Approximation solvability of a class of A-monotone implicit variational inclusion problems in semi-inner product spaces
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 is r-strongly monotone and is H-monotone, then the generalized resolvent operator is - Lipschitz continuous. Proof For any , we haveThis implies thatSince H is r-strongly monotone and M is H-monotone, we have
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 , generate the sequence such thatwhere are A-monotone set valued mappings for
Acknowledgments
The authors are thankful to the referees for their valuable suggestions.
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