# One-Signed Periodic Solutions of First-Order Functional Differential Equations with a Parameter

Authors: Ruyun Ma; Yanqiong Lu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study one-signed periodic solutions of the first-order functional differential equationu′(t)=-a(t)u(t)+λb(t)f(u(t-τ(t))),t∈ℝ by using global bifurcation techniques. Where a,b∈C(ℝ,[0,∞)) are ω-periodic functions with ∫0ωa(t)dt>0, ∫0ωb(t)dt>0, τ is a continuous ω-periodic function, and λ>0 is a parameter. f∈C(ℝ,ℝ) and there exist two constants s2<0<s1 such that f(s2)=f(0)=f(s1)=0, f(s)>0 for s∈(0,s1)∪(s1,∞) and f(s)<0 for s∈(-∞,s2)∪(s2,0).


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## Body

## 1. Introduction

In recent years, there has been considerable interest in the existence of periodic solutions of the following equation:(1.1)u′(t)=-a(t)u(t)+λb(t)f(u(t-τ(t))),
where a,b∈C(ℝ,[0,∞)) are ω-periodic functions, and ∫0ωa(t)dt>0,∫0ωb(t)dt>0, τ is a continuous ω-periodic function, λ>0 is a parameter. (1.1) has been proposed as a model for a variety of physiological processes and conditions including production of blood cells, respiration, and cardiac arrhythmias; see, for example, [1–12] and the references therein. Roughly speaking, u(t) represents the number of adult (sexually mature) members in a population at time t,a(t) is the per capita death rate, and f(u(t-τ(t))) is the rate at which new members are recruited into the population at time t ( τ is the age at which members mature, and it is assumed that the birth rate at a given time depends only on the adult population size). The most famous models of this type are(i)
the Nicholson's blowflies equation proposed in [1] to explain the oscillatory population fluctuations observed by A. J. Nicholson in 1957 in his studies of the sheep blowfly Lucilia cuprina:
(1.2)u′(t)=-au(t)+p⋅u(t-h)e-γu(t-h),a,p,γ,h>0;(ii)
the model for blood cell populations proposed by Mackey and Glass in [2]
(1.3)u′(t)=-au(t)+pu(t-h)1+[u(t-h)]n,a,p,γ,h>0,n>1;(iii)
the model for the survival of red blood cells in an animal proposed by Wazewska-Czyzewska and Lasota in [3]
(1.4)u′(t)=-au(t)+p⋅e-γu(t-h),a,p,γ,h>0.Recently, Cheng and Zhang [7] studied the existence of positive ω-periodic solutions of the functional equation (1.1) under the assumptions:(H1)
f∈C([0,∞),[0,∞)), and f(s)>0 for s>0;(H2)
a,b∈C(ℝ,[0,∞)) are ω-periodic functions, ∫0ωa(t)dt>0, ∫0ωb(t)dt>0, τ∈C(ℝ,ℝ) is a ω-periodic function;(H3)
there existf0,f∞∈(0,∞) such that
(1.5)f0=lim⁡|s|→0f(s)s,f∞=lim⁡|s|→∞f(s)s.
They proved the following.Theorem A.
Assume (H1)–(H3)hold. Then for eachλ satisfying (1.6)1σBf∞<λ<1Af0,or1σAf0<λ<1Bf∞,
equation (1.1) has a positive periodic solution, where
(1.7)A=max⁡t∈[0,ω]∫0ωG(t,s)b(s)ds,B=min⁡t∈[0,ω]∫0ωG(t,s)b(s)ds,σ=e∫0ωa(t)dt.However, the condition used in [7] is not sharp, and the main results in [7] give no any information about the global structure of the set of positive periodic solutions. Moreover, f satisfied (H1) in [7], so a natural question is what would happen if f is allowed to have some zeros in ℝ? The purpose of this work is to study the global behavior of the components of one-signed solutions of (1.1) under the condition(H4)
f∈C(ℝ,ℝ); there exist two constants s2<0<s1 such that f(s2)=f(0)=f(s1)=0, f(s)>0 for s∈(0,s1)∪(s1,∞), and f(s)<0 for s∈(-∞,s2)∪(s2,0).The rest of this paper is organized as follows. In Section2, we give some notations and the main results. Section 3 is devoted to proving the main results.


## 2. Statement of the Main Results

LetY={u∈C(ℝ,ℝ):u(t)=u(t+ω)} with the norm (2.1)‖u‖∞=max⁡t∈[0,ω]|u(t)|.
Then (Y,∥·∥∞) is a Banach space. Let (2.2)E={u∈C1(R,R):u(t)=u(t+ω)}
be the Banach space with the norm ∥u∥=max{∥u∥∞,∥u′∥∞}.It is well known that (1.1) is equivalent to
(2.3)u(t)=λ∫tt+ωG(t,s)b(s)f(u(s-τ(s)))ds:=(Au)(t),
where
(2.4)G(t,s)=e∫tsa(θ)dθe∫0ωa(θ)dθ-1,s∈[t,t+ω].
Notice that ∫0wa(t)dt>0, we have
(2.5)1σ-1≤G(t,s)≤σσ-1,
where σ=e∫0ωa(t)dt, and 0<1/σ<1.Define thatK is a cone in Y by (2.6)K={u∈Y:u(t)≥0,u(t)≥1σ‖u‖}.
It is not difficult to prove that A(K)⊂K and A:K→K is completely continuous.Let us consider the spectrum of the linear eigenvalue problem(2.7)u′(t)=-a(t)u(t)+λb(t)u(t-τ(t)),t∈R.Lemma 2.1.
Assume that (H2)  holds. Then the linear problem (2.7) has a unique eigenvalue λ1, which is positive and simple, and the corresponding eigenfunction φ is of one sign.Proof.
It is a direct consequence of the Krein-Rutman Theorem [13, Theorem 19.3].In the rest of the paper, we always assume that(2.8)‖φ‖=1,φ(t)>0,t∈R.DefineL:E→Y by setting (2.9)Lu:=u′(t)+a(t)u(t),u∈E.
Then L-1:Y→E is completely continuous.Letζ,ξ∈C(ℝ,ℝ) be such that (2.10)f(s)=f0s+ζ(s),f(s)=f∞s+ξ(s).
Clearly, (2.11)lim⁡|s|→0ζ(s)s=0,lim⁡|s|→∞ξ(s)s=0.
Let us consider (2.12)Lu(t)-λb(t)f0u(t-τ(t))=λb(t)ζ(u(t-τ(t)))
as a bifurcation problem from the trivial solution u≡0 and (2.13)Lu(t)-λb(t)f∞u(t-τ(t))=λb(t)ξ(u(t-τ(t)))
as a bifurcation problem from infinity. We note that (2.12) and (2.13) are the same and each of them is equivalent to (1.1).Let𝔼=ℝ×E under the product topology. We add the points {(λ,∞)∣λ∈ℝ} to our space 𝔼. Let S+ denote the set of positive functions in E and S-=-S+, and S=S-∪S+. They are disjoint and open in E. Finally, let Φ±=ℝ×S± and Φ=ℝ×S.Remark 2.2.
It is worth remaking that ifu is a nontrivial solution of (1.1) and a,b, and f satisfy (H2)–(H4), then u∈Sν for some ν={+,-}. To see this, define
(2.14)q(t)={f(u(t))u(t),u(t)≠0,f0,u(t)=0.
Thus (1.1) is equivalent to
(2.15)u′(t)=-a(t)u(t)+λb(t)q(t-τ(t))u(t-τ(t)),t∈R.
Obviously, b(·)q(·-τ(·)) satisfies (H2). From Lemma 2.1, the nontrivial solution u∈Sν for some ν∈{+,-}.The result of Rabinowitz [14] for (2.12) can be stated as follows: for each ν∈{+,-}, there exists a continuum 𝒞ν of solutions of (2.12) joining (λ1/f0,0) to infinity, and 𝒞ν∖{(λ1/f0,0)}⊂Φν.The result of Rabinowitz [15] for (2.13) can be stated as follows: for each ν∈{+,-}, there exists a continuum 𝒟ν of solutions of (2.13) meeting (λ1/f∞,∞), and 𝒟ν∖{(λ1/f∞,∞)}⊂Φν.Our main result is the following.Theorem 2.3.
Assume (H2)–(H4) hold. Moreover, suppose that(H5)
f satisfies the Lipschitz condition in [s2,s1].  Then(i)
for(λ,u)∈𝒞+∪𝒞-,
(2.16)s2<u(t)<s1,t∈[0,ω];(ii)
for(λ,u)∈𝒟+∪𝒟-, we have that either(2.17)max⁡t∈[0,ω]u(t)>s1
or
(2.18)min⁡t∈[0,ω]u(t)<s2.Corollary 2.4.
Let (H2)–(H5) hold. Then(i)
ifλ∈(λ1/f∞,λ1/f0], then (1.1) has at least two solutions u∞+ and u∞-, such that u∞+ is positive on [0,ω] and u∞- is negative on [0,ω];(ii)
ifλ∈(λ1/f0,∞), then (1.1) has at least four solutions u∞+, u∞-, u0+, and u0-, such that u∞+, u0+ are positive on [0,ω] and u∞-, u0- are negative on [0,ω].Corollary 2.5.
Let (H2)–(H5) hold. Then(i)
ifλ∈(λ1/f0,λ1/f∞], then (1.1) has at least two solutions u0+ and u0-, such that u0+ is positive on [0,ω] and u0- is negative on [0,ω];(ii)
ifλ∈(λ1/f∞,∞), then (1.1) has at least four solutions u∞+, u∞-, u0+, and u0-, such that u∞+, u0+ are positive on [0,ω] and u∞-, u0- are negative on [0,ω].


## 3. Proof of the Main Results

To prove Theorem2.3, we give a Proposition.Proposition 3.1.
(i) The first-order boundary value problem(3.1)u′(t)+a(t)u(t)=h(t),u(0)=u(ω)
has a unique solution for all h∈L1[0,ω] if and only if ∫0ωa(s)ds≠0.
(ii) Assume thatu is a solution of (3.1). If h≥0 and h(·)≢0 on any subinterval of [0,ω], then u(t)∫0ωa(s)ds>0 on [0,ω].Proof.
(i) The equationu′(t)+a(t)u(t)=0 has a solution u(t)=Ce-∫0ta(s)ds, where C is a constant. If u(t) is a nontrivial solution, then by u(0)=C,u(ω)=Ce-∫0ωa(s)ds, we can get that ∫0ωa(s)ds=0.
On the other hand, from∫0ωa(s)ds=0, we can get that u′(t)+a(t)u(t)=0 has a nontrivial solution u(t)=Ce-∫0ta(s)ds, where C∈ℝ∖{0}.
(ii) We claim thatu(t)≠0,t∈[0,ω]. Suppose on the contrary that there exists t0∈[0,ω], such that u(t0)=0; it is not difficult to compute that
(3.2)u′(t)+a(t)u(t)=h(t),u(t0)=u0
has a solution
(3.3)u(t)=∫t0th(s)e∫tsa(τ)dτds.
Since h≥0, we have
(3.4)u(0)≤u(t0)≤u(ω).
Ifh(t̂)>0,t̂∈[0,t0), then there exists a neighborhood U(t̂)⊂[0,t0) of t̂, such that h(t)>0 on U(t̂). Thus, u(0)=∫t00h(s)e∫0sa(τ)dτds<0; this contradicts with u(0)=u(ω).
Ifh(t̅)>0,t̅∈(t0,ω], then there exists a neighborhood U(t̅)⊂(t0,ω] of t̅, such that h(t)>0 on U(t̂). By using a similar way, we can prove that u(ω)>0, which also contradicts with u(0)=u(ω).
Henceu(t)≠0 on [0,ω]. Moreover, it follows that
(3.5)∫0ωu′(t)u(t)dt+∫0ωa(t)dt=∫0ωh(t)u(t)dt,
that is,
(3.6)(ln⁡⁡u(t))∣0ω+∫0ωa(t)dt=∫0ωh(t)u(t)dt.
Thus ∫0ωa(t)dt=∫0ωh(t)/u(t)dt, that is u∫0ωa(s)ds>0.Next, we prove Theorem2.3 and Corollaries 2.4 and 2.5.Proof of Theorem2.3.
Suppose on the contrary that there exists(λ,u)∈𝒞+∪𝒞-∪𝒟+∪𝒟- such that either
(3.7)max⁡⁡{u(t)∣t∈[0,ω]}=s1
or
(3.8)min⁡⁡{u(t)∣t∈[0,ω]}=s2.
We divide the proof into two cases.
Case 1 (max{u(t)|t∈[0,ω]}=s1).
In this case, we know that(3.9)0≤u(t)≤s1,0≤u(t-τ(t))≤s1,t∈[0,ω].
Let us consider the functional differential equation
(3.10)u′(t)+a(t)u(t)=λb(t)f(u(t-τ(t))),t∈R.
By (H2), (H4) and (H5), there exists m≥0 such that b(t)f(s)+ms is strictly increasing on s for s∈[s2,s1]. Then (3.10) can be rewritten to the form
(3.11)Lu+λmu(t-τ(t))=λ[b(t)f(u(t-τ(t)))+mu(t-τ(t))],
and since Ls1-a(t)s1=0=f(s1),
(3.12)Ls1-a(t)s1+λms1=λ[b(t)f(s1)+ms1].
Subtracting, we get
(3.13)L(s1-u)+λm(s1-u(t-τ(t)))-a(t)s1≥0.
That is,
(3.14)L(s1-u)+λms1≥0,t∈[0,ω],s1-u(0)=s1-u(ω)>0.
From Proposition 3.1, we deduce that s1>u(t),t∈[0,ω], which contradicts with that max{u(t)|t∈[0,ω]}=s1. Hence,
(3.15)u(t)<s1,t∈[0,ω].
Case 2 (min{u(t)|t∈[0,ω]}=s2).
In this case, we know that(3.16)s2≤u(t)≤0,s2≤u(t-τ(t))≤0,t∈[0,ω].
Let us consider (3.10); by (H2), (H4), and (H5), there exists m≥0 such that b(t)f(s)+ms is strictly increasing in s for s∈[s2,s1]. Then
(3.17)Lu+λmu(t-τ(t))=λ[b(t)f(u(t-τ(t)))+mu(t-τ(t))]
and since Ls2-a(t)s2=0=f(s2),
(3.18)Ls2-a(t)s2+λms2=λ[b(t)f(s2)+ms2].
Subtracting, we get
(3.19)L(s2-u)+λm(s2-u(t-τ(t)))-a(t)s2≤0.
That is
(3.20)L(s2-u)+λms2≤0,t∈[0,ω],s2-u(0)=s2-u(ω)<0.
From Proposition 3.1, we deduce that s2-u(t)<0,t∈[0,ω], this contradicts with that min{u(t)∣t∈[0,ω]}=s2. Therefore,
(3.21)s2<u(t),t∈[0,ω].Proof of Corollaries2.4 and 2.5.
Since boundary value problem(3.22)u′(t)+a(t)u(t)=0,u(0)=u(ω)
has a unique solution u≡0, we get
(3.23)(C+∪C-∪D+∪D-)⊂{(λ,u)∈R×E∣λ≥0}.
Take Λ∈ℝ as an interval such that Λ∩{λ1/f∞}={λ1/f∞} and ℳ as a neighborhood of (λ1/f∞,∞) whose projection on ℝ lies in Λ and whose projection on E is bounded away from 0. Then by [15, Theorem 1.6, and Corollary 1.8], we have that for each ν∈{+,-}, either(1)
𝒟ν∖ℳ is bounded in ℝ×E in which case 𝒟ν∖ℳ meets {(λ,0)|λ∈ℝ}, or(2)
𝒟ν∖ℳ is unbounded.
Moreover, if (1) occurs and𝒟ν∖ℳ has a bounded projection on ℝ, then 𝒟ν∖ℳ meets (λk/f∞,∞), where λk≠λ1 is another eigenvalue of (2.7).
Obviously, Theorem2.3 (ii) implies that (1) does not occur. So 𝒟+∖ℳ is unbounded.
Remark2.2 guarantees that 𝒟+ is a component of solutions of (2.12) in S+ which meets (λ1/f∞,∞), and consequently Projℝ(𝒟+∖ℳ) is unbounded. Thus
(3.24)ProjR(D+)⊃(λ1f∞,+∞).
Similarly, we get
(3.25)ProjR(D-)⊃(λ1f∞,+∞).
By Theorem 2.3, for any (λ,u)∈(𝒞+∪𝒞-),
(3.26)‖u‖∞<max⁡⁡{s1,|s2|}:=s*.
(3.26) and (2.12) imply that
(3.27)‖u‖<max⁡⁡{s*,‖a‖∞s*+λ‖b‖∞max⁡|s|≤s*|f(s)|},
which means that the sets {(λ,u)∈𝒞+∣λ∈[0,d]} and {(λ,u)∈𝒞-∣λ∈[0,d]} are bounded for any fixed d∈(0,∞). This together with the fact that 𝒞+ and 𝒞- join (λ1/f0,0) to infinity yields, respectively, that
(3.28)ProjR(C+)⊃(λ1f0,+∞),ProjR(C-)⊃(λ1f0,+∞).
Combining (3.24), (3.25), and (3.28), we conclude the desired results.Remark 3.2.
The methods used in the proof of Theorem2.3, Corollaries 2.4, and 2.5 have been used in the study of other kinds of boundary value problems; see [16–18] and the references therein.Remark 3.3.
The conditions in Corollaries2.4 and 2.5 are sharp. Let us take
(3.29)a(t)≡a>0,λ=a,b(t)=1,f(s)=s+h(s),τ(t)≡0.
Let
(3.30)h(s)={-2ss2+1,s∈(-∞,-1)∪(1,+∞),-2s3s2+1,s∈[-1,1],
and consider problem
(3.31)u′(t)=-a(t)u(t)+a[u(t)+h(u(t))],t∈[0,ω],u(0)=u(ω).
It is easy to see that λ1=a,f0=f∞=1. Since
(3.32)λ1f∞=a=λ1f0,
the conditions of Corollaries 2.4 and 2.5 are not valid. In this case, (3.31) has no nontrivial solution. In fact, if u is a nontrivial solution of (3.31), then
(3.33)0=∫0ωu′(t)dt=a∫0ωh(u(t))dt≠0,
which is a contradiction.



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