HOMOLOGICAL LOCAL LINKING

We generalize the notion of local linking to include certain cases where the functional does not have a local splitting near the origin. Applications to second-order Hamiltonian systems are given.


Introduction
The notion of local linking introduced by Li and Liu [4] plays a useful role in a wide variety of problems in the Calculus of Variations.Let F be a real C 1 function defined on a Banach space X.We say that F has a local linking near the origin if X has a direct sum decomposition X = X 1 ⊕ X 2 with j = dim X 1 < ∞, F (0) = 0, and, for some r > 0, Then it is clear that 0 is a critical point of F .
In this paper we give the following more general definition of "homological" local linking: Definition 1.1.Assume that 0 is an isolated critical point of F with F (0) = 0 and let q, β be positive integers.We say that F has a local (q, β)-linking near the origin if there exist a neighborhood U of 0 and subsets A, S, B of U with A ∩ S = ∅, 0 ∈ A, A ⊂ B such that 1. 0 is the only critical point of F in F 0 ∩ U where F 0 is the sublevel set {u ∈ X : F (u) ≤ 0}, 2. denoting by i 1 * : H q−1 (A) −→ H q−1 (U \ S) and i 2 * : H q−1 (A) −→ H q−1 (B) the embeddings of the singular homology groups induced by inclusions, rank i 1 * − rank i 2 * ≥ β, 3. F ≤ 0 on B, 4. F > 0 on S \{0}.
If F satisfies the condition (1) with j ≥ 1 and 0 is an isolated critical point of F , taking U to be a sufficiently small closed ball B ρ centered at the origin, A = ∂B ρ ∩ X 1 , S = B ρ ∩ X 2 , and B = B ρ ∩ X 1 , we see that F has a local (j, 1)-linking near the origin.The following example in R shows that our definition is, in fact, weaker than (1): Note that the critical groups of F at 0 are given by Chang [2] or Mawhin and Willem [8]).It was proved in Liu [7] that if F satisfies (1) and 0 is an isolated critical point of F , then C j (F, 0) = 0.This fact was used in Perera [9] to obtain a nontrivial critical point u with either C j+1 (F, u) = 0 or C j−1 (F, u) = 0, under an additional assumption on the behavior of F at infinity.Here we extend these results to the case where F satisfies the weaker conditions given in Definition 1.1 near the origin.As an application we prove the existence of nontrivial time-periodic solutions of a system of ordinary differential equations, under different hypotheses on the behavior of the nonlinearity at infinity.
For the existence of nontrivial critical points under the usual definition of local linking and various assumptions at infinity see Li and Liu [4], Li and Liu [5], Liu [7], Silva [10], Brézis and Nirenberg [1], Li and Willem [6], and their references.

An Example
As an example of homological local linking, we prove the following proposition: Proposition 2.1.Assume that 0 is an isolated critical point of F and where P is homogeneous of degree s > 1, i.e., Assume also that there are disjoint subsets Ã, S of S ∞ , the unit sphere in X, such that 1. the rank of the embedding Then F has a local (q, β)-linking near the origin.
Proof.Fix > 0 so that sup Ã P + < 0 < inf S P − and take ρ > 0 sufficiently small such that 0 is the only critical point of F in U = B ρ = {u ∈ X : u ≤ ρ} and Then take On B, and on S \{0}, Now we verify the condition 2 of Definition 1.1.Set S ρ = ∂B ρ and Sρ = ρ u : u ∈ S .By the assumption 1, the rank of the embedding On the other hand, B is contractible to 0 via

Critical Groups of the Origin and Nontrivial Critical Points
The following theorem extends Theorem 2.1 of Liu [7]: Proof.Consider the following portion of the exact sequence of the pair (F 0 ∩ U, F 0 ∩ U \{0}): Consider the following commutative diagram induced by inclusions: Now we assume that F satisfies the Palais-Smale compactness condition (PS) and has only isolated critical values, with each critical value corresponding to a finite number of critical points, and set The main result of this section is the following: Theorem 3.2.Suppose that F has a local (q, β)-linking near the origin and assume that there are regular values a, b of F such that a < 0 < b and In particular, F has a (nontrivial) critical point u with either a < F (u) < 0 and C q−1 (F, u) = 0, or The proof of Lemma 3.3 makes use of the following topological lemma: Proof.Consider the following portions of the exact sequences of the triples (A, B, B ) and (A , A, B ), respectively: But, by Chapter I, Theorem 4.3, Corollary 4.1, and Theorem 4.2 of Chang [2], The following corollary generalizes Theorem 2.2 of Liu [7] and Theorem 5 of Brézis and Nirenberg [1].See Remark 2.3 of Liu [7] and the remarks following Theorem 4 and proof of Theorem 5 of Brézis and Nirenberg [1] for the history of this result: Corollary 3.5 (Three Critical Point Theorem).Suppose that F has a local (q, β)-linking near the origin and assume that F is bounded below.If q = 1 and β ≥ 2, or q ≥ 2, then F has at least two nontrivial critical points.If q ≥ 2, then F has a nontrivial critical point which is not a local minimizer.
Proof.F achieves its minimum at some point u 0 with F (u 0 ) ≤ 0 and rank C j (F, u 0 ) = δ j0 (for the critical groups of an isolated local minimum point see Example 1 in Chapter I, Section 4 of Chang [2]).By Theorem 3.1, C q (F, 0) = 0 and hence u 0 = 0. Supposing 0 and u 0 to be the only critical points and taking a < inf X F and b = +∞ in Theorem 3.2, we have If q ≥ 2, the critical point u = 0 with either obtained in Theorem 3.2 is not a local minimizer.

Second-order Hamiltonian systems
Consider the second-order nonautonomous system ẍ = ∇ V (t, x) (2) where V ∈ C 1 (R × R , R) is 2π-periodic in t and satisfies (V 1 ): there are constants µ > 2 and R > 0 such that , and the following condition on P : (P ): there are disjoint subsets Ã, S of S n−1 such that 1. for some positive integers q ≤ n and β, the rank of the embedding Then ( 2) has at least one nonzero 2π-periodic solution.
Remark 4.2.Our assumption (P ) generalizes the condition (P 4 ) of Felmer and Silva [3].See Theorem 7 and the remark following it in Li and Willem [6] for related results.
Proof of Theorem 4.1.We seek solutions of (2) as critical points of the functional defined on the Hilbert space X of vector functions x(t) having period 2π and belonging to H 1 on [0, 2π], with the norm It is well-known that F satisfies (PS).As in the proof of Lemma 3.2 of Wang [11], (V 1 ) also implies that for a < 0 and |a| sufficiently large.The conclusion follows from Theorem 3.2 and the Lemma 4.3 below.
Lemma 4.3.If V satisfies (V 2 ) with P as in Theorem 4.1, then F has a local (q, β)-linking near the origin.
Proof.We have the splitting X = X 1 ⊕ X 2 where X 1 is the space of constant functions, identified with R , and X 2 is the space of functions in X whose integral is zero.We take As in the proof of Proposition 2.1, F ≤ 0 on B for ρ > 0 sufficiently small.On S \{0}, By the mean value theorem and the Young's inequality, where p, p are conjugate exponents with 2 < p < s.It follows that on S for small ρ.
As in the proof of Proposition 2.1, the rank of the embedding is a strong deformation retraction of U \S onto U 1 \S 1 , and hence the embedding H q−1 (U 1 \S 1 ) −→ H q−1 (U \S) is an isomorphism.
Remark 4.4.Note that Theorem 3.2 gives a critical point x with either F (x) < 0 and C q−1 (F, x) = 0, or F (x) > 0 and C q+1 (F, x) = 0, yielding Morse index estimates for x via the Shifting theorem when V , and hence F , is C 2 (see Chapter I, Theorem 5.4 of Chang [2]): either where m(x) and m * (x) = m(x) + dim ker d 2 F (x) denote the Morse index and the large Morse index of x, respectively.This additional information can sometimes be used to distinguish x from the constant solutions, when they exist.Now we replace (V 1 ) by the condition (V 1 ) : V (t, x) → +∞ as |x| → ∞ uniformly in t, which implies that F satisfies (PS) and is bounded below.Then Corollary 3.5 yields Theorem 4.5.Assume (V 1 ) , (V 2 ), and (P ).If q = 1 and β ≥ 2, or q ≥ 2, then (2) has at least two nonzero 2π-periodic solutions.Remark 4.6.See Theorems 7 and 7' in Brézis and Nirenberg [1] for related results.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Lemma 3 . 3 .
Theorem 3.2 follows from Lemma 3.3 below and Theorem 3.1.If a < b are regular values of F , c ∈ (a, b), and q

Proof of Lemma 3 . 3 .
Take > 0 such that a < c − < c + < b and c is the only critical value of F in [c − , c + ].Applying Lemma 3.4 to