Ram U. Verma

Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods

AbstractLet K be a nonempty closed convex subset of a real Hilbert space H. The approximate solvability of a system of nonlinear variational inequality problems, based on the convergence of projection methods, is discussed as follows: find an element (x*, y*)∈K×K such that

Nonlinear relaxed cocoercive variational inclusions involving (A, η )-accretive mappings in banach spaces

\begin{gathered} \left\langle {\rho {\rm T}(y^* ,x^* ) + x^* - y^* ,x - x^* } \right\rangle \geqslant 0,{\text{ }}\forall x \in K{\text{ and }}\rho > 0, \hfill \\ \left\langle {\eta {\rm T}(x^* ,y^* ) + y^* - x^* ,x - x^* } \right\rangle \geqslant 0,{\text{ }}\forall x \in K{\text{ and }}\eta > 0, \hfill \\ \end{gathered}

A-monotone nonlinear relaxed cocoercive variational inclusions

where T: K×K→H is a nonlinear mapping on K×K.

General nonlinear variational inclusion problems involving A-monotone mappings

A new notion of the A -monotonicity is introduced, which generalizes the H -monotonicity. Since the A -monotonicity originates from hemivariational inequalities, and hemivariational inequalities are connected with nonconvex energy functions, it turns out to be a useful tool proving the existence of solutions of nonconvex constrained problems as well.

A hybrid proximal point algorithm based on the (A,η)-maximal monotonicity framework

In this paper, we introduce a new concept of (A, @h)-accretive mappings, which generalizes the existing monotone or accretive operators. We study some properties of (A, @h)-accretive mappings and define resolvent operators associated with (A, @h)-accretive mappings. By using the new resolvent operator technique, we also construct a new perturbed iterative algorithm with mixed errors for a class of nonlinear relaxed Cocoercive variational inclusions involving (A, @h)-accretive mappings and study applications of (A, @h)-accretive mappings to the approximation-solvability of this class of nonlinear relaxed Cocoercive variational inclusions in q-uniformly smooth Banach spaces. Our results improve and generalize the corresponding results of recent works.

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

Abstract Sensitivity analysis for generalized strongly monotone variational inclusions based on the ( A , η ) -resolvent operator technique is investigated. The results obtained encompass a broad range of results.

A general framework for the over-relaxed A-proximal point algorithm and applications to inclusion problems

Based on the notion of A — monotonicity, a new class of nonlinear variational inclusion problems is presented. Since A — monotonicity generalizes H — monotonicity (and in turn, generalizes maximal monotonicity), results thus obtained, are general in nature.

The generalized relaxed proximal point algorithm involving A-maximal-relaxed accretive mappings with applications to Banach spaces

Based on the notion of the A-monotonicity, the solvability of a class of nonlinear variational inclusions using the generalized resolvent operator technique is given. The results obtained are general in nature.