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A strong convergence theorem for contraction semigroups in Banach spaces

  • Hong-Kun Xu (a1)

Extract

We establish a Banach space version of a theorem of Suzuki [8]. More precisely we prove that if X is a uniformly convex Banach space with a weakly continuous duality map (for example, lp for 1 < p < ∞), if C is a closed convex subset of X, and if F = {T (t): t ≥ 0} is a contraction semigroup on C such that Fix(F) ≠ ∅, then under certain appropriate assumptions made on the sequences {αn} and {tn} of the parameters, we show that the sequence {xn} implicitly defined by

for all n ≥ 1 converges strongly to a member of Fix(F).

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References

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[1]Browder, F.E., Fixed point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 12721276.
[2]Browder, F.E., ‘Convergence theorems for sequences of nonlinear operators in Banach spaces’, Math. Z. 100 (1967), 201225.
[3]Bruck, R.E., ‘A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces’, Israel J. Math. 32 (1979), 107116.
[4]Goebel, K. and Reich, S., Uniform convexity, hyperbolic geometry, and nonexpansive mappings (Marcel Dekker, New York, 1984).
[5]Lim, T.C. and Xu, H.K., ‘Fixed point theorems for asymptotically nonexpansive mappings’, Nonlinear Anal. 22 (1994), 13451355.
[6]Reich, S., ‘Strong convergence theorems for resolvents of accretive operators in Banach spaces’, J. Math. Anal. Appl. 75 (1980), 287292.
[7]Shioji, N. and Takahashi, W., ‘Strong convergence theorems for asymptotically nonexpansive mappings in Hilbert spaces’, Nonlinear Anal. 34 (1998), 8799.
[8]Suzuki, T., ‘On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces’, Proc. Amer. Math. Soc. 131 (2002), 21332136.
[9]Xu, H.K., ‘Approximations to fixed points of contraction semigroups in Hilbert spaces’, Numer. Funct. Anal. Optim. 19 (1998), 157163.
[10]Xu, H.K., ‘Iterative algorithms for nonlinear operators’, J. London Math. Soc. 66 (2002), 240256.
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