Fixed point approximation for \(SKC\)-mappings in hyperbolic spaces. (English) Zbl 1338.47096
Summary: In this paper, we introduce the class of \(SKC\)-mappings, which is a generalization of the class of Suzuki-generalized nonexpansive mappings, and we prove strong and \(\Delta\)-convergence theorems of the \(S\)-iteration process which is generated by \(SKC\)-mappings [E. Karapınar and K. Taş, Comput. Math. Appl. 61, No. 11, 3370–3380 (2011; Zbl 1223.47063)] in uniformly convex hyperbolic spaces. As uniformly convex hyperbolic spaces contain Banach spaces as well as \(\operatorname{CAT}(0)\) spaces, our results can be viewed as an extension and generalization of several well-known results in Banach spaces as well as \(\operatorname{CAT}(0)\) spaces.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 54E50 | Complete metric spaces |
Keywords:
\(SKC\)-mappings; \(\Delta\)-convergence; uniformly convex hyperbolic spaces; \(S\)-iteration processCitations:
Zbl 1223.47063References:
| [1] | Suzuki, T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 340, 1088-1095 (2008) · Zbl 1140.47041 · doi:10.1016/j.jmaa.2007.09.023 |
| [2] | Dhompongsa, S, Inthakon, W, Kaewkhao, A: Edelsteins method and fixed point theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 350, 12-17 (2009) · Zbl 1153.47046 · doi:10.1016/j.jmaa.2008.08.045 |
| [3] | Karapinar, E, Tas, K: Generalized (C)-conditions and related fixed point theorems. Comput. Math. Appl. 61, 3370-3380 (2011) · Zbl 1223.47063 · doi:10.1016/j.camwa.2011.04.035 |
| [4] | Abbas, M, Khan, SH, Postolache, M: Existence and approximation results for SKC mappings in CAT(0) spaces. J. Inequal. Appl. 2014, 212 (2014) · Zbl 1310.54029 · doi:10.1186/1029-242X-2014-212 |
| [5] | Nanjaras, B, Panyanak, B, Phuengrattana, W: Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT(0) spaces. Nonlinear Anal. Hybrid Syst. 4, 25-31 (2010) · Zbl 1225.54021 · doi:10.1016/j.nahs.2009.07.003 |
| [6] | Kim, JK, Dashputre, S, Hyun, HG: Convergence theorems of S-iteration process for quasi-contractive mappings in Banach spaces. Commun. Appl. Nonlinear Anal. 21(4), 89-99 (2014) · Zbl 1315.47066 |
| [7] | Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-610 (1953) · Zbl 0050.11603 · doi:10.1090/S0002-9939-1953-0054846-3 |
| [8] | Ishikawa, S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147-150 (1974) · Zbl 0286.47036 · doi:10.1090/S0002-9939-1974-0336469-5 |
| [9] | Chidume, CE: Global iteration schemes for strongly pseudocontractive maps. Proc. Am. Math. Soc. 126, 2641-2649 (1998) · Zbl 0901.47046 · doi:10.1090/S0002-9939-98-04322-6 |
| [10] | Deng, L: Convergence of the Ishikawa iteration process for nonexpansive mappings. J. Math. Anal. Appl. 199, 769-775 (1996) · Zbl 0856.47041 · doi:10.1006/jmaa.1996.0174 |
| [11] | Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301-308 (1993) · Zbl 0895.47048 · doi:10.1006/jmaa.1993.1309 |
| [12] | Zeng, LC: A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 245-250 (1998) · Zbl 0916.47047 · doi:10.1006/jmaa.1998.6053 |
| [13] | Zhou, HY, Jia, YT: Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption. Proc. Am. Math. Soc. 125, 1705-1709 (1997) · Zbl 0871.47045 · doi:10.1090/S0002-9939-97-03850-1 |
| [14] | Agarwal, RP, O’Regan, D, Sahu, DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Convex Anal. 8(1), 61-79 (2007) · Zbl 1134.47047 |
| [15] | Saluja, GS, Kim, JK: On the convergence of modified S-iteration process for asymptotically quasi-nonexpansive type mappings in a CAT(0) space. Nonlinear Funct. Anal. Appl. 19, 329-339 (2014) · Zbl 1318.54035 |
| [16] | Leuştean, L: A quadratic rate of asymptotic regularity for CAT(0) spaces. J. Math. Anal. Appl. 325(1), 386-399 (2007) · Zbl 1103.03057 · doi:10.1016/j.jmaa.2006.01.081 |
| [17] | Kohlenbach, U: Some logical metatheorems with applications in functional analysis. Trans. Am. Math. Soc. 357(1), 89-128 (2005) · Zbl 1079.03046 · doi:10.1090/S0002-9947-04-03515-9 |
| [18] | Zhao, LC, Chang, SS, Kim, JK: Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces. Fixed Point Theory Appl. 2013, 363 (2013) · Zbl 1476.47098 |
| [19] | Kim, JK, Pathak, RP, Dashputre, S, Diwan, SD, Gupta, R: Fixed point approximation of generalized nonexpansive mappings in hyperbolic spaces. Int. J. Math. Math. Sci. 2015, Article ID 368204 (2015) · Zbl 1481.47101 · doi:10.1155/2015/368204 |
| [20] | Takahashi, WA: A convexity in metric space and nonexpansive mappings, I. Kodai Math. Semin. Rep. 22, 142-149 (1970) · Zbl 0268.54048 · doi:10.2996/kmj/1138846111 |
| [21] | Kim, JK, Kim, KS, Nam, YM: Convergence of stability of iterative processes for a pair of simultaneously asymptotically quasi-nonexpansive type mappings in convex metric spaces. J. Comput. Anal. Appl. 9(2), 159-172 (2007) · Zbl 1136.47044 |
| [22] | Goebel, K.; Kirk, WA; Singh, SP (ed.); Thomeier, S. (ed.); Watson, B. (ed.), Iteration processes for nonexpansive mappings, Toronto, 1982, Providence · Zbl 0525.47040 · doi:10.1090/conm/021/729507 |
| [23] | Itoh, S: Some fixed point theorems in metric spaces. Fundam. Math. 102, 109-117 (1979) · Zbl 0412.54054 |
| [24] | Reich, S, Shafrir, I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15, 537-558 (1990) · Zbl 0728.47043 · doi:10.1016/0362-546X(90)90058-O |
| [25] | Kirk, WA: Krasnosel’skiı̌’s iteration process in hyperbolic spaces. Numer. Funct. Anal. Optim. 4, 371-381 (1982) · Zbl 0505.47046 · doi:10.1080/01630568208816123 |
| [26] | Goebel, K, Reich, S: Uniformly Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984) · Zbl 0537.46001 |
| [27] | Gromov, M: Metric Structure of Riemannian and Non-Riemannian Spaces. Proge. Math., vol. 152. Birkhäuser, Boston (1984) |
| [28] | Dhompongsa, S, Panyanak, B: On Δ-convergence theorems in CAT(0) spaces. Comput. Math. Appl. 56, 2572-2579 (2008) · Zbl 1165.65351 · doi:10.1016/j.camwa.2008.05.036 |
| [29] | Khan, SH, Abbas, M: Strong and Δ-convergence of some iterative schemes in CAT(0) spaces. Comput. Math. Appl. 61, 109-116 (2011) · Zbl 1207.65069 · doi:10.1016/j.camwa.2010.10.037 |
| [30] | Laokul, T, Panyanak, B: Approximating fixed points of nonexpansive mappings in CAT(0) spaces. Int. J. Math. Anal. 3(25-28), 1305-1315 (2009) · Zbl 1196.54077 |
| [31] | Leuştean, L.; Leizarowitz, A. (ed.); Mordukhovich, BS (ed.); Shafrir, I. (ed.); Zaslavski, A. (ed.), Nonexpansive iteration in uniformly convex W-hyperbolic spaces, No. 513, 193-210 (2010), Providence · Zbl 1217.47117 · doi:10.1090/conm/513/10084 |
| [32] | Chang, SS, Agarwal, RP, Wang, L: Existence and convergence theorems of fixed points for multi-valued SCC-, SKC-, KSC-, SCS- and C-type mappings in hyperbolic spaces. Fixed Point Theory Appl. 2015, 83 (2015) · Zbl 1345.54041 · doi:10.1186/s13663-015-0339-9 |
| [33] | Khan, AR, Fukhar-ud-din, H, Khan, MA: An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Fixed Point Theory Appl. 2012, 54 (2012) · Zbl 1345.54055 · doi:10.1186/1687-1812-2012-54 |
| [34] | Senter, HF, Doston, WG Jr.: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44(2), 375-380 (1974) · Zbl 0299.47032 · doi:10.1090/S0002-9939-1974-0346608-8 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
