Thai Journal of Mathematics : (2018) 1-16 Special Issue (ACFPTO2018) on : Advances in fixed point theory towards real world optimization problems http://thaijmath.in.cmu.ac.th Online ISSN 1686-0209 Modified Proximal Point Algorithms for Solving Fixed Point Problem and Convex Minimization Problem in Non-Positive Curvature Metric Spaces Kamonrat Sombut †, 1 , Nuttapol Pakkaranang ‡ and Plern Saipara ♮ † Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Thanyaburi, Pathumthani 12110, Thailand e-mail : kamonrat s@rmutt.ac.th ‡ Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi(KMUTT), 126 Pracha-Uthit Road Bang Mod, Thung Khru, Bangkok 10140 Thailand e-mail : nuttapol.pak@mail.kmutt.ac.th ♮ Division of Mathematics, Department of Science, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Nan, 59/13 Fai Kaeo, Phu Phiang, Nan 55000 Thailand e-mail : splernn@gmail.com Abstract : In this paper, we aim to introduce new four steps of proximal point algorithm for nonexpansive mappings in non-positive curvature metric spaces, namely CAT(0) spaces and also prove that the sequence generated by the pro- posed algorithm converges to a minimizer of a convex function and common fixed point of such mappings. Keywords : CAT(0) space; convex minimization problem; fixed point problem; proximal point algorithm; iteration process 2000 Mathematics Subject Classification : 47H09; 47H10 (2000 MSC ) 1 Corresponding author email: kamonrat s@rmutt.ac.th (K. Sombut) Copyright c ⃝ 2018 by the Mathematical Association of Thailand. All rights reserved.