|
Strong convergence of path for continuous pseudo-contractive mappings
Author(s):
Claudio
H.
Morales
Journal:
Proc. Amer. Math. Soc.
135
(2007),
2831-2838.
MSC (2000):
Primary 47H10;
Secondary 65J15
Posted:
February 9, 2007
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
The purpose of this paper is to study the convergence of a path that begins at the unique fixed point of a strongly pseudo-contractive operator defined on a closed and convex subset of a reflexive Banach space and converges to a fixed point of a pseudo-contractive mapping. Primarily, it is proven that a convex combination of these two operators is indeed strongly pseudo-contractive under the weakly inward condition. This fact generalizes a result of Barbu for accretive operators.
References:
-
- 1.
- V. Barbu, ``Continuous perturbations of nonlinear
 -accretive operators in Banach spaces", Bollettino U. M. I. 6 (1972), 270-278.MR 0326514 (48:4858) - 2.
- V. Barbu, ``Nonlinear semigroups and differential equations in Banach spaces", Editura Academiei R.S.R, Bucharest, 1976.MR 0390843 (52:11666)
- 3.
- V. Barbu and Th. Preocupanu, ``Convexity and optimization in Banach spaces", Editura Academiei R.S.R, Bucharest, 1978.MR 0513634 (58:23939)
- 4.
- F. F. Browder, ``Nonlinear operators and nonlinear equations of evolution in Banach spaces", Proc. Symp. Pure Math. 18 (1976), part 2. Amer. Math. Soc., Providence, R.I.MR 0405188 (53:8982)
- 5.
- F. E. Browder, ``Nonlinear mappings of non-expansive and accretive type in Banach spaces", Bull. Amer. Math. Soc. 73 (1967), 875-882.MR 0232255 (38:581)
- 6.
- C. E. Chidume and H. Zegeye, ``Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps", Proc. Amer. Math. Soc. 132 (2004), 831-840. MR 2019962 (2004j:47114)
- 7.
- R. Chen, P. K. Lin, and Y. Song, ``An approximation method for strictly pseudocontractive mappings", Nonlinear Anal., 64 (2006), 2527-2535.MR 2215825 (2006k:47114)
- 8.
- M. G. Crandall, ``Differential equations on convex sets", J. Math. Soc. Japan , 22 (1970), 443-455. MR 0268491 (42:3388)
- 9.
- K. Deimling ``Zeros for accretive operators", Manuscripta Math. 13 (1974), 365-374.MR 0350538 (50:3030)
- 10.
- T. Kato, ``Nonlinear semigroups and evolution equations", J. Math. Soc. Japan 19 (1967), 508-520. MR 0226230 (37:1820)
- 11.
- W. A. Kirk `` A fixed point theorem for local pseudo-contractions in uniformly convex Banach spaces", Manuscripta Math. , 30 (1979), 98-102. MR 0552364 (80k:47064)
- 12.
- W. A. Kirk and R. Schöneberg, ``Mapping theorems for local expansions in metric and Banach spaces", J. Math. Anal. Appl. 72 (1979), 114-121. MR 0552327 (81c:47053)
- 13.
- Y. Kobayashi, ``Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups", J. Math. Soc. Japan 27 (1975), 640-665.MR 0399974 (53:3812)
- 14.
- R. H. Martin, ``Differential equations on closed subsets of a Banach space", Trans. Amer. Math. Soc. 179 (1973), 399-414.MR 0318991 (47:7537)
- 15.
- M. M. Day, ``Reflexive Banach spaces not isomorphic to uniformly convex spaces", Bull. Amer. Math. Soc., 47 (1941), 313-317.MR 0003446 (2:221b)
- 16.
- C. H. Morales, ``On the fixed-point theory for local
 -pseudocontractions", Proc. Amer. Math. Soc. 81 (1981), 71-74.MR 0589138 (82c:47072) - 17.
- C. H. Morales and Jong S. Jung, ``Convergence of paths for pseudo-contractive mappings in Banach spaces", Proc. Amer. Math. Soc. 128 (2000), 3411-3419. MR 1707528 (2001b:47090)
- 18.
- A. Moudafi, ``Viscosity approximation methods for fixed-points problems", J. Math. Anal. Appl. 241 (2000), 46-55.MR 1738332 (2000k:47085)
- 19.
- H. K. Xu, ``Iterative algorithms for nonlinear operators", J. London Math. Soc. 66 (2002), 240-256.MR 1911872 (2003e:47114)
- 20.
- H. K. Xu, ``Viscosity approximation methods for nonexpansive mappings", J. Math. Anal. Appl. 298 (2004), 279-297.MR 2086546 (2005e:47133)
- 21.
- V. Zizler, ``On some rotundity and smoothness properties of Banach spaces", Dissert. Math. 87 (1971), 5-33. MR 0300060 (45:9108)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
47H10,
65J15
Retrieve articles in all Journals with MSC
(2000):
47H10,
65J15
Additional Information:
Claudio
H.
Morales
Affiliation:
Department of Mathematics, University of Alabama in Huntsville, Huntsville, Alabama 35899
Email:
morales@math.uah.edu
DOI:
10.1090/S0002-9939-07-08910-1
PII:
S 0002-9939(07)08910-1
Keywords:
Pseudo-contractive operators,
weakly inward condition,
reflexive Banach spaces,
uniformly G\^{a}taux differentiable norm.
Received by editor(s):
May 23, 2006
Posted:
February 9, 2007
Communicated by:
Joseph A. Ball
Copyright of article:
Copyright
2007,
American Mathematical Society
|
Comments: webmaster@ams.org