A Minimization Theorem in Quasi-metric Spaces and Its Applications

We prove a new minimization theorem in quasi-metric spaces, which improves the results of Takahashi (1993). Further, this theorem is used to generalize Caristi's fixed point theorem and Ekeland's ε-variational principle. 1. Introduction. Caristi [1] proved a fixed point theorem on complete metric spaces which generalizes the Banach contraction principle. Ekeland [3] also obtained a non-convex minimization theorem, often called the ε-variational principle, for a proper lower semicontinuous function, bounded from below, on complete metric spaces. Later Takahashi [4] proved the following minimization theorem: let X be a complete metric space and let f : X → (−∞, ∞] be a proper lower semicontinuous function, bounded from below. Suppose that, for each u ∈ X with f (u) > inf x∈X f (x), there exists v ∈ X such that v ≠ u and f (v)+ d(u, v) ≤ f (u). Then there exists x 0 ∈ X such that f (x 0) = inf x∈X f (x). These theorems are very useful tools in nonlinear analysis, control theory, economic theory, and global analysis.