# The Existence of Periodic Solutions of Delay Differential Equations byE+-Conley Index Theory

Authors: Huafeng Xiao
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, theE+-Conley index theory has been used to study the existence of periodic solutions of nonautonomous delay differential equations (in short, DDEs). The variational structure for DDEs is built, and the existence of periodic solutions of DDEs is transferred to that of critical points of the associated function. When DDEs are 2π-nonresonant, some sufficient conditions are obtained to guarantee the existence of periodic solutions. When the system is 2π-resonant at infinity, by making use a second disturbing of the original functional, some sufficient conditions are obtained to guarantee the existence of periodic solutions to DDEs.


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## Body

## 1. Introduction

In the last several decades, there has been increasing interest in the dynamical properties of delay differential equations due to their important applications in many fields such as biological, physical, and social sciences. Many rich mathematical investigations and interesting results on DDEs have been available in the literature. In particular, the existence of periodic solutions is of great interest. As far as the author’s knowledge, it was Jones who first studied the existence of periodic solutions of DDEs. Since then, some methods had been developed to search for periodic solutions of DDEs, such as fixed point theorems (cf. [1–4]), cone mapping method (cf. [5]), coincidence degree theory (cf. [6, 7]), Poincaré-Bendixson theorem (cf. [8–10]), and Hopf bifurcation theorem (cf. [11, 12]).In 1973, Rabinowitz built the variational approach. It had been developed quickly. The variational approach had been used in two different ways to study the existence of periodic solutions of DDEs. Firstly, it can be used indirectly. More specially, in 1974, Kaplan and Yorke studied the following equation:(1)x′t=−fxt−1,x∈R.They (cf. [13]) translated the existence of periodic solutions of (1) to that of an associated plane ordinary differential equations. Li and his cooperators developed Kaplan and Yorke’s technique and applied to the following equation:
(2)x′t=−fxt−1−fxt−2−⋯−fxt−n,x∈R.They showed that the associated equation of (2) is a Hamiltonian system (resp., general Hamiltonian system) (cf. [14–16]). By noting the symmetric property, Fei (cf. [17, 18]) made use of variational methods to study the existence of periodic solutions of the associated Hamiltonian system and obtain the existence of multiple periodic solutions of (2). In 2019, Li et al. (cf. [19]) proved the multiplicity of periodic solutions to the following equation:
(3)x2n+1t=−∑i=12kfxt−i,a.e.t∈0,22k+1,xt−xt−22k+1=0,x∈R.For more results in this direction, we refer to [20] and the reference therein.In 2005, Guo and Yu firstly established a variational structure for DDEs below:(4)x′t=−fxt−1,x∈Rn.They converted the existence of periodic solutions of (4) to that of critical points of the associated variational functional. By using of pseudo-index theory, they (cf. [21]) proved multiple periodic solutions for (4). Later, Guo and Yu and Zheng and Guo studied the following systems:
(5)x′t=−fxt−1−fxt−2−⋯−fxt−n,x∈Rn.They (cf. [22, 23]) proved the existence of multiple periodic solutions of (5). In 2009, C.J. Guo and Z.M. Guo studied second order system as follows:
(6)x′′t=−fxt−1,x∈Rn.They (cf. [24]) obtained some sufficient conditions to guarantee the existence of multiple periodic solutions of (6). For more results on this direction, we refer to [25] and the reference therein.As for nonautonomous DDEs, Yu and Xiao studied the following system:(7)x′t=−ft,xt−1,x∈Rn.By making use of the Maslov-type index theory, they (cf. [26]) proved the existence of multiple periodic solutions. For more reference on this direction, we refer to [27].It is well-known that there are many index theories in variational approach. However, only a few had been used to study the existence and multiplicity of periodic solutions to DDEs. A nature question is that whether those index theories can be used to study the existence of periodic solutions to DDEs. It is well known thatE+-Conley index, which is a relative Morse index, was introduced by Abbondandolo [28, 29]. It has been used to study periodic solutions of Hamiltonian systems by proving the Morse-Conley relations. Motivated by [21, 28, 29], we use E+-Conley index to study the existence of periodic solutions of the following equations:
(8)x′t=−ft,xt−π2,where x∈Rn,f∈CR×Rn,Rn. Assume that(F1) f is odd with respect to variable x and π-periodic with respect to variable t, i.e., for any x∈Rn, ft,−x=−ft,x uniformly for t∈0,π and ft+π,x=ft,x uniformly for x∈Rn(F2) there exists a twice continuously differentiable function F, such that the gradient of F is f, i.e., ∇xFt,x=ft,x(F3) there exists a π-periodic continuous loop of symmetric matrices B∞t and a function G∈C2ℝ×ℝn such that
(9)ft,x=B∞tx+∇Gt,x,where ∇Gt,x=o∣x∣ as ∣x∣⟶∞, uniformly with respect to t∈0,π(F4) D2G is boundedTheorem 1.
Assume thatF satisfies F1-F4. Assume that (8) is 2π-nonresonant at infinity (see Definition 31). If all the 2π-periodic solutions of (8) are 2π-nonresonant, then there is an odd number of them and the following relation holds:
(10)∑λE+−dimW−=λE+−dimV∞−+1+λQλ,where the sum is taken over all the 2π-periodic solutions and Q is a Laurent polynomial with positive coefficients.Assume the following:(F5) ∣D2Gt,x∣⟶0 as ∣x∣⟶∞, uniformly with respect to t(F6) ∇Gt,x is boundedTheorem 2.
Assume that (F1)-(F3), (F5), and (F6) hold. Assume that x0 is a 2π-nonresonant periodic solution of the system (8). If
(11)E+−dimWx0∉E+−dimV∞−−1,E+−dimV∞−+E+−dimV∞0+1,then (8) has at least another 2π-periodic solution.The paper is organized as follows: in Section2, we present the space which the variational functional is built; in Section 3, we summarize some basic knowledge on the E+-Conley index; in Section 4, we will study system which is 2π-nonresonant system at infinity; in Section 5, we will study system which is 2π-resonant system at infinity.


## 2. Preparation

Denote byS1=R/2πZ. A similar discussion as reference [21], we can build the Hilbert space H=H1/2S1,Rn. If x∈H, it has Fourier expansion
(12)xt=a02π+1π∑j=1∞ajcosjt+bjsinjt,where a0,aj,bj∈Rn,j∈N∗. The space H equips with the following norm and inner product:
(13)x2=a02+∑j=1∞jaj2+bj2,x,y=a0,c0+∑j=1∞jaj,cj+bj,dj,where y=c0/2π+1/π∑j=1∞cjcosjt+djsinjt,c0,cj,dj∈Rn,j∈N∗.Proposition 3 (see [30]).
For everyp∈1,+∞, H is compactly embedded into the Banach space LpR/2πZ;Rn.
Letx∈L2S1,Rn. If for every z∈C∞S1,Rn,
(14)∫02πxt,z′tdt=−∫02πyt,ztdt,then y is called a weak derivative of x, denoted by ẋ.Now, the variational function defined onH is
(15)Jx=∫02π12ẋt+π2,xt+Ft,xtdt.Defined an operator by extending the bilinear forms(16)Ax,y=∫02πẋt+π2,ytdt,x,y∈H.It is easily to check thatA is a linear bounded operator on H.Define(17)φx=∫02πFt,xtdt,∀x∈H.Then,J can be rewritten as
(18)Jx=12Ax,x+φx,∀x∈H.Now we state a useful lemma, which can be proved as Proposition B.37 in the book of Rabinowitz [30] and Appendix A3 in the book of Hofer and Zehnder [31].Lemma 4.
Assume that (F1)-(F3) are satisfied; then, φ∈C2H and
(19)φ′x,y=∫02πft,xt,ytdt,d2bxy,z=D2φxy,z=∫02πD2Ft,xtyt,ztdt.Moreover, the mapφ′:H⟶H is completely continuous.Define an operatorΓ:H⟶H(20)Γxt=xt+π2,forallx∈H.Clearly,Γ is a bounded linear operator and satisfies ∥Γx∥=∥x∥.Next, we set(21)E=x∈H∣Γ2x=−x.Then,E is a closed space of H. It is easily to check that A is a linear bounded self-adjoint operator on E. If x∈E, it has the following Fourier expansion:
(22)xt=1π∑j=1∞ajcos2j−1t+bjsin2j−1t.Lemma 5.
Assume that (F1)-(F4) are satisfied. Then, J is twice continuously differentiable. The existence of 2π-periodic solutions xt for (1) is equivalent to the existence of critical points of functional J restricted to E. Moreover, if x is a critical point of J, the linear operator D2φx is compact.Proof.
Based on the definition ofJ and Lemma 4, J is twice continuously differentiable on E.By a standard argument as in [32] (cf. Proposition 4.1.3) and [21] (cf. Lemma 4), we can prove the second and third assertions.Remark 6.
Since (F5) and (F6) imply that of (F4), Lemma 5 holds when F satisfies (F1)-(F3), (F5), and (F6).Now, we restrict our discussion on the spaceE. Then, A can be computed explicitly in terms of the Fourier expansion of x:
(23)Ax=1π∑j=1∞−1jajcos2j−1t+bjsin2j−1.Define the following family of subspaces ofE.
(24)Ej=ajcos2j−1t+bjsin2j−1t.Then,Ax=x on E2j and Ax=−x on E2j−1 for all j∈N∗.Set(25)E−=⊕j=1∞E2j−1,E+=⊕j=1∞E2j.Then,E=E−⊕E+. Notice that the null space of A is 0. Then, A is a self-adjoint invertible Fredholm operator.


## 3.E+-Conley Index Theory

### 3.1. TheE+-Dimension and the E+-Morse Index

LetE be a real Hilbert space. We denote by ⋅ and ⋅,⋅ the Hilbert norm and scalar product on E, respectively. Assume that E has an orthogonal splitting
(26)E=E−⊕E+,where both E+ and E− are infinite dimensional.Definition 7.
Two closed subspacesV and W of E are called commensurable if the quotient projections W⟶E/V and V⟶E/W are compact.LetPV denote the orthogonal projection onto V and V⊥ denote the orthogonal complement of V. Two closed subspaces V and W of E are commensurable if and only if both PV⊥ restricted to W and PW⊥ restricted to V are compact. Subsequently, both dimV∩W⊥ and dimV⊥∩W are finite.Definition 8.
LetV be a closed subspace of E commensurable with E−. The E+-dimension of V is defined as
(27)E+−dimV=dimV∩E+−codimV+E+=dimV∩E+−dimV⊥∩E−.
LetJ be a twice differentiable functional on E and d2Jx be its second order differential at x. Denote by D2Jx the associated self-adjoint operator:
(28)D2Jxy,z=d2Jxy,z.Definition 9.
A critical pointx of J is called nondegenerate if D2Jx is invertible. A critical point x of J is called weakly nondegenerate if D2Jx is a Fredholm operator.Ifx is a weakly nondegenerate critical point of J, E splits into three closed subspaces.
(29)E=V+⊕V0⊕V−,where V+, V0, and V− are positive, null, and negative eigenspaces of D2Jx, respectively. Obviously, V0 is finite dimensional. And there exists α>0,β>0 such that
(30)D2Jxx,x≥αx2,∀x∈V+,D2Jxx,x≤−βx2,∀x∈V−.Definition 10.
Letx be a weakly nondegenerate critical point of J. Assume moreover that the negative eigenspace V− of D2Jx is commensurable with E−. The E+-Morse index is the finite relative integer
(31)E+−mx=E+−mx,J=E+−dimV−.
The largeE+-Morse index is the finite relative integer
(32)E+−m∗x=E+−m∗x,J=E+−dimV−⊕V0.Proposition 11 (see [29]).
LetL be a Fredholm operator whose negative eigenspace V− is commensurable with E−. Then,
(33)E+−dimV−=maxE+−dimW,where the maximum is taken over the family of all closed subspaces of W of E which are commensurable with E− and such that L is strictly negative on W.Lemma 12 (see [29]).
Denote byVL− the negative eigenspace of the self-adjoint operator L. Let LE− be the set of invertible self-adjoint operators whose negative eigenspace is commensurable with E−. Then, LE− is relatively closed in the set of invertible operators. The function
(34)LE−∍L↦E+−dimVL−∈Zis continuous.Lemma 13 (see [32]).
Assume thatL is a self-adjoint Fredholm operator, and that K is a self-adjoint compact operator. Then, the negative eigenspaces of L+K and of L are commensurable.


### 3.2. TheE+-Cohomology

LetτE+ be the weakest topology on E such that all the bounded linear functions and the quotient projection π:E⟶E/E+ are continuous. Equivalently, τE+ is the product topology between the weak topology of E+ and the strong topology of E−. If V− is commensurable with E−, then the topologies τE+ and τV+ coincide. Thus, the topology τE+ depends only on the commensurability class of E+.Definition 14.
A subsetX of E is E+-locally compact if X∩π−1α is τE+-locally compact, for every finite dimensional subspace α of E/E+.Obviously, every boundedτE+-closed subset X of E is E+-locally compact.Definition 15.
AnE+-pair X,A is a topological pair of subset of E such that X and A are τE+-closed and E+-locally compact.Definition 16.
AnτE+-continuous map Φ:X,A↦Y,B is an E+-morphism if
(1)
It has the form(35)Φx=Tx+Kx,where K maps bounded sets into τE+-precompact sets and T is a linear automorphism of E such that TE+=E+.
(2)
Φ−1U is bounded for every bounded UDefinition 17.
AnτE+-continuous map Ψ:0,1×X,0,1×A↦Y,B is an E+-homotopy if
(1)
It has the form(36)Ψt,x=Ttx+Kt,x,where K maps bounded sets into τE+-precompact sets and Tt is a linear automorphism of E such that TtE+=E+.
(2)
Ψ−1U is bounded for every bounded UTheorem 18 (see [29]).
There exists a generalised cohomologyHE+∗, with coefficients in Z2, which acts on the category of E+-pairs and E+-morphisms. More precisely,
(1)
Controvariant Functoriality. If I:X,A↦X,A is the identity map, HE+∗I is the identity homomorphism on HE+∗X,A. If Φ:X,A↦Y,B and Φ′:Y,B↦Z,C are E+-morphisms, then HE+∗Φ′∘Φ=HE+∗Φ∘HE+∗Φ′.(2)
Homotopy Invariance. If two E+-morphisms Φ,Φ′ are E+-homotopic, then HE+∗Φ=HE+∗Φ′.(3)
Strong Excision. If X and Y are τE+-closed E+-locally compact subsets of E and i:X,X∩Y↣X∪Y,Y is the inclusion map, then HE+∗i is an isomorphism.(4)
Naturality of the Coboundary. Given three τE+-closed and E+-locally compact sets X,Y,Z such that X⊂Y⊂Z, there exists a family of coboundary homomorphisms(37)δE+qZ,Y,Z:HE+qY,X⟶HE+q+1Z,Y.
IfΦ:Z,Y↦Z′,Y′ is a E+-morphism such that ΦX⊂X′⊂Y′, then
(38)HE+q+1∘δE+qZ′,Y′,X′=δE+qZ,Y,X∘HE+qΦY,X.(5)
Long Exact Sequence. Given three τE+-closed and E+-locally compact sets X,Y,Z such that X⊂Y⊂Z, denote by iY,X↣Z,X and j:Z,X↣Z,Y the inclusion maps. Then, the following sequence of homomorphisms is exact(39)⋯⟶HE+qZ,X⟶HE+qiHE+qY,X⟶δE+qHE+q+1Z,Y⟶HE+q+1jHE+q+1Z,X⟶⋯`.(6)
Dimension Property. Let V be a closed subspace of E, commensurable with E−. Let B be a closed ball in V and ∂B be its relative boundary in V. Then,(40)HE+qB,∂B=Z2,ifq=E+−dimV,0,otherwise.Proposition 19 (see [29]).
Assume thatA⊂Y⊂X are τE+-closed E+-locally compact subsets of E.
(1)
If there exists aE+-homotopy(41)Ψ:0,1×X,0,1×A↦X,A,such that Φ0=id, Φ1X⊂Y, and ΨtY⊂Y for every t, then
(42)HE+∗X,A≅HE+∗Y,A.(2)
If there exists aE+-homotopy(43)Φ:0,1×X,0,1×Y↦X,Y,such that Φ0=id, Φ1Y⊂A, and ΦtA⊂A for every t, then
(44)HE+∗X,Y≅HE+∗X,A.


### 3.3. TheE+-Conley Theory

IfJ is a twice continuously differentiable real valued function on E. Let K be the critical set of J, i.e., the set K=x∈E∣dJx=0. Denote by Ja the set x∈E∣Jx≤a.Recall that the Palais-Smale sequence (denoted by (PS) sequence) is a sequencexn of elements of E such that Jxn is bounded and ∥∇Jxn∥ converges to zero.We assume that(A1) Each sublevelJa is τE+-closed and τE+-locally compact(A2) Every bounded (PS) sequence has a converging subsequence(A3)∇J is globally Lipschitz(A4) The flowη defined by
(45)∂∂tηt,x=−J′ηt,x,η0,x=x,is an E+-homotopy.Since∇J is globally Lipschitz, the flow η exists globally. If t∈R and x∈E, we will use both notations ηt,x=ηtx. Besides, (A3) implies that J is bounded on bounded sets. Recall that an isolated critical set is an isolated subset of K.Definition 20.
LetK0 be a compact isolated critical set. An E+-index pair for K0 is a topological pair X,A in E such that
(1)
X,A is a bounded E+-pair(2)
X,A∩K=K0⊂IntX and K\K0∩X⊂IntXA. Here IntX is the interior part of X with respect to the topology of E, and IntXA is the interior part of A with respect to the topology induced on X by the strong topology of E(3)
A is strongly positively invariant with respect to X; if x∈A and ηt,x∈X for t>0, then ηt,x∈A(4)
A is an exit set for X: if x∈X and η0,t×x is not contained in X, then there exists t∗∈0,t such that ηt∗,x∈A
Moreover, anE+-index pair X,A for K0 is called strict if K∩X=K0.Definition 21.
An elementary critical set is a compact isolated critical setK0 such that J is constant on K0.Lemma 22 (see [29]).
LetK0 be an elementary critical set and let U be a neighborhood of K0 which is τE+-closed. Then, there exists a strict E+-index pair X,A for K0 such that X⊂U.Lemma 23 (see [29]).
LetK0 be a compact isolated critical set. Let X,A be an E+-index pair for K0 and X0,A0 a strict E+-index pair for K0 such that X0⊂X. Then,
(46)HE+∗X,A≅HE+∗X0,A0.Lemma 24 (see [29]).
LetK0 be an elementary critical set. If X,A and Y,B are E+-index pairs for K0, then
(47)HE+∗X,A≅HE+∗Y,B.
We recall that theE+-Poincaré polynomial of an E+-pair X,A is defined as
(48)PE+X,A=∑q∈ZdimHE+qX,Aλq.
In general,PE+X,A is a formal Laurent series, whose coefficients are nonnegative integers or +∞.Definition 25.
LetK0 be an elementary critical set of J. The E+-critical group of K0 is the vector space
(49)cE+qK0=HE+qX,A,where X,A is an E+-index pair for K0. The E+-Morse polynomial of K0 is the E+-Poincaré polynomial of X,A:
(50)ME+K0=∑q∈ZdimcE+qK0λq.Definition 26.
Assume thatf∈C2E. An elementary critical set K0 is called E+-nondegenerate critical manifold if it is a finite dimensional compact C1-manifold embedded in E and for every x∈K0:
(1)
x is a weak nondegenerate critical point, and the kernel of D2fx coincides with the tangent space of K0 in x(2)
The negative eigenspace ofD2fx is commensurable with E−Proposition11 easily implies that if the E+-nondegenerate critical manifold K0 is connected, the E+-Morse index E+−mx is constant on K0. In this case, we set
(51)E+−mK0=E+−mK0;J=E+−mx;J,forx∈K0.Proposition 27 (see [29]).
IfK0 is a connected E+-nondegenerate critical manifold of a functional of class C2, then
(52)ME+K0=PK0λmK0,where PK0 is the standard Poincaré polynomial of K0.Now we state a result which follows immediately from Morse-Conley relations.Theorem 28 (see [29]).
LetK be an isolated critical set of J, and let X,A be an E+-index pair for K. Assume that K=K1∪⋯∪Km, where K1,⋯,Km are pairwise disjoint elementary critical sets. Then, there exists a Laurent series Q with positive or infinite coefficients such that
(53)∑i=1mME+Ki=PE+X,A+1+λQλ.
IfK consists of E+-nondegenerate critical manifolds, the above relation is an equality between true Laurent polynomials with finite coefficients.In the case of a Morse function, we have the following immediate corollary.Corollary 29 (see [32]).
Assume thatx1,⋯,xk are nondegenerate critical points of J with finite E+-Morse index. If X,A is an E+-index pair for the critical set x1,⋯,xk, there exists a Laurent polynomial Q with positive coefficients such that
(54)∑i=1kλE+−mxi,J=PE+X,A+1+λQλ.
Now we state a useful lemma.Lemma 30 (see [29]).
LetE=V+⊕V− be an orthogonal splitting such that V+ is commensurable with E+ (and therefore, V− is commensurable with E−). Let P+ and P− be the orthogonal projections onto V+ and V−, respectively. Assume that there exists R¯ such that for every R>R¯, the following inequalities hold:
(55)ddtP−ηt,x2t=0>0∀x∈BV+R×∂BV−R,(56)ddtP+ηt,x2t=0<0∀x∈∂BV+R×BV−R.
Then, the critical setK of J is compact, and
(57)X,A=BV+R¯×BV−R¯,BV+R¯×∂BV−R¯is an E+-index pair for K.


## 3.1. TheE+-Dimension and the E+-Morse Index

LetE be a real Hilbert space. We denote by ⋅ and ⋅,⋅ the Hilbert norm and scalar product on E, respectively. Assume that E has an orthogonal splitting
(26)E=E−⊕E+,where both E+ and E− are infinite dimensional.Definition 7.
Two closed subspacesV and W of E are called commensurable if the quotient projections W⟶E/V and V⟶E/W are compact.LetPV denote the orthogonal projection onto V and V⊥ denote the orthogonal complement of V. Two closed subspaces V and W of E are commensurable if and only if both PV⊥ restricted to W and PW⊥ restricted to V are compact. Subsequently, both dimV∩W⊥ and dimV⊥∩W are finite.Definition 8.
LetV be a closed subspace of E commensurable with E−. The E+-dimension of V is defined as
(27)E+−dimV=dimV∩E+−codimV+E+=dimV∩E+−dimV⊥∩E−.
LetJ be a twice differentiable functional on E and d2Jx be its second order differential at x. Denote by D2Jx the associated self-adjoint operator:
(28)D2Jxy,z=d2Jxy,z.Definition 9.
A critical pointx of J is called nondegenerate if D2Jx is invertible. A critical point x of J is called weakly nondegenerate if D2Jx is a Fredholm operator.Ifx is a weakly nondegenerate critical point of J, E splits into three closed subspaces.
(29)E=V+⊕V0⊕V−,where V+, V0, and V− are positive, null, and negative eigenspaces of D2Jx, respectively. Obviously, V0 is finite dimensional. And there exists α>0,β>0 such that
(30)D2Jxx,x≥αx2,∀x∈V+,D2Jxx,x≤−βx2,∀x∈V−.Definition 10.
Letx be a weakly nondegenerate critical point of J. Assume moreover that the negative eigenspace V− of D2Jx is commensurable with E−. The E+-Morse index is the finite relative integer
(31)E+−mx=E+−mx,J=E+−dimV−.
The largeE+-Morse index is the finite relative integer
(32)E+−m∗x=E+−m∗x,J=E+−dimV−⊕V0.Proposition 11 (see [29]).
LetL be a Fredholm operator whose negative eigenspace V− is commensurable with E−. Then,
(33)E+−dimV−=maxE+−dimW,where the maximum is taken over the family of all closed subspaces of W of E which are commensurable with E− and such that L is strictly negative on W.Lemma 12 (see [29]).
Denote byVL− the negative eigenspace of the self-adjoint operator L. Let LE− be the set of invertible self-adjoint operators whose negative eigenspace is commensurable with E−. Then, LE− is relatively closed in the set of invertible operators. The function
(34)LE−∍L↦E+−dimVL−∈Zis continuous.Lemma 13 (see [32]).
Assume thatL is a self-adjoint Fredholm operator, and that K is a self-adjoint compact operator. Then, the negative eigenspaces of L+K and of L are commensurable.


## 3.2. TheE+-Cohomology

LetτE+ be the weakest topology on E such that all the bounded linear functions and the quotient projection π:E⟶E/E+ are continuous. Equivalently, τE+ is the product topology between the weak topology of E+ and the strong topology of E−. If V− is commensurable with E−, then the topologies τE+ and τV+ coincide. Thus, the topology τE+ depends only on the commensurability class of E+.Definition 14.
A subsetX of E is E+-locally compact if X∩π−1α is τE+-locally compact, for every finite dimensional subspace α of E/E+.Obviously, every boundedτE+-closed subset X of E is E+-locally compact.Definition 15.
AnE+-pair X,A is a topological pair of subset of E such that X and A are τE+-closed and E+-locally compact.Definition 16.
AnτE+-continuous map Φ:X,A↦Y,B is an E+-morphism if
(1)
It has the form(35)Φx=Tx+Kx,where K maps bounded sets into τE+-precompact sets and T is a linear automorphism of E such that TE+=E+.
(2)
Φ−1U is bounded for every bounded UDefinition 17.
AnτE+-continuous map Ψ:0,1×X,0,1×A↦Y,B is an E+-homotopy if
(1)
It has the form(36)Ψt,x=Ttx+Kt,x,where K maps bounded sets into τE+-precompact sets and Tt is a linear automorphism of E such that TtE+=E+.
(2)
Ψ−1U is bounded for every bounded UTheorem 18 (see [29]).
There exists a generalised cohomologyHE+∗, with coefficients in Z2, which acts on the category of E+-pairs and E+-morphisms. More precisely,
(1)
Controvariant Functoriality. If I:X,A↦X,A is the identity map, HE+∗I is the identity homomorphism on HE+∗X,A. If Φ:X,A↦Y,B and Φ′:Y,B↦Z,C are E+-morphisms, then HE+∗Φ′∘Φ=HE+∗Φ∘HE+∗Φ′.(2)
Homotopy Invariance. If two E+-morphisms Φ,Φ′ are E+-homotopic, then HE+∗Φ=HE+∗Φ′.(3)
Strong Excision. If X and Y are τE+-closed E+-locally compact subsets of E and i:X,X∩Y↣X∪Y,Y is the inclusion map, then HE+∗i is an isomorphism.(4)
Naturality of the Coboundary. Given three τE+-closed and E+-locally compact sets X,Y,Z such that X⊂Y⊂Z, there exists a family of coboundary homomorphisms(37)δE+qZ,Y,Z:HE+qY,X⟶HE+q+1Z,Y.
IfΦ:Z,Y↦Z′,Y′ is a E+-morphism such that ΦX⊂X′⊂Y′, then
(38)HE+q+1∘δE+qZ′,Y′,X′=δE+qZ,Y,X∘HE+qΦY,X.(5)
Long Exact Sequence. Given three τE+-closed and E+-locally compact sets X,Y,Z such that X⊂Y⊂Z, denote by iY,X↣Z,X and j:Z,X↣Z,Y the inclusion maps. Then, the following sequence of homomorphisms is exact(39)⋯⟶HE+qZ,X⟶HE+qiHE+qY,X⟶δE+qHE+q+1Z,Y⟶HE+q+1jHE+q+1Z,X⟶⋯`.(6)
Dimension Property. Let V be a closed subspace of E, commensurable with E−. Let B be a closed ball in V and ∂B be its relative boundary in V. Then,(40)HE+qB,∂B=Z2,ifq=E+−dimV,0,otherwise.Proposition 19 (see [29]).
Assume thatA⊂Y⊂X are τE+-closed E+-locally compact subsets of E.
(1)
If there exists aE+-homotopy(41)Ψ:0,1×X,0,1×A↦X,A,such that Φ0=id, Φ1X⊂Y, and ΨtY⊂Y for every t, then
(42)HE+∗X,A≅HE+∗Y,A.(2)
If there exists aE+-homotopy(43)Φ:0,1×X,0,1×Y↦X,Y,such that Φ0=id, Φ1Y⊂A, and ΦtA⊂A for every t, then
(44)HE+∗X,Y≅HE+∗X,A.


## 3.3. TheE+-Conley Theory

IfJ is a twice continuously differentiable real valued function on E. Let K be the critical set of J, i.e., the set K=x∈E∣dJx=0. Denote by Ja the set x∈E∣Jx≤a.Recall that the Palais-Smale sequence (denoted by (PS) sequence) is a sequencexn of elements of E such that Jxn is bounded and ∥∇Jxn∥ converges to zero.We assume that(A1) Each sublevelJa is τE+-closed and τE+-locally compact(A2) Every bounded (PS) sequence has a converging subsequence(A3)∇J is globally Lipschitz(A4) The flowη defined by
(45)∂∂tηt,x=−J′ηt,x,η0,x=x,is an E+-homotopy.Since∇J is globally Lipschitz, the flow η exists globally. If t∈R and x∈E, we will use both notations ηt,x=ηtx. Besides, (A3) implies that J is bounded on bounded sets. Recall that an isolated critical set is an isolated subset of K.Definition 20.
LetK0 be a compact isolated critical set. An E+-index pair for K0 is a topological pair X,A in E such that
(1)
X,A is a bounded E+-pair(2)
X,A∩K=K0⊂IntX and K\K0∩X⊂IntXA. Here IntX is the interior part of X with respect to the topology of E, and IntXA is the interior part of A with respect to the topology induced on X by the strong topology of E(3)
A is strongly positively invariant with respect to X; if x∈A and ηt,x∈X for t>0, then ηt,x∈A(4)
A is an exit set for X: if x∈X and η0,t×x is not contained in X, then there exists t∗∈0,t such that ηt∗,x∈A
Moreover, anE+-index pair X,A for K0 is called strict if K∩X=K0.Definition 21.
An elementary critical set is a compact isolated critical setK0 such that J is constant on K0.Lemma 22 (see [29]).
LetK0 be an elementary critical set and let U be a neighborhood of K0 which is τE+-closed. Then, there exists a strict E+-index pair X,A for K0 such that X⊂U.Lemma 23 (see [29]).
LetK0 be a compact isolated critical set. Let X,A be an E+-index pair for K0 and X0,A0 a strict E+-index pair for K0 such that X0⊂X. Then,
(46)HE+∗X,A≅HE+∗X0,A0.Lemma 24 (see [29]).
LetK0 be an elementary critical set. If X,A and Y,B are E+-index pairs for K0, then
(47)HE+∗X,A≅HE+∗Y,B.
We recall that theE+-Poincaré polynomial of an E+-pair X,A is defined as
(48)PE+X,A=∑q∈ZdimHE+qX,Aλq.
In general,PE+X,A is a formal Laurent series, whose coefficients are nonnegative integers or +∞.Definition 25.
LetK0 be an elementary critical set of J. The E+-critical group of K0 is the vector space
(49)cE+qK0=HE+qX,A,where X,A is an E+-index pair for K0. The E+-Morse polynomial of K0 is the E+-Poincaré polynomial of X,A:
(50)ME+K0=∑q∈ZdimcE+qK0λq.Definition 26.
Assume thatf∈C2E. An elementary critical set K0 is called E+-nondegenerate critical manifold if it is a finite dimensional compact C1-manifold embedded in E and for every x∈K0:
(1)
x is a weak nondegenerate critical point, and the kernel of D2fx coincides with the tangent space of K0 in x(2)
The negative eigenspace ofD2fx is commensurable with E−Proposition11 easily implies that if the E+-nondegenerate critical manifold K0 is connected, the E+-Morse index E+−mx is constant on K0. In this case, we set
(51)E+−mK0=E+−mK0;J=E+−mx;J,forx∈K0.Proposition 27 (see [29]).
IfK0 is a connected E+-nondegenerate critical manifold of a functional of class C2, then
(52)ME+K0=PK0λmK0,where PK0 is the standard Poincaré polynomial of K0.Now we state a result which follows immediately from Morse-Conley relations.Theorem 28 (see [29]).
LetK be an isolated critical set of J, and let X,A be an E+-index pair for K. Assume that K=K1∪⋯∪Km, where K1,⋯,Km are pairwise disjoint elementary critical sets. Then, there exists a Laurent series Q with positive or infinite coefficients such that
(53)∑i=1mME+Ki=PE+X,A+1+λQλ.
IfK consists of E+-nondegenerate critical manifolds, the above relation is an equality between true Laurent polynomials with finite coefficients.In the case of a Morse function, we have the following immediate corollary.Corollary 29 (see [32]).
Assume thatx1,⋯,xk are nondegenerate critical points of J with finite E+-Morse index. If X,A is an E+-index pair for the critical set x1,⋯,xk, there exists a Laurent polynomial Q with positive coefficients such that
(54)∑i=1kλE+−mxi,J=PE+X,A+1+λQλ.
Now we state a useful lemma.Lemma 30 (see [29]).
LetE=V+⊕V− be an orthogonal splitting such that V+ is commensurable with E+ (and therefore, V− is commensurable with E−). Let P+ and P− be the orthogonal projections onto V+ and V−, respectively. Assume that there exists R¯ such that for every R>R¯, the following inequalities hold:
(55)ddtP−ηt,x2t=0>0∀x∈BV+R×∂BV−R,(56)ddtP+ηt,x2t=0<0∀x∈∂BV+R×BV−R.
Then, the critical setK of J is compact, and
(57)X,A=BV+R¯×BV−R¯,BV+R¯×∂BV−R¯is an E+-index pair for K.


## 4. Systems with Nonresonant at Infinity

In this section, we consider system (8) satisfying F1-F4. If zt is a 2π-periodic solutions of (8), we can linearise system (8) near z obtaining the 2π-periodic system
(58)ẋt=−D2Ft,ztxt−π2,where D2F is the Hessian of F with respect to variable x.Definition 31.
A2π-periodic solution z of system (8) is called 2π-resonant if the linearised system (58) has nontrivial 2π-periodic solutions. Otherwise, it is called 2π-nonresonant.The corresponding functional is quadratic(59)Jx=12Ax,x+12Bzx,x,where Bz is the self-adjoint operator such that
(60)Bzx,y=∫02πD2Ft,ztxt,ytdt.Lemma5 implies that Bz is a compact operator. Let Wz+, Wz0, and Wz− be the positive, kernel, and negative eigenspaces of A+Bz.The linearisation of (8) at infinity is
(61)ẋt=−B∞txt−π2.Definition 32.
The linear system (61) is said 2π-resonant if it has nontrivial 2π-periodic solutions. Otherwise, it is said 2π-nonresonant. The asymptotically linear system (8) is said 2π-resonant at infinity if system (61) is 2π-resonant, 2π-nonresonant at infinity in the opposite case.LetB∞ be the self-adjoint operator on E defined by extending bilinear form
(62)B∞x,y=∫02πB∞txt,ytdt,and then, B∞ is a compact operator. Define
(63)ϕx=∫02πGt,xtdt.Therefore,(64)Jx=12A+B∞x,x+ϕx.LetV∞+,V∞0,V∞− be the positive, kernel, and negative eigenspaces of A+B∞, respectively. Let P+,P0,P− be the orthogonal projections onto V∞+,V∞0,V∞−, respectively.Now, we want to check that (A1)-(A4) hold under assumptions (F1)-(F4), so that it is possible to apply theE+-Conley theory to J.Lemma 33 (see [32]).
LetV be a finite codimensional subspace of E+. If σ,θ,λ+,λ− are positive constants, the functional
(65)hu=λ+PVuσ−λ−PV⊥uθhas E+-locally compact sublevels.Lemma 34.
Assume thatF satisfies (F2) and (F3). Then, the sublevels of J are τE+-closed and also E+-locally compact. So (A1) holds.Proof.
Notice that the variational functional is(66)Jx=12Ax,x+φx.
Its quadratic part is lower semicontinuous and convex onE+ and thus weakly lower semicontinuous on E+. Since it is strongly continuous on E−, it is τE+-lower semicontinuous. By (F3), F has quadratic growth. It follows that φ is continuous on Lp, if p is large enough. By Proposition 3, φ is weakly continuous on E, and therefore, it is also τE+-continuous. We conclude that J is τE+-lower semicontinuous.
By (F3), φx≥−C1∥x∥L22−C2.Since E embeds compactly into L2, for every ε>0, we can find a finite codimensional subspace V of E+ such that
(67)xL22≤εx2,∀x∈V.
Then,(68)Jx=12PV+x2−12PV−x2+φx≥12PV+x2−12PV−x2−C1xL22−C2≥12PVx2+12PV⊥V+x2−12PV−x2−C1εPVx2+PV⊥x2−C2≥12−C1εPVx2−12+C1PV⊥x2−C2.
SinceV is finite codimensional in V+, Lemma 33 implies that the right side of (68) has E+-locally compact sublevels. Then, J also has E+-locally compact sublevels.Lemma 35.
Assume thatF satisfies (F2) and (F3). Then, all the bounded (PS) sequences are precompact. So (A2) holds.Proof.
Assume thatxn⊂E is a bounded (PS) sequence. The boundedness of xn yields that there exists a subsequence xnk which converges weakly to some x0.
According to Lemma4, φ′ is completely continuous. Then, there exists a subsequence xnk such that the sequence φ′xnk converges strongly. Since J′xnk=Axnk+φ′xnk⟶0, then Axnk converges strongly. Together with the fact that A is invertible implies that xnk must converge. Thus, (A2) is satisfied.Lemma 36.
Assume thatF satisfies F2 and F3. Then, ∇J is globally Lipschitz, and (A3) holds.Proof.
SinceJ′x−J′y≤Ax−Ay+φ′x−φ′y, we need to only check that the second part in the right side of inequality is globally Lipschitz. By a directly computation, we have
(69)φ′x−φ′y≤supz=1∫02πft,xt−ft,yt,ztdt≤ft,xt−ft,ytL2supz=1ztL2≤M1x−y,where M1 is a positive constant. The last inequality is guaranteed by F3 and Proposition 3. This finishes the proof of this lemma.Lemma 37.
Assume thatF satisfies (F2) and (F3); then, the flow defined by (48) is an E+-homotopy, and (A4) holds.Proof.
By induction, we can define a sequence of flows(70)η0,x=x,ηnt,x=x−∫0tJ′ηn−1s,xds,n≥1.
It is a standard fact in the theory of ordinary differential equations in Banach spaces thatJ′ is locally Lipschitz, and ηn converges uniformly on the bounded subsets of R×E to the solution η:R×E⟶E of the Cauchy problem
(71)∂∂tηt,x=−J′ηt,x,η0,x=x.
SinceJ′ maps bounded sets into bounded sets, so does ηn. Therefore, also η maps bounded sets into bounded sets.
SinceJ′ is τE+-continuous, also ηn is τE+-continuous. Thus, both PE−∘ηn and gynt,x=ηnt,x,y,y∈E are τE+-continuous. Moreover, PE−∘ηn and gyn converge uniformly on bounded sets to PE−∘η and gy, respectively. Therefore, PE−∘η and gy are τE+-continuous, for every y∈E. Then, η is τE+-continuous.
Sinceη solves the following nonhomogeneous equation
(72)∂∂tηt,x+Lηt,x=−φ′ηt,x,it can be represented as
(73)ηt,x=e−tAx−∫0tes−tAφ′ηs,xds.
e−tA is a continuous path of invertible operators which preserve the splitting E=E+⊕E−. Set
(74)Kt,x=−∫0tes−tAφ′ηs,xds.
IfX⊂E is bounded and T>0, ηX×−T,T is bounded, as we have shown before. Therefore, φ′η−T,T×X is τE+-continuous, and we conclude that K−T,T×X is τE+-continuous.
Finally, sinceηt,⋅ is a diffeomorphism whose inverse is ηt,⋅,
(75)η−T,T×E−1X=η−T,T×Xmust be bounded for every bounded X.Proposition 38.
Assume thatF satisfies (F1)-(F3). If the system (8) is 2π-nonresonant at infinity, the critical set K is compact and
(76)BV∞+¯R×BV∞−¯R,BV∞+¯R×∂BV∞−Ris an E+-index pair for K, provided R is large enough.Proof.
Since the system is2π-nonresonant at infinity, the operator A+B∞ is invertible and V∞0=0. Let γ>0 be such that A+B∞≥γI on V∞+ and A+B∞≤−γI on V∞−.
Lemma35 implies that all the bounded (PS) sequences are precompact: in order to prove that K is compact, it is enough to show that it is bounded.
Now we check conditions (55) and (56) of Lemma 30.
Letx∈BV∞+¯R×∂BV∞−R. Then,
(77)ddtP∞−ηt,x2t=0=−2<J′x,P∞−x>=−2A+B∞P∞−x,P∞−x−2ϕ′x,P∞−x≥2γ∥P∞−x∥2−2ϕ′x,P∞−x.
By (F3), there existsr such that
(78)∣gt,x∣≤γ4∣x∣if∣x∣≥r.
LetM2>0 be such that
(79)∣gt,x∣≤M2if∣x∣≤r.
Then,(80)ϕ′x,y≤∫t∣∣xt∣≥rgt,xytdt+∫t∣∣xt∣≤rgt,xytdt≤γ4xL2yL2+M2yL1≤γ4xy+M2y.
Therefore,(81)ddtP∞−ηt,x2t=0≥2γP∞−x2−2ϕ′x,P∞−x≥2γP∞−x2−γ2xP∞−x−2M2P∞−x≥3γ2P∞−x2−2M2P∞−x−γ2P∞−xP∞+x≥γR2−2M2R,which is positive when R is large enough. A similar discussion can prove (56). Then, we can use Lemma 30.Proof of Theorem 39.
Notice that (F3) and (F4) imply that F has quadratic growth. Therefore, Lemmas 34–37 show that we can apply the E+-Conley theory to the functional J.
By Proposition38, if R is large enough, the pair
(82)BV∞+¯R¯×BV∞−¯R¯,BV∞+¯R¯×∂BV∞−R¯is an E+-index pair for the critical set of J, which is compact. Then, Theorem 18 is
(83)∑i=1kλE+−mxi;J=PE+X,A+1+λQλ.
Then, both (48) and Definition 10 imply
(84)λE+−mxi;J=λE+−dimW−,PE+X,A=λE+−dimV∞−.
Theorem1 follows from the above two equalities.


## 5. Systems with Resonant at Infinity

In this section, we want to study the existence of periodic solutions of asymptotically linear systems which is resonant at infinity.Since we consider resonant system in this section, the variational functional does not satisfy (PS) condition. So we first make a perturbation. We prove that the perturbable functional satisfies (PS) condition. Since critical points of first perturbable functional may not to be nondegenerate, we make a second perturbation and critical points of the secondly perturbable functional are nondegenerate. Finally, we prove the critical points of the secondly perturbable functional are the critical points of the original functional.Letθ∈C∞R be a nondecreasing function such that θs=0 for s≤0 and θs=s for s≥1. For R>∥x0∥, we define two new functionals on E, JR+ and JR−, of class of C2 as
(85)JR±x=Jx±θP∞0x2−R2.The gradients of these functionals are(86)JR±′x=A+B∞x+ϕ′x±2θ′P∞0x2−R2P∞0x.Critical pointsx of JR± such that ∥P∞0x∥≤R are also critical points of J, and thus 2π-periodic solutions of system (8). Let us check that the perturbed functionals JR± satisfy the (PS) condition.Lemma 40.
For everyR>0, every sequence xn⊂E such that JR±′xn⟶0 contains a convergent subsequence.Proof.
Letxn=xn++xn0++xn−∈V∞+⊕V∞0⊕V∞−. Since the proofs of two cases are the same, we only prove this lemma for JR+. Multiplying both sides of (86) by x+ and integrating over 0,2π, we have
(87)JR+′xn,xn+=A+B∞xn,xn++ϕ′xn,xn++2θ′P∞0xn2−R2P∞0xn,xn+.
By (F6) and Proposition 3, there exist two positive constants M and a1 such that
(88)∣ϕ′xn∣≤M3,xL1≤a1x,∀x∈E.
According to the assumption, there are small enoughε>0 such that
(89)εxn+≥γxn+2−M3a1xn+,which implies that the sequence xn+ is bounded. Arguing similarly, we can prove that the sequence xn− is bounded.
Next, we prove the sequencexn0 is bounded. Since
(90)o1∥xn0∥=JR+′xn,xn0=A+B∞xn,xn0+ϕ′xn,xn0+2θ′∥P∞0xn∥2−R2P∞0xn,xn0=ϕ′xn,xn0+2θ′P∞0xn2−R2xn02.
Suppose thatθ⟶+∞ as n⟶∞. For large n, we have θ′∥P∞0xn∥2−R2=1. It follows that
(91)o1xn0=<ϕ′xn,xn0>+2xn02.
Subsequently, we have(92)2xn02=o1xn0−ϕ′xn,xn0≤o1xn0+ϕ′xn,xn0≤o1xn0+M3a1xn0,which contradicts with the fact that xn0 is unbounded. Thus, the sequence xn0 is bounded and so is xn. Up to subsequence, it converges weakly to x0. Standard arguments show that this convergence is strong.Now we can check that critical points ofJR+′xn have priori boundedness in V∞+ and V∞−.Lemma 41.
There existsN>0, independent of R, such that, for every critical point x of JR±′, we have ∥P∞±x∥≤N.Proof.
Letx be a critical point of JR+′. Set x=x++x0+x−∈V∞+⊕V∞0⊕V∞−. Multiplying both sides of (86) by x+ and integrating over 0,2π, we have
(93)A+B∞x,x++ϕ′x,x+=0.
So(94)γ∥x+∥2≤A+B∞x,x+=ϕ′x,x+≤M3a1∥x+∥,and thus, ∥x+∥ must be bounded. Similarly, we can prove the boundedness of ∥x−∥.Lemma 42.
Assume that (F1)-(F3), (F5), and (F6) hold. For every N>0, there exists Q>0, independent of R, such that the following property holds: for every critical point x of JR±′, if ∥P∞+x∥≤N, ∥P∞−x∥≤N, and ∥P∞0x∥≥Q, we have
(95)E+−dimVx−≥E+−dimV∞−,E+−dimVx−⊕Vx0≤E+−dimV∞−⊕V∞0,where Vx0 and Vx− are the kernel and negative eigenspace of D2JR±x.Proof.
We firstly prove the second inequality of this lemma. Ify∈V∞+, we have
(96)JR+′′xy,y=J′′xy,y+2θ′P∞0x2−R2P∞0y,y+4θ′′P∞0x2−R2P∞0x,y2=J′′xy,y=A−B∞y,y−∫02πG′′t,xy,ydt≥γy2−∫02πG′′t,xy,ydt.
We claim that∫02πG′′t,xy,ydt<γ∥y∥2. By (F5) and (F6), there exist M4>0, large enough r>0 and small enough ε>0 such that
(97)∣G′′t,x∣≤M4,∀t,x∈0,2π×ℝn,∣G′′t,x∣≤ε,∀∣x∣>r.
SetΓ1=t∈0,2π∣∣xt∣≤r. By Proposition 3, there exists ai>0 such that yLi2≤aiy2, where ⋅Li denotes the Li-norm and i=2,4. Computing directly, we obtain
(98)∫02πG′′t,xy,ydt≤εa2y2−M4a4y2measΓ11/2.
By assumptions∥P∞+x+P∞−x∥≤2N. Let s>0 be such that for every x∈V∞+⊕V∞− with ,∥x∥≤2N, there holds
(99)meast∈0,2π∣∣xt∣≥s≤ε2.
Then,(100)measΓ1≤meast∈0,2π∣∣P∞+xt+P∞−xt∣≥s+meast∈0,2π∣∣P∞0xt∣≤r+s.
The subspaceV∞0 is finite dimensional. Subsequently, there exists q>0 such that
(101)∣yt∣>r+s,∀y∈V∞0suchthaty∞≥q.
Since⋅∞ and ⋅ are equivalent on V∞0, there exists l>1 such that
(102)1ly∞≤y≤ly∞,∀y∈V∞0.
Denote byQ=lq. If ∥P∞0x∥≥Q, then ∥P∞0x∥∞≥q. It follows that
(103)measΓ11/2≤ε.
Substituting (103) into (98), we get
(104)∫02πG′′t,xy,ydt≤εa2+M4a4y2.
The arbitrary ofε induces that ∫02πG′′t,xy,ydt≤γ∥y∥2. Therefore, if x is a critical point of JR+′x, then for all y∈V∞+\0, we have
(105)JR+′′xy,y>0.
Subsequently,JR+′′x is strictly positive on V∞+. Therefore, E+−dimVx−⊕Vx0≤E+−dimV∞−⊕V∞0.
Repeating the above argument, we conclude thatJR−′′x is strictly negative on V∞−. Thus, E+−dimVx−≥E+−dimV∞−.Lemma 43.
The mapsJR+′ and JR−′ are proper Fredholm maps of index 0.Proof.
Notice thatJR±′ can be written as
(106)JR±′x=A+B∞±2P∞0x+ϕ′x±2θ′P∞0x2−R2−1P∞0x.
The first term is an invertible linear operator. The last two terms are compact operators. Assuming thatJR±′xn converges, it is easy to prove that xn has a converging subsequence. Thus, JR±′ are proper operator. Moreover, the differentials of the last two terms are compact self-adjoint operators. Therefore, D2JR± is a Fredholm operator of index 0, for every u∈E.Consider the constantQ, fixed in Lemma 42. We assume, by contradiction, that the functionals JQ±′ have no critical points x such that ∥P∞0x∥≤Q, apart from x0.According to Lemma43, the critical set of JQ± is compact. Since JQ± has no critical points x such that ∥P∞0x∥=Q, there exists ε>0 such that there are no critical points x such that
(107)Q≤P∞0x≤Q+ε.As we can see, the perturbable functionals satisfy (PS) condition. However, the critical points ofJR±x maybe not to be a nondegenerate set. Thus, we need a second perturbation to grantee the critical points of perturbable functional are nondegenerate.Letω~∈C∞R be a nondecreasing function such that ω~s=0 for s≤Q2 and ω~s=1 for s≥Q+ε2. Set ωx=ω~∥P∞0x∥2 for all x∈E. ω is a smooth functional on E. ω′x always belongs to V∞0, and there exists a constant D such that ∥ω′x∥≤D for all x∈E. For z∈E, we define two new functional hz± on E as
(108)hz±x=JQ±x−ωx<x,z>.The gradients of these functional are(109)hz±′x=JQ±′x−ω′x<x,z>−ωxz.Lemma 44.
If∥z∥ is small enough, for every critical point x of hz±, there holds
(110)∥P∞±x∥≤N=2M3γ.Proof.
From (86) and (109), we get
(111)0=dhz±xP∞+x=A+B∞P∞+x,P∞+x+dϕxP∞+x−ωxz,P∞+x≥γ∥P∞+x∥2−M3∥P∞+x∥−∥z∥∥P∞+x∥.
Therefore,(112)∥P∞+x∥≤M3+∥z∥γ,and it is enough to assume ∥z∥≤M3. By a similar argument, we estimate the bound for ∥P∞−x∥.Lemma 45.
If∥z∥ is small enough, there are no critical points x of hz± such that
(113)Q≤∥P∞0x∥≤Q+ε.Proof.
By Lemma44, we can assume that ∥x∥ is small enough that
(114)∥P∞±x∥≤N,for every critical point x of hz+.
SinceJQ+ has no critical point x with Q≤∥P∞0x∥≤Q+ε, by Lemma 43, there exists m>0 such that
(115)∥JQ+′x∥≥mifQ≤∥P∞0x∥≤Q+ε.
Therefore, ifx is a critical point of hz+ with Q≤∥P∞0x∥≤Q+ε, we have
(116)0=hz±′x=JQ+′x−ω′xx,z−ωxz≥JQ+′x−zω′xx+ωx≥m−zD2N+Q+ε+1.
This implies that(117)z≥mD2N+Q+ε+1,which is contradiction, provided ∥z∥ is small enough.Now we can work with second perturbation functionals. Firstly, we check some proposition of those functionals.Lemma 46.
Every bounded (PS) sequence forhz± admits a converging subsequence; thus, (A2) holds.Proof.
Letxn be a bounded (PS) sequence for Hz±. Up to its subsequence, it converges weakly. Since ω is weakly continuous and ω′ is completely continuous, by (109), JQ+′ converges strongly. By Lemma 43, xn converges strongly.Lemma 47.
The functionalshz± satisfy the assumption (A1), (A3), and (A4).Proof.
Since(118)hz±x=Jx±θ∥P∞0x∥2−R2±ωxx,y,
Lemma34 implies that J is τE+-lower semicontinuous. The last two terms are weakly continuous and thus also τE+-continuous. So the sublevels of hz± are τE+-closed. Arguing similarly as in the proof of Lemma 34, the sublevels of hz± are E+-locally compact and (A1) holds.
Computing directly, we obtain(119)hz±′x=J′x±2θ′∥P∞0x∥2−Q2P∞0x±ω′xx,z±.
According to Lemma36, J′ is globally Lipschitz. Since θ and ω are both C∞ functions, the second and third items are globally Lipschitz. We need only the last item is globally Lipschitz. But,
(120)∥ωxz−ωx¯z∥≤D∥z∥∥∥x−x¯∥,which yields that ω is globally Lipschitz when z is fixed. Thus, (A3) holds.
Sincehz± can be seen as the sum of the quadratic form 1/2A−B∞x,x and of a function which is continuous from the weak to the strong topology of E, then (A4) follows from Proposition 15.1 in [28].Now we are in a position to useE+-Conley index theory. We claim that the critical points of hz± are compact and prove neighborhoods of zero are their E+-Conley index pairs.Lemma 48.
The critical setK+ of hz+ is compact, and
(121)BV∞+⊕V∞0R×BV∞−R,BV∞+⊕V∞0R×∂BV∞−Ris an E+-index pair for K+, with respect to the functional hz+, provided R is large enough and ∥y∥ is small enough. The same conclusion also works where the E+-index pair is replaced by
(122)BV∞+R×BV∞−⊕V∞0R,BV∞+R×∂BV∞−⊕V∞0R.Proof.
We just prove the first claim, since the second one can be proved with analogous arguments.
Letη+ be the flow of the vector field −hz+′. We use Lemma 30 to prove this lemma. In order to use Lemma 30, we should check conditions (55) and (56).
Letx∈BV∞+⊕V∞0R×∂BV∞−R. Then,
(123)ddt∥P∞−η+t,x∥2t=0=−2hz+′x,P∞−x=−2A+B∞P∞−x,P∞−x−2ϕ′x,P∞−x−ωxy,P∞−x≥2γ∥P∞−x∥2−2C+∥y∥∥P∞−x∥=R2γR−2C−∥y∥.
Then, (55) holds when R>2C−∥y∥/2γ.
Now letx∈∂BV∞+⊕V∞0R×BV∞−R. Then,
(124)ddt∥P∞++P∞0η+t,x∥2t=0=−2hz+′x,P∞++P∞0x=−2A+B∞P∞+x,P∞+x−2ϕ′x,P∞++P∞0x−4θ′∥P∞0x∥2−Q2∥P∞0x∥2−ωxz,P∞++P∞0x−ω′x,P∞++P∞0xx,z≤−2γ∥P∞+x∥2−4θ′∥P∞0x∥2−Q2∥P∞0x∥2+2C+∥z∥∥P∞++P∞0x∥+D∥z∥∥x∥∥P∞0x∥.
Since∥x∥≤2R and ∥P∞++P∞0x∥=R, we have that
(125)∥P∞+x∥2≥12R2or∥P∞0x∥2≥12R2.
In the first case, noticing that∥y∥ is small enough, we have
(126)ddtP∞++P∞0η+t,x2t=0≤−γR2+2CR=−RγR−2C,which is negative for R>2C/γ. In the second case, we can assume that ∥P∞0x∥2≥1/2R2>Q2+1>Q2. So θ′∥P∞0x∥2−Q2=1. We have
(127)ddtP∞++P∞0η+t,x2t=0≤−2R2+2C+zR+2DzR2=−R2−2DzR−2C−z,
which is negative if(128)∥z∥<22D,R>2C+∥z∥2−2D∥z∥.
Then, (56) holds. Applying Lemma 30, we prove this lemma.Proof of Theorem 49.
We claim that we can choosez0 and z1, such that ∥z0∥ and ∥z1∥ are so small that these of Lemmas 44, 45, and 48 hold and such that all critical points of hz± are nondegenerate.
Since we assume the system is2π-nonresonant periodic solution, the critical point x0 is nondegenerate. Lemma 45 implies all the other critical points x such that ∥P∞0x∥>Q+ε. If y is in a neighborhood of such a critical point x, then
(129)hz±′y=JQ±′y−z.
Therefore,(130)D2hz±y=D2JQ±y.
JQ±′ is a continuously differentiable Fredholm map of index 0. By an infinite dimensional Sard-Smale theorem, the set of its critical values has first category. Therefore, we can choose z0 and z1, such that ∥z0∥ and ∥z1∥ are so small that the theses of Lemmas 44, 45, and 48 hold and such that y+ and y− are regular values for JQ+′ and JQ−′, respectively.
Ifx is a critical point of hz0+ different from x0, then
(131)JQ±′y=z0,and the linear map D2hz0+x=D2JQ+x is invertible. The same result holds for hz1−.
Lemmas46 and 47 imply that we can use E+-Conley index theory and get the following Morse relations:
(132)λE+−mx0,hz0++∑μλE+−mx,hz0+=λE+−dimV∞−+1+λQλ,where the sum is taken over the finite set of all critical points x of hz0+ such that ∥P∞0x∥>Q. If E+−dimWx0<E+−dimV∞−−1, there is existence of a critical point x′ such that
(133)∣E+−mx′,hz0+−E+−mx0,hz0+∣=1.
Therefore,(134)E+−mx′,hz0+≤E+−mx0,hz0++1<E+−dimV∞−.
But by Lemmas42 and 44, every critical point x has E+-Morse index E+−mx,hz0+≥E+−dimV∞− which is a contradiction.
IfE+−mx,hz1−>E+−dimV∞−+E+−dimV∞0+1, arguing similarly as the previews case can result a contradiction which finishes our proof.



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*Source: 3396716-2022-04-20.xml*