×
Uddin, Izhar; Garodia, Chanchal; Nieto, Juan Jose

Mann iteration for monotone nonexpansive mappings in ordered CAT(0) space with an application to integral equations. (English) Zbl 1478.65057

J. Inequal. Appl. 2018, Paper No. 339, 13 p. (2018).
Summary: In this paper, we establish some convergence results for a monotone nonexpansive mapping in a \(\operatorname{CAT}(0)\) space. We prove the \(\Delta\)- and strong convergence of the Mann iteration scheme. Further, we provide a numerical example to illustrate the convergence of our iteration scheme, and also, as an application, we discuss the solution of integral equation. Our results extend some of the relevant results.

Cited in 10 Documents

MSC:

65J15 Numerical solutions to equations with nonlinear operators
54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
45G10 Other nonlinear integral equations

Keywords:

\(\operatorname{CAT}(0)\) space; fixed point; \(\delta\)-convergence; monotone nonexpansive mapping
Cite Review PDF
Full Text: DOI
OA License

References:

[1] Banach, S.: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fundam. Math. 3, 133-181 (1922) · JFM 48.0201.01 · doi:10.4064/fm-3-1-133-181
[2] Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435-1443 (2004) · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4
[3] Nieto, J.J., Rodriguez-Löpez, R.: Contractive mapping theorems in partially ordered sets and application to ordinary differential equations. Order 22, 223-239 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5
[4] Browder, F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54, 1041-1044 (1965) · Zbl 0128.35801 · doi:10.1073/pnas.54.4.1041
[5] Göhde, D.: Zum Prinzip der kontraktiven Abbildung. Math. Nachr. 30, 251-258 (1965) · Zbl 0127.08005 · doi:10.1002/mana.19650300312
[6] Kirk, W.A.: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004-1006 (1965) · Zbl 0141.32402 · doi:10.2307/2313345
[7] Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-510 (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] Bachar, M., Khamsi, M.A.: On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces. Fixed Point Theory Appl. 2015, 160 (2015) · Zbl 1346.47023 · doi:10.1186/s13663-015-0405-3
[10] Abdullatif, B., Dehaish, B., Khamsi, M.A.: Browder and Göhde fixed point theorem for monotone nonexpansive mappings. Fixed Point Theory Appl. 2016, 20 (2016) · Zbl 1436.47018 · doi:10.1186/s13663-016-0505-8
[11] Song, Y., Kumam, P., Cho, Y.J.: Fixed point theorems and iterative approximations for monotone nonexpansive mappings in ordered Banach spaces. Fixed Point Theory Appl. 2016, 73 (2016) · Zbl 1510.47071 · doi:10.1186/s13663-016-0563-y
[12] Dehaish, B.A.B., Khamsi, M.A.: Mann iteration process for monotone nonexpansive mappings. Fixed Point Theory Appl. 2015, 177 (2015) · Zbl 1448.47069 · doi:10.1186/s13663-015-0416-0
[13] Bridson, M., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999) · Zbl 0988.53001 · doi:10.1007/978-3-662-12494-9
[14] Burago, D., Burago, Y., Ivanov, S.: In: A Course in Metric Geometry. Graduate Studies in Math., vol. 33. Am. Math. Soc., Providence (2001) · Zbl 0981.51016
[15] Kumam, W., Pakkaranang, N., Kumam, P.: Modified viscosity type iteration for total asymptotically nonexpansive mappings in CAT \((0)\operatorname{CAT}(0)\) spaces and its application to optimization problems. J. Nonlinear Sci. Appl. 11(2), 288-302 (2018) · Zbl 1438.47109 · doi:10.22436/jnsa.011.02.10
[16] Pakkaranang, N., Kumam, P., Cho, Y.J., Saipara, P., Padcharoen, A., Khaofong, C.: Strong convergence of modified viscosity implicit approximation methods for asymptotically nonexpansive mappings in complete CAT \((0)\operatorname{CAT}(0)\) spaces. J. Math. Comput. Sci. 17(3), 345-354 (2017) · Zbl 1427.47023 · doi:10.22436/jmcs.017.03.01
[17] Kumam, W., Pakkaranang, N., Kumam, P.: Convergence analysis of modified Picard-S hybrid iterative algorithms for total asymptotically nonexpansive mappings in Hadamard spaces. Int. J. Comput. Math. (2018) · Zbl 1398.65112
[18] Pakkaranang, N., Kumam, P., Cho, Y.J.: Proximal point algorithms for solving convex minimization problem and common fixed points problem of asymptotically quasi-nonexpansive mappings in CAT \((0)\operatorname{CAT}(0)\) spaces with convergence analysis. Numer. Algorithms 78(3), 827-845 (2018) · Zbl 1398.65112 · doi:10.1007/s11075-017-0402-1
[19] Uddin, I., Dalal, S., Imdad, M.: Approximating fixed points for generalized nonexpansive mapping in CAT \((0)\operatorname{CAT}(0)\) spaces. J. Inequal. Appl. 2014, 155 (2014) · Zbl 1472.47094 · doi:10.1186/1029-242X-2014-155
[20] Laowang, W., Panyanak, B.: Strong and Delta convergence theorems for multivalued mappings in CAT \((0)\operatorname{CAT}(0)\) spaces. J. Inequal. Appl. 2009, Article ID 730132 (2009) · Zbl 1176.47056 · doi:10.1155/2009/730132
[21] Dhompongsa, S., Panyanak, B.: On Δ-convergence theorems in CAT \((0)\operatorname{CAT}(0)\) spaces. Comput. Math. Appl. 56, 2572-2579 (2008) · Zbl 1165.65351 · doi:10.1016/j.camwa.2008.05.036
[22] Kirk, W.A., 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
[23] Uddin, I., Nieto, J.J., Ali, J.: One-step iteration scheme for multivalued nonexpansive mappings in CAT \((0)\operatorname{CAT}(0)\) spaces. Mediterr. J. Math. 13, 1211-1225 (2017) · Zbl 1350.54038 · doi:10.1007/s00009-015-0531-5
[24] Uddin, I., Ali, J., Nieto, J.J.: An iteration scheme for a family of multivalued mapping in CAT \((0)\operatorname{CAT}(0)\) spaces with an application to image recovery. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112(2), 373-384 (2018) · Zbl 1393.54022 · doi:10.1007/s13398-017-0382-y
[25] Dhompongsa, S., Kirk, W.A., Sims, B.: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 65, 762-772 (2006) · Zbl 1105.47050 · doi:10.1016/j.na.2005.09.044
[26] Lim, T.C.: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 60, 179-182 (1976) · Zbl 0346.47046 · doi:10.1090/S0002-9939-1976-0423139-X
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.