Characterizations of reflexivity and compactness via the strong Ekeland variational principle
Introduction
Ekeland [1], [2] proved the following very useful and interesting existence theorem, which is called the Ekeland variational principle.
Theorem 1 Ekeland [1], [2] Let be a complete metric space. Let be a function from into which is proper lower semicontinuous and bounded from below. Then for and , there exists satisfying the following: . for every .
Georgiev [3] proved the following theorem.
Theorem 2 Georgiev [3] Let , and be as inTheorem 1. Then for , and , there exists satisfying the following: . for every . If a sequence in satisfies , then converges to .
Theorem 2 is called the strong Ekeland variational principle. But, since (P)′ is weaker than (P), Theorem 2 is not stronger than Theorem 1. Indeed, Theorem 2 is one of variants of Theorem 1. The author guesses that there is no theorem which is stronger than Theorem 1 because the conclusion of Theorem 1 is equivalent to the completeness of ; see [4], [5], [6]. Motivated by both theorems, the author proved the following.
Theorem 3 [7] Assume either of the following: is a reflexive Banach space and is a quasiconvex function from into which is proper lower semicontinuous and bounded from below. is a compact metric space and is a function from into which is proper and lower semicontinuous.
Then for and , there exists satisfying (P)–(R).
It is a very natural question as to whether the reflexivity and compactness of in Theorem 3 are appropriate. In this paper, we give an answer to this question. Our answer is that the reflexivity in Theorem 3(A) is appropriate and the compactness in Theorem 3(B) is a little strong. We also give two characterizations of the compactness of .
Section snippets
Reflexivity
Throughout this paper, we denote by , and the sets of positive integers, real numbers and complex numbers, respectively.
Let be a normed linear space and let be a function from into . We recall that is called quasiconvex if is a convex subset of for every . It is obvious that convexity implies quasiconvexity. is called Lipschitz continuous with constant if there exists a real number such that for all .
In this section, we prove
Bounded compactness
In this section, we characterize the bounded compactness. A metric space is said to be boundedly compact or said to have the Heine–Borel property if every closed bounded subset of is compact; see [9], [10], [11], [12], [13]. It is obvious that every boundedly compact metric space is complete. If is a normed linear space, then is finite-dimensional if and only if is boundedly compact.
In order to compare our result with Sullivan’s theorem [5] directly, we modify Sullivan’s theorem again.
Compactness
In this section, using Theorem 8, we give two characterizations of the compactness.
Theorem 10 Let be a metric space. Then the following are equivalent: is compact. If is a proper lower semicontinuous function from into , is a -distance on such that is lower semicontinuous in its second variable, satisfies and , then there exists satisfying (P)p–(R)p . There exists such that if is a Lipschitz continuous function from into , and , then
Acknowledgements
The author is very grateful to the referee for his/her careful reading. Also the referee gives Corollary 9.
The author is supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.
References (15)
On the variational principle
J. Math. Anal. Appl.
(1974)The strong Ekeland variational principle, the strong drop theorem and applications
J. Math. Anal. Appl.
(1988)The strong Ekeland variational principle
J. Math. Anal. Appl.
(2006)- et al.
Boundedly connected sets and the distance to the intersection of two sets
J. Math. Anal. Appl.
(2007) Generalized distance and existence theorems in complete metric spaces
J. Math. Anal. Appl.
(2001)Nonconvex minimization problems
Bull. Amer. Math. Soc.
(1979)Caristi’s fixed point theorem and metric convexity
Colloq. Math.
(1976)
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