Skip to main content

Theory and Modern Applications

Existence of positive periodic solutions for third-order differential equation with strong singularity

Abstract

Sufficient conditions are presented for the existence of positive periodic solutions for a third-order nonlinear differential equation with singularity. Besides, an example is given to illustrate the results.

MSC:34K13, 34B16, 34B18.

1 Introduction

In this paper, we consider the following third-order differential equation with singularity:

x +f ( t , x ) x +h(t,x) x =g(t,x),
(1.1)

where f, h are continuous function and T-periodic about t, h(t,x)0, g:[0,T]×(0,)R is an L 2 -Carathéodory function, i.e., it is measurable in the first variable and continuous in the second variable, and for every 0<r<s there exists h r , s L 2 [0,ω] such that |f(t,x(t))| h r , s for all x[r,s] and a.e. t[0,ω], f is ω-periodic function about t. Equation (1.1) is singular at 0, which means that g(t,x) becomes unbounded when x 0 + . We say that (1.1) is of repulsive type (resp. attractive type) if g(t,x)+ (resp. g(t,x)) when x0.

The study of singular differential equations began with the paper of Taliaferro. In 1979, Taliaferro [1] discussed the model equation with singularity

y + q ( t ) y α =0,0<t<1,
(1.2)

subject to

y(0)=0=y(1),

and obtained the existence of a solution for the problem. Here α>0, qC(0,1) with q>0 on (0,1) and 0 1 t(1t)q(t)dt<. We call the equation a strong force condition if α1 and we call it a weak force condition if 0<α<1.

Taliaferro’s work has attracted the attention of many specialists in differential equations and they have contributed to the research of singular differential equations (see, e.g., [210]). Among these results, some are obtained for a second-order equation with strong force condition; see, e.g., [5, 9]. With a strong singularity, the energy near the origin becomes infinite and this fact is helpful for obtaining either a priori bounds, which are needed for a classical application of the degree theory, or the fast rotation, which is needed in recent versions of the Poincaré-Birkhoff theorem. Afterwards, in 2007 Torres [10] considered the periodic problem for a singular second-order equation with the weak force condition and showed that weak singularities may help periodic solutions to exist, which has driven the study of weak singularities (see [7]).

At the beginning, most of work concentrated on second-order singular differential equation, as in the references we mentioned above. Recently there have been published some results on third-order singular differential equation (see [1117]). For example, in [13], Sun and Liu considered the singular nonlinear third-order periodic boundary value problem

u + ρ 3 u=f(t,u),0t2π
(1.3)

with u ( i ) (0)= u ( i ) (2π), i=0,1,2, where ρ(0,1/ 3 ) and f is singular at t=0, t=1, and u=0. Under suitable growth conditions, it is proved by constructing a special cone in C[0,2π] and employing fixed point index theory that the problem has at least one solution or at least two positive solutions. Afterwards, Li [15] investigated the third-order ordinary differential equation

u (t)=f ( t , u ( t ) , u ( t ) , u ( t ) ) ,tR,
(1.4)

where fC(R×(0,)×R×R) is ω-periodic in t, and f(t,u,v,w) may be singular at u=0. By applying of a fixed point theorem in cones, the author obtained that existence results of positive ω-periodic solutions for (1.4). Recently, Ren et al. [17] studied the third-order nonlinear singular differential equation

x (t)+a x (t)+b x (t)+cx(t)=f ( t , x ( t ) ) +e(t).
(1.5)

Using Green’s function for a third-order differential equation and some fixed point theorems, i.e., the Leray-Schauder alternative principle and Schauder’s fixed point theorem, they established three new existence results of periodic solutions for (1.5).

Based on the above work, in this paper we will study (1.1) and obtain the existence of periodic solutions by using topological degree theorem. The rest of this paper is organized as follows. In Section 2, we give some lemmas. In Section 3, by using topological degree theorem by Mawhin [18], some sufficient conditions are obtained for the existence of positive periodic solutions of (1.1). We, respectively, consider repulsive type and attractive type. In Section 4, an example is given to show the feasibility of the main result of this paper.

2 Some lemmas

Lemma 2.1 [[18], Theorem 2.4]

Let X, Y be real normed spaces and L:D(L)XY a linear Fredholm map of index zero. Assume that ΩX is an open bounded set and N: Ω ¯ Y is an L-compact mapping. Assume that the following conditions are satisfied:

  1. (i)

    Lx+λNx0, for each (x,λ)[(D(L)kerL)Ω]×(0,1);

  2. (ii)

    NxImL, for each xkerLΩ;

  3. (iii)

    D 0 (QN | ker L ,ΩkerL)0, where Q:YY is a continuous projector such that kerQ=ImL and D 0 is the Brouwer degree,

then the equation Lx+Nx=0 has at least one solution in D(L) Ω ¯ .

For the sake of convenience, throughout this paper we will adopt the following notation:

| u | = max t [ 0 , T ] | u ( t ) | , | u | 0 = min t [ 0 , T ] | u ( t ) | , | u | p = ( 0 T | u | p d t ) 1 p , h ¯ = 1 T 0 T h ( t ) d t .

Lemma 2.2 [19]

If ω C 1 (R,R) and ω(0)=ω(T)=0, then

0 T | ω ( t ) | p dt ( T π p ) p 0 T | ω ( t ) | p dt,

where 1p<, π p =2 0 ( p 1 ) / p d s ( 1 s p p 1 ) 1 / p = 2 π ( p 1 ) 1 / p p sin ( π / p ) .

Remark 2.1 When p=2, π 2 =2 0 ( 2 1 ) / 2 d s ( 1 s 2 2 1 ) 1 / 2 = 2 π ( 2 1 ) 1 / 2 2 sin ( π / 2 ) =π.

Lemma 2.3 [20]If x C 2 (R,R) with x(t+T)=x(t), then

| x ( t ) | 2 2 ( T 2 π ) 2 | x ( t ) | 2 2 .

Lemma 2.4 If x C 1 (R,R) with x(t+T)=x(t), and t 0 [0,T] such that |x( t 0 )|<d, then

( 0 T | x ( t ) | p d t ) 1 p ( T π p ) ( 0 T | x ( t ) | p d t ) 1 p +d T 1 p .

Proof Let ω(t)=x(t+ t 0 )x( t 0 ), and then ω(0)=ω(T)=0. By Lemma 2.2 and Minkowski’s inequality, we have

( 0 T | x ( t ) | p d t ) 1 p = ( 0 T | ω ( t ) + x ( t 0 ) | p d t ) 1 p ( 0 T | ω ( t ) | p d t ) 1 p + ( 0 T | x ( t 0 ) | p d t ) 1 p ( T π p ) ( 0 T | ω ( t ) | p d t ) 1 p + d T 1 p = ( T π p ) ( 0 T | x ( t ) | p d t ) 1 p + d T 1 p .

This completes the proof of Lemma 2.4. □

Lemma 2.5 If x C 2 (R,R) with x(t+T)=x(t), and t 0 [0,T] such that |x( t 0 )|<d, then

( 0 T | x ( t ) | 2 d t ) 1 2 ( T 2 2 π 2 ) ( 0 T | x ( t ) | 2 d t ) 1 2 +d T 1 2 .

Proof From Lemma 2.3 and Lemma 2.4, we know that, when p=2, we have

( 0 T | x ( t ) | 2 d t ) 1 2 ( T π ) ( 0 T | x ( t ) | 2 d t ) 1 2 + d T 1 2 ( T π ) ( T 2 π ) ( 0 T | x ( t ) | 2 d t ) 1 2 + d T 1 2 = ( T 2 2 π 2 ) ( 0 T | x ( t ) | 2 d t ) 1 2 + d T 1 2 .

This completes the proof of Lemma 2.5. □

3 Main results

First we consider (1.1) when g(t,x) is of attractive type. Assume that

φ(t)= lim x + sup g ( t , x ) x
(3.1)

exists uniformly a.e. t[0,T], i.e., for any ε>0 there is g ε L 2 (0,T) such that

g(t,x) ( φ ( t ) + ε ) x+ g ε (t)
(3.2)

for all x>0 and a.e. t[0,T]. Moreover, φC(R,R) and φ(t+T)=φ(t).

For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:

(H1) There exist two positive constants D 1 < D 2 such that

g(t,x)<0,for all 0<x< D 1 ;andg(t,x)>0,for all x> D 2 .

(H2) (Decomposition condition) g(t,x)= g 0 (x)+ g 1 (t,x), where g 0 C((0,);R) and g 1 :[0,T]×[0,)R is an L 2 -Carathéodory function, i.e., it is measurable in the first variable and continuous in the second variable, and for any b>0 there is h b L 2 (0,T; R + ) such that

| g 1 ( t , x ) | h b (t),a.e. t[0,T],0xb.

(H3) (Strong force condition at x=0) 0 1 g 0 (x)dx=.

(H4) There exists a positive constant A such that |f(t,u)|A, for all (t,u)[0,T]×R.

Theorem 3.1 Assume that (3.2), h(t,x)0, and (H1)-(H4) hold. We have the following condition:

(H5) A( T 2 π )+ | φ + | ( T 3 π 2 )<1.

Then (1.1) has at least one positive T-periodic solution.

Proof Let X={x:RR is  C 2  and satisfies x(t+T)x(t)}, endowed with the C 2 -norm. Let Y= L 2 (0,T;R) with the L 2 -norm.

Let D(L)={xX: x  is absolutely continuous on R} and let L:D(L)Y be the operator defined by

(Lx)(t)= x (t),tR.

Define a nonlinear mapping N:YY by

(Nx)(t)=f ( t , x ) x (t)+h(t,x) x (t)g ( t , x ( t ) ) .
(3.3)

Then (1.1) can be converted to the abstract equation Lx+Nx=0. Define the projectors P:XX and Q:YY by

Px= 1 T 0 T x(s)ds;Qy= 1 T 0 T y(s)ds.
(3.4)

The real number Px and Qy are seen as elements of X and Y inasmuch constant function. It is easy to see that kerL=R, ImL={yY: 0 T y(t)dt=0}, kerQ=ImL, ImP=kerL, and then L is a Fredholm linear mapping with zero index.

Let K denote the inverse of L | ker P D ( L ) . Then we have

[ K y ] ( t ) = t 2 0 T ( T s ) y ( s ) d s t 2 T 0 T ( T s ) 2 y ( s ) d s t 2 2 T 0 T ( T s ) y ( s ) d s + 1 2 0 t ( t s ) 2 y ( s ) d s .
(3.5)

From (3.3), (3.4), and (3.5), it follows that QN and K(IQ)N are continuous, and QN( Ω ¯ ) is bounded and then K(IQ)N( Ω ¯ ) is compact for any open bounded ΩX, which means N is L-compact on Ω ¯ .

Now we consider the following (homotopy) family of (1.1):

x +λf ( t , x ) x +λh(t,x) x =λg(t,x),λ[0,1],
(3.6)

i.e., the abstract equation Lx+λNx=0. We need to show that the set of all possible solutions of the family of (3.6) is, a priori, bounded in C 2 (R,R) by a constant independent of λ[0,1].

Suppose that x is a solution to (3.6) for some λ[0,1]. Let t , t be, respectively, the global maximum point and global minimum point of x (t) on [0,T]; Firstly, we consider x ( t )= max t R x (t)= max t [ 0 , T ] x (t). Since 0 T x (t)dt=0, we know that there exist two points t 1 , t 2 such that x ( t 1 )<0, x ( t 2 )>0. So, we get x ( t )> x ( t 2 )>0. Because x ( t ) is the maximum value of x (t), x ( t )=0. Furthermore, we conclude

x ( t ) 0.
(3.7)

So, we have

x ( t ) +λh ( t , x ( t ) ) x ( t ) =λg ( t , x ( t ) ) ,

since x ( t )0 and h( t ,x( t )) x ( t )0, and we get

g ( t , x ( t ) ) 0.

From (H1) we obtain

x ( t ) D 2 .
(3.8)

Similarly, we get

g ( t , x ( t ) ) 0.

From (H1) we obtain

x( t ) D 1 .
(3.9)

From (3.8) and (3.9), we know that there exists a point ξ[0,T] such that

D 1 x(ξ) D 2 .
(3.10)

Therefore, we have

| x ( t ) | =|x(ξ)+ ξ t x (s)ds| D 2 + 0 T | x ( s ) | ds.
(3.11)

Multiplying by x (t) on both sides of (3.6) and integrating from 0 to T, we have

0 T x ( t ) x ( t ) d t + λ 0 T f ( t , x ) x ( t ) x ( t ) d t + λ 0 T h ( t , x ) ( x ( t ) ) 2 d t = λ 0 T g ( t , x ( t ) ) x ( t ) d t .

Since 0 T x (t) x (t)= 0 T | x ( t ) | 2 dt, from (H4) and h(t,x)0, we have

0 T | x ( t ) | 2 d t = λ 0 T f ( t , x ) x ( t ) x ( t ) d t + λ 0 T h ( t , x ) | x ( t ) | 2 d t λ 0 T g ( t , x ( t ) ) x ( t ) d t λ 0 T f ( t , x ) x ( t ) x ( t ) d t λ 0 T g ( t , x ( t ) ) x ( t ) d t 0 T | f ( t , x ) | | x ( t ) | | x ( t ) | d t + 0 T | g ( t , x ( t ) ) | | x ( t ) | d t A 0 T | x ( t ) | | x ( t ) | d t + 0 T | g ( t , x ( t ) ) | | x ( t ) | d t A 0 T | x ( t ) | | x ( t ) | d t + | x | 0 T | g ( t , x ( t ) ) | d t .

For any ε>0, let g ε L 2 (0,T) be as in (3.2). Thus we have

g + (t,x) ( φ + ( t ) + ε ) x(t)+ g ε + (t).

Therefore,

0 T | g + ( t , x ) | dt ( | φ + | + ε ) 0 T | x ( t ) | dt+ 0 T | g ε + ( t ) | dt.

Since 0 T g(t,x)dt=0, we can get 0 T |g(t,x)|dt=2 0 T | g + (t,x)|dt. So, we have

0 T | g ( t , x ) | dt2 ( | φ + | + ε ) 0 T | x ( t ) | dt+2 0 T | g ε + ( t ) | dt.
(3.12)

From x(0)=x(T), we know that there exists a point ξ 1 [0,T] such that x ( ξ 1 )=0. So, we have

| x | 0 T | x ( t ) | dt.

From (3.12) and the Hölder inequality, we have

0 T | x ( t ) | 2 d t A 0 T | x ( t ) | | x ( t ) | d t + 2 ( | φ + | + ε ) 0 T | x ( t ) | d t 0 T | x ( t ) | d t + 2 0 T | g ε + ( t ) | d t 0 T | x ( t ) | d t A ( 0 T | x ( t ) | 2 d t ) 1 2 ( 0 T | x ( t ) | 2 d t ) 1 2 + 2 ( | φ + | + ε ) T ( 0 T | x ( t ) | 2 d t ) 1 2 ( 0 T | x ( t ) | 2 d t ) 1 2 + 2 T ( 0 T | x ( t ) | 2 d t ) 1 2 ( 0 T | g ε + ( t ) | 2 d t ) 1 2 .

From (3.10) and Lemma 2.5, we have

( 0 T | x ( t ) | 2 d t ) 1 2 ( T 2 2 π 2 ) ( 0 T | x ( t ) | 2 d t ) 1 2 + D 2 T .
(3.13)

From (3.13) and Lemma 2.3, we have

0 T | x ( t ) | 2 d t A ( 0 T | x ( t ) | 2 d t ) 1 2 ( T 2 π ) ( 0 T | x ( t ) | 2 d t ) 1 2 + 2 T ( | φ + | + ε ) [ ( T 2 2 π 2 ) ( 0 T | x ( t ) | 2 d t ) 1 2 + D 2 T ] ( 0 T | x ( t ) | 2 d t ) 1 2 + 2 T ( 0 T | x ( t ) | 2 d t ) 1 2 ( 0 T | g ε + ( t ) | 2 d t ) 1 2 = A ( T 2 π ) 0 T | x ( t ) | 2 d t + ( | φ + | + ε ) ( T 3 π 2 ) 0 T | x ( t ) | 2 d t + [ 2 T ( | φ + | + ε ) D 2 T + 2 T | g ε + | 2 ] ( 0 T | x ( t ) | 2 d t ) 1 2 = [ A ( T 2 π ) + ( | φ + | + ε ) ( T 3 π 2 ) ] 0 T | x ( t ) | 2 d t + 2 T [ ( | φ + | + ε ) D 2 T + | g ε + | 2 ] ( 0 T | x ( t ) | 2 d t ) 1 2 .

Since ε sufficiently small, from (H5) we know that A( T 2 π )+ | φ + | ( T 3 π 2 )<1. Thus, it is easy to see that there exists a positive constant M 1 such that

0 T | x ( t ) | 2 dt M 1 .

So, by the Hölder inequality, we have

| x | 0 T | x ( t ) | dt T ( 0 T | x ( t ) | 2 ) 1 2 T M 1 1 2 := M 2 .
(3.14)

From (3.11), we have

| x | D 2 + 0 T | x ( t ) | ds D 2 + M 2 T:= M 1 .
(3.15)

On the other hand, from x (0)= x (T), we know that there exists a point ξ 2 [0,T] such that x ( ξ 2 )=0. From (3.2), (3.14), and (3.15), and by (3.6), we have

| x | max t R | ξ 2 t x ( s ) d s | λ 0 T | f ( t , x ) | | x ( t ) | d t + λ 0 T | h ( t , x ) | | x ( t ) | d t + λ 0 T | g ( t , x ) | d t λ ( | f | M 2 T M 1 1 2 + | h | M 1 T M 2 + 2 ( | φ + | + ε ) T M 1 + 2 T | g ε + | 2 ) : = λ M 3 ,
(3.16)

where | h | M 1 = max | x | M 1 |h(t,x)|, | f | M 2 = max | u | M 2 |f(t,u)|.

Next, multiplying (3.6) by x (t) we get

x ( t ) x ( t ) + λ f ( t , x ) x ( t ) x ( t ) + λ h ( t , x ) ( x ( t ) ) 2 = λ ( g 0 ( x ( t ) ) + g 1 ( t , x ( t ) ) ) x ( t ) .
(3.17)

Let τ[0,T], for any τtT, we integrate (3.17) on [τ,t] and get

λ x ( τ ) x ( t ) g 0 ( u ) d u = λ τ t g 0 ( x ( s ) ) x ( s ) d s = x ( t ) x ( t ) x ( τ ) x ( τ ) τ t | x ( s ) | 2 d t + λ τ t f ( s , x ) x ( s ) x ( s ) d s + λ τ t h ( s , x ) ( x ( t ) ) 2 d t λ τ t g 1 ( s , x ( s ) ) x ( s ) d s .
(3.18)

By (3.14), (3.15), and (3.16) we have

x ( t ) x ( t ) λ M 3 M 2 , | τ t | x ( s ) | 2 d s | λ 2 M 3 2 T , | τ t f ( s , x ) x ( s ) x ( s ) d s | λ | f | M 2 M 3 M 2 T , | τ t h ( s , x ) x ( s ) x ( s ) d s | | h | M 1 M 2 2 T , | τ t g 1 ( s , x ( s ) ) x ( s ) d s | T M 2 | g M 1 | 2 ,

where g M 1 = max 0 x M 1 | g 1 (t,x)| L 2 (0,T) is as in (H2).

From these inequalities we can derive form (3.18) that

| x ( τ ) x ( t ) g 0 (u)du| M 4 ,
(3.19)

for some constant M 4 which is independent on λ, x and t. In view of the strong force condition (H3), we know that there exists a constant M 4 >0 such that

x(t) M 4 ,t[τ,T].
(3.20)

The case t[0,τ] can be treated similarly.

From (3.8), (3.14), (3.15), (3.16), and (3.20), we let

Ω= { x X : E 1 < x ( t ) < E 2 , | x ( t ) | < E 3  and  | x ( t ) | < E 4 , t [ 0 , T ] } ,
(3.21)

where 0< E 1 <min( M 4 , D 1 ), E 2 >max( M 1 , D 2 ), E 3 > M 2 and E 4 > M 3 . Then the conditions (i) and (ii) of Lemma 2.1 are satisfied. For a constant xkerL, x>0, we have

QNx= 1 T 0 T g(t,x)dt.

The degree condition (H1) shows that

D 0 (QN | ker L ,ΩkerL)=1.

Thus (iii) of Lemma 2.1 is also verified. Therefore Lx+Nx=0 has at least one solution in  Ω ¯ , which means (1.1) has at least one positive T-periodic solution. □

Next we consider (1.1) when g(t,x) is of repulsive type.

Theorem 3.2 Assume that (3.2), h(t,x)0, (H2), and (H4), (H5) are satisfied. We have the following condition:

( H 1 ) there exist two positive constants D 1 < D 2 such that

g ( t , x ( t ) ) >0,for all0<x(t)< D 1 ;andg ( t , x ( t ) ) <0,for allx(t)> D 2 .

( H 3 ) (Strong force condition at x=0) 0 1 g 0 (x)dx=+.

Then (1.1) has at least one positive T-periodic solution.

Proof Let λ[0,1] and consider the following:

x +λf ( t , x ) x +λh(t,x) x =λg(t,x).
(3.22)

Let t , t be, respectively, the global maximum point and global minimum point of x (t) on [0,T]. First, we consider x ( t )= min t R x (t)= min t [ 0 , T ] x (t). Since 0 T x (t)dt=0, we know that there exist two points t 1 , t 2 such that x ( t 1 )<0, x ( t 2 )>0. So, we get x ( t )< x ( t 1 )<0. Because x ( t ) is the minimum value of x (t), x ( t )=0. Furthermore, we can conclude

x ( t )0.
(3.23)

So, we have

x ( t )+λh ( t , x ( t ) ) x ( t )=λg ( t , x ( t ) ) ,

since x ( t )0 and h( t ,x( t )) x ( t )0, we get

g ( t , x ( t ) ) 0.

Hence, from ( H 1 ) we know that there exists a positive constant D 2 such that

x( t ) D 2 .
(3.24)

Similarly, we get

g ( t , x ( t ) ) 0.

Hence, from ( H 1 ) we know that there exists a positive constant D 1 such that

x ( t ) D 1 .
(3.25)

From (3.24) and (3.25), we know that there exists a point ζ[0,T] such that

D 1 x(ζ) D 2 .

Therefore, we have

| x ( t ) | =|x(ζ)+ ζ t x (s)ds| D 2 + 0 T | x ( s ) | ds.
(3.26)

The rest of the proof is the same as that of Theorem 3.1. □

4 Examples

Finally, we present some examples to illustrate our result.

Example 4.1 Consider the three-order differential equation with singularity:

x ( t ) + 1 5 ( 2 sin 2 t cos x ( t ) + 1 ) x ( t ) + 1 10 ( x 2 ( t ) 3 ) x ( t ) = 1 25 ( sin 2 t + 3 ) x ( t ) 1 x ( t ) κ ,
(4.1)

where κ1.

It is clear that T=π, f(t, x )= 1 5 (2sin2tcos x +1), h(t,x)= 1 10 ( x 2 3)<0, g(t,x)= 1 25 (sin2t+3)x 1 x r , φ(t)= 1 25 (sin2t+3). It is obvious that (H1)-(H4) hold. Now we consider the assumption (H5). Since A 3 5 , | φ + | 4 25 , we have

A ( T 2 π ) + | φ + | ( T 3 π 2 ) 3 5 × π 2 π + 4 25 × ( π 3 π 2 ) = 3 5 × 1 2 + 4 25 × π = 15 + 8 π 50 < 1 .

So by Theorem 3.1, we know (4.1) has at least one positive π-periodic solution.

Author’s contributions

ZBC worked together in the derivation of the mathematical results. The author read and approved the final manuscript.

References

  1. Taliaferro S: A nonlinear singular boundary value problem. Nonlinear Anal. TMA 1979, 3: 897–904. 10.1016/0362-546X(79)90057-9

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal RP, O’Regan D: Existence theory for single and multiple solutions to singular positone boundary value problems. J. Differ. Equ. 2001, 175: 393–414. 10.1006/jdeq.2001.3975

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheng ZB, Ren JL: Periodic and subharmonic solutions for Duffing equation with a singularity. Discrete Contin. Dyn. Syst. 2012, 32: 1557–1574.

    Article  MathSciNet  MATH  Google Scholar 

  4. Chu JF, Torres P, Zhang MR: Periodic solution of second order non-autonomous singular dynamical systems. J. Differ. Equ. 2007, 239: 196–212. 10.1016/j.jde.2007.05.007

    Article  MathSciNet  MATH  Google Scholar 

  5. Fonda A, Manásevich R: Subharmonics solutions for some second order differential equations with singularities. SIAM J. Math. Anal. 1993, 24: 1294–1311. 10.1137/0524074

    Article  MathSciNet  MATH  Google Scholar 

  6. Martins RF: Existence of periodic solutions for second-order differential equations with singularities and the strong force condition. J. Math. Anal. Appl. 2006, 317: 1–13. 10.1016/j.jmaa.2004.07.016

    Article  MathSciNet  MATH  Google Scholar 

  7. Li X, Zhang ZH: Periodic solutions for damped differential equations with a weak repulsive singularity. Nonlinear Anal. TMA 2009, 70: 2395–2399. 10.1016/j.na.2008.03.023

    Article  MathSciNet  MATH  Google Scholar 

  8. del Poin M, Manásevich R: Infinitely many T -periodic solutions for a problem arising in nonlinear elasticity. J. Differ. Equ. 1993, 103: 260–277. 10.1006/jdeq.1993.1050

    Article  MathSciNet  MATH  Google Scholar 

  9. del Poin M, Manásevich R, Murua A: On the number of 2 π -periodic solutions for u +g(u)=s(1+h(t)) using the Poincaré-Birkhoff theorem. J. Differ. Equ. 1992, 95: 240–258. 10.1016/0022-0396(92)90031-H

    Article  MATH  Google Scholar 

  10. Torres P: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 2007, 232: 277–284. 10.1016/j.jde.2006.08.006

    Article  MathSciNet  MATH  Google Scholar 

  11. Sun YP: Positive solution of singular third-order three-point boundary value problem. J. Math. Anal. Appl. 2005, 306: 586–603.

    Article  Google Scholar 

  12. Chu JF, Zhou ZC: Positive solutions for singular non-linear third-order periodic boundary value problems. Nonlinear Anal. TMA 2006, 64: 1528–1542. 10.1016/j.na.2005.07.005

    Article  MathSciNet  MATH  Google Scholar 

  13. Sun JX, Liu YS: Multiple positive solutions of singular third-order periodic boundary value problem. Acta Math. Sci. 2005, 25: 81–88.

    MathSciNet  MATH  Google Scholar 

  14. Palamides AP, Smyrlis G: Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Green’s function. Nonlinear Anal. TMA 2008, 68: 2014–2118.

    MathSciNet  MATH  Google Scholar 

  15. Li YX: Positive periodic solutions for fully third-order ordinary differential equations. Comput. Math. Appl. 2010, 59: 3464–3471. 10.1016/j.camwa.2010.03.035

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu ZQ, Ume JS, Kang SM: Positive solutions of a singular nonlinear third order two-point boundary value problem. J. Math. Anal. Appl. 2007, 326: 589–601. 10.1016/j.jmaa.2006.03.030

    Article  MathSciNet  MATH  Google Scholar 

  17. Ren JL, Cheng ZB, Chen YL: Existence results of periodic for third-order nonlinear singular differential equation. Math. Nachr. 2013, 286: 1022–1042. 10.1002/mana.200910173

    Article  MathSciNet  MATH  Google Scholar 

  18. Mawhin J: Topological degree and boundary value problems for nonlinear differental equations. Lecture Notes in Math. 1537. In Topological Methods for Ordinary Differential Equations. Springer, Berlin; 1993:74–142.

    Chapter  Google Scholar 

  19. Zhang MR: Nonuniform nonresonance at the first eigenvalue of the p -Laplacian. Nonlinear Anal. TMA 1997, 29: 41–51. 10.1016/S0362-546X(96)00037-5

    Article  MathSciNet  MATH  Google Scholar 

  20. Ren JL, Cheng ZB: On high-order delay differential equation. Comput. Math. Appl. 2009, 57: 324–331. 10.1016/j.camwa.2008.10.071

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (No. 11326124, 11271339).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhibo Cheng.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, Z. Existence of positive periodic solutions for third-order differential equation with strong singularity. Adv Differ Equ 2014, 162 (2014). https://doi.org/10.1186/1687-1847-2014-162

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1847-2014-162

Keywords