Algorithm for Solving a New System of Generalized Variational Inclusions in Hilbert Spaces

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.

-Mixed Cocoercive Operators with an Application for Solving Variational Inclusions in Hilbert Spaces

“…Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as 2 Journal of Function Spaces and Applications an extension and generalization of some known results [14][15][16][17][18][19][20][21][22]. For illustration of Definitions 4 and 7 and Theorem 20, Examples 5, 8, and 21 are given, respectively.…”

“…Motivated by the recent work going in this direction, we consider a class of ((⋅, ⋅), (⋅, ⋅))-mixed cocoercive operators, a natural generalization of monotone (accretive) operators in Hilbert (Banach) spaces. For details, we refer to see [8,9,[12][13][14][18][19][20][21][22]. We extend the concept of resolvent operators associated with (⋅, ⋅)-cocoercive operators to the ((⋅, ⋅), (⋅, ⋅))-mixed cocoercive operators and prove that the resolvent operator of ((⋅, ⋅), (⋅, ⋅))-mixed cocoercive operator is single valued and Lipschitz continuous.…”

Algorithm for Solving a New System of Generalized Nonlinear Quasi-Variational-Like Inclusions in Hilbert Spaces

“…The method based on the resolvent operator technique is a generalization of the projection method and has been widely used to solve variational inclusions. For details, we refer to see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”