General Variational Inclusion Problem Research Articles

On over-relaxed ( A , η , m ) -proximal point algorithm frameworks with errors and applications to general variational inclusion problems

The purpose of this paper is to provide some remarks for the main results of the paper Verma (Appl. Math. Lett. 21:142-147, 2008). Further, by using the generalized proximal operator technique associated with the -monotone operators, we discuss the approximation solvability of general variational inclusion problem forms in Hilbert spaces and the convergence analysis of iterative sequences generated by the over-relaxed -proximal point algorithm frameworks with errors, which generalize the hybrid proximal point algorithm frameworks due to Verma. MSC:47H05, 49J40.

Existence Results for General Inclusions Using Generalized KKM Theorems with Applications to Minimax Problems

Applying generalized KKM-type theorems established in our previous paper (Khanh et al. in Nonlinear Anal. 71:1227–1234, 2009), we prove the existence of solutions to a general variational inclusion problem, which contains most of the existing results of this type. As applications, we obtain minimax theorems in various settings and saddle-point theorems in particular. Examples are given to explain advantages of our results.

General implicit variational inclusion problems based on A-maximal ( m)-relaxed monotonicity (AMRM) frameworks

A general framework for an algorithmic procedure based on the variational convergence of operator sequences involving A-maximal ( m)-relaxed monotone (AMRM) mappings in a Hilbert space setting is developed, and then it is applied to approximating the solution of a general class of nonlinear implicit inclusion problems involving A-maximal ( m)-relaxed monotone mappings. Furthermore, some specializations of interest on existence theorems and corresponding approximation solvability theorems on H-maximal monotone mappings are included that may include several other results for general variational inclusion problems on general maximal monotonicity in the literature.

The solution existence of general variational inclusion problems

We propose general variational inclusion problems which are slightly different from corresponding problems considered in several recent papers in the literature and show that they are advantageous. Sufficient conditions for the solution existence are established. As applications we derive consequences for several special cases of variational inclusion problems, quasioptimization problems, equilibrium problems and implicit variational inequalities and show that they improve the results of some recent existing papers.

Existence conditions in variational inclusions with constraints

In this article we study a general variational inclusion problem with constraints. An existence theorem is given for the scalar problem which allows us to derive several results on existence of solutions of variational inequality and variational inclusion models in which the set-valued data are present.