AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 310469 10.1155/2013/310469 310469 Research Article Positive Periodic Solution for Second-Order Singular Semipositone Differential Equations Xing Xiumei Chu Jifeng School of Mathematics and Statistics Yili Normal University Yining City 835000 China ylsy.edu.cn 2013 3 02 2013 2013 07 11 2012 20 12 2012 2013 Copyright © 2013 Xiumei Xing. 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.

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.

1. Introduction

In this paper, we study the existence of positive T-periodic solutions for the following singular semipositone differential equation: (1)x′′+h(t)x+a(t)x=f(t,x,x), where h,aC(R/TZ,R) and the nonlinearity fC((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)limx0+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 . Some strong force conditions introduced by Gordon  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 , degree theory , some fixed point theorem in cones for completely continuous operators , 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 Section 2, 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>γ0,  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: given φL1[0,T], we write φ0 if φ0 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 pL1[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.

The Green function G(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(/T) and the unique T-periodic solution xl(t)>0 for all t if l0.

Under hypothesis (A), we denote (8)A=min0s,tTG(t,s),B=max0s,tTG(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)=(1)(t). In particular, TAw(t)TB.

With the help of [18, 19], the authors give a sufficient condition to ensure that (4) admits the antimaximum principle in . 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 [<xref ref-type="bibr" rid="B3">14</xref>, Corollary 2.6]).

Assume that a0 and the following two inequalities are satisfied: (10)0Ta(s)σ(h)(s)σ1(-h)(s)ds0,sup0tT{tt+Tσ(-h)(s)dstt+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  and has been used by Meehan and O’Regan in .

Lemma 3.

Assume Ω is an open subset of a convex set K in a normed linear space X and pΩ. Let T:Ω¯K be a compact and continuous map. Then one of the following two conclusions holds:

T has at least one fixed point in Ω¯.

There exists xΩ and 0<λ<1 such that x=λTx+(1-λ)p.

In applications below, we take K=CT1={x:x,xC(R/TZ,R)}X with the norm x=maxt[0,T]|x(t)| and define Ω={xCT1: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:

F(t,x,y)=f(t,x,y)+M0 for all (t,x,y)[0,T]×(0,r]×R.

f(t,x,y)g0(x) for (t,x,y)[0,T]×(0,R0]×R, where the nonincreasing continuous function g0(x)>0 satisfies limx0+g0(x)=+ and limx0+xR0g0(u)du=+.

0F(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].

(12)rg(ιr-Mw*)(1+h(r)/g(r))ϱ((r+M)L)>w*,

where (13)L=20Ta(t)σ(h)(t)dtmin0tTσ(h)(t).

Then (1) has at least one positive periodic solution u(t) with 0<u+Mwr.

Proof.

For convinence, let us write Z(t)=x(t)-Mw(t), Zn(t)=xn(t)-Mw(t), where w(t)=(1)(t). Let (14)Ah=min0tTσ(h)(t),Bh=max0tTσ(h)(t),ιh=BhAh,(15)M¯h=(r+M)L·max0tT|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<xr 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+Mwr.

Choose n0{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],  nN0,  xBr={x:x<r} and Fn(t,x,y)=F(t,max{1/n,x},y).

A T-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)ds1.

We claim that for any T-periodic solution xn(t) of (17) satisfies (21)xnLr. 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 have x(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=20Tσ(h)(s)a(s)xn(s)ds2r0Tσ(h)(s)a(s)ds,where we have used the assumption (11) and xn<r. Therefore, (25)(min0tTσ(h)(t))|xn(t)|2r0Tσ(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.

Choose n1N0 such that 1/n1R1, and then let N1={n1,n1+1,}. The following lemma holds.

Lemma 5.

There exists an integer n2>n1 large enough such that, for all nN2={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)xg~0(x)>max{M+M¯,ιhra1} for all (t,x,y)[0,T]×(0,R1]×R, where g~0(x) satisfies also the strong force condition like in (H2).

For nN1, let αn=min0tTZn(t),  βn=max0tTZn(t).

If αnR1, due to nN1, (27) holds.

If αn<R1, we claim that, for all nN1, (29)βn>R1. Otherwise, suppose that βnR1 for some nN1. Then it is easy to verify (30)Fn(t,Zn(t),Zn(t))>ιhra1. In fact, if 1/nZn(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))>ιhra1. 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)>ιhra1. 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 to αn<R1, that is, αn=min0tT[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 antcn.

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, if t[an,cn] is such that 1/nZn(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¯,ιhrα1}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¯ha(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)na(t)n0.   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 nN1, Zn(t)<1/n<R1. Then there would exist bn(an,cn) such that Zn(bn)=1/n and (38)Zn(t)1nforantbn,1nZn(t)R1forbntcn. 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,  xnLr,  wr 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))+Mbncnxn(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)dZB1. On the other hand, by (H2), we can choose n2N1 large enough such that (42)1/nR1F(ξn(Z),Z,Z)dZ1/nR1g0(Z)dZ>B1 for all nN2={n2,n2+1,}. So (27) holds.

Furthermore, we can prove Zn(t) has a uniform positive lower bound δ.

Lemma 6.

There exist a constant δ>0 such that, for all nN2, (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 nN2, by (H2), (45)αnR1F(ξn(Z),Z,Z)dZαnR1g0(Z)dZ+M(R1-αn)+ if αn0+. Thus we know that there exists a constant δ>0 such that αnδ. Hence (43) holds.

Next, we will prove (17) has periodic solution xn(t).

For ιr>0, we can choose n3N2 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 nN3, consider (17).

Next we claim that any fixed point xn of (18) for any λ[0,1] must satisfy xnr. 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=λ0TG(t,s)Fn(s,Zn(s),Zn(s))dsλA0TFn(s,Zn(s),Zn(s))ds=ιBλ0TFn(s,Zn(s),Zn(s))dsιmaxt[0,T]{λ0TG(t,s)Fn(s,Zn(s),Zn(s))ds}=ιxn-1n. For nN3, we have (48)xn(t)ιxn-1n+1nι(xn-1n)+1nιr. By (27) and assumption (H3), for all t[0,T] and nN3, we have (49)xn(t)=λ0TG(t,s)Fn(s,Zn(s),Zn(s))ds+1n=λ0TG(t,s)F(s,Zn(s),Zn(s))ds+1n0TG(t,s)F(s,Zn(s),Zn(s))ds+1n0TG(t,s)(g(Zn(s))+h(Zn(s)))ϱ(|Zn(s)|)ds+1n0TG(t,s)g(Zn(s))(1+h(Zn(s))g(Zn(s)))ϱ(|Zn(s)|)ds+1n0TG(t,s)g(ιr-Mw*)(1+h(r)g(r))ϱ((r+M)L)ds+1ng(ιr-Mw*)(1+h(r)g(r))ϱ((r+M)L)w*+1n3. Therefore, (50)r=xg(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}nN3 is a bounded and equicontinuous family on [0,T]. Now Arzela-Ascoli Theorem guarantees that {xn}nN3 has a subsequence {xnk}k, converging uniformly on [0,T] to a function xC[0,T]. From the fact xn<r and xn(t)>δ, x satisfies δx(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>γ0,  μ>0 is a positive parameter, e(t) is a T-periodic function.

If β+γ<1, then (1) has at least one positive periodic solution for each μ>0.

If β+γ1, then (1) has at least one positive periodic solution for each 0<μ<μ1, where μ1 is some positive constant.

Proof.

We will apply Theorem 4 with M=max0tT|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+γ1. We have the desired results.

Acknowledgments

The research of X. Xing is supported by the Fund of the Key Disciplines in the General Colleges and Universities of Xin Jiang Uygur Autonomous Region (Grant no. 2012ZDKK13). It is a pleasure for the author to thank Professor J. Chu for his encouragement and helpful suggestions.

Bravo J. L. Torres P. J. Periodic solutions of a singular equation with indefinite weight Advanced Nonlinear Studies 2010 10 4 927 938 MR2683689 ZBL1232.34064 Chu J. Torres P. J. Zhang M. Periodic solutions of second order non-autonomous singular dynamical systems Journal of Differential Equations 2007 239 1 196 212 10.1016/j.jde.2007.05.007 MR2341553 ZBL1127.34023 Jiang D. Chu J. Zhang M. Multiplicity of positive periodic solutions to superlinear repulsive singular equations Journal of Differential Equations 2005 211 2 282 302 10.1016/j.jde.2004.10.031 MR2125544 ZBL1074.34048 Meehan M. O'Regan D. Existence theory for nonlinear Volterra integrodifferential and integral equations Nonlinear Analysis: Theory, Methods & Applications 1998 31 3-4 317 341 10.1016/S0362-546X(96)00313-6 MR1487548 ZBL0891.45004 Gordon W. B. Conservative dynamical systems involving strong forces Transactions of the American Mathematical Society 1975 204 113 135 MR0377983 10.1090/S0002-9947-1975-0377983-1 ZBL0276.58005 del Pino M. A. Manásevich R. F. Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity Journal of Differential Equations 1993 103 2 260 277 10.1006/jdeq.1993.1050 MR1221906 Zhang M. A relationship between the periodic and the Dirichlet BVPs of singular differential equations Proceedings of the Royal Society of Edinburgh A 1998 128 5 1099 1114 10.1017/S0308210500030080 MR1642144 ZBL0918.34025 Chu J. Torres P. J. Applications of Schauder's fixed point theorem to singular differential equations Bulletin of the London Mathematical Society 2007 39 4 653 660 10.1112/blms/bdm040 MR2346946 ZBL1128.34027 Franco D. Torres P. J. Periodic solutions of singular systems without the strong force condition Proceedings of the American Mathematical Society 2008 136 4 1229 1236 10.1090/S0002-9939-07-09226-X MR2367097 ZBL1129.37033 Bonheure D. de Coster C. Forced singular oscillators and the method of lower and upper solutions Topological Methods in Nonlinear Analysis 2003 22 2 297 317 MR2036378 ZBL1108.34033 Zhang M. Periodic solutions of equations of Emarkov-Pinney type Advanced Nonlinear Studies 2006 6 1 57 67 MR2196891 ZBL1107.34037 Torres P. J. Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem Journal of Differential Equations 2003 190 2 643 662 10.1016/S0022-0396(02)00152-3 MR1970045 ZBL1032.34040 Torres P. J. Existence and stability of periodic solutions for second-order semilinear differential equations with a singular nonlinearity Proceedings of the Royal Society of Edinburgh A 2007 137 1 195 201 10.1017/S0308210505000739 MR2359779 ZBL1190.34050 Chu J. Fan N. Torres P. J. Periodic solutions for second order singular damped differential equations Journal of Mathematical Analysis and Applications 2012 388 2 665 675 10.1016/j.jmaa.2011.09.061 MR2869777 ZBL1232.34065 Chu J. Li M. Positive periodic solutions of Hill's equations with singular nonlinear perturbations Nonlinear Analysis: Theory, Methods & Applications 2008 69 1 276 286 10.1016/j.na.2007.05.016 MR2417870 ZBL1148.34025 Li X. Zhang Z. Periodic solutions for damped differential equations with a weak repulsive singularity Nonlinear Analysis: Theory, Methods & Applications 2009 70 6 2395 2399 10.1016/j.na.2008.03.023 MR2498337 ZBL1165.34349 Zhang M. Periodic solutions of damped differential systems with repulsive singular forces Proceedings of the American Mathematical Society 1999 127 2 401 407 10.1090/S0002-9939-99-05120-5 MR1637460 ZBL0908.34024 Hakl R. Torres P. J. Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign Applied Mathematics and Computation 2011 217 19 7599 7611 10.1016/j.amc.2011.02.053 MR2799774 ZBL1235.34064 Zhang M. Optimal conditions for maximum and antimaximum principles of the periodic solution problem Boundary Value Problems 2010 2010 26 410986 MR2659774 ZBL1200.34001 10.1155/2010/410986 Granas A. Guenther R. B. Lee J. W. Some general existence principles in the Carathéodory theory of nonlinear differential systems Journal de Mathématiques Pures et Appliquées 1991 70 2 153 196 MR1103033 ZBL0687.34009