Abstract
Due to the importance of Yosida approximation operator, we generalized the variational inequality problem and its equivalent problems by using Yosida approximation operator. The aim of this work is to introduce and study a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation. We prove an existence result together with convergence analysis for Yosida proximal operator equation involving XOR-operation. For this purpose, we establish an algorithm based on fixed point formulation. Our approach is based on a proximal operator technique involving a subdifferential operator. As an application of our main result, we provide a numerical example using the MATLAB program R2018a. Comparing different iterations, a computational table is assembled and some graphs are plotted to show the convergence of iterative sequences for different initial values.
Highlights
Stampacchia [1] and Ficchera [2] originated the study of variational inequalities, separately
D thesis introduced nonlinear complementarity problem which is closely related to Hartman and Stampacchia variational inequality problem. e proximal operator technique is useful to establish equivalence between variational inequalities and proximal operator equations. e proximal operator equation approach is used to solve variational inequalities and related optimization problems
We introduce and study three new problems, that is, a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation
Summary
Stampacchia [1] and Ficchera [2] originated the study of variational inequalities, separately. E proximal operator equation approach is used to solve variational inequalities and related optimization problems. E possible strategy of solving stochastic notion of multivalued differential equation in finite dimensional space is based on Yosida approximation approach. E existence of multivalued stochastic differential equation in finite dimensional space with a time-independent, deterministic maximal monotone operator through Yosida approximation approach was first discussed by Petterson [14]. For more details and recent past developments about complementarity problems, variational inequalities, proximal operator equations, Yosida approximation operator, and related topics, we refer to [15–28] and references therein. Motivated by all the above discussed concepts, in this paper, we consider and study a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation. To obtain the solution of Yosida proximal operator equation involving XOR-operation, we define an algorithm based on fixed point formulation. A comparison of different iterations is assembled in the form of a computational table, and the convergence of the iterative sequences is shown by some graphs for different initial values
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