Document Type : Research Paper

Authors

1 Department of Mathematics, Bandar Abbas Branch, Islamic Azad University, P.O.Box 79158-93144, Bandar Abbas, Iran.

2 Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Gava Zang, P.O.Box 45137-66731, Zanjan, Iran

Abstract

In this paper, we first introduce a monotone mapping and its resolvent in general metric spaces.
Then, we give two new iterative methods  by combining the resolvent method with Halpern's iterative method and viscosity approximation method for  finding a fixed point of monotone mappings and a solution of variational inequalities. We prove convergence theorems of the proposed iterations  in CAT(0) metric spaces.

Keywords

[1] S. Alizadeha, H. Dehghan, and F. Moradlou, $Delta$-convergence theorems for inverse-strongly monotone mappings in CAT(0) spaces, Fixed Point Theory, 19(1) (2018), pp. 45-56.

[2] M. Asadi, Fixed points and common fixed points of mappings on CAT(0) spaces, Fixed Point Theory, 14(1) (2013), pp. 29-38.

[3] M. Asadi, Some notes on fixed point sets in CAT(0) spaces, Adv. Fixed Point Theory, 4(3) (2014), pp. 395-401.

[4] M. Asadi, S.M. Vaezpour, and H. Soleimani, $alpha$-Nonexpansive Mappings on CAT(0) spaces, World Applied Sciences Journal, 11(10) (2010), pp. 1303-1306.

[5] W. Ballmann, Lectures on Spaces of Nonpositive Curvature, in: DMV Seminar Band, vol. 25, Birkhauser, Basel, (1995)

[6] I.D. Berg and I.G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata 133 (2008), pp. 195-218.

[7] M. Bridson and A. Haefliger, Metric Spaces of Nonpositive Curvature, Springer-Verlag, Berlin, Heidelberg, New York, (1999).

[8] A. Cuntavepanit and B. Panyanak, Strong convergence of modified Halpern iterations in CAT(0) spaces. Fixed Point Theory Appl. 869458, (2011) doi:10.1155/2011/869458

[9] H. Dehghan and J. Rooin, A characterization of metric projection in CAT(0) spaces, arXiv:1311.4174v1, 2013.

[10] S. Dhompongsa, W.A. Kirk, and B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear Anal., 65 (2006), pp. 762-772.

[11] S. Dhompongsa, W.A. Kirk, and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear and Convex Anal., 8 (2007), pp. 35-45.

[12] S. Dhompongsa and B. Panyanak, On $Delta$-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), pp. 2572-2579.

[13] B.A. Kakavandi and M. Amini, Duality and subdifferential for convex functions on complete CAT(0) metric spaces, Nonlinear Anal., 73 (2010), pp. 3450-3455.

[14] B.A. Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc., 141 (2013), pp. 1029-1039.

[15] J.K. Kim and T.M. Tuyen, New iterative methods for finding a common zero of a finite family of monotone operators in Hilbert spaces, Bull. Korean Math. Soc., 54 (2017), pp. 1347-1359.

[16] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), pp. 3689-3696.

[17] W.A. Kirk, Geodesic geometry and fixed point theory, in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), vol. 64 of Colecc. Abierta, pp. 195-225, University of Seville, Secretary Publication, Seville, Spain, (2003).

[18] T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), pp. 179-182.

[19] A. Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature, IRMA Lectures in Mathematics and Theoretical Physics, 6. European Mathematical Society (EMS), Zurich, 2005.

[20] S. Ranjbar, Strong convergence of a composite Halpern type iteration for a family of nonexpansive mappings in CAT(0) spaces, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 3 (2017).

[21] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), pp. 240-256.