Convergence Analysis of a Three-Step Iterative Algorithm for Generalized Set-Valued Mixed-Ordered Variational Inclusion Problem

This manuscript aims to study a generalized, set-valued, mixed-ordered, variational inclusion problem involvingH(·, ·)-compression XOR-αM-non-ordinary difference mapping and relaxed cocoercive mapping in real-ordered Hilbert spaces. The resolvent operator associated withH(·, ·)compression XOR-αM-non-ordinary difference mapping is defined, and some of its characteristics are discussed. We prove existence and uniqueness results for the considered generalized, set-valued, mixed-ordered, variational inclusion problem. Further, we put forward a three-step iterative algorithm using a ⊕ operator, and analyze the convergence of the suggested iterative algorithm under some mild assumptions. Finally, we reconfirm the existence and convergence results by an illustrative numerical example.

[1]  A. Nagurney.  Network Economics: A Variational Inequality Approach , 1992 .

[2]  Ram U. Verma,et al.  Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods , 2004 .

[3]  M. Akram,et al.  Graph Convergence for the H ( · , · ) -Co-accretive Mapping with an Application , 2013 .

[4]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[5]  Mijanur Rahaman,et al.  Convergence analysis and stability of perturbed three-step iterative algorithm for generalized mixed ordered quasi-variational inclusion involving XOR operator , 2020, Optimization.

[6]  Mike Smith,et al.  The existence, uniqueness and stability of traffic equilibria , 1979 .

[7]  Xinlong Weng,et al.  Fixed point iteration for local strictly pseudo-contractive mapping , 1991 .

[8]  Mohammad Akram,et al.  System of generalized set-valued parametric ordered variational inclusion problems with operator $\oplus$ in ordered Banach spaces , 2021 .

[10]  R. Agarwal,et al.  Fixed point theorems in partially ordered Banach spaces with applications to nonlinear fractional evolution equations , 2017 .

[11]  H. H. Schaefer.  Banach Lattices and Positive Operators , 1975 .

[12]  Jen-Chih Yao,et al.  On the generalized complementarity problem , 1994, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[13]  M. Noor,et al.  New System Of General Nonconvex Variational Inequalities , 2010 .

[14]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[15]  Mao-Ming Jin,et al.  Stability for a New Class of GNOVI with (γG,λ)-Weak-GRD Mappings in Positive Hilbert Spaces , 2016 .

[16]  Yihong Du.  Fixed points of increasing operators in ordered Banach spaces and applications , 1990 .

[17]  Yuehu Wang,et al.  Applications of order-theoretic fixed point theorems to discontinuous quasi-equilibrium problems , 2015 .

[18]  Sjur Didrik Flåm,et al.  A Continuous Approach to Oligopolistic Market Equilibrium , 1990, Oper. Res..

[20]  Jinde Cao,et al.  A Recurrent Neural Network Based on Projection Operator for Extended General Variational Inequalities , 2010, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[21]  C. E. Lemke,et al.  Bimatrix Equilibrium Points and Mathematical Programming , 1965 .

[22]  Hong-gang Li,et al.  A nonlinear inclusion problem involving (α, λ)-NODM set-valued mappings in ordered Hilbert space , 2012, Appl. Math. Lett..

[24]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[25]  R. U. Verma,et al.  Projection methods, algorithms, and a new system of nonlinear variational inequalities , 2001 .

[26]  Yeol Je Cho,et al.  ALGORITHMS FOR SYSTEMS OF NONLINEAR VARIATIONAL INEQUALITIES , 2004 .