Approximation solvability of a class of A-monotone implicit variational inclusion problems in semi-inner product spaces

https://doi.org/10.1016/j.amc.2014.02.095Get rights and content

Abstract

This paper deals with the existence of solutions for a class of nonlinear implicit variational inclusion problems in semi-inner product spaces. We construct an iterative algorithm for approximating the solution for the class of implicit variational inclusions problems involving A-monotone and H-monotone operators by using the generalized resolvent operator technique.

Section snippets

Introduction and preliminaries

Variational inequalities have been the subject of considerable research and have profound contributions in a large variety of problems arising in mechanics, physics, optimization and control theory, economics and transportation equilibrium problems, and engineering sciences. In [16], Stampacchia introduced the classical variational inequality problem. Because of its wide applications, the classical variational inequality problem has been generalized in different directions. Variational

Main results

We need the following lemmas for the proof of our results.

Lemma 2.1

If H:XX is r-strongly monotone and M:X2X is H-monotone, then the generalized resolvent operator Jρ,HM is 1r- Lipschitz continuous.

Proof

For any x,yX, we haveJρ,HM(x)=(H+ρM)-1(x),Jρ,HM(y)=(H+ρM)-1(y).This implies that1ρx-HJρ,HM(x)MJρ,HM(x),1ρy-HJρ,HM(y)MJρ,HM(y).Since H is r-strongly monotone and M is H-monotone, we havex-yJρ,HM(x)-Jρ,HM(y)x-y,Jρ,HM(x)-Jρ,HM(y)=x-y-HJρ,HM(x)-HJρ,HM(y),Jρ,HM(x)-Jρ,HM(y)+HJρ,HM(x)-HJρ,HM(y),Jρ,HM(x)-Jρ,HM(

Approximation solvability

In Theorem 2.2, we have discussed the existence of solution of the variational inclusion problem (2.3). In this section we outline an iterative method to approximate a solution of the variational inclusion problem (2.3). On the basis of Theorem 2.1, we develop an iterative algorithm to find the solution.

Algorithm 3.1

For an arbitrarily chosen initial point a0X, generate the sequence {ak} such thatak+1=(1-αk)ak+αkak-g(ak)+Jρ,AMkA(g(ak))-ρ(S-T)(ak)+ρx,where Mk:X2X are A-monotone set valued mappings for k=0,1,

Acknowledgments

The authors are thankful to the referees for their valuable suggestions.

View full text