Volume 2009, Article ID 540863,20pages doi:10.1155/2009/540863
Research Article
Recent Existence Results for Second-Order Singular Periodic Differential Equations
Jifeng Chu
1, 2and Juan J. Nieto
31Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China
2Department of Mathematics, Pusan National University, Busan 609-735, South Korea
3Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain
Correspondence should be addressed to Jifeng Chu,jifengchu@126.com Received 12 February 2009; Accepted 29 April 2009
Recommended by Donal O’Regan
We present some recent existence results for second-order singular periodic differential equations.
A nonlinear alternative principle of Leray-Schauder type, a well-known fixed point theorem in cones, and Schauder’s fixed point theorem are used in the proof. The results shed some light on the differences between a strong singularity and a weak singularity.
Copyrightq2009 J. Chu and J. J. Nieto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The main aim of this paper is to present some recent existence results for the positive T- periodic solutions of second order differential equation
xatxft, x et, 1.1
where at, et are continuous and T-periodic functions. The nonlinearity ft, x is continuous int, xandT-periodic int. We are mainly interested in the case thatft, xhas a repulsive singularity atx0:
xlim→0ft, x ∞, uniformly int. 1.2
It is well known that second order singular differential equations describe many problems in the applied sciences, such as the Brillouin focusing system1and nonlinear elasticity 2. Therefore, during the last two decades, singular equations have attracted many researchers, and many important results have been proved in the literature; see, for
example, 3–10. Recently, it has been found that a particular case of 1.1, the Ermakov- Pinney equation
xatx 1
x3 1.3
plays an important role in studying the Lyapunov stability of periodic solutions of Lagrangian equations11–13.
In the literature, two different approaches have been used to establish the existence results for singular equations. The first one is the variational approach 14–16, and the second one is topological methods. Because we mainly focus on the applications of topological methods to singular equations in this paper, here we try to give a brief sketch of this problem. As far as the authors know, this method was started with the pioneering paper of Lazer and Solimini17. They proved that a necessary and sufficient condition for the existence of a positive periodic solution for equation
x 1
xλ et 1.4
is that the mean value of e is negative, e < 0, here λ ≥ 1, which is a strong force condition in a terminology first introduced by Gordon18. Moreover, if 0 < λ < 1, which corresponds to a weak force condition, they found examples of functionse with negative mean values and such that periodic solutions do not exist. Since then, the strong force condition became standard in the related works; see, for instance,2,8–10,13,19–21, and the recent review22. With a strong singularity, the energy near the origin becomes infinity and this fact is helpful for obtaining the a priori bounds needed for a classical application of the degree theory. Compared with the case of a strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity by topological methods is more recent but has also attracted many researchers4,6,23–28. In 27, for the first time in this topic, Torres 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.
Among topological methods, the method of upper and lower solutions6,29,30, degree theory8,20,31, some fixed point theorems in cones for completely continuous operators 25,32–34, and Schauder’s fixed point theorem27,35,36are the most relevant tools.
In this paper, we select several recent existence results for singular equation 1.1 via different topological tools. The remaining part of the paper is organized as follows. In Section 2, some preliminary results are given. InSection 3, we present the first existence result for 1.1 via a nonlinear alternative principle of Leray-Schauder. In Section 4, the second existence result is established by using a well-known fixed point theorem in cones. The condition imposed onatin Sections 3and 4is that the Green function Gt, sassociated with the linear periodic equations is positive, and therefore the results cannot cover the critical case, for example, whenais a constant,at k2, 0 < k <
λ1 π/T, andλ1 is the first eigenvalue of the linear problem with Dirichlet conditionsx0 xT 0. Different from Sections3and4, the results obtained inSection 5, which are established by Schauder’s fixed point theorem, can cover the critical case because we only need that the Green function Gt, sis nonnegative. All results in Sections3–5shed some lights on the differences between a strong singularity and a weak singularity.
To illustrate our results, in Sections 3–5, we have selected the following singular equation:
x”atxx−αμxβet, 1.5
herea, e ∈C0, T,α, β >0,andμ ∈Ris a given parameter. The corresponding results are also valid for the general case
x”atx bt
xα μctxβet, 1.6
withb, c∈C0, T. Some open problems for1.5or1.6are posed.
In this paper, we will use the following notation. Givenψ ∈L10, T, we writeψ 0 ifψ ≥0 for a.e.t∈0, T,and it is positive in a set of positive measure. For a given function p∈L10, Tessentially bounded, we denote the essential supremum and infimum ofpbyp∗ andp∗, respectively.
2. Preliminaries
Consider the linear equation
xatxpt 2.1
with periodic boundary conditions
x0 xT, x0 xT. 2.2
In Sections3and4, we assume that
Athe Green function Gt, s, associated with 2.1–2.2, is positive for all t, s ∈ 0, T×0, T.
InSection 5, we assume that
Bthe Green functionGt, s,associated with2.1–2.2, is nonnegative for allt, s∈ 0, T×0, T.
Whenat k2,conditionAis equivalent to 0< k2< λ1 π/T2and conditionB is equivalent to 0< k2≤λ1. In this case, we have
Gt, s
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
sinkt−s sinkT−ts
2k1−coskT , 0≤s≤t≤T, sinks−t sinkT−st
2k1−coskT , 0≤t≤s≤T.
2.3
For a nonconstant functionat, there is anLp-criterion proved in37, which is given in the following lemma for the sake of completeness. Let Kq denote the best Sobolev constant in the following inequality:
Cu2q≤u2
2, ∀u∈H010, T. 2.4 The explicit formula forKqis
K
q
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩ 2π qT12/q
2 2q
1−2/q Γ1/q
Γ1/21/q 2
if 1≤q <∞, 4
T, ifq∞,
2.5
whereΓis the Gamma function; see21,38
Lemma 2.1. Assume thatat0 anda∈Lp0, Tfor some 1≤p≤ ∞. If ap<K
2p , 2.6
then the condition (A) holds. Moreover, condition (B) holds if ap≤K
2p . 2.7
When the hypothesisAis satisfied, we denote
m min
0≤s,t≤TGt, s, M max
0≤s,t≤TGt, s, σ m
M. 2.8
Obviously,M > m >0 and 0< σ <1.
Throughout this paper, we define the functionγ:R → Rby
γt T
0
Gt, sesds, 2.9
which corresponds to the uniqueT-periodic solution of
xatxet. 2.10
3. Existence Result (I)
In this section, we state and prove the first existence result for1.1. The proof is based on the following nonlinear alternative of Leray-Schauder, which can be found in39. This part can be regarded as the scalar version of the results in4.
Lemma 3.1. AssumeΩis a relatively compact subset of a convex setKin a normed spaceX. Let T :Ω → Kbe a compact map with 0∈Ω. Then one of the following two conclusions holds:
aT has at least one fixed point inΩ;
bthereexistx∈∂Ωand 0< λ <1 such thatxλTx.
Theorem 3.2. Suppose thatatsatisfies (A) andft, xsatisfies the following.
H1There exist constantsσ >0 andν≥1 such that
ft, x≥σx−ν, ∀t∈0, T, ∀0< x1. 3.1
H2There exist continuous, nonnegative functionsgxandhxsuch that
0≤ft, x≤gx hx ∀t, x∈0, T×0,∞, 3.2
gx>0 is nonincreasing andhx/gxis nondecreasing inx∈0,∞.
H3There exists a positive numberrsuch thatσrγ∗>0,and r
g
σrγ∗ 1h
rγ∗ /g
rγ∗ > ω∗, hereωt T
0
Gt, sds. 3.3
Then for eache ∈ CR/TZ,R,1.1has at least one positive periodic solutionxwith xt> γtfor alltand 0<x−γ< r.
Proof. The existence is proved using the Leray-Schauder alternative principle, together with a truncation technique. The idea is that we show that
xatxf
t, xt γt 3.4
has a positive periodic solutionxsatisfyingxt γt > 0 fortand 0 < x < r.If this is true, it is easy to see thatut xt γtwill be a positive periodic solution of1.1with 0<u−γ< rsince
uatuxγ”atxatγf
t, xγ et ft, u et. 3.5
SinceH3holds, we can choosen0∈ {1,2,· · · }such that 1/n0< σrγ∗and
ω∗g σrγ∗
1 h
rγ∗ g
rγ∗
1 n0
< r. 3.6
LetN0{n0, n01,· · · }. Consider the family of equations
xatxλfn
t, xt γt at
n , 3.7
whereλ∈0,1, n∈N0,and
fnt, x
⎧⎪
⎪⎨
⎪⎪
⎩
ft, x, ifx≥ 1 n, f
t,1
n
, ifx≤ 1 n.
3.8
Problem3.7is equivalent to the following fixed point problem:
xλTnx1
n, 3.9
whereTnis defined by
Tnxt λ T
0
Gt, sfn
s, xs γs ds 1
n. 3.10
We claim that any fixed point x of 3.9 for any λ ∈ 0,1 must satisfy x/r.
Otherwise, assume that x is a fixed point of3.9 for someλ ∈ 0,1such that x r.
Note that
xt− 1 n λ
T
0
Gt, sfn
s, xs γs ds
≥λm T
0
fn
s, xs γs ds
σMλ T
0
fn
s, xs γs ds
≥σmax
t∈0,T
λ
T
0
Gt, sfn
s, xs γs ds
σ x− 1
n .
3.11
By the choice ofn0, 1/n≤1/n0< σrγ∗. Hence, for allt∈0, T, we have
xt≥σ x− 1
n 1
n ≥σ
x −1 n
1
n ≥σr. 3.12
Therefore,
xt γt≥σrγ∗> 1
n. 3.13
Thus we have from conditionH2, for allt∈0, T,
xt λ
T
0
Gt, sfn
s, xs γs ds 1 n λ
T
0
Gt, sf
s, xs γs ds1 n
≤ T
0
Gt, sf
s, xs γs ds 1 n
≤ T
0
Gt, sg
xs γs
1 h
xs γs
g
xs γs
ds 1
n
≤g σrγ∗
1h
rγ∗ g
rγ∗ T
0
Gt, sds1 n
≤g σrγ∗
1h
rγ∗ g
rγ∗
ω∗ 1 n0
.
3.14
Therefore,
r x ≤g σrγ∗
1 h
rγ∗ g
rγ∗
ω∗ 1
n0. 3.15
This is a contradiction to the choice ofn0,and the claim is proved.
From this claim, the Leray-Schauder alternative principle guarantees that xTnx 1
n 3.16
has a fixed point, denoted byxn, inBr {x∈X:x< r}, that is, equation xatxfn
t, xt γt at
n 3.17
has a periodic solutionxn withxn < r. Since xnt ≥ 1/n > 0 for allt ∈ 0, Tandxn is actually a positive periodic solution of3.17.
In the next lemma, we will show that there exists a constantδ >0 such that
xnt γt≥δ, ∀t∈0, T, 3.18
fornlarge enough.
In order to pass the solutionsxnof the truncation equations3.17to that of the original equation3.4, we need the following fact:
xn≤H 3.19
for some constantH >0 and for alln≥n0. To this end, by the periodic boundary conditions, xnt0 0 for somet0∈0, T. Integrating3.17from 0 toT, we obtain
T
0
atxntdt T
0
fn
t, xnt γt at n
dt. 3.20
Therefore
xnmax
0≤t≤Txntmax
0≤t≤T
t
t0
xnsds max
0≤t≤T
t
t0
fn
s, xns γs as
n −asxns
ds
≤ T
0
fn
s, xns γs as n
ds
T
0
asxnsds
2 T
0
asxnsds <2ra1H.
3.21
The factxn < r and 3.19show that {xn}n∈N0 is a bounded and equicontinuous family on0, T. Now the Arzela-Ascoli Theorem guarantees that{xn}n∈N0has a subsequence, {xnk}k∈N, converging uniformly on0, T to a function x ∈ X. Moreover, xnk satisfies the integral equation
xnkt T
0
Gt, sf
s, xnks γs ds 1
nk. 3.22
Lettingk → ∞, we arrive at
xt T
0
Gt, sf
s, xs γs ds, 3.23
where the uniform continuity offt, xon0, T×δ, rγ∗is used. Therefore,xis a positive periodic solution of3.4.
Lemma 3.3. There exist a constantδ >0 and an integern2> n0such that any solutionxnof 3.17 satisfies3.18for alln≥n2.
Proof. The lower bound in3.18is established using the strong force conditionH1offt, x.
By conditionH1, there existsc0 ∈0,1small enough such that ft, x≥σc0−ν>max
ra1, a∗
rγ∗ e∗
, ∀0≤t≤T, 0< x≤c0. 3.24
Taken1∈N0such that 1/n1≤c0and letN1{n1, n11,· · · }. Forn∈N1, let
αnmin
0≤t≤T
xnt γt
, βnmax
0≤t≤T
xnt γt
. 3.25
We claim first thatβn > c0 for alln ∈ N1. Otherwise, suppose that βn ≤ c0 for some n∈N1. Then from3.24, it is easy to verify
fn
t, xnt γt > ra1. 3.26
Integrating3.17from 0 toT, we deduce that
0 T
0
x”nt atxnt−fn
t, xnt γt −at n
dt
T
0
atxntdt− 1
n T
0
atdt− T
0
fn
t, xnt γt dt
<
T
0
atxntdt−ra1≤0.
3.27
This is a contradiction. Thusβn> c0 for n∈N1.
Now we consider the minimum valuesαn. Letn ≥n1. Without loss of generality, we assume thatαn< c0, otherwise we have3.18. In this case,
αnmin
0≤t≤T
xnt γt
xntn γtn< c0 3.28
for sometn∈0, T. Asβn> c0, there existscn ∈0,1 without loss of generality, we assume tn < cnsuch thatxncn γcn c0andxnt γt≤c0 fortn ≤t≤cn.By3.24, it can be checked that
fn
t, xnt γt > at
xnt γt et, ∀t∈tn, cn. 3.29
Thus fort∈tn, cn, we havexn”t γ”t>0.Asxntn γtn 0,xnt γt>0 for allt ∈tn, cnand the functionyn :xnγis strictly increasing ontn, cn. We useξn to denote the inverse function ofynrestricted totn, cn.
In order to prove3.18in this case, we first show that, forn∈N1,
xnt γt≥ 1
n. 3.30
Otherwise, suppose thatαn<1/nfor somen∈N1. Then there would existbn∈tn, cn such thatxnbn γbn 1/nand
xnt γt≤ 1
n fortn≤t≤bn, 1
n ≤xnt γt≤c0 forbn≤t≤cn. 3.31 Multiplying3.17byxnt γtand integrating frombntocn, we obtain
R1
1/n
f ξn
y , y dy cn
bn
f
t, xnt γt xnt γt dt
cn
bn
fn
t, xnt γt xnt γt dt
cn
bn
xnt atxnt− at n
xnt γt dt
cn
bn
xnt
xnt γt dt
cn
bn
atxnt− at n
xnt γt dt.
3.32
By the factsxn < r and xn ≤ H,one can easily obtain that the right side of the above equality is bounded. As a consequence, there existsL >0 such that
R1
1/n
f ξn
y , y dy≤L. 3.33
On the other hand, by the strong force conditionH1, we can choose n2 ∈ N1 large enough such that
c0
1/n
f ξn
y , y dy≥σ c0
1/n
y−νdy > L 3.34
for alln∈N2 {n2, n21,· · · }. So3.30holds forn∈N2.
Finally, multiplying3.17byxnt γtand integrating fromtntocn, we obtain c0
αn
f ξn
y , y dy cn
tn
f
t, xnt γt xnt γt dt
cn
tn
fn
t, xnt γt xnt γt dt
cn
tn
xnt atxnt−at n
xnt γt dt.
3.35
We notice that the estimate3.30is used in the second equality above. In the same way, one may readily prove that the right-hand side of the above equality is bounded. On the other hand, ifn∈N2,byH1,
c0
αn
f ξn
y , y dy≥σ c0
αn
y−νdy−→∞ 3.36
ifαn → 0.Thus we know thatαn≥δfor some constantδ >0.
From the proof ofTheorem 3.2andLemma 3.3, we see that the strong force condition H1is only used when we prove3.18. From the next theorem, we will show that, for the caseγ∗ ≥0, we can remove the strong force conditionH1, and replace it by one weak force condition.
Theorem 3.4. Assume that (A) and (H2)–( H3) are satisfied. Suppose further that
H4for each constantL >0, there exists a continuous functionφL0 such thatft, x≥φLt for allt, x∈0, T×0, L.
Then for eachetwithγ∗≥0,1.1has at least one positive periodic solutionxwithxt> γtfor alltand 0<x−γ< r.
Proof. We only need to show that3.18is also satisfied under conditionH4andγ∗≥0.The rest parts of the proof are in the same line ofTheorem 3.2. SinceH4holds, there exists a continuous functionφrγ∗ 0 such thatft, x≥φrγ∗tfor allt, x∈0, T×0, rγ∗. Let xrγ∗be the unique periodic solution to the problems2.1–2.2withhφrγ∗. That is
xrγ∗t T
0
Gt, sφrγ∗sds. 3.37
Then we have
xrγ∗t γt T
0
Gt, sφrγ∗sdsγt≥Φ∗γ∗>0, 3.38
here
Φt T
0
Gt, sφrγ∗sds. 3.39
Corollary 3.5. Assume thatatsatisfies (A) andα >0, β≥0, μ >0. Then
iif α ≥ 1, β < 1,then for eache ∈ CR/TZ,R,1.5has at least one positive periodic solution for allμ >0;
iiif α ≥ 1, β ≥ 1, then for eache ∈ CR/TZ,R,1.5has at least one positive periodic solution for each 0< μ < μ1,hereμ1is some positive constant.
iiiifα >0, β <1, then for eache ∈CR/TZ,Rwithγ∗ ≥0,1.5has at least one positive periodic solution for allμ >0;
ivifα >0, β ≥1, then for eache ∈CR/TZ,Rwithγ∗ ≥0,1.5has at least one positive periodic solution for each 0< μ < μ1.
Proof. We apply Theorems3.2and3.4. Take
gx x−α, hx μxβ, 3.40
thenH2is satisfied, and the existence conditionH3becomes
μ < r
σrγ∗ α−ω∗ ω∗
rγ∗ αβ 3.41
for somer >0. Note that conditionH1is satisfied whenα≥1, whileH4is satisfied when α >0. So1.5has at least one positive periodic solution for
0< μ < μ1:sup
r>0
r
σrγ∗ α−ω∗ ω∗
rγ∗ αβ . 3.42
Note thatμ1∞ifβ <1 andμ1<∞ifβ≥1. Thus we havei–iv.
4. Existence Result (II)
In this section, we establish the second existence result for1.1using a well-known fixed point theorem in cones. We are mainly interested in the superlinear case. This part is essentially extracted from24.
First we recall this fixed point theorem in cones, which can be found in40. LetKbe a cone inXandDis a subset ofX, we writeDKD∩Kand∂KD ∂D∩K.
Theorem 4.1see40. LetX be a Banach space andK a cone inX. Assume Ω1,Ω2 are open bounded subsets ofXwithΩ1K/∅,Ω1K⊂Ω2K.Let
T :Ω2K−→K 4.1
be a completely continuous operator such that aTx ≤ xforx∈∂KΩ1,
bThere existsυ∈K\ {0}such thatx /Txλυ for allx∈∂KΩ2and allλ >0.
ThenT has a fixed point inΩ2K\Ω1K.
In applications below, we takeXC0, Twith the supremum norm · and define
K
x∈X:xt≥0 ∀t∈0, T, min
0≤t≤Txt≥σx
. 4.2
Theorem 4.2. Suppose thatatsatisfies (A) andft, xsatisfies (H2)–(H3). Furthermore, assume that
H5there exist continuous nonnegative functionsg1x, h1xsuch that
ft, x≥g1x h1x, ∀t, x∈0, T×0,∞, 4.3
g1x>0 is nonincreasing andh1x/g1xis nondecreasing inx;
H6there existsR >0 withσR > rsuch that
σR g1
Rγ∗ 1h1
σRγ∗ /g1
σRγ∗ ≤ω∗. 4.4
Then1.1has one positive periodic solutionxwithr <x−γ ≤R.
Proof. As in the proof ofTheorem 3.2, we only need to show that3.4has a positive periodic solutionu∈Xwithut γt>0 andr <u ≤R.
LetKbe a cone inXdefined by4.2. Define the open sets
Ω1{x∈X:x< r}, Ω2{x∈X :x< R}, 4.5
and the operatorT :Ω2K → Kby
Txt T
0
Gt, sf
s, xs γs ds, 0≤t≤T. 4.6
For eachx∈Ω2K\Ω1K, we haver ≤ ||x|| ≤R. Thus 0< σrγ∗ ≤xt γt≤ Rγ∗ for allt∈0, T.Sincef :0, T×σrγ∗, Rγ∗ → 0,∞is continuous, then the operator T : Ω2K\Ω1K → Kis well defined and is continuous and completely continuous. Next we claim that:
iTx ≤ xforx∈∂KΩ1,and
iithere existsυ∈K\ {0}such thatx /Txλυ for allx∈∂KΩ2and allλ >0.
We start withi. In fact, ifx∈∂KΩ1,thenxrandσrγ∗≤xt γt≤rγ∗for allt∈0, T.Thus we have
Txt T
0
Gt, sf
s, xs γs ds
≤ T
0
Gt, sg
xs γs
1 h
xs γs
g
xs γs
ds
≤g σrγ∗
1h
rγ∗ g
rγ∗ T
0
Gt, sds
≤g σrγ∗
1h
rγ∗ g
rγ∗
ω∗< r x.
4.7
Next we considerii. Letυt ≡ 1,thenυ ∈K\ {0}.Next, suppose that there exists x∈∂KΩ2andλ >0 such thatxTxλυ.Sincex∈∂KΩ2,thenσRγ∗≤xt γt≤Rγ∗ for allt∈0, T.As a result, it follows fromH5andH6that, for allt∈0, T,
xt Txt λ
T
0
Gt, sf
s, xs γs dsλ
≥ T
0
Gt, sg1
xs γs
1h1
xs γs g1
xs γs
dsλ
≥g1 Rγ∗
1 h1
σRγ∗ g1
σRγ∗ T
0
Gt, sdsλ
≥g1 Rγ∗
1 h1
σRγ∗ g1
σRγ∗
ω∗λ
> g1 Rγ∗
1 h1
σRγ∗ g1
σRγ∗
ω∗≥σR.
4.8
Hence min
0≤t≤Txt> σR,this is a contradiction and we prove the claim.
NowTheorem 4.1guarantees thatT has at least one fixed pointx ∈ Ω2K\Ω1K with r≤ x ≤R.Notex/rby4.7.
Combined Theorem 4.2 with Theorems 3.2 or 3.4, we have the following two multiplicity results.
Theorem 4.3. Suppose thatatsatisfies (A) andft, xsatisfies (H1)–( H3) and (H5)–( H6). Then 1.1has two different positive periodic solutionsxandxwith 0<x−γ< r <x−γ ≤R.
Theorem 4.4. Suppose thatatsatisfies (A) andft, xsatisfies (H2)–( H6). Then1.1has two different positive periodic solutionsxandxwith 0<x−γ< r <x−γ ≤R.
Corollary 4.5. Assume thatatsatisfies (A) andα >0, β >1, μ >0. Then
iifα≥1, then for eache ∈ CR/TZ,R,1.5has at least two positive periodic solutions for each 0< μ < μ1;
iiifα >0, then for eache∈CR/TZ,Rwithγ∗≥0,1.5has at least two positive periodic solutions for each 0< μ < μ1.
Proof. Takeg1x x−α, h1x μxβ.ThenH5is satisfied and the existence conditionH6 becomes
μ≥ σR
Rγ∗ α−ω∗ ω∗
σRγ∗ αβ . 4.9
Sinceβ >1, it is easy to see that the right-hand side goes to 0 asR → ∞. Thus, for any given 0 < μ < μ1, it is always possible to find suchR rthat4.9is satisfied. Thus,1.5has an additional positive periodic solutionx.
5. Existence Result (III)
In this section, we prove the third existence result for1.1by Schauder’s fixed point theorem.
We can cover the critical case because we assume that the conditionBis satisfied. This part comes essentially from35, and the results for the vector version can be found in4.
Theorem 5.1. Assume that conditions (B) and (H2), (H4) are satisfied. Furthermore, suppose that
H7there exists a positive constantR >0 such thatR >Φ∗,Φ∗γ∗>0 andR≥gΦ∗γ∗{1 hRγ∗/gRγ∗}ω∗, hereΦ∗mintΦt, Φt T
0Gt, sφRγ∗sds.
Then1.1has at least one positiveT-periodic solution.
Proof. AT-periodic solution of1.1is just a fixed point of the mapT : X → Xdefined by 4.6. Note thatT is a completely continuous map.
LetRbe the positive constant satisfyingH7andr Φ∗>0.Then we haveR > r >0.
Now we define the set
Ω {x∈X :r≤xt≤R ∀t}. 5.1
Obviously,Ωis a closed convex set. Next we proveTΩ⊂Ω.
In fact, for eachx∈Ω, using thatGt, s≥0 and conditionH4,
Txt≥ T
0
Gt, sφRγ∗sds≥Φ∗r >0. 5.2
On the other hand, by conditionsH2andH7, we have
Txt≤ T
0
Gt, sg
xs γs
1 h
xs γs
g
xs γs
ds
≤g Φ∗γ∗
1h
Rγ∗ g
Rγ∗
ω∗≤R.
5.3
In conclusion,TΩ⊂Ω. By a direct application of Schauder’s fixed point theorem, the proof is finished.
As an application ofTheorem 5.1, we consider the caseγ∗0. The following corollary is a direct result ofTheorem 5.1.
Corollary 5.2. Assume that conditions (B) and (H2), (H4) are satisfied. Furthermore, assume that H8there exists a positive constantR >0 such thatR >Φ∗and
R≥gΦ∗
1 h Rγ∗ g
Rγ∗
ω∗. 5.4
Ifγ∗0,then1.1has at least one positiveT-periodic solution.
Corollary 5.3. Suppose thatasatisfies (B) and 0< α <1,β≥0, then for eachet∈CR/TZ,R withγ∗0,one has the following:
iifαβ <1−α2,then1.5has at least one positive periodic solution for eachμ≥0.
iiifαβ≥1−α2,then1.5has at least one positiveT-periodic solution for each 0≤μ < μ2, whereμ2is some positive constant.
Proof. We applyCorollary 3.5and follow the same notation as in the proof ofCorollary 3.5.
ThenH2andH4are satisfied, and the existence conditionH8becomes
μ < RΦα∗−ω∗ ω∗
Rγ∗ αβ, 5.5
for someR >0 withR >Φ∗. Note that Φ∗
Rγ∗ −αω∗. 5.6
Therefore,5.5becomes
μ < R
Rγ∗ −α2ωα∗−ω∗ ω∗
Rγ∗ αβ , 5.7
for someR >0.
So1.5has at least one positiveT-periodic solution for
0< μ < μ2sup
R>0
R
Rγ∗ −α2ωα∗−ω∗ ω∗
Rγ∗ αβ . 5.8
Note thatμ2∞ifαβ <1−α2andμ2<∞ifαβ≥1−α2. We have the desired resultsi andii.
Remark 5.4. The validity of ii inCorollary 5.3 under strong force conditions remains still open to us. Such an open problem has been partially solved byCorollary 3.5. However, we do not solve it completely because we need the positivity ofGt, sinCorollary 3.5, and therefore it is not applicable to the critical case. The validity for the critical case remains open to the authors.
The next results explore the case whenγ∗>0.
Theorem 5.5. Suppose that at satisfies (B) and ft, x satisfies condition (H2). Furthermore, assume that
H9there existsR > γ∗such that
g γ∗
1h
Rγ∗ g
Rγ∗
ω∗≤R. 5.9
Ifγ∗>0,then1.1has at least one positiveT-periodic solution.
Proof. We follow the same strategy and notation as in the proof ofTheorem 5.1. LetRbe the positive constant satisfyingH9and r γ∗,thenR > r > 0 sinceR > γ∗. Next we prove TΩ⊂Ω.
For eachx∈Ω, by the nonnegative sign ofGt, sandft, x, we have
Txt T
0
Gt, sfs, xsdsγt≥γ∗r >0. 5.10
On the other hand, byH2andH9, we have
Txt≤ T
0
Gt, sg
xs γs
1 h
xs γs
g
xs γs
ds
≤g γ∗
1 h
Rγ∗ g
Rγ∗
ω∗≤R.
5.11
In conclusion,TΩ⊂Ω,and the proof is finished by Schauder’s fixed point theorem.
Corollary 5.6. Suppose that at satisfies (B) andα, β ≥ 0, then for each e ∈ CR/TZ,Rwith γ∗>0, one has the following:
iifαβ <1,then1.5has at least one positiveT-periodic solution for eachμ≥0;
iiifαβ≥ 1, then1.5has at least one positiveT-periodic solution for each 0≤ μ < μ3, whereμ3is some positive constant.
Proof. We applyTheorem 5.5and follow the same notation as in the proof ofCorollary 3.5.
ThenH2is satisfied, and the existence conditionH9becomes
μ < Rγ∗α−ω∗ ω∗
Rγ∗ αβ 5.12
for someR >0. So1.5has at least one positiveT-periodic solution for
0< μ < μ3sup
R>0
Rγ∗α−ω∗ ω∗
Rγ∗ αβ. 5.13
Note thatμ3 ∞ifαβ < 1 andμ3 < ∞ifαβ ≥ 1. We have the desired resultsiand ii.
Acknowledgments
The authors express their thanks to the referees for their valuable comments and suggestions.
The research of J. Chu is supported by the National Natural Science Foundation of China Grant no. 10801044 and Jiangsu Natural Science Foundation Grant no. BK2008356.
The research of J. J. Nieto is partially supported by Ministerio de Education y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PR.
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