# Positive Periodic Solution for Second-Order Singular Semipositone Differential Equations

Authors: Xiumei Xing
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study the existence of a positive periodic solution for second-order singular semipositone differential equation by a nonlinear alternative principle of Leray-Schauder. Truncation plays an important role in the analysis of the uniform positive lower bound for all the solutions of the equation. Recent results in the literature (Chu et al., 2010) are generalized.


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## Body

## 1. Introduction

In this paper, we study the existence of positiveT-periodic solutions for the following singular semipositone differential equation:
(1)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) satisfies f(t,x,x′)≥-M for some M>0. In particular, the nonlinearity may have a repulsive singularity at x=0, which means that
(2)limx→0+f(t,x,y)=+∞,uniformlyin(t,y)∈R2.
Electrostatic or gravitational forces are the most important examples of singular interactions.During the last two decades, the study of the existence of periodic solutions for singular differential equations has attracted the attention of many researchers [1–4]. Some strong force conditions introduced by Gordon [5] are standard in the related earlier works [6, 7]. 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 [2, 8, 9], but has also attracted many researchers. Some classical tools have been used to study singular differential equations in the literature, including the method of upper and lower solutions [10], degree theory [11], some fixed point theorem in cones for completely continuous operators [12], Schauder’s fixed point theorem [8, 9, 13], and a nonlinear Leray-Schauder alternative principle [2, 3, 14, 15].However the singular differential equations, in which there is the damping term, that is, the nonlinearity is dependent on the derivative, has not attracted much attention in the literature. Several existence results can be found in [14, 16, 17].The aim of this paper is to further show that the nonlinear Leray-Schauder alternative principle can be applied to (1) in  the semipositone cases, that is, f(t,x,x′)≥-M for some M>0.The remainder of the paper is organized as follows. In Section2, we state some known results. In Section 3, the main results of this paper are stated and proved. To illustrate our result, we select the following system:
(3)x′′+h(t)x′+a(t)x=(1+|x|γ)(x-α+μxβ)+e(t),
where α>1,  β>0,  1>γ≥0,  μ>0 is a positive parameter, e(t) is a T-periodic function.In this paper, let us fix some notations to be used in the following: givenφ∈L1[0,T], we write φ≻0 if φ≥0 for almost everywhere t∈[0,T] and it is positive in a set of positive measure. The usual Lp-norm is denoted by ∥·∥p. p* and p* the essential supremum and infinum of a given function p∈L1[0,T], if they exist.


## 2. Preliminaries

We say that(4)x′′+h(t)x′+a(t)x=0,
associated to the periodic boundary conditions
(5)x(0)=x(T),x′(0)=x′(T),
is nonresonant when its unique solutions is the trival one. When (4)-(5) is nonresonant, as a consequence of Fredholm’s alternative, the nonhomogeneous equation
(6)x′′+h(t)x′+a(t)x=l(t)
admits a unique T-periodic solution, which can be written as
(7)x(t)=∫0T‍G(t,s)l(s)ds,
where G(t,s) is the Green’s function of problem (4)-(5). Throughout this paper, we assume that the following standing hypothesis is satisfied.(A)
The Green functionG(t,s), associated with (4)-(5), is positive for all (t,s)∈[0,T]×[0,T].In other words, the strict antimaximum principle holds for (4)-(5).Definition 1.
We say that (4) admits the antimaximum principle if (6) has a unique T-periodic solution for any l∈ℂ(ℝ/Tℤ) and the unique T-periodic solution xl(t)>0 for all t if l≻0.Under hypothesis (A), we denote(8)A=min0≤s,t≤TG(t,s),B=max0≤s,t≤TG(t,s),ι=AB.
Thus B>A>0 and 0<ι<1. We also use w(t) to denote the unique periodic solution of (6) with l(t)=1 under condition (5), that is, w(t)=(ℒ1)(t). In particular, TA≤w(t)≤TB.With the help of [18, 19], the authors give a sufficient condition to ensure that (4) admits the antimaximum principle in [14]. In order to state this result, let us define the functions
(9)σ(h)(t)=exp(∫0t‍h(s)ds),σ1(h)(t)=σ(h)(T)∫0t‍σh(s)ds+∫tT‍σ(h)(s)ds.Lemma 2 (see [14, Corollary 2.6]).
Assume thata≢0 and the following two inequalities are satisfied:
(10)∫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 the Green’s function G(t,s), associated with (5), is positive for all (t,s)∈[0,T]×[0,T].Next, recall a well-known nonlinear alternative principle of Leray-Schauder, which can be found in [20] and has been used by Meehan and O’Regan in [4].Lemma 3.
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.In applications below, we takeK=CT1={x:x,x′∈C(R/TZ,R)}⊂X with the norm ∥x∥=maxt∈[0,T]|x(t)| and define Ω={x∈CT1:∥x∥<r}.


## 3. Main Results

In this section, we prove a new existence result of (1).Theorem 4.
Suppose that (4) satisfies (A) and
(11)a(t)≻0.
Furthermore, assume that there exist three constants M,R0,r>Mw*/ι such that:(H1)
F(t,x,y)=f(t,x,y)+M≥0 for all (t,x,y)∈[0,T]×(0,r]×R.(H2)
f(t,x,y)≥g0(x) for (t,x,y)∈[0,T]×(0,R0]×R, where the nonincreasing continuous function g0(x)>0 satisfies limx→0+g0(x)=+∞ and limx→0+∫xR0‍g0(u)du=+∞.(H3)
0≤F(t,x,y)≤(g(x)+h(x))ϱ(|y|),  for  all  (t,x,y)∈[0,T]×(0,r]×R, where g(·)>0 is nonincreasing in (0,r] and h(·)/g(·)≥0,  ϱ(·)≥0 are nondecreasing in (0,r].(H4)
(12)rg(ιr-Mw*)(1+h(r)/g(r))ϱ((r+M)L)>w*,where(13)L=2∫0T‍a(t)σ(h)(t)dtmin0≤t≤Tσ(h)(t).Then (1) has at least one positive periodic solution u(t) with 0<∥u+Mw∥≤r.Proof.
For convinence, let us writeZ(t)=x(t)-Mw(t), Zn(t)=xn(t)-Mw(t), where w(t)=(ℒ1)(t). Let
(14)Ah=min0≤t≤Tσ(h)(t),Bh=max0≤t≤Tσ(h)(t),ιh=BhAh,(15)M¯h=(r+M)L·max0≤t≤T|h(t)|.
First we show that
(16)x′′+h(t)x′+a(t)x=F(t,Z(t),Z′(t))
has a solution x satisfying (5), 0<∥x∥≤r and Z(t)>0 for t∈[0,T]. If this is true, it is easy to see that Z(t) will be a positive solution of (1)–(5) with 0<∥Z+Mw∥≤r.
Choosen0∈{1,2,…} such that 1/n0<r, and then let N0={n0,n0+1+⋯}.
Consider the family of equations(17)x′′+h(t)x′+a(t)x=λFn(t,Z(t),Z′(t))+a(t)n,
where λ∈[0,1],  n∈N0,  x∈Br={x:∥x∥<r} and Fn(t,x,y)=F(t,max{1/n,x},y).
AT-periodic solution of (17) is just a fixed of the operator equation
(18)x=λTn(x)+(1-λ)p,
where p=1/n and Tn is a completely continuous operator defined by
(19)(Tnx)(t)=∫0T‍G(t,s)Fn(s,Z(s),Z′(s))ds+1n,
where we have used the fact
(20)∫0T‍G(t,s)a(s)ds≡1.
We claim that for anyT-periodic solution xn(t) of (17) satisfies
(21)∥xn′∥≤Lr.
Note that the solution xn(t) of (17) is also satisfies the following equivalent equation
(22)(σ(h)(t)xn′)′+a(t)σ(h)(t)xn=σ(h)(t)(λFn(t,Zn(t),Zn′(t))+a(t)n).
Integrating (22) from 0 to T, we obtain
(23)∫0T‍a(t)σ(h)(t)xn(t)dt=∫0T‍σ(h)(t)(λFn(t,Zn(t),Zn′(t))+a(t)n)dt.
By the periodic boundary conditions, we havex′(t0)=0 for some t0∈[0,T]. Therefore,(24)|σ(h)(t)xn′(t)|=|∫t0t‍(σ(h)(s)xn′(s))′ds|=|∫t0t‍σ(h)(s)(λFn(s,Zn(s),Zn′(s))+a(s)n-a(s)xn(s))|ds≤|∫0T‍σ(h)(s)(λFn(s,Zn(s),Zn′(s))+a(s)n+a(s)xn(s))|ds=2∫0T‍σ(h)(s)a(s)xn(s)ds≤2r∫0T‍σ(h)(s)a(s)ds,where we have used the assumption (11) and ∥xn∥<r. Therefore,
(25)(min0≤t≤Tσ(h)(t))|xn′(t)|≤2r∫0T‍σ(h)(s)a(s)ds,
which implies that (21) holds. In particular, let λFn(t,Z(t),Z′(t))+a(t)/n=1 in (17), we have
(26)∥w′(t)∥≤L.
Choosen1∈N0 such that 1/n1≤R1, and then let N1={n1,n1+1,…}. The following lemma holds.Lemma 5.
There exists an integern2>n1 large enough such that, for all n∈N2={n2,n2+1,…},
(27)Zn(t)=xn(t)-Mw(t)≥1n.Proof.
The lower bound in (27) is established by using the strong force condition of f(t,x,y). By condition (H2), there exists R1∈(0,R0) and a continuous function g~0(x) such that
(28)F(t,x,y)-a(t)x≥g~0(x)>max{M+M¯,ιhr∥a∥1}
for all (t,x,y)∈[0,T]×(0,R1]×R, where g~0(x) satisfies also the strong force condition like in (H2).
Forn∈N1, let αn=min0≤t≤TZn(t),  βn=max0≤t≤TZn(t).
Ifαn≥R1, due to n∈N1, (27) holds.
Ifαn<R1, we claim that, for all n∈N1,
(29)βn>R1.
Otherwise, suppose that βn≤R1 for some n∈N1. Then it is easy to verify
(30)Fn(t,Zn(t),Zn′(t))>ιhr∥a∥1.
In fact, if 1/n≤Zn(t)≤R1, we obtain from (28)
(31)Fn(t,Zn(t),Zn′(t))=F(t,Zn(t),Zn′(t))≥a(t)Zn(t)+g~0(Zn(t))≥g~0(Zn(t))>ιhr∥a∥1.
and, if Zn(t)≤1/n, we have
(32)Fn(t,Zn(t),Zn′(t))=F(t,1n,Zn′(t))≥a(t)n+g~0(1n)≥g~0(1n)>ιhr∥a∥1.
Integrating (22) (with λ=1) from 0 to T, we deduce that
(33)0=∫0T‍{(Fn(t,Zn(t),Zn′(t))+a(t)n)(σ(h)(t)xn′)′+a(t)σ(h)(t)xn-σ(h)(t)(Fn(t,Zn(t),Zn′(t))+a(t)n)}dt=∫0T‍a(t)σ(h)(t)xndt-∫0T‍σ(h)(t)Fn(t,Zn(t),Zn′(t))dt-∫0T‍σ(h)(t)a(t)ndt<∫0T‍a(t)σ(h)(t)xndt-∫0T‍σ(h)(t)Fn(t,Zn(t),Zn′(t))dt<0,
where estimation (30) and the fact ∥xn∥<r are used. This is a contradiction. Hence (29) holds.
Due toαn<R1, that is, αn=min0≤t≤T[xn(t)-Mw(t)]=xn(an)-Mw(an)<R1 for some an∈[0,T]. By (29), there exists cn∈[0,T] (without loss of generality, we assume an<cn.) such that xn(cn)=Mw(cn)+R1 and xn(t)≤Mw(t)+R1 for an≤t≤cn.
It can be checked that(34)Fn(t,Zn(t),Zn′(t))>a(t)Zn(t)+M+M¯h,
where M¯h is defined by (15).
In fact, ift∈[an,cn] is such that 1/n≤Zn(t)≤R1, we have
(35)Fn(t,Zn(t),Zn′(t))=F(t,Zn(t),Zn′(t))≥a(t)Zn(t)+g~0(x)>max{M+M¯,ιhr∥α∥1}≥a(t)Zn(t)+M+M¯.
and, if t∈[an,cn] is such that Zn(t)≤1/n, we have
(36)Fn(t,Zn(t),Zn′(t))=F(t,1n,Zn′(t))≥a(t)n+g~0(1n)>a(t)n+M+M¯h≥a(t)Zn(t)+M+M¯h.
So (34) holds.
Using (17) (with λ=1) for xn(t) and the estimation (34), we have, for t∈[an,cn](37)Zn′′(t)=-h(t)Zn′(t)-a(t)Zn(t)-M+Fn(t,Zn(t),Zn′(t))+a(t)n>-h(t)Zn′(t)-a(t)Zn(t)-M+a(t)Zn(t)+M+M¯h+a(t)n≥-M¯h-a(t)Zn(t)-M+a(t)Zn(t)+M+M¯h+a(t)n≥a(t)n≥0.
As Zn′(an)=0, Zn′(t)>0 for all t∈[an,cn], so Zn(t) is strictly increasing on [an,cn]. We use ξn to denote the inverse function of Zn restricted to [an,cn].
Suppose that (27) does not hold, that is, for some n∈N1, Zn(t)<1/n<R1. Then there would exist bn∈(an,cn) such that Zn(bn)=1/n and
(38)Zn(t)≤1nforan≤t≤bn,1n≤Zn(t)≤R1forbn≤t≤cn.
Multiplying (17) (with λ=1) by Zn′(t) and integrating from bn to cn, we obtain
(39)∫1/nR1‍F(ξn(Z),Z,Z′)dZ=∫bncn‍F(t,Zn(t),Zn′(t))Zn′(t)dt=∫bncn‍Fn(t,Zn(t),Zn′(t))Zn′(t)dt=∫bncn‍(xn′′(t)+h(t)xn′(t)+a(t)xn(t)-a(t)n)Zn′(t)dt=∫bncn‍xn′′(t)(xn′(t)-Mw′(t))dt+∫bncn‍h(t)xn′(t)Zn′(t)dt+∫bncn‍(a(t)xn(t)-a(t)n)Zn′(t)dt.
By the facts∥xn∥<r,  ∥xn′∥≤Lr,  ∥w′∥≤r and the definition of Zn(t), we can obtain |Zn(t)|≤r+TB,|Zn′(t)|≤(r+M)L, together with ∥xn∥<r, implies that the second term and the third term are bounded. The first term is
(40)([xn′(cn)]2-[xn′(bn)]2)2-M(xn′(cn)w′(cn)-xn′(bn)w′(bn))+M∫bncn‍xn′(t)w′′(t)dt,
which is also bounded. As a consequence, there exists a B1>0 such that
(41)∫1/nR1‍F(ξn(Z),Z,Z′)dZ≤B1.
On the other hand, by (H2), we can choose n2∈N1 large enough such that
(42)∫1/nR1‍F(ξn(Z),Z,Z′)dZ≥∫1/nR1‍g0(Z)dZ>B1
for all n∈N2={n2,n2+1,…}. So (27) holds.Furthermore, we can proveZn(t) has a uniform positive lower bound δ.Lemma 6.
There exist a constantδ>0 such that, for all n∈N2,
(43)Zn(t)≥δ.Proof.
Multiplying (17) (with λ=1) by Zn′(t) and integrating from an to cn, we obtain
(44)∫αnR1‍F(ξn(Z),Z,Z′)dZ=∫ancn‍F(t,Zn(t),Zn′(t))Zn′(t)dt=∫ancn‍Fn(t,Zn(t),Zn′(t))Zn′(t)dt=∫ancn‍(xn′′(t)+h(t)xn′(t)+a(t)xn(t)-a(t)n)Zn′(t)dt=∫ancn‍xn′′(t)(xn′(t)-Mw′(t))dt+∫ancn‍h(t)xn′(t)Zn′(t)dt+∫ancn‍(a(t)xn(t)-a(t)n)Zn′(t)dt.
In the same way as in the proof of (41), one way readily prove that the right-hand side of the above equality is bounded. On the other hand, if n∈N2, by (H2),
(45)∫αnR1‍F(ξn(Z),Z,Z′)dZ≥∫αnR1‍g0(Z)dZ+M(R1-αn)→+∞
if αn→0+. Thus we know that there exists a constant δ>0 such that αn≥δ. Hence (43) holds.
Next, we will prove (17) has periodic solution xn(t).
Forιr>0, we can choose n3∈N2 such that 1/n3<ιr, which together with (H4) imply
(46)w*g(ιr-Mw*)(1+h(r)g(r))ϱ((r+M)L)+1n3<r.
Let N3={n3,n3+1,…}. For n∈N3, consider (17).
Next we claim that any fixed pointxn of (18) for any λ∈[0,1] must satisfy ∥xn∥≠r. So, by using the Leray-Schauder alternative principle, (17) (with λ=1) has a periodic solution xn(t). Otherwise, assume that xn is a fixed point xn of (18) for some λ∈[0,1] such that ∥xn∥=r. Note that
(47)xn(t)-1n=λ∫0T‍G(t,s)Fn(s,Zn(s),Zn′(s))ds≥λA∫0T‍Fn(s,Zn(s),Zn′(s))ds=ιBλ∫0T‍Fn(s,Zn(s),Zn′(s))ds≥ιmaxt∈[0,T]{λ∫0T‍G(t,s)Fn(s,Zn(s),Zn′(s))ds}=ι∥xn-1n∥.
For n∈N3, we have
(48)xn(t)≥ι∥xn-1n∥+1n≥ι(∥xn∥-1n)+1n≥ιr.
By (27) and assumption (H3), for all t∈[0,T] and n∈N3, we have
(49)xn(t)=λ∫0T‍G(t,s)Fn(s,Zn(s),Zn′(s))ds+1n=λ∫0T‍G(t,s)F(s,Zn(s),Zn′(s))ds+1n≤∫0T‍G(t,s)F(s,Zn(s),Zn′(s))ds+1n≤∫0T‍G(t,s)(g(Zn(s))+h(Zn(s)))ϱ(|Zn′(s)|)ds+1n≤∫0T‍G(t,s)g(Zn(s))(1+h(Zn(s))g(Zn(s)))ϱ(|Zn′(s)|)ds+1n≤∫0T‍G(t,s)g(ιr-Mw*)(1+h(r)g(r))ϱ((r+M)L)ds+1n≤g(ιr-Mw*)(1+h(r)g(r))ϱ((r+M)L)w*+1n3.
Therefore,
(50)r=∥x∥≤g(ιr-Mw*)(1+h(r)g(r))ϱ((r+M)L)w*+1n3.
This is a contradiction to the choice of n3 and the claim is proved.
The fact∥xn∥<r and ∥xn′(t)∥<Lr show that {xn}n∈N3 is a bounded and equicontinuous family on [0,T]. Now Arzela-Ascoli Theorem guarantees that {xn}n∈N3 has a subsequence {xnk}k∈ℕ, converging uniformly on [0,T] to a function x∈C[0,T]. From the fact ∥xn∥<r and xn(t)>δ, x satisfies δ≤x(t)≤r for all t. Moreover, {xnk} satisfies the integral equation
(51)xnk(t)=∫0T‍G(t,s)F(s,Znk(s),Znk′(s))ds+1nk.
Letting k→∞, we arrive at
(52)x(t)=∫0T‍G(t,s)F(s,x(s)-Mw(s),x′(s)-Mw′(s))ds,
where the uniform continuity of F(t,x,y) on [0,T]×[δ,r]×[-(r+M)L,(r+M)L] is used. Therefore, x is a positive periodic solution of (16) and Z(t)=x(t)-Mw(t)≥δ. Thus we complete the prove of Theorem 4.Corollary 7.
Let the nonlinearity in (1) be
(53)f(t,x,y)=(1+|y|γ)(x-α+μxβ)+e(t),
where α>1,  β>0,  1>γ≥0,  μ>0 is a positive parameter, e(t) is a T-periodic function. (i)
Ifβ+γ<1, then (1) has at least one positive periodic solution for each μ>0.(ii)
Ifβ+γ≥1, then (1) has at least one positive periodic solution for each 0<μ<μ1, where μ1 is some positive constant.Proof.
We will apply Theorem4 with M=max0≤t≤T|e(t)| and g(x)=x-α,  h(x)=μxβ+2M,  ϱ(y)=1+|y|γ. Then condition (H1)–(H3) are satisfied and existence condition (H4) becomes
(54)μ<r(ιr-Mw*)α-w*(1+(r+M)γLγ)(1+2Mrα)w*(1+(r+M)γLγ)rα+β.
So (1) has at least one positive periodic solution for
(55)0<μ<μ1=supr>Mw*/ι(r(ιr-Mw*)α-w*(1+(r+M)γLγ)×(1+2Mrα)r(ιr-Mw*)α)×(w*(1+(r+M)γLγ)rα+β)-1.
Note that μ1=∞ if β+γ<1 and μ1<∞ if β+γ≥1. We have the desired results.



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