Feasibility Problem In Hilbert Spaces Research Articles

New strong convergence theorems for split variational inclusion problems in Hilbert spaces

The split variational inclusion problem is an important problem, and it is a generalization of the split feasibility problem. In this paper, we present a descent-conjugate gradient algorithm for the split variational inclusion problems in Hilbert spaces. Next, a strong convergence theorem of the proposed algorithm is proved under suitable conditions. As an application, we give a new strong convergence theorem for the split feasibility problem in Hilbert spaces. Finally, we give numerical results for split variational inclusion problems to demonstrate the efficiency of the proposed algorithm.

A general convergence theorem for multiple-set split feasibility problem in Hilbert spaces

We establish strong convergence result of split feasibility problem for a family of quasi-nonexpansive multi-valued mappings and a total asymptotically strict pseudo-contractive mapping in infinite dimensional Hilbert spaces.

A gradient projection method for solving split equality and split feasibility problems in Hilbert spaces

In this paper, we first study in a Hilbertian framework the weak convergence of a general Gradient Projection Algorithm for minimizing a convex function of class over a convex constraint set. The way of selecting the stepsizes corresponds to the one used by López et al. for the particular case of the Split Feasibility Problem. This choice allows us to avoid the computation of operator norms. Afterwards, a relaxed version of the Gradient Projection Algorithm is considered where the feasible set is approximated by half-spaces making the projections explicit. Finally, to get the strong convergence, each step of the general Gradient Projection Method is combined with a viscosity step. This is done by adapting Halpern’s algorithm to our problem. The general scheme is then applied to the Split Equality Problem, and also to the Split Feasibility Problem.

Weak Convergence Theorems for Families of Nonlinear Mappings with Generalized Parameters

In this article, we first introduce new families of nonlinear mappings in a Hilbert space. Next, we consider iterative processes for the families with generalized parameters and then prove weak convergence theorems for finding common fixed points of the families. As applications, we obtain weak convergence theorems for finding solutions of variational inclusion problems, equilibrium problems, and split feasibility problems in Hilbert spaces. Our results generalize and improve related results in the literatures.

Iterative Methods for Generalized Split Feasibility Problems in Hilbert Spaces

Generalized split feasibility problems governed by generalized hybrid mappings are studied via iterative methods. Several algorithms are introduced to solve them. In particular, weak convergence of these algorithms is proved. As tools, averaged mappings and resolvents of maximal monotone operators are technically maneuvered to facilitate the argument of the proofs to the main results. Applications to Mann’s iteration method for nonexpansive mappings and to equilibrium problems are included.

A strong convergence theorem for equilibrium problems and split feasibility problems in Hilbert spaces

Abstract The main purpose of this paper is to introduce an iterative algorithm for equilibrium problems and split feasibility problems in Hilbert spaces. Under suitable conditions we prove that the sequence converges strongly to a common element of the set of solutions of equilibrium problems and the set of solutions of split feasibility problems. Our result extends and improves the corresponding results of some others. MSC:90C25, 90C30, 47J25, 47H09.

Weak convergence theorems of the modified relaxed projection algorithms for the split feasibility problem in Hilbert spaces

In this paper, we introduce a modified relaxed projection algorithm and a modified variable-step relaxed projection algorithm for the split feasibility problem in infinite-dimensional Hilbert spaces. The weak convergence theorems under suitable conditions are proved. Finally, some numerical results are presented, which show the advantage of the proposed algorithms.

General Split Feasibility Problems in Hilbert Spaces

Introducing a general split feasibility problem in the setting of infinite-dimensional Hilbert spaces, we prove that the sequence generated by the purposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.

Cyclic algorithms for split feasibility problems in Hilbert spaces

Cyclic algorithms for split feasibility problems in Hilbert spaces

Approximation of Common Solutions to System of Mixed Equilibrium Problems, Variational Inequality Problem, and Strict Pseudo-Contractive Mappings

AbstractWe introduce an iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions mapping, the set of common solutions of a system of two mixed equilibrium problems and the set of common solutions of the variational inequalities with inverse strongly monotone mappings. Strong convergence theorems are established in the framework of Hilbert spaces. Finally, we apply our results for solving convex feasibility problems in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.

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.

Solving stochastic convex feasibility problems in hilbert spaces

In a stochastic convex feasibility problem connected with a complete probability space (Ω,A,μ) and a family of closed convex sets (Cω)ωεΩ in a real Hilbert space H, one wants to find a point that belongs to Cω for μ almost all ω ε Ω. We present a projection based method where the variable relaxation parameter is defined by a geometrical condition, leading to an iteration sequence that is always weakly convergent to a μ almost common point. We then give a general condition assuring norm convergence of this equation to that μ almost common point