of nonexpansive mappings and the set of solutions of variational inequalities for an inverse-strongly monotone mapping in Hilbert spaces. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions. Our results extend and improve the recent results of Cai and Hu [G. Cai, C.S. Hu, A hybrid approximation method for equilibrium and fixed point problems for a family of infinitely nonexpansive mappings and a monotone mapping, Nonlinear Anal. Hybrid Syst., 3(2009) 395–407], Kangtunyakarn and Suantai [A. Kangtunyakarn, S. Suantai, A new mapping for finding common solution of equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Anal., 71(2009) 4448–4460] and Thianwan [S. Thianwan, Strong convergence theorems by hybrid methods for a finite family of nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. Hybrid Syst., 3(2009) 605–614] and many others.

A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings

In this paper, we introduce and analyze a new general iterative scheme by the viscosity approximation method for finding the common element of the set of equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings and the set solutions of the variational inequality problems for an ξ-inverse-strongly monotone mapping in Hilbert spaces. We show that the sequence converge strongly to a common element of the above three sets under some parameters controlling conditions. The result extends and improves a recent result of Chang et al. (Nonlinear Anal. 70:3307–3319, 2009) and many others.

A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings

In this paper, we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α -inverse-strongly monotone mapping in Hilbert spaces. We prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main result improve and extend Plubtieng and Punpaeng’s corresponding result [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Applied Mathematics and Computation 197 (2008), 548–558]. Using this theorem, we obtain three corollaries.

General system of [formula omitted]-maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms

In this paper, a new system of nonlinear (set-valued) variational inclusions involving ( A,η) -maximal relaxed monotone and relative ( A,η) -maximal monotone mappings in Hilbert spaces is introduced and its approximation solvability is examined. The notion of ( A,η) -maximal relaxed monotonicity generalizes the notion of general η-maximal monotonicity, including ( H,η) -maximal monotonicity (also referred to as ( H,η) -monotonicity in literature). Using the general ( A,η) -resolvent operator method, approximation solvability of this system based on a generalized hybrid iterative algorithm is investigated. Furthermore, for the nonlinear variational inclusion system on hand, corresponding nonlinear Yosida regularization inclusion system and nonlinear Yosida approximations are introduced, and as a result, it turns out that the solution set for the nonlinear variational inclusion system coincides with that of the corresponding Yosida regularization inclusion system. Approximation solvability of the Yosida regularization inclusion system is based on an existence theorem and related Yosida approximations. The obtained results are general in nature.

An Iterative Method for Generalized Equilibrium Problems, Fixed Point Problems and Variational Inequality Problems

AbstractWe introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for "Equation missing"-inverse-strongly monotone mappings in Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. (2007).

A Hybrid Extragradient Viscosity Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings

AbstractWe introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems, the set of solutions of fixed points of an infinitely many nonexpansive mappings, and the set of solutions of the variational inequality problems for "Equation missing"-inverse-strongly monotone mapping in Hilbert spaces. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality, which is the optimality condition for a minimization problem. Results obtained in this paper improve the previously known results in this area.

Viscosity approximation methods for nonexpansive mappings and inverse-strongly monotone mappings in Hilbert spaces

Let C be a closed convex subset of a Hilbert space H. Let f is a contraction on C. Let S be a nonexpansive mapping of C into itself and A be an α-inverse-strongly monotone mapping of C into H. Assuming that F(S)∩VI(C,A)≠φ, and x0=x∈C, in this paper we introduce the iterative process xn+1=αnf(xn)+βnxn+γn(μSxn+(1−μ)(PC(I−λnA)yn)), where yn=PC(I−λnA)xn. We prove that {xn} and {yn} converge strongly to the same point z∈F(S)∩VI(C,A). As its application, we give a strong convergence theorem for nonexpansive mapping and strictly pseudo-contractive mapping in a Hilbert space.

ON A SYSTEM OF GENERALIZED NONLINEAR VARIATIONAL INEQUALITIES

In this paper a new class of system of generalized nonlinear variational inequalities involving strongly monotone, relaxed co coercive and relaxed generalized monotone mappings in Hilbert spaces is introduced and studied. Based on the projection method, an equivalence between the system of generalized nonlinear variational inequalities and the fixed point problem is established, which is used to suggest some new iterative algorithms for computing approximate solutions of the system of generalized nonlinear variational inequalities. A few sufficient conditions which ensure the existence and uniqueness of solution of the system of generalized nonlinear variational inequalities are given, and the convergence analysis of iterative sequences generated by the algorithms are also discussed.

General system of [formula omitted]-monotone variational inclusion problems based on generalized hybrid iterative algorithm

First a new system of nonlinear set-valued variational inclusions involving ( A , η ) -monotone mappings in Hilbert spaces is introduced and then its solvability is explored. Based on the general resolvent operator method associated with ( A , η ) -monotone mappings, approximation solvability of this system of nonlinear set-valued variational inclusions is established. The convergence analysis is discussed in detail. The obtained results generalize a number of results on nonlinear variational inclusion systems.

Generalized Nonlinear Variational Inclusions Involving "Equation missing" -Monotone Mappings in Hilbert Spaces

AbstractA new class of generalized nonlinear variational inclusions involving "Equation missing"-monotone mappings in the framework of Hilbert spaces is introduced and then based on the generalized resolvent operator technique associated with "Equation missing"-monotonicity, the approximation solvability of solutions using an iterative algorithm is investigated. Since "Equation missing"-monotonicity generalizes "Equation missing"-monotonicity and "Equation missing"-monotonicity, results obtained in this paper improve and extend many others.

Graph-distance convergence and uniform local boundedness of monotone mappings

In this article we study graph-distance convergence of monotone operators. First, we prove a property that has been an open problem up to now: the limit of a sequence of graph-distance convergent maximal monotone operators in a Hilbert space is a maximal monotone operator. Next, we show that a sequence of maximal monotone operators converging in the same sense in a reflexive Banach space is uniformly locally bounded around any point from the interior of the domain of the limit mapping. The result is an extension of a similar one from finite dimensions. As an application we give a simplified condition for the stability (under graph-distance convergence) of the sum of maximal monotone mappings in Hilbert spaces.

Proximal point algorithm with errors for generalized strongly nonlinear quasivariational inclusions

In this paper, a class of generalized strongly nonlinear quasivariational inclusions are studied. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert space, an existence theorem of solutions for generalized strongly nonlinear quasivariational inclusion is established and a new proximal point algorithm with errors is suggested for finding approximate solutions which strongly converge to the exact solution of the generalized strongly nonlinear quasivariational inclusion. As special cases, some known results in this field are also discussed.

Perturbed Proximal Point Algorithms for Generalized Quasivariational Inclusions

In this paper, we study a class of generalized quasivariational inclusions. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert space, we have established an existence theorem of solutions for generalized quasivariational inclusions, suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized quasivariational inclusions. As special cases, some known results in this field are also discussed.

On the constructive solution of nonlinear functional equations

In the recent development of the theory of nonlinear operators in Banach spaces, a number of general existence theorems have been established for various classes of mappings by the use of compactness, convexity, or topological arguments that do not give a constructive procedure for the generation of the solutions thus proved to exist. Even in cases when the solutions are unique and one obtains them as limits of a precisely determined sequence of approximants, they often fail to satisfy an important and useful principle of constructivity in that the procedures have no effective control provided for the error at each state of the approximation. This is particularly the case for the Galerkin approximations used in the theory of operators of monotone type and for various fixed-point methods used in the theory of nonlinear accretive operators. About a decade ago, the writer developed existence results which satisfied these principles of constructivity for the case of a broad class of continuous monotone mappings in Hilbert spaces, and more generally, for continuous accretive mappings in Banach spaces X whose conjugate spaces X* are uniformly convex. These results were stated in the rather inaccessible paper [l] and developed in detail in the middle of a lengthy treatment of accretive operators in the writer’s paper [2]. It is our object here to develop these results explicitly and in detail in connection with the problem of constructivity. Our renewed interest in this question was stimulated by the recent paper by Bruck [3], who has developed an iteration procedure to obtain the solution of the equation (I + T)(u) = 0 for a continuous, bounded monotone operator T on a Hilbert space H with explicit control of the error. Bruck’s result has the curious feature that it depends on the assumption of the prior existence of the solution. Although the procedure we give is not an iteration procedure it corresponds to a simpler intuitive picture of the situation in the general context of accretive operators and the error control does not depend upon the assumption of the existence of a solution.