On generalizing Takahashi’s nonconvex minimization theorem

We present a simple proof of vectorial Takahashi’s nonconvex minimization theorem based on Gopfert, Tammer and Zalinescu A. Gopfert, C. Tammer, C. Zalinescu, On the vectorial Ekeland’s variational principle and minimal points in product spaces, Nonlinear Anal. 39 (2000) 909–922; C. Tammer, A variational principle and a fixed point theorem, in: System Modelling and Optimization (Compiegne, 1993), in: Lecture Notes in Control and Inform. Sci., vol. 197, Springer, London, 1994, pp. 248–257].