Structure of the fixed point set of asymptotically nonexpansive mappings in Banach spaces with weak uniformly normal structure

• Autorzy: Sahu D. R. , Agarwal R. P. , O'Regan D.
• Języki publikacji: EN

Abstrakty

EN

This paper is concerned with weak uniformly normal structure and the structure of the set of fixed points of Lipschitzian mappings. It is shown that in a Banach space X with weak uniformly normal structure, every asymptotically regular Lipschitzian semigroup of self-mappings defined on a weakly compact convex subset of X satisfies the (ω)-fixed point property. We show that if X has a uniformly Gâteaux differentiable norm, then the set of fixed points of every asymptotically nonexpansive mapping is nonempty and sunny nonexpansive retract of C. Our results improve several known fixed point theorems for the class of Lipschitzian mappings in a general Banach space.

Słowa kluczowe

EN

nonexpansive mapping; asymptotically nonexpansive mapping; retraction; sunny nonexpansive retraction; common fixed point; weak uniformly normal structure

PL

odwzorowania nieekspansywne; retrakcja; retrakcja nieekspansywna

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2011
• Tom: Vol. 17, nr 1
• Strony: 51--68
• Opis fizyczny: Bibliogr. 33 poz.

Twórcy

autor

Sahu D. R.

autor

Agarwal R. P.

autor

O'Regan D.

• Department of Mathematics, Banaras Hindu University, Varanasi - 221005, India,

Bibliografia

• [1] R. P. Agarwal, D. O'Regan and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Topological Fixed Point Theory and Its Applications 6, Springer, New York, 2009.
• [2] B. Beauzamy, Projections contractantes dans les espaces de Banach, Bull. Sci. Math. 102 (1978), 43-47.
• [3] T D. Benavides, A. G. Lopez and H. K. Xu, Weak uniform normal structure and iterative fixed points of nonexpansive mappings, Colloq. Math. 68 (1995), 17-23.
• [4] T. D. Benavides and P. L. Ramirez, Weak uniform normal structure and iterative fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 129 (2001), 3549-3557.
• [5] F. E. Browder, Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. Ration. Mech. Anal. 24 (1967), 82-90.
• [6] R. E. Brack, Nonexpansive projections on subsets of Banach spaces. Pacific J. Math. 47 (1973), 341-355.
• [7] R. E. Brack, Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Trans. Amer. Math. Soc. 179 (1973), 251-262.
• [8] R. E. Brack, T. Kuczumow and S. Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. Math. 65(1993), 169-179.
• [9] W. L. Bynum, Normal structure coefficient for Banach spaces. Pacific J. Math., 86 (1980), 427-436.
• [10] E. Casini and E. Maluta, Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure. Nonlinear Anal. 9 (1985), 103-108.
• [11] C. E. Chidume, J. Liu and A. Udomene, Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 113 (2005), 473-480.
• [12] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.
• [13] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, Inc., New York, 1984.
• [14] J. Górnicki, Fixed points of asymptotically regular semigroups in Banach spaces. Rend. Circ. Mat. Palermo (2) 46 (1997), 89-118.
• [15] T. H. Kim and H. K. Xu, Remarks on asymptotically nonexpansive mappings, Nonlinear Anal. 41 (2000), 405-415.
• [16] E. Kopeckd and S. Reich, Nonexpansive retracts in Banach spaces, Banach Center Publ. 77(2007), 161-174.
• [17] T.C. Lim and H. K. Xu, Fixed point theorems for asymptotically nonexpansaive mappings. Nonlinear Anal. 22(1994), 1345-1355.
• [18] P. K. Lin, K. K. Tan and H. K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal. 22 (1995), 929-946.
• [19] E. MatouSkova and S. Reich, The Hundal example revisited, J. Nonlinear Convex Anal. 4 (2003), 411^127.
• [20] C. H. Morales and J. S. Jung, Convergence of paths for pseudo-contractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128 (2000), 3411-3419.
• [21] S. Pras, Banach spaces with the uniform Opial property, Nonlinear Anal. 18 (1992), 697-704.
• [22] S. Reich, Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl. 44(1973),57-70.
• [23] S. Reich, Extension problems for accretive sets in Banach spaces, J. Funct. Anal. 26 (1977), 378-395.
• [24] S. Reich, Product formulas, nonlinear semigroups, and accretive operators, J. Funct. Anal. 36(1980), 147-168.
• [25] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287-292.
• [26] S. Reich, On the asymptotic behavior of nonlinear semigroups and the range of accretive operators, J. Math. Anal. Appl. 79 (1981), 113-126.
• [27] S. Reich and H. K. Xu, Nonlinear ergodic theory for semigroups of Lipschitizian mappings, Comm. Appl. Nonlinear Anal. 1 (1994), 47-60.
• [28] D. R. Sahu, Z. Liu and S. M. Kang, Iterative approaches to common fixed points of asymptotically nonexpansive mappings. Rocky Mountain J. Math. 39 (2009), 281-304.
• [29] D. R. Sahu and D. O'Regan, Convergence theorems for semigroup-type families of non-self mappings. Rend. Circ. Mat. Palermo (2) 57 (2008), 305-329.
• [30] J. Schu, Approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 112 (1991), 143-151.
• [31] N. Shahzad and A. Udomeen, Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal. 64 (2006), 558-567.
• [32] N. Shioji and W. Takahashi, Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces, J. Approx. Theory 97 (1999), 53-64.
• [33] N. C. Wong, D. R. Sahu, and J. C. Yao, Solving variational inequalities involving nonexpansive type mappings. Nonlinear Anal. 69 (2008), 4732-4753.