## Strong convergence for hybrid implicit $$S$$-iteration scheme of nonexpansive and strongly pseudocontractive mappings.(English)Zbl 1472.47073

Summary: Let $$K$$ be a nonempty closed convex subset of a real Banach space $$E$$, let $$S : K \rightarrow K$$ be nonexpansive, and let $$T : K \rightarrow K$$ be Lipschitz strongly pseudocontractive mappings such that $$p \in F \left(S\right) \cap F \left(T\right) = \left\{x \in K : S x = T x = x\right\}$$ and $$\| x-Sy\| \leq \| x-Sy\|$$ and $$\|x-Ty\| \leq \|Tx-Ty\|$$ for all $$x, y \in K$$. Let $$\left\{\beta_n\right\}$$ be a sequence in $$\left[0, 1\right]$$ satisfying (i) $$\sum_{n = 1}^\infty \beta_n = \infty$$; (ii) $$\lim_{n \rightarrow \infty} \beta_n = 0$$. For arbitrary $$x_0 \in K$$, let $$\left\{x_n\right\}$$ be a sequence iteratively defined by $$x_n = S y_n, y_n = \left(1 - \beta_n\right) x_{n - 1} + \beta_n T x_n, n \geq 1$$. Then the sequence $$\left\{x_n\right\}$$ converges strongly to a common fixed point $$p$$ of $$S$$ and $$T$$.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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### References:

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