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Strong convergence of path for continuous pseudo-contractive mappings. (English) Zbl 1121.47055
The path (or viscosity) fixed point approximation originates in a series of papers of Browder, starting from the seminal paper [{\it F. E.\thinspace Browder}, Proc. Nat. Acad. Sci. USA 56, 1080--1086 (1966; Zbl 0148.13502)]. The interest for these kind of fixed point iterative methods seems to have been reawakened by two recent papers, namely, [{\it C. H.\thinspace Morales} and {\it J. S.\thinspace Jung}, Proc. Am. Math. Soc. 128, 3411--3419 (2000; Zbl 0970.47039)], devoted to the study of path convergence for pseudo-contractive mappings and [{\it A. Moudafi}, J. Math. Anal. Appl. 241, 46--55 (2000; Zbl 0957.47039)], devoted to viscosity approximation of fixed points for nonexpansive mappings. The main idea of viscosity (path) methods is to approximate fixed points of a mapping $T$ that has a “rich” set of fixed points, by means of a path defined by a convex combination $U_{\lambda}$ of $T$ and of a certain contractive type function $f$ which has a unique fixed point (e.g., a strict contraction, or a strongly pseudo-contraction). As the resulting mapping $U_{\lambda}$, defined by a parameter $\lambda\in (0,1)$, is itself a strict contraction (or a strongly pseudo-contraction, respectively) and therefore has a unique fixed point, the desired path is obtained as this unique fixed point, $\{x_\lambda\}$. The paper under review is intended to obtain path convergence for approximating fixed points of a pseudo-contraction $T$ by means of a path defined by the convex combination of $T$ and of a certain strongly pseudo-contraction $h$.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
65J15Equations with nonlinear operators (numerical methods)
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