*(English)*Zbl 1121.47055

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 $T$ 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, $\left\{{x}_{\lambda}\right\}$.

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 $h$.

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47H09 | Mappings defined by “shrinking” properties |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |

65J15 | Equations with nonlinear operators (numerical methods) |