Convergence of SP-iteration for generalized nonexpansive mapping in Hadamard space

Yıl 2018, Cilt: 47 Sayı: 6, 1595 - 1604, 12.12.2018

İzhar Uddin Mohammad Imdad

Öz

In this paper, we study the convergence of  SP-iteration scheme for a class of mappings satisfying the condition (C) and prove $\Delta$-convergence as well as strong convergence theorems in Hadamard spaces. Our results generalize and improve several relevant results of the existing literature.

Anahtar Kelimeler

Hadamard spaces, $\Delta$-convergence

Kaynakça

• Agarwal, R. P., O'Regan, D., Sahu, D. R., Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex. Anal. 8 (1), 61-79, 2007.
• Berinde, V., Iterative Approximation of Fixed Points, Lecture Notes Math., vol. 1912, Springer, Berlin, 2007.
• Bridson, M., Haefliger, A., Metric Spaces of Nonpositive Curvature, Springer-Verlag, Berlin, 1999.
• Brown, K. S., Buildings, Springer-Verlag, New York, 1989.
• Bruhat, F., Tits, J., Groupes reductifs sur un corps local, I. Donnees radicielles valuees Inst Hautes Etudes Sci Publ Math. 41, 1972.
• Burago, D., Burago, Y., Ivanov, S., A Course in Metric Geometry, in: Graduate Studies in Math., vol. 33, Amer. Math. Soc., Providence, RI, 2001.
• Dhompongsa, S., Kirk, W. A., Panyanak, B., Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal. 8, 35-45, 2007.
• Dhompongsa, S., Kirk, W. A., Sims, B., Fixed points of uniformly lipschitzian mappings, Nonlinear Anal. 65, 762-772, 2006.
• Dhompongsa, S., Panyanak, B., On $\Delta$-convergence theorems in CAT(0) space, Comput. Math. Appl. 56, 2572-2579, 2008. Goebel, K., Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, Inc., New York, 1984.
• Ishikawa, S., Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44, 147-150, 1974.
• Ishikawa, S., Fixed point and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59, 65-71, 1976.
• Kirk, W. A., Panyanak, B., A concept of convergence in geodesic spaces, Nonlinear Anal. 68, 3689-3696, 2008.
• Lim, T. C., Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60, 179-182, 1976.
• Mann, W. R., Mean value methods in iterations, Proc. Amer. Math. Soc., 4, 506-510, 1953.
• Phuengrattana, W., Suantai, S., On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235, 3006-3014, 2011.
• Razani, A., Salahifrd, H., Approximating fixed points of generalized non-expansive mappings, Bull. Iranian Math. Soc. 37 (1), 235-246, 2011.
• Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67, 274-276, 1979.
• Şahin, A., Başarır, M., On the strong and $\Delta$-convergence of SP-iteration on CAT(0) space, J. Inequal. Appl., 311, 10pp., 2013.
• Senter, H. F., Dotson, W. G., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44, 375-380, 1974.
• Suzuki, T., Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340, 1088-1095, 2008.
• Tan, K. K., Xu, H. K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178, 301-308, 1993.
• Thianwan, S., Common fixed points of new iterations for two asymptotically nonexpansive nonself mappings in a Banach space, J. Comput. Appl. Math. 224, 688-695, 2009.
• Uddin, I., Imdad, M., On certain convergence of S-iteration scheme in CAT(0) spaces Kuwait J. Sci. 42 (2), 93-106, 2015.
• Uddin, I., Imdad, M., Some convergence theorems for a hybrid pair of generalized nonexpansive mappings in CAT (0) spaces, J. Nonlinear Convex Anal. 16 (3), 447-457, 2015.
• Uddin, I., Imdad, M., Ali, J., Convergence Theorems for a Hybrid Pair of Generalized Nonexpansive Mappings in Banach Spaces, Bull. Malays. Math. Sci. Soc., 38 (2), 695-705, 2015.
• Wangkeeree, R., Preechasilp, P., $\Delta$-Convergence for Generalized Hybrid Mappings in CAT(0) Spaces, Bull. Malays. Math. Sci. Soc., 38 (3), 1127-1141, 2015.
• Xu, B., Noor, M. A., Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267, 444-453, 2002.