# Infinitely Many Periodic Solutions to Delay Differential Equations via Critical Point Theory

Authors: Xiaosheng Zhang; Duo Wang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

By the critical point theory, infinitely many 4σ-periodic solutions are obtained for the system of delay differential equations ẋt=-f(x(t-σ)), where σ∈(0,+∞) and f∈C(ℝn,ℝn). It is shown that all the periodic solutions derived here are brought about by the time delay.


---

## Body

## 1. Introduction

This paper is concerned with the existence of periodic solutions to the system of delay differential equations(1)x˙(t)=-f(x(t-σ)),
where σ∈(0,+∞) and f∈C(ℝn,ℝn).Delay differential equations have widely been applied to describe the dynamics phenomena in both natural and manmade processes such as chemistry, physics, engineering, and economics. The existence of the periodic solutions for delay differential equations has been extensively investigated by using various methods, including fixed point theorems [1–5], Hopf bifurcation theorems [6–8], variational methods [9–14], the methods of differential inequalities [15–21], and other effective approaches (e.g., see [22–24]). In [25–31], the minimal periods of the periodic solutions to Lipschitzian differential equations are estimated through the Lipschitz constants (see Remark 4).The use of variational methods in the study of4σ-periodic solutions of system (1) having a variational structure was introduced in 2005 by Guo and Yu [9]. Assume that(F1)
f is odd in x; that is, f(-x)=-f(x), for all x∈ℝn;(F2)
there existsF∈C1(ℝn,ℝ) such that Fx(x)=f(x), for all x∈ℝn, where Fx denotes the gradient of F.In [9], the authors obtained the multiplicity results for periodic solutions to (1) in the case that f is asymptotically linear. Later, the existence of the periodic solution of (1) was investigated by using Morse theory and Galerkin methods [10]. For the other relative investigations, we refer the reader to [11–14].Many practical problems, such as nonlinear population growth models and control systems working with potentially explosive chemical reactions, can be transformed into the form of (1). For example, by the change of variables y=atanh(ax), the following generalized food-limited population model
(2)y˙(t)=-θsign(y(t-1))|y(t-1)|γ(a2-y2(t))
is transformed equivalently into (1) with n=1, f(x)=θaγsign(x)|tanh(ax)|γ, and σ=1, where θ and a are positive numbers. When γ=1, f′(0)=θa2. It is known from [24] that, with the slope f′(0) increasing and tending to infinity, the number of the periodic solutions of (2) increases and tends to infinity. Naturally, one would conjecture that when 0<γ<1, (2) possesses infinitely many periodic solutions, since in this case limx→0f′(x)=+∞.Motivated by the above observation, in this paper, we study the existence of infinitely many periodic solutions to the system (1) under the assumptions (F1), (F2), and(F3)
there  are1<α, β<2 and d1,r0>0 such that
(i)
0<(f(x),x)⩽αF(x), for all x∈Br0∖{0};(ii)
d1|f(x)|β′⩽F(x), for all x∈Br0,where1/β+1/β′=1, Br0={x∈ℝn:|x|⩽r0}.Here and subsequently,(·,·), |·| denote the inner product and the standard norm in ℝn, respectively, and the bold face 0 represents the coordinate origin of Rn. The main result of this paper is stated as follows.Theorem 1.
Assume that(F1)–(F3) hold. Then (1) possesses a sequence of nonconstant 4σ-periodic solutions {xm} satisfying ∥xm∥∞→0 as m→∞.Example 2.
When0<γ<1, it is easy to check that f(x)=θasign(x)|tanh(ax)|γ satisfies (F1)–(F3) with α=β=1+γ; then (2) has a sequence of nonconstant 4-periodic solutions {xm} satisfying ∥xm∥∞→0 as m→∞.Remark 3.
Let us compare the result here with that in the case of ordinary differential equations (ODE). Without the time delay, (1) reduces to the following system of ODE
(3)x′(t)=f(x(t)).
Let x(t)=x(t;x0) be the solution of (3) satisfying the initial condition x(0)=x0≠0. Then the derivative of the Lyapunov function V(x)=|x|2 along x(t) reads
(4)dVdt|(3)=(-f(x(t)),x(t)).
From (F3)-(i), we see that dV/dt|(3)<0 for 0<|x|<r0, which implies that there is no any periodic orbit of (3) across Br0∖{0}; that is, the trivial solution is an isolated periodic solution. However, by the above theorem, with the time delay, the system (1) possesses infinitely many periodic solutions in any neighborhood of the origin.Remark 4.
Consider the following system ofmth order functional differential equations:
(5)x(m)(t)=f(x(τ(t))),t∈ℝ,
where f:ℝn↦ℝn satisfies the Lipschitz condition and τ:ℝ1↦ℝ1 is a measurable function. The lower bounds for the periods of the periodic solutions to (5) and their special forms are estimated in [25–31]. From this perspective, Theorem 1 complements the information in the case of non-Lipschitzian differential equations. For the unique solvability of the periodic problems on functional differential equations, we refer the reader to [1, 15–21].The remainder of this paper is divided into two parts. In the next section, we state the preliminaries on the variational structure for (1). In the final section, the proof of Theorem 1 will be given via the ℤ2-genus theory, together with an approximating argument.


## 2. Preliminaries

LetL2(S1,ℝn) denote the set of n-tuples of 2π-periodic functions which are square integrable. If x∈L2(S1,ℝn), it has a Fourier expansion
(6)x(t)=a0+∑j∈ℕ‍(ajcosjt+bjsinjt),
where ai,bj∈ℝn and the series converges in the space L2(S1,ℝn). For x∈L2(S1,ℝn) with its expansion (6), set H:={x∈L2(S1,ℝn)∣∥x∥H<∞}, where
(7)∥x∥H∶=|a0|2+∑j∈ℕ‍(1+j)(|aj|2+|bj|2).
Then H, equipped with the norm ∥·∥H, is a Sobolev space.On the other hand, forx∈H with its expansion (6), set
(8)∥x∥∶=|a0|2+∑j∈ℕ‍j(|aj|2+|bj|2).
Then H possesses another norm ∥·∥ which is equivalent to ∥·∥H. In the following, we always employ ∥·∥ as the norm of H. The associated inner product with ∥·∥ is denoted by 〈·,·〉.Now set(9)E∶={x∈H∣x(·+π)=-x(·)}.
Then E is a closed subspace of H and the Fourier expansion of x∈E reduces to
(10)x(t)=∑j=1∞‍[a2j-1cos(2j-1)t+b2j-1sin(2j-1)t].
Thus with x1,x2∈E being expanded as
(11)xi(t)=∑j=1∞‍[a2j-1(i)cos(2j-1)t+b2j-1(i)sin(2j-1)t],∑j=1∞‍a2j-1(i)cos(2j-1)t+sin(2j-1)ti=1,2,
we have
(12)〈x1,x2〉=∑j=1∞‍(2j-1){(a2j-1(1),a2j-1(2))+(b2j-1(1),b2j-1(2))}.Forx,y∈E, we call y a weak derivative of x and denote it by x˙=y if
(13)∫02π‍(x(t),z′(t))dt=-∫02π‍(y(t),z(t))dt,y(t),∀z∈C∞(S1,ℝn).
Further, for x∈C∞(S1,ℝn)∩E with its expansion (10), define
(14)A(x)∶=12∫02π‍(x˙(t+π2),x(t))dt=12∑j=1∞‍(-1)j(2j-1)(|a2j-1|2+|b2j-1|2).
Then it is easy to check that |A(x)|⩽∥x∥2 for x∈C∞(S1,ℝn)∩E. Therefore A extends to all of E as a continuous quadratic form. This extension will still be denoted by A.LetF~∈C1(ℝn,ℝ) and satisfy
(15)F~(-x)=F~(x),|F~(x)|⩽C1+C2|x|s,x∈ℝn
for some s∈[1,∞). Define
(16)Φ(x)∶=∫02π‍F~(x(t))dt,x∈E
and I(x)=A(x)+Φ(x), x∈E. The following lemma is derived from [9, Lemma  2.2].Lemma 5 (see [9]).
LetF~∈C1(ℝn,ℝ) and satisfy (15). Then I∈C1(E,ℝ) and
(17)I′(x)y=A′(x)y+∫02π‍(F~x(x(t)),y(t)),y∈E,
where
(18)A′(x)y=∫02π‍(x˙(t+π2),y(t))dt,y∈E.
Moreover, the existence of 2π-periodic solutions x(t) for
(19)x′(t)=-F~x(x(t-π2))
satisfying x∈E is equivalent to the existence of critical points of functional I.Let{e1,…,en} be the orthonormal basis of ℝn. For k∈ℕ, set
(20)E+(k)∶=span{cos[(4k-1)t]ei,sin[(4k-1)t]ei:i=1,2,…,nsin[(4k-1)t]ei},E-(k)∶=span{cos[(4k-3)t]ei,sin[(4k-3)t]ei:i=1,2,…,nsin[(4k-3)t]ei}.
For l,m∈ℕ∪{+∞}, define
(21)Vl±=⊕k=1lE±(k)¯,Vlm=Vl-⊕Vm+,
where the closure is of E sense. Set V±∶=V+∞±; then E=V+⊕V-. In the rest of this paper, this decomposition will always be referred to when a point x∈E is written as x=x++x-, where x±∈V±.Remark 6.
In view of (12), (14), and (18), we see that
(22)A(x)=12(∥x+∥2-∥x-∥2)
and that
(23)A′(x)y=〈x+,y+〉-〈x-,y-〉,x,y∈E.
The following lemma is derived from [32, Lemma  2.1].Lemma 7 (see [32]).
For eachs∈[1,∞) there is γs>0 such that
(24)∥x∥s⩽γsm-1/s∥x∥
for all x∈(Vm-1m-1)⊥ with m⩾2, the orthogonal complement in E, where (and below) ∥·∥s denotes the usual Ls-norm.


## 3. Proof of Theorem1

Without loss of generality we assume thatσ=π/2 since, under the change of variables y(t)=x(2σt/π), (1) can be transformed into the system
(1′)y˙(t)=-f~(y(t-π2)),
where f~(y)=(2σ/π)f(y) still satisfies (F1–F3) with f being replaced by f~.Letχ∈C∞(ℝ,[0,1]) be such that χ(s)=0 for s⩽r0/2, χ(s)=1 for s⩾r0, and χ′(s)>0 for s∈(r0/2,r0). Define F~:ℝn↦ℝ by
(25)F~(x)∶=(1-χ(|x|))F(x)+χ(|x|)M0|x|α,
where M0=inf{F(x)/r0α:r0/2⩽|x|⩽r0}.Letα′>0 be such that 1/α+1/α′=1. By (F3) we get
(26)0<(F~x(x),x)⩽αF~(x),∀x∈ℝn,(27)F~(x)⩾{C1|F~x(x)|β′,|x|⩽1,C1|F~x(x)|α′,|x|>1,
where (and below) Cj's stand for positive constants.Lemma 8.
LetF~:ℝn↦ℝ be defined by (25); then 1<β<α<2, F~∈C1(ℝn,ℝ), and
(28)C2|x|α⩽F~(x)⩽{C3|x|β,|x|⩽1,C3|x|α,|x|>1.Proof.
From (25), it is easy to see that F~∈C1(ℝn,ℝ). Now we start to prove (28). Let M be such a constant that |lnF~(x)-αln|x||⩽M for x∈S1≡∂B1. For x∈ℝn, |x|⩽1, set x0=x/|x|; then x0∈S1. Define g(t)=lnF~(tx0)-αln|tx0|, t∈(0,1]. Then, by (26),
(29)g′(t)=(F~x(tx0)F~(tx0),x0)-α⩽0,
which implies that g(|x|)⩾g(1); that is,
(30)lnF~(x)-αln|x|⩾lnF~(x0)-αln|x0|⩾-M.
It follows that F~(x)⩾e-M|x|α for |x|⩽1, which, combining with (25), leads to the inequality on the left hand of (28) with C2 being chosen adequately.
Again, forx∈ℝn, |x|⩽1, set x0=x/|x| and define h(t)=(F~(tx0))1/β-t|x0|/(βC11/β′), t∈[0,1]. Then by the first inequality in (27),
(31)h′(t)=1β((F~x(tx0),x0)(F~(tx0))1/β′-1C11/β′|x0|)⩽|x0|β(|F~x(tx0)|(F~(tx0))1/β′-1C11/β′)<0.
Thus h(|x|)⩽h(0)=0, which leads to F~(x)⩽C3′|x|β, where C3′=1/(βC11/β′)β. In the same way, from the second inequality in (27), we can arrive at F~(x)⩽C3′′|x|α for |x|>1, where the constant C3′′ only depends on α and C1. With C3=max{C3′,C3′′}, the inequalities on the right hand of (28) hold. Thus we get (28), which implies that C2|x|α⩽C3|x|β for |x|⩽1 and that 1<β<α<2. The proof is complete.Now we consider the functional(32)I(x)=A(x)+∫02π‍F~(x)dt,x∈E.Lemma 9.
I satisfies (PS) condition; that is, every sequence {xk}⊂E such that {I(xk)} is bounded and I′(xk)→0 as k→∞ has a convergent subsequence.Proof.
By Lemma5, for x∈E, I′(x) is defined by
(33)I′(x)y=A′(x)y+∫02π‍(F~x(x),y)dt,∀y∈E.
To verify that I satisfies  (PS) condition, we suppose |I(xk)|⩽C4 and I′(xk)→0 as k→∞. Note that, for large k, |I′(xk)x|⩽∥x∥. Thus for large k and x=xk, from (32) and (33),
(34)C4+∥x∥⩾I(x)-12I′(x)x=∫02π‍[F~(x)-12(F~x(x),x)]dt.
Noticing that 1<β<α<2, we see from (25) that, for all x∈ℝn,
(35)(F~x(x),x)⩾C5max{|x|α,|x|β}-C6,
which, combining with (26) and (34), implies
(36)C4+∥x∥⩾(α-1-2-1)∫02π‍(F~x(x),x)dt⩾C7max{∥x∥αα,∥x∥ββ}-C8,(37)max{∥x∥αα,∥x∥ββ}⩽C9(∥x∥+1).
Next for largek, taking x=xk and ς=xk+ in
(38)|∫02π‍(F~x(x),ς)dt+A′(x)ς|=|I′(x)ς|⩽∥ς∥
and using (23), (27), and (28) and the Hölder inequality (1/α+1/α′=1, 1/β+1/β′=1), we get
(39)∥x+∥2⩽|∫02π‍(F~x(x),x+)dt|+∥x+∥⩽C10(∫|x(t)|>1‍|x|α/α′|x+|dt+∫|x(t)|⩽1‍|x|β/β′|x+|dt)+∥x+∥⩽C10(∥x∥αα/α′∥x+∥α+∥x∥ββ/β′∥x+∥β)+∥x+∥⩽C11(∥x∥αα/α′+∥x∥ββ/β′+1)∥x+∥,
where the last inequality holds since E is compactly embedded in Ls(S1,ℝn) for s⩾1. It follows from (36) that
(40)∥x+∥⩽C12(∥x∥1/α′+∥x∥1/β′+1).
Similarly, (40) works with x+ being replaced by x-. Combining these inequalities shows
(41)∥x∥⩽C13(∥x∥1/α′+∥x∥1/β′+1),
which implies that {xk} is bounded in E.
LetΦ be defined by (16). By [33, Proposition B.37], {Φ′(xk)} is precompact in E. Moreover, from (23) and (33),
(42)I′(xk)=xk+-xk-+Φ′(xk).
It follows that {xk} has a convergent subsequence. The proof is complete.Lemma 10.
For eachl∈ℕ, there are ρl>0,  al>0, and 0<bl→0 such that  (a)
I(x)⩾0,   for all x∈Bρl∩Vl+∞ and infI(∂Sρl∩Vl+∞)⩾al, where Bρl={x∈E:∥x∥⩽ρl};(b)
supI((Vl-1+∞)⊥)⩽bl.Proof.
Noticing thatVl+∞=Vl-⊕V+∞+ and that dim(Vl-)<∞, we have, for x∈Bρl∩Vl+∞,
(43)∥x∥α=sup{∫S1‍(x(t),y(t))dt∣y∈Lα′(S1,ℝn),∥y∥α′=1}⩾sup{∫S1‍(x(t),y(t))dt∣y∈Vl-,∥y∥α′=1}=sup{∫S1‍(x-(t),y(t))dt∣y∈Vl-,∥y∥α′=1}=∥x-∥α⩾ηl∥x-∥,
where ηl is a positive constant depending on l. It follows by (28) that, for x∈Bρl∩Vl+∞,
(44)I(x)⩾12∥x+∥2+C2ηl∥x-∥α-12∥x-∥2=12∥x+∥2+(C2ηl-12∥x-∥2-α)∥x-∥α,
which implies (a) by setting ρl∶=min{1,(C2ηl)1/(2-α)} and al=ρl2(1+C2ηl)/2.
Letx∈(Vl-1+∞)⊥. By (28) and Lemma 7,
(45)I(x)⩽C3(∥x∥αα+∥x∥ββ)-12∥x∥2⩽C16l-1(∥x∥α+∥x∥β)-12∥x∥2⩽bl∶=sups≥0g(s),
where g(s)∶=C16l-1(sα+sβ)-s2/2. Noticing 1<β⩽α<2, one can see that bl→0 as l→∞ and (b) follows. The proof is complete.In the following, letΣ denote the family of closed (in E) subsets of E∖{0} symmetric with respect to the origin, and
(46)γ:Σ↦ℕ∪{0,∞},
the ℤ2-genus map (see [33]). For l,m∈ℕ, set
(47)Σlm={A∈Σ:A⊂V+∞m,γ(A)⩾n(l+m)},
and define
(48)cl,m=supA∈Σlminfx∈AI(u).Lemma 11.
For alll,m∈ℕ, cl,m is a critical value of I and
(49)al⩽cl,m⩽bl.Proof.
We first prove that (49) holds. For each m∈ℕ, let ρl be chosen as that in Lemma 10; then it follows by Lemma 10(a) that infI(∂Sρl∩Vl+∞)⩾al. Denote A~=∂Sρl∩Vlm; then γ(A~)=n(m+l) and A~∈Σlm. Since A~⊂∂Sρl∩Vl+∞, we have
(50)cl,m⩾infx∈A~I(x)⩾infI(∂Sρl∩Vl+∞)⩾al.
On the other hand, for everyA∈Σlm, by the property of genus, A∩(Vl-1+∞)⊥≠⌀, which, from Lemma 10(b), leads to infx∈AI(x)⩽bl for every A∈Σlm. Thus cl,m⩽bl and (49) holds.
By(F1) and (25), F~(x) is even with respect to x, which implies that I is even. We claim that c=cl,m is a critical point of I. Otherwise, there exists ϵ>0, such that there is no any critical point in the interval (c-ϵ,c+ϵ). By the definition of cl,m, there exists A∈Σlm, such that
(51)infx∈AI(x)>c-ϵ.
For a∈ℝ, denote Ia={x∈E:I(x)⩾a}. Use a positive rather than a negative gradient flow [33, Remark  A.17], we get η∈C([0,1]×E,E) such that η(1,·) is odd and
(52)η(1,Ic-ϵ)⊂Ic+ϵ.
Since A⊂Ic-ϵ, we have η(1,A)⊂Ic+ϵ; that is,
(53)infx∈η(1,A)I(x)⩾c+ϵ.
On the other hand, by the property of genus, we know thatγ(η(1,A))∈Σlm, which, by the definition of c, leads to
(54)c⩾infx∈η(1,A)I(x)⩾c+ϵ.
This contradiction implies that cl,m is a critical value of I. The proof is complete.Now we are in a position to give the following proof.Proof of Theorem1.
In view of Lemma11, let xl,m∈V+∞m be such that
(55)I(xl,m)=cl,m,I′(xl,m)=0.
Then by (PS) condition, along a subsequence as m→∞, xl,m→xl∈E such that
(56)al⩽I(xl)⩽bl,I′(xl)=0,
which implies that xl is nonzero. Moreover, by Lemma 5,
(57)xl′(t)=-F~x(xl(t-π2)).
We claim that, for sufficiently largel, xl solves (1). In fact, from (26) and (56)
(58)bl⩾I(xl)=I(xl)-12I′(xl)xl⩾(1-α2)∫02π‍F~(xl)dt.
By (27), (58), and Hölder inequality
(59)∥xl+∥2=∫02π‍(F~x(xl),xl+)dt⩽∥xl+∥α(∫02π‍|F~x(xl)|α′dt)1/α′+∥xl+∥β(∫02π‍|F~x(xl)|β′dt)1/β′⩽C17∥xl+∥(∫02π‍F~(xl)dt)1/α′+C18∥xl+∥(∫02π‍F~(xl)dt)1/β′⩽(C17bl1/α′+C18bl1/β′)∥x+∥.
Similarly, the above inequality works with xl+ replaced by xl-. These inequalities yield
(60)∥xl∥≤C19bl1/α′+C20bl1/β′.
Since bl→0 as l→∞, it follows that
(61)∥xl∥⟶0asl⟶∞.
Furthermore, from (27) and (57), we have
(62)∫02π‍|x˙l(t)|2dt=∫02π‍|F~x(xl)|2dt⩽C21(∫02π‍[F~(xl)]2/β′dt+∫02π‍[F~(xl)]2/α′dt).
It follows from (58) that ∥x˙l∥2→0 as l→∞. Recalling (61), we get
(63)∥xl∥W1,2⟶0asl⟶∞,
which implies that ∥xl∥∞→0 as l→0. Thus for m sufficiently large, ∥xl∥∞<r0/2 and therefor F~x(xl)=Fx(xl). It follow from (57) that, for l sufficiently large, xl solves (1). In addition, by (1) and (F3)(i), the only constant solution of (1) is the trivial solution. Then (56) yields that xl is nonconstant and the proof of Theorem 1 is complete.



---

*Source: 526350-2013-12-05.xml*