Homotopy invariants methods in the global dynamics of strongly damped wave equation

We are interested in the following differential equation $\ddot u(t) = -A u(t) - c A \dot u(t) + \lambda u(t) + F(u(t))$ where $c>0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that $\lambda$ is an eigenvalue of $A$ and $F$ is a bounded map. We provide geometrical conditions for the nonlinearity $F$ and determine the Conley index of the set $K_\infty$, that is the union of the bounded orbits of this equation.


Introduction
We are concerned with the strongly damped hyperbolic equations of the form u(t) = −Au(t) − cAu(t) + λu(t) + F (u(t)), t > 0 (1.1) where c > 0 is a damping factor, λ is a real number, A : X ⊃ D(A) → X is a sectorial operator on a Banach space X and F : X α → X is a continuous map, where X α for α ∈ (0, 1), is a fractional power space associated with A. This equation is an abstract formulation of many partial differential equations including nonlinear heat equation u tt (x, t) = ∆u(x, t) + c∆u t (x, t) + λu(x, t) + f (x, u(x, t)), t ≥ 0, x ∈ Ω (1. 2) where Ω is an open subset of R n (n ≥ 1), ∆ is a Laplace operator with the Dirichlet boundary conditions and f : Ω × R → R is a continuous map. To see this, it is enough to take Au := −∆u and F (u) = f ( · , u(·)). The dynamics of strongly damped wave equation has been investigated by many authors in the last years (see e.g. [5], [6], [7], [1], [2], [3], [4], [8], [9], [10], [11], [12]). All of these papers concerns the existence of global attractor: the maximal compact invariant set K with the property that any the limit set is arbitrary close to K. The existence of global attractor is proved under dissipation assumption which roughly speaking says that there is a bounded set into which every orbit eventually enters and remains.
In this paper we intend to study the existence of compact invariant sets for the equation (1.1) being at resonance at infinity, that is, Ker (λI − A) = {0} and F is a bounded map. The main difficulty lies in the fact that, in the presence of resonance, the dissipation condition is not satisfied and the problem of existence of compact invariant sets may not have solution for general nonlinearity F . This fact has been explained in detail in Remark 4.1. To handle with this problem we impose geometrical assumptions (G1) and (G2) on the nonlinearity F (see page 12), that allow to obtain, the main result of this paper, Conley index formulas stating that the Conley index the associated semiflow with respect to large ball is equal to the suspension of the sphere with dimension depending on what of the two geometrical assumptions is satisfied. The obtained abstract results are applied to derive criteria on existence of compact invariant sets for strongly damped wave equation (1.2).
To explain our methods more precisely observe that the second order equation (1.1) can be written as the first order equatioṅ w(t) = −Aw(t) + F(w(t)), t > 0, (1.3) where A : E ⊃ D(A) → E is a linear operator on the space E := X α × X given by In this paper we deal with an approach (see [35]) for finding compact invariant sets for (1.1) that relies on seeking of Conley index of the semiflow Φ associated with the equation (1.3). The Conley index, that is the main topological tool that we use in this paper, was initially constructed for semiflows acting on finite dimensional spaces, see [19], [38], [40]. However in the study of partial differential equations the semiflow acts usually on the infinite dimensional functional spaces which is no longer locally compact. In [35] and [36] the index theory were extended on arbitrary metric space, which gave a rise to study the dynamics of partial differential equations. The paper is organized as follows. In Section 2, we provides some spectral properties of the hyperbolic operator A. We prove that the elements of spectrum of A with negative real part are actually eigenvalues of this operator. Subsequently we describe spectral decomposition of the operator A in the case when λ is a eigenvalue of the operator A. Here the main difficulties are caused by the fact that the operator A does not have compact resolvents, despite the fact that A has compact resolvents as we assumed. Crucial point for our considerations is to see the relationship between spectral decomposition of operators A and A (see Theorem 2.6 (iii)). Section 3 is devoted to the mild solutions for (1.1). First we provide the standard facts concerning the existence and uniqueness for this equation. Furthermore, we focus on continuity and compactness properties for the semiflow Φ.
In Section 4 we provide geometrical assumptions for the nonlinearity F and use them to prove the Conley index formula that is the main result of this paper.
Finally, in section 6 we provide applications of the obtained abstract results partial differential equations. First of all, in Theorems 5.3 and 5.5, we prove that if F is a Niemytzki operator associated with a map f , then the well known Landesman-Lazer (see [30]) and strong resonance conditions (see [16]) are actually particular case of introduced geometrical assumption. Then, we provide the criteria on the existence of T -periodic solution for the strongly damped wave equation in terms of Landesman-Lazer and strong resonance type conditions.

Spectral properties of hyperbolic operator
Let A : X ⊃ D(A) → X be a positive sectorial operator on a real Banach space X with norm · such that following assumptions are satisfied: (A1) the operator A has compact resolvents, (A2) there is a Hilbert space H endowed with a scalar product · , · H and norm · H and a continuous injective map i : X ֒→ H, where the graph inclusion is understood in the sense of product map X × X i×i − − → H × H. As we are working on a real space X, by the spectrum σ(A) of the operator A we mean the sense of its complexification. To be more precise we put X C := X × X and, denoting x + iy := (x, y) for (x, y) ∈ X C , we define the operations of addition and multiplication by complex scalar for (x, y), (x 1 , y 1 ), (x 2 , y 2 ) ∈ X C and λ = (λ 1 + λ 2 i) ∈ C. The function By the spectrum of the operator A we mean the spectrum of its complexification: σ(A) := σ(A C ). Similarly by the eigenvalue of A we mean eigenvalue of A C .
Remark 2.1. The spectrum σ(A) consists of the sequence (possibly finite) of real eigenvalues. Indeed, the operator A has compact resolvents which implies that and this set is finite or |λ i | → +∞ when n → +∞. Furthermore, if λ ∈ C is a complex eigenvalue of A, then, by (A3), it is also a complex eigenvalue of the symmetric operator A and hence λ is a real number. It follows that the spectrum of A can be exhibited as the increasing sequence of eigenvalues Let X α be a fractional space associated with A a let A : E ⊃ D(A) → E be a linear operator on the space E := X α × X given by where c > 0, and λ is a real number. We assume that the space E is equipped with the norm (x, y) E := x α + y for (x, y) ∈ E.
Theorem 2.2. The following assertions hold.
(i) The set σ(A) \ {1/c} consists of the eigenvalues of the operator A.
It can be easily checked that A C is adjoint with the operator B : for (x, y) ∈ D(B).
Then, it is not difficult to see that U A C = BU . Hence, without loos of generality one can tak A C := B. In the proof we will use the following lemma.
We return to the proof of Theorem 2.2. For the point (i), take µ ∈ σ(A) \ {1/c} such that Ker (µI − A C ) = {0}. We show that for every (f, g) ∈ X α C × X C there is (x, y) ∈ D(A C ) such that µ(x, y) − A C (x, y) = (f, g). Assume for the moment that this is true. Then there is inverse operator (µI − A C ) −1 . Since A C is closed, the inverse (µI − A C ) −1 is bounded on X α × X. Therefore µ ∈ ̺(A), which contradicts the assumption and proves that Ker (µI − A C ) = {0}. Take (f, g) ∈ X α C × X C and consider the following equations Multiplying the former equation by cλ − µ and later by 1 − µc we have which after adding gives where h = (cλ − µ)/(1 − cµ)f + g, and hence Note that (λ − µ 2 )/(1 − cµ) is an element from the resolvent set of the operator which contradicts the assumption because (w, −µw) = 0. Therefore which allows us to define , which implies that (x, y) ∈ D(A C ). Therefore, it is enough to check that the equations (2.5) are satisfied. To this end observe that Furthermore, we have the sequence of equivalent equalities The last equality is true and the proof of the point (i) is completed.
(iii) The spaces X 0 , X − are X + mutually orthogonal, that is, i(u l ), i(u m ) H = 0 for u l ∈ X l and u m ∈ X m where l, m ∈ {0, −, +}, l = m.
Remark 2.5. Let P, Q ± : X → X be projections given for any x ∈ X by Therefore the projections P and Q ± can be also considered as continuous maps P, Q ± : X α → X α given for any x ∈ X α by (2.6).
We proceed to the spectral decomposition of the operator A. Theorem 2.6. Let λ = λ k for some k ≥ 1 and let E 0 := Ker (λI−A)×Ker (λI−A). Then there are closed subspaces E + , E − of E such that E := E − ⊕ E 0 ⊕ E + and the following assertions hold.
respectively and Q := Q − + Q − , then P(x, y) = (P x, P y) and Q(x, y) = (Qx, Qy) for (x, y) ∈ E. (2.7) In the proof we use the following lemmata . It follows that x + cy ∈ D(A) and where the last equality follows from the fact that µ ± i are the roots of the equation and the proof is completed.
We will also need the following lemmata Proof. It is enough to prove that σ(B) ⊂ {ν i | 1 ≤ i ≤ l}. The opposite inclusion is obvious. Let ν ∈ C be such that νz = B C z for some z := x + iy ∈ V C , z = 0.
Proof of Theorem 2.6. If k = 1 then we define If k ≥ 2 we put M 1 := K + 1 ⊕ . . . ⊕ K + k−1 , M 2 := (X α ∩ X + ) × X + and define it is not difficult to check that the assertion (iii) holds. Observe i is contained in eigenspace of the operator A. Hence, for the proof of (i), it is enough to verify the inclusion A(D(A) ∩ E + ) ⊂ E + . Similarly as before Lemma 2.7 says that, for any 1 ≤ i ≤ k − 1, the elements of K + i are contained in the eigenspace of A and therefore To this end take (x, y) ∈ D(A) ∩ M 2 . Then x + cy ∈ D(A) and x ∈ X α , so we have y ∈ X α which yields y ∈ X α ∩ X + . Observe that A(x + cy) − λx ∈ X + , as a consequence of the fact that x + cy ∈ D(A) ∩ X + and A(D(A) ∩ X + ) ⊂ X + . Therefore A(x, y) ∈ M 2 as desired and proof of (i) is completed. To see Suppose to the contrary that µ ∈ σ(A + ) and Re µ ≤ 0. Let A 1 + and A 2 + be the parts of the operator A + in M 1 and M 2 , respectively. By inclusion A(M 1 ) ⊂ M 1 and Lemma 2.8, it follows that the spectrum of µI − A 1 + consists of its eigenvalues Since Re µ ≤ 0 and µ + i > 0 for 1 ≤ i ≤ k we deduce that the complexification of the operator µI − A 1 + is bijection. This implies that µ ∈ σ(A 2 + ). Observe that the operator A 2 + can be given by the formula for (x, y) ∈ D(A 2 + ). Now we can apply Theorem 2.2 with operator A := A + and derive that that the set σ(A 2 + ) \ {1/c} consists of the eigenvalues of the operator A 2 Lemma 2.8. This completes the proof of (ii).
It is known that the operator A is sectorial (see e.g. [31], [21]) and hence −A generates an equicontinuous The following corollary is a simple consequence of Theorem 2.6 and [25, Theorem Corollary 2.9. Let λ = λ k for some k ≥ 1 and let E = E − ⊕ E 0 ⊕ E + be a direct sum decomposition obtained in Theorem 2.6. Then can be uniquely extended to a C 0 group on E − and there are constants M, c > 0 such that

Solutions and compactness properties of hyperbolic equations
Assume that A : X ⊃ D(A) → X is a positively defined sectorial operator with compact resolvents on a separable Banach space X. Consider the second order differential equation where λ is a real number, c > 0, s ∈ [0, 1] is a parameter and F : [0, 1] × X α → X is a continuous map satisfying the following assumptions The second order equation (3.1) may br written in the following forṁ Remark 3.2. (a) Since X is separable, it is known that X α is also separable and hence E is separable as well.
(b) Consider the direct sum decomposition E := E − ⊕E 0 ⊕E + obtained in Theorem 2.6 and let P : E → E, Q − : E → E and Q + : E → E be projections on spaces E 0 , E − and E + , respectively. Since the components are closed subspaces, the projections are continuous. Furthermore, Corollary 2.9 gives where L > 0 is a constant. Writing W := U × X for the neighborhood of (x, y), we infer that, for any ( and we see that F satisfies assumption (F 1). Since F satisfies assumption (F 2), which shows that F satisfies this assumption as well.
(d) Observe that assumption (F 3) implies that the map F is completely continuous, that is, for any bounded By assumption (F 3), the set {0} × F (S × Ω 1 ) is relatively compact in E, which proves that F is completely continuous.
The following theorem is quite standard and its proof is a consequence of[25,  Our aim is to prove the following theorem concerning the continuity of mild solution with respect to parameter and initial data.
for t ≥ 0, and this convergence in uniform for t from bounded subsets of [0, +∞).
In the proof we will use the following theorem We will also use the following lemma which is a simple consequence of Remark 3.2 (c) and the Gronwall inequality. Lemma 3.6. Let (s n ) be a sequence in [0, 1] and let (x n , y n ) be a bounded sequence in E. Then the set {w(t; s n , (x n , y n )) | t ∈ [0, t 0 ], n ≥ 1} is bounded in E.
Proof of Theorem 3.4. Step 1. Write w n := w( · ; s n , (x n , y n )) for n ≥ 1 and let β be a Hausdorff measure of noncompactness on the space E. We show that the sequence (w n ) is relatively compact in C([0, T ], E).
From the integral formula, we see that Hence, for any t ≥ 0, we have where From Lemma 3.6 it follows that the set is bounded in E. Hence, (see e.g. [22], [26]), we find that the map ϕ : By Remark 3.2 (d), the map F is completely continuous. Hence in view of the fact that the set in (3.7) is bounded we infer that the set {F(s n , w n (τ )) | n ≥ 1} is relatively compact in E which implies that ϕ(τ ) = 0 for τ ∈ [0, t]. Hence, by (3.6) and (3.8), we obtain where the last inequality follows from that fact that (x n , y n ) n≥1 is convergent. Hence, Theorem 3.5 says that, for any T > 0, the sequence (w n ) is relatively compact in C([0, T ], E).
Step 2. Take an arbitrary subsequence (w n k ) k≥1 of (w n ) n≥1 . By Step 1, we deduce that it contains a convergent subsequence (w n k l ).
Theorem 3.7. Let N ⊂ E be a bounded subset and let sequences (s n ) in [0, 1], (t n ) in [0, +∞) and (z n ) in E are such that t n → +∞ as n → +∞ and w([0, t n ]×{s n }× {z n }) ⊂ N for n ≥ 1. Then the set {w(t n ; s n , z n ) | n ≥ 1 is relatively compact.
Proof. For any n ≥ 1 write w n := w( · ; s n , z n ) and R := sup z∈N z E . By Corollary 2.9 there are constants δ, M > 0 such that In view of the fact that w n (t n ) = Φ sn (t 0 , w n (t n − t 0 )) for n ≥ n 0 , it follows that (3.10) By Remark 3.2 (c) the map F satisfies assumptions (F 1) and (F 2). Since the set is contained in N , it is bounded and hence is also bounded. Similarly as before the function ϕ : [0, t] → R given by Therefore, in view of boundedness of (3.11) and the fact that F is completely Put P − := P + Q − . By the standard properties of the measure β, we have for t ≥ 0, where the last inequality follows from the fact that the set S A (t)P − N is relatively compact as a bounded subset of finite dimensional space E 0 ⊕ E − . Let β E+ be a Hausdorff measure of noncompactness on the space E + . By (3.9) for t ≥ 0. Combining this with (3.10) and (3.12), we find that which completes the proof.

Conley index formula for invariant sets
We are interested in the study of the existence of bounded orbits for the differential equations of the form where c > 0, λ is an eigenvalue of the operator A : X ⊃ D(A) → X defined on a separable Banach space X and F : X α → X is a continuous map. Assume that (A1), (A2), (A3), (F 1), (F 3) are satisfied and the following condition holds The second order equation (4.1) can be written in the following forṁ where A : E ⊃ D(A) → E is an operator given by (2.2) and F : E → E is defined by F(x, y) := (0, F (x)) for (x, y) ∈ E = X α × X. From Remark 3.2 (c) and Theorem 3.3 it follows that, for any (x, y) ∈ E, there is a mild solution w( · ; (x, y)) : [0, +∞) → E of the equation (4.1) starting at (x, y). Let Φ : [0, +∞) × E → E be a semiflow associated with this equation given by , (x, y)) := w(t; (x, y)) for t ∈ [0, +∞), (x, y) ∈ E.
Then, Theorems 3.4 and 3.7 imply that the semiflow Φ is a continuous map and any bounded subset of E is admissible with respect to Φ.
If w : R → E is a mild solution of (4.2), then Consider the direct sum decomposition E := E − ⊕ E 0 ⊕ E + obtained in Theorem 2.6 along with projections P, Q + , Q − on its components. Acting on the equation by the operator P and using (3.3), (3.4) and (2.7) we infer that which together with (3.4) and (2.7) implies that Since A 0 is bounded, we infer that the map (u 0 , v 0 ) : R → E 0 given by From the above equation it follows that d dt for t ∈ R. Therefore, if w is bounded, then the left side of (4.3) is also bounded, a contradiction with y 0 H > 0.
Recalling that X α + and X α − are subspaces from Remark 2.5, we overcome these difficulties by introducing the following geometric conditions for F which will guarantee the existence of T -periodic solutions for the equation (4.2): Now we proceed to the main result of this section, namely the index formula for bounded orbits. It is a tool to determining the Conley index for the maximal invariant set contained in appropriately large ball in terms of geometrical conditions (G1) and (G2). This theorem will be used to prove the existence of bounded orbits for the equation (4.2).
Let G : [0, 1] × X α → X be a map given by the formula In the proof of above theorem we will use the following differential equationṡ Remark 4.3. (a) It is not difficult to see that G satisfies assumption (F 1) and (F 2). In view of Remark 3.2 c, this implies that G also satisfies assumptions (F 1) and (F 2). The later means that there is m 0 > 0 such that Since F is completely continuous, it is not difficult to see that G is also completely continuous. Hence, from Remark 3.2 (d) we deduce that G satisfies assumption (F 3), that is, the set G([0, 1] × [0, +∞) × Ω) is a relatively compact in E, for any bounded Ω ⊂ E.

Lemma 4.4.
There is a constant R > 0 such that the following assertions hold.
(i) If w = w s : R → E, where s ∈ [0, 1], is a full solution of (4.5) such that the set {Q + w(t) | t ≤ 0} is bounded in E, then , is a full solution of (4.5) such that the set Proof. If w is a full solution, then Ψ s (t − t ′ , w(t ′ )) = w(t) for t, t ′ ∈ R, t ≥ t ′ , which implies that To verify point (i) we act on the above equation by Q + . Then, by (3.3) for t ≥ t ′ . Form (2.8), we infer that there are constants c, M > 0 such that By (2.8) and (4.6), we obtain Letting with t ′ → −∞ and using (4.12), we deduce that (4.9) is satisfied with R := m 0 M Q + L(E) /c.
To prove (ii), we act on (4.11) by the operator Q − . Then we use (3.3) and obtain which implies that, for t, t ′ ∈ R and t ≥ t ′ , we have because the semigroup {S A (t)} t≥0 can be extended on the space E − to a C 0 group. Then, by (2.9) and therefore the boundedness of {Q − w(t) | t ≥ 0} implies that (4.14) Hence, using (4.6) and (2.9), we derive that Letting t → +∞ and using (4.14), we have Let {f 1 , f 2 , . . . , f 2n } be a basis on the space E 0 given by Then W (x, y) = (w 1 , w 2 ) where w 1 := a(cλx 1 + y 1 , cλx 2 + y 2 , . . . , cλx n + y n ) and Observe that |w 1 | = a cλx + y H and where | · | is the Euclidean norm in R n . Step By assumption (F 4) there is a constant m 1 > 0 such that Choose R 2 > 0 such that − cλR 2 2 + m 1 R 2 < 0, (4.20) and define the following sets Using geometrical conditions (G1), (G2) and orthogonality from Theorem 2.4 (iii), one can find R 3 > 0 such that
Step 3. Now, we prove that, if condition (G1) is satisfied, then the set B := N 2 is an isolating block for ϕ 2 and the sets of strict egress points, strict ingress points and bounce off points are Furthermore, we show that, if condition (G2) is satisfied, then the set B := N 2 is an isolating block for the semiflow ϕ 2 and the boundary ∂B consists of strict ingress points. Assume that condition (G1) is satisfied and let (u, v) : [−δ 2 , δ 1 ) → E 0 , where δ 1 > 0 and δ 2 ≥ 0, be a solution of the semiflow ϕ 2 such that (u(0), v(0)) ∈ W −1 M i . Let (w 1 , w 2 ) : R → R 2n be a map given by for t ∈ [−δ 2 , δ 1 ).
Step 4. For any s ∈ [0, 1] write K s := Inv (Ψ s , N ). By Step 2 and the homotopy invariance of the Conley index we have (4.36) Let K ′ 1 := Inv (ϕ 1 , N 1 ) and K ′ 2 := Inv (ϕ 2 , N 2 ). In view of (2.8), (2.9) and Theorem 11.1 from [36] it follows that where the last equality is a consequence of Theorem 2.6 (i). Further, by Step 3, we infer that K ′ 2 ∈ S(E 0 ) is an isolated invariant set. Hence the multiplication property of the homotopy index yields and (4.8) implies that U (K ′ ) = Inv (Ψ 0 , N ) = K 0 . Therefore, by the topological invariance of Conley index we find that which together with (4.40) gives 42) and the proof of (i) is completed. If condition (G2) is satisfied, then B := N 2 is an isolating block for the semiflow ϕ 2 with that boundary ∂B consisting of strict ingress points. In this case the pair (B, B − ) is homeomorphic with (M, ∅), which implies that h(ϕ 2 , K ′ 2 ) = Σ 0 . Combining this with (4.40) we find that and the point (ii) follows.

Applications
We assume that Ω ⊂ R n where n ≥ 1, is an open bounded set with the boundary ∂Ω of class C ∞ . Let A be a second order differential operator with a Dirichlet boundary conditions: such that a ij = a ji ∈ C 2 (Ω) for 1 ≤ i, j ≤ n and there is c 0 > 0 such that 1≤i,j≤n Furthermore, assume that f : Ω × R × R n → R is a continuous map satisfying: (E1) there is L > 0 such that for x ∈ Ω, s 1 , s 2 ∈ R and y 1 , y 2 ∈ R n we have Write X := L p (Ω) where p ≥ 1. With the operator A we can associate the operator It is known (see e.g. [18,41]) that A p is positively defined sectorial operator on X. Let X α := D(A α p ) for (α ∈ (0, 1)) be a fractional space with the norm ū α := A α pū forū ∈ X α .
Remark 5.1. (a) Observe that A p satisfies assumptions (A1), (A2) and (A3). Since A p has compact resolvent (see e.g. [18,41]), the assumption (A1) holds. Take H := L 2 (Ω) equipped with the standard inner product and norm and put A := A 2 . Then we see that the boundedness of Ω and the fact that p ≥ 2 imply that there is a continuous embedding i : L p (Ω) ֒→ L 2 (Ω) and the assumption (A2) is satisfied. Furthermore we have D(A p ) ⊂ D( A) and Aū = A pū andū ∈ D(A p ). This shows that A p ⊂ A in the sense of the inclusion i × i. Since the operator A is self-adjoint (see e.g. [18]) we see that the assumption (A3) is also satisfied.
(b) Remark 2.1 shows that the spectrum σ(A p ) of the operator A p consists of sequence of positive eigenvalues and furthermore (λ i ) is finite or λ i → +∞ when i → +∞.
(c) Note that the following inclusion is continuous Indeed, according to assumption (E3) we have α ∈ (3/4, 1) and p ≥ 2n, and hence 2α − n p > 1. Therefore, the assertion is a consequence of [25, Theorem 1.6.1]. (d) If 1 ≥ α > β ≥ 0 then the inclusion X α ⊂ X β is continuous and compact as [25,Theorem 1.4.8] says. By Remark 5.1 (c) we can define a map F : X α → X given, for anyū ∈ X α , as We call F the Niemytzki operator associated with f . Inclusion (5.2) along with simple calculations can be used to obtain the following lemma. for x ∈ Ω, uniformly for y ∈ R n . Let B 1 ⊂ X α + ⊕ X α − and B 2 ⊂ X 0 be bounded subsets in norms · α and · L 2 , respectively.
(i) Assume that Then there is R > 0 such that for any (w,v,ū) ∈ B 1 × B 2 × X 0 , with ū L 2 ≥ R, we have the following inequality: (ii) Assume that Then there is R > 0 such that for any (w,v,ū) ∈ B 1 × B 2 × X 0 , with ū L 2 ≥ R, we have the following inequality: Proof. Since the proofs of points (i) and (ii) are analogous, we focus only on the first one. Suppose, contrary to the point (i), that there are sequences (w n ) in B 1 , (v n ) in B 2 and (ū n ) in X 0 such that ū n L 2 → ∞ when n → ∞ and F (w n +ū n ),ū n L 2 ≤ − F (w n +ū n ),v n L 2 for n ≥ 1.
(i) If the following condition is satisfied (ii) If the following condition is satisfied Proof of Theorem 5.5. It suffices to prove the first point, as the proof of the second one goes analogously. We argue by contradiction and assume that there are sequences (w n ) in B 1 , (v n ) in B 2 and (ū n ) in X 0 such that ū n L 2 → +∞ and F (w n +ū n ),ū n L 2 ≤ − F (w n +ū n ),v n L 2 for n ≥ 1. (5.12) Since B 1 ⊂ X α is a bounded set and the inclusion X α ⊂ X is compact, passing if necessary to subsequence, we can assume that there isw 0 ∈ X such thatw n →w 0 in X andw n (x) →w 0 (x) for a.a. x ∈ Ω as n → +∞. For any n ≥ 1, definē z n :=ū n / ū n L 2 . Since X 0 is a finite dimensional space we can also assume that there isz 0 ∈ X 0 such thatz n →z 0 andz n (x) →z 0 (x) for a.a. x ∈ Ω as n → +∞. Putc n :=w n +ū n for n ≥ 1 and take x ∈ Ω + := {x ∈ Ω |z 0 (x) > 0}. Then when n → +∞. If we take x ∈ Ω − := {x ∈ Ω |z 0 (x) < 0} we infer that c n (x) =w n (x) +ū n (x) =w n (x) + ū n L 2z n (x) → −∞ (5.14) when n → +∞. Using (5.12) we derive that F (w n +ū n ),w n +ū n L 2 ≤ F (w n +ū n ),w n −v n L 2 (5.15) for any n ≥ 1. Note that for the both conditions (SR1) and (SR2) we have and f (x,c n (x), ∇c n (x))c n (x) dx ≥ − h L1 for n ≥ 1.
Since Ω 0 is of Lebesgue measure zero, the boundedness of f (assumption (E2)) and dominated convergence theorem imply that

5.2.
Criteria on existence of bounded orbits. We consider the second order differential equation of the form where c > 0, λ is a real number and f : Ω × R → R is a continuous map satisfying assumptions (E1), (E2) and (E3). Then the second order equation ( for x ∈ Ω. If λ = λ k for some k ≥ 1, then there is a compact full solution w : R → E of (5.21) provided either (SR1) or (SR2) holds.
If σ is defined on R, then σ is called a full solution for Φ. Let K ⊂ E be a subset. We say that K is invariant with respect to Φ, if for every (x, y) ∈ K there is full solution σ of Φ such that σ(0) = (x, y) and σ(R) ⊂ K. If N ⊂ E then we define maximal invariant set as Inv (N, Φ) := (x, y) ∈ N there is a solution σ : R → E of Φ such that σ(0) = (x, y) and σ(R) ⊂ N .
A closed invariant set K ⊂ E is called isolated, if there is a closed set N ⊂ E such that K = Inv (N ) ⊂ int N . In this case N is called isolating neighborhood for K. Let Φ s : [0, +∞) × E → E for s ∈ [0, 1], be a family of semiflows. We say that N ⊂ E is admissible with respect to {Φ s } s∈[0,1] , if for every sequences s n ∈ [0, 1], z n in E and (t n ) in [0, +∞) such that t n → +∞ when n → ∞, the inclusion Φ sn ([0, t n ] × {z n }) ⊂ N for n ≥ 1, implies that the set {Φ sn n (t n , z n ) | n ≥ 1} is relatively compact in E. Furthermore the family {Φ s } s∈[0,1] is continuous provided Φ sn (t n , z n ) → Φ s0 (t 0 , z 0 ) as long as s n → s 0 , t n → t 0 and z n → z 0 as n → +∞.
A subset N ⊂ E is admissible with respect to the single semiflow Φ, if it is admissible with respect to family consisting from constant semiflow Φ. From now on we write S(E) = S(E, Φ) for a class of invariant sets admitting an admissible isolated neighborhood with respect to Φ. A special case of isolated neighborhood is an isolating block. To define it assume that B ⊂ E is a closed set and let (x, y) ∈ ∂B. We say that (x, y) is a strict egress point (resp. strict ingress point, resp. bounce off point ), if for any solution σ : [−δ 1 , δ 2 ) → E, where δ 1 ≥ 0 and δ 2 > 0, of the semiflow Φ such that σ(0) = (x, y) the following holds: (a) there is ε 2 ∈ (0, δ 2 ] such that σ(t) / ∈ B (resp. σ(t) ∈ int B, resp. σ(t) / ∈ B) for t ∈ (0, ε 2 ]; (b) if δ 1 > 0 then there is ε 1 ∈ (0, δ 1 ) such that σ(t) ∈ int B (resp. σ(t) / ∈ B, resp. σ(t) / ∈ B) for t ∈ [−ε 1 , 0). We write B e , B i and B b for the sets of strict egress points, strict ingress points and bounce off points, respectively. Furthermore, put B − := B e ∪ B b .  where B/B − is the quotient space and B∪{c} is a disjoint sum of B and the one point space {c}. It is known that the homotopy index is independent from the choice of isolating block of K has the following properties: (H1) (Existence) If K ∈ S(E) and h(Φ, K) = 0, then K = ∅.