# Estimate of Number of Periodic Solutions of Second-Order Asymptotically Linear Difference System

Authors: Honghua Bin; Zhenkun Huang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We investigate the number of periodic solutions of second-order asymptotically linear difference system. The main tools are Morse theory and twist number, and the discussion in this paper is divided into three cases. As the system is resonant at infinity, we use perturbation method to study the compactness condition of functional. We obtain some new results concerning  the lower bounds of the nonconstant periodic solutions for discrete system.


---

## Body

## 1. Introduction

In this paper we are interested in the lower bound of the number of periodic solutions for second-order autonomous difference system(1)Δ2xn-1+f(xn)=0,n∈ℤ,
where xn∈ℝN, f=(f1,f2,…,fN)T∈C1(ℝN,ℝN), Δxn=xn+1-xn,Δ2xn=Δ(Δxn), and N is a fixed positive integer.Discrete systems have been investigated by many authors using various methods, and many interesting results have obtained; see [1–7] and references therein. The critical point theory [8, 9] is a useful tool to investigate differential equations, which is developed to study difference equations. Using minimax methods in critical point theory, Guo and Yu [10, 11] investigated the existence of periodic and subharmonic solutions of system (1), where nonlinearity f is either sublinear or superlinear. In this paper, we assume that(P1)
there exist a functiong∈C1(ℝN,ℝN) and a N×N symmetric matrix A∞ such that f(xn)=A∞xn+g(xn), xn∈ℝN, and
(2)|g(xn)|=o(|xn|)as|xn|⟶∞,where |·| denotes the usual norm in ℝN. Moreover there exist functions F, G such that F′(xn)=(∂F/∂xn1,…,∂F/∂xnN)T=f(xn)=(f1,f2,…,fN)T, G′(xn)=g(xn), where ′ denotes the gradient of function.System (1) can be regarded as discrete analogous of the following differential system:
(3)-Δu=f(u).
A great deal of research has been devoted to (3). For example, by using minimax theory, Rabinowitz [12] has given some interesting results, and Mawhin and Willem [9] obtained some results using the critical point theory. Moreover, there is a vast literature on the problems concerning periodic solutions, BVP, asymptotically behavior of solutions, and so forth.Morse theory [8, 9, 13–16] has been used to solve the asymptotically linear problem. Chang [17], Amann and Zehnder [18] obtained the existence of three distinct solutions via Morse theory, where (3) was nonresonant at infinity. Moreover, the resonant case has been considered in [19–23]. The estimate of number of periodic solutions of (3) was established in [24]. Motivated by [24], we will use Morse theory to consider the lower bound of number of periodic solutions for system (1).Throughout this paper we employ some standard notations. Denote byℝ,ℤ the real number and the integer sets, respectively. ℝN is the real space with dimension N. Z[a,b]={a,a+1,…,b} if a≤b and a,b∈ℤ. AT or xT denotes the transpose of matrix A or vector x.Ifg(t) and G(t) are bounded on ℝN, and system (1) is p-resonant at ∞, then functional J does not satisfy the compactness condition of the Palais-Smale type. Therefore our discussion will be divided into three cases. Moreover, we assume that(P2)
J has a finite number of nondegenerated critical points;(P3)
allp-periodic solutions of system (1) are not p-resonant;(P4)
form∈Z[0,r], σ(A∞)⊂(λm,λm+1], where λm=4sin2(mπ/p) and r=[p/2].Now we state the main results as follows.Theorem 1.
Assume that (P1)–(P4) hold, and system (1) is not p-resonant at ∞. Then
(4)n(p)≥12Θp-h(pN+12)+12,
where n(p) is the number of the nonconstant p-periodic solutions of system (1), Θ is the global twist number (see (32)), and h will be defined in Section 3.Theorem 2.
Assume that (P1)–(P4) hold, system (1) is p-resonant at ∞, and g(t) is bounded in ℝN, lim|t|→+∞G(t)=-∞. Then (4) is valid.Theorem 3.
Assume that (P1)–(P4) hold, system (1) is p-resonant at ∞, and g(t),G(t) are bounded in ℝN. Then
(5)n(p)≥12Θp-h(pN+1).Remark 4.
Benci and Fortunato [24] studied asymptotically linear equation (3). Theorem 1 extends and generalizes the analogous results in [24], and Theorems 2-3 are new results.The organization of this paper is organized as follows. In Section2 we study the compactness condition for functional J. Some facts about Morse theory and necessary preliminaries are given in Section 3. In Section 4 the main results are proved.


## 2. (PS) Condition

We say that aC1-functional ϕ on Hilbert space X satisfies the Palais-Smale (PS) condition, if every sequence {x(j)} in X such that {ϕ(x(j))} is bounded and ϕ′(x(j))→0 as j→∞, contains a convergent subsequence.Here we first introduce spaceEp.LetEp={x={xn}∈S∣xn+p=xn,n∈ℤ}, where S={x={xn}∣xn∈ℝN,n∈ℤ}. For any x,y∈S,a,b∈ℝ,ax+by={axn+byn}n∈ℤ. Then S is a linear space. Let Ep equip with inner product and norm as follows:
(6)〈x,y〉=∑n=1p‍(xn,yn),∥x∥=(∑n=1p‍|xn|2)1/2,∀x,y∈Ep,
where (·,·) and |·| are the usual inner product and norm in ℝN, respectively. Obviously, Ep is a Hilbert space with dimension pN and homeomorphism to ℝpN.By the variational method, thep-periodic solutions of (1) are same as the critical points of the C2-functional (7)J(x)=∑n=1p‍[12|Δxn|2-F(xn)],x∈Ep.
By assumption (P1), the functional J can be rewritten as
(8)J(x)=12∑n=1p‍[|Δxn|2-(A∞xn,xn)]-∑n=1p‍G(xn),
and we write I(x)=∑n=1p[|Δxn|2-(A∞xn,xn)].Consider eigenvalue problem(9)-Δ2xn-1=λxn,xn+p=xn,xn∈ℝN,
that is, xn+1+(λ-2)xn+xn-1=0, xn+p=xn. By the periodicity, the difference system has complexity solution xn=einθc for c∈ℂN, where θ=2kπ/p, k∈ℤ. Moreover, λ=2-e-iθ-eiθ=2(1-cosθ)=4sin2(kπ/p).Letηk denote the real eigenvector corresponding to the eigenvalues λk=4sin2(kπ/p), k∈Z[0,r], and r=[p/2], where [·] stands for the greatest-integer function. In terms of eigenvalue λm=4sin2(mπ/p) for some m∈Z[0,r], we can split space Ep as follows:
(10)Ep=W-⊕W0⊕W+,
where
(11)W-=span{ηk∣k∈Z[0,m-1]},W0=span{ηm},W+=span{ηk∣k∈Z[m+1,r]}.
Moreover, there exists δ>0 such that
(12)I(u)≥δ∥u∥2foru∈W+,I(v)≤-δ∥v∥2forv∈W-,I(w)=0forw∈W0.Let us recall the definition of resonance (see [24]).Ap-periodic solution {xn} of (1) is called p-resonance, if there exists λk=4sin2(kπ/p)∈σ(F′′(xn)), where F′′ denotes the Hessian matrix of F and σ(·) is the spectrum of matrix. We say that (1) is p-resonant at ∞, if there exists λk=4sin2(kπ/p)∈σ(A∞).Lemma 5.
Assume that (P1) and (P4) hold, and system (1) is not p-resonant at ∞. Then functional J (see (8)) satisfies the (PS) condition.Proof.
Let{x(j)}⊂Ep be the (PS) sequence for functional J; that is, {J(x(j))} is bounded, and J′(x(j))→0 as j→∞. Therefore, for any φ∈Ep, we have
(13)〈J′(x(j)),φ〉=o(∥φ∥)asj⟶∞.
ByW0={0}, we write x(j)=u(j)+v(j) with u(j)∈W+,v(j)∈W-. To show that J satisfies (PS) condition, it is enough to prove that {x(j)} is bounded in Ep. That is, we need only to prove that {u(j)} and {v(j)} are bounded in Ep. By contradiction, without loss of generality, there exists k∈Z[1,p] such that
(14)|xn(j)|⟶∞asj⟶∞forn∈Z[1,k],xn(j)areboundedforn∈Z[k+1,p].
Therefore, for all n∈Z[1,p], by assumption (P1), there exist ɛ>0 and c1>0 such that
(15)|G(xn(j))|≤ɛ|xn(j)|2+c1,|g(xn(j))|≤ɛ|xn(j)|+c1
for large j. Thus there is c>0, |∑n=1p(g(xn(j)),xn(j))|≤∑n=1p|g(xn(j))||xn(j)|≤ɛ∥x(j)∥2+c∥x(j)∥. Taking φ=u(j)-v(j) in (13), by previous argument,
(16)o(∥u(j)∥+∥v(j)∥)=〈J′(x(j)),u(j)-v(j)〉=I(u(j))-I(v(j))-∑n=1p‍(g(xn(j)),un(j)-vn(j))≥δ(∥u(j)∥2+∥v(j)∥2)-ɛ(∥u(j)∥2+∥v(j)∥2)-c(∥u(j)∥+∥v(j)∥),
it follows a contradiction. Therefore {u(j)} and {v(j)} are bounded in Ep. This completes the proof.Here and in the sequel, the letterδ will be indiscriminately used to denote various positive constants whose exact values are irrelevant, and ɛ∈(0,1) is arbitrarily small. Moreover we also denote by c the various positive constants in this paper.Lemma 6.
Assume that (P1) and (P4) hold. System (1) is p-resonant at ∞, g(t) is bounded in ℝN, and lim|t|→+∞G(t)=-∞. Then J satisfies the (PS) condition.Proof.
Let{x(j)}⊂Ep be the (PS) sequence for functional J; that is, {J(x(j))} is bounded, and J′(x(j))→0 as j→∞.
Since system (1) is p-resonant at ∞, W0≠{0}. Similarly, let x(j)=u(j)+v(j)+w(j) with u(j)∈W+, v(j)∈W-, andw(j)∈W0. By the same method as proof of Lemma 5, it also follows that {u(j)} and {v(j)} are bounded in Ep. Next we prove that {w(j)} is bounded in Ep.
J
(
x
(
j
)
)
=
(
1
/
2
)
I
(
u
(
j
)
)
+
(
1
/
2
)
I
(
v
(
j
)
)
-
∑
n
=
1
p
G
(
x
n
(
j
)
)
, by {u(j)}, {v(j)}, and J(x(j)) are bounded in Ep, and it follows that ∑n=1pG(xn(j)) is bounded. On the other hand, |∑n=1pG(xn(j))-∑n=1pG(wn(j))|≤supt∈Ep∥g(t)∥(∥u(j)∥+∥v(j)∥), so ∑n=1pG(wn(j)) is bounded. It is easy to see from assumption lim|t|→+∞G(t)=-∞ that {w(j)} is bounded. The proof is completed.If we assume thatG(t),g(t) are bounded and system (1) is p-resonant at ∞, then functional J does not satisfy the (PS) condition. In order to overcome the difficult arising from the lack of compactness condition, we use a suitable penalization technique (one can refer to [20, 24]) and add a perturbation term to the functional J. Define
(17)φR(t)={(t-R)4,ift>R,0,ift≤R,
where R is a positive real number and the penalized functional is given by
(18)JR(x)=J(x)+φR(∥w∥2),
where x=u+v+w∈W+⊕W-⊕W0. Obviously, if x∈Ep is a critical point of JR with ∥w∥2≤R, then x is also the critical point of J.Lemma 7.
Assume that (P1) and (P4) hold,G(t),g(t) are bounded in ℝN, and system (1) is p-resonant at ∞. Then JR satisfies the (PS) condition. Moreover, for any critical point x of JR, there exists M>0 such that ∥u+v∥≤M, where x=u+v+w∈Ep,u∈W+,v∈W-, andw∈W0.Proof.
Let{x(j)}⊂Ep be the (PS) sequence for functional JR; that is, {JR(x(j))} is bounded in Ep, and for any φ∈Ep,
(19)〈JR′(x(j)),φ〉=o(∥φ∥)asj⟶∞.
Similarly to the proof of Lemma 5, we need only to prove that {w(j)} is bounded in Ep.
Takingφ=w(j) in (19), it follows that o(∥w(j)∥)=〈JR′(x(j)),w(j)〉≥-c∥w∥+2∥w∥2φR′(∥w∥2). By the definition of φR, it follows that {w(j)} is bounded. Therefore the penalized functional JR satisfies the (PS) condition.
Letx be the critical point of JR, then
(20)0=〈JR′(x),u-v〉=I(u)-I(v)-∑n=1p‍(g(xn),un-vn)≥δ∥u+v∥2-ɛ∥u+v∥2-c∥u+v∥.
So there is a M>0 such that ∥u+v∥≤M, and the proof is completed.


## 3. Preliminaries

LetE be a real Hilbert space, and let ϕ be a C2-functional on E. We denote by crit(ϕ)={x∈E∣ϕ′(x)=0} the set of critical points of ϕ, ϕc={x∈E∣ϕ(x)≤c} the level set of ϕ, and ϕab={x∈E∣a≤ϕ(x)≤b}. In the following we suppose that ϕ is a C2-functional on E which satisfies the (PS) condition.Definition 8 (see [9, 14]).
Letx be a critical point of ϕ. The Morse index of x by m(x,ϕ) is defined as the supremum of the dimensions of the vector subspace of E on which ϕ″(x) is negative definite. The nullity of x by ν(x,ϕ) is defined as the dimension of Kerϕ″(x). A critical point x will be said to be nondegenerate if ϕ″(x) is invertible.
Denote bym∞, ν∞ the Morse index and nullity of ∞ for functional J. By (10), m∞=dimW-,ν∞=dimW0.
A setK⊂E is called critical set if K⊂ϕ-1(c)∩crit(ϕ) for some c∈ℝ. A critical set K is called discrete nondegenerate critical manifold, if K is connected and m(x,ϕ) does not depend on x∈K.Definition 9.
The Poincare polynomial of the pair(ϕb,ϕa) is defined by Pλ(ϕb,ϕa)=∑n=0∞dimHn(ϕb,ϕa;Γ)λn, where Hn(ϕb,ϕa;Γ) denotes the nth singular relative homology of the pair (ϕb,ϕa) with coefficients in field Γ. Define the topological Morse index of critical set K as iλ(K)=∑n=0∞dimHn(ϕc,ϕc∖K;Γ)λn.
For simplicity, we writem(x) and m(K) instead of m(x,ϕ) and m(K,ϕ), respectively. It is well known that if x is a nondegenerate critical point and m(x) is finite, then iλ(x)=λm(x). If K is a nondegenerate critical manifold and m(K) is finite, then iλ(K)=λm(K)Q(λ), where Q(λ) is a polynomial with nonnegative integer coefficients (see [13, 15]).Next we investigatePλ(E,ϕa) and use functional J (see (8)) or JR (see (18)) instead of ϕ, Ep instead of E.Lemma 10 (see [19, 24]).
Assume that (P1) and (P4) hold, and system (1) is not p-resonant at ∞. Then there exists a∈ℝ, a<J(crit(J)) such that
(21)Pλ(Ep,Ja)=λm(∞).Lemma 11.
Assume that (P1) and (P4) hold, system (1) is p-resonant at ∞, lim|t|→+∞G(t)=-∞, and g(t) is bounded in ℝN. Then there exists a∈ℝ, and (21) is valid.Proof.
Writex=u+v+w∈Ep with u∈W+,v∈W-,w∈W0. Then there exist M1>0,M2>0 such that 〈J′(x),u〉≥δ∥u∥2-c∥u∥>0 as ∥u∥>M1, 〈J′(x),v〉≤-δ∥v∥2+c∥v∥<0 as ∥v∥>M2. Let BM1={x∈Ep∣∥u∥≤M1}, BM2={x∈Ep∣∥v∥≤M2}. By previous argument, it follows that J has no critical points in Ep∖(BM1∪BM2).
On the other hand, for allx∈BM1∪BM2,
(22)J(x)≥c-∑n=1p‍G(un+vn+∥wn∥·wn∥wn∥)⟶+∞,as∥w∥⟶∞.
Therefore there exists a1∈ℝ, such that a1<J(crit(J)). For x∈BM2, we have
(23)J(x)≥12δ∥u∥2-c-∑n=1p‍G(∥un∥·un∥un∥+vn+∥wn∥·wn∥wn∥),
hence J(x)→+∞ as ∥u+w∥→∞, which implies that J is bounded from the following in BM2. Let a<min{a1,infx∈BM2J(x)}, then Ja⊂Ep∖BM2, and Ja is a strong deformation retraction of Ep∖BM2. By Lemma 6, J satisfies (PS) condition, and we have
(24)Hn(Ep,Ja)≅Hn(Ep,Ep∖BM2)≅Hn(W-,W-∖BM2)≅δn,m(∞)Γ.
So we obtain (21).Lemma 12.
Under the assumption of Theorem3, there exists a∈ℝ such that Pλ(Ep,JRa)=λm(∞).Proof.
Letx=u+v+w∈Ep with u∈W+, v∈W-, and w∈W0. Then there exist R1>R+1 such that all critical points of JR are in BM1∩BM2∩BM3, where BM1 and BM2 are the same as in proof of Lemma 11, and BM3={x∈Ep∣∥w∥2≤R1}. In fact,
(25)〈JR′(x),u〉=〈J′(x),u〉>0,x∉BM1,〈JR′(x),v〉=〈J′(x),v〉<0,x∉BM2,〈JR′(x),w〉≥-c∥w∥+8∥w∥2(∥w∥2-R)3≥8∥w∥2-c∥w∥>0,x∉BM3.
Similarly, forx∈BM2, JR(x)≥(1/2)δ∥u∥2+φR(∥w∥2)-c-∑n=1pG(xn), and JR(x)→+∞ as ∥u+w∥→∞, which implies that JR(x) is bounded from the following in BM2. Let a0=infx∈BM2JR(x). If a<min{a0,JR(crit(JR))}, by JR satisfies (PS) condition, and methods of strong deformation retract, we have Pλ(Ep,JRa)=λm(∞). The proof is completed.Assume that on Hilbert spaceE there is an action of discrete group G, and denote by fix(G) the fixed points set for the G action; that is, fix(G)={x∈E∣gx=x,∀g∈G}. The functional ϕ is called G invariant, if ϕ(gx)=ϕ(x), ∀x∈E, and ∀g∈G. In the following, Zp denotes a cyclic group of p order. In terms of Proposition 8.2 and Proposition  8.5 in [13], we have following lemma.Lemma 13.
Assume thatϕ is a C2-functional on an Hilbert space E and satisfies (PS) condition. Let a,b  (b possible ∞) be two regular values of ϕ. Assume that crit(ϕab)=crit(ϕ)∩ϕ-1(a,b) consists only of critical sets, and then the following Morse relation holds:
(26)∑K⊂crit(ϕab)iλ(K)=Pλ(ϕb,ϕa)+(1+λ)Q(λ),
where Q(λ) is a polynomial with nonnegative integer coefficients. If all the critical points of ϕ in ϕab are nondegenerate and have finite Morse index, then (26) can be written as
(27)∑x∈crit(ϕab)‍λm(x)=Pλ(ϕb,ϕa)+(1+λ)Q(λ).
Now if ϕ is Zp invariant, and crit(ϕab)∩fix(Zp) consists only of nondegenerate critical points having finite Morse index, then (26) becomes
(28)∑x∈crit(ϕab)∩fix(Zp)‍λm(x)+(1+λ)Z(λ)=Pλ(ϕb,ϕa)+(1+λ)Q(λ),
where Z(λ) is a formal series with nonnegative integer coefficients. Moreover if crit(ϕab)-fix(Zp) consists only of nondegenerate critical manifolds having finite Morse index, then
(29)Z(λ)=∑K⊂crit(ϕab)-fix(Zp)‍λm(K).Remark 14.
By (29), our main goal in this paper is to estimate Z(1) which gives a lower bound of the number of the nonconstant critical points of J in Ep.Lemma 15.
Letz={zn} be a critical point of functional J. Denote by τ12,τ22,…,τl2 the positive eigenvalues (repeated according to their multiplicity) of F″(zn), where τj>0,j∈Z[1,l]. Under the assumption (P2), we have ♯(z,J)=l+2∑j=1l[(p/π)arcsin(τj/2)], where [·] denotes the greatest-integer function and ♯(z,J) is the number of eigenvalues λ<0 such that 〈J″(z)u,u〉=λ∥u∥2.Proof.
By〈J″(z)u,u〉=∑n=1p[|Δun|2-(F″(zn)un,un)]=-∑n=1p[(Δ2un-1+F″(zn)un,un)], we consider the equation Δ2yn-1+F″(zn)yn=-λyn, yn+p=yn, where z={zn} is the critical point of J. It is easy to see that λk,j=4sin2(kπ/p)-τj2 are eigenvalues of Δ2yn-1+F″(zn)yn on ℝN, where n,k∈Z[1,p],j∈Z[1,l]. Therefore the number of negative eigenvalues λk,j is just what we are looking for; the proof is completed.Definition 16.
For any critical pointz of J, there are l positive eigenvalues (repeated according to their multiplicity) of F″(zn), which will be denoted by τ12,…,τl2. The number ρ(z)=(2p/π)∑j=1larcsin(τj/2) is called twist number of z. Moreover the twist number of ∞ is defined by ρ(∞)=(2p/π)∑j=1l(∞)arcsin(τj/2), where l(∞) is the number of the positive eigenvalues (repeated according to their multiplicity) of A∞.Letz={zn} be a constant critical point of functional J; that is, z1=z2=⋯=zp. By Lemma 15 and Definition 16, it is easy to deduce the following relation between the Morse index and the twist number as follows:
(30)ρ(z)p-pN≤m(z,J)≤ρ(z)p+pN.In view of the numberl or l(∞) of the positive eigenvalues (repeated according to their multiplicity) of F″(z) or A∞, the constant critical point z is called τ-positive (resp., τ-negative) if l is even (resp., odd). On the contrary, the virtual critical point ∞ is called τ-positive (resp., τ-negative) if l(∞) is odd (resp., even), see [24].We denote byh1 and h2 the number of τ-positive and τ-negative critical points of J. If A∞ is invertible, then h1-h2=(-1)l(∞). Thus, if we consider ∞ as a virtual critical point, we have that the number of τ-positive critical points equals the number of τ-negative critical points. However, if A∞ is singular, the result is not hold in general. If we introduce |h1-h2| virtual critical points having twist number zero, where they are considered as τ-positive if h1<h2 and as τ-negative if h1>h2, then the number of τ-positive critical points is also equal to the number of τ-negative critical points.Leth=max{h1,h2}, which has been used in (4) and (5). We denote by x1,…,xh the τ-positive critical points and by y1,…,yh the τ-negative critical points  such that
(31)ρ(x1)≤⋯≤ρ(xh),ρ(y1)≤⋯≤ρ(yh).
Then the global twist number Θ of the system (1) is defined by
(32)Θ=∑i=1h‍|ρ(xi)-ρ(yi)|.


## 4. Proof of Main Results

Proof of Theorem1.
The argument is analogous to one used by Benci and Fortunato in [24]. Set m(z)=m(z,J). Under the assumption (P2), let z1,…,zn be the nondegenerate constant critical points of J.
By Lemmas5 and 10, functional J satisfies (PS) condition, and there exists sufficiently small a∈ℝ such that Pλ(Jb,Ja)=Pλ(Ep,Ja)=λm(∞), where b=∞. Since J is C2 and Zp invariant functional on Ep, then by assumption (P3), we have ∑i=1nλm(zi)+(1+λ)Z(λ)=λm(∞)+(1+λ)Q(λ); that is,
(33)∑i=1n‍λm(zi)-λm(∞)=(1+λ)(Q(λ)-Z(λ)).
Letmi, fi (i∈Z[1,h]) denote the Morse indices of the τ-positive and τ-negative critical points (including ∞) of J, and without loss of generalities, assume that ∞ is τ-negative. So m(∞)=fj for some j∈Z[1,h], where h is referred to (31). Then (33) becomes
(34)∑i=1,i≠jh‍λmi+λfi1+λ+λmj-λm(∞)1+λ=Q(λ)-Z(λ).
Set Q(λ)=∑sqsλs, Z(λ)=∑szsλs, and B(λ)=Q(λ)-Z(λ)=∑sbsλs, where qs,zs are nonnegative integer and bs=qs-zs.
By Remark14, the lower bound of the number of nonconstant p-periodic solutions for system (1) is to estimate Z(1). Since qs≥0,zs≥0, then
(35)n(p)=∑s‍zs≥∑bs<0‍zs=∑bs<0‍(qs-bs)≥-∑bs<0‍bs.
Let B-=-∑bs<0bs. By (35), we turn our attention to estimate B-.
Ifl is even (resp., odd), by Lemma 15, m(z,J) is also even (resp., odd). Therefore by the definition of τ-positive and τ-negative critical points of J, mi (i∈Z[1,h]) are even numbers, fi are odd numbers for i≠j, i∈Z[1,h], and fj=m(∞) is a even number.
SetM1={r∣mr>fr,r∈Z[1,h],r≠j}, M2={r∣mr<fr,r∈Z[1,h],r≠j}, and
(36)C(λ)=∑r∈M1‍λmr+λfr1+λ,D(λ)=∑r∈M2‍λmr+λfr1+λ,E(λ)=λmj-λfj1+λ.
By (34), we have B(λ)=Q(λ)-Z(λ)=∑sbsλs=C(λ)+D(λ)+E(λ), and
(37)C(λ)=∑r∈M1‍∑i=frmr-1‍cr,iλi,D(λ)=∑r∈M2‍∑i=mrfr-1‍dr,iλi,
where cr,i=(-1)i+1, dr,i=(-1)i. Meanwhile, if mj>fj, E(λ)=∑i=fjmj-1ei,1λi, ei,1=(-1)i+1. If mj<fj, E(λ)=∑i=mjfj-1ei,2λi, ei,2=(-1)i. Clearly E(λ)=0 if mj=fj.
A straight analysis shows thatB-=(1/2)∑r=1h|mr-fr|-((h-1)/2). By (30) and the definition of global twist number that refer to (32), we have n(p)≥(1/2)Θp-h(pN+(1/2))+(1/2). It completes the proof of Theorem 1.Proof of Theorem2.
Under the assumptions of Theorem2, by Lemmas 6 and 11, functional J satisfies (PS) condition, and there exists a∈ℝ such that Pλ(Jb,Ja)=Pλ(Ep,Ja)=λm(∞), for b=∞.
Similarly, we have∑i=1nλm(zi)+(1+λ)Z(λ)=λm(∞)+(1+λ)Q(λ), where zi (i∈Z[1,n]) are nondegenerate critical points of J. The remainder is the same as that of Theorem 1.The following lemma is needed to prove Theorem3.Lemma 17.
If all assumptions in Theorem3 hold, then there exists Q>0 (independent of R) such that
(38)m(x,JR)+ν(x,JR)≤m(∞)+ν(∞),
where x=u+v+w with u∈W+,v∈W-,w∈W0,∥w∥≥Q, and m(x,JR),ν(x,JR) denote the Morse index and nullity of critical point x for functional JR, respectively.Proof.
Letx=u+v+w∈Ep be a critical point of JR. By Lemma 7, we have ∥u+v∥≤M. Therefore ∥x∥→∞ if and only if ∥w∥→∞. Since
(39)〈JR″(x)u,u〉=∑n=1p‍[(g′(xn)un,un)|Δun|2-(A∞un,un)-(g′(xn)un,un)]≥δ∥u∥2-∑n=1p‍(g′(xn)un,un),
by assumption (P1), there exists ɛ∈(0,δ) such that
(40)∑n=1p‍(g′(xn)un,un)=∑n=1p‍(g′(∥w∥·wn∥w∥+un+vn)un,un)≤ɛ∥u∥2,
as ∥w∥→∞. Therefore there exists Q>0 such that 〈JR″(x)u,u〉>0 as ∥w∥≥Q. It follows that relation (38) is valid, and the proof is completed.Proof of Theorem3.
LetL=m(∞)+ν(∞), and denote by z1,…,zn the constant critical points of J. We assume, without loss of generalities, m(zi,J)≥L+1 as i∈Z[1,r], m(zi,J)<L as i∈Z[r+1,n]. Clearly L≤pN.
SetH1={x∈crit(JR)∣∥w∥>Q}, H2={x∈crit(JR)∩fix(Zp)∣∥w∥≤Q}, and H3=crit(JR)-(H1∪H2), where x=u+v+w is the decomposition of x∈Ep with w∈W0 and Q is large enough.
By Lemma17, H1 contains only critical points of JR which have m(x,JR)+ν(x,JR)≤L. H2={zi∣i∈Z[1,n]}, since JR(x)=J(x) as ∥w∥≤Q. H3⊂crit(J)-fix(Zp), since φR(x)=0 when ∥w∥≤Q. Moreover by assumption (P3), H3 contains only nondegenerate critical manifolds.
SinceJR satisfies (PS) condition, by Lemma 13, relation (28) reads
(41)∑x∈H1‍iλ(x)+∑x∈H2‍iλ(x)+∑K⊂H3‍iλ(K)=Pλ(Ep,JRa)+(1+λ)Q(λ),
that is,
(42)∑x∈H1‍iλ(x)+∑i=1r‍iλ(zi)+∑i=r+1n‍iλ(zi)+(1+λ)Z(λ)=λm(∞)+(1+λ)Q(λ),
where Z(λ)=∑K⊂H3λm(K). For Z(λ)=∑i=0∞ziλi, we set Zl=∑i=l∞ziλi, where l∈ℕ. And analogous notation can be introduced for Q(λ). Then, considering the terms of degree ≥L+1 in (42), we have
(43)∑i=1r‍λm(zi,J)+bLλL+1=(1+λ)B(λ),
where bL=zL-qL,B(λ)=QL+1(λ)-ZL+1(λ). Clearly
(44)n(p)=∑s‍zs≥∑bs≤0,s≥L+1‍zs=∑bs≤0,s≥L+1‍(qs-bs)≥-∑bs≤0,s≥L+1‍bs=B-,
that is, B- is the absolute value of the sum of the negative coefficients of B(λ). Next we estimate the number B-.
Letx1,…,xh1 and y1,…,yh2(h1+h2=r) be the τ-positive and τ-negative critical points of J with nonzero twist numbers, whose order satisfies (31), and xi,yj∈{z1,…,zr}, i∈Z[1,h1], j∈Z[1,h2]. Without loss of generalities, assume h1≥h2, and introduce h3(=h1-h2) virtual τ-negative critical points y¯i (i∈Z[1,h3]) having twist number 0 and Morse index 0; that is,
(45)ρ(y¯i)=0,fi=m(y¯i,J)=0,i∈Z[1,h3].
For i∈Z[1,h1], set mi=m(xi,J),fi=m(y¯i,J), where y¯j+h3=yj, j∈Z[1,h2]. Then (43) can be written as ∑i=1h1λmi+∑i=h3+1h1λfi+bLλL+1=(1+λ)B(λ). Setting λ=-1, then bL=-h3 if L is odd, and bL=h3 if L is even. So
(46)B(λ)=∑i=1h3‍λmi+λL+11+λ+∑i=h3+1h1‍λmi+λfi1+λ,ifLiseven,B(λ)=∑i=1h3‍λmi-λL+11+λ+∑i=h3+1h1‍λmi+λfi1+λ,ifLisodd.
A straight analysis shows that B-=∑i=1h3[(1/2)(mi-L)-1]+∑i=h3+1h(1/2)|mi-fi| if L is even, and B-=∑i=1h3(1/2)(mi-L)+∑i=h3+1h(1/2)(|mi-fi|-1) if L is odd. Therefore
(47)B-≥∑i=1h3‍12(mi-L-2)+∑i=h3+1h‍12(|mi-fi|-1).
By (30) and (45), we have
(48)mi=mi-fi≥(ρ(xi)-ρ(y¯i))p-pN,i∈Z[1,h3],|mi-fi|≥|ρ(xi)-ρ(y¯i)|p-2pN,i∈Z[h3+1,h].
In view of (45), (47), and (48), we have
(49)B-≥12∑i=1h‍p|ρ(xi)-ρ(yi)|-12pNh3-12Lh3-pNh2-h3-12h2≥12Θp-h(pN+1).
The proof is completed.Remark 18.
AlthoughA∞ is invertible under the assumptions of Theorem 3, we do not make use of (42) directly, because we consider only the terms of degree ≥L+1 in proof of Theorem 3.



---

*Source: 707686-2013-06-25.xml*