Abstract
In this paper, we prove a generalization of the strong Ekeland variational principle for a generalized distance (i.e., u-distance) on complete metric spaces. The result present in this paper extends and improves the corresponding result of Georgiev (J. Math. Anal. Appl. 131:1-21, 1988) and Suzuki (J. Math. Anal. Appl. 320:788-794, 2006).
Highlights
In, Ekeland [ ] proved the following, which is called the Ekeland variational principle.Theorem . [ ] Let (X, d) be a complete metric space with metric d and f be a function from X into
For u ∈ X and λ >, there exists v ∈ X such that (P) f (v) ≤ f (u) – λd(u, v); (Q) f (w) > f (v) – λd(v, w) for every w = v
For all u ∈ X, λ > and δ >, there exists v ∈ X satisfying the following: (P) f (v) < f (u) – λd(u, v) + δ; (Q) f (w) > f (v) – λd(v, w) for every w ∈ X \ {v}; (R) if a sequence {xn} in X satisfies limn→∞(f + λd(v, xn)) = f (v), {xn} converges to v
Summary
In , Ekeland [ ] proved the following, which is called the Ekeland variational principle (for short, EVP).Theorem . [ ] Let (X, d) be a complete metric space with metric d and f be a function from X into For all u ∈ X, λ > and δ > , there exists v ∈ X satisfying the following: (P) f (v) < f (u) – λd(u, v) + δ; (Q) f (w) > f (v) – λd(v, w) for every w ∈ X \ {v}; (R) if a sequence {xn} in X satisfies limn→∞(f (xn) + λd(v, xn)) = f (v), {xn} converges to v. Ume [ ] introduced a more generalized concept than τ -distance, which is called u-distance, and proved a new minimization and a new fixed point theorem by using u-distance on a complete metric space.
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