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Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces. (English) Zbl 1132.47056

Let \(E\) be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let \(K\) be a nonempty closed convex subset of \(E\), and let \(T:K\to E\) be a continuous pseudocontraction which satisfies the weakly inward condition. For \(f:K\to K\) any contraction map on \(K\) and every nonempty closed convex and bounded subset of \(K\) having the fixed point property for nonexpansive self-mappings, it is shown that the path \(x\mapsto x_t\), \(t\in [0,1)\) in \(K\), defined by \(x_t=tTx_t+(1- t)f(x_t)\), is continuous and strongly converges to the fixed point of \(T\), which is the unique solution of some co-variational inequality. If, in particular, \(T\) is a Lipschitz pseudocontractive self-mapping of \(K\), it is also shown, under appropriate conditions on the sequences of real numbers \(\{\alpha_n\}, \{\mu_n\}\), that the iteration process:
\[ z_1 \in K,\quad z_{n+1}=\mu_n(\alpha_n Tz_n+(1-\alpha_n)z_n)+(1-\mu_n)f(z_n),\quad n\in \mathbb N, \]
strongly converges to the fixed point of \(T\), which is the unique solution of the same co-variational inequality. Our results propose viscosity approximation methods for Lipschitz pseudocontractions.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J05 Equations involving nonlinear operators (general)
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H10 Fixed-point theorems
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References:

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