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A hybrid method for monotone variational inequalities involving pseudocontractions. (English) Zbl 1215.49021
Summary: We use strongly pseudocontraction to regularize the following ill-posed monotone variational inequality: finding a point $$x^*$$ with the property $$x^*\in \text{Fix}(T)$$ such that $$\langle(I-S)x^*,x-x^*\rangle\geq 0$$, $$x\in\text{Fix}(T)$$ where $$S,T$$ are two pseudocontractive self-mappings of a closed convex subset $$C$$ of a Hilbert space with the set of fixed points $$\text{Fix}(T)\neq\emptyset$$. Assume the solution set $$\Omega$$ of (VI) is nonempty. In this paper, we introduce one implicit scheme which can be used to find an element $$x^*\in\Omega$$. Our results improve and extend a recent result of X. Lu, H.-K. Xu and X. Yin [Nonlinear Anal., Theory Methods Appl. 71, No. 3–4, A, 1032–1041 (2009; Zbl 1176.90462)].

##### MSC:
 49J40 Variational inequalities 47J20 Variational and other types of inequalities involving nonlinear operators (general) 47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics 49N60 Regularity of solutions in optimal control
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##### References:
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