## Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces.(English)Zbl 1222.47092

Summary: Let $$E$$ be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, $$D$$ a nonempty closed convex subset of $$E$$, and $$T:D\rightarrow K(E)$$ a nonself multimap such that $$F(T)\neq \varnothing$$ and $$P_T$$ is nonexpansive, where $$F(T)$$ is the fixed point set of $$T$$, $$K(E)$$ is the family of nonempty compact subsets of $$E$$ and $$P_T(x)=\{u_x\in T_X: \|x-u_x\| = d(x,Tx)\}$$. Suppose that $$D$$ is a nonexpansive retract of $$E$$ and that, for each $$v\in D$$ and $$t\in (0,1)$$, the contraction $$S_t$$ defined by $$S_tx=tP_Tx+(1 - t)v$$ has a fixed point $$x_t\in D$$. Let $$\{\alpha _n\},\{\beta _n\}$$ and $$\{\gamma _n\}$$ be three real sequences in (0,1) satisfying approximate conditions. Then, for fixed $$u\in D$$ and arbitrary $$x_{0}\in D$$, the sequence $$\{x_n\}$$ generated by
$x_n\in \alpha _nu+\beta _nx_{n-1}+\gamma _nP_T(x_n), \quad n\geq 0,$
converges strongly to a fixed point of $$T$$.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H05 Monotone operators and generalizations 47H10 Fixed-point theorems
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