# Fixed point theorems for (L)-type mappings in complete CAT(0) spaces

Volume 10, Issue 3, pp 964--974 Publication Date: March 20, 2017       Article History
• 928 Views

### Authors

Jing Zhou - Department of Mathematics, Harbin Institute of Technology, Harbin 150080, P. R. China. Yunan Cui - Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, P. R. China.

### Abstract

In this paper, fixed point properties for a class of more generalized nonexpansive mappings called (L)-type mappings are studied in geodesic spaces. Existence of fixed point theorem, demiclosed principle, common fixed point theorem of single-valued and set-valued are obtained in the third section. Moreover, in the last section, $\Delta$-convergence and strong convergence theorems for (L)-type mappings are proved. Our results extend the fixed point results of Suzuki’s results in 2008 and Llorens-Fuster’s results in 2011.

### Keywords

• (L)-type mappings
• geodesic spaces
• fixed point theorems
• common fixed point theorems
• three-step iteration scheme.

•  47H09
•  47H10
•  54E40

### References

• [1] A. Abkar, M. Eslamian, Fixed point theorems for Suzuki generalized nonexpansive multivalued mappings in Banach spaces, Fixed Point Theory Appl., 2010 (2010 ), 10 pages.

• [2] A. Abkar, M. Eslamian, A fixed point theorem for generalized nonexpansive multivalued mappings, Fixed Point Theory, 12 (2011), 241–246.

• [3] A. Abkar, M. Eslamian, Generalized nonexpansive multivalued mappings in strictly convex Banach spaces, Fixed Point Theory, 14 (2013), 269–280.

• [4] M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1999)

• [5] F. E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc., 74 (1968), 660–665.

• [6] K. S. Brown , Buildings, Springer-Verlag , New York (1989)

• [7] S. S. Chang, L. Wang, H. W. J. Lee, C. K. Chan, L. Yang, Demiclosed principle and $\Delta$-convergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces, Appl. Math. Comput., 219 (2012), 2611–2617.

• [8] P. Chaoha, A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl., 320 (2006), , 983–987.

• [9] S. Dhompongsa, A. Kaewcharoen, Fixed point theorems for nonexpansive mappings and Suzuki-generalized nonexpansive mappings on a Banach lattice, Nonlinear Anal., 71 (2009), 5344–5353.

• [10] S. Dhompongsa, W. A. Kirk, B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), 762–772.

• [11] S. Dhompongsa, B. Panyanak , On $\Delta$-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572– 2579.

• [12] R. Espínola, P. Lorenzo, A. Nicolae, Fixed points, selections and common fixed points for nonexpansive-type mappings, J. Math. Anal. Appl., 382 (2011), 503–515.

• [13] J. García-Falset, E. Llorens-Fuster, E. Moreno-Gálvez, Fixed point theory for multivalued generalized nonexpansive mappings, Appl. Anal. Discrete Math., 6 (2012), 265–286.

• [14] J. García-Falset, E. Llorens-Fuster, T. Suzuki , Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl., 375 (2011), 185–195.

• [15] J. García-Falset, B. Sims, M. A. Smyth, The demiclosedness principle for mappings of asymptotically nonexpansive type, Houston J. Math., 158 (1996), 101–108.

• [16] K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (1984)

• [17] A. Kaewcharoen, B. Panyanak, Fixed point theorems for some generalized multivalued nonexpansive mappings, Nonlinear Anal., 74 (2011), 5578–5584.

• [18] W. A. Kirk, Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory Appl., 4 (2004), 309–316.

• [19] W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689–3696.

• [20] U. Kohlenbach, L. Leuştean, Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces, J. Eur. Math. Soc. (JEMS), 12 (2007), 71–92.

• [21] T. C. Lim , Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), 179–182.

• [22] P.-K. Lin, K.-K. Tan, H.-K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal., 24 (1995), 929–946.

• [23] E. Llorens-Fuster, E. Moreno Gálvez, The fixed point theory for some generalized nonexpansive mappings, Abstr. Appl. Anal., 2011 (2011), 15 pages.

• [24] E. Moreno Gálvez, E. Llorens-Fuster, The fixed point property for some generalized nonexpansive mappings in a nonreflexive Banach space, Fixed Point Theory, 14 (2013), 141–150.

• [25] B. Nanjaras, B. Panyanak, W. Phuengrattana, Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT(0) spaces, Nonlinear Anal. Hybrid Syst., 4 (2010), 25–31.

• [26] A. Razani, H. Salahifard, Invariant approximation for CAT(0) spaces, Nonlinear Anal., 72 (2010), 2421–2425.

• [27] T. Shimizu, W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal., 8 (1996), 197–203.

• [28] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340 (2008), 1088–1095.

• [29] B. S. Thakur, D. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147–155.

• [30] Z.-F. Zuo, Y.-N. Cui, Iterative approximations for generalized multivalued mappings in Banach spaces, Thai J. Math., 9 (2011), 333–342.