Coupled fixed point theorems for asymptotically nonexpansive mappings. (English) Zbl 1423.47018
Summary: We introduce the theory of asymptotical nonexpansiveness of mappings defined in the algebraic product \(E \times E\) and with values in the space \(E\). We then prove the existence of coupled fixed points of such mappings when \(E\) is a uniformly convex Banach space. This paper is an extension of some recent results in the literature.
MSC:
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H10 | Fixed-point theorems |
References:
| [1] | Bruck RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces.Isr. J. Math. 1979, 32:107-116. · Zbl 0423.47024 · doi:10.1007/BF02764907 |
| [2] | Bruck RE: On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces.Isr. J. Math. 1981,38(4): 304-314. · Zbl 0475.47037 · doi:10.1007/BF02762776 |
| [3] | Zarantonello, EH, Protections on convex sets in Hilbert space and spectral theory, 237-244 (1971), New York |
| [4] | Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings.Proc. Am. Math. Soc. 1972, 35:171-174. · Zbl 0256.47045 · doi:10.1090/S0002-9939-1972-0298500-3 |
| [5] | Chang SS, Cho YJ, Zhao H: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings.J. Korean Math. Soc. 2001,38(6): 1245-1260. · Zbl 1020.47059 |
| [6] | Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications.Nonlinear Anal. 2006, 65:1379-1393. · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017 |
| [7] | Chang SS, Cho YJ, Huang NJ: Coupled fixed point theorems with applications.J. Korean Math. Soc. 1996,33(3): 575-585. · Zbl 0859.47037 |
| [8] | Guo D, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications.Nonlinear Anal. TMA 1987, 11:623-632. · Zbl 0635.47045 · doi:10.1016/0362-546X(87)90077-0 |
| [9] | Oka H: An ergodic theorem for asymptotically nonexpansive mappings in the intermediate sense.Proc. Am. Math. Soc. 1997, 125:1693-1703. · Zbl 0871.47041 · doi:10.1090/S0002-9939-97-03745-3 |
| [10] | Giesy DP: On a convexity condition in normed linear spaces.Trans. Am. Math. Soc. 1966, 125:114-146. · Zbl 0183.13204 · doi:10.1090/S0002-9947-1966-0205031-7 |
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