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asymptotically regular mappings; uniformly normal structure; fixed points
It is proved that: for every Banach space $X$ which has uniformly normal structure there exists a $k>1$ with the property: if $A$ is a nonempty bounded closed convex subset of $X$ and $T:A\rightarrow A$ is an asymptotically regular mapping such that $$ \liminf _{n\rightarrow \infty } |\kern -0.8pt|\kern -0.8pt|T^n|\kern -0.8pt|\kern -0.8pt|< k, $$ where $|\kern -0.8pt|\kern -0.8pt|T|\kern -0.8pt|\kern -0.8pt|$ is the Lipschitz constant (norm) of $T$, then $T$ has a fixed point in $A$.
[1] Browder F.E., Petryshyn V.W.: The solution by iteration of nonlinear functional equations in Banach spaces. Bull. AMS 72 (1966), 571-576. MR 0190745 | Zbl 0138.08202
[2] Bynum W.L.: Normal structure coefficients for Banach spaces. Pacific J. Math. 86 (1980), 427-436. MR 0590555 | Zbl 0442.46018
[3] Casini E., Maluta E.: Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure. Nonlinear Anal., TMA 9 (1985), 103-108. MR 0776365 | Zbl 0526.47034
[4] Daneš J.: On densifying and related mappings and their applications in nonlinear functional analysis. in: Theory of Nonlinear Operators (Proc. Summer School, October 1972, GDR), Akademie-Verlag, Berlin, 1974, 15-56. MR 0361946
[5] Downing D.J., Turett B.: Some properties of the characteristic convexity relating to fixed point theory. Pacific J. Math. 104 (1983), 343-350. MR 0684294
[6] Edelstein M., O'Brien C.R.: Nonexpansive mappings, asymptotic regularity and successive approximations. J. London Math. Soc. (2) 17 (1978), 547-554. MR 0500642 | Zbl 0421.47031
[7] Gillespie A.A., Williams B.B.: Fixed point theorem for nonexpansive mappings on Banach spaces with uniformly normal structure. Appl. Anal. 9 (1979), 121-124. MR 0539537
[8] Górnicki J.: A fixed point theorem for asymptotically regular mappings. to appear. MR 1201441
[9] Krüppel M.: Ein Fixpunktsatz für asymptotisch reguläre Operatoren in gleichmäßig konvexen Banach-Räumen. Wiss. Z. Pädagog. Hochsch. ``Liselotte Herrmann'' Güstrow, Math.-naturwiss. Fak. 25 (1987), 241-246. MR 0971250
[10] Lin P.K.: A uniformly asymptotically regular mapping without fixed points. Canad. Math. Bull. 30 (1987), 481-483. MR 0919440 | Zbl 0645.47050
[11] Yu X.T.: On uniformly normal structure. Kexue Tongbao 33 (1988), 700-702. Zbl 0681.46020
[12] Yu X.T.: A geometrically aberrant Banach space with uniformly normal structure. Bull. Austral. Math. Soc. 38 (1988), 99-103. MR 0968233 | Zbl 0646.46017
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