Characterizations of reflexivity and compactness via the strong Ekeland variational principle

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Abstract

It is well known that the famous Ekeland variational principle characterizes the metric completeness of underlying spaces. In this paper, we prove that some versions of the strong Ekeland variational principle characterize the reflexivity and the compactness of underlying spaces.

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 (X,d) be a complete metric space. Let f be a function from X into (,+] which is proper lower semicontinuous and bounded from below. Then for uX and λ>0 , there exists vX satisfying the following:

  • (P)

    f(v)f(u)λd(u,v) .

  • (Q)

    f(w)>f(v)λd(v,w) for every wX{v} .

Georgiev [3] proved the following theorem.

Theorem 2 Georgiev [3]

Let X , d and f be as inTheorem 1. Then for uX , λ>0 and δ>0 , there exists vX satisfying the following:

  • (P)′

    f(v)<f(u)λd(u,v)+δ .

  • (Q)

    f(w)>f(v)λd(v,w) for every wX{v} .

  • (R)

    If a sequence {xn} in X satisfies limn(f(xn)+λd(v,xn))=f(v) , then {xn} converges to v .

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 X; see [4], [5], [6]. Motivated by both theorems, the author proved the following.

Theorem 3 [7]

Assume either of the following:

  • (A)

    X is a reflexive Banach space and f is a quasiconvex function from X into (,+] which is proper lower semicontinuous and bounded from below.

  • (B)

    X is a compact metric space and f is a function from X into (,+] which is proper and lower semicontinuous.

Then for uX and λ>0 , there exists vX satisfying (P)(R).

It is a very natural question as to whether the reflexivity and compactness of X 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 X.

Section snippets

Reflexivity

Throughout this paper, we denote by N, R and C the sets of positive integers, real numbers and complex numbers, respectively.

Let X be a normed linear space and let f be a function from X into (,+]. We recall that f is called quasiconvex if {xX:f(x)α} is a convex subset of X for every αR. It is obvious that convexity implies quasiconvexity. f is called Lipschitz continuous with constant L if there exists a real number L such that |f(x)f(y)|Lxy for all x,yX.

In this section, we prove

Bounded compactness

In this section, we characterize the bounded compactness. A metric space X is said to be boundedly compact or said to have the Heine–Borel property if every closed bounded subset of X is compact; see [9], [10], [11], [12], [13]. It is obvious that every boundedly compact metric space is complete. If X is a normed linear space, then X is finite-dimensional if and only if X 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 (X,d) be a metric space. Then the following are equivalent:

  • (i)

    X is compact.

  • (ii)

    If f is a proper lower semicontinuous function from X into (,+] , p is a τ-distance on X such that p is lower semicontinuous in its second variable, uX satisfies p(u,u)=0 and λ>0 , then there exists vX satisfying (P)p–(R)p .

  • (iii)

    There exists c>0 such that if f is a Lipschitz continuous function from (X,d) into [0,) , uX and λ>0 , 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.

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