Abstract and Applied Analysis
Volume 2013, Issue 1 268650
Research Article
Open Access

Coercivity Properties for Sequences of Lower Semicontinuous Functions on Metric Spaces

D. Motreanu, 

D. Motreanu

Département de Mathématiques, Université de Perpignan Via Domitia, 66860 Perpignan, France univ-perp.fr

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V. V. Motreanu, 

Corresponding Author

V. V. Motreanu

Department of Mathematics, Ben Gurion University of the Negev, 84105 Be′er Sheva, Israel bgu.ac.il

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First published: 02 December 2013
https://doi.org/10.1155/2013/268650
Academic Editor: Salvador Hernandez

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 X be a complete metric space endowed with the metric d. We recall from De Giorgi et al. [1] the notion of strong slope of a lower semicontinuous function f : X → ℝ ∪ {+∞} (which is not identically +∞) at a point u ∈ dom (f): = {w ∈ X : f(w)<+∞}:
()
If X is a Banach space and f ∈ C1(X, ℝ), then |∇f | (u) = ∥f′(u)∥ for all u ∈ X.

Let F : X → ℝ be a function satisfying the property.

H(F): there exist constants γ1, γ2 > 0 such that
()
Let fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be a sequence of lower semicontinuous functions. This paper develops a general approach for studying the asymptotic behavior of this sequence with respect to F. An aspect which makes our approach general and natural is that we do not require the sequence {fn} 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 r > 0 we denote
()
while and int  [F > r] stand for the closure and the interior of [F > r], respectively. The expression αF({fn}) always exists, generally belonging to ℝ ∪ {±∞}. Equivalent expressions to αF({fn}) can be given, for instance replacing by [F > r] or by [F ≥ r]: = {w ∈ X : F(w) ≥ r} (see Lemma 16 below).

In what follows, we will always assume that.

HF({fn}): there holds αF({fn})>−∞.

For instance, HF({fn}) is satisfied if {fn} 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 X be a complete metric space, let F : X → ℝ be a function satisfying H (F), and let fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be a sequence of lower semicontinuous functions satisfying αF({fn}) ∈ ℝ. Then, for every ε > 0, there exist a subsequence (depending on ε) of {fn} and a number kε ∈ ℕ such that for each k ≥ kε one finds satisfying

()
()
()
In particular, there are a subsequence of {fn} and elements such that
()

The proof of Theorem 1 is done in Section 2.

Note that, due to the hypothesis that αF({fn}) ∈ ℝ, at least a subsequence of functions fn is not identically +∞ and the sets [F > r] are nonempty for all r ∈ ℝ.

We say that a sequence {un} ⊂ X is F-bounded if the sequence {F(un)} ⊂ X is bounded. We introduce the following notions of Palais-Smale condition and coercivity relative to the function F.

Definition 2. Let fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be lower semicontinuous functions which are not identically +∞. We say that the sequence {fn} satisfies the Palais-Smale condition relative to F (condition (PS) F, for short) if whenever is a subsequence of {fn} and {uk} ⊂ X is a sequence such that is bounded and as k → ∞, then {uk} is F-bounded.

Definition 3. Assume that the function F satisfies in addition the requirement that sup XF = +∞. We say that the sequence fn : X → ℝ ∪ {+∞} (n ∈ ℕ) is F-coercive if αF({fn}) = +∞.

If (X, ∥·∥) is a Banach space, F = ∥·∥, and fn ≡ f for all n ∈ ℕ, then we retrieve the usual notion of coercivity.

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

Corollary 4. Let X be a complete metric space, let F : X → ℝ be a function satisfying H (F) and sup XF = +∞, and let fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be a sequence of lower semicontinuous functions satisfying H F({fn}). If {fn} satisfies condition ( PS ) F, then the sequence {fn} is F-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 (X, ∥·∥) be a Banach space, and let fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be a sequence of lower semicontinuous functions satisfying H ∥·∥({fn}). If {fn} satisfies condition ( PS ) ∥·∥, then the sequence {fn} is ∥·∥-coercive.

Consider now the particular case in Theorem 1 when the number αF({fn}) in H F({fn}) is also a liminf  of a given lower semicontinuous function f : X → ℝ ∪ {+∞}. Setting
()
the hypothesis αF({fn}) ∈ ℝ 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.
HF({fn}, f): there exists r0 ∈ ℝ such that for every r ≥ r0 one has
  • (i)

    for every , there exists a sequence {un} such that un → u and fn(un) → f(u) as n → ∞;

  • (ii)

    .

Concerning the second condition in (10), we have the following simple characterization: given c ∈ ℝ, we have that c = αF(f) 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 X be a complete metric space, let F : X → ℝ be a function bounded on bounded subsets of X and satisfying H (F), and let f, fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be lower semicontinuous functions satisfying that,

()
()
Assume (11) for some c ∈ ℝ. Then there exist a subsequence of {fn} and a sequence {uk} ⊂ X such that , , and F(uk)→+∞ as k → ∞.

Remark 7. The number c ∈ ℝ in Corollary 6 is necessarily c = αF(f) = αF({fn}). Moreover, hypotheses (12)-(13) are particular cases of HF({fn}, f) (i)-(ii) that involve only sets of the form . Hence the hypotheses of Corollary 6, namely, (11), (12), (13), and c ∈ ℝ, imply that αF({fn}) ∈ ℝ. Therefore, [2, Theorem 1′] (i.e., Corollary 6) is retrieved as a consequence of Theorem 1 (and then the hypothesis that F is bounded on bounded subsets of X 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 f in its hypotheses.

Next, we note that the hypothesis HF({fn}) in Corollary 4 is satisfied if we assume the conditions

()
As noticed above, the first condition in (14) is satisfied if HF({fn}, f) holds, which in turn is satisfied if (12) and (13) are assumed. The second condition in (14) is equivalent to the property [F > b]⊂[f > a] for some a, b ∈ ℝ; that is,
()
(see Lemma 18). We thus have the following consequence of Corollary 4. Here, as in [2], it is said that f satisfies condition (PSB) * if whenever is a subsequence of {fn} and {uk} ⊂ X is a sequence such that is bounded and as k → ∞, then {uk} is bounded.

Corollary 8 (Corvellec [2, Theorem 1]). Let X be a complete metric space, let F : X → ℝ be a function bounded on bounded subsets of X satisfying H (F) and sup XF = +∞, and let f, fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be lower semicontinuous functions satisfying (12) and (13). If f satisfies (15) and condition ( PSB ) *, then f is F-coercive (i.e., f(u)→+∞ as F(u)→+∞).

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 f. In fact, on the one hand to study the coercivity of a function f we do not need to look for a sequence {fn} as in Corollary 8 (in applications it seems to be more difficult to prove the existence of a sequence {fn} related to the function f as in Corollary 8 than to prove the coercivity of f itself). On the other hand, while studying the coercivity of a sequence {fn}, the interest of Corollary 4 is to give sufficient conditions for the coercivity of the sequence {fn} without using an auxiliary function f. Finally, we note that in addition to the F-coercivity of f, the hypotheses of Corollary 8 imply also the F-coercivity of {fn}, and so αF({fn}) = αF(f) = +∞.

We also recall the following.

Corollary 10 (Corvellec [2, Corollary 1]). Let (X, ∥·∥) be a Banach space, let f, fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be bounded below, lower semicontinuous functions satisfying (12) and (13). Then f is ∥·∥-coercive if and only if f satisfies condition ( PSB ) *.

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 {fn} and a sequence {uk} ⊂ X such that is bounded, and ∥uk∥ → +∞ as k → ∞, then f(uk)→+∞ as k → ∞ (by the ∥·∥-coercivity of f). Then, for every n ∈ ℕ, using (13) with the closed set Yn : = {un, un+1, …}, we have
()
which contradicts the boundedness of the sequence .

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

Example 12. (a) Let f : ℝ → ℝ ∪ {+∞} be a lower semicontinuous, even (i.e., f(x) = f(−x) for all x ∈ ℝ) function which is not identically +∞ on ℝ∖{0}, let the lower semicontinuous functions fn : ℝ → ℝ ∪ {+∞} (n ∈ ℕ) be given by

()
and let F(x) = |x| for all x ∈ ℝ. Then we have αF({fn}) = αF(f). Condition HF({fn}, f) (i) is not satisfied (thus, (12) is not satisfied) since, if x ∈ dom (f), x < 0, then for every sequence xn → x we have fn(xn) = +∞ > f(x) for n ∈ ℕ large enough. So Corollaries 6, 8, and 10 cannot be applied, while Theorem 1 can be applied whenever αF(f) ∈ ℝ. Corollaries 4 and 5 can also be applied.

(b) Let f : ℝ → ℝ be defined by for all x ∈ ℝ, let fn : ℝ → ℝ (n ∈ ℕ) be defined by for all x ∈ ℝ, and let F(x) = |x| for all x ∈ ℝ. Then we have αF({fn}) = αF(f) = 0. Condition HF({fn}, f) (ii) is not satisfied (so neither (13)) since for every r > 0 we have

()
Hence we may apply Theorem 1 to the sequence {fn}, but not Corollary 6 with {fn} and the chosen f. Besides being more general, the advantage of Theorem 1 is to study the asymptotic behavior of a sequence {fn} without an auxiliary function f (if it exists) as in Corollary 6.

(c) Let f : ℝ → ℝ be defined by f(x) = e|x| for all x ∈ ℝ, let fn : ℝ → ℝ (n ∈ ℕ) be defined by fn(x) = (x2/2)−(1/n) for all x ∈ ℝ, and let F(x) = |x| for all x ∈ ℝ. Then αF({fn}) = αF(f) = +∞. Condition HF({fn}, f) (ii) is not satisfied (so neither (13)) since for every r > 0 we have

()
Corollary 4 (or Corollary 5) can be applied to {fn} (note that {fn} satisfies condition (PS) F), while Corollaries 8 and 10 cannot be applied to {fn} and f (however, f is coercive).

(d) Let (X, ∥·∥) be a Banach space and F : X → ℝ be a continuous function satisfying H(F). Let g : ℝ → ℝ ∪ {+∞} and h : ℝ → ℝ be nondecreasing, lower semicontinuous functions with the property that there exists r0 > 0 such that h(r) < g(r) for all r ≥ r0 and lim r→+∞h(r) = lim r→+∞g(r)>−∞. Define f = g∘F and let fn : ℝ → ℝ (n ∈ ℕ) be given by

()
where {sn} ⊂ ℝ. Suppose liminf n→∞sn = 0. Then αF({fn}) = αF(f), but condition HF({fn}, f) (ii) is not satisfied (so neither (13)) since for every r > r0 we have
()
So, neither Corollary 6 nor Corollary 8 can be applied. Theorem 1 can be applied whenever αF({fn}) = lim r→+∞h(r) + liminf n→∞sn ∈ ℝ (e.g., we do not need that liminf n→∞sn = 0, and then αF({fn}) = αF(f) + liminf n→∞sn) and its application is not related to some auxiliary function f. For Corollary 4, we must necessarily have αF({fn}) = +∞.

Example 13. (a) As examples of functions F : X → ℝ satisfying H(F), we can consider any Lipschitz continuous function on a metric space X, or any uniformly continuous function. For example, if X is a metric space endowed with the metric d, then the function

()
for some nonempty subset A of X, satisfies hypothesis H(F) with any γ1 = γ2, and if X∖A is unbounded, then F satisfies also that sup XF = +∞ (so both Theorem 1 and Corollary 4 can be applied in this case). In particular, in the case where (X, ∥·∥) is a Banach space, the function F = ∥·∥ satisfies H(F) and that sup XF = +∞.

(b) Note that if A is a bounded subset of the metric space X, then the expression αd(·,A)({fn}) (i.e., αF({fn}) for F(·) = d(·, A)) and the notion of d(·, A)-coercivity do not depend on the choice of the set A (for this reason, we refer to d-coercivity in place of d(·, A)-coercivity). If A is unbounded, then it is not anymore the case: for example, if X = ℝ2, A = {(x, y) ∈ ℝ2 : y = 0}, and f : ℝ2 → ℝ given by f(x, y) = |y|, then f is d(·, A)-coercive, but it is not norm coercive (taking the Euclidean norm and denoting by d the induced distance).

(c) As another example of F (which is not even continuous), let X = ℝ, F : ℝ → ℝ given by

()
The function F satisfies H(F) with γ1 = 1, γ2 = 2. Note that for every r ∈ ℝ, we have F(r + 1) > r (so the sets [F > r] are nonempty).

Remark 14. (a) If F : X → ℝ and are two functions satisfying H(F) and that , and if is bounded, then the F-coercivity and the -coercivity of a sequence {fn} as in Corollary 4 are equivalent.

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

(c) Let X be a metric space endowed with the metric d, let F : X → ℝ be a function satisfying H(F) and sup XF = +∞, and let us define a new metric:

()
The topology induced by is finer than the topology induced by d (if F is Lipschitz continuous with respect to d, then they are equivalent). Applying Corollary 4 to the metric , if d is bounded, then the F-coercivity is equivalent to the -coercivity. In this case, Corollary 4 yields that if {fn} is a sequence of lower semicontinuous functions with respect to d (thus with respect to ) satisfying HF({fn}) and such that condition (PS) F holds with respect to the metric , then {fn} 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 HF({fn}) in the case where liminf  in the definition of αF({fn}) 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 f on a Banach space under a Palais-Smale type condition relative to a sequence of Galerkin approximations of f. 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 HF({fn}) we consider the case where the limits in αF({fn}) 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 (X, d) be a complete metric space and let f : X → ℝ ∪ {+∞} be a lower semicontinuous function which is bounded below and not identically +∞. Then for every ε > 0 and every v ∈ X there exists u ∈ X such that

()

We need the following preliminary lemma.

Lemma 16. If F : X → ℝ satisfies H (F), then for every r ∈ ℝ one has,

  • (a)

    ,

  • (b)

    [F > r] ⊂ int [F > r − γ2].

Proof. (a) Let , and let {un}⊂[F > r] be a sequence such that un → u as n → ∞. Since, for n sufficiently large, d(un, u) < γ1, by hypothesis H(F) we have that F(u) > F(un) − γ2 > r − γ2.

(b) Let u ∈ [F > r], and let v ∈ X be such that d(v, u) < γ1. Then hypothesis H(F) yields F(v) > F(u) − γ2 > r − γ2, which completes the proof.

Proof of Theorem 1. We denote α : = αF({fn}) ∈ ℝ. Fix ε ∈ (0, γ1/2). By the definition of α, it follows that there exists rε > 0 such that

()
This implies that for every r ≥ rε we find a number nr,ε ∈ ℕ such that
()
and a subsequence (depending on r) of {fn} and a number kr,ε ∈ ℕ 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 {fn} 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 d 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 με ∈ (0, γ1 − 2ε). To get (40) it suffices to prove that

()
To see this, let w satisfy d(w, uk,ε) < με. By (39), we have that d(uk,ε, vk,ε) < 2ε, which implies that d(w, vk,ε) ≤ d(w, uk,ε) + d(uk,ε, vk,ε) < με + 2ε < γ1. In view of H(F), this leads to |F(w) − F(vk,ε)| < γ2. 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 limsup  as w → uk,ε in (42) to obtain
()
where the limsup  is taken for w ∈ X (using (41)). If uk,ε is not a local minimum of , then inequality (43) means that , while if uk,ε is a local minimum of , then we know that . Hence (6) is proved. Since there is no loss of generality in taking ε ∈ (0, γ1/2), the proof of the first part of the conclusion is complete.

For every fixed integer ℓ ≥ 1, applying the first part of the conclusion with ε = 1/ℓ, we find a subsequence (depending on ℓ) of {fn} and a number kℓ ∈ ℕ such that for all k ≥ kℓ 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 {fn} is not F-coercive; that is, α : = αF({fn})<+∞. Combining this with hypothesis HF({fn}), we infer that α ∈ ℝ. 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 (PS) F yield that the sequence {uℓ} is F-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 X be a metric space and let F : X → ℝ satisfy H (F). Given c ∈ ℝ, for every f : X → ℝ ∪ {+∞} one has that c = αF(f) if and only if (11) holds.

Proof. Suppose that c = αF(f). Then, we see from (46) that if a < c then there exists r ∈ ℝ such that inf [F>r]f > a (i.e., [f ≤ a]⊂[F ≤ r]), and if a > c then inf [F>r]f < a for all r ∈ ℝ (i.e., for every r ∈ ℝ there exists vr ∈ [f ≤ a]∖[F ≤ r]).

Conversely, suppose that c satisfies (11). From the first part of (11), we have that if a < c, then there exists r ∈ ℝ such that [F > r]⊂[f > a]. It follows that αF(f) ≥ inf [F>r]f ≥ a (see (46)), whence αF(f) ≥ c. From the second part of (11), we have that if a > c, then for every r ∈ ℝ, there exists vr ∈ [F > r] such that f(vr) ≤ a; hence inf [F>r]f ≤ a. We conclude that αF(f) ≤ a for all a > c, whence αF(f) ≤ c.

Lemma 18. Let X be a metric space and let F : X → ℝ satisfy H (F). Then for every f : X → ℝ ∪ {+∞} one has that αF(f)>−∞ if and only if [F > b]⊂[f > a] for some a, b ∈ ℝ.

Proof. Suppose that αF(f)>−∞. Then, by (46), there exist a, b ∈ ℝ with a < αF(f) such that inf [F>b]f > a; that is, [F > b]⊂[f > a]. Conversely, if [F > b]⊂[f > a] for some a, b ∈ ℝ, then αF(f) ≥ inf [F>b]f ≥ a > −∞ (see (46)).

Lemma 19. Let X be a metric space, let F : X → ℝ satisfy H (F), and let f, fn : X → ℝ ∪ {+∞} (n ∈ ℕ) satisfy H F({fn}, f). Then

()

Proof. Fix r ≥ r0, with r0 given in HF({fn}, f), and let . By HF({fn}, f) (i), there exists a sequence {un} such that un → u and fn(un) → f(u) as n → ∞. By Lemma 16, we have . Then un ∈ [F > r − 2γ2] for all n ∈ ℕ sufficiently large, say n ≥ n0. It follows that

()
Passing to liminf  as n → ∞ and using HF({fn}, f) (ii), we infer that
()
Since this inequality holds for every , we obtain
()
Since r ≥ r0 is arbitrary, letting r → +∞ establishes the lemma.

3. Special Cases and Further Remarks

3.1. Case of the Existence of Limit in αF({fn})

Let X be a Banach space, let F : X → ℝ be a function satisfying H(F), and let fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be a sequence of lower semicontinuous functions which are not identically +∞. We assume the following.

: there exists r0 ∈ ℝ such that for all r > r0, 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 F.

Definition 20. Let fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be lower semicontinuous functions which are not identically +∞. We say that the sequence {fn} satisfies the generalized Palais-Smale condition relative to F (condition (gPS) F, for short) if there exists a subsequence of {fn} such that whenever {uk} ⊂ X is a sequence such that is bounded and as k → ∞, then {uk} is F-bounded.

Remark 21. Condition (gPS) F is more general than condition (PS) F of Definition 2.

Corollary 22. Assume that H F({fn}) and hold.

  • (i)

    Assume αF({fn})<+∞. Then for every ε > 0, there exists nε ∈ ℕ such that for each n ≥ nε one finds un,ε ∈ dom (fn) satisfying

    ()
    In particular, there exists un ∈ dom (fn) (n ∈ ℕ) satisfying
    ()

  • (ii)

    Assume that {fn} satisfies condition ( gPS ) F and that sup XF = +∞. Then the sequence {fn} is F-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 {fn} is not F-coercive; that is, α : = αF({fn})<+∞; thus α ∈ ℝ (see HF({fn})). Then, by part (i), we can find un ∈ dom (fn) satisfying (52). Let be the subsequence of {fn} that satisfies condition (gPS) F. Then the convergences and yield that is F-bounded, which contradicts the third convergence in (52).

3.2. Case of Functionals with Special Structure

Let (X, ∥·∥) be a Banach space, F : X → ℝ be a function satisfying H(F), and fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be of the form
()
with Φn : X → ℝ locally Lipschitz and Ψn : X → ℝ ∪ {+∞} 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 {fn} as in (54) satisfies the Palais-Smale condition in the sense of Motreanu and Panagiotopoulos relative to F (condition (PS+) F for short) if whenever is a subsequence of {fn} and {uk} ⊂ X is a sequence such that is bounded and for which there exists a sequence {εk} ⊂ ℝ+, εk ↓ 0, such that

()
then {uk} is F-bounded.

Hereafter, the notation Φ0(u; v) stands for the generalized directional derivative of a locally Lipschitz functional Φ : X → ℝ at the point u ∈ X in the direction v ∈ X (see Clarke [6]) given by
()

Remark 24. Condition (PS+) F in the above definition generalizes the Palais-Smale conditions of Chang [7] (for the case where Φn ≡ Φ is locally Lipschitz, Ψ = 0, and F = ∥·∥) and Szulkin [8] (for the case where Φn ≡ Φ ∈ C1(X, ℝ), Ψn ≡ Ψ is lower semicontinuous, convex, not identically +∞, and F = ∥·∥).

Lemma 25. (a) Let Φ : X → ℝ be a locally Lipschitz functional, let Ψ : X → ℝ ∪ {+∞} be a convex, lower semicontinuous function which is not identically +∞, and let f = Φ + Ψ. Then

()

(b) For a sequence {fn} as in Definition 23, one has

()

Proof. (a) Using the convexity of Ψ, for every u, v ∈ X, v ≠ u, we have

()
If u is not a local minimum of f, then the desired inequality follows. If u is a local minimum of f, then it is a critical point of f in the sense of Motreanu-Panagiotopoulos [5, Definition 3.1]; that is, Φ0(u; v − u) + Ψ(v) − Ψ(u) ≥ 0 for all v ∈ X, and so again we are done.

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

Corollary 26. Let X be a Banach space and let F : X → ℝ satisfy H (F). Let fn = Φn + Ψn : X → ℝ ∪ {+∞} (n ∈ ℕ) as in (54). Assume that H F({fn}) holds.

  • (i)

    Assume αF({fn})<+∞. Then, for every ε > 0, there exist a subsequence (depending on ε) of {fn} and a number kε ∈ ℕ such that for each k ≥ kε one finds satisfying

    ()

  • (ii)

    Assume that {fn} satisfies condition ( PS +) F and that sup XF = +∞. Then the sequence {fn} is F-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 {fn} coincide (and F = ∥·∥), 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 (X, ∥·∥) be a Banach space, let {Xn} be a sequence of closed vector subspaces of X (not necessarily increasing) such that , and let F : X → ℝ be a function satisfying H(F). Let f ∈ C1(X, ℝ) and let
()
We consider the following Palais-Smale condition (see Li and Willem [12]).

Definition 28. The function f ∈ C1(X, ℝ) satisfies the Palais-Smale condition in the sense of Li-Willem relative to F (condition (PS*) F for short) if every sequence {un} ⊂ X with , αn → +∞, {f(un)} bounded and is F-bounded.

For all n ∈ ℕ, define the functions by
()
It is clear that the functions are lower semicontinuous and not identically +∞.

Lemma 29. (a) for all u ∈ Xn and all n ∈ ℕ.

(b) ( PS *) F (of Definition 28 for f ∈ C1(X, ℝ)) ⇔( PS ) F (of Definition 2 for ).

Proof. (a) Using that fn ∈ C1(Xn, ℝ), for every u ∈ Xn which is not a local minimizer of (equivalently, nor of fn) we have

()
The case where u ∈ Xn is a local minimizer of is straightforward.

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

Denote .

Corollary 30. Let X be a Banach space, let {Xn} be a sequence of closed vector subspaces of X such that , and let F : X → ℝ be a function satisfying H (F). Let f ∈ C1(X, ℝ) and (n ∈ ℕ). Assume that .

  • (i)

    Assume . Then, for every ε > 0, there exist a subsequence (depending on ε) of {fn} and a number kε ∈ ℕ such that for each k ≥ kε one finds satisfying

    ()

  • (ii)

    Assume that f satisfies condition (PS*) F and that sup XF = +∞. 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 {fn} coincide, say fn = f, for all n, Theorem 1 and Corollary 4 yield the following.

Corollary 31. Let X be a complete metric space and let F : X → ℝ satisfy H (F). Let f : X → ℝ ∪ {+∞} be a lower semicontinuous function with the property

()
  • (i)

    If αF(f)<+∞, then for every ε > 0 there exists uε ∈ dom (f) such that

    ()

  • (ii)

    Assume that f satisfies the following Palais-Smale condition: every sequence {uk} ⊂ X such that {f(uk)} is bounded and |∇f | (uk) → 0 as k → ∞ is F-bounded. Assume also that sup XF = +∞. Then f is F-coercive; that is, αF(f) = +∞, or, equivalently, f(v)→+∞ as F(v)→+∞.

Proof. Part (i) follows from Theorem 1 applied with fn = f for all n ∈ ℕ (note that αF(f) = αF({f})), while part (ii) follows from Corollary 4 in the same way.

Remark 32. (a) The condition that αF(f) ∈ ℝ ensures that the function f is not identically +∞ and that the sets {v ∈ X : F(v) > r} are nonempty for all r ∈ ℝ.

(b) In Motreanu et al. [13], results concerning the asymptotic behavior as in Corollary 31(i) and the F-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 F = d(·, A) (see Example 13(a)).

Corollary 33. Let X be a complete metric space, let A be a nonempty subset of X, and let f : X → ℝ ∪ {+∞} be a lower semicontinuous function with the property

()
  • (i)

    If αd(·,A)(f)<+∞, then for every ε > 0 there is uε ∈ dom (f) such that

    ()

  • (ii)

    Assume that f satisfies the following Palais-Smale condition: every sequence {uk} ⊂ X such that {f(uk)} is bounded and |∇f | (uk) → 0 as k → ∞ satisfies that {d(uk, A)} is bounded. If X∖A is unbounded, then f is coercive in the sense that f(v)→+∞ as d(v, A)→+∞.

Proof. This readily follows from Corollary 31 applied to F = d(·, A).

Remark 34. In the case where A is bounded, the expression αd(·,A)(f) in Corollary 33 and the coercivity property in (ii) do not depend on the set A ⊂ X.

In particular, we can consider A = {u0}, for some u0 ∈ X. If X is a Banach space and u0 = 0, then Corollary 33 becomes the following.

Corollary 35. Let (X, ∥·∥) be a Banach space and let f : X → ℝ ∪ {+∞} be a lower semicontinuous function with the property α∥·∥(f): = liminf ∥v∥→+∞f(v)>−∞.

  • (i)

    If α∥·∥(f)<+∞, then for every ε > 0 there exists uε ∈ dom (f) such that

    ()

  • (ii)

    Assume that f satisfies the following Palais–Smale condition: every sequence {uk} ⊂ X such that {f(uk)} is bounded and |∇f | (uk) → 0 as k → ∞ is bounded. Then f is coercive in the sense that f(v)→+∞ as ∥v∥ → +∞.

Remark 36. Corollary 35 extends the corresponding result in the smooth case, that is, for f of class C1 (see Brezis and Nirenberg [14]).

3.5. Case Where the Limits in αF({fn}) Are Interchanged

It is natural to ask what happens when the expression αF({fn}) in HF({fn}) is replaced with
()
that is, when the limits in HF({fn}) are interchanged (we work directly with a sequence {fn} for which the limit as n → ∞ in (70) exists in place of considering liminf ).

Proposition 37. Let X be a complete metric space, let F : X → ℝ be a function satisfying H (F), and let fn : X → ℝ ∪ {+∞} (n ∈ ℕ) be a sequence of lower semicontinuous functions with the property .

  • (i)

    Assume . Then for every ε > 0, there exists nε ∈ ℕ such that for each n ≥ nε one finds un,ε ∈ dom (fn) satisfying

    ()

  • (ii)

    Assume that {fn} satisfies condition ( gPS ) F (see Definition 20) and that sup XF = +∞. Then .

Proof. (i) We denote . Fix ε > 0. By (70), there exists nε ∈ ℕ such that

()
Denote . In view of (72) we have αn ∈ ℝ for all n ≥ nε. Then, for each n ≥ nε we can apply Corollary 31(i) to the function f : = fn and the number . Thus, we find un,ε ∈ dom (fn) such that
()

(ii) Arguing by contradiction, suppose that . Let a subsequence of {fn}, denoted again by {fn}, with the property in condition (gPS) F. Applying part (i) of the proposition to the subsequence {fn} and ε = 1/k (k ∈ ℕ), we find a number nk ∈ ℕ and an element with the properties

()
Consequently, we have , , and F(uk)→+∞ as k → ∞. The first two convergences in conjunction with condition (gPS) F yield that the sequence {uk} is F-bounded, which contradicts the third convergence.

Remark 38. The above proof shows that Proposition 37(i) is a consequence of Corollary 31(i). On the other hand, Corollary 31(i) can be obtained from Proposition 37(i) by taking fn = f, for all n ∈ ℕ. So, Proposition 37(i) and Corollary 31(i) are conceptually equivalent, in the sense that

()
Thus our main result, Theorem 1 (involving the expression αF({fn}) in HF({fn}) is more general than Proposition 37(i) (involving the expression in (70)).

Acknowledgment

V. V. Motreanu is supported by a Marie Curie Intra-European Fellowship for Career Development within the European Community’s 7th Framework Program (Grant Agreement no. PIEF-GA-2010-274519).

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