REMARKS ON "EQUILIBRIUM PROBLEMS IN HADAMARD MANIFOLDS" BY V. COLAO ET AL.

Abstract

In 2012, Colao, Lopez, Marino, and Martin-Marquez [J. Math. Anal. Appl. 388 (2012) 61-77] developed an equilibrium theory in Hadamard manifolds. In this paper, we show that three of their key results (the KKM lemma, the Ky Fan type minimax inequality, and Nash equilibrium theorem) on Hadamard manifolds can be extended to hyperbolic spaces and are particular ones for abstract convex spaces in the sense of ours in [7,8]. Similarly, most of main theorems in the KKM theory on abstract convex spaces can be applied to hyperbolic spaces and Hadamard manifolds.