Strong convergence of path for continuous pseudo-contractive mappings. (English) Zbl 1121.47055

Morales, Claudio H.

Proc. Am. Math. Soc. 135, No. 9, 2831-2838 (2007).

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 \(h\).

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

Citations:

Zbl 0148.13502; Zbl 0970.47039; Zbl 0957.47039

References:

[1]	V. Barbu, Continuous perturbations of nonlinear \?-accretive operators in Banach spaces, Boll. Un. Mat. Ital. (4) 6 (1972), 270 – 278 (English, with Italian summary). · Zbl 0256.47053
[2]	Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. · Zbl 0328.47035
[3]	V. Barbu and Th. Precupanu, Convexity and optimization in Banach spaces, Revised edition, Editura Academiei, Bucharest; Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1978. Translated from the Romanian. · Zbl 0379.49010
[4]	Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R. I., 1976, pp. 1 – 308.
[5]	Felix E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces., Bull. Amer. Math. Soc. 73 (1967), 875 – 882. · Zbl 0176.45302
[6]	C. E. Chidume and H. Zegeye, Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps, Proc. Amer. Math. Soc. 132 (2004), no. 3, 831 – 840. · Zbl 1051.47041
[7]	Rudong Chen, Pei-Kee Lin, and Yisheng Song, An approximation method for strictly pseudocontractive mappings, Nonlinear Anal. 64 (2006), no. 11, 2527 – 2535. · Zbl 1237.47069
[8]	Michael G. Crandall, Differential equations on convex sets, J. Math. Soc. Japan 22 (1970), 443 – 455. · Zbl 0224.34006
[9]	Klaus Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974), 365 – 374. · Zbl 0288.47047
[10]	Tosio Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508 – 520. · Zbl 0163.38303
[11]	W. A. Kirk, A fixed point theorem for local pseudocontractions in uniformly convex spaces, Manuscripta Math. 30 (1979/80), no. 1, 89 – 102. · Zbl 0422.47032
[12]	W. A. Kirk and R. Schöneberg, Mapping theorems for local expansions in metric and Banach spaces, J. Math. Anal. Appl. 72 (1979), no. 1, 114 – 121. · Zbl 0415.54008
[13]	Yoshikazu Kobayashi, Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Math. Soc. Japan 27 (1975), no. 4, 640 – 665. · Zbl 0313.34068
[14]	R. H. Martin Jr., Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179 (1973), 399 – 414. · Zbl 0293.34092
[15]	Mahlon M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), 313 – 317. · Zbl 0024.41401
[16]	Claudio Morales, On the fixed-point theory for local \?-pseudocontractions, Proc. Amer. Math. Soc. 81 (1981), no. 1, 71 – 74. · Zbl 0479.47050
[17]	Claudio H. Morales and Jong Soo Jung, Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3411 – 3419. · Zbl 0970.47039
[18]	A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000), no. 1, 46 – 55. · Zbl 0957.47039
[19]	Hong-Kun Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), no. 1, 240 – 256. · Zbl 1013.47032
[20]	Hong-Kun Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), no. 1, 279 – 291. · Zbl 1061.47060
[21]	V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. Rozprawy Mat. 87 (1971), 33 pp. (errata insert). · Zbl 0231.46036