General Iterative Scheme Research Articles

Convergence Properties and Fixed Points of Two General Iterative Schemes with Composed Maps in Banach Spaces with Applications to Guaranteed Global Stability

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems.

General iterative scheme based on the regularization for solving a constrained convex minimization problem

It is well known that the regularization method plays an important role in solving a constrained convex minimization problem. In this article, we introduce implicit and explicit iterative schemes based on the regularization for solving a constrained convex minimization problem. We establish results on the strong convergence of the sequences generated by the proposed schemes to a solution of the minimization problem. Such a point is also a solution of a variational inequality. We also apply the algorithm to solve a split feasibility problem. MSC:47H09, 47H05, 47H06, 47J25, 47J05.

A general iterative method for two maximal monotone operators and 2-generalized hybrid mappings in Hilbert spaces

Let C be a closed and convex subset of a real Hilbert space H. Let T be a 2-generalized hybrid mapping of C into itself, let A be an α-inverse strongly-monotone mapping of C into H, and let B and F be maximal monotone operators on and respectively. The purpose of this paper is to introduce a general iterative scheme for finding a point of which is a unique solution of a hierarchical variational inequality, where is the set of fixed points of T, and are the sets of zero points of and F, respectively. A strong convergence theorem is established under appropriate conditions imposed on the parameters. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to mixed equilibrium problems and the set of fixed points of a 2-generalized hybrid mapping in a real Hilbert space.

Scalar correction method for finding least-squares solutions on Hilbert spaces and its applications

We use the idea of two-point stepsize gradient methods, developed to solve unconstrained minimization problems on Rn, for computing least-squares solutions of a given linear operator equation on Hilbert spaces. Among them we especially pay attention to corresponding modification of the scalar correction method. An application of this approach is presented related to computation of {1,3} inverses and the Moore–Penrose inverse of a given complex matrix. Convergence properties of the general gradient iterative scheme for computation of various pseudoinverses are investigated. The efficiency of the presented algorithm is theoretically verified and approved by selected test matrices.

The general iterative methods for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces

In this paper, the researcher introduces the general iterative scheme for finding a common element of the set of equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others. MSC: 47H10; 47H09

Convergence theorems for generalized equilibrium problems and nonexpansive mappings

We propose a general iterative scheme for solving a generalized equilibrium problem and fixed points of nonexpansive mappings in a Hilbert space. We then prove strong convergence theorems. Mathematics Subject Classification: 47H09, 47H10

A General Iterative Scheme Based on Regularization for Solving Equilibrium and Constrained Convex Minimization Problems

The present paper is divided into two parts. First, we introduce implicit and explicit iterative schemes based on the regularization for solving equilibrium and constrained convex minimization problems. We establish results on the strong convergence of the sequences generated by the proposed schemes to a common solution of minimization and equilibrium problem. Such a point is also a solution of a variational inequality. In the second part, as applications, we apply the algorithm to solve split feasibility problem and equilibrium problem.

Viscosity iterative scheme for generalized mixed equilibrium problems and nonexpansive semigroups

In this paper, we propose a new general iterative scheme based on the viscosity approximation method for finding a common element of the set of solutions of the generalized mixed equilibrium problem and the set of all common fixed points of a finite family of nonexpansive semigroups. Then, we prove the strong convergence of the iterative scheme to find a unique solution of the variational inequality that is the optimality condition for the minimization problem. Our results extend and improve some recent results of Cianciaruso et al. (J. Optim. Theory Appl. 146:491–509, 2010), Kamraksa and Wangkeeree (J. Glob. Optim. 51:689–714, 2011), and many others.

Averaging Over General Random Networks

This technical note studies the distributed averaging problem over general random networks, by means of augmenting state space. A general iterative scheme (with a certain structure) is proposed that is discrete-time, linear, and stochastic; its generality compared to the literature lies in that the weight matrices corresponding to the networks need not be column-stochastic, and the random process generating the update matrices need not be ergodic or i.i.d. It is then justified that the scheme achieves average consensus in the mean-square sense, which, in a special case, also implies averaging with probability one. A key technique to the justification is a matrix perturbation result, which describes the behavior of eigenvalues perturbed simultaneously by multiple parameters.

The general iterative scheme for semigroups of nonexpansive mappings and variational inequalities with applications

In this paper, we introduce the implicit and explicit viscosity iteration schemes for nonexpansive semigroups {T(t):t≥0}. Additionally, it proves that the proposed iterative schemes converge strongly to a unique common fixed point of {T(t):t≥0} in the framework of reflexive and strictly convex Banach space, which solves some variational inequality. The main results of this paper improve and generalize recent known results in the current literature.

A viscosity scheme for mixed equilibrium problems, variational inequality problems and fixed point problems

In this paper, we study a new general iterative scheme for finding a common element of the set of solutions of finite general mixed equilibrium problems, the set of solutions of finite variational inequalities for cocoercive mappings, the set of solutions of common fixed points of an infinite family of nonexpansive mappings and the set of solutions of fixed points of a nonexpansive semigroup in Hilbert space. Strong convergence theorems are obtained under appropriate conditions.

A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces

In this paper, we introduce a general iterative algorithm for finding a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of solutions of systems of variational inequalities for two inverse strongly accretive mappings in a q-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as improvement, supplementation, development and extension of the corresponding results in some references to a great extent.

Convergence theorems of a general approximation method for common fixed points of a finite family of asymptotically-quasi nonexpansive mappings in Banach spaces

In this paper, we consider a general iterative scheme to approximate a common fixed point for a finite family of asymptotically quasi-nonexpansive mappings. Several strong and weak convergence results are presented in Banach spaces and an finite family of asymptotically quasi-nonexpansive mappings is constructed. Our results generalize and extend many known results in the current literature.

About an analytical application of the spherical symmetric diffusion equation

A time-dependent solution of the radiative diffusion energy equation in a hot plutonium sphere close to radiative equilibrium is derived. Based on a general iteration scheme, the analytical solutions correct to first and second order are derived. The outer boundary condition has been adopted from neutron transport theory.

An Iterative Method for Solving a System of Mixed Equilibrium Problems, System of Quasivariational Inclusions, and Fixed Point Problems of Nonexpansive Semigroups with Application to Optimization Problems

We introduce a general implicit iterative scheme base on viscosity approximation method with a ϕ‐strongly pseudocontractive mapping for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed point for a nonexpansive semigroup, and the set of solutions of system of variational inclusions with set‐valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator.

Shrinking Projection Method of Fixed Point Problems for Asymptotically Pseudocontractive Mapping in the Intermediate Sense and Mixed Equilibrium Problems in Hilbert Spaces

This paper is concerned with a common element of the set of fixed point for an asymptotically pseudocontractive mapping in the intermediate sense and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method, which extends and improves that of Qin et al. (2010) and many others.

Shrinking Projection Method of Common Fixed Point Problems for Discrete Asymptotically Strictly Pseudocontractive Semigroups and Mixed Equilibrium Problems in Hilbert Spaces

This paper is concerned with a common element of the set of common fixed point for a discrete asymptotically strictly pseudocontractive semigroup and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method which extends and improves the corresponding ones due to Kim [Proceedings of the Asian Conference on Nonlinear Analysis and Optimization (Matsue, Japan, 2008), 139–162].

A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems

In this article, we introduce a new general composite iterative scheme for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality problem for an inverse-strongly monotone mapping in Hilbert spaces. It is shown that the sequence generated by the proposed iterative scheme converges strongly to a common element of the above three sets under suitable control conditions, which solves a certain optimization problem. The results of this article substantially improve, develop, and complement the previous well-known results in this area.2010 Mathematics Subject Classifications: 49J30; 49J40; 47H09; 47H10; 47J20; 47J25; 47J05; 49M05.

Shrinking Projection Method for Fixed Point Problems of an Infinite Family of Strictly Pseudocontractive Mappings and the System of Cocoercive Quasivariational Inclusions Problems in Hilbert Spaces

This paper is concerned with a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

Convergence Theorems of a General Composite Iterative Method for Nonexpansive Semigroups in Banach Spaces

We introduce a general composite iterative scheme for nonexpansive semigroups in Banach spaces. We establish some strong convergence theorems of the general iteration scheme under different control conditions. The results presented in this paper improve and extend the corresponding results of Marino and Xu (2006), and others, from Hilbert spaces to Banach spaces.