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Two step algorithm for solving regularized generalized mixed variational inequality problem. (English) Zbl 1196.49008

Summary: We consider a new class of regularized (nonconvex) generalized mixed variational inequality problems in real Hilbert space. We give the concepts of partially relaxed strongly mixed monotone and partially relaxed strongly \(\theta\)-pseudomonotone mappings, which are extension of the concepts given by F. Q. Xia and X. P. Ding [Appl. Math. Comput. 188, No. 1, 173–179 (2007; Zbl 1123.65065)], M. A. Noor [J. Appl. Math. Comput. 23, No. 1–2, 183–191 (2007; Zbl 1111.49005)] and K. R. Kazmi, A. Khaliq and A. Raouf [Math. Inequal. Appl. 10, No. 3, 677–691 (2007; Zbl 1127.47052)]. Further we use the auxiliary principle technique to suggest a two-step iterative algorithm for solving regularized (nonconvex) generalized mixed variational inequality problems. We prove that the convergence of the iterative algorithm requires only the continuity, partially relaxed strongly mixed monotonicity and partially relaxed strongly \(\theta\)-pseudomonotonicity. The theorems presented in this paper improve and generalize the previously known results for solving equilibrium problems and variational inequality problems involving the nonconvex (convex) sets, see for example Noor [loc. cit.], L.-P. Pang, J. Shen and H.-S. Song [Comput. Math. Appl. 54, No. 3, 319–325 (2007; Zbl 1131.49010)], and Xia and Ding [loc. cit.].

MSC:

49J40 Variational inequalities
90C30 Nonlinear programming
49M30 Other numerical methods in calculus of variations (MSC2010)
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