Abstract
We introduce and study a new system of generalized variational inclusions involving -cocoercive and relaxed -cocoercive operators, which contain the systems of variational inclusions and the systems of variational inequalities, variational inclusions, and variational inequalities as special cases. By using the resolvent technique for the -cocoercive operators, we prove the existence of solutions and the convergence of a new iterative algorithm for this system of variational inclusions in Hilbert spaces. An example is given to justify the main result. Our results can be viewed as a generalization of some known results in the literature.
Highlights
Variational inclusions have been widely studied in recent years
We introduce and study a new system of generalized variational inclusions involving H(⋅, ⋅)-cocoercive and relaxed (p,q)-cocoercive operators, which contain the systems of variational inclusions and the systems of variational inequalities, variational inclusions, and variational inequalities as special cases
By using the resolvent technique for the H(⋅, ⋅)-cocoercive operators, we prove the existence of solutions and the convergence of a new iterative algorithm for this system of variational inclusions in Hilbert spaces
Summary
Variational inclusions have been widely studied in recent years. The theory of variational inclusions includes variational, quasi-variational, variational-like inequalities as special cases. By using the resolvent technique for the H(⋅, ⋅)-cocoercive operators, we prove the existence of solutions and the convergence of a new iterative algorithm for this system of variational inclusions in Hilbert spaces. (iii) H(A, ⋅) is said to be δ1-Lipschitz continuous with respect to A if there exists a constant δ1 > 0 such that
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