# The Existence of Multiple Periodic Solutions of Nonautonomous Delay Differential Equations

Authors: Huafeng Xiao; Bo Zheng
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

We study the multiplicity of periodic solutions of nonautonomous delay differential equations which are asymptotically linear both at zero and at
infinity. By making use of a theorem of Benci, some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions.


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## Body

## 1. Introduction

The existence and multiplicity of periodic solutions of delay differential equations have received a great deal of attention. In 1962, Jones [1] firstly investigated the existence of periodic solutions to the following scalar equation:(1.1)u′(t)=-au(t-1)[1+u(t)].
By making use of Browder fixed point theorem, the author showed that there exist periodic solutions of (1.1) for each a>π/2. Since then, various fixed point theorems have been used to study the existence of periodic solutions of delay differential equations(cf. [2]). As pointed out in [3], by making change of variable 1+u=ex, (1.1) turns into(1.2)x′(t)=-f(x(t-1)).
In 1974, Kaplan and Yorke [4] studied the following more general form of  (1.2)(1.3)x′(t)=-f(x(t-1))-f(x(t-2))-⋯-f(x(t-n)).
They introduced a technique which translates the problem of the existence of periodic solutions of a scalar delay differential equation to that of the existence of critical points of an associated ordinary differential system. Using this method, they proved that (1.3) has a periodic solution with minimal period 4 (resp., 6) when (1.3) has one delay (resp., two delays). In this direction, Fei, Li and He did some excellent work and got some signification results (cf. [5–8]).Many other approaches, such as coincidence degree theory, the Hopf bifurcation theorem, and the Poincaré-Bendixson theorem, have also been used to study the existence of periodic solutions of delay differential equations (cf. [9, 10]). However, most of those results are concerned with scalar delay equations. In 2005, Guo and Yu [3] studied vector delay differential system (1.2). They built a variational structure for (1.2) on certain suitable spaces. Then they reduced the existence of periodic solutions of (1.2) to that of critical points of an associated variational functional. By making use of pseudoindex theory, they obtained some sufficient conditions to guarantee the existence of multiple periodic solutions.In spite of so many papers on periodic solutions of delay differential equations, there are a quite few researches on nonautonomous case (see for example [11]). The main goal of this paper is to investigate the following nonautonomous system:(1.4)x′(t)=-f(t,x(t-π2)).We assume that(f1) there exists F∈C1([0,π/2]×ℝn,ℝ) such that f is the gradient of F with respect to x, and(1.5)F(t,x)=F(t,-x),F(t+π2,x)=F(t,x),∀(t,x)∈R×Rn,(f2)f(t,x)=B0(t)x+o(|x|) as |x|→0 uniformly for t∈[0,π/2],(f3)f(t,x)=B∞(t)x+o(|x|) as |x|→∞ uniformly for t∈[0,π/2],whereB0,B∞ are n×n symmetric continuous π/2-periodic matrix functions.Hypothesis(f3) is known as asymptotically linear condition at infinity. Hypothesis (f2) is an asymptotically linear condition at zero, which implies that 0 is a trivial solution of (1.4). We are interested in nontrivial periodic solutions of (1.4). Similar to [3], we build a variational structure for (1.4) and convert the existence of periodic solutions to that of critical points of variational functional. Since the asymptotically linear hypothesis at infinity is given by a periodic loop of symmetric matrix, it will be more difficult to deal with more than a constant matrix. However, we can prove the existence of multiple periodic solutions by making use of a multiple critical points theorem of Benci (cf. [12]).The rest of this paper is organized as follows: in Section2, we build the variational functional and state some useful lemmas; in Section 3, the main results will be proved.


## 2. Variational Tools

DenoteS1=ℝ/(2πℤ). The space H=H1/2(S1,ℝn) has been introduced in [3]. The space H can be equipped with inner product as follows: (2.1)〈x,y〉=(a0,c0)+∑j=1∞(1+j)[(aj,cj)+(bj,dj)],
where x=a0/2π+1/π∑j=1∞(ajcosjt+bjsinjt), y=c0/2π+1/π∑j=1∞(cjcosjt+djsinjt), a0,c0∈ℝn, aj,cj,bj,dj∈ℝn, j∈ℕ.Set(2.2)E={x∈H∣x(t+π)=-x(t),∀t∈R}.
Then E is a closed subspace of H. If x∈E, it has Fourier expansion (2.3)x(t)=1π∑j=1∞[ajcos⁡⁡(2j-1)t+bjsin⁡(2j-1)t].Letx∈L2(S1,ℝn). If for every z∈C∞(S1,ℝn)(2.4)∫02π(x(t),z′(t))dt=-∫02π(y(t),z(t))dt,
then y is called a weak derivative of x, denoted by ẋ.The variational functional defined onH, corresponding to (1.4), is(2.5)J(x)=∫02π[12(x(t+π2),ẋ(t))-F(t,x(t))]dt.
Define a linear bounded operator A:H→H by setting(2.6)〈Ax,y〉=∫02π(x(t+π2),ẏ(t))dt.
It is easy to prove that E is an invariant subspace of H with respect to A and A is self-adjoint if it is restricted to E.Lemma 2.1 (see [3]).
The essential spectrum of the operatorA restricted to E is just {2,-2}.Define(2.7)φ(x)=-∫02πF(t,x(t))dt,∀x∈H.
Then J can be rewritten as(2.8)J(x)=12〈Ax,x〉+φ(x),∀x∈H.Similar to the argument as in [3], we can prove the following two basic lemmas.Lemma 2.2.
Assume thatf satisfies (f1)–(f3). Then J is continuous differentiable on H and
(2.9)〈J′(x),h〉=∫02π[12(ẋ(t-π2)-ẋ(t+π2),h(t))-(f(t,x(t)),h(t))]dt,∀h∈H.
Moreover, φ′:H→H* is a compact mapping defined as follows:
(2.10)〈φ′(x),h〉=-∫02π(f(t,x(t)),h(t))dt,∀x,h∈H.Lemma 2.3.
The existence of2π-periodic solutions of (1.4) belonging to E is equivalent to the existence of critical points of functional J restricted to E.Lemma2.3 implies that we can restrict our discussion on space E. At the end of this section, we recall a useful embedding theorem.Lemma 2.4 (see [13]).
For everyp∈[1,+∞), H is compactly embedded into the Banach space Lp(S1,ℝn). In particular, there is an αp such that
(2.11)‖x‖Lp≤αp‖x‖,∀x∈H.Remark 2.5.
Here and hereafter,αp(p∈[1,∞)) denotes the real number satisfying (2.11).


## 3. Main Results

LetB(t) be an n×n symmetric continuous π/2-periodic matrix function. We define a bounded self-adjoint linear operator B∈L(E) by extending the bilinear forms(3.1)〈Bx,y〉=∫02π(B(t)x(t),y(t))dt,∀x,y∈E.
It is well known that B is compact (cf. [14]).Denote byB0,B∞ the operators defined by (3.1), corresponding to B0(t),B∞(t), respectively. Set (3.2)n0=dim Ker(A-B0),n∞=dim Ker(A-B∞),Gi(t,x)=F(t,x)-12(Bi(t)x,x),ψi(x)=∫02πGi(t,x)dt,i=0,∞.
Then the functional J defined by (2.5) can be rewritten as(3.3)J(x)=12〈(A-Bi)x,x〉-ψi(x),∀x∈E,i=0,∞.Lemma 3.1.
Suppose thatf satisfies (f1)–(f3). Then
(3.4)lim⁡‖x‖→0‖ψ0′(x)‖‖x‖=0,lim⁡‖x‖→+∞‖ψ∞′(x)‖‖x‖=0.The proof uses the same arguments of [5].In order to prove our results, we need an abstract theorem by Benci [12].Proposition 3.2.
Letχ∈C1(E,ℝ) satisfy the following:(J1)
χ(x)=1/2〈Lx,x〉+ω(x), where L is a bounded linear self-adjoint operator and ω′ is compact, where ω′ denotes the Frechét derivative of ω;(J2)
every sequence{xj} such that χ(xj)→c<φ(0) and ∥χ′(xj)∥→0 as j→+∞ has a convergent subsequence;(J3)
ω(x)=ω(-x), x∈E;(J4)
there are two closed subspaces ofE, E+, and E-, and some constant c0,c∞,ρ with c0<c∞<ω(0) and ρ>0 such that
(a)
χ(x)>c0 for x∈E+,(b)
χ(x)<c∞<ω(0) for u∈E-∩Sρ(Sρ={u∈E∥x∥=ρ}).Then the number of pairs of nontrivial critical points of χ is greater than or equal to dim(E+∩E-)-codim(E-+E+). Moreover, the corresponding critical values belong to [c0,c∞].Definition 3.3.
LetB1(t) and B2(t) be symmetric matrices function in ℝn, continuous and π/2-periodic in t. A index I of B1(t) and B2(t) is defined as follows:
(3.5)I(B1(t),B2(t))=dim⁡(M+(A-B1)⋂M-(A-B2))-dim⁡[(M-(A-B1)⊕M0(A-B1))⋂(M+(A-B2)⊕M0(A-B2))],
where Bi(i=1,2) are the operators, defined by (3.1), corresponding to Bi(t)(i=1,2) and M+(A-Bi) (resp., M-(A-Bi(t)), M0(A-Bi)) denotes the subspace of E on which A-Bi is positive definite (resp., negative definite, null).Lemma 3.4.
Iff satisfies (f1)–(f3), then J, defined by (2.8), satisfies (J1), (J3) and (J4).Proof.
Hypothesis(f1), (2.8), and Lemma 2.2 imply both (J1) and (J3). By definition of ψ0 and Lemma 3.1, we have
(3.6)ψ0(x)=-φ(0)+o(‖x‖2),for‖x‖⟶0.
Since B0 and B∞ are compact operators from E to E, it follows from Lemma 2.1 and a well-known theorem (cf. [15]) that the essential spectrum of A-B0 and A-B∞ is {2,-2}. Thus 0 is either an isolated eigenvalue of finite multiplicity or it belongs to the resolvent. Hence, we decompose E as follows:
(3.7)E=M+(A-B0)⊕M0(A-B0)⊕M-(A-B0)=M+(A-B∞)⊕M0(A-B∞)⊕M-(A-B∞).
Setting E+=M+(A-B∞), E-=M-(A-B0), there exists positive constant α,β such that
(3.8)〈(A-B0)x,x〉≤-α‖x‖2,∀x∈E-,〈(A-B∞)x,x〉≥β‖x‖2,∀x∈E+.
It follows that, for any x∈E-, it is
(3.9)J(x)≤-α2‖x‖2+φ(0)-o(‖x‖2),as‖x‖⟶0.
Then there exist constants ρ>0 and γ>0 such that
(3.10)J(x)<-γ+φ(0),∀x∈E-∩Sρ.
Setting c∞=-γ/2+φ(0), (J4)(b) is satisfied.
By(f3), there exists R0>0 such that
(3.11)G∞(t,x)≤β4α22|x|2,∀|x|>R0.
Since G∞(t,x) is continuous with respect to (t,x), denote by M=max0≤t≤π/2,|x|=R0{G∞(t,x)}. Then M is finite. Thus(3.12)|ψ∞(x)|≤∫02π|G∞(t,x)|dt≤∫02π[β4α22|x|2+M]dt≤β4‖x‖2+2πM.
Then, for every x∈E+,
(3.13)J(x)=12〈(A-B∞)x,x〉-ψ∞(x)≥β2‖x‖2-|ψ∞(x)|≥β4‖x‖2-2πM.
Thus J is bounded from below on E+. Setting
(3.14)c0=inf⁡x∈E+J(x)-ω
with ω>0 such that c0<c∞, then (J4)(a) is satisfied.Remark 3.5.
Supposing that0∉σe(A-B∞), any bounded sequence has a convergent subsequence (cf. [12]).Theorem 3.6.
Suppose thatf satisfies (f1)–(f3), and n0=n∞=0, then (1.4) has at least |I(B∞,B0)| pairs of nonconstant 2π-periodic solutions if |I(B∞,B0)|>0.Proof.
Sincen∞=0, dimM0(A-B∞)=0. By Proposition 3.2 and Lemma 3.4, we only need to check (J2). Let {xj} be a sequence such that
(3.15)J′(xj)→0,J(xj)⟶c,
where c∈ℝ, c<φ(0). Suppose to the contrary that we can choose ∥xj∥→+∞ as j→+∞. Clearly, xj can be written as xj=xj++xj-∈M+(A-B∞)⊕M-(A-B∞). On one hand,
(3.16)|〈J′(xj),xj+-xj-〉||〈xj,xj〉|≤‖J′(xj)‖‖xj‖‖xj‖2,
then we have
(3.17)0≤limsup⁡j→+∞|〈J′(xj),xj+-xj-〉||〈xj,xj〉|≤limsup⁡j→+∞‖J′(xj)‖‖xj‖‖xj‖2=0.
Thus
(3.18)limsup⁡j→+∞|〈J′(xj),xj+-xj-〉||〈xj,xj〉|=0.
On the other hand,(3.19)〈J′(xj),xj+-xj-〉=〈(A-B∞)xj,xj+-xj-〉-〈ψ∞′(t,xj),xj+-xj-〉.
Since
(3.20)|〈ψ∞′(xj),xj+-xj-〉||〈xj,xj〉|≤‖ψ∞′(xj)‖‖xj‖‖xj‖2=‖ψ∞′(xj)‖‖xj‖,
it follows by Lemma 3.1 that
(3.21)lim⁡j→+∞|〈ψ∞′(xj),xj+-xj-〉||〈xj,xj〉|=0.
Using a similar discussion as (3.8), there exists β0>0 such that 〈(A-B∞)x,x〉≤-β0∥x∥2 for all x∈M-(A-B∞). Choosing β′=min(β,β0)>0, we have
(3.22)〈(A-B∞)xj,xj+-xj-〉=〈(A-B∞)xj+,xj+〉-〈(A-B∞)xj-,xj-〉≥β′‖x‖2.
Thus,
(3.23)lim⁡inf⁡j→+∞|〈J′(xj),xj+-xj-〉||〈xj,xj〉|=lim⁡inf⁡j→+∞|〈(A-B∞)xj,xj+-xj-〉-〈g∞′(xj),xj+-xj-〉||〈xj,xj〉|=lim⁡inf⁡j→+∞|〈(A-B∞)xj,xj+-xj-〉||〈xj,xj〉|≥β′>0
which contradicts (3.18). This proves (J2). By Lemma 3.1, (1.4) has at least dim(E+∩E-)-codim(E-+E+)=I(B∞,B0) pairs of nontrivial solutions if I(B∞,B0)>0. Since the Sobolev space E does not contain ℝn as its subspace, all nontrivial periodic solutions are nonconstant periodic solutions.
IfI(B∞,B0)<0, then I(B0,B∞)=-I(B∞,B0)>0. In this case, we replace J by -J and let E+=M-(A-B0) and E-=M+(A-B∞). It is easy to see that (J1)–(J4) are satisfied. Similarly, we can show that (1.4) has at least I(B0,B∞) pairs of nonconstant solutions.Remark 3.7.
When Theorem3.6 is applied to autonomous delay differential equations, we obtain the same number of periodic solutions as that in [3].Theorem 3.8.
Supposef satisfies (f1)–(f3) and (f4)
G∞′(t,x) is bounded, where G∞′ denotes the derivative of G∞ with respect to x,(f5)±
G∞(t,x)→±∞ as |x|→+∞, uniformly for t∈[0,π/2].
Then (1.4) has at least I(B∞,B0) pairs of nonconstant 2π-periodic solutions provided I(B∞,B0)>0.Proof.
By Proposition3.2 and Lemma 3.4, it suffices to check condition (J2). Let {xj} be a sequence satisfying (3.15). Suppose to the contrary that {xj} is unbounded. Clearly, xj can be written as xj=xj++xj0+xj-∈M+(A-B∞)⊕M0(A-B∞)⊕M-(A-B∞). Since J′(xj)→0, for j large enough, we get
(3.24)|〈(A-B∞)xj,xj+〉-∫02π(G∞′(t,xj),xj+)dt|≤‖xj+‖.
By (f4), there exists c1>0 such that |G∞′(t,x)|≤c1. Then the above inequality and (3.8) imply
(3.25)β‖xj+‖2≤|〈(A-B∞)xj,xj+〉|≤‖xj+‖+c1α22π‖xj+‖.
This gives a uniform bound for {xj+}. In the same manner, one gets a uniform bound for {xj-}. Since {J(xj)} is convergent, it is bounded and there exist positive constants c2,c3,c4 such that
(3.26)c2≤J(xj)≤-ψ∞(xj)+12|〈(A-B∞)xj,xj〉|≤-ψ∞(xj0)+(ψ∞(xj0)-ψ∞(xj))+c3≤-ψ∞(xj0)+c1∫02π|xj0-xj|dt+c3≤-ψ∞(xj0)+c4.
Therefore, ψ∞(xj0) is bounded from above. (f5)+ implies that ∥xj0∥ is bounded. Otherwise, since the kernel of A-B∞ is a finite dimensional space, thus ψ∞(xj0)=∫02πG∞(t,xj0)dt→∞ as j→∞, which contradicts to (3.26).
If(f5)- holds, we replace (3.26) by
(3.27)c2≥J(xj)≥-ψ∞(xj)-|〈(A-B∞)xj,xj〉|.
Arguing as above, we can get a contradiction and complete our proof.Theorem 3.9.
Suppose thatf satisfies (f1)–(f3) and (f6)
there exist constantsr>0, p∈(1,2), a1>0, and a2>0 such that(3.28)pG∞(t,x)≥(x,G∞′(t,x))>0for|x|≥r,t∈[0,π2];G∞(t,x)≥a1|x|p-a2,∀x∈Rn,t∈[0,π2].
Then (1.4) has at least I(B∞,B0) pairs of nonconstant 2π-periodic solutions provided I(B∞,B0)>0.Proof.
Let{xj} be a sequence satisfying (3.15). We want to show that {xj} is a bounded sequence in E. Decompose xj as xj=xj++xj0+xj-∈M+(A-B∞)⊕M0(A-B∞)⊕M-(A-B∞). Then
(3.29)〈J′(xj),xj+〉=〈(A-B∞)xj+,xj+〉-〈ψ∞′(xj),xj+〉≥β‖xj+‖2-‖ψ∞′(xj)‖⋅‖xj+‖.
Combining the above inequality with (3.15) and Lemma 3.1, we have
(3.30)‖xj+‖‖xj‖→0,asj→∞.
Similarly, we have
(3.31)‖xj-‖‖xj‖⟶0,asj⟶∞.
Then by (3.30) and (3.31), there exists a positive integer j0 such that for j≥j0(3.32)‖xj0‖≥‖xj++xj-‖.
It follows that
(3.33)‖xj‖=‖xj++xj-+xj0‖≤‖xj++xj-‖+‖xj0‖≤2‖xj0‖.
By(f6), there exist positive constants M1,M2,M3,M4 such that for j large
(3.34)M1+12‖xj‖≥12〈J′(xj),xj〉-J(xj)=∫02π[G∞(t,xj)-12(G∞′(t,xj),xj)]dt≥(1-p2)∫02πG∞(t,xj)dt-M2≥M3‖xj‖Lpp-M4.
Let q be such that p-1+q-1=1. Since E⊂Lq(S1,ℝn), the embedding being continuous, the dual space E* of E, contains Lp(S1,ℝn) with continuous embedding. Therefore, by (3.34)
(3.35)M5(1+‖xj‖)≥‖xj‖E*p.
Since ∥xj∥E*=sup∥w∥E≤1(xj,w)L2=sup∥w∥E≤1[(xj0,w0)L2+(xj-,w-)L2+(xj+,w+)L2], taking w=xj0/∥xj0∥E, it follows that
(3.36)‖xj‖E*≥1‖xj0‖E‖xj0‖L22.
Owing to the fact that M0(A-B∞) is a finite dimensional subspace of E, there exist two positive constants c1 and c2 such that
(3.37)c1∥xj0∥E≤∥xj0∥L2≤c2∥xj0∥E.
Therefore by (3.35), (3.36), and (3.37),
(3.38)M6(1+‖xj‖)≥‖xj0‖p.
Both (3.33) and (3.38) imply that there exists M8>0 such that
(3.39)M7(1+‖xj0‖)≥‖xj0‖p
which yields a bound for ∥xj0∥ and hence xj via (3.33). Thus (J2) holds.Theorem 3.10.
Suppose thatf satisfies (f1)–(f3) and
(f7)± there exist positive constants c1,c2>0 such that
(3.40)±[2G∞(t,x)-(G∞′(t,x),x)]≥c1|x|-c2∀x∈Rn,t∈[0,π2].
Then (1.4) has at least I(B∞,B0) pairs of nonconstant 2π-periodic solutions provided I(B∞,B0)>0.Proof.
Let{xj} be a sequence satisfying (3.15). We want to prove that {xj} is bounded in E. Suppose, to the contrary, {xj} is unbounded in E. Decompose xj as xj=xj++xj0+xj-∈M+(A-B∞)⊕M0(A-B∞)⊕M-(A-B∞). Clearly, (3.30)–(3.33) still hold.
Assume that(f7)+ holds. Since M0(A-B∞) is a finite dimensional subspace of E, we have
(3.41)〈J′(xj),xj〉-2J(xj)=∫02π[2G∞(t,xj)-(G∞′(t,xj),xj)]dt≥c1∫02π|xj|dt-2πc2≥c1∫02π|xj0|dt-c1∫02π(|xj+|+|xj-|)dt-2πc2≥c3‖xj0‖-c4(‖xj+‖+‖xj-‖+1).
Combining the above inequality with (3.30), (3.31), we have
(3.42)‖xj0‖‖xj‖⟶0asj⟶∞.
But this implies the following contradiction:
(3.43)1=‖xj‖‖xj‖=‖xj0‖+‖xj-‖+‖xj+‖‖xj‖⟶0asj⟶+∞,
therefore, {xj} must be a bounded sequence.
If(f7)- holds, using a similar argument, we can get a contradiction which completes our proof.Theorem 3.11.
Suppose thatf satisfies (f1)–(f3) and (f8)±
there exist constants1≤γ<2, 0<δ<γ/2, and b1,b2,L>0 such that(3.44)|G∞′(t,x)|≤b1|x|δ,±G∞(t,x)≥b2|x|γ,∀|x|≥L,t∈[0,π2].
Then (1.4) has at least I(B∞,B0) pairs of nonconstant 2π-periodic solutions provided I(B∞,B0)>0.Proof.
Let{xj} be a sequence satisfying (3.15). Suppose, to the contrary, ∥xj∥→+∞ as j→+∞. Decompose xj as xj=xj++xj0+xj-∈M+(A-B∞)⊕M0(A-B∞)⊕M-(A-B∞). First, we show that for j large enough
(3.45)‖xj++xj-‖≤b3‖xj0‖δ+η,
where b3>0 and η>0 are constants independent of j. Since |xj|≥L for sufficiently large j, therefore, |G∞′(t,xj)|≤b1|xj|δ, and we have
(3.46)|G∞′(t,xj)|2≤b12|xj|2δ+b4;|〈ψ∞′(xj),y〉|≤∫02π|G∞′(t,xj)||y|dt≤(∫02π|G∞′(t,xj)|2dt)1/2‖y‖L2≤α2[b12(2π)1-δα22δ‖xj‖2δ+2πb4]1/2‖y‖,for anyy∈E.
This implies that for j large enough
(3.47)‖ψ∞′(xj)‖‖xj‖δ≤b5.
By (3.15), (3.22), (3.33) and (3.47), for j large enough, we have
(3.48)|〈J′(xj),xj+-xj-〉|=|〈(A-B∞)xj,xj+-xj-〉-〈ψ∞′(xj),xj+-xj-〉|≥β′‖xj++xj-‖2-b5‖xj‖δ‖xj+-xj-‖≥β′‖xj++xj-‖2-b52δ‖xj0‖δ‖xj+-xj-‖.
Therefore, for sufficiently large j,
(3.49)‖J′(xj)‖≥β′‖xj++xj-‖-b52δ‖xj0‖δ.
This implies that (3.45) holds, where b3=b52δ/β′.
By (3.15) and (3.45), for j large enough, there exist positive constants b6,b7,b7′,b8 such that
(3.50)ψ∞(xj)=12〈(A-B∞)(xj++xj-),xj++xj-〉-J(xj)≤b6‖xj++xj-‖2+b7′≤b8‖xj0‖2δ+b7.
Now, we claim that there exists b9>0 such that, for j large enough,
(3.51)∫02π|xj|γdt≥b9‖xj0‖γ,
In fact, for γ>1, by (3.45) and the fact that δ<1, we have
(3.52)∫02π(xj,xj0)dt≤(∫02π|xj|γdt)1/γ(∫02π|xj0|γ/γ-1dt)γ-1/γ≤b10(∫02π|xj|γdt)1/γ‖xj0‖;∫02π(xj,xj0)dt=∫02π(xj0,xj0)dt+∫02π(xj++xj-,xj0)dt≥∫02π|xj0|2dt-‖xj++xj-‖L2‖xj0‖L2≥b11‖xj0‖2-b3α22‖xj0‖1+δ-α22η‖xj0‖≥b12‖xj0‖2,
for j large enough. This implies (3.51) for γ>1.
Forγ=1, since M0(A-B∞) is a finite dimensional subspace of E, we know that for any j,
(3.53)b13‖xj0‖≤‖xj0‖∞≤b14‖xj0‖.
where b13,b14>0 are constants independent of j. Now we have
(3.54)∫02π(xj,xj0)dt≤∫02π|xj||xj0|dt≤(∫02π|xj|dt)‖xj0‖∞≤b15‖xj0‖(∫02π|xj|dt).
Combining (3.52) with (3.54), we get (3.51) for γ=1.
On the other hand, by(f8)+(3.55)ψ∞(xj)=∫02πG∞(t,xj)dt≥∫02πb2|xj|γdt-2πb16≥b17‖xj0‖γ-2πb16.
Since that γ>2δ, we get a contradiction from (3.50) and (3.55). Therefore, {xj} is bounded.
If(f8)- holds, using a similar argument as above, we get a contradiction and completes our proof.Theorem 3.12.
Supposef satisfies (f1)–(f3) and (f9)±
there exist positive constants1≤γ<2, 0<δ<γ/2, and b1,b2,L such that(3.56)|G∞′(t,x)|≤b1|x|δ,±〈G∞′(t,x),x〉≥b2|x|γ,∀|x|≥L,t∈[0,π2].
Then (1.4) has at least I(B∞,B0) pairs of nonconstant 2π-periodic solutions provided I(B∞,B0)>0.Proof.
If(f9)+ holds, for j large enough, by (3.45) and (3.51), we have
(3.57)∫02π(G∞′(t,xj),xj)dt≤|-〈J′(xj),xj〉+〈(A-B∞)(xj++xj-),(xj++xj-)〉|≤‖xj‖+M1‖xj++xj-‖2≤‖xj0‖+M2‖xj0‖δ+M3‖xj0‖2δ+M4;∫02π(G∞′(t,xj),xj)dt≥b2∫02π|xj|γdt-M5≥M6‖xj0‖γ-M5.
Since γ>2δ, {xj0} is bounded, so is {xj}. Therefore, J satisfies (J2).
In the case that(f9)- holds, using a similar argument, we can verify (J2). This completes the proof.Example 3.13.
Consider the following nonautonomous delay differential equation(3.58)x′(t)=-Mx(t-π2)a+b(t)|x(t-(π/2))|5/2+c|x(t-(π/2))|41+|x(t-(π/2))|4,
where M is a 4×4 matrix, a,c are constants, b∈C([0,π/2],ℝ+).
Case 1.
LetA=diag(0.3,2.7,7.3,9.3), a=1, c=2, and b arbitrary. Computing directly, we have I(B0(t),B∞(t))=10. Applying Theorem 3.6, equation (3.58) has at least 10 pairs of 2π-periodic solutions.
Case 2.
LetA=diag(0.3,2.7,5,10.5), a=2, c=1, and b arbitrary. Then by Theorem 3.8, (3.58) has at least 8 pairs of nonconstant 2π-periodic solutions.



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*Source: 829107-2011-07-07.xml*