# Existence for Singular Periodic Problems: A Survey of Recent Results

Authors: Jifeng Chu; Juntao Sun; Patricia J. Y. Wong
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We present a survey on the existence of periodic solutions of singular
differential equations. In particular, we pay our attention to singular
scalar differential equations, singular damped differential equations, singular
impulsive differential equations, and singular differential systems.


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## Body

## 1. Introduction

During the last two decades, singular differential equations have attracted many researchers [1–11] because such equations describe many problems in the applied sciences, such as the Brillouin focusing system [12–14], nonlinear elasticity [15], and gravitational forces [3]. Besides these important applications, it has been found that a particular case of singular equations, the Ermakov-Pinney equation, plays an important role in studying the Lyapunov stability of periodic solutions of Lagrangian equations [16–18].In the literature, two different approaches have been used to establish the existence results for singular equations. The first one is the variational approach [3, 4, 6, 19, 20] and the second one is topological methods [1, 10, 21–28]. In our opinion, the first important result was proved in the pioneering paper of Lazer and Solimini [29]. They proved that a necessary and sufficient condition for the existence of a positive periodic solution for
(1)x′′=1xλ+e(t)
is that the mean value of e is negative; that is, e-<0, here λ≥1, which corresponds to a strong force condition, according to a terminology first introduced by Gordon [30]. Moreover, if 0<λ<1, which corresponds to a weak force condition, they found examples of functions e with negative mean values and yet no periodic solutions exist. Therefore, there is an essential difference between a strong singularity and a weak singularity. Since the work of Lazer and Solimini, the strong force condition became standard in related work, see, for instance, [8, 15, 18, 27, 28]. Compared with the case of a strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity is more recent, but it has also attracted many researchers [31–39]. In [39], for the first time in this topic, Torres et al. proved an existence result which is valid for a weak singularity, whereas the validity of such results under a strong force assumption remains as an open problem, which was partially solved in [32].The main aim of this survey is to present some recent existence results for singular differential equations. In particular, we will consider the scalar singular equations, singular damped equations, singular impulsive equations, and singular differential systems. We will also include some examples to illustrate the results presented.The rest of this paper is organized as follows. In Section2, we will state some important results for the second-order scalar singular differential equations. Singular damped equations will be considered in Section 3. In Section 4, singular impulsive differential equations will be studied. Finally in Section 5, we will focus on the singular differential systems. Sections 2 and 3 are mainly written by the first author. Section 4 is mainly written by the second author, and Section 5 is mainly completed by the third author. All the results presented in Sections 3–5 shed some lights on the differences between a strong singularity and a weak singularity.Finally in this section, we must note that besides the results presented in this survey, many interesting and important results on singular differential equations have been obtained by other researchers, see, for example, [9, 40–45] and the references cited therein.In this paper, we denote the essential supremum and infimum ofp by p* and p*, respectively, for a given function p∈L1[0,T] essentially bounded.


## 2. Second-Order Scalar Singular Equations

In this section, we recall some results for second-order singular differential equations(2)x′′+a(t)x=f(t,x)+e(t),
here a(t),e(t) are continuous, T-periodic functions. The nonlinearity f(t,x) is continuous in (t,x) and T-periodic in t and has a singularity at x=0.First we need to present some preliminary results on the linear equation(3)x′′+a(t)x=p(t)
with periodic boundary conditions
(4)x(0)=x(T),x′(0)=x′(T).
We assume the following:(A)
the Green functionG(t,s), associated with (3)-(4), is positive for all (t,s)∈[0,T]×[0,T], or(B)
the Green functionG(t,s), associated with (3)-(4), is nonnegative for all (t,s)∈[0,T]×[0,T].Whena(t)=k2, condition (A) is equivalent to 0<k2<λ1=(π/T)2 and condition (B) is equivalent to 0<k2≤λ1. In this case, we have
(5)G(t,s)={sink(t-s)+sink(T-t+s)2k(1-coskT),0≤s≤t≤T,sink(s-t)+sink(T-s+t)2k(1-coskT),0≤t≤s≤T.For a nonconstant functiona(t), there is an Lp-criterion proved in [46], which is given in Lemma 1 for the sake of completeness. Let K(q) denote the best Sobolev constant in the following inequality:
(6)C∥u∥q2≤∥u′∥22,∀u∈H01(0,T).
The explicit formula for K(q) is
(7)K(q)={2πqT1+2/q(22+q)1-2/q(Γ(1/q)Γ(1/2+1/q))2if1≤q<∞,4Tifq=∞,
where Γ is the gamma function, see [47, 48].Lemma 1 (see [46, Corollary 2.3]).
Assume thata(t)≻0 and a∈Lp[0,T] for some 1≤p≤∞. If
(8)∥a∥p<K(2p~),
then the condition (A) holds. Moreover, condition (B) holds if
(9)∥a∥p≤K(2p~).When the hypothesis (A) is satisfied, we denote(10)m=min0≤s,t≤TG(t,s),M=max0≤s,t≤TG(t,s),σ=mM.
Obviously, M>m>0 and 0<σ<1.The first existence result deals with the case of a strong singularity and the proof is based on the following nonlinear alternative of Leray-Schauder, which can be found in [49] or [50, pages 120–130].Lemma 2.
AssumeΩ is an open subset of a convex set K in a normed linear space X and p∈Ω. Let T:Ω¯→K be a compact and continuous map. Then one of the following two conclusions holds.(I)
T has at least one fixed point in Ω¯.(II)
There existsx∈∂Ω and 0<λ<1 such that x=λTx+(1-λ)p.Theorem 3 (see [37, Theorem 4.1]).
Suppose thata(t) satisfies (A) and f(t,x) satisfies the following.(H1)
There exists a nonincreasing positive continuous functiong0(x) on (0,∞) and a constant R0>0 such that f(t,x)≥g0(x) for (t,x)∈[0,T]×(0,R0], where g0(x) satisfies
(11)limx→0+g0(x)=+∞,limx→0+∫xR0‍g0(u)du=+∞.(H2)
There exist continuous, nonnegative functionsg(x) and h(x) such that
(12)0≤f(t,x)≤g(x)+h(x)∀(t,x)∈[0,T]×(0,∞),g(x)>0 is nonincreasing and h(x)/g(x) is nondecreasing in x∈(0,∞).(H3)
There exists a positive numberr such that σr+γ*>0 and
(13)rg(σr+γ*){1+(h(r+γ*)/g(r+γ*))}>ω*,here(14)γ(t)=∫0T‍G(t,s)e(s)ds,ω(t)=∫0T‍G(t,s)ds.Then for each e∈ℂ(ℝ/Tℤ,ℝ), (2) has at least one positive periodic solution x with x(t)>γ(t) for all t and 0<∥x-γ∥<r.Note that the study in [37, Theorem 4.1] is slightly different from the above presentation. However, the proof of the above theorem follows from that of [37, Theorem 4.1] with some minor necessary changes. Condition (H1) corresponds to the classical strong force condition, which was first introduced by Gordon in [30]. In fact, condition (H1) is only used when we try to obtain a prior lower bound. In Theorem 4, we will show that, for the case γ*≥0, we can remove the strong force condition (H1) and replace it by one weak force condition.Theorem 4 (see [33, Theorem 3.1]).
Assume that (A) and (H2)-(H3) are satisfied. Suppose further the following condition.(H4)
For each constantL>0, there exists a continuous function ϕL≻0 such that f(t,x)≥ϕL(t) for all (t,x)∈[0,T]×(0,L].
Then for eache(t) with γ*≥0,  (2) has at least one positive periodic solution x with x(t)>γ(t) for all t and 0<∥x-γ∥<r.For the superlinear case, we can establish the multiplicity result. The proof is based on a well-known fixed point theorem in cones, which can be found in [51]. Let K be a cone in X and D is a subset of X, we write DK=D∩K and ∂KD=(∂D)∩K.Theorem 5 (see [51]).
LetX be a Banach space and K a cone in X. Assume Ω1,Ω2 are open bounded subsets of X with ΩK1≠∅,Ω¯K1⊂ΩK2. Let
(15)T:Ω¯K2→K
be a completely continuous operator such that(a)
∥Tx∥≤∥x∥ for x∈∂KΩ1,(b)
there existsυ∈K∖{0} such that x≠Tx+λυforallx∈∂KΩ2 and all λ>0.
ThenT has a fixed point in Ω¯K2∖ΩK1.Theorem 6 (see [33, Theorem 3.2]).
Suppose thata(t) satisfies (A) and f(t,x) satisfies (H2)-(H3). Furthermore, assume the following conditions.(H5)
There exist continuous, nonnegative functionsg1(x),h1(x) such that
(16)f(t,x)≥g1(x)+h1(x),∀(t,x)∈[0,T]×(0,∞),g1(x)>0 is nonincreasing and h1(x)/g1(x) is non-decreasing in x.(H6)
There existsR>0 with σR>r such that
(17)σRg1(R+γ*){1+(h1(σR+γ*)/g1(σR+γ*))}≤ω*.Then  (2) has one positive periodic solution x~ with r<∥x~-γ∥≤R.Combined Theorems3 and 4 with Theorem 6, we can get the following two multiplicity results.Theorem 7.
Suppose thata(t) satisfies (A) and f(t,x) satisfies (H1)–(H3) and (H5)-(H6). Then  (2) has two different positive periodic solutions x and x~ with 0<∥x-γ∥<r<∥x~-γ∥≤R.Theorem 8.
Suppose thata(t) satisfies (A) and f(t,x) satisfies (H2)–(H6). Then  (2) has two different positive periodic solutions x and x~ with 0<∥x-γ∥<r<∥x~-γ∥≤R.To illustrate our results, we have selected the following singular equation:(18)x′′+a(t)x=x-α+μxβ+e(t),
here a,e∈ℂ[0,T], α,β>0, and μ∈ℝ is a given parameter. The corresponding results are also valid for the general case
(19)x′′+a(t)x=b(t)xα+μc(t)xβ+e(t),
with b,c∈ℂ[0,T].Corollary 9.
Assume thata(t) satisfies (A) and α>0,β≥0,μ>0. Then one has the following results.(i)
Ifα≥1,β<1, then for each e∈ℂ(ℝ/Tℤ,ℝ), (18) has at least one positive periodic solution for all μ>0.(ii)
Ifα≥1,β≥1, then for each e∈ℂ(ℝ/Tℤ,ℝ), (18) has at least one positive periodic solution for each 0<μ<μ1; here μ1 is some positive constant.(iii)
Ifα≥1,β>1, then for each e∈ℂ(ℝ/Tℤ,ℝ), (18) has at least two positive periodic solutions for each 0<μ<μ1.(iv)
Ifα>0,β<1, then for each e∈ℂ(ℝ/Tℤ,ℝ), with γ*≥0, (18) has at least one positive periodic solution for all μ>0.(v)
Ifα>0,β≥1, then for each e∈ℂ(ℝ/Tℤ,ℝ), with γ*≥0, (18) has at least one positive periodic solution for each 0<μ<μ1.(vi)
Ifα>0,β>1, then for each e∈ℂ(ℝ/Tℤ,ℝ), with γ*≥0, (18) has at least two positive periodic solutions for each 0<μ<μ1.All the above results require that the linear equation satisfies (A), which cannot cover the critical case. The next few results deal with the case when the condition (B) is satisfied and the proof is based on Schauder's fixed point theorem.Theorem 10 (see [31, Theorem 3.1]).
Assume that conditions (B) and (H2) and (H4) are satisfied. Furthermore, suppose that(H7)
there exists a positive constantR>0 such that R>Φ*,Φ*+γ*>0 and
(20)R≥g(Φ*+γ*){1+h(R+γ*)g(R+γ*)}ω*,hereΦ*=mintΦ(t),Φ(t)=∫0T‍G(t,s)ϕR+γ*(s)ds.
Then  (2) has at least one positive T-periodic solution.As an application of Theorem10, we consider the case γ*=0. Corollary 11 is a direct result of Theorem 10.Corollary 11 (see [31, Corollary 3.2]).
Assume that conditions (B) and (H2) and (H4) are satisfied. Furthermore, assume that (H8)
there exists a positive constantR>0 such that R>Φ* and
(21)R≥g(Φ*){1+h(R+γ*)g(R+γ*)}ω*.  If γ*=0, then  (2) has at least one positive T-periodic solution.Corollary 12 (see [31, Example 3.5]).
Suppose thata satisfies (B) and 0<α<1, β≥0, then for each e(t)∈ℂ(ℝ/Tℤ,ℝ), with γ*=0, one has the following:(i)
ifα+β<1-α2, then (18) has at least one positive periodic solution for each μ≥0,(ii)
ifα+β≥1-α2, then (18) has at least one positive T-periodic solution for each 0≤μ<μ2, where μ2 is some positive constant.The next results explore the case whenγ*>0.Theorem 13 (see [31, Theorem 3.6]).
Suppose thata(t) satisfies (B) and f(t,x) satisfies condition (H2). Furthermore, assume that(H9)
there existsR>γ* such that
(22)g(γ*){1+h(R+γ*)g(R+γ*)}ω*≤R.  If γ*>0, then  (2) has at least one positive T-periodic solution.Corollary 14 (see [31, Example 3.8]).
Suppose thata(t) satisfies (B) and α,β≥0, then for each e∈ℂ(ℝ/Tℤ,ℝ), with γ*>0, one has the following:(i)
ifα+β<1, then (18) has at least one positive T-periodic solution for each μ≥0,(ii)
ifα+β≥1, then (18) has at least one positive T-periodic solution for each 0≤μ<μ3, where μ3 is some positive constant.


## 3. Singular Damped Equations

In this section, we recall some results on second-order singular damped differential equations(23)x′′+h(t)x′+a(t)x=f(t,x,x′),
where h,a∈ℂ(ℝ/Tℤ,ℝ) and the nonlinearity f∈ℂ((ℝ/Tℤ)×(0,∞)×ℝ,ℝ). In particular, the nonlinearity may have a repulsive singularity at x=0, which means that
(24)limx→0+f(t,x,y)=+∞,uniformlyin(t,y)∈ℝ2.
First we recall some results on the linear damped equation
(25)x′′+h(t)x′+a(t)x=0,
associated to periodic boundary conditions (4). As in the last section, we say that (25)-(4) is nonresonant when its unique T-periodic solution is the trivial one. When (25)-(4) is nonresonant, as a consequence of Fredholm's alternative, the nonhomogeneous equation
(26)x′′+h(t)x′+a(t)x=l(t)
admits a unique T-periodic solution which can be written as
(27)x(t)=∫0T‍G2(t,s)l(s)ds,
where G2(t,s) is the Green's function of problem (25)-(4). We also assume that the following standing hypothesis is satisfied.(C)
The Green's functionG2(t,s), associated with (25)-(4), is positive for all (t,s)∈[0,T]×[0,T].To guarantee that (C) is satisfied, we require the antimaximum principle for (25)-(4) proved by Hakl and Torres in [52]. To do this, let us define the functions
(28)σ(h)(t)=exp(∫0t‍h(s)ds),σ1(h)(t)=σ(h)(T)∫0t‍σ(h)(s)ds+∫tT‍σ(h)(s)ds.Lemma 15 (see [52, Theorem 2.2]).
Assume thata≢0 and the following two inequalities are satisfied:
(29)∫0Ta(s)σ(h)(s)σ1(-h)(s)ds≥0,sup0≤t≤T{∫tt+Tσ(-h)(s)ds∫tt+T[a(s)]+σ(h)(s)ds}≤4,
where [a(s)]+=max{a(s),0}. Then (C) holds.For the special case∫0T‍a(t)σ(h)(t)dt>0 and h∈ℂ~(ℝ/Tℤ):={h∈ℂ(ℝ/Tℤ):h-=0}, one criterion has been developed by Cabada and Cid in [40].Theorem 16 (see [40, Theorem 5.1]).
Assume thath∈ℂ~(ℝ/Tℤ) and ∫0T‍a(t)σ(h)(t)dt>0. Suppose further that there exists 1≤p≤∞ such that
(30)(B(T))1+1/q∥𝒜+∥p,T<M2(2q),
where
(31)B(T)=∫0T‍σ(-h)(t)dt,𝒜+(t)=a+(t)(σ(h)(t))2-1/p.
Then (C) holds.Theorem 17 (see [35, Theorem 3.2]).
Suppose that  (25)  satisfies (C) and
(32)∫0T‍a(t)σ(h)(t)dt>0.
Furthermore, assume that there exists a constant r>0 such that(G1)
there exists a continuous functionϕr≻0 such that f(t,x,y)≥ϕr(t) for all (t,x,y)∈[0,T]×(0,r]×(-∞,∞),(G2)
there exist continuous, nonnegative functionsg(·), h(·), and ϱ(·) such that
(33)0≤f(t,x,y)≤(g(x)+h(x))ϱ(|y|),∀(t,x,y)∈[0,T]×(0,r]×ℝ,whereg(·)>0 is nonincreasing, h(·)/g(·) is non-decreasing in (0,r], and ϱ(·) is non-decreasing in (0,∞),(G3)
the following inequality holds:(34)rg(ιr){1+(h(r)/g(r))}ϱ(Lr)>ω*,where(35)ω(t)=∫0T‍G(t,s)ds,L=2∫0T‍a(t)σ(h)(t)dtmin0≤t≤Tσ(h)(t),ι=mM,m=min0≤s,t≤TG(t,s),M=max0≤s,t≤TG(t,s),
then   (23)  has at least one positive T-periodic solution x with 0<∥x∥≤r.Corollary 18 (see [35, Corollary 3.3]).
Let the nonlinearity in   (23)  be
(36)f(t,x,y)=(1+|y|γ)(x-α+μxβ),
where α>0, β,γ≥0, μ>0 is a positive parameter.(i)
Ifβ+γ<1, then   (23)  has at least one positive periodic solution for each μ>0.(ii)
Ifβ+γ≥1, then   (23)  has at least one positive periodic solution for each 0<μ<μ1*, where μ1* is some positive constant.Corollary 19 (see [35, Corollary 3.4]).
Let the nonlinearity in   (23)  be
(37)f(t,x,y)=(1+|y|γ)(1xα-μxβ),
where α>β>0,γ≥0 with γ<α+1, μ>0 is a positive parameter. Then there exists a positive constant μ2* such that   (23)  has at least one positive T-periodic solution for each 0≤μ<μ2*.Corollary19 is interesting because the singularity on the right-hand side combines attractive and repulsive effects. The analysis of such differential equations with mixed singularities is at this moment very incomplete, and few references can be cited [22, 44]. Therefore, the results in Corollary 19 can be regarded as one contribution to the literature trying to fill partially this gap in the study of singularities of mixed type.As in the last section, if we assume that the linear equation  (25)-(4) has a nonnegative Green's function, we can also get some results based on Schauder's fixed point theorem, and the results can cover the critical case.


## 4. Singular Impulsive Differential Equations

In this section, we will study the existence of periodic solutions for some singular differential equations with impulsive effects by using variational methods.Firstly, we consider the following second-order nonautonomous singular problem:(38)u′′-b(t)uα=e(t),a.e.t∈(0,T),u(0)-u(T)=u′(0)-u′(T)=0,
under the impulse conditions
(39)Δu′(tj)=Ij(u(tj)),j=1,2,…,p-1,
where tj,j=1,2,…,p-1 are the instants where the impulses occur and 0=t0<t1<t2<⋯<tp-1<tp=T, Ij:ℝ→ℝ(j=1,2,…,p-1) are continuous.Our result is presented as follows.Theorem 20 (see [19, Theorem 1.1]).
Assume thatα>1 and the following conditions hold.(S1)
b∈ℂ1([0,T],(0,∞)) is T-periodic and b′(t)≥0 for all t∈[0,T].(S2)
e∈L2([0,T],ℝ) is T-periodic and ∫0T‍e(t)dt<0.(S3)
There exist two constantsm,M such that for any t∈ℝ,
(40)m≤Ij(t)≤M,j=1,2,…,p-1,where m<0 and 0≤M<(-1/(p-1))∫0T‍e(t)dt.(S4)
For anyt∈ℝ,
(41)∫0t‍Ij(s)ds≥0,j=1,2,…,p-1.Then problem (38)-(39) has at least one solution.Remark 21.
In fact, it is not difficult to find some functionsIj satisfying (S3) and (S4). For example,
(42)Ij(t)=sint,t∈ℝ.Let(43)HT1={L2([0,T],ℝ)u:[0,T]→ℝ∣uisabsolutelycontinuous,u(0)=u(T)andu′∈L2([0,T],ℝ)},
with the inner product
(44)(u,v)=∫0T‍u(t)v(t)dt+∫0T‍u′(t)v′(t)dt,∀u,v∈HT1.
The corresponding norm is defined by
(45)∥u∥HT1=(∫0T‍|u(t)|2dt+∫0T‍|u′(t)|2dt)1/2,∀u∈HT1.
Then HT1 is a Banach space (in fact it is a Hilbert space).Ifu∈HT1, then u is absolutely continuous and u′∈L2([0,T],ℝ). In this case, Δu′(t)=u′(t+)-u′(t-)=0 is not necessarily valid for every t∈(0,T) and the derivative u′ may exist some discontinuities. It may lead to impulse effects.Following the ideas of [53], take v∈HT1 and multiply the two sides of the equality
(46)-u′′+b(t)uα+e(t)=0
by v and integrate from 0 to T, so we have
(47)∫0T‍[-u′′+b(t)uα+e(t)]vdt=0.
Note that since u′(0)-u′(T)=0, one has
(48)∫0T‍u′′(t)v(t)dt=∑j=0p-1‍∫tjtj+1‍u′′(t)v(t)dt=∑j=0p-1‍(u′(tj+1-)v(tj+1-)-u′(tj+)v(tj+))-∑j=0p-1‍∫tjtj+1‍u′(t)v′(t)dt=u′(T)v(T)-u′(0)v(0)-∑j=1p-1‍Δu′(tj)v(tj)-∫0T‍u′(t)v′(t)dt=-∑j=1p-1‍Ij(u(tj))v(tj)-∫0T‍u′(t)v′(t)dt.
Combining with (47), we get (49)∫0T‍u′(t)v′(t)dt+∑j=1p-1‍Ij(u(tj))v(tj)+∫0T‍b(t)uαv(t)dt+∫0T‍e(t)v(t)dt=0.
As a result, we introduce the following concept of a weak solution for problem (38)-(39).Definition 22.
One says that a functionu∈HT1 is a weak solution of problem (38)-(39) if
(50)∫0T‍u′(t)v′(t)dt+∑j=1p-1‍Ij(u(tj))v(tj)+∫0T‍b(t)uαv(t)dt+∫0T‍e(t)v(t)dt=0
holds for any v∈HT1.Define the functionalΦ:HT1→ℝ by
(51)Φ(u)≔12∫0T‍|u′(t)|2dt+∑j=1p-1‍∫0u(tj)‍Ij(s)ds+∫0T‍b(t)(∫1u(t)‍1sαds)dt+∫0T‍e(t)u(t)dt,
for every u∈HT1. Clearly, Φλ is well defined on HT1, continuously Gáteaux differentiable functional whose Gáteaux derivative is the functional Φλ′(u), given by
(52)Φλ′(u)v=∫0T‍u′(t)v′(t)dt+∑j=1p-1‍Ij(u(tj))v(tj)-∫0T‍b(t)uαv(t)dt+∫0T‍e(t)v(t)dt,
for any v∈HT1. Moreover, it is easy to verify that Φλ is weakly lower semicontinuous. Indeed, if {un}⊂HT1,u∈HT1, and un⇀u, then {un} converges uniformly to u on [0,T] and un→u on L2([0,T]), and combining the fact that liminfn→∞∥un∥HT1≥∥u∥HT1, one has
(53)liminfn→∞Φλ(un)=liminfn→∞(∑j=1p-1‍12∥un∥HT12-12∫0T‍|un(t)|2dt+∑j=1p-1‍∫0un(tj)‍Ij(s)ds-∫0T‍b(t)(∫1un(t)‍1sαds)dt+∫0T‍e(t)un(t)dt∑j=1p-1‍)≥12∫0T‍|u′(t)|2dt+∑j=1p-1‍∫0u(tj)‍Ij(s)ds-∫0T‍b(t)(∫1u(t)‍1sαds)dt+∫0T‍e(t)u(t)dt=Φλ(u).
By the standard discussion, the critical points of Φλ are the weak solutions of problem (38)-(39), see [53, 54].The following version of the mountain pass theorem will be used in our argument.Theorem 23 (see [55, Theorem 4.10]).
LetX be a Banach space and let φ∈C1(X,ℝ). Assume that there exist x0,x1∈X and an open neighborhood Ω of x0 such that x1∈X∖Ω- and
(54)max{φ(x0),φ(x1)}<infx∈∂Ωφ(x).
Let
(55)Γ={h∈C([0,1],X):h(0)=x0,h(1)=x1},c=infh∈Γmaxs∈[0,1]φ(h(s)).
If φ satisfies the (PS)-condition, that is, a sequence {un} in X satisfying φ(un) is bounded and φ′(un)→0 as n→∞ has a convergent subsequence, then c is a critical value of φ and c>max{φ(x0),φ(x1)}.Next we considerT-periodic solution for another impulsive singular problem:
(56)u′′(t)-1uα(t)=e(t),
under impulsive conditions
(57)Δu′(tj)=Ij(u(tj)),j=1,2,…,p-1,
where α≥1, e∈L1([0,T],ℝ) is T-periodic, Δu′(tj)=u′(tj+)-u′(tj-) with u′(tj±)=limt→tj±u′(t); tj,j=1,2,…,p-1 are the instants where the impulses occur, and 0=t0<t1<t2<⋯<tp-1<tp=T,tj+p=tj+T;Ij:ℝ→ℝ(j=1,2,…,p-1) are continuous and Ij+p≡Ij.In 1987, Lazer and Solimini [29] proved a famous result as follows.Theorem 24 (see [29]).
Assume thate∈L1([0,T],ℝ) is T-periodic. Then problem (56) has a positive T-periodic weak solution if and only if ∫0T‍e(t)dt<0.From Theorem24, if ∫0T‍e(t)dt≥0, then problem (52) does not have a positive T-periodic weak solution. However, if the impulses happen, for this singular problem may exist a positive T-periodic weak solution. Inspired by the above facts, our aim is to reveal a new existence result on positive T-periodic solution for singular problem (56) when impulsive effects are considered, that is, problem (56)-(57). Indeed, this periodic solution is generated by impulses. Here, we say a solution is generated by impulses if this solution is nontrivial when Ij≢0 for some 1<j<p-1, but it is trivial when Ij≡0 for all 1<j<p-1. For example, if problem (56)-(57) does not possess positive periodic solution when Ij≡0 for all 1<j<p-1, then a positive periodic solution u of problem (56)-(57) with Ij≢0 for some 1<j<p-1 is called a positive periodic solution generated by impulses.Our result is presented as follows.Theorem 25 (see [35, Theorem 1.2]).
Assume the following:(S1)
e∈L1([0,T],ℝ) is T-periodic and ∫0T‍e(t)dt≥0;(S2)
there exist two constantsm,M such that for any s∈ℝ,
(58)m≤Ij(s)≤M,j=1,2,…,p-1,where m≤M<(-1/(p-1))∫0T‍e(t)dt≤0.
Then problem (56)-(57) has at least a positive T-periodic solution.


## 5. Singular Differential Systems

In this section, we will consider the system of Hill's equations(59)ui′′(t)+ai(t)ui(t)=Fi(t,u1(t),u2(t),…,un(t)),ui′′(t)+ai(t)ui(t)=ui′′(t)(t)ui(t)1≤i≤n.
Here, ai and Fi are T-periodic in the variable t, ai∈L1[0,T], and the nonlinearities Fi(t,x1,x2,…,xn) can be singular at xj=0 where j∈{1,2,…,n}.Throughout, letu=(u1,u2,…,un). We are interested in establishing the existence of continuous T-periodic solutions u of the system (59), that is, u∈(C(ℝ))n and u(t)=u(t+T) for all t∈ℝ. Moreover, we are concerned with constant-sign solutions u, by which we mean θiui(t)≥0 for all t∈ℝ and 1≤i≤n, where θi∈{1,-1} is fixed. Note that positive solution, the usual consideration in the literature, is a special case of constant-sign solution when θi=1 for 1≤i≤n.We will employ the Schauder's fixed point theorem to establish the existence of solutions. Indeed, in Section5.1 we will first tackle a particular case of (59) when
(60)Fi(t,u(t))=∂2hi(t,12|u(t)|2)ui(t)+fi(t).
Here, ∂2hi is the partial derivative of hi with respect to the second variable, and |·| is a norm in ℝn. The particular case (60) occurs in the problem [36]
(61)u¨(t)+∇uP(t,u(t))=f(t),
where the potential
(62)P(t,u)=12a(t)|u|2-h(t,12|u|2),
and h presents a singularity of the repulsive type, that is, lim|x|→0h(t,x)=∞ uniformly in t. The general problem (59) will be investigated in Section 5.2; here the singularities are not necessarily generated by a potential as in the case of (60). To illustrate our results, several examples will be presented.In [45], the authors use a nonlinear alternative of the Leray-Schauder type and a fixed point theorem in cones to establish the existence of two positive periodic solutions for the system
(63)u¨(t)+a(t)u(t)=G(u(t)),
where G can be expressed as a sum of two positive functions satisfying certain monotone conditions. Therefore, the results in [45] are not applicable to (59) with Fi as in (60). In [45] it is also shown that the system
(64)u1′′(t)+a1(t)u1(t)=(u12+u22)-β+ν(u12+u22)γ,u2′′(t)+a2(t)u2(t)=(u12+u22)-β+ν(u12+u22)γ
has a solution when β>0,γ∈[0,1), and ν>0. We will generalize the system (64) in Examples 46–48 to allow ν to be zero or negative. The improvement is possible probably due to the fact that we do not need to make a technical truncation to get compactness when we employ the Schauder fixed point theorem as compared to when the Leray-Schauder alternative is used. In fact, the set that we work on excludes the singularities. The results presented in this section not only generalize the papers [36, 39, 45] to systems and existence of constant-sign solutions, but also improve and/or complement the results in these earlier work as well as other research papers [56–60]. This section is based on the work in [61].


### 5.1. Existence Results for (60)

In this section we will consider the system of Hill's equations(65)ui′′(t)+ai(t)ui(t)=∂2hi(t,12|u(t)|2)ui(t)+fi(t),ui′′(t)+ai(t)ui(t)=(t,12|u(t)|2)ui(t)1≤i≤n.
Here, ∂2hi(t,s)≡(∂/∂s)hi(t,s) and |·| is a norm in ℝn. Moreover, ai(t), ∂2hi(t,s), and fi(t) are T-periodic in t, ai∈L1[0,T],fi∈L1[0,T], and ∂2hi(t,s) can be singular at s=0.To seek aT-periodic solution uT=(u1T,u2T,…,unT) of the system (65), we first obtain a solution u*=(u1*,u2*,…,un*) of the following system of boundary value problems:
(66)ui′′(t)+ai(t)ui(t)=∂2hi(t,12|u(t)|2)ui(t)+fi(t),t∈[0,T],ui(0)=ui(T),ui′(0)=ui′(T),1≤i≤n.
Then, set
(67)uT(t)=u*(t-mT),t∈[mT,(m+1)T],m∈ℤ.Our main tool is Schauder's fixed point theorem, which is stated below for completeness.Theorem 26 (see [62]).
LetΩ be a convex subset of a Banach space B and S:Ω→Ω a continuous and compact map. Then S has a fixed point.To begin, letgi be Green's function of the boundary value problem
(68)x′′(t)+ai(t)x(t)=0,t∈[0,T],x(0)=x(T),x′(0)=x′(T).
Throughout, we will assume that the functions ai∈L1[0,T] are such that  (C1)
the Hill's equationx′′(t)+ai(t)x(t)=0 is nonresonant (i.e., the unique periodic solution is the trivial solution), and gi(t,s)≥0 for all (t,s)∈[0,T]×[0,T].Note that Torres [46] has a result on ai(t) that ensures that condition (C1) is satisfied. In fact, if ai(t)=k2, then (C1) holds if k∈(0,π/T]; if ai(t) is not a constant, then (C1) is valid if the Lp norm of ai(t) is bounded above by some specific constant.Letθi∈{1,-1},1≤i≤n be fixed. Define
(69)ϕi(t)=∫0T‍gi(t,s)θifi(s)ds,t∈[0,T],1≤i≤n.
We also let
(70)ϕimin=mint∈[0,T]ϕi(t),ϕimax=maxt∈[0,T]ϕi(t).We now present our main result which tackles (65) when the norm |·| in ℝn is the lp norm or the l∞ norm.Theorem 27.
Assume that the following conditions hold for each1≤i≤n:(C1),  (C2)
ϕimin>0;(C3)
letHi(t,s)=(∂/∂s)hi(t,s); for any numbers b,b′ with b′≥b>0, the function Hi:[0,T]×[b,b′]→ℝ is an L1-Carathéodory function, that is,
(i)
the maps↦Hi(t,s) is continuous for almost all t∈[0,T],(ii)
the mapt↦Hi(t,s) is measurable for all s∈[b,b′],(iii)
for anyr>0, there exists μr,i∈L1[0,T] such that |s|≤r(s∈[b,b′]) implies |Hi(t,s)|≤μr,i(t) for almost all t∈[0,T];(C4)
(∂/∂s)hi(t,s)≥0 for t∈[0,T] and s>0;(C5)
there existci>0 and αi>0 such that
(71)∂∂shi(t,s)≤cis-αi,t∈[0,T],s>0;(C6)
the norm|·| is the lp norm where 1≤p≤∞ is fixed, and
(72)∫0T‍gi(t,s)ds<Aip(ci2αi)-1,t∈[0,T],where(73)Aip={[∑k=1n‍(ϕkmin)p]2αi/p,1≤p<∞,[max1≤k≤nϕkmin]2αi,p=∞.Then, (65) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(74)ϕimin≤θiuiT(t)≤Ri,t∈ℝ,1≤i≤n,
where
(75)Ri≥ϕimin,Ri≥ϕimax[1-ci2αiAipmaxt∈[0,T]∫0T‍gi(t,s)ds]-1,1≤i≤n.Theorem27 is proved using Theorem 26; in fact we will seek a constant-sign solution of (66) in (C[0,T])n and then extend it to a T-periodic constant-sign solution of (65) as in (67). Here, let Ω be the closed convex set given by
(76)Ω={u∈(C[0,T])n∣ϕimin≤θiui(t)≤Ri,t∈[0,T],1≤i≤n(C[0,T])n(C[0,T])n∣ϕimin(C[0,T])n},
where Ri(≥ϕimin>0) is chosen as in (75), and define the operator S:Ω→(C[0,T])n as
(77)Su(t)=(S1u(t),S2u(t),…,Snu(t)),t∈[0,T],
where
(78)Siu(t)=∫0T‍gi(t,s)[∂2hi(s,12|u(s)|2)ui(s)+fi(s)]ds,t∈[0,T],1≤i≤n.
Clearly, a fixed point of Su=u is a solution of (66). We can show that S(Ω)⊆Ω; that is, Si(Ω)⊆Ω for each 1≤i≤n. Further, we can prove that S:Ω→Ω is continuous and compact; that is, Siu is bounded and is equicontinuous for any u∈Ω and 1≤i≤n. By Theorem 26, the system (66) has a constant-sign solution u*∈Ω. Now, a T-periodic constant-sign solution uT of (65) can be obtained as in (67).Remark 28.
The constantsci that appear in (C5) determine the upper bounds Ri of the solution uiT,1≤i≤n. Noting (75), we see that a smaller (bigger) ci gives a smaller (bigger) Ri, and hence a smaller (bigger) set Ω where the solution lies.In the next result, we will relax the condition (C6). The tradeoff is the upper boundsRi of the solution that may be bigger than those in (75). Also the bounds Ri do not depend on p (p as in lp norm) and so the information of p is not utilized. This result is obtained by following the main arguments in the derivation of Theorem 27 but modify the proof of θiSiu(t)≤Ri,t∈[0,T].Theorem 29.
Assume that (C1)–(C5) hold for each1≤i≤n. The norm |·| is the lp norm where 1≤p≤∞ is fixed. Then (65) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(79)ϕimin≤θiuiT(t)≤Ri,t∈ℝ,1≤i≤n,
where, for 1≤i≤n we have Ri≥ϕimin,
(80)Ri2αi>ci2αimaxt∈[0,T]∫0T‍gi(t,s)ds,ifαi∈(0,12),(81)Ri[1-ci2αiRi-2αimaxt∈[0,T]∫0T‍gi(t,s)ds]≥ϕimax,ifαi∈(0,12),(82)Ri≥ci2αi(ϕimin)1-2αimaxt∈[0,T]∫0T‍gi(t,s)ds+ϕimax,ifαi≥12.Remark 30.
A similar remark as Remark28 also holds for Theorem 29. Moreover, we note that the upper bounds Ri that fulfill (80)–(82) are independent of p, thus the information of |·| being a particular lp norm is not used. On the other hand, in Theorem 27, the upper bounds Ri that satisfy (75) depend on p. The sharpness of the bounds in both theorems cannot be compared in general; however, we will give an example at the end of this section to illustrate the results.In the next result, we will relax the condition (C2). Here, we allowϕi(t)≤0 for some i∈{1,2,…,n} and some t∈[0,T].Theorem 31.
Suppose that(C7)
there existsj∈{1,2,…,n} such that ϕjmin>0.
LetJ={j∈{1,2,…,n}|ϕjmin>0} and let J′={1,2,…,n}∖J. Assume that the following conditions hold for each 1≤i≤n: (C1), (C3), (C4), and  (C8)
there existci>0 such that
(83)∂∂shi(t,s)≤cis-αi,t∈[0,T],s>0,whereαj>0 for j∈J and αk∈(0,1/2) for k∈J′.
Further, let the following hold for eachj∈J:(C9)
the norm|·| is the lp norm where 1≤p≤∞ is fixed, and
(84)∫0T‍gj(t,s)ds<A-jp(cj2αj)-1,t∈[0,T],where(85)A-jp={[∑ℓ∈J‍(ϕℓmin)p]2αj/p,1≤p<∞,[maxℓ∈Jϕℓmin]2αj,p=∞.Then, (65) has a T-periodic solution uT∈(C(ℝ))n such that
(86)ϕjmin≤θjujT(t)≤Rj,t∈ℝ,j∈J,|ukT(t)|≤Rk,t∈ℝ,k∈J′,
where
(87)Rj≥ϕjmin,Rj≥ϕjmax[1-cj2αj(A-jp)-1maxt∈[0,T]∫0T‍gj(t,s)ds]-1,j∈J,(88)Rk2αk>ck2αkmaxt∈[0,T]∫0T‍gk(t,s)ds,k∈J′,(89)Rk[1-ck2αkRk-2αkmaxt∈[0,T]∫0T‍gk(t,s)ds]≥maxt∈[0,T]|ϕk(t)|,k∈J′.To derive Theorem31, we let the closed convex set Ω* be
(90)Ω*={u∈(C[0,T])n∣ϕjmin≤θjuj(t)≤Rj,t∈[0,T],j∈J;|uk(t)|≤Rk,t∈[0,T],k∈J′},
where Rj(≥ϕjmin>0) and Rk are chosen as in (87)–(89). Next, we define the operator S:Ω*→(C[0,T])n as in (78) and show that Theorem 26 is applicable.Remark 32.
From the conclusion of Theorem29, we see that the solution uT is “partially” of constant sign, in the sense that θjujT(t)≥0 for j∈J, but may not be so for j∈J′. Further, the constants ci that appear in (C8) determine the upper bounds Ri of the solution uiT,1≤i≤n. From (87) and (88), we see that a smaller (bigger) ci gives a smaller (bigger) Ri, and hence a smaller (bigger) set Ω* where the solution lies.Using similar arguments as in the derivation of Theorems31 and 29 (in getting Sju∈Ω* for j∈J and u∈Ω*), we obtain the following result.Theorem 33.
Suppose that (C7) hold. LetJ={j∈{1,2,…,n}|ϕjmin>0} and let J′={1,2,…,n}∖J. Assume the following conditions hold for each 1≤i≤n: (C1), (C3), (C4), and (C8). Then, (65) has a T-periodic solution uT∈(C(ℝ))n such that (91)ϕjmin≤θjujT(t)≤Rj,t∈ℝ,j∈J,|ukT(t)|≤Rk,t∈ℝ,k∈J′,
where
(92)Rj≥ϕjmin,j∈J,Rj2αj>cj2αjmaxt∈[0,T]∫0T‍gj(t,s)ds,ifαj∈(0,12),j∈J,Rj[1-cj2αjRj-2αjmaxt∈[0,T]∫0T‍gj(t,s)ds]≥ϕjmax,maxt∈[0,T]∫0T‍∫0T‍gj(t,s)ifαj∈(0,12),j∈J,Rj≥cj2αj(ϕjmin)1-2αjmaxt∈[0,T]∫0T‍gj(t,s)ds+ϕjmax,ifαj≥12,j∈J,(93)Rk2αk>ck2αkmaxt∈[0,T]∫0T‍gk(t,s)ds,k∈J′,Rk[1-ck2αkRk-2αkmaxt∈[0,T]∫0T‍gk(t,s)ds]≥maxt∈[0,T]|ϕk(t)|,k∈J′.Remark 34.
A similar remark as Remark32 holds for Theorem 33. Also, we observe once again that the upper bounds Rj that fulfill (92) are independent of p, thus the information of |·| being a particular lp norm is not used. On the other hand, in Theorem 31, the upper bounds Rj that satisfy (87) depend on p.We will now present an example that illustrates Theorems27 and 29.Example 35.
Consider (65) when
(94)T=2π,n=2,a1(t)=a2(t)=14,f1(t)=1,f2(t)=12,h1(t,s)=ln(s+1)|sint|+1,h2(t,s)=ln(s+1)3(|cost|+1),|·|=lpnorm(1≤p≤∞).Fixθi=1,1≤i≤n, that is, we are seeking positive solutions. The corresponding Green's function has the explicit expression [36]
(95)g1(t,s)=g2(t,s)={cos12(t-s-π),0≤s≤t≤2π,cos12(s-t-π),0≤t≤s≤2π.
Condition (C1) is satisfied. By direct computation, we get ϕ1(t)=4 and ϕ2(t)=2 for t∈[0,2π]. Thus, (C2) is fulfilled with
(96)ϕ1min=ϕ1max=4,ϕ2min=ϕ2max=2.
Moreover, we have
(97)∂∂sh1(t,s)=1|sint|+11s+1≤1s+1≤1s,∂∂sh2(t,s)=13(|cost|+1)1s+1≤13(s+1)≤13s
and so it is clear that (C4) and (C5) are satisfied with
(98)α1=1,c1=1,α2=1,c2=13.
Finally, we compute
(99)A1p=A2p=(4p+2p)2/p,1≤p<∞,A1∞=A2∞=16.
Since ∫02π‍gi(t,s)ds=4 for t∈[0,2π] and i=1,2, we check that (C6) holds for all 1≤p≤∞.All the conditions of Theorem27 are satisfied, thus we conclude that the problem (65) with (94) has a positive 2π-periodic solution u=(u1,u2) such that
(100)ϕimin≤ui(t)≤Ri,t∈ℝ,i=1,2,
where (from (75))
(101)Ri≥ϕimax[1-8ciAip]-1≡Lip,1≤p≤∞,i=1,2.We can also apply Theorem29 to conclude that the problem (65) with (94) has a positive 2π-periodic solution u=(u1,u2) satisfying (100) and (from (82))
(102)Ri≥8ci(ϕimin)-1+ϕimax≡Mi,i=1,2.As mentioned in Remark30, in general we cannot compare Lip and Mi. In fact, a direct calculation gives
(103)p=1:L11=5.14<M1=6,L21=2.16<M2=3.33,p=2:L12=6.67>M1=6,L22=2.31<M2=3.33,p=∞:L1∞=8>M1=6,L2∞=2.4<M2=3.33.


### 5.2. Existence Results for (59)

In this section we will consider the general system of Hill's equations(104)ui′′(t)+ai(t)ui(t)=Fi(t,u(t)),1≤i≤n.
Here, ai and Fi are T-periodic in the variable t, ai∈L1[0,T], and the nonlinearities Fi(t,x1,x2,…,xn) can be singular at xj=0 where j∈{1,2,…,n}.Once again, to obtain aT-periodic solution uT=(u1T,u2T,…,unT) of the system (104), we first seek a solution u*=(u1*,u2*,…,un*) of the following system of boundary value problems:
(105)ui′′(t)+ai(t)ui(t)=Fi(t,u(t)),t∈[0,T],ui(0)=ui(T),ui′(0)=ui′(T),1≤i≤n.
The periodic solution is then given by
(106)uT(t)=u*(t-mT),t∈[mT,(m+1)T],m∈ℤ.Withgi being the Green's function of the boundary value problem (68), throughout we will assume that (C1) is satisfied. Moreover, for fixed θi∈{-1,1} and T-periodic functions qi∈L1[0,T],1≤i≤n, we define
(107)ηi(t)=∫0T‍gi(t,s)θiqi(s)ds,t∈[0,T],1≤i≤n
and also
(108)ηimin=mint∈[0,T]ηi(t),ηimax=maxt∈[0,T]ηi(t).
For b≥b′≥0 and 1≤i≤n, we denote the interval
(109)[b,b′]i={[b,b′],ifθi=1,[-b′,-b],ifθi=-1.
A similar definition is valid for (b,b′)i.Using Schauder's fixed point theorem, we will establish existence results for the system (104).Theorem 36.
Assume the following conditions hold for each1≤i≤n: (C1);(C10)
for any numbersbj,bj′,1≤j≤n with bj′≥bj>0, the function Fi:[0,T]×∏j=1n‍[bj,bj′]j→ℝ is a L1-Carathéodory function, that is,
(i)
the mapu↦Fi(t,u) is continuous for almost all t∈[0,T],(ii)
the mapt↦Fi(t,u) is measurable for all u∈∏j=1n‍[bj,bj′]j,(iii)
for anyr>0, there exists μr,i∈L1[0,T] such that |u|≤r(u∈∏j=1n‍[bj,bj′]j) implies |Fi(t,u)|≤μr,i(t) for almost all t∈[0,T];(C11)
there existβi>0,γi∈[0,1), and T-periodic functions wi,qi with wi∈L1[0,T],qi∈L1[0,T] and wi(t)>0 for a.e.t∈[0,T] such that
(110)θiqi(t)|u|γi≤θiFi(t,u)≤θiqi(t)|u|γi+wi(t)|u|-βi,t∈[0,T],u∈∏k=1n‍(0,∞)k(here|·| is the lp norm where 1≤p≤∞ is fixed);(C12)
ηimin>0.
Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(111)ri≤θiuiT(t)≤Ri,t∈ℝ,1≤i≤n,
where, for 1≤i≤n one has
(112)0<ri≤Ri,ri≤(ηimin)1/(1-γi),(113)Ri≥ηimax|R|pγi+|r|p-βimaxt∈[0,T]∫0T‍gi(t,s)wi(s)ds,
(here |R|p is the lp norm of (R1,R2,…,Rn), likewise |r|p is the lp norm of (r1,r2,…,rn)).In proving Theorem36, we actually seek a constant-sign solution of (105) in (C[0,T])n and then extend it to a T-periodic constant-sign solution of (104) as in (106). Let Ω be the closed convex set given by
(114)Ω={u∈(C[0,T])n∣ri≤θiui(t)≤Ri,t∈[0,T],1≤i≤n(C[0,T])n},
where Ri≥ri>0 are chosen as in (112) and (113), and define the operator S:Ω→(C[0,T])n as
(115)Su(t)=(S1u(t),S2u(t),…,Snu(t)),t∈[0,T],
where
(116)Siu(t)=∫0T‍gi(t,s)Fi(s,u(s))ds,t∈[0,T],1≤i≤n.
Clearly, a fixed point of Su=u is a solution of (105). The conditions of Theorem 26 are then shown to be satisfied.Remark 37.
As seen from (112) and (113), the functions wi and qi that appear in (C11) determine the lower and upper bounds of the solution uiT,1≤i≤n.Theorem 38.
Assume that the following conditions hold for each1≤i≤n: (C1), (C10), (C11), and (C12). Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(117)r≤θiuiT(t)≤R,t∈ℝ,1≤i≤n,
where 0<r≤R, and for all 1≤i≤n,
(118)r≤{(ηiminnγi/p)1/(1-γi),1≤p<∞,(ηimin)1/(1-γi),p=∞,(119)R≥{Rγiηimaxnγi/p+r-βin-βi/p[maxt∈[0,T]∫0T‍gi(t,s)wi(s)ds],+r-βin-βi/p+r-βin-βi/p1≤p<∞,Rγiηimax+r-βi[maxt∈[0,T]∫0T‍gi(t,s)wi(s)ds],Rγiηimax[maxt∈[0,T]∫0T‍gi(t,s)wi(s)ds]p=∞.Theorem38 is obtained by similar arguments used in the derivation of Theorem 36, with a new Ω defined as
(120)Ω={u∈(C[0,T])n∣r≤θiui(t)≤R,t∈[0,T],1≤i≤n},
where R≥r>0 are chosen as in (118) and (119).Remark 39.
Remark37 also holds for Theorem 38. Further, comparing the bounds ri, Ri, 1≤i≤n in Theorem 36 (see (112), (113)) with the bounds r,R in Theorem 38 (see (118), (119)), we note that ri and Ri are lower and upper bounds for a particular θiuiT, whereas r and R are uniform lower and upper bounds for allθiuiT,1≤i≤n. However, the computation of Ri from (113) is more difficult than calculating R from (119).Our next result tackles the case whenηimin=0.Theorem 40.
Assume that the following conditions hold for each1≤i≤n: (C1), (C10),  (C13)
there existβi∈(0,1),γi∈[0,1), and T-periodic functions wi,vi,qi with wi∈L1[0,T],vi∈L1[0,T],qi∈L1[0,T], and wi(t),vi(t)>0 for a.e.t∈[0,T] such that
(121)θiqi(t)|u|γi+vi(t)|u|-βi≤θiFi(t,u)≤θiqi(t)|u|γi+wi(t)|u|-βi,t∈[0,T],u∈∏k=1n(0,∞)k(here|·| is the lp norm where 1≤p≤∞ is fixed);(C14)
ηimin=0.
Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(122)1R≤θiuiT(t)≤R,t∈ℝ,1≤i≤n,
where R≥1, and for all 1≤i≤n,
(123)R≥{nβi/p(1-βi)[mint∈[0,T]∫0T‍gi(t,s)vi(s)ds]-1/(1-βi),[mint∈[0,T]∫0T‍gi(t,s)vi(s)]-1/1-βi1≤p<∞,[mint∈[0,T]∫0T‍gi(t,s)vi(s)ds]-1/(1-βi),[mint∈[0,T]∫0T‍gi(s)ds]-1/1-βip=∞,R≥{Rγiηimaxnγi/p+Rβin-βi/p[maxt∈[0,T]∫0T‍gi(t,s)wi(s)ds],Rγiηimaxnγi/p+Rβin-βi/p11111111111111≤p<∞,Rγiηimax+Rβi[maxt∈[0,T]∫0T‍gi(t,s)wi(s)ds],Rγiηimaxnγi/p+1111111Rβin-βi/pp=∞.The closed convex set used to get Theorem40 is given by
(124)Ω={u∈(C[0,T])n∣r≤θiui(t)≤R,t∈[0,T],1≤i≤n(C[0,T])n},
where r=1/R and R≥1 satisfies (123).Remark 41.
As seen from (123), the functions wi,vi, and qi that appear in (C13) determine the lower and upper bounds of the solution uiT,1≤i≤n.Finally, the next result tackles the case whenηimax<0.Theorem 42.
Assume that the following conditions hold for each1≤i≤n: (C1), (C10),  (C15)
there existβ∈(0,1) and T-periodic functions wi,vi,qi with wi∈L1[0,T],vi∈L1[0,T],qi∈L1[0,T], and wi(t),vi(t)>0 for a.e.t∈[0,T] such that
(125)θiqi(t)+vi(t)|u|-β≤θiFi(t,u)≤θiqi(t)+wi(t)|u|-β,t∈[0,T],u∈∏k=1n‍(0,∞)k(here|·| is the lp norm where 1≤p≤∞ is fixed);(C16)
ηimax<0;(C17)
ηimin≥n-β/(1+β)pW-β/(1-β2)(Vβ2)1/(1-β2)(1-1/β2) where
(126)W=max1≤k≤n[maxt∈[0,T]∫0T‍gk(t,s)wk(s)ds],V=min1≤k≤n[mint∈[0,T]∫0T‍gk(t,s)vk(s)ds].Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(127)r≤θiuiT(t)≤R,t∈ℝ,1≤i≤n,
where 0<r≤R are given by
(128)r={n-β/(1+β)pW-β/(1-β2)(Vβ2)1/(1-β2),1≤p<∞W-β/(1-β2)(Vβ2)1/(1-β2),p=∞,R={n-β/(1+β)pW1/(1-β2)(Vβ2)-β/(1-β2),1≤p<∞W1/(1-β2)(Vβ2)-β/(1-β2),p=∞.Theorem42 is obtained by considering the closed convex set
(129)Ω={u∈(C[0,T])n∣r≤θiui(t)≤R,t∈[0,T],1≤i≤n},
where R≥r>0 are determined later as those given in (128).Remark 43.
As seen from (128), the functions wi and vi that appear in (C15) determine the lower and upper bounds of the solution uiT,1≤i≤n.We have so far established the results when (i)ηimin>0, (ii) ηimin=0, and (iii) ηimax<0 for all1≤i≤n. However, it could be that we only have ηi(t)≥0 for some i and ηj(t)<0 for some j, which results in ηimin≥0 and ηjmax<0 for some1≤i,j≤n. We present two results for such a case as follows. Note that Theorem 44 is obtained by applying Theorems 38–42, while Theorem 45 is obtained by applying Theorems 36, 40, and 42.Theorem 44.
Let (C1) and (C10) hold for each1≤i≤n. Assume the following:(C18)
conditions (C11) and (C12) hold for somei∈I⊆{1,2,…,n};(C19)
conditions (C13) and (C14) hold for somei∈J⊆{1,2,…,n};(C20)
conditions (C15), (C16), and (C17) hold for somei∈K⊆{1,2,…,n};
whereI∪J∪K={1,2,…,n}. Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(130)r≤θiuiT(t)≤R,t∈ℝ,1≤i≤n,
where 0<r≤R satisfy  (a)
(118) and (119) for i∈I;(b)
r=1/R,R≥1, (123) for i∈J;(c)
(128) for i∈K.Theorem 45.
Let (C1) and (C10) hold for each1≤i≤n. Assume that (C18)–(C20) hold with I∪J∪K={1,2,…,n}. Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(131)ri≤θiuiT(t)≤Ri,t∈ℝ,i∈I,
where 0<ri≤Ri satisfy (112) and (113) for i∈I, and
(132)r≤θiuiT(t)≤R,t∈ℝ,i∈J∪K,
where 0<r≤R satisfy conclusions (b) and (c) of Theorem 44.We will now apply the results obtained to the following system of Hill's equations, a particular form of it (see (64)) that has been discussed in [45],
(133)u1′′(t)+a1(t)u1(t)=(u12+u22)-β1+ν1(u12+u22)γ1,u2′′(t)+a2(t)u2(t)=(u12+u22)-β2+ν2(u12+u22)γ2.
Clearly, the system (133) corresponds to (104) where n=2 and
(134)Fi(t,u)=(u12+u22)-βi+νi(u12+u22)γi,i=1,2.
We will assume that a1,a2∈L1[0,T] satisfy (C1). Note that condition (C10) is clearly satisfied. Further, let θ1=θ2=1, that is, we are interested in positive periodic solutions of (133).Example 46.
Consider the system (133) with
(135)νi>0,βi>0,γi∈[0,1),i=1,2.Clearly, (C11) is satisfied withp=2,qi=νi and wi=1,i=1,2. Thus, (C12) also holds since
(136)ηimin=mint∈[0,T]∫0T‍gi(t,s)θiqi(s)ds=νimint∈[0,T]∫0T‍gi(t,s)ds>0.
Theorem 38 (or Theorem 36) is applicable and we conclude that the system (133) with (135) has a T-periodic positive solution uT∈(C(ℝ))2 such that
(137)r≤uiT(t)≤R,t∈ℝ,i=1,2,
where 0<r≤R are such that
(138)r≤mini=1,2{(ηimin2γi/2)1/(1-γi)},(139)R≥maxi=1,2{Rγiηimax2γi/2+r-βi2-βi/2[maxt∈[0,T]∫0T‍gi(t,s)ds]}.To illustrate numerically, suppose(140)a1(t)=a2(t)=14,T=2π,ν1=14,ν2=1,γ1=γ2=12,β1=β2=1.
Green's function is given in (95) and
(141)ηimin=νimint∈[0,T]∫0T‍gi(t,s)ds=4νi.
Hence, (138) yields r≤2. Let r=2, then (139) reduces to
(142)R≥maxi=1,2{R1/24νi21/4+r-12-1/24νi}=R1/229/4+2,
which is satisfied by R≥26.48. Let R=26.48, then from (137) we conclude that the system (133) with (140) has a 2π-periodic positive solution u∈(C(ℝ))2 such that
(143)2≤ui(t)≤26.48,t∈ℝ,i=1,2.Example 47.
Consider the system (133) with
(144)νi=0,βi∈(0,1),γi∈[0,1),i=1,2.Here, (C13) is satisfied withp=2,qi=νi=0 and wi=vi=1,i=1,2. Subsequently, (C14) also holds since
(145)ηimin=mint∈[0,T]∫0T‍gi(t,s)θiqi(s)ds=νimint∈[0,T]∫0T‍gi(t,s)ds=0.
Employing Theorem 40, we conclude that the system (133) with (144) has a T-periodic positive solution uT∈(C(ℝ))2 such that
(146)1R≤uiT(t)≤R,t∈ℝ,i=1,2,
where R≥1, and from (123), we have for i=1,2,
(147)R≥2βi/2(1-βi)[mint∈[0,T]∫0T‍gi(t,s)ds]-1/(1-βi),R≥{2-βi/2[maxt∈[0,T]∫0T‍gi(t,s)ds]}1/(1-βi).
Combining the inequalities, we see that R should satisfy
(148)R≥max{{2-βi/2[maxt∈[0,T]∫0T‍gi(t,s)ds]}1/1-βi1,maxi=1,22βi/2(1-βi)×[mint∈[0,T]∫0T‍gi(t,s)ds]-1/(1-βi),maxi=1,2{2-βi/2[maxt∈[0,T]∫0T‍gi(t,s)ds]}1/(1-βi)}.Example 48.
Consider the system (133) with
(149)νi<0,βi=β∈(0,1),γi=0,i=1,2,(150)νimint∈[0,T]∫0T‍gi(t,s)ds≥2-β/2(1+β)W-β/(1-β2)(Vβ2)1/(1-β2)×(1-1β2),i=1,2,
where
(151)W=maxk=1,2[maxt∈[0,T]∫0T‍gk(t,s)ds],V=mink=1,2[mint∈[0,T]∫0T‍gk(t,s)ds].Obviously, (C15) is satisfied withp=2,qi=νi<0 and wi=vi=1,i=1,2. Then, (C16) also holds since
(152)ηimax=maxt∈[0,T]∫0T‍gi(t,s)θiqi(s)ds=νimaxt∈[0,T]∫0T‍gi(t,s)ds<0.
Moreover, condition (C17) is simply (150). Hence, we conclude from Theorem 42 that the system (133) with (149) and (150) has a T-periodic positive solution uT∈(C(ℝ))2 such that
(153)r≤uiT(t)≤R,t∈ℝ,i=1,2,
where 0<r≤R are given by
(154)r=2-β/2(1+β)W-β/(1-β2)(Vβ2)1/(1-β2),R=2-β/2(1+β)W1/(1-β2)(Vβ2)-β/(1-β2).Remark 49.
In [45], it is shown that (64) has a solution when β>0,γ∈[0,1) and ν>0. As seen from Examples 46–48, we have generalized the system (64) to allow ν to be zero or negative.


## 5.1. Existence Results for (60)

In this section we will consider the system of Hill's equations(65)ui′′(t)+ai(t)ui(t)=∂2hi(t,12|u(t)|2)ui(t)+fi(t),ui′′(t)+ai(t)ui(t)=(t,12|u(t)|2)ui(t)1≤i≤n.
Here, ∂2hi(t,s)≡(∂/∂s)hi(t,s) and |·| is a norm in ℝn. Moreover, ai(t), ∂2hi(t,s), and fi(t) are T-periodic in t, ai∈L1[0,T],fi∈L1[0,T], and ∂2hi(t,s) can be singular at s=0.To seek aT-periodic solution uT=(u1T,u2T,…,unT) of the system (65), we first obtain a solution u*=(u1*,u2*,…,un*) of the following system of boundary value problems:
(66)ui′′(t)+ai(t)ui(t)=∂2hi(t,12|u(t)|2)ui(t)+fi(t),t∈[0,T],ui(0)=ui(T),ui′(0)=ui′(T),1≤i≤n.
Then, set
(67)uT(t)=u*(t-mT),t∈[mT,(m+1)T],m∈ℤ.Our main tool is Schauder's fixed point theorem, which is stated below for completeness.Theorem 26 (see [62]).
LetΩ be a convex subset of a Banach space B and S:Ω→Ω a continuous and compact map. Then S has a fixed point.To begin, letgi be Green's function of the boundary value problem
(68)x′′(t)+ai(t)x(t)=0,t∈[0,T],x(0)=x(T),x′(0)=x′(T).
Throughout, we will assume that the functions ai∈L1[0,T] are such that  (C1)
the Hill's equationx′′(t)+ai(t)x(t)=0 is nonresonant (i.e., the unique periodic solution is the trivial solution), and gi(t,s)≥0 for all (t,s)∈[0,T]×[0,T].Note that Torres [46] has a result on ai(t) that ensures that condition (C1) is satisfied. In fact, if ai(t)=k2, then (C1) holds if k∈(0,π/T]; if ai(t) is not a constant, then (C1) is valid if the Lp norm of ai(t) is bounded above by some specific constant.Letθi∈{1,-1},1≤i≤n be fixed. Define
(69)ϕi(t)=∫0T‍gi(t,s)θifi(s)ds,t∈[0,T],1≤i≤n.
We also let
(70)ϕimin=mint∈[0,T]ϕi(t),ϕimax=maxt∈[0,T]ϕi(t).We now present our main result which tackles (65) when the norm |·| in ℝn is the lp norm or the l∞ norm.Theorem 27.
Assume that the following conditions hold for each1≤i≤n:(C1),  (C2)
ϕimin>0;(C3)
letHi(t,s)=(∂/∂s)hi(t,s); for any numbers b,b′ with b′≥b>0, the function Hi:[0,T]×[b,b′]→ℝ is an L1-Carathéodory function, that is,
(i)
the maps↦Hi(t,s) is continuous for almost all t∈[0,T],(ii)
the mapt↦Hi(t,s) is measurable for all s∈[b,b′],(iii)
for anyr>0, there exists μr,i∈L1[0,T] such that |s|≤r(s∈[b,b′]) implies |Hi(t,s)|≤μr,i(t) for almost all t∈[0,T];(C4)
(∂/∂s)hi(t,s)≥0 for t∈[0,T] and s>0;(C5)
there existci>0 and αi>0 such that
(71)∂∂shi(t,s)≤cis-αi,t∈[0,T],s>0;(C6)
the norm|·| is the lp norm where 1≤p≤∞ is fixed, and
(72)∫0T‍gi(t,s)ds<Aip(ci2αi)-1,t∈[0,T],where(73)Aip={[∑k=1n‍(ϕkmin)p]2αi/p,1≤p<∞,[max1≤k≤nϕkmin]2αi,p=∞.Then, (65) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(74)ϕimin≤θiuiT(t)≤Ri,t∈ℝ,1≤i≤n,
where
(75)Ri≥ϕimin,Ri≥ϕimax[1-ci2αiAipmaxt∈[0,T]∫0T‍gi(t,s)ds]-1,1≤i≤n.Theorem27 is proved using Theorem 26; in fact we will seek a constant-sign solution of (66) in (C[0,T])n and then extend it to a T-periodic constant-sign solution of (65) as in (67). Here, let Ω be the closed convex set given by
(76)Ω={u∈(C[0,T])n∣ϕimin≤θiui(t)≤Ri,t∈[0,T],1≤i≤n(C[0,T])n(C[0,T])n∣ϕimin(C[0,T])n},
where Ri(≥ϕimin>0) is chosen as in (75), and define the operator S:Ω→(C[0,T])n as
(77)Su(t)=(S1u(t),S2u(t),…,Snu(t)),t∈[0,T],
where
(78)Siu(t)=∫0T‍gi(t,s)[∂2hi(s,12|u(s)|2)ui(s)+fi(s)]ds,t∈[0,T],1≤i≤n.
Clearly, a fixed point of Su=u is a solution of (66). We can show that S(Ω)⊆Ω; that is, Si(Ω)⊆Ω for each 1≤i≤n. Further, we can prove that S:Ω→Ω is continuous and compact; that is, Siu is bounded and is equicontinuous for any u∈Ω and 1≤i≤n. By Theorem 26, the system (66) has a constant-sign solution u*∈Ω. Now, a T-periodic constant-sign solution uT of (65) can be obtained as in (67).Remark 28.
The constantsci that appear in (C5) determine the upper bounds Ri of the solution uiT,1≤i≤n. Noting (75), we see that a smaller (bigger) ci gives a smaller (bigger) Ri, and hence a smaller (bigger) set Ω where the solution lies.In the next result, we will relax the condition (C6). The tradeoff is the upper boundsRi of the solution that may be bigger than those in (75). Also the bounds Ri do not depend on p (p as in lp norm) and so the information of p is not utilized. This result is obtained by following the main arguments in the derivation of Theorem 27 but modify the proof of θiSiu(t)≤Ri,t∈[0,T].Theorem 29.
Assume that (C1)–(C5) hold for each1≤i≤n. The norm |·| is the lp norm where 1≤p≤∞ is fixed. Then (65) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(79)ϕimin≤θiuiT(t)≤Ri,t∈ℝ,1≤i≤n,
where, for 1≤i≤n we have Ri≥ϕimin,
(80)Ri2αi>ci2αimaxt∈[0,T]∫0T‍gi(t,s)ds,ifαi∈(0,12),(81)Ri[1-ci2αiRi-2αimaxt∈[0,T]∫0T‍gi(t,s)ds]≥ϕimax,ifαi∈(0,12),(82)Ri≥ci2αi(ϕimin)1-2αimaxt∈[0,T]∫0T‍gi(t,s)ds+ϕimax,ifαi≥12.Remark 30.
A similar remark as Remark28 also holds for Theorem 29. Moreover, we note that the upper bounds Ri that fulfill (80)–(82) are independent of p, thus the information of |·| being a particular lp norm is not used. On the other hand, in Theorem 27, the upper bounds Ri that satisfy (75) depend on p. The sharpness of the bounds in both theorems cannot be compared in general; however, we will give an example at the end of this section to illustrate the results.In the next result, we will relax the condition (C2). Here, we allowϕi(t)≤0 for some i∈{1,2,…,n} and some t∈[0,T].Theorem 31.
Suppose that(C7)
there existsj∈{1,2,…,n} such that ϕjmin>0.
LetJ={j∈{1,2,…,n}|ϕjmin>0} and let J′={1,2,…,n}∖J. Assume that the following conditions hold for each 1≤i≤n: (C1), (C3), (C4), and  (C8)
there existci>0 such that
(83)∂∂shi(t,s)≤cis-αi,t∈[0,T],s>0,whereαj>0 for j∈J and αk∈(0,1/2) for k∈J′.
Further, let the following hold for eachj∈J:(C9)
the norm|·| is the lp norm where 1≤p≤∞ is fixed, and
(84)∫0T‍gj(t,s)ds<A-jp(cj2αj)-1,t∈[0,T],where(85)A-jp={[∑ℓ∈J‍(ϕℓmin)p]2αj/p,1≤p<∞,[maxℓ∈Jϕℓmin]2αj,p=∞.Then, (65) has a T-periodic solution uT∈(C(ℝ))n such that
(86)ϕjmin≤θjujT(t)≤Rj,t∈ℝ,j∈J,|ukT(t)|≤Rk,t∈ℝ,k∈J′,
where
(87)Rj≥ϕjmin,Rj≥ϕjmax[1-cj2αj(A-jp)-1maxt∈[0,T]∫0T‍gj(t,s)ds]-1,j∈J,(88)Rk2αk>ck2αkmaxt∈[0,T]∫0T‍gk(t,s)ds,k∈J′,(89)Rk[1-ck2αkRk-2αkmaxt∈[0,T]∫0T‍gk(t,s)ds]≥maxt∈[0,T]|ϕk(t)|,k∈J′.To derive Theorem31, we let the closed convex set Ω* be
(90)Ω*={u∈(C[0,T])n∣ϕjmin≤θjuj(t)≤Rj,t∈[0,T],j∈J;|uk(t)|≤Rk,t∈[0,T],k∈J′},
where Rj(≥ϕjmin>0) and Rk are chosen as in (87)–(89). Next, we define the operator S:Ω*→(C[0,T])n as in (78) and show that Theorem 26 is applicable.Remark 32.
From the conclusion of Theorem29, we see that the solution uT is “partially” of constant sign, in the sense that θjujT(t)≥0 for j∈J, but may not be so for j∈J′. Further, the constants ci that appear in (C8) determine the upper bounds Ri of the solution uiT,1≤i≤n. From (87) and (88), we see that a smaller (bigger) ci gives a smaller (bigger) Ri, and hence a smaller (bigger) set Ω* where the solution lies.Using similar arguments as in the derivation of Theorems31 and 29 (in getting Sju∈Ω* for j∈J and u∈Ω*), we obtain the following result.Theorem 33.
Suppose that (C7) hold. LetJ={j∈{1,2,…,n}|ϕjmin>0} and let J′={1,2,…,n}∖J. Assume the following conditions hold for each 1≤i≤n: (C1), (C3), (C4), and (C8). Then, (65) has a T-periodic solution uT∈(C(ℝ))n such that (91)ϕjmin≤θjujT(t)≤Rj,t∈ℝ,j∈J,|ukT(t)|≤Rk,t∈ℝ,k∈J′,
where
(92)Rj≥ϕjmin,j∈J,Rj2αj>cj2αjmaxt∈[0,T]∫0T‍gj(t,s)ds,ifαj∈(0,12),j∈J,Rj[1-cj2αjRj-2αjmaxt∈[0,T]∫0T‍gj(t,s)ds]≥ϕjmax,maxt∈[0,T]∫0T‍∫0T‍gj(t,s)ifαj∈(0,12),j∈J,Rj≥cj2αj(ϕjmin)1-2αjmaxt∈[0,T]∫0T‍gj(t,s)ds+ϕjmax,ifαj≥12,j∈J,(93)Rk2αk>ck2αkmaxt∈[0,T]∫0T‍gk(t,s)ds,k∈J′,Rk[1-ck2αkRk-2αkmaxt∈[0,T]∫0T‍gk(t,s)ds]≥maxt∈[0,T]|ϕk(t)|,k∈J′.Remark 34.
A similar remark as Remark32 holds for Theorem 33. Also, we observe once again that the upper bounds Rj that fulfill (92) are independent of p, thus the information of |·| being a particular lp norm is not used. On the other hand, in Theorem 31, the upper bounds Rj that satisfy (87) depend on p.We will now present an example that illustrates Theorems27 and 29.Example 35.
Consider (65) when
(94)T=2π,n=2,a1(t)=a2(t)=14,f1(t)=1,f2(t)=12,h1(t,s)=ln(s+1)|sint|+1,h2(t,s)=ln(s+1)3(|cost|+1),|·|=lpnorm(1≤p≤∞).Fixθi=1,1≤i≤n, that is, we are seeking positive solutions. The corresponding Green's function has the explicit expression [36]
(95)g1(t,s)=g2(t,s)={cos12(t-s-π),0≤s≤t≤2π,cos12(s-t-π),0≤t≤s≤2π.
Condition (C1) is satisfied. By direct computation, we get ϕ1(t)=4 and ϕ2(t)=2 for t∈[0,2π]. Thus, (C2) is fulfilled with
(96)ϕ1min=ϕ1max=4,ϕ2min=ϕ2max=2.
Moreover, we have
(97)∂∂sh1(t,s)=1|sint|+11s+1≤1s+1≤1s,∂∂sh2(t,s)=13(|cost|+1)1s+1≤13(s+1)≤13s
and so it is clear that (C4) and (C5) are satisfied with
(98)α1=1,c1=1,α2=1,c2=13.
Finally, we compute
(99)A1p=A2p=(4p+2p)2/p,1≤p<∞,A1∞=A2∞=16.
Since ∫02π‍gi(t,s)ds=4 for t∈[0,2π] and i=1,2, we check that (C6) holds for all 1≤p≤∞.All the conditions of Theorem27 are satisfied, thus we conclude that the problem (65) with (94) has a positive 2π-periodic solution u=(u1,u2) such that
(100)ϕimin≤ui(t)≤Ri,t∈ℝ,i=1,2,
where (from (75))
(101)Ri≥ϕimax[1-8ciAip]-1≡Lip,1≤p≤∞,i=1,2.We can also apply Theorem29 to conclude that the problem (65) with (94) has a positive 2π-periodic solution u=(u1,u2) satisfying (100) and (from (82))
(102)Ri≥8ci(ϕimin)-1+ϕimax≡Mi,i=1,2.As mentioned in Remark30, in general we cannot compare Lip and Mi. In fact, a direct calculation gives
(103)p=1:L11=5.14<M1=6,L21=2.16<M2=3.33,p=2:L12=6.67>M1=6,L22=2.31<M2=3.33,p=∞:L1∞=8>M1=6,L2∞=2.4<M2=3.33.


## 5.2. Existence Results for (59)

In this section we will consider the general system of Hill's equations(104)ui′′(t)+ai(t)ui(t)=Fi(t,u(t)),1≤i≤n.
Here, ai and Fi are T-periodic in the variable t, ai∈L1[0,T], and the nonlinearities Fi(t,x1,x2,…,xn) can be singular at xj=0 where j∈{1,2,…,n}.Once again, to obtain aT-periodic solution uT=(u1T,u2T,…,unT) of the system (104), we first seek a solution u*=(u1*,u2*,…,un*) of the following system of boundary value problems:
(105)ui′′(t)+ai(t)ui(t)=Fi(t,u(t)),t∈[0,T],ui(0)=ui(T),ui′(0)=ui′(T),1≤i≤n.
The periodic solution is then given by
(106)uT(t)=u*(t-mT),t∈[mT,(m+1)T],m∈ℤ.Withgi being the Green's function of the boundary value problem (68), throughout we will assume that (C1) is satisfied. Moreover, for fixed θi∈{-1,1} and T-periodic functions qi∈L1[0,T],1≤i≤n, we define
(107)ηi(t)=∫0T‍gi(t,s)θiqi(s)ds,t∈[0,T],1≤i≤n
and also
(108)ηimin=mint∈[0,T]ηi(t),ηimax=maxt∈[0,T]ηi(t).
For b≥b′≥0 and 1≤i≤n, we denote the interval
(109)[b,b′]i={[b,b′],ifθi=1,[-b′,-b],ifθi=-1.
A similar definition is valid for (b,b′)i.Using Schauder's fixed point theorem, we will establish existence results for the system (104).Theorem 36.
Assume the following conditions hold for each1≤i≤n: (C1);(C10)
for any numbersbj,bj′,1≤j≤n with bj′≥bj>0, the function Fi:[0,T]×∏j=1n‍[bj,bj′]j→ℝ is a L1-Carathéodory function, that is,
(i)
the mapu↦Fi(t,u) is continuous for almost all t∈[0,T],(ii)
the mapt↦Fi(t,u) is measurable for all u∈∏j=1n‍[bj,bj′]j,(iii)
for anyr>0, there exists μr,i∈L1[0,T] such that |u|≤r(u∈∏j=1n‍[bj,bj′]j) implies |Fi(t,u)|≤μr,i(t) for almost all t∈[0,T];(C11)
there existβi>0,γi∈[0,1), and T-periodic functions wi,qi with wi∈L1[0,T],qi∈L1[0,T] and wi(t)>0 for a.e.t∈[0,T] such that
(110)θiqi(t)|u|γi≤θiFi(t,u)≤θiqi(t)|u|γi+wi(t)|u|-βi,t∈[0,T],u∈∏k=1n‍(0,∞)k(here|·| is the lp norm where 1≤p≤∞ is fixed);(C12)
ηimin>0.
Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(111)ri≤θiuiT(t)≤Ri,t∈ℝ,1≤i≤n,
where, for 1≤i≤n one has
(112)0<ri≤Ri,ri≤(ηimin)1/(1-γi),(113)Ri≥ηimax|R|pγi+|r|p-βimaxt∈[0,T]∫0T‍gi(t,s)wi(s)ds,
(here |R|p is the lp norm of (R1,R2,…,Rn), likewise |r|p is the lp norm of (r1,r2,…,rn)).In proving Theorem36, we actually seek a constant-sign solution of (105) in (C[0,T])n and then extend it to a T-periodic constant-sign solution of (104) as in (106). Let Ω be the closed convex set given by
(114)Ω={u∈(C[0,T])n∣ri≤θiui(t)≤Ri,t∈[0,T],1≤i≤n(C[0,T])n},
where Ri≥ri>0 are chosen as in (112) and (113), and define the operator S:Ω→(C[0,T])n as
(115)Su(t)=(S1u(t),S2u(t),…,Snu(t)),t∈[0,T],
where
(116)Siu(t)=∫0T‍gi(t,s)Fi(s,u(s))ds,t∈[0,T],1≤i≤n.
Clearly, a fixed point of Su=u is a solution of (105). The conditions of Theorem 26 are then shown to be satisfied.Remark 37.
As seen from (112) and (113), the functions wi and qi that appear in (C11) determine the lower and upper bounds of the solution uiT,1≤i≤n.Theorem 38.
Assume that the following conditions hold for each1≤i≤n: (C1), (C10), (C11), and (C12). Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(117)r≤θiuiT(t)≤R,t∈ℝ,1≤i≤n,
where 0<r≤R, and for all 1≤i≤n,
(118)r≤{(ηiminnγi/p)1/(1-γi),1≤p<∞,(ηimin)1/(1-γi),p=∞,(119)R≥{Rγiηimaxnγi/p+r-βin-βi/p[maxt∈[0,T]∫0T‍gi(t,s)wi(s)ds],+r-βin-βi/p+r-βin-βi/p1≤p<∞,Rγiηimax+r-βi[maxt∈[0,T]∫0T‍gi(t,s)wi(s)ds],Rγiηimax[maxt∈[0,T]∫0T‍gi(t,s)wi(s)ds]p=∞.Theorem38 is obtained by similar arguments used in the derivation of Theorem 36, with a new Ω defined as
(120)Ω={u∈(C[0,T])n∣r≤θiui(t)≤R,t∈[0,T],1≤i≤n},
where R≥r>0 are chosen as in (118) and (119).Remark 39.
Remark37 also holds for Theorem 38. Further, comparing the bounds ri, Ri, 1≤i≤n in Theorem 36 (see (112), (113)) with the bounds r,R in Theorem 38 (see (118), (119)), we note that ri and Ri are lower and upper bounds for a particular θiuiT, whereas r and R are uniform lower and upper bounds for allθiuiT,1≤i≤n. However, the computation of Ri from (113) is more difficult than calculating R from (119).Our next result tackles the case whenηimin=0.Theorem 40.
Assume that the following conditions hold for each1≤i≤n: (C1), (C10),  (C13)
there existβi∈(0,1),γi∈[0,1), and T-periodic functions wi,vi,qi with wi∈L1[0,T],vi∈L1[0,T],qi∈L1[0,T], and wi(t),vi(t)>0 for a.e.t∈[0,T] such that
(121)θiqi(t)|u|γi+vi(t)|u|-βi≤θiFi(t,u)≤θiqi(t)|u|γi+wi(t)|u|-βi,t∈[0,T],u∈∏k=1n(0,∞)k(here|·| is the lp norm where 1≤p≤∞ is fixed);(C14)
ηimin=0.
Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(122)1R≤θiuiT(t)≤R,t∈ℝ,1≤i≤n,
where R≥1, and for all 1≤i≤n,
(123)R≥{nβi/p(1-βi)[mint∈[0,T]∫0T‍gi(t,s)vi(s)ds]-1/(1-βi),[mint∈[0,T]∫0T‍gi(t,s)vi(s)]-1/1-βi1≤p<∞,[mint∈[0,T]∫0T‍gi(t,s)vi(s)ds]-1/(1-βi),[mint∈[0,T]∫0T‍gi(s)ds]-1/1-βip=∞,R≥{Rγiηimaxnγi/p+Rβin-βi/p[maxt∈[0,T]∫0T‍gi(t,s)wi(s)ds],Rγiηimaxnγi/p+Rβin-βi/p11111111111111≤p<∞,Rγiηimax+Rβi[maxt∈[0,T]∫0T‍gi(t,s)wi(s)ds],Rγiηimaxnγi/p+1111111Rβin-βi/pp=∞.The closed convex set used to get Theorem40 is given by
(124)Ω={u∈(C[0,T])n∣r≤θiui(t)≤R,t∈[0,T],1≤i≤n(C[0,T])n},
where r=1/R and R≥1 satisfies (123).Remark 41.
As seen from (123), the functions wi,vi, and qi that appear in (C13) determine the lower and upper bounds of the solution uiT,1≤i≤n.Finally, the next result tackles the case whenηimax<0.Theorem 42.
Assume that the following conditions hold for each1≤i≤n: (C1), (C10),  (C15)
there existβ∈(0,1) and T-periodic functions wi,vi,qi with wi∈L1[0,T],vi∈L1[0,T],qi∈L1[0,T], and wi(t),vi(t)>0 for a.e.t∈[0,T] such that
(125)θiqi(t)+vi(t)|u|-β≤θiFi(t,u)≤θiqi(t)+wi(t)|u|-β,t∈[0,T],u∈∏k=1n‍(0,∞)k(here|·| is the lp norm where 1≤p≤∞ is fixed);(C16)
ηimax<0;(C17)
ηimin≥n-β/(1+β)pW-β/(1-β2)(Vβ2)1/(1-β2)(1-1/β2) where
(126)W=max1≤k≤n[maxt∈[0,T]∫0T‍gk(t,s)wk(s)ds],V=min1≤k≤n[mint∈[0,T]∫0T‍gk(t,s)vk(s)ds].Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(127)r≤θiuiT(t)≤R,t∈ℝ,1≤i≤n,
where 0<r≤R are given by
(128)r={n-β/(1+β)pW-β/(1-β2)(Vβ2)1/(1-β2),1≤p<∞W-β/(1-β2)(Vβ2)1/(1-β2),p=∞,R={n-β/(1+β)pW1/(1-β2)(Vβ2)-β/(1-β2),1≤p<∞W1/(1-β2)(Vβ2)-β/(1-β2),p=∞.Theorem42 is obtained by considering the closed convex set
(129)Ω={u∈(C[0,T])n∣r≤θiui(t)≤R,t∈[0,T],1≤i≤n},
where R≥r>0 are determined later as those given in (128).Remark 43.
As seen from (128), the functions wi and vi that appear in (C15) determine the lower and upper bounds of the solution uiT,1≤i≤n.We have so far established the results when (i)ηimin>0, (ii) ηimin=0, and (iii) ηimax<0 for all1≤i≤n. However, it could be that we only have ηi(t)≥0 for some i and ηj(t)<0 for some j, which results in ηimin≥0 and ηjmax<0 for some1≤i,j≤n. We present two results for such a case as follows. Note that Theorem 44 is obtained by applying Theorems 38–42, while Theorem 45 is obtained by applying Theorems 36, 40, and 42.Theorem 44.
Let (C1) and (C10) hold for each1≤i≤n. Assume the following:(C18)
conditions (C11) and (C12) hold for somei∈I⊆{1,2,…,n};(C19)
conditions (C13) and (C14) hold for somei∈J⊆{1,2,…,n};(C20)
conditions (C15), (C16), and (C17) hold for somei∈K⊆{1,2,…,n};
whereI∪J∪K={1,2,…,n}. Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(130)r≤θiuiT(t)≤R,t∈ℝ,1≤i≤n,
where 0<r≤R satisfy  (a)
(118) and (119) for i∈I;(b)
r=1/R,R≥1, (123) for i∈J;(c)
(128) for i∈K.Theorem 45.
Let (C1) and (C10) hold for each1≤i≤n. Assume that (C18)–(C20) hold with I∪J∪K={1,2,…,n}. Then, (104) has a T-periodic constant-sign solution uT∈(C(ℝ))n such that
(131)ri≤θiuiT(t)≤Ri,t∈ℝ,i∈I,
where 0<ri≤Ri satisfy (112) and (113) for i∈I, and
(132)r≤θiuiT(t)≤R,t∈ℝ,i∈J∪K,
where 0<r≤R satisfy conclusions (b) and (c) of Theorem 44.We will now apply the results obtained to the following system of Hill's equations, a particular form of it (see (64)) that has been discussed in [45],
(133)u1′′(t)+a1(t)u1(t)=(u12+u22)-β1+ν1(u12+u22)γ1,u2′′(t)+a2(t)u2(t)=(u12+u22)-β2+ν2(u12+u22)γ2.
Clearly, the system (133) corresponds to (104) where n=2 and
(134)Fi(t,u)=(u12+u22)-βi+νi(u12+u22)γi,i=1,2.
We will assume that a1,a2∈L1[0,T] satisfy (C1). Note that condition (C10) is clearly satisfied. Further, let θ1=θ2=1, that is, we are interested in positive periodic solutions of (133).Example 46.
Consider the system (133) with
(135)νi>0,βi>0,γi∈[0,1),i=1,2.Clearly, (C11) is satisfied withp=2,qi=νi and wi=1,i=1,2. Thus, (C12) also holds since
(136)ηimin=mint∈[0,T]∫0T‍gi(t,s)θiqi(s)ds=νimint∈[0,T]∫0T‍gi(t,s)ds>0.
Theorem 38 (or Theorem 36) is applicable and we conclude that the system (133) with (135) has a T-periodic positive solution uT∈(C(ℝ))2 such that
(137)r≤uiT(t)≤R,t∈ℝ,i=1,2,
where 0<r≤R are such that
(138)r≤mini=1,2{(ηimin2γi/2)1/(1-γi)},(139)R≥maxi=1,2{Rγiηimax2γi/2+r-βi2-βi/2[maxt∈[0,T]∫0T‍gi(t,s)ds]}.To illustrate numerically, suppose(140)a1(t)=a2(t)=14,T=2π,ν1=14,ν2=1,γ1=γ2=12,β1=β2=1.
Green's function is given in (95) and
(141)ηimin=νimint∈[0,T]∫0T‍gi(t,s)ds=4νi.
Hence, (138) yields r≤2. Let r=2, then (139) reduces to
(142)R≥maxi=1,2{R1/24νi21/4+r-12-1/24νi}=R1/229/4+2,
which is satisfied by R≥26.48. Let R=26.48, then from (137) we conclude that the system (133) with (140) has a 2π-periodic positive solution u∈(C(ℝ))2 such that
(143)2≤ui(t)≤26.48,t∈ℝ,i=1,2.Example 47.
Consider the system (133) with
(144)νi=0,βi∈(0,1),γi∈[0,1),i=1,2.Here, (C13) is satisfied withp=2,qi=νi=0 and wi=vi=1,i=1,2. Subsequently, (C14) also holds since
(145)ηimin=mint∈[0,T]∫0T‍gi(t,s)θiqi(s)ds=νimint∈[0,T]∫0T‍gi(t,s)ds=0.
Employing Theorem 40, we conclude that the system (133) with (144) has a T-periodic positive solution uT∈(C(ℝ))2 such that
(146)1R≤uiT(t)≤R,t∈ℝ,i=1,2,
where R≥1, and from (123), we have for i=1,2,
(147)R≥2βi/2(1-βi)[mint∈[0,T]∫0T‍gi(t,s)ds]-1/(1-βi),R≥{2-βi/2[maxt∈[0,T]∫0T‍gi(t,s)ds]}1/(1-βi).
Combining the inequalities, we see that R should satisfy
(148)R≥max{{2-βi/2[maxt∈[0,T]∫0T‍gi(t,s)ds]}1/1-βi1,maxi=1,22βi/2(1-βi)×[mint∈[0,T]∫0T‍gi(t,s)ds]-1/(1-βi),maxi=1,2{2-βi/2[maxt∈[0,T]∫0T‍gi(t,s)ds]}1/(1-βi)}.Example 48.
Consider the system (133) with
(149)νi<0,βi=β∈(0,1),γi=0,i=1,2,(150)νimint∈[0,T]∫0T‍gi(t,s)ds≥2-β/2(1+β)W-β/(1-β2)(Vβ2)1/(1-β2)×(1-1β2),i=1,2,
where
(151)W=maxk=1,2[maxt∈[0,T]∫0T‍gk(t,s)ds],V=mink=1,2[mint∈[0,T]∫0T‍gk(t,s)ds].Obviously, (C15) is satisfied withp=2,qi=νi<0 and wi=vi=1,i=1,2. Then, (C16) also holds since
(152)ηimax=maxt∈[0,T]∫0T‍gi(t,s)θiqi(s)ds=νimaxt∈[0,T]∫0T‍gi(t,s)ds<0.
Moreover, condition (C17) is simply (150). Hence, we conclude from Theorem 42 that the system (133) with (149) and (150) has a T-periodic positive solution uT∈(C(ℝ))2 such that
(153)r≤uiT(t)≤R,t∈ℝ,i=1,2,
where 0<r≤R are given by
(154)r=2-β/2(1+β)W-β/(1-β2)(Vβ2)1/(1-β2),R=2-β/2(1+β)W1/(1-β2)(Vβ2)-β/(1-β2).Remark 49.
In [45], it is shown that (64) has a solution when β>0,γ∈[0,1) and ν>0. As seen from Examples 46–48, we have generalized the system (64) to allow ν to be zero or negative.



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