Existence ofixed points for pointwise eventually asymptotically nonexpansive mappings. (English) Zbl 1475.47034
Summary: Kirk introduced the notion of pointwise eventually asymptotically nonexpansive mappings and proved that uniformly convex Banach spaces have the fixed point property for pointwise eventually asymptotically nonexpansive maps [W. A. Kirk, J. Nonlinear Convex Anal. 18, No. 1, 1–15 (2017; Zbl 1477.47042)]. Further, Kirk raised the following question: “Does a Banach space \(X\) have the fixed point property for pointwise eventually asymptotically nonexpansive mappings whenever \(X\) has the fixed point property for nonexpansive mappings?”. In this paper, we prove that a Banach space \(X\) has the fixed point property for pointwise eventually asymptotically nonexpansive maps if \(X\) has uniform normal structure or \(X\) is uniformly convex in every direction with the Maluta constant \(D(X) < 1\). Also, we study the asymptotic behavior of the sequence \(\{T^n x\}\) for a pointwise eventually asymptotically nonexpansive map \(T\) defined on a nonempty weakly compact convex subset \(K\) of a Banach space \(X\) whenever \(X\) satisfies the uniform Opial condition or \(X\) has a weakly continuous duality map.
MSC:
| 47H10 | Fixed-point theorems |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
fixed points; pointwise eventually asymptotically nonexpansive mappings; uniform normal structure; uniform Opial condition; duality mappingsCitations:
Zbl 1477.47042References:
| [1] | A. G. Aksoy and M. A. Khamsi, Nonstandard Methods in Fixed Point Theory, Springer-Verlag, New York, 1990. https://doi.org/10.1007/978-1-4612-3444-9 · Zbl 0713.47050 |
| [2] | J. B. Baillon, R. E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston J. Math. 4 (1978), 1-9. · Zbl 0431.47034 |
| [3] | M. S. Brodskii and D. P. Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948), 837-840. · Zbl 0030.39603 |
| [4] | F. E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z. 100 (1967), 201-225. https://doi.org/10.1007/BF01109805 · Zbl 0149.36301 |
| [5] | W. L. Bynum, Normal structure coefficients for Banach spaces, Pac. J. Math. 86 (1980), 427-436. https://doi.org/10.2140/pjm.1980.86.427 · Zbl 0442.46018 |
| [6] | E. Casini and E. Maluta, Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure, Nonlinear Anal. 9 (1985), 103-108. https://doi.org/10.1016/0362-546X(85)90055-0 · Zbl 0526.47034 |
| [7] | G. Emmanuele, Asymptotic behavior of iterates of nonexpansive mappings in Banach spaces with Opial’s condition, Proc. Amer. Math. Soc. 94 (1985), 103-109. https://doi.org/10.2307/2044960 · Zbl 0578.47046 |
| [8] | G. Li and B. Sims, Fixed point theorems for mappings of asymptotically nonexpansive type, Nonlinear Anal. 50 (2002), 1085-1091. https://doi.org/10.1016/S0362-546X(01)00744-1 · Zbl 1004.47038 |
| [9] | K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174. https://doi.org/10.1090/S0002-9939-1972-0298500-3 · Zbl 0256.47045 |
| [10] | K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511526152 · Zbl 0708.47031 |
| [11] | J. P. Gossez and E. Lami Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pac. J. Math. 40 (1972), 565-573. https://doi.org/10.2140/pjm.1972.40.565 · Zbl 0223.47025 |
| [12] | T. H. Kim and H. K. Xu, Remarks on asymptotically nonexpansive mappings, Nonlinear Anal. 41 (2000), 405-415. https://doi.org/10.1016/S0362-546X(98)00284-3 · Zbl 0957.47037 |
| [13] | W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. https://doi.org/10.2307/2313345 · Zbl 0141.32402 |
| [14] | W. A. Kirk, Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math. 17 (1974), 339-346. https://doi.org/10.1007/BF02757136 · Zbl 0286.47034 |
| [15] | W. A. Kirk, Remarks on nonexpansive mappings and related asymptotic conditions, J. Nonlinear Convex Anal. 18 (2017), 1-15. · Zbl 1477.47042 |
| [16] | W. A. Kirk and H. K. Xu, Asymptotic pointwise contraction, Nonlinear Anal. 68 (2008), 4706-4712. https://doi.org/10.1016/j.na.2007.11.023 · Zbl 1172.47038 |
| [17] | T. C. Lim and H. K. Xu, Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal. 22 (1994), 1345-1355. https://doi.org/10.1016/0362-546X(94)90116-3 · Zbl 0812.47058 |
| [18] | P. K. Lin, K. K. Tan and H. K. Xu, Demiclosed principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal. 24 (1995), 929-946. https://doi.org/10.1016/0362-546X(94)00128-5 · Zbl 0865.47040 |
| [19] | E. Maluta, Uniformly normal structure and related coefficients, Pac. J. Math. 111 (1984), 357-369. https://doi.org/10.2140/pjm.1984.111.357 · Zbl 0495.46012 |
| [20] | Z. Opial, Weak convergence of the sequences of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 595-597. https://doi.org/10.1090/S0002-9904-1967-11761-0 · Zbl 0179.19902 |
| [21] | S. Prus, Banach spaces with the uniform Opial property, Nonlinear Anal. 18 (1992), 697-704. https://doi.org/10.1016/0362-546X(92)90165-B · Zbl 0786.46023 |
| [22] | H. K. Xu, Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear Anal. 16 (1991), 1139-1146. https://doi.org/10.1016/0362-546X(91)90201-B · Zbl 0747.47041 |
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