40 research outputs found

    Existence and convergence results for a class of nonexpansive type mappings in hyperbolic spaces

    Get PDF
    [EN] We consider a wider class of nonexpansive type mappings and present some fixed point results for this class of mappingss in hyperbolic spaces. Indeed, first we obtain some existence results for this class of mappings. Next, we present some convergence results for an iteration algorithm for the same class of mappings. Some illustrative non-trivial examples have also been discussed.Pant, R.; Pandey, R. (2019). Existence and convergence results for a class of nonexpansive type mappings in hyperbolic spaces. Applied General Topology. 20(1):281-295. https://doi.org/10.4995/agt.2019.11057SWORD281295201M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik 66, no. 2 (2014), 223-234.R. P. Agarwal, D. O'Regan and D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8, no. 1 (2007), 61-79.A. Amini-Harandi, M. Fakhar and H. R. Hajisharifi, Weak fixed point property for nonexpansive mappings with respect to orbits in Banach spaces, J. Fixed Point Theory Appl. 18, no. 3 (2016), 601-607. https://doi.org/10.1007/s11784-016-0310-3K. Aoyama and F. Kohsaka, Fixed point theorem for α-nonexpansive mappings in Banach spaces, Nonlinear Anal. 74, no. 13 (2011), 4387-4391. https://doi.org/10.1016/j.na.2011.03.057B. A. Bin Dehaish and M. A. Khamsi, Browder and Göhde fixed point theorem for monotone nonexpansive mappings, Fixed Point Theory Appl. 2016:20 (2016). https://doi.org/10.1186/s13663-016-0505-8H. Busemann, Spaces with non-positive curvature, Acta Math. 80 (1948), 259-310. https://doi.org/10.1007/BF02393651T. Butsan, S. Dhompongsa and W. Takahashi, A fixed point theorem for pointwise eventually nonexpansive mappings in nearly uniformly convex Banach spaces, Nonlinear Anal. 74, no. 5 (2011), 1694-1701. https://doi.org/10.1016/j.na.2010.10.041J. GarcĂ­a-Falset, E. Llorens-Fuster and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375, no. 1 (2011), 185-195. https://doi.org/10.1016/j.jmaa.2010.08.069H. Fukhar-ud-din and M. A. Khamsi, Approximating common fixed points in hyperbolic spaces, Fixed Point Theory Appl. 2014:113 (2014). https://doi.org/10.1186/1687-1812-2014-113K. Goebel and M. JapĂłn-Pineda, A new type of nonexpansiveness, Proceedings of 8-th international conference on fixed point theory and applications, Chiang Mai, 2007.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/S0002-9939-1972-0298500-3K. Goebel, T. Sekowski and A. Stachura, Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball, Nonlinear Anal. 4, no. 5 (1980), 1011-1021. https://doi.org/10.1016/0362-546X(80)90012-7K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982), Contemp. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1983, pp. 115-123. https://doi.org/10.1090/conm/021/729507K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. https://doi.org/10.1112/blms/17.3.293M. Gregus, Jr., A fixed point theorem in Banach space, Boll. Un. Mat. Ital. A (5) 17, no. 1 (1980), 193-198.M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, english ed., Modern BirkhĂ€user Classics, BirkhĂ€user Boston, Inc., Boston, MA, 2007, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates. https://doi.org/10.1007/978-0-8176-4583-0B. Gunduz and S. Akbulut, Strong convergence of an explicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces, Miskolc Math. Notes 14 (2013), no. 3, 905-913. https://doi.org/10.18514/mmn.2013.641S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59, no. 1 (1976), 65-71. https://doi.org/10.1090/S0002-9939-1976-0412909-XM. A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106, no. 3 (1989), 723-726. https://doi.org/10.1090/S0002-9939-1989-0972234-4M. A. Khamsi and A. R. Khan, Inequalities in metric spaces with applications, Nonlinear Anal. 74 (2011), no. 12, 4036-4045. https://doi.org/10.1016/j.na.2011.03.034S. H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl. 2013:69 (2013), 10. https://doi.org/10.1186/1687-1812-2013-69S. H. Khan, D. Agbebaku and M. Abbas, Three step iteration process for two multivalued nonexpansive maps in hyperbolic spaces, J. Math. Ext. 10, no. 4 (2016), 87-109.S. H. Khan and M. Abbas, Common fixed point results for a Banach operator pair in CAT(0) spaces with applications, Commun. Fac. Sci. Univ. Ank. S'{e}r. A1 Math. Stat. 66 (2017), no. 2, 195-204. https://doi.org/10.1501/commua1_0000000811S. H. Khan, M. Abbas and T. Nazir, Existence and approximation results for skc mappings in busemann spaces, Waves Wavelets Fractals Adv. Anal. 3 (2017), 48-60.https://doi.org/10.1515/wwfaa-2017-0005S. H. Khan and H. Fukhar-ud din, Convergence theorems for two finite families of some generalized nonexpansive mappings in hyperbolic spaces, J. Nonlinear Sci. Appl. 10, no. 2 (2017), 734-743. https://doi.org/10.22436/jnsa.010.02.34A. R. Khan, H. Fukhar-ud din and M. A. A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 2012:54 (2012), 12. https://doi.org/10.1186/1687-1812-2012-54W. 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/BF02757136W. A. Kirk, Fixed point theory for nonexpansive mappings, Fixed point theory (Sherbrooke, Que., 1980), Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 484-505. https://doi.org/10.1007/bfb0092201W. A. Kirk, Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory Appl. 2004:4 (2004), 309-316. https://doi.org/10.1155/S1687182004406081W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68, no. 12 (2008), 3689-3696. https://doi.org/10.1016/j.na.2007.04.011U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357, no. 1 (2005), 89-128. https://doi.org/10.1090/S0002-9947-04-03515-9L. Leustean, Nonexpansive iterations in uniformly convex W-hyperbolic spaces, Nonlinear analysis and optimization I. Nonlinear analysis, Contemp. Math., vol. 513, Amer. Math. Soc., Providence, RI, 2010, pp. 193-210. https://doi.org/10.1090/conm/513/10084L. Leustean, A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl. 325, no. 1 (2007), 386-399. https://doi.org/10.1016/j.jmaa.2006.01.081T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179-182. https://doi.org/10.1090/S0002-9939-1976-0423139-XE. Llorens-Fuster, Orbitally nonexpansive mappings, Bull. Austral. Math. Soc. 93, no. 3 (2016), 497-503. https://doi.org/10.1017/S0004972715001318W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3K. Menger, Untersuchungen ĂŒber allgemeine metrik, Math. Ann. 100, no. 1 (1928), 75-163. https://doi.org/10.1007/BF01448840S. A. Naimpally, K. L. Singh and J. H. M. Whitfield, Fixed points in convex metric spaces, Math. Japon. 29, no. 4 (1984), 585-597.A. Nicolae, Generalized asymptotic pointwise contractions and nonexpansive mappings involving orbits, Fixed Point Theory Appl. (2010), Art. ID 458265, 19. https://doi.org/10.1155/2010/458265M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251, no. 1 (2000), 217-229. https://doi.org/10.1006/jmaa.2000.7042R. Pant and R. Shukla, Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim. 38, no. 2 (2017), 248-266. https://doi.org/10.1080/01630563.2016.1276075S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), no. 6, 537-558. https://doi.org/10.1016/0362-546x(90)90058-oRitika and S. H. Khan, Convergence of Picard-Mann hybrid iterative process for generalized nonexpansive mappings in CAT(0) spaces, Filomat 31, no. 11 (2017), 3531-3538. https://doi.org/10.2298/FIL1711531RRitika and S. H. Khan, Convergence of RK-iterative process for generalized nonexpansive mappings in CAT(0) spaces, Asian-European Journal of Mathematics, to appear. https://doi.org/10.1142/s1793557119500773H. F. Senter and W. G. Dotson, Jr., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-380. https://doi.org/10.1090/S0002-9939-1974-0346608-8T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340, no. 2 (2008), 1088-1095. https://doi.org/10.1016/j.jmaa.2007.09.023W. Takahashi, A convexity in metric space and nonexpansive mappings. I, Kodai Math. Sem. Rep. 22 (1970), 142-149. https://doi.org/10.2996/kmj/1138846111B. S. Thakur, D. Thakur and M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, Appl. Math. Comput. 275 (2016), 147-155. https://doi.org/10.1016/j.amc.2015.11.06

    Nonexpansive mappings and monotone vector fields in Hadamard manifolds

    Get PDF
    This paper briefly surveys some recent advances in the investigation of nonexpansive mappings and monotone vector fields, focusing in the extension of basic results of the classical nonlinear functional analysis from Banach spaces to the class of nonpositive sectional curvature Riemannian manifolds called Hadamard manifolds. Within this setting, we first analyze the problem of finding fixed points of nonexpansive mappings. Later on, different classes of monotonicity for set-valued vector fields and the relationship between some of them will be presented, followed by the study of the existence and approximation of singularities for such vector fields. We will discuss about variational inequality and minimization problems in Hadamard manifolds, stressing the fact that these problems can be solved by means of the iterative approaches for monotone vector fields

    Results on a faster iterative scheme for a generalized monotone asymptotically

    Get PDF
    This article devoted to present results on convergence of  Fibonacci-Halpern scheme (shortly, FH) for monotone asymptotically αn-nonexpansive  mapping (shortly, ma αn-n mapping) in partial ordered Banach space (shortly, POB space). Which are auxiliary theorem for demi-close's proof of this type of mappings, weakly convergence of increasing FFH-scheme to a fixed point with aid monotony of a norm and  Σn+=∞1 λn= +∞, λn =min{hn , (1-hn)} where hn ➊ (0,1)   where is associated with FH-scheme for an integer n>0 more than that, convergence amounts to be strong by using Kadec-Klee property and finally, prove that this scheme is weak-w2 stable up on suitable status

    Hybrid algorithm with perturbations for total asymptotically non-expansive mappings in CAT(0) space

    Get PDF
    In this paper, we establish strong and Δ-convergence theorems of the modified hybrid-CR three steps iteration with perturbations for total asymptotically non-expansive mapping in CAT(0) spaces. Our results improve and extend the corresponding results from the current literature. We also provide three examples to illustrate the convergence behaviour of the proposed algorithm and numerically compare the convergence of the proposed iteration scheme with the existing schemes

    Alternative iterative methods for nonexpansive mappings, rates of convergence and application

    Get PDF
    Alternative iterative methods for a nonexpansive mapping in a Banach space are proposed and proved to be convergent to a common solution to a fixed point problem and a variational inequality. We give rates of asymptotic regularity for such iterations using proof-theoretic techniques. Some applications of the convergence results are presented
    corecore