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Jung, Jong Soo

Convergence of a viscosity iterative method for multivalued nonself-mappings in Banach spaces. (English) Zbl 1275.47124

Abstr. Appl. Anal. 2013, Article ID 369412, 7 p. (2013).
Summary: Let \(E\) be a reflexive Banach space having a weakly sequentially continuous duality mapping \(J_\varphi\) with gauge function \(\varphi\), let \(C\) be a nonempty closed convex subset of \(E\), and let \(T : C \to \mathcal{K}(E)\) be a multivalued nonself-mapping such that \(P_T\) is nonexpansive, where \(P_T(x) = \{u_x \in Tx : ||x - u_x|| = d(x, Tx)\}\). Let \(f : C \to C\) be a contraction with constant \(k\). Suppose that, for each \(v \in C\) and \(t \in (0, 1)\), the contraction defined by \(S_tx = tP_Tx + (1 - t)v\) has a fixed point \(x_t \in C\). Let \(\{\alpha_n\}, \{\beta_n\}\) and \(\{\gamma_n\}\) be three sequences in \((0, 1)\) satisfying approximate conditions. Then, for arbitrary \(x_0 \in C\), the sequence \(\{x_n\}\) generated by \(x_n \in \alpha_nf(x_{n-1}) + \beta_nx_{n-1} + \gamma_nP_T(x_n)\) for all \(n \geq 1\) converges strongly to a fixed point of \(T\).

Cited in 1 Document

MSC:

47J25 Iterative procedures involving nonlinear operators
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

viscosity iterative method multivalued nonself-mappings Banach space; strong convergence
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References:

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