Acta Appl Math https://doi.org/10.1007/s10440-019-00242-8 Existence and Convergence Theorems for Global Minimization of Best Proximity Points in Hilbert Spaces Raweerote Suparatulatorn 1 · Watcharaporn Cholamjiak 2 · Suthep Suantai 1 Received: 5 June 2018 / Accepted: 25 January 2019 © Springer Nature B.V. 2019 Abstract In order to solve global minimization problems involving best proximity points, we introduce general Mann algorithm for nonself nonexpansive mappings and then prove weak and strong convergence of the proposed algorithm under some suitable conditions in real Hilbert spaces. Furthermore, we also provide numerical experiment to illustrate the convergence behavior of our proposed algorithm. Keywords General Mann algorithm · Global minimization problem · Best proximity point problem · Nonexpansive mapping Mathematics Subject Classification (2010) 41A29 · 90C26 · 47H09 1 Introduction Various problems arising in different areas of science, applied science, economics, physics and engineering, can be modeled as fixed point equations of the form x = Tx , where T : X → X is a nonlinear operator. So fixed point theory plays very important role in solv- ing existence and uniqueness of solutions of those problems. In the case that T is nonself mapping, the above fixed point equation may have no solution. For more precisely, suppose T : A → B with A ∩ B =∅, where A and B are two subsets of a metric space (X,d), in this case, d(A,B) ≤ d(x,Tx) for all x ∈ A, where d(A,B) is the gap distance between A and B , i.e., d(A,B) := inf{d(a,b) : a ∈ A and b ∈ B}. It is natural to ask how can we find a B S. Suantai suthep.s@cmu.ac.th R. Suparatulatorn raweerote.s@gmail.com W. Cholamjiak c-wchp007@hotmail.com 1 Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand 2 School of Science, University of Phayao, Phayao 56000, Thailand