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Existence and algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities in Banach spaces. (English) Zbl 1291.49004

Summary: We introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities. First, the existence of solutions of the auxiliary problems for the generalized strongly nonlinear mixed variational-like inequalities is shown. Second, an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequalities is constructed by using this existence result. Finally, not only the existence of solutions of the generalized strongly nonlinear mixed variational-like inequalities is shown, but also the convergence of iterative sequences generated by the algorithm is also proven.

MSC:

49J40 Variational inequalities
65K10 Numerical optimization and variational techniques
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