## A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense.(English)Zbl 0981.47037

Let $$X$$ be a uniformly convex real Banach space, $$C$$ a nonempty bounded closed convex subset of $$X$$. If the inequality $\limsup_{t\to\infty} \sup_{x,y\in C}(\|T(t)x- T(t)y\|-\|x-y\|)\leq 0,$ holds, then the $$C_0$$-semigroup $$\{T(t)\}_{t\geq 0}$$ of (nonlinear) selfmappings of $$C$$ is called “asymptotically nonexpansive in the intermediate sense”. One says that a function $$u: [0,\infty)\to C$$ is an “almost orbit” of $$\{T(t)\}_{t\geq 0}$$ if $\lim_{s\to \infty} \Biggl(\sup_{t\in [0,\infty)}\|u(t+ s)- T(t) u(s)\|\Biggr)= 0.$ A long part of the paper is devoted to proving that if $$X^*$$ has the Kadec-Klee property and $$\{T(t)\}_{t\geq 0}$$ is asymptotically nonexpansive in the intermediate sense, then every continuous almost orbit $$u$$ of $$\{T(t)\}_{t\geq 0}$$ is weakly almost convergent to some $$y\in \text{Fix}(T)$$ (the set of common fixed points of $$T(t)$$), i.e. $$w$$-$$\lim_{t\to\infty} {1\over t} \int^t_0 u(h+\tau) d\tau= y$$ uniformly in $$h\geq 0$$.

### MSC:

 47H20 Semigroups of nonlinear operators 47D06 One-parameter semigroups and linear evolution equations 46B20 Geometry and structure of normed linear spaces
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### References:

 [1] Baillon, J.-B., Un théorème de type ergodique pour LES contractions nonlinéaires dans un espace de Hilbert, C. R. acad. sci. Paris, 280, 1511-1514, (1975) · Zbl 0307.47006 [2] J.-B. Baillon, Comportement asymptotique des contractions et semi-groupes de contractions; Equations de Schrödinger nonlinéaires et divers, Thèses présentées à, l’Université Paris VI, 1978. [3] Baillon, J.-B.; Brezis, H., Une remarque sur le comportement asymptotique des semigroupes non linéaires, Houston J. math., 2, 5-7, (1976) · Zbl 0318.47039 [4] Browder, F.E., Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. symposia in pure mathematics, 18, (1976), Amer. Math. Soc Providence · Zbl 0176.45301 [5] Bruck, R.E., On the almost-convergence of iterates of a nonexpansive mapping in Hilbert space and the structure of the weak ω-limit set, Israel J. math., 29, 1-16, (1978) · Zbl 0367.47037 [6] Bruck, R.E., A simple proof of the Mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. math., 32, 107-116, (1979) · Zbl 0423.47024 [7] Bruck, R.E., On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. math., 38, 304-314, (1981) · Zbl 0475.47037 [8] Bruck, R.E., Asymptotic behavior of nonexpansive mappings, Contemp. math., 18, 1-47, (1983) · Zbl 0528.47039 [9] Bruck, R.E.; Kuczumow, T.; Reich, S., Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. math., 65, 169-179, (1993) · Zbl 0849.47030 [10] J. Garcı́a, Falset, W. Kaczor, T. Kuczumow, and, S. Reich, Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal, in press. · Zbl 0983.47040 [11] Goebel, K.; Kirk, W.A., Topics in metric fixed point theory, (1990), Cambridge Univ. Press Cambridge · Zbl 0708.47031 [12] Hirano, N.; Kido, K.; Takahashi, W., Asymptotic behavior of commutative semigroups of nonexpansive mappings in Banach spaces, Nonlinear anal., 10, 229-249, (1986) · Zbl 0594.47053 [13] Hirano, N.; Kido, K.; Takahashi, W., Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces, Nonlinear anal., 12, 1269-1281, (1988) · Zbl 0679.47031 [14] Hirano, N.; Takahashi, W., Nonlinear ergodic theorems for nonexpansive mappings in Hilbert spaces, Kodai math. J., 2, 11-25, (1979) · Zbl 0404.47031 [15] Kaczor, W.; Kuczumow, T., A remark on a lemma due to Oka, Ann. univ. mariae Curie skłodowska, 52, 67-70, (1998) · Zbl 1012.47025 [16] Kirk, W.A., Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. math., 17, 339-346, (1974) · Zbl 0286.47034 [17] Kobayashi, K.; Miyadera, I., On the strong convergence of the Cesàro means of contractions in Banach spaces, Proc. Japan. acad. ser. A. math. sci., 56, 245-249, (1980) · Zbl 0472.47039 [18] Lau, A.T.M.; Nishiura, K.; Takahashi, W., Nonlinear ergodic theorems for semigroups of nonexpansive mappings and left ideals, Nonlinear anal., 26, 1411-1427, (1996) · Zbl 0880.47048 [19] Lorentz, G.G., A contribution to the theory of divergent sequences, Acta. math., 80, 167-190, (1948) · Zbl 0031.29501 [20] Miyazaki, K.I., On an analogue of zarantonello’s inequality and a nonlinear Mean ergodic theorem, J. math. anal. appl., 134, 186-195, (1988) · Zbl 0674.47044 [21] Oka, H., Nonlinear ergodic theorems for commutative semigroups of asymptotically nonexpansive mappings, Nonlinear anal., 18, 619-635, (1992) · Zbl 0753.47044 [22] Oka, H., An ergodic theorem for asymptotically nonexpansive mappings in the intermediate sense, Proc. amer. math. soc., 125, 1693-1703, (1997) · Zbl 0871.47041 [23] Park, J.Y.; Takahashi, W., On the asymptotic behavior of almost-orbits of commutative semigroups in Banach spaces, Nonlinear and convex analysis, (1987), Dekker New York, p. 271-293 [24] Reich, S., Nonlinear evolution equations and nonlinear ergodic theorems, Nonlinear anal., 1, 319-330, (1977) · Zbl 0359.34059 [25] Reich, S., Almost convergence and nonlinear ergodic theorems, J. approx. theory, 24, 269-272, (1978) · Zbl 0404.47032 [26] Reich, S., A note on the Mean ergodic theorem for nonlinear semigroups, J. math. anal. appl., 91, 547-551, (1983) · Zbl 0521.47034 [27] Reich, S.; Xu, H.K., Nonlinear ergodic theory for semigroups of Lipschitzian mappings, Comm. appl. nonlinear anal., 1, 47-60, (1994) · Zbl 0859.47043 [28] Rodé, G., An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. math. anal. appl., 85, 172-178, (1982) · Zbl 0485.47041 [29] Takahashi, W.; Zhang, P.J., Asymptotic behavior of almost-orbits of semigroups of Lipschitzian mappings in Banach spaces, Kodai math. J., 11, 129-140, (1988) · Zbl 0673.47046 [30] Tan, K.K.; Xu, H.K., The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces, Proc. amer. math. soc., 114, 399-404, (1992) · Zbl 0781.47045 [31] Tan, K.K.; Xu, H.K., An ergodic theorem for nonlinear semigroups of Lipschitzian mappings in Banach spaces, Nonlinear anal., 19, 805-813, (1992) · Zbl 0782.47058 [32] Tan, K.K.; Xu, H.K., Asymptotic behavior of almost-orbits of nonlinear semigroups of non-Lipschitzian mappings in Hilbert spaces, Proc. amer. math. soc., 117, 385-393, (1993) · Zbl 0807.47055 [33] Zarantonello, E.H., Projections on convex sets in Hilbert spaces and spectral theory, (), 237-424
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