Minimax Results with Respect to Different Altitudes in the Situation of Linking

Abstract

Consider a continuous function on a metric space. In the presence of linking between a compact pair and a closed set, depending on the different behaviors of the function on the linking sets, we establish minimax results guaranteeing existence of Palais-Smale sequences or providing gradient estimates. Our approach relies on deformation techniques.

1. Introduction and Statement of Main Results

Minimax theorems play a central role in critical point theory: in this respect, we refer to celebrated minimax results as the mountain pass theorem (see Ambrosetti and Rabinowitz [1]) and the saddle point theorem (see Rabinowitz [2]). A practical way to define minimax values is by means of a linking condition in a topological space adapted to the problem. In the above-cited results, there is a specific linking based on the geometry of the involved problem. In this paper, we provide linking results for continuous functions on metric spaces, which extend the results in [3, 4] in this setting. Beyond this general framework, the main novelty that we emphasize with our approach is that we provide a systematic study related to the involved geometry, being concerned with situations which are not covered by the “classical geometry.” This approach is original in the case of Banach spaces and smooth functions too.

Recall that when (𝑋,‖⋅‖) is a Banach space and 𝑓∶𝑋→ℝ is continuously differentiable, an element 𝑢∈𝑋 is called a critical point of 𝑓 if 𝑓^′(𝑢)=0. In this paper, more generally, we assume that 𝑋 is a metric space endowed with the metric 𝑑 and 𝑓 is a continuous function. In this context, the following notion of critical point has been introduced by Degiovanni and Marzocchi (see [4]).

Definition 1. Let 𝑓∶𝑋→ℝ be a continuous function defined on a metric space 𝑋. An element 𝑢∈𝑋 is called a critical point of 𝑓 if |𝑑𝑓|(𝑢)=0, where by |𝑑𝑓|(𝑢) we denote the supremum of the real numbers 𝜎≥0 such that there exist 𝛿>0 and a continuous map ℋ∶𝐵_𝛿(𝑢)×[0,𝛿]→𝑋 satisfying 𝑑(ℋ())𝑓(ℋ((𝑣)∀()𝑣,𝑡,𝑣≤𝑡,𝑣,𝑡))≤𝑓−𝜎𝑡,𝑣,𝑡∈𝐵_𝛿(𝑢)×[0,𝛿]. (1)

Remark 2. (a) Note that |𝑑𝑓|(𝑢)∈[0,+∞] (take 𝜎=0, any 𝛿>0, and ℋ(𝑣,𝑡)≡𝑣 in (1)) and the map 𝑋→[0,+∞], 𝑢↦|𝑑𝑓|(𝑢) is lower semicontinuous. The quantity |𝑑𝑓|(𝑢) is usually called the weak slope of 𝑓 at 𝑢.
(b) If (𝑋,‖⋅‖) is a real Banach space, 𝑑 is the distance induced by the norm ‖⋅‖, and 𝑓∈𝐶¹(𝑋), then |𝑑𝑓|(𝑢) coincides with the norm of the differential 𝑓^′(𝑢). Thus, Definition 1 reduces to the usual notion of critical point in this context.

If 𝑎∈ℝ, we denote by 𝐾_𝑎 the set of critical points of 𝑓 at level 𝑎; that is, 𝐾_𝑎∶=𝑢∈𝑋∶𝑑𝑓(𝑢)=0,𝑓(𝑢)=𝑎. (2)

The essential tool in our approach to critical point theory is the linking condition presented in the next definition. We say that (𝐷,𝑆) is a compact pair in 𝑋 if 𝑆,𝐷 are compact subsets of 𝑋 with 𝑆⊂𝐷 and 𝐷≠∅.

Definition 3. A compact pair (𝐷,𝑆) in 𝑋 and a closed subset 𝐴 of 𝑋 are linking if the following property holds: 𝑆∩𝐴=∅,𝛾(𝐷)∩𝐴≠∅,∀𝛾∈Γ, (3) where Γ∶={𝛾∈𝐶(𝐷,𝑋)∶𝛾_|𝑆=id_𝑆}.

Given a continuous function 𝑓∶𝑋→ℝ, a compact pair (𝐷,𝑆) in 𝑋, and a nonempty, closed subset 𝐴 of 𝑋, we set 𝑎∶=inf_𝐴𝑓,𝑏∶=sup_𝑆𝑓,𝑐∶=inf_𝛾∈Γsup_𝐷𝑓∘𝛾. (4) Clearly, 𝑐≥𝑏. Moreover, if (3) holds, then we have 𝑐≥𝑎. Note that 𝑎,𝑏,𝑐∈ℝ∪{−∞}. If 𝑆≠∅, then 𝑏,𝑐∈ℝ.

The “classical” minimax principles are usually based on the assumption that 𝑏<𝑎 (i.e., the “altitude” 𝑎−𝑏 is positive; actually, refined results involve the weaker condition 𝑏≤𝑎), establishing in this situation the existence of a critical point for 𝑓. Our approach covers the situation where 𝑏>𝑎 (“negative altitude”). We will distinguish the cases: 𝑐=𝑎 and 𝑐>𝑎. In the situation 𝑐>𝑎, we will also distinguish the cases 𝑐=𝑏>𝑎 and 𝑐>𝑏.

Now we formulate our main results. In all these results, unless otherwise stated, 𝑋 is a metric space, 𝑓∶𝑋→ℝ is a continuous function, (𝐷,𝑆) is a compact pair in 𝑋 and 𝐴 is a closed subset of 𝑋 satisfying (3).

The first statement is a preliminary result providing existence and location of critical points.

Proposition 4. Assume that 𝑎>−∞ (see (4)) and that there exist 𝛾∈Γ and a number 𝜌>0 such that 𝑓𝛾(𝑢)≤𝑎,∀𝑢∈𝐷𝑤𝑖𝑡ℎ𝛾(𝑢)∈𝐵_𝜌(𝐴). (5) Then 𝐾_𝑎∩𝐴≠∅.

Hereafter, 𝐵_𝜌(𝐴)={𝑢∈𝑋∶𝑑(𝑢,𝐴)<𝜌} (the open 𝜌-neighborhood of 𝐴). The proof of this proposition will be done in Section 2.

In the following results, the metric space 𝑋 is supposed to be complete.

Here are our main results in the cases 𝑐=𝑎 and 𝑐>𝑎.

Theorem 5. Assume that 𝑐=𝑎>−∞. Then inf𝑑𝑓(𝑢)∶𝑐≤𝑓(𝑢)<𝑐+𝛿,𝑢∈𝐵_𝛿(𝐴)=0,∀𝛿>0. (6) In particular, there exists a sequence (𝑢_𝑛)⊂𝑋 such that 𝑑𝑓𝑢_𝑛⟶0,𝑓𝑢_𝑛𝑑𝑢↓𝑐,_𝑛,𝐴⟶0𝑎𝑠𝑛⟶∞. (7)

Theorem 6. Assume that 𝑐>𝑎>−∞. Then(a)for all 𝛿>0 and 𝜌≥2𝑑(𝑆,𝐴), one has inf𝑑𝑓(𝑢)∶𝑎≤𝑓(𝑢)<𝑐+𝛿,𝑢∈𝐵_𝜌(𝐴)≤𝑐−𝑎𝑑();𝑆,𝐴 (8)(b)for all 𝛿>0 and 𝜌≥2𝑑(𝑆∩[𝑓>𝑎],𝐴), one has inf𝑑𝑓(𝑢)∶𝑎≤𝑓(𝑢)<𝑐+𝛿,𝑢∈𝐵_𝜌(𝐴≤)𝑐−𝑎𝑑.𝑆∩𝑓>𝑎,𝐴 (9)

Hereafter, [𝑓>𝑎]={𝑢∈𝑋∶𝑓(𝑢)>𝑎}. The notation 𝑑(𝑆,𝐴) stands for inf{𝑑(𝑢,𝑣)∶𝑢∈𝑆,𝑣∈𝐴}.

Remark 7. The estimate in part (b) of Theorem 6 is better than the one in part (a) when 𝜌∈[2𝑑(𝑆∩[𝑓>𝑎],𝐴),+∞). Nevertheless, Theorem 6(a) provides an estimate for all 𝜌∈[2𝑑(𝑆,𝐴),+∞).

The next result distinguishes the situations 𝑐>𝑏 and 𝑐=𝑏 in (4). Since Theorem 5 treats the case 𝑐=𝑎, the next result is meaningful when 𝑐>𝑎.

Theorem 8. (a) Let (𝐷,𝑆) be a compact pair in 𝑋. Assume that 𝑐>𝑏. Then inf𝑑𝑓(𝑢)∶𝑐≤𝑓(𝑢)<𝑐+𝛿=0,∀𝛿>0. (10) In particular, there exists a sequence (𝑢_𝑛)⊂𝑋 such that 𝑑𝑓𝑢_𝑛⟶0,𝑓𝑢_𝑛↓𝑐as𝑛⟶∞. (11)
(b) Let (𝐷,𝑆) be a compact pair in 𝑋 and let 𝐴 be a closed subset of 𝑋 satisfying (3). Assume that 𝑐=𝑏>𝑎>−∞. Then 𝑢)(𝑢)≤inf𝑑𝑓(∶𝑎≤𝑓<𝑐+𝛿𝑏−𝑎𝑑𝑆∩𝑓>𝑎,𝐴,∀𝛿>0. (12)

As a consequence of Theorems 5, 6(b), and 8, we have the next result, which studies the situations 𝑏≤𝑎 and 𝑏>𝑎 in (4).

Corollary 9. (a) Assume that −∞<𝑏≤𝑎. Then (10) holds. If 𝑐=𝑎, then (6) holds.
(b) Assume that 𝑏>𝑎>−∞. Then (12) holds. If 𝑐>𝑏, then (10) holds.

Theorems 5 and 8(a) and Corollary 9(a) lead to the construction of a Palais-Smale sequence at level 𝑐 (i.e., a sequence (𝑢_𝑛)⊂𝑋 such that 𝑓(𝑢_𝑛)→𝑐 and |𝑑𝑓|(𝑢_𝑛)→0 as 𝑛→∞). Moreover, in the case of Theorem 5 (referring to the limiting case 𝑐=𝑎), the Palais-Smale sequence is located near 𝐴 (i.e., 𝑑(𝑢_𝑛,𝐴)→0 as 𝑛→∞). Recall from [3] the following Palais-Smale condition.

Definition 10. We say that a continuous function 𝑓∶𝑋→ℝ satisfies the Palais-Smale condition at level ℓ (condition (PS)_ℓ for short), with ℓ∈ℝ, if every sequence (𝑢_𝑛)⊂𝑋 such that 𝑓(𝑢_𝑛)→ℓ and |𝑑𝑓|(𝑢_𝑛)→0 as 𝑛→∞ has a convergent subsequence in 𝑋.

If we assume condition (PS)_𝑐 (see (4)), the previous results guarantee the existence of critical points.

Corollary 11. If conditions (3) and (PS)_𝑐 hold, then one has the following.(a)In the case 𝑐=𝑎>−∞, then 𝐾_𝑐∩𝐴≠∅.(b)In the case 𝑐>𝑎, if one assumes that 𝑐>𝑏 (for instance, if 𝑏≤𝑎), then 𝐾_𝑐≠∅.

Remark 12. (a) Corollary 11 actually holds under a weaker condition than (PS)_𝑐, namely, (PS)_𝑐+: every sequence (𝑢_𝑛)⊂𝑋 such that 𝑓(𝑢_𝑛)↓𝑐 and |𝑑𝑓|(𝑢_𝑛)→0 as 𝑛→∞ has a convergent subsequence in 𝑋.
(b) Corollary 11(a) holds under a weaker condition than (PS)_𝑐+, namely, (PS)_𝑐+,𝐴: every sequence (𝑢_𝑛)⊂𝑋 such that 𝑓(𝑢_𝑛)↓𝑐, 𝑑(𝑢_𝑛,𝐴)→0, and |𝑑𝑓|(𝑢_𝑛)→0 as 𝑛→∞ has a convergent subsequence in 𝑋. The conclusion in Corollary 11(a) is also more precise since it establishes the location of the critical point in the set 𝐴.

Theorems 6 and 8(b) and Corollary 9(b) go beyond the “classical geometry” 𝑏≤𝑎 in the situation of linking, allowing the case 𝑏>𝑎 and providing an estimate of the infimum of |𝑑𝑓| at the points 𝑢 with 𝑓(𝑢)∈[𝑎,𝑐] in terms of 𝑏−𝑎 and of 𝑑(𝑆,𝐴) or of 𝑑(𝑆∩[𝑓>𝑎],𝐴).

We may still obtain a Palais-Smale sequence from Theorem 8(b) (the case of “nonclassical geometry”) under appropriate hypotheses for a sequence of sets (in the spirit of [5]) which are linking.

Corollary 13. Let 𝑓∶𝑋→ℝ be a continuous function on the metric space 𝑋, and let (𝐴_𝑛), (𝐷_𝑛), and (𝑆_𝑛) be sequences of subsets of 𝑋 such that for all 𝑛≥1, 𝐴_𝑛 is closed, (𝐷_𝑛,𝑆_𝑛) is a compact pair in 𝑋, (𝐷_𝑛,𝑆_𝑛) and 𝐴_𝑛 satisfy (3) with Γ replaced by Γ_𝑛∶={𝛾∈𝐶(𝐷_𝑛,𝑋)∶𝛾_|𝑆𝑛=id_𝑆𝑛}. Assume that 𝑎_𝑛∶=inf_𝐴𝑛𝑓∈ℝ,𝑏_𝑛∶=sup_𝑆𝑛𝑐𝑓∈ℝ,_𝑛∶=inf_{𝛾∈Γ𝑛}sup_𝐷𝑛𝑓∘𝛾∈ℝ,∀𝑛≥1,lim_𝑛→∞max𝑏_𝑛−𝑎_𝑛,0𝑑𝑆_𝑛∩𝑓>𝑎_𝑛,𝐴_𝑛=0. (13) Then, denoting 𝑎∶=liminf_𝑛→∞𝑎_𝑛 and 𝑐∶=limsup_𝑛→∞𝑐_𝑛, there exists a sequence (𝑢_𝑛)⊂𝑋 satisfying 𝑎≤liminf_𝑛→∞𝑓𝑢_𝑛≤limsup_𝑛→∞𝑓𝑢_𝑛≤𝑐,lim_𝑛→∞𝑑𝑓𝑢_𝑛=0. (14) In particular, if 𝑓 satisfies condition (PS)_ℓ for all ℓ∈[𝑎,𝑐], then there exists 𝑢∈𝑋 such that 𝑓(𝑢)∈[𝑎,𝑐] and |𝑑𝑓|(𝑢)=0.

Remark 14 (assume that 𝑏≤𝑎). Considering particular situations of linking as in (3) in a Banach space (𝑋,‖⋅‖), we obtain through Corollary 11 generalizations of classical minimax results in smooth critical point theory.(a)If 𝐷={𝑡𝑒∶𝑡∈[0,1]}, 𝑆={0,𝑒}, and 𝐴={𝑢∈𝑋∶‖𝑢‖=𝜌} (with 𝜌>0 and 𝑒∈𝑋 such that ‖𝑒‖>𝜌), then we get the mountain pass theorem (see [1]).(b)If 𝑋=𝑋₁⊕𝑋₂, with dim𝑋₁<+∞, 𝐷={𝑢∈𝑋₁∶‖𝑢‖≤1}, 𝑆={𝑢∈𝑋₁∶‖𝑢‖=1}, and 𝐴=𝑋₂, then we get the saddle point theorem (see [2]).(c)If 𝑋=𝑋₁⊕𝑋₂, with dim𝑋₁<+∞, 𝐷={𝑢∈𝑋₁∶‖𝑢‖≤𝑟}  +  {𝑡𝑒∶𝑡∈[0,𝑟]},𝑆=𝜕𝐷, and 𝐴={𝑢∈𝑋₂∶‖𝑢‖=𝜌} (with 𝑟>𝜌>0 and 𝑒∈𝑋₂ such that ‖𝑒‖=1), we get the generalized mountain pass theorem (see [2]).
It is worth noting that generalizations of the minimax results cited in (a)–(c), relative to critical point theories for certain classes of nondifferentiable functionals, can already be found in the literature and have numerous applications to partial differential equations, differential inclusions, variational inequalities, hemivariational inequalities, and variational-hemivariational inequalities. In this respect, motivated by the study of existence of solutions of the so-called variational-hemivariational inequalities, introduced by Motreanu-Panagiotopoulos [6], we mention the critical point theory developed by these authors in [6] for functionals 𝑓=𝜑+𝜓∶𝑋→ℝ∪{+∞}, with 𝜑∶𝑋→ℝ locally Lipschitz and 𝜓∶𝑋→ℝ∪{+∞} lower semicontinuous, convex, and not identically +∞ (see also Chang [7] for the case when 𝜓=0, and see Szulkin [8] for the case when 𝜑∶𝑋→ℝ is of class 𝐶¹ and 𝜓∶𝑋→ℝ∪{+∞} is lower semicontinuous, convex, and not identically +∞).

The rest of the paper is organized as follows. Section 2 contains the proof of Proposition 4. Section 3 deals with the proofs of Theorems 5, 6, and 8 and Corollaries 9, 11, and 13. The method of proofs is based on suitable deformation results given in Sections 2, 3.

2. Proof of Proposition 4

We start with stating the following deformation result.

Theorem 15. Let 𝑋 be a metric space endowed with the metric 𝑑, and let 𝑓∶𝑋→ℝ be a continuous function. For all 𝜌>0, there exists a deformation 𝜂∶𝑋×[0,1]→𝑋 (i.e., 𝜂 is continuous and satisfies 𝜂(𝑢,0)=𝑢 for all 𝑢∈𝑋) such that for all 𝑢∈𝑋 and 𝑡∈[0,1], one has(a)𝑡∈(0,1]⇒𝑑(𝜂(𝑢,𝑡),𝑢)<𝜌𝑡; (b)|𝑑𝑓|(𝑢)=0⇒𝜂(𝑢,𝑡)=𝑢; (c)|𝑑𝑓|(𝑢)>0, 𝑡∈(0,1]⇒𝑓(𝜂(𝑢,𝑡))<𝑓(𝑢).

Proof. First, assume that |𝑑𝑓|≢+∞. We consider the continuous function 𝜎∶𝑋→[0,+∞) given by 𝜎(𝑢)1∶=2inf𝑑(𝑢,𝑣)+𝑑𝑓(𝑣)∶𝑣∈𝑋. (15) We have 𝜎(𝑢)=0 for all 𝑢∈𝑋 with |𝑑𝑓|(𝑢)=0. Using that 𝑢↦|𝑑𝑓|(𝑢) is lower semicontinuous (see Remark 2(a)), we see that 𝑑𝑓(𝑢)≠0⟹𝑑𝑓(𝑢)>𝜎(𝑢)>0. (16) Applying [3, Theorem (2.8)], we find a deformation 𝜂∶𝑋×[0,1]→𝑋 and a continuous map 𝜏∶𝑋→[0,+∞) with the properties (for all 𝑢∈𝑋 and 𝑡∈[0,1]): (i) 𝜏(𝑢)=0⇔|𝑑𝑓|(𝑢)=0; (ii) 𝑑(𝜂(𝑢,𝑡),𝑢)≤𝑡; (iii) 0≤𝑡≤𝜏(𝑢)⇒𝑓(𝜂(𝑢,𝑡))≤𝑓(𝑢)−𝜎(𝑢)𝑡. Then the deformation 𝜂∶𝑋×[0,1]→𝑋 defined by 𝜂()𝜂1𝑢,𝑡∶=𝑢,2min𝜏(𝑢),𝜌𝑡,∀𝑢∈𝑋,∀𝑡∈[0,1], (17) has properties (a)–(c) of the statement. If |𝑑𝑓|≡+∞, we make the same reasoning taking an arbitrary positive constant in place of 𝜎.

Proof of Proposition 4. Let 𝜌∶=min{𝑑(𝑆,𝐴),𝜌}>0, and let 𝜂∶𝑋×[0,1]→𝑋 be a deformation satisfying properties (a)–(c) of Theorem 15 (with 𝜌 in place of 𝜌). Note that 𝑆⊂𝑋⧵𝐵_𝜌(𝐴). Let 𝜗∶𝑋→[0,1] be a continuous function such that 𝜗(𝑢)=0⟺𝑢∈𝑋⧵𝐵_𝜌(𝐴). (18) Defining 𝛾∶𝐷→𝑋 by 𝛾(𝑢)∶=𝜂𝛾(𝑢),𝜗𝛾(𝑢),∀𝑢∈𝐷, (19) with 𝛾 as in the statement of Proposition 4, we have 𝛾∈Γ. Fix 𝑢₀∈𝐷 with 𝛾(𝑢₀)∈𝐴 (see (3)). Then, using property (a) in Theorem 15, we infer that 𝑑𝛾𝑢₀,𝐴≤𝑑𝛾𝑢₀,𝛾𝑢₀<𝜌≤𝜌. (20) Hence, using properties (b), (c) of Theorem 15, relation (5), and the assumption that 𝑎∈ℝ, we obtain that 𝑎≤𝑓𝛾𝑢₀=𝑓𝜂𝛾𝑢₀,𝜗𝛾𝑢₀≤𝑓𝛾𝑢₀≤𝑎 (21) and |𝑑𝑓|(𝛾(𝑢₀))=0. This combined with property (b) of Theorem 15 yields 𝛾(𝑢₀)=𝛾(𝑢₀)∈𝐾_𝑎∩𝐴, which completes the proof.

3. Proofs of Theorems 5, 6, and 8 and Corollaries 9, 11, and 13

The following deformation result was shown in [9].

Theorem 16. Let 𝑋 be a metric space endowed with the metric 𝑑, let 𝑓∶𝑋→ℝ be a continuous function, let 𝐶 be a subset of 𝑋, and let ℓ∈ℝ,𝜎>0, and 𝜌>0 be real numbers. Assume that the metric space 𝑓⁻¹([ℓ,𝑠]) (endowed with the induced metric) is complete for all 𝑠∈(ℓ,ℓ+𝜎𝜌) and that 𝑑𝑓(𝑢)>𝜎,∀𝑢∈𝐵_𝜌(𝐶)∩ℓ<𝑓<ℓ+𝜎𝜌. (22) Then there exist a continuous function 𝜏∶𝐶∩[𝑓<ℓ+𝜎𝜌]→[0,𝜌) and a continuous map 𝜂∶(𝐶∩[𝑓<ℓ+𝜎𝜌])  ×  [0,1]  →  [𝑓<ℓ+𝜎𝜌] such that for all 𝑢∈𝐶∩[𝑓<ℓ+𝜎𝜌] and 𝑡∈[0,1], one has(a)𝜏(𝑢)≤(1/𝜎)max{𝑓(𝑢)−ℓ,0}; (b)𝑑(𝜂(𝑢,𝑡),𝑢)≤𝜏(𝑢)𝑡; (c)𝑓(𝜂(𝑢,𝑡))≤𝑓(𝑢)−𝜎𝜏(𝑢)𝑡; (d)𝑓(𝑢)≥ℓ⇒𝑓(𝜂(𝑢,1))=ℓ.

Hereafter, we denote [𝑓<𝑠]={𝑢∈𝑋∶𝑓(𝑢)<𝑠} and [ℓ<𝑓<𝑠]={𝑢∈𝑋∶ℓ<𝑓(𝑢)<𝑠}.

Proof of Theorem 5. For a fixed 𝛿>0, it suffices to prove that inf𝑑𝑓(𝑢)∶𝑢∈𝑐<𝑓<𝑐+𝛿∩𝐵_𝛿(𝐴)>0⟹𝐾_𝑎∩𝐴≠∅. (23) To this end, let 𝜎 be a number such that 𝑑𝑓(𝑢)>𝜎,∀𝑢∈𝑐<𝑓<𝑐+𝛿∩𝐵_𝛿(𝐴). (24) Let 𝜌>0 such that 𝜎𝜌≤𝛿,4𝜌≤min{𝑑(𝑆,𝐴),𝛿}. (25) In particular, these choices combined with (24) guarantee that 𝑑𝑓(𝑢)>𝜎,∀𝑢∈𝐵_𝜌𝐵_3𝜌(𝐴)∩𝑐<𝑓<𝑐+𝜎𝜌. (26) Hereafter, we denote 𝐵_3𝜌(𝐴)={𝑢∈𝑋∶𝑑(𝑢,𝐴)≤3𝜌}. Thus, applying Theorem 16 with the numbers 𝜌 and 𝜎, 𝐶∶=𝐵_3𝜌(𝐴), and ℓ∶=𝑐, we find a deformation 𝜂∶(𝐶∩[𝑓<𝑐+𝜎𝜌])×[0,1]→[𝑓<𝑐+𝜎𝜌] such that 𝑑𝜂()𝜂()[].𝑢,𝑡,𝑢<𝜌,𝑓𝑢,1≤𝑐=𝑎,∀𝑢∈𝐶∩𝑓<𝑐+𝜎𝜌,∀𝑡∈0,1 (27)
Using that 3𝜌≤𝑑(𝑆,𝐴), we see that 𝑆⊂(𝑋⧵𝐵_3𝜌(𝐴))∩[𝑓<𝑐+𝜎𝜌]. Let 𝜗∶[𝑓<𝑐+𝜎𝜌]→[0,1] be a continuous function such that 𝜗(𝑢)=0 if 𝑢∈𝑋⧵𝐵_3𝜌(𝐴) and 𝜗(𝑢)=1 if 𝑢∈𝐵_2𝜌(𝐴). Define the deformation 𝜂∶[𝑓<𝑐+𝜎𝜌]  ×  [0,1]  →  [𝑓<𝑐+𝜎𝜌] by 𝜂()𝜂((𝑢)𝑡)𝑢,𝑡∶=𝑢,𝜗if𝑢∈𝐵_3𝜌(𝐴),𝑢otherwise. (28) To prove the continuity of 𝜂, we note that the restriction of 𝜂 to the sets ([𝑓<𝑐+𝜎𝜌]⧵𝐵_3𝜌(𝐴))×[0,1] and ([𝑓<𝑐+𝜎𝜌]∩𝐵_3𝜌(𝐴))×[0,1], which are closed in [𝑓<𝑐+𝜎𝜌], is continuous. Indeed, the restriction of 𝜂 to ([𝑓<𝑐+𝜎𝜌]⧵𝐵_3𝜌(𝐴))×[0,1] coincides with the projection (𝑢,𝑡)↦𝑢, whereas the restriction of 𝜂 to ([𝑓<𝑐+𝜎𝜌]×𝐵_3𝜌(𝐴))×[0,1] coincides with the map (𝑢,𝑡)↦𝜂(𝑢,𝜗(𝑢)𝑡) (since 𝜗≡0 on 𝜕𝐵_3𝜌(𝐴)∶=𝐵_3𝜌(𝐴)⧵𝐵_3𝜌(𝐴)) which is continuous (by the continuity of 𝜗).
Let 𝛾₀∈Γ with max_𝐷(𝑓∘𝛾₀)<𝑐+𝜎𝜌. Then the map 𝛾∶𝐷→𝑋 defined by 𝛾(𝑢)∶=𝜂𝛾₀(𝑢),1 (29) is well defined and 𝛾∈Γ. Finally, let 𝑢∈𝐷 such that 𝛾(𝑢)∈𝐵_𝜌(𝐴). Since 𝑑𝛾(𝑢),𝛾₀(𝑢)=𝑑𝜂𝛾₀(𝑢),1,𝛾₀(𝑢)<𝜌 (30) (by the first part of (27)), we have that 𝛾₀(𝑢)∈𝐵_2𝜌(𝐴), so 𝜗(𝛾₀(𝑢))=1; thus 𝑓𝛾(𝑢)=𝑓𝜂𝛾₀(𝑢),1=𝑓𝜂𝛾₀(𝑢),1≤𝑎 (31) (by the second part of (27)). Consequently, we can apply Proposition 4. Therefore, 𝐾_𝑎∩𝐴≠∅. This concludes the proof.

Proof of Theorem 6. (a) By (3), we see that 𝑑(𝑆,𝐴)>0. For a fixed 𝛿>0 and 𝜌≥2𝑑(𝑆,𝐴), it suffices to prove that inf𝑑𝑓(𝑢)∶𝑢∈𝑎<𝑓<𝑐+𝛿∩𝐵_𝜌(𝐴>)𝑐−𝑎𝑑(𝑆,𝐴)⟹𝐾_𝑎∩𝐴≠∅. (32) To this end, let 𝜎 be a number such that 𝑑𝑓(𝑢)>𝜎>𝑐−𝑎𝑑(𝑆,𝐴),∀𝑢∈𝑎<𝑓<𝑐+𝛿∩𝐵_𝜌(𝐴). (33) Let 𝜌,𝜀 be positive numbers such that 𝑐−𝑎𝜎<𝜌<𝑐−𝑎+𝛿𝜎,𝜌+2𝜀<𝑑(𝑆,𝐴). (34) Setting 𝜃∶=𝑎+𝜎𝜌−𝑐∈(0,𝛿), since 2(𝜌+𝜀)<𝜌, we have 𝑑𝑓(𝑢)>𝜎,∀𝑢∈𝐵_𝜌𝐵_𝜌+2𝜀(𝐴)∩𝑎<𝑓<𝑐+𝜃. (35) This allows us to apply Theorem 16 with the numbers 𝜌 and 𝜎, 𝐶∶=𝐵_𝜌+2𝜀(𝐴), and ℓ∶=𝑎. Hence we find a deformation 𝜂∶(𝐶∩[𝑓<𝑐+𝜃])  ×  [0,1]  →  [𝑓<𝑐+𝜃] such that 𝑑𝜂()𝜂()[].𝑢,𝑡,𝑢<𝜌,𝑓𝑢,1≤𝑎,∀𝑢∈𝐶∩𝑓<𝑐+𝜃,∀𝑡∈0,1 (36)
The fact that 𝜃>0 in conjunction with the inequality 𝜌+2𝜀<𝑑(𝑆,𝐴) yields 𝑆⊂(𝑋⧵𝐵_𝜌+2𝜀(𝐴))∩[𝑓<𝑐+𝜃]. Let 𝜗∶[𝑓<𝑐+𝜃]→[0,1] be a continuous function such that 𝜗(𝑢)=0 if 𝑢∈𝑋⧵𝐵_𝜌+2𝜀(𝐴) and 𝜗(𝑢)=1 if 𝑢∈𝐵_𝜌+𝜀(𝐴). Define the deformation 𝜂∶[𝑓<𝑐+𝜃]  ×  [0,1]  →[𝑓<𝑐+𝜃] by 𝜂()𝜂((𝑢)𝑡)𝑢,𝑡∶=𝑢,𝜗if𝑢∈𝐵_𝜌+2𝜀(𝐴)𝑢otherwise. (37) The continuity of 𝜂 can be shown as in the proof of Theorem 5. Let 𝛾₀∈Γ with max_𝐷(𝑓∘𝛾₀)<𝑐+𝜃. Then the map 𝛾∶𝐷→𝑋 given by 𝛾(𝑢)∶=𝜂𝛾₀(𝑢),1,∀𝑢∈𝐷, (38) is well defined, and we have 𝛾∈Γ. Let 𝑢∈𝐷 such that 𝛾(𝑢)∈𝐵_𝜀(𝐴). Since 𝑑𝛾(𝑢),𝛾₀(𝑢)=𝑑𝜂𝛾₀(𝑢),1,𝛾₀(𝑢)<𝜌, (39) it follows that 𝛾₀(𝑢)∈𝐵_𝜌+𝜀(𝐴), which yields 𝑓𝛾(𝑢)=𝑓𝜂𝛾₀(𝑢),1=𝑓𝜂𝛾₀(𝑢),1≤𝑎. (40) Consequently, Proposition 4 (with 𝜌 replaced by 𝜀) can be applied, and we conclude that 𝐾_𝑎∩𝐴≠∅.
(b) We see that 𝑑(𝑆∩[𝑓>𝑎],𝐴)≥𝑑(𝑆,𝐴)>0. For a fixed 𝛿>0 and 𝜌≥2𝑑(𝑆∩[𝑓>𝑎],𝐴), it suffices to show that inf𝑑𝑓(𝑢)∶𝑢∈𝑎<𝑓<𝑐+𝛿∩𝐵_𝜌(𝐴>)𝑐−𝑎𝑑𝑆∩𝑓>𝑎,𝐴⟹𝐾_𝑎∩𝐴≠∅. (41) To this end, let 𝜎 be a number such that 𝑑𝑓(𝑢)>𝜎>𝑐−𝑎𝑑,𝑆∩𝑓>𝑎,𝐴∀𝑢∈𝑎<𝑓<𝑐+𝛿∩𝐵_𝜌(𝐴). (42) Let 𝜌,𝜀 be positive numbers such that 𝑐−𝑎𝜎<𝜌<𝑐−𝑎+𝛿𝜎,𝜌+2𝜀<𝑑𝑆∩𝑓>𝑎,𝐴. (43) Setting 𝜃∶=𝑎+𝜎𝜌−𝑐∈(0,𝛿), we have 𝑑𝑓(𝑢)>𝜎,∀𝑢∈𝐵_𝜌𝐵_𝜌+2𝜀(𝐴)∩𝑎<𝑓<𝑐+𝜃. (44) Thus, we can apply Theorem 16 with the numbers 𝜌 and 𝜎, 𝐶∶=𝐵_𝜌+2𝜀(𝐴), and ℓ∶=𝑎. So we find a deformation 𝜂∶(𝐶∩[𝑓<𝑐+𝜃])×[0,1]→[𝑓<𝑐+𝜃] such that 𝜂()()∈×[](𝑢)𝑑𝜂()𝜂()[].𝑢,𝑡=𝑢,∀𝑢,𝑡𝐶∩𝑓<𝑐+𝜃0,1with𝑓≤𝑎,𝑢,𝑡,𝑢<𝜌,𝑓𝑢,1≤𝑎,∀𝑢∈𝐶∩𝑓<𝑐+𝜃,∀𝑡∈0,1 (45)(46)
Let 𝜗∶[𝑓<𝑐+𝜃]→[0,1] be a continuous function such that 𝜗(𝑢)=0 if 𝑢∈𝑋⧵𝐵_𝜌+2𝜀(𝐴) and 𝜗(𝑢)=1 if 𝑢∈𝐵_𝜌+𝜀(𝐴). Define the deformation 𝜂∶[𝑓<𝑐+𝜃]  ×  [0,1]  →  [𝑓<𝑐+𝜃] by 𝜂()𝜂((𝑢)𝑡)𝑢,𝑡∶=𝑢,𝜗if𝑢∈𝐵_𝜌+2𝜀(𝐴),𝑢otherwise. (47) The continuity of 𝜂 can be shown as in the proof of Theorem 5.
Let 𝛾₀∈Γ with max_𝐷(𝑓∘𝛾₀)<𝑐+𝜃. Define 𝛾∶𝐷→𝑋 by 𝛾(𝑢)𝜂𝛾∶=⁰(𝑢),1if𝑓𝛾₀(𝑢)𝛾>𝑎,₀(𝑢)otherwise. (48) We show that 𝛾∈Γ. To prove that 𝛾 is continuous, it suffices to note that the restrictions of 𝛾 to the closed sets {𝑢∈𝐷∶𝑓(𝛾₀(𝑢))≥𝑎} and {𝑢∈𝐷∶𝑓(𝛾₀(𝑢))≤𝑎} are continuous. This is immediate for the latter closed set. Concerning the former, this follows from the fact that the equality 𝛾(𝑢)=𝜂(𝛾₀(𝑢),1) holds whenever 𝑓(𝛾₀(𝑢))=𝑎 (which is a consequence of the definition of 𝜂 and relation (45)). Whence 𝛾 is continuous. Using that 𝑆∩[𝑓>𝑎]⊂𝑋⧵𝐵_𝜌+2𝜀(𝐴), we infer that 𝛾∈Γ.
Let an arbitrary point 𝑢∈𝐷 with 𝑑(𝛾(𝑢),𝐴)<𝜀. If 𝑓(𝛾₀(𝑢))>𝑎, then 𝑑𝛾(𝑢),𝛾₀(𝑢)=𝑑𝜂𝛾₀(𝑢),1,𝛾₀(𝑢)<𝜌 (49) (by the first part of (46)), so 𝛾₀(𝑢)∈𝐵_𝜌+𝜀(𝐴), and thus 𝑓(𝛾(𝑢))=𝑓(𝜂(𝛾₀(𝑢),1))=𝑓(𝜂(𝛾₀(𝑢),1))≤𝑎 (by the second part of (46)). If 𝑓(𝛾₀(𝑢))≤𝑎, then it is clear that 𝑓(𝛾(𝑢))≤𝑎. Consequently, in both cases, we have 𝑓(𝛾(𝑢))≤𝑎. This allows us to apply Proposition 4 (with 𝜌 replaced by 𝜀) and thus to obtain that 𝐾_𝑎∩𝐴≠∅, which completes the proof.

Proof of Theorem 8. (a) The hypotheses allow us to apply Theorem 5 with 𝐴∶={𝑢∈𝑋∶𝑓(𝑢)≥𝑐}. Indeed, since 𝑐>𝑏, we have that 𝑆∩𝐴=∅. Moreover, since 𝐷 is compact, for each 𝛾∈Γ, there exists 𝑢_𝛾∈𝐷 with max_𝐷(𝑓∘𝛾)=𝑓(𝛾(𝑢_𝛾))≥𝑐. It follows that condition (3) is satisfied and 𝑐=inf_𝛾∈Γ𝑓𝛾𝑢_𝛾≥inf_𝐴𝑓≥𝑐, (50) so inf_𝐴𝑓=𝑐>−∞. The conclusion follows now from Theorem 5.
(b) This follows from Theorem 6(b).

Proof of Corollary 9. (a) If 𝑐>𝑎, then we have 𝑐>𝑏. Thus, we can apply Theorem 8(a). In the case 𝑐=𝑎, we apply Theorem 5.
(b) If 𝑐>𝑏, then we obtain from Theorem 8(a) that (10) holds (so that a fortiori (12) holds since 𝑏−𝑎≥0). If 𝑐=𝑏, then we apply Theorem 8(b).

Proof of Corollary 11. Parts (a) and (b) follow from Theorems 5 and 8(a), respectively, taking into account the lower semicontinuity of the map 𝑢↦|𝑑𝑓|(𝑢) (see Remark 2(a)).

Proof of Corollary 13. Fix an arbitrary 𝑛≥1. If 𝑏_𝑛≤𝑎_𝑛, then applying Corollary 9(a), we have inf𝑑𝑓(𝑢)∶𝑐_𝑛≤𝑓(𝑢)<𝑐_𝑛+1𝑛=0. (51) If 𝑏_𝑛>𝑎_𝑛, then using Corollary 9(b), we have inf𝑑𝑓(𝑢)∶𝑎_𝑛≤𝑓(𝑢)<𝑐_𝑛+1𝑛≤𝑏_𝑛−𝑎_𝑛𝑑𝑆_𝑛∩𝑓>𝑎_𝑛,𝐴_𝑛. (52) Let 𝜎_𝑛𝑏∶=max_𝑛−𝑎_𝑛,0𝑑𝑆_𝑛∩𝑓>𝑎_𝑛,𝐴_𝑛. (53) Then we have 𝜎_𝑛→0 as 𝑛→∞ and inf𝑑𝑓(𝑢)∶𝑎_𝑛≤𝑓(𝑢)<𝑐_𝑛+1𝑛≤𝜎_𝑛,∀𝑛≥1. (54) This completes the proof.

Acknowledgments

The author is grateful to Jean-Noël Corvellec for his suggestion to study this problem and his valuable advices. This work was funded 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).