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Let C be a convex compact subset of a uniformly convex Banach space. Let {Tt}t≥0 be a strongly-continuous nonexpansive semigroup on C. Consider the iterative process defined by the sequence of equations xk+1=ckTtk+1(xk+1)+(1−ck)xk. We prove that, under certain conditions on {ck} and {tk}, the sequence {xk}∞n=1 converges strongly to a common fixed point of the semigroup {Tt}t≥0. There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, and also weak convergence results in Banach spaces that are simultaneously uniformly convex and uniformly smooth. In this paper, we do not assume the Opial property or uniform smoothness of the norm.
Wydawca
Czasopismo
Rocznik
Tom
Strony
203--208
Opis fizyczny
Bibliogr. 15 poz.
Twórcy
autor
- School of Mathematics and Statistics, University of New South Wales Sydney, NSW 2052, Australia
Bibliografia
- [1] F.E. Browder, Fixed point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1272 - 1276.
- [2] F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1041 - 1044.
- [3] G. E. Kim, and W. Takahashi Approximating common fixed points of nonexpansive semigroups in Banach spaces, Sci. Math. Japon., 63 (2006), 31 - 36.
- [4] W.M. Kozlowski, Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 377 (2011), 43 - 52.
- [5] W.M. Kozlowski, Common fixed points for semigroups of pointwise Lipschitzian mappings in Banach spaces, Bull. Austral. Math Soc., 84 (2011), 353 - 361.
- [6] W.M. Kozlowski, On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces, Comment. Math., 52.2 (2012), 113 - 136.
- [7] W.M. Kozlowski, Pointwise Lipschitzian mappings in uniformly convex and uniformly smooth Banach spaces, Nonlinear Analysis, 84 (2013), 50 - 60.
- [8] W.M. Kozlowski, On convergence of iteration processes for nonexpansive semigroups in uniformly convex and uniformly smooth Banach spaces, Preprint, 2014.
- [9] S. Reich, Strong convergence theorems for for resolvents of accretive operators in Banach in Banach spaces, J. Math. Anal. Appl., 75 (1980), 287 - 292.
- [10] S. Saejung, Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals, Fixed Point Theory and Applications, 2008:745010 (2008).
- [11] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153 - 159.
- [12] T. Suzuki, On strong convergence to common fixed points of nonexpansive mappings in Hilbert spaces, Proc. Amer. Math. Soc., 131.7 (2002), 2133 - 2136.
- [13] D.V. Thong, An implicit iteration process for nonexpansive semigroups, Nonlinear Anal., 74 (2011), 6116 - 6120.
- [14] H-K. Xu, A strong convergence theorem for contraction semigroups in Banach spaces, Bull. Austral. Math. Soc., 72 (2005), 371 - 379.
- [15] E. Zeidler, Nonlinear Functional Analysis and its Applications I, Fixed Points Theorems, Springer-Verlag, New York/Heidelberg, 1986.
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Bibliografia
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