Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces

Abstract

In this paper, we introduce the concept of τ -function which generalizes the concept of w-distance studied in the literature. We establish a generalized Ekeland's variational principle in the setting of lower semicontinuous from above and τ -functions. As applications of our Ekeland's variational principle, we derive generalized Caristi's (common) fixed point theorems, a generalized Takahashi's nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces. We also prove that these theorems also imply our Ekeland's variational principle.

On generalized weakly directional contractions and approximate fixed point property with applications

“…Recall that a function p : X × X [0, ∞) is called a w-distance [1,[25][26][27][28][29][30], if the following are satisfied:…”

“…is said to be a τ-function [14,26,[28][29][30], first introduced and studied by Lin and Du, if the following conditions hold:…”

“…Note that not either of the implications p(x, y) = 0 ⇔ x = y necessarily holds and p is nonsymmetric in general. It is well known that the metric d is a w-distance and any w-distance is a τ-function, but the converse is not true; see [26] for more detail.…”

“…By the completeness of X and the lower semicontinuity of f and p(u, ·), we know that (Y, d) is a nonempty complete metric space. Applying a generalization version of Ekeland's variational principle due to Lin and Du (see, for instance, [26,28] …”

The existence of fixed points for new nonlinear multivalued maps and their applications

“…Recently, Lin and Du introduced and studied τ-functions [5,9,[18][19][20][21][22]. A function p: X × X [0, ∞) is said to be a τ-function, if the following conditions hold: (τ1) p(x, z) ≤ p(x, y) + p(y, z) for all x, y, z X; (τ2) If x X and {y n } in X with lim n ∞ y n = y such that p(x, y n ) ≤ M for some M = M(x) > 0, then p(x, y) ≤ M;…”