Existence and convergence theorems of fixed points for multi-valued SCC-, SKC-, KSC-, SCS- and C-type mappings in hyperbolic spaces. (English) Zbl 1345.54041
Summary: The purpose of this paper is to introduce the concepts of multi-valued SCC-, SKC-, KSC-, SCS- and C-type mappings and propose a classical Kuhfittig-type iteration [P. K. F. Kuhfittig, Pac. J. Math. 97, 137–139 (1981; Zbl 0478.47036)] for finding a common fixed point of the SKC-, KSC-, SCS- and C-type multi-valued mappings in the setting of hyperbolic spaces. Under suitable conditions, some \(\Delta\)-convergence theorems and strong convergence theorems for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a finite family of SKC-, KSC-, SCS- and C-type multi-valued mappings are proved. The results presented in the paper extend and improve some recent results.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54C60 | Set-valued maps in general topology |
| 54E40 | Special maps on metric spaces |
Keywords:
hyperbolic space; multi-valued SCC-, SKC-, KSC-, SCS- and C-type mappings; Kuhfittig-type iteration; common fixed point; \(\Delta\)-convergence theoremCitations:
Zbl 0478.47036References:
| [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] | Nanjaras, B, Panyanak, B, Phuengrattana, W: Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT \((0)\operatorname{CAT}(0)\) spaces. Nonlinear Anal. Hybrid Syst. 4, 25-31 (2010) · Zbl 1225.54021 · doi:10.1016/j.nahs.2009.07.003 |
| [3] | Dhompongsa, S, Kaewkhaoa, A, Panyanaka, B: On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on CAT \((0)\operatorname{CAT}(0)\) spaces. Nonlinear Anal. 75, 459-468 (2012) · Zbl 1443.47062 · doi:10.1016/j.na.2011.08.046 |
| [4] | 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 |
| [5] | Ghoncheh, SJH, Razani, A: Multi-valued version of SCC, SKC, KSC, and CSC conditions in Ptolemy metric spaces. J. Inequal. Appl. 2014, 471 (2014) · Zbl 1338.54168 · doi:10.1186/1029-242X-2014-471 |
| [6] | Kuhfitting, PKF: Common fixed points of nonexpansive mappings by iteration. Pac. J. Math. 97(1), 137-139 (1981) · Zbl 0478.47036 · doi:10.2140/pjm.1981.97.137 |
| [7] | Chang, S, Tang, Y, Wang, L, Xu, Y, Zhao, Y, Wang, G: Convergence theorems for some multi-valued generalized nonexpansive mappings. Fixed Point Theory Appl. 2014, 33 (2014) · Zbl 1345.47023 · doi:10.1186/1687-1812-2014-33 |
| [8] | Khan, AR, Khamsi, MA, Fukhar-ud-din, H: Strong convergence of a general iteration scheme in CAT \((0)\operatorname{CAT}(0)\) spaces. Nonlinear Anal. 74, 783-791 (2011) · Zbl 1202.47076 · doi:10.1016/j.na.2010.09.029 |
| [9] | Kirk, W, Panyanak, B: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689-3696 (2008) · Zbl 1145.54041 · doi:10.1016/j.na.2007.04.011 |
| [10] | Sahin, A, Basarir, M: On the strong convergence of a modified S-iteration process for asymptotically quasi-nonexpansive mappings in a CAT \((0)\operatorname{CAT}(0)\) space. Fixed Point Theory Appl. 2013, 12 (2013). doi:10.1186/1687-1812-2013-12 · Zbl 1298.47077 · doi:10.1186/1687-1812-2013-12 |
| [11] | Agarwal, RP, O’Regan, D, Sahu, DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8(1), 61-79 (2007) · Zbl 1134.47047 |
| [12] | Chang, SS, Cho, YJ, Zhou, H: Demiclosed principal and weak convergence problems for asymptotically nonexpansive mappings. J. Korean Math. Soc. 38(6), 1245-1260 (2001) · Zbl 1020.47059 |
| [13] | Chang, SS, Wang, L, Lee, HWJ: Total asymptotically nonexpansive mappings in a CAT \((0)\operatorname{CAT}(0)\) space demiclosed principle and a convergence theorems for total asymptotically nonexpansive mappings in a CAT \((0)\operatorname{CAT}(0)\) space. Appl. Math. Comput. 219, 2611-2617 (2012) · Zbl 1308.47060 · doi:10.1016/j.amc.2012.08.095 |
| [14] | Chang, S, Wang, G, Wang, L, Tang, YK, Ma, ZL: Δ-Convergence theorems for multi-valued nonexpansive mappings in hyperbolic spaces. Appl. Math. Comput. 249, 535-540 (2014) · Zbl 1338.47083 · doi:10.1016/j.amc.2014.10.076 |
| [15] | Khan, AR, Fukhar-ud-din, H, Kuan, MAA: An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Fixed Point Theory Appl. 2012, 54 (2012). doi:10.1186/1687-1812-2012-54 · Zbl 1345.54055 · doi:10.1186/1687-1812-2012-54 |
| [16] | Reich, S, Shafrir, I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal., Theory Methods Appl. 15, 537-558 (1990) · Zbl 0728.47043 · doi:10.1016/0362-546X(90)90058-O |
| [17] | Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984) · Zbl 0537.46001 |
| [18] | Bridson, MR, Haefliger, A: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999) · Zbl 0988.53001 · doi:10.1007/978-3-662-12494-9 |
| [19] | Dhompongsa, S, Panyanak, B: On Δ-convergence theorem in CAT \((0)\operatorname{CAT}(0)\) spaces. Comput. Math. Appl. 56(10), 2572-2579 (2008) · Zbl 1165.65351 · doi:10.1016/j.camwa.2008.05.036 |
| [20] | Leustean, L.; Leizarowitz, A. (ed.); Mordukhovich, BS (ed.); Shafrir, I. (ed.); Zaslavski, A. (ed.), Nonexpansive iteration in uniformly convex W-hyperbolic spaces, No. 513, 193-209 (2010), Providence · doi:10.1090/conm/513/10084 |
| [21] | Abkar, A, Eslamian, M: Generalized nonexpansive multivalued mappings in strictly convex Banach spaces. Fixed Point Theory 14, 269-280 (2013) · Zbl 1302.47080 |
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.
