## On a two-step algorithm for hierarchical fixed point problems and variational inequalities.(English)Zbl 1180.47040

The paper is concerned with the variational inequality problem of finding $$x^* \in \text{Fix}(T)$$ with $$\langle (I-S)x^*, x-x^* \rangle \geq 0$$ for all $$x\in \text{Fix}(T)$$, where $$T,S: C\to C$$ are nonexpansive mappings such that Fix$$(T)$$, the set of fixed points set of $$T$$, is nonempty, and $$C$$ is a closed convex subset of a Hilbert space $$H$$. Let $$f: C \to C$$ be a contraction. The authors study convergence properties of the iterative process $$x_{n+1}=\alpha_n f(x_n)+(1-\alpha_n) T y_n$$, $$y_n=\beta_n S x_n+(1-\beta_n) x_n$$, where $$\alpha_n,\beta_n \in [0,1]$$.

### MSC:

 47J20 Variational and other types of inequalities involving nonlinear operators (general) 49J40 Variational inequalities 47J25 Iterative procedures involving nonlinear operators
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### References:

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