The path (or viscosity) fixed point approximation originates in a series of papers of Browder, starting from the seminal paper [F. E. 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, [C. H. Morales and J. S. Jung, Proc. Am. Math. Soc. 128, 3411–3419 (2000; Zbl 0970.47039)], devoted to the study of path convergence for pseudo-contractive mappings and [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 that has a “rich” set of fixed points, by means of a path defined by a convex combination of and of a certain contractive type function which has a unique fixed point (e.g., a strict contraction, or a strongly pseudo-contraction). As the resulting mapping , defined by a parameter , 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, .
The paper under review is intended to obtain path convergence for approximating fixed points of a pseudo-contraction by means of a path defined by the convex combination of and of a certain strongly pseudo-contraction .