#### Abstract

The paper presents various results studying the asymptotic behavior of a sequence of lower semicontinuous functions on a metric space. In particular, different coercivity properties are obtained extending and refining previous results. The specific features and the structure of the terms of the sequence are used to construct appropriate quantities relevant in the verification of Palais-Smale compactness type conditions.

#### 1. Introduction and Main Results

Let be a complete metric space endowed with the metric . We recall from De Giorgi et al. [1] the notion of strong slope of a lower semicontinuous function (which is not identically ) at a point : If is a Banach space and , then for all .

Let be a function satisfying the property.

: there exist constants such that Let () be a sequence of lower semicontinuous functions. This paper develops a general approach for studying the asymptotic behavior of this sequence with respect to . An aspect which makes our approach general and natural is that we do not require the sequence to admit a limit (in any sense); see also Remarks 7, 9, and 11 below. We introduce the notation

Here and throughout the paper, for all we denote while and stand for the closure and the interior of , respectively. The expression always exists, generally belonging to . Equivalent expressions to can be given, for instance replacing by or by (see Lemma 16 below).

In what follows, we will always assume that.

: there holds .

For instance, is satisfied if is uniformly bounded below.

In the following we state our main result, which studies the asymptotic behavior of a sequence of lower semicontinuous functions.

Theorem 1. *Let be a complete metric space, let be a function satisfying , and let () be a sequence of lower semicontinuous functions satisfying . Then, for every , there exist a subsequence (depending on ) of and a number such that for each one finds satisfying
**
In particular, there are a subsequence of and elements such that
*

The proof of Theorem 1 is done in Section 2.

Note that, due to the hypothesis that , at least a subsequence of functions is not identically and the sets are nonempty for all .

We say that a sequence is *-bounded* if the sequence is bounded. We introduce the following notions of Palais-Smale condition and coercivity relative to the function .

*Definition 2. *Let () be lower semicontinuous functions which are not identically . We say that the sequence satisfies the Palais-Smale condition relative to (condition , for short) if whenever is a subsequence of and is a sequence such that is bounded and as , then is -bounded.

*Definition 3. *Assume that the function satisfies in addition the requirement that . We say that the sequence () is -*coercive* if .

If is a Banach space, , and for all , then we retrieve the usual notion of coercivity.

We state the following result on the -coercivity of a sequence of lower semicontinuous functions.

Corollary 4. *Let be a complete metric space, let be a function satisfying and , and let () be a sequence of lower semicontinuous functions satisfying . If satisfies condition , then the sequence is -coercive.*

The proof of Corollary 4 is given in Section 2.

As an immediate consequence of Corollary 4, in a Banach space we have the following.

Corollary 5. *Let be a Banach space, and let () be a sequence of lower semicontinuous functions satisfying . If satisfies condition , then the sequence is -coercive.*

Consider now the particular case in Theorem 1 when the number in is also a of a given lower semicontinuous function . Setting the hypothesis in Theorem 1 is obviously satisfied if we assume the conditions It will be noted in Lemma 19 that the first condition in (10) is satisfied if the following is assumed.

: there exists such that for every one has(i)for every , there exists a sequence such that and as ;(ii).

Concerning the second condition in (10), we have the following simple characterization: given , we have that if and only if, (see Lemma 17 below). With the above comments, the following result is a consequence of Theorem 1.

Corollary 6 (Corvellec [2, Theorem 1′]). *Let be a complete metric space, let be a function bounded on bounded subsets of and satisfying , and let () be lower semicontinuous functions satisfying that,
**
Assume (11) for some . Then there exist a subsequence of and a sequence such that , , and as .*

*Remark 7. *The number in Corollary 6 is necessarily . Moreover, hypotheses (12)-(13) are particular cases of (i)-(ii) that involve only sets of the form . Hence the hypotheses of Corollary 6, namely, (11), (12), (13), and , imply that . Therefore, [2, Theorem 1′] (i.e., Corollary 6) is retrieved as a consequence of Theorem 1 (and then the hypothesis that is bounded on bounded subsets of is not even needed). As seen from Example 12, Theorem 1 is actually more general than [2, Theorem 1′], and besides it does not need an auxiliary function in its hypotheses.

Next, we note that the hypothesis in Corollary 4 is satisfied if we assume the conditions
As noticed above, the first condition in (14) is satisfied if holds, which in turn is satisfied if (12) and (13) are assumed. The second condition in (14) is equivalent to the property for some ; that is,
(see Lemma 18). We thus have the following consequence of Corollary 4. Here, as in [2], it is said that satisfies condition if whenever is a subsequence of and is a sequence such that is bounded and as , then is bounded.

Corollary 8 (Corvellec [2, Theorem 1]). *Let be a complete metric space, let be a function bounded on bounded subsets of satisfying and , and let () be lower semicontinuous functions satisfying (12) and (13). If satisfies (15) and condition , then is -coercive (i.e., as ).*

*Remark 9. *From the above discussion, Corollary 8 (i.e., [2, Theorem 1]) is obtained as a consequence of Corollary 4. In fact, Corollary 4 is more general (see Example 12) and does not rely on an auxiliary function . In fact, on the one hand to study the coercivity of a function we do not need to look for a sequence as in Corollary 8 (in applications it seems to be more difficult to prove the existence of a sequence related to the function as in Corollary 8 than to prove the coercivity of itself). On the other hand, while studying the coercivity of a sequence , the interest of Corollary 4 is to give sufficient conditions for the coercivity of the sequence without using an auxiliary function . Finally, we note that in addition to the -coercivity of , the hypotheses of Corollary 8 imply also the -coercivity of , and so .

We also recall the following.

Corollary 10 (Corvellec [2, Corollary 1]). *Let be a Banach space, let () be bounded below, lower semicontinuous functions satisfying (12) and (13). Then is -coercive if and only if satisfies condition .*

The sufficiency in Corollary 10 follows from Corollary 5 (or from Corollary 8). The necessity can be proved arguing by contradiction in the following way. If there are a subsequence of and a sequence such that is bounded, and as , then as (by the -coercivity of ). Then, for every , using (13) with the closed set , we have which contradicts the boundedness of the sequence .

*Remark 11. *Hypotheses (12) and (13) imply the first part of (10); that is, . Hence, in view of Corollary 5, in place of assuming that and are bounded below in [2, Corollary 1] it would have been enough to assume that , which in fact is implied just by the boundedness below of . Corollary 5 is more general than Corollary 10 and its advantage is that it studies the coercivity of a sequence without dealing with an auxiliary function . For the study of the coercivity of a function we do not need to involve a sequence of functions (see Corollary 31 below).

*Example 12. *(a) Let be a lower semicontinuous, even (i.e., for all ) function which is not identically on , let the lower semicontinuous functions () be given by
and let for all . Then we have . Condition (i) is not satisfied (thus, (12) is not satisfied) since, if , , then for every sequence we have for large enough. So Corollaries 6, 8, and 10 cannot be applied, while Theorem 1 can be applied whenever . Corollaries 4 and 5 can also be applied.

(b) Let be defined by for all , let () be defined by for all , and let for all . Then we have . Condition (ii) is not satisfied (so neither (13)) since for every we have
Hence we may apply Theorem 1 to the sequence , but not Corollary 6 with and the chosen . Besides being more general, the advantage of Theorem 1 is to study the asymptotic behavior of a sequence without an auxiliary function (if it exists) as in Corollary 6.

(c) Let be defined by for all , let () be defined by for all , and let for all . Then . Condition (ii) is not satisfied (so neither (13)) since for every we have
Corollary 4 (or Corollary 5) can be applied to (note that satisfies condition ), while Corollaries 8 and 10 cannot be applied to and (however, is coercive).

(d) Let be a Banach space and be a continuous function satisfying . Let and be nondecreasing, lower semicontinuous functions with the property that there exists such that for all and . Define and let () be given by
where . Suppose . Then , but condition (ii) is not satisfied (so neither (13)) since for every we have
So, neither Corollary 6 nor Corollary 8 can be applied. Theorem 1 can be applied whenever (e.g., we do not need that , and then ) and its application is not related to some auxiliary function . For Corollary 4, we must necessarily have .

*Example 13. *(a) As examples of functions satisfying , we can consider any Lipschitz continuous function on a metric space , or any uniformly continuous function. For example, if is a metric space endowed with the metric , then the function
for some nonempty subset of , satisfies hypothesis with any , and if is unbounded, then satisfies also that (so both Theorem 1 and Corollary 4 can be applied in this case). In particular, in the case where is a Banach space, the function satisfies and that .

(b) Note that if is a bounded subset of the metric space , then the expression (i.e., for ) and the notion of -coercivity do not depend on the choice of the set (for this reason, we refer to -coercivity in place of -coercivity). If is unbounded, then it is not anymore the case: for example, if , , and given by , then is -coercive, but it is not norm coercive (taking the Euclidean norm and denoting by the induced distance).

(c) As another example of (which is not even continuous), let , given by
The function satisfies with , . Note that for every , we have (so the sets are nonempty).

*Remark 14. *(a) If and are two functions satisfying and that , and if is bounded, then the -coercivity and the -coercivity of a sequence as in Corollary 4 are equivalent.

(b) If is a metric space endowed with two metrics and which induce the same topology, then a sequence as in Corollary 4 may be -coercive and non--coercive.

(c) Let be a metric space endowed with the metric , let be a function satisfying and , and let us define a new metric:
The topology induced by is finer than the topology induced by (if is Lipschitz continuous with respect to , then they are equivalent). Applying Corollary 4 to the metric , if is bounded, then the -coercivity is equivalent to the -coercivity. In this case, Corollary 4 yields that if is a sequence of lower semicontinuous functions with respect to (thus with respect to ) satisfying and such that condition holds with respect to the metric , then is -coercive.

The rest of the paper is organized as follows.

Section 2 contains the proofs of the results stated in Section 1, based on the Ekeland variational principle. Our approach in showing Theorem 1 relies on the ideas in the proof of Motreanu-Motreanu [3, Theorem 3.1], which is a different approach from the one of Corvellec [2].

Section 3 contains further applications of Theorem 1 and Corollary 4 to special classes of sequences of lower semicontinuous functions. Section 3.1 is concerned with the coercivity of a sequence of lower semicontinuous functions fulfilling in the case where in the definition of is actually a limit. Section 3.2 studies the coercivity of a sequence of lower semicontinuous functionals which can be written as a sum of a locally Lipschitz function and a convex, lower semicontinuous function which is not identically . Section 3.3 deals with the coercivity of a continuously differentiable functional on a Banach space under a Palais-Smale type condition relative to a sequence of Galerkin approximations of . Section 3.4 deals with the case of a constant sequence of lower semicontinuous functions. Section 3.5 discusses what happens if in place of hypothesis we consider the case where the limits in are interchanged.

#### 2. Proofs of Theorem 1, Corollary 4, and Additional Lemmas

A basic ingredient in proving our results is the following version of the Ekeland variational principle (see Ekeland [4]).

Theorem 15. *Let be a complete metric space and let be a lower semicontinuous function which is bounded below and not identically . Then for every and every there exists such that
*

We need the following preliminary lemma.

Lemma 16. *If satisfies , then for every one has,*(a)*,*(b)*.*

*Proof. *(a) Let , and let be a sequence such that as . Since, for sufficiently large, , by hypothesis we have that .

(b) Let , and let be such that . Then hypothesis yields , which completes the proof.

*Proof of Theorem 1. *We denote . Fix . By the definition of , it follows that there exists such that
This implies that for every we find a number such that
and a subsequence (depending on ) of and a number such that

Denote
In particular, from (27), for , we find a number such that
while from (28), for we find a subsequence (which is the subsequence ) of and a number such that
Fix . Using (31) we find a point
for which one has
Corresponding to the set (see (29)), consider the function as follows:
It is clear that the set is a complete metric space with respect to the metric induced by and that the function is lower semicontinuous. In addition, is not identically since by (32), (33), and (34) we have that . By (34), (30) and using that , we see that
hence is bounded from below. Therefore we are in a position to apply Theorem 15 to the function on the complete metric space . Then there exists
such that
In view of (33), we see that , which combined with (37) ensures that .

By (36) and Lemma 16 (a) we have that , which, in view of (29), yields , so (7) holds true.

Using (35), (36), (37), and (33) we have
which proves (5).

Let us show that

Let . To get (40) it suffices to prove that
To see this, let satisfy . By (39), we have that , which implies that . In view of , this leads to . On the other hand, by (32) and Lemma 16 (a), we know that . It follows that , where we have used (29). This yields (41), which proves (40).

On the other hand, (38) yields
Taking (40) into account, we can pass to as in (42) to obtain
where the is taken for (using (41)). If is not a local minimum of , then inequality (43) means that , while if is a local minimum of , then we know that . Hence (6) is proved. Since there is no loss of generality in taking , the proof of the first part of the conclusion is complete.

For every fixed integer , applying the first part of the conclusion with , we find a subsequence (depending on ) of and a number such that for all there exists satisfying
In particular, we obtain a subsequence of (setting ) and with the properties stated in the second part of the conclusion.

*Proof of Corollary 4. *Arguing by contradiction, suppose that the sequence is not -coercive; that is, . Combining this with hypothesis , we infer that . With all the hypotheses of Theorem 1 being satisfied, we then obtain a subsequence of and satisfying
The first two convergences in (45) in conjunction with condition yield that the sequence is -bounded, which contradicts the last convergence in (45).

We conclude this section with the proof of some assertions stated in Section 1. First, recall that where the last equality is true in view of Lemma 16(a).

Lemma 17. *Let be a metric space and let satisfy . Given , for every one has that if and only if (11) holds.*

*Proof. *Suppose that . Then, we see from (46) that if then there exists such that (i.e., ), and if then for all (i.e., for every there exists ).

Conversely, suppose that satisfies (11). From the first part of (11), we have that if , then there exists such that . It follows that (see (46)), whence . From the second part of (11), we have that if , then for every , there exists such that ; hence . We conclude that for all , whence .

Lemma 18. *Let be a metric space and let satisfy . Then for every one has that if and only if for some .*

*Proof. *Suppose that . Then, by (46), there exist with such that ; that is, . Conversely, if for some , then (see (46)).

Lemma 19. *Let be a metric space, let satisfy , and let () satisfy . Then
*

*Proof. *Fix , with given in , and let . By (i), there exists a sequence such that and as . By Lemma 16, we have . Then for all sufficiently large, say . It follows that
Passing to as and using (ii), we infer that
Since this inequality holds for every , we obtain
Since is arbitrary, letting establishes the lemma.

#### 3. Special Cases and Further Remarks

##### 3.1. Case of the Existence of Limit in

Let be a Banach space, let be a function satisfying , and let () be a sequence of lower semicontinuous functions which are not identically . We assume the following.

: there exists such that for all , exists.

Note that is satisfied by all the functions in Example 12(a), (b), and (c). We consider another notion of Palais-Smale condition relative to .

*Definition 20. *Let () be lower semicontinuous functions which are not identically . We say that the sequence satisfies the generalized Palais-Smale condition relative to (condition , for short) if there exists a subsequence of such that whenever is a sequence such that is bounded and as , then is -bounded.

*Remark 21. *Condition is more general than condition of Definition 2.

Corollary 22. *Assume that and hold.*(i)*Assume . Then for every , there exists such that for each one finds satisfying
In particular, there exists () satisfying
*(ii)*Assume that satisfies condition and that . Then the sequence is -coercive.*

*Proof. *(i) We argue as in the proof of Theorem 1 noting that, due to the assumption, in place of (27) and (28) we have
and choosing .

(ii) Arguing by contradiction, suppose that the sequence is not -coercive; that is, ; thus (see ). Then, by part (i), we can find satisfying (52). Let be the subsequence of that satisfies condition . Then the convergences and yield that is -bounded, which contradicts the third convergence in (52).

##### 3.2. Case of Functionals with Special Structure

Let be a Banach space, be a function satisfying , and () be of the form with locally Lipschitz and convex, lower semicontinuous, not identically . In this setting we consider an appropriate version of Palais-Smale condition (see Motreanu and Panagiotopoulos [5, Chapter 3]).

*Definition 23. *The sequence of functionals as in (54) satisfies the Palais-Smale condition in the sense of Motreanu and Panagiotopoulos relative to (condition for short) if whenever is a subsequence of and is a sequence such that is bounded and for which there exists a sequence , , such that
then is -bounded.

Hereafter, the notation stands for the generalized directional derivative of a locally Lipschitz functional at the point in the direction (see Clarke [6]) given by

*Remark 24. *Condition in the above definition generalizes the Palais-Smale conditions of Chang [7] (for the case where is locally Lipschitz, , and ) and Szulkin [8] (for the case where , is lower semicontinuous, convex, not identically , and ).

Lemma 25. *(a) Let be a locally Lipschitz functional, let be a convex, lower semicontinuous function which is not identically , and let . Then
**(b) For a sequence as in Definition 23, one has
*

*Proof. *(a) Using the convexity of , for every , , we have
If is not a local minimum of , then the desired inequality follows. If is a local minimum of , then it is a critical point of in the sense of Motreanu-Panagiotopoulos [5, Definition 3.1]; that is, for all , and so again we are done.

(b) This is an immediate consequence of part (a).

Corollary 26. *Let be a Banach space and let satisfy . Let () as in (54). Assume that holds. *(i)*Assume . Then, for every , there exist a subsequence (depending on ) of and a number such that for each one finds satisfying
*(ii)*Assume that satisfies condition and that . Then the sequence is -coercive.*

*Proof. *Part (i) follows from Theorem 1 by using Lemma 25(a), while part (ii) follows from Corollary 4 by using Lemma 25(b).

*Remark 27. *When all the terms of the sequence coincide (and ), an extension of Corollary 26 has been obtained in Motreanu et al. [9] by means of a general Palais-Smale condition incorporating the Palais-Smale conditions in the sense of Cerami [10] and Zhong [11].

##### 3.3. Case of Galerkin Approximations

Let be a Banach space, let be a sequence of closed vector subspaces of (not necessarily increasing) such that , and let be a function satisfying . Let and let We consider the following Palais-Smale condition (see Li and Willem [12]).

*Definition 28. *The function satisfies the Palais-Smale condition in the sense of Li-Willem relative to (condition for short) if every sequence with , , bounded and is -bounded.

For all , define the functions by It is clear that the functions are lower semicontinuous and not identically .

Lemma 29. *(a) for all and all .**(b) (of Definition 28 for ) (of Definition 2 for ).*

*Proof. *(a) Using that , for every which is not a local minimizer of (equivalently, nor of ) we have
The case where is a local minimizer of is straightforward.

(b) This easily follows from part (a).

Denote .

Corollary 30. *Let be a Banach space, let be a sequence of closed vector subspaces of such that , and let be a function satisfying . Let and (). Assume that .*(i)*Assume . Then, for every , there exist a subsequence (depending on ) of and a number such that for each one finds satisfying
*(ii)*Assume that satisfies condition and that . Then .*

*Proof. *Part (i) follows by applying Theorem 1 to the sequence in (62), noting that and using Lemma 29(a). To prove part (ii), apply Corollary 4 to and use Lemma 29(b).

##### 3.4. Case of a Constant Sequence of Functions

When all the terms of the sequence coincide, say , for all , Theorem 1 and Corollary 4 yield the following.

Corollary 31. *Let be a complete metric space and let satisfy . Let be a lower semicontinuous function with the property
*(i)*If , then for every there exists such that
*(ii)*Assume that satisfies the following Palais-Smale condition: every sequence such that is bounded and as is -bounded. Assume also that . Then is -coercive; that is, , or, equivalently, as .*

*Proof. *Part (i) follows from Theorem 1 applied with for all (note that ), while part (ii) follows from Corollary 4 in the same way.

*Remark 32. *(a) The condition that ensures that the function is not identically and that the sets are nonempty for all .

(b) In Motreanu et al. [13], results concerning the asymptotic behavior as in Corollary 31(i) and the -coercivity as in Corollary 31(ii) are given in the more general setting of a metric space endowed with a quasiorder ≤ and of ≤-lower semicontinuous functions, by means of an appropriate notion of strong slope. Corollary 31 can be obtained from [13, Theorems 6.1, and 6.2] in the case where the quasiorder is the trivial one.

We apply Corollary 31 to the special situation (see Example 13(a)).

Corollary 33. *Let be a complete metric space, let be a nonempty subset of , and let be a lower semicontinuous function with the property
*(i)*If , then for every there is such that
*(ii)*Assume that satisfies the following Palais-Smale condition: every sequence such that is bounded and as satisfies that is bounded. If is unbounded, then is coercive in the sense that as .*

*Proof. *This readily follows from Corollary 31 applied to .

*Remark 34. *In the case where is bounded, the expression in Corollary 33 and the coercivity property in (ii) do not depend on the set .

In particular, we can consider , for some . If is a Banach space and , then Corollary 33 becomes the following.

Corollary 35. *Let be a Banach space and let be a lower semicontinuous function with the property .*(i)*If , then for every there exists such that
*(ii)*Assume that satisfies the following Palais–Smale condition: every sequence such that is bounded and as is bounded. Then is coercive in the sense that as .*

*Remark 36. *Corollary 35 extends the corresponding result in the smooth case, that is, for of class (see Brezis and Nirenberg [14]).

##### 3.5. Case Where the Limits in Are Interchanged

It is natural to ask what happens when the expression in is replaced with