# zbMATH — the first resource for mathematics

Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces. (English) Zbl 1102.47048
Summary: 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 $$T: K\rightarrow K$$ be a uniformly continuous pseudocontraction. If $$f: K\rightarrow K$$ is any contraction map on $$K$$ and if every nonempty closed convex and bounded subset of $$K$$ has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers $$\{\alpha_n\}$$, $$\left\{ {\mu _n } \right\}$$, that the iteration process $$z_1\in K$$, $$z_{n+1}=\mu_n(\alpha_nTz_n+(1-\alpha_n)z_n)+(1-\mu_n)f(z_n)$$, $$n\in\mathbb{N}$$, strongly converges to the fixed point of $$T$$, which is the unique solution of some variational inequality, provided that $$K$$ is bounded.

##### MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
Full Text:
##### References:
 [1] Deimling, K, Zeros of accretive operators, Manuscripta Mathematica, 13, 365-374, (1974) · Zbl 0288.47047 [2] Halpern, B, Fixed points of nonexpanding maps, Bulletin of the American Mathematical Society, 73, 957-961, (1967) · Zbl 0177.19101 [3] Kato, T, Nonlinear semigroups and evolution equations, Journal of the Mathematical Society of Japan, 19, 508-520, (1967) · Zbl 0163.38303 [4] Lim, T-C, On characterizations of Meir-Keeler contractive maps, Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods, 46, 113-120, (2001) · Zbl 1009.54044 [5] Martin, RH, Differential equations on closed subsets of a Banach space, Transactions of the American Mathematical Society, 179, 399-414, (1973) · Zbl 0293.34092 [6] Morales, CH, On the fixed-point theory for local [inlineequation not available: see fulltext.]-pseudocontractions, Proceedings of the American Mathematical Society, 81, 71-74, (1981) · Zbl 0479.47050 [7] Morales, CH; Jung, JS, Convergence of paths for pseudo-contractive mappings in Banach spaces, Proceedings of the American Mathematical Society, 128, 3411-3419, (2000) · Zbl 0970.47039 [8] Moudafi, A, Viscosity approximation methods for fixed-points problems, Journal of Mathematical Analysis and Applications, 241, 46-55, (2000) · Zbl 0957.47039 [9] Schu, J, Approximating fixed points of Lipschitzian pseudocontractive mappings, Houston Journal of Mathematics, 19, 107-115, (1993) · Zbl 0804.47057 [10] Udomene A: Path convergence, approximation of fixed points and variational solutions of pseudocontractions in Banach spaces. submitted to Nonlinear Analysis, TMA · Zbl 1061.47060 [11] Xu, H-K, Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society. Second Series, 66, 240-256, (2002) · Zbl 1013.47032 [12] Xu, H-K, Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 279-291, (2004) · Zbl 1061.47060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.