Abstract
In this paper, the auxiliary principle technique is extended to study a system of generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities problem in Hilbert spaces. First, we establish the existence of solutions of the corresponding system of auxiliary variational inequalities problem. Then, using the existence result, we construct a new iterative algorithm. Finally, both the existence of solutions of the original problem and the convergence of iterative sequences generated by the algorithm are proved. We give an affirmative answer to the open problem raised by Noor et al. (Korean J. Comput. Appl. Math. 1:73-89, 1998; J. Comput. Appl. Math. 47:285-312, 1993). Our results improve and extend some known results.
Highlights
Throughout the paper, let I = {, } be an index set, let, for each i ∈ I, Hi be a Hilbert space with inner product ·, · i and norm · i, and CB(Hi) be the family of all nonempty bounded closed subsets of Hi
We extend the auxiliary principle technique to study the generalized setvalued strongly nonlinear mixed implicit quasi-variational-like inequalities problem ( . ) in Hilbert spaces
Using the existence result, we construct a new iterative algorithm. Both the existence of solutions of the original problem and the convergence of iterative sequences generated by the algorithm are proved
Summary
We consider the following generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities problem: Find (x, y) ∈ K (x) × K (y), u ∈ A x, v ∈ B x, u ∈ A y, v ∈ B y, and wi ∈ Fi(x, y) such that In the system of the generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities problem
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