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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 [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, \(\{x_\lambda\}\).

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 \(h\).

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 \(h\).

Reviewer: Vasile Berinde (Baia Mare)

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

47H10 | Fixed-point theorems |

65J15 | Numerical solutions to equations with nonlinear operators |

### Keywords:

reflexive Banach space; pseudocontractive mapping; accretive mapping; fixed point; path; convergence theorem; weakly inward condition
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\textit{C. H. Morales}, Proc. Am. Math. Soc. 135, No. 9, 2831--2838 (2007; Zbl 1121.47055)

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### References:

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