Generalized -Cocoercive Operators and Generalized Set-Valued Variational-Like Inclusions

Abstract

We investigate a new class of cocoercive operators named generalized -cocoercive operators in Hilbert spaces. We prove that generalized -cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with -cocoercive operators to the generalized -cocoercive operators. Some examples are given to justify the definition of generalized -cocoercive operators. Further, we consider a generalized set-valued variational-like inclusion problem involving generalized -cocoercive operator. In terms of the new resolvent operator technique, we give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of iterative algorithm under some suitable conditions. 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.…”

“…In order to study various variational inequalities and variational inclusions, Fang and Huang, Kazmi and Khan, and Lan et al investigated many generalized operators such as -monotone [8], -accretive [9], ( , )-proximal point [10], ( , )-accretive [11], ( , )monotone [12], and ( , )-accretive mappings [13]. Recently, Zou and Huang [14] introduced and studied (⋅, ⋅)-accretive operators; Kazmi et al [15][16][17] introduced and studied generalized (⋅, ⋅)-accretive operators and (⋅, ⋅)--proximal point mapping; Xu and Wang [18] introduced and studied ( (⋅, ⋅), )-monotone operators; Ahmad et al [19] introduced and studied (⋅, ⋅)-cocoercive operators and Husain and Gupta [20,21] introduced and studied (⋅, ⋅)-mixed operator and generalized (⋅, ⋅, ⋅)--cocoercive operators.…”

“…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.…”

Performance on ICI Self-Cancellation in FFT-OFDM and DCT-OFDM System

“…Deepmala [21] and Mishra [22] have discussed approximations of signals (functions) using fixed point theorems (PD-operator) and summability operators as double digital filter. Acar et al [23] and Husain et al [24] also studied approximation properties of certain operators and generalized (⋅, ⋅, ⋅)--cocoercive operators and generalized set-valued variational-like inclusions and their applications in engineering fields. Deepmala [25] studied the existence theorems for solvability of a functional equation arising in dynamic programming.…”

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].…”