1 Introduction

The study of singular differential equations began with the paper of Lazer and Solimini. The authors [17] in 1987 investigated the model equation with singularity of repulsive type

$$\begin{aligned} {} u''=\frac{\vartheta }{u^\zeta }+d(t), \end{aligned}$$
(1.1)

where \(\zeta \) and \(\vartheta \) are two positive constants, \(d\in C(\mathbb (R),\mathbb (R))\) is a T-periodic function. They proved the existence of a periodic solution of Eq. (1.1) if and only if \({\overline{d}}:=\frac{1}{T}\int ^T_0d(t)\hbox {d}s>0\). We call this equation with strong singularity if \(\zeta \ge 1\) and we call it with weak singularity if \(0<\zeta <1\).

Lazer and Solimini’s work has attracted the attention of many scholars in singular equations. More recently, method of lower and upper solutions [11], Poincar\(\acute{\text{ e }}\)–Birkhoff twist theorem [1, 10], Leray–Schauder alternative principle [2, 5, 8, 15, 18], Schauder’s fixed point theorem [3, 6, 19, 20, 22], coincidence degree theory [16, 28,29,30], Leray–Schauder degree theory [12, 13] and Krasnoselskii’s fixed point theorem [25, 26] have been employed to investigate the existence of positive periodic solutions of singular equations.

Among these results, there have been published some nice results on the following second-order singular equation (see [6, 11, 15, 20, 22, 25, 26])

$$\begin{aligned} {} u''+q(t)u=f(t,u)+e(t), \end{aligned}$$
(1.2)

where the external force \(e\in L^1({\mathbb {R}})\) is a T-periodic function, the nonlinear term \(f\in C({\mathbb {R}}\times {\mathbb {R}}^+,{\mathbb {R}})\) is a T-periodic function about t and has a singularity of repulsive type at the origin, i.e.,

$$\begin{aligned} \lim _{u\rightarrow 0^+} f(t,u)=+\infty , \quad \text{ uniformly } \text{ in } ~ t. \end{aligned}$$

Using Leray–Schauder alternative principle, Jiang et al. [15] in 2005 investigated the existence of a positive periodic solution for Eq. (1.2) with the external force \(e(t)\equiv 0\). Torres [22] in 2007 discussed the existence of a positive periodic solution for Eq. (1.2) with weak singularity by applications of Schauder’s fixed point theorem. Afterward, Chu and Torres [6] improved the above results and gave a new condition which is weaker than the singular condition in [22]. Applying Krasnoselskii’s fixed point theorem, Wang [25] in 2010 studied the existence and multiplicity of positive periodic solutions for Eq. (1.2) with super-linearity or sub-linearity condition. Recently, in [2, 3, 18], the authors discussed the existence of a positive periodic solution for a damped differential equation with singularity of repulsive type

$$\begin{aligned} {} u''+p(t)u'+q(t)u=f(t,u)+e(t). \end{aligned}$$
(1.3)

We are mainly motivated by the recent work [2, 3, 6, 15, 18, 22, 25] and focus on Eq. (1.3), where \(p,~q\in C({\mathbb {R}},{\mathbb {R}}^+)\) are T-periodic functions, \(e\in C({\mathbb {R}},{\mathbb {R}})\) is a T-periodic function, the nonlinear term \(f\in C({\mathbb {R}}\times {\mathbb {R}}^+,{\mathbb {R}})\) is a T-periodic function about t and has a singularity of repulsive type at the origin and may satisfy sub-linearity, semi-linearity and super-linearity conditions at infinity. The aim of this paper is to show that fixed point theorem in cones can be applied to the singular equations. Using the positivity of Green’s function and the external force e(t), we obtain the existence of a positive periodic solution to Eq. (1.3) with weak and strong singularities.

Besides, as applications, we consider nonlinear Elasticity modeling radial oscillations of an elastic spherical membrane made up of a Neo-Hookean material, and subjected to an internal pressure. The oscillations are governed by the scalar equation (see [23], P.101)

$$\begin{aligned} u''+\frac{4C_1}{\rho r_1^2}\left( u-\frac{1}{u^5}\right) =\frac{3}{\rho r_1^2}h(t)u^2, \end{aligned}$$
(1.4)

where \(\rho \) is the density of the incompressible material, \(r_1\) is an inner radii, \(C_1\) is a positive constant, \(h\in C({\mathbb {R}},{\mathbb {R}})\) is a T-periodic function, h models the difference of pressures between the inner and the outer side of the membrane, and so it may be positive, negative. Del Pino and Manásevich [21] in 1993 proved that Eq. (1.4) possessed positive periodic solutions if \(h(t)<0\). Motivated by [21], the purpose of present paper is to study the existence of a positive periodic solution for Eq. (1.4) if \(h(t)\ge 0\). Our proof is based on the existence result (Corollary 2.3). Afterward, we study the existence of a positive periodic solution for Ermakov–Pinney equation ([23], P. 30)

$$\begin{aligned} {} u''+{\tilde{a}}(t)u=\frac{\mu ^2}{u^3}, \end{aligned}$$
(1.5)

where \(\mu \) is the angular momentum of u, which is a constant of motion, \({\tilde{a}}\in C({\mathbb {R}},\mathbb {R^+})\) is a T-periodic function.

Remark 1.1

In [2, 15, 18], using Leray–Schauder alternative principle, the authors obtained that singular equations have at least one positive periodic solution, and they only gave upper bounds of existence interval of periodic solutions of these equations. However, in this paper, we obtain the lower and upper bounds of existence interval of periodic solution of Eq. (1.3). Our new results generalize some recent results contained in [2, 15, 18].

Remark 1.2

It is worth mentioning that in [3, 6, 22], applying Schauder’s fixed theorem, the authors obtained the existence of positive periodic solutions for Eqs. (1.2) and (1.3) in the case of sub-linearity condition. Wang [25] proved the existence of a positive periodic solution for Eq. (1.2) in the case of sub-linearity or super-linearity condition. In this paper, we establish the existence of a positive periodic solution for equation (1.3) in the cases of sub-linearity, semi-linearity and super-linearity conditions. Therefore, our result can be more general.

2 Preliminaries

Firstly, we consider the following nonhomogeneous linear differential equation,

$$\begin{aligned} {} {\left\{ \begin{array}{ll} u''(t)+p(t)u'(t)+q(t)u(t)=b(t),\\ u(0)=u(T),~u'(0)=u'(T), \end{array}\right. } \end{aligned}$$
(2.1)

where \(b\in C({\mathbb {R}},{\mathbb {R}}^+) \) is a T-periodic function. Equation (2.1) has only one T-periodic solution which can be written as

$$\begin{aligned} u(t)=\int ^T_0 G(t,s)b(s)\hbox {d}s. \end{aligned}$$

where G(t, s) is the Green’s function of Eq. (2.1). Throughout this paper, we assume that

(A) The Green’s function G(t, s) of Eq. (2.1) is positive for all \((t,s)\in [0,T]\times [0,T]\).

For the general case, it is difficult to verify that condition (A) holds. In 2005, Wang, Lian and Ge [27] investigated the positivity of the Green’s function G(t, s), they proved the Green’s function \(G(t,s)> 0\) for all \((t,s)\in [0,T]\times [0,T]\) if the following conditions hold:

\( (A_1)\):

There are continuous T-periodic functions \(a_i(t)\) with \(\int ^T_0 a_i(s)\hbox {d}s>0\), \(i=1,2\) such that

$$\begin{aligned} a_1(t)+a_2(t)=p(t),~~~~a_1(t)a_2(t)+a_2'(t)=q(t),~~~~\text{ for }~~t\in {\mathbb {R}}. \end{aligned}$$
\((A_2)\):

\( 2T\exp \left( \frac{1}{T}\int ^T_0 \ln q(s)\hbox {d}s\right) ^{\frac{1}{2}}\le \int ^T_0p(s)\hbox {d}s.\) Obviously, the condition \((A_2)\) is hard restrictive for the positivity of Green’s function G(t, s). In the following, we study the positivity of Green’s function G(t, s) for all \((t,s)\in [0,T]\times [0,T]\) if the condition \((A_1)\) is only satisfied. From condition \((A_1)\), Eq. (2.1) is transformed into

$$\begin{aligned} {} {\left\{ \begin{array}{ll} v'(t)+a_1(t)v(t)=b(t),\\ v(0)=v(T), \end{array}\right. } \end{aligned}$$
(2.2)

and

$$\begin{aligned} {} {\left\{ \begin{array}{ll} u'(t)+a_2(t)u(t)=v(t),\\ u(0)=u(T). \end{array}\right. } \end{aligned}$$
(2.3)

The solution of Eqs. (2.2) and (2.3) is written as

$$\begin{aligned} v(t)=\int ^{T}_{0}G_2(t,s)b(s)\hbox {d}s \quad \text{ and }\quad u(t)=\int ^{T}_{0}G_1(t,s)v(s)\hbox {d}s , \end{aligned}$$

where

$$\begin{aligned} G_i(t,s)={\left\{ \begin{array}{ll} \frac{e^{-\int ^t_s a_i(\tau )\text {d}\tau }}{1-e^{-\int ^T_0a_i(\tau )\text {d}\tau }}, \quad 0\le s\le t\le T,\\ \frac{e^{-\int ^{T+t}_s a_i(\tau )\text {d}\tau }}{1-e^{-\int ^T_0a_i(\tau )\text {d}\tau }}, \quad 0\le t<s \le T, \end{array}\right. }\quad i=1,2. \end{aligned}$$

the Green’s function \(G_i(t,s)\) can be found in [3]. Obviously, the Green’s function \(G_{i}(t,s)>0\) for all \((t,s)\in [0,T]\times [0,T]\) if \(\int ^T_0a_i(t)\hbox {d}t>0,\, i=1,2\). Therefore, we deduce

$$\begin{aligned} u(t)&=\int ^{T}_{0}G_{1}(t,\tau )\int ^{T}_{0}G_2(\tau ,s)b(s)\hbox {d}s \hbox {d}\tau \\&=\int ^{T}_{0}\int ^{T}_{0}G_{1}(t,\tau )G_2(\tau ,s)b(s)\hbox {d}s \hbox {d}\tau \\&=\int ^{T}_{0}\left[ \int ^{T}_{0}G_{1}(t,\tau )G_2(\tau ,s)d\tau \right] b(s)\hbox {d}s\\&=\int ^T_0G(t,s)b(s)\hbox {d}s, \end{aligned}$$

where \( G(t,s)=\int ^{T}_{0}G_1(t,\tau )G_2(\tau ,s)\hbox {d}\tau . \)

Lemma 2.1

(See [3]) Assume that the condition \((A_1)\) holds. Then, the Green’s function \(G(t,s)>0\) for all \((t,s)\in [0,T]\times [0,T]\), i.e., condition (A) holds.

From condition \((A_1)\), it is easy to see that \(\int ^T_0p(t)\hbox {d}t>0\). In the following, we consider that the damped term p(t) does not require \(\int ^T_0p(t)\hbox {d}t>0\), the condition (A) holds. Define the functions

$$\begin{aligned} \sigma (p)(t)=\exp \left( \int ^t_0p(s)\hbox {d}s\right) , \end{aligned}$$

and

$$\begin{aligned} \sigma _1(p)(t)=\sigma (p)(T)\int ^t_0\sigma (p)(s)\hbox {d}s+\int ^T_t\sigma (p)(s)\hbox {d}s. \end{aligned}$$

Lemma 2.2

(See [7]) Assume that \(q(t)\ge 0\) and \(q\not \equiv 0\) and the following two inequalities are satisfied

$$\begin{aligned} {} \int ^T_0q(s)\sigma (p)(s)\sigma _1(-p)(s)\mathrm{d}s\ge 0, \end{aligned}$$
(2.4)

and

$$\begin{aligned} {} \sup _{0\le t\le T}\left\{ \int ^{t+T}_t\sigma (-p)(s)\mathrm{d}s,\int ^{t+T}_tq(s)\sigma (p)(s)\mathrm{d}s\right\} \le 4. \end{aligned}$$
(2.5)

Then, condition (A) holds.

Next, we consider two special cases.

Remark 2.1

In the case \(p(t)\equiv 0\), define

$$\begin{aligned} M(\iota )={\left\{ \begin{array}{ll} \frac{2\pi }{\iota T^{1+2/\iota }}\left( \frac{2}{2+\iota }\right) ^{1-2/\iota }\left( \frac{\Gamma \left( \frac{1}{\iota }\right) }{\Gamma \left( \frac{1}{2}+\frac{1}{\iota }\right) }\right) ^2,&{}\quad \text{ if }~1\le \iota <\infty ,\\ \frac{4}{T},&{}\quad \text{ if }~\iota =\infty , \end{array}\right. } \end{aligned}$$

where \(\Gamma \) is the Gamma function, i.e., \(\Gamma (t)=\int ^{+\infty }_0u^{t-1}e^{-u}\hbox {d}u\). Assume that \(q(t)\succ 0\) (i.e., \(q\ge 0\) for almost every \(t\in [0,T]\)) and \(q\in L^\xi (0,\omega )\) for some \(1\le \xi \le \infty \). If

$$\begin{aligned} {} \Vert q\Vert _\xi :=\left( \int ^T_0|q(t)|^\xi \hbox {d}t\right) ^{\frac{1}{\xi }}\le M(2\xi ^*), \end{aligned}$$
(2.6)

where \(\xi ^*=\frac{\xi }{\xi -1}\) if \(1\le \xi <\infty \) and \(\xi ^*=1\) if \(\xi =+\infty \), then the Green’s function \(G(t,s)> 0\) for all \((t,s)\in [0,T]\times [0,T].\) See [24].

Remark 2.2

In the case \(p(t)\equiv 0\) and \(q(t)=\delta ^2\) with \(\delta >0\), the Green’s function has the form

$$\begin{aligned} {} G_3(t,s)=\left\{ \begin{aligned}&\frac{\cos \delta \left( t-s-\frac{T}{2}\right) }{2\delta \sin \frac{\delta T}{2}},\quad 0\le s\le t\le T,\\&\frac{\cos \delta \left( t-s+\frac{T}{2}\right) }{2\delta \sin \frac{\delta T}{2}},\quad 0\le t<s\le T. \end{aligned} \right. \end{aligned}$$
(2.7)

If \(\delta <\frac{\pi }{T}\), then the Green’s function \(G_3(t,s)>0\) for all \((t,s)\in [0,T]\times [0,T].\) See [4, 14].

From condition (A), we denote

$$\begin{aligned} A:=\min _{0\le s,t\le T} G(t,s),\quad B:=\max _{0\le s,t\le T}G(t,s),\quad \sigma :=\frac{A}{B}. \end{aligned}$$

It is clear that \(0<A\le B\) and \(0<\sigma \le 1\). Moreover, the following equality holds

$$\begin{aligned} \int ^T_0G(t,s)q(s)\hbox {d}s\equiv 1. \end{aligned}$$

Define function

$$\begin{aligned} \gamma (t):=\int ^T_0G(t,s)e(s)\hbox {d}s. \end{aligned}$$

Denote

$$\begin{aligned} \gamma ^*:=\sup _{t\in [0,T]}\gamma (t)\quad \text{ and }\quad \gamma _*:=\inf _{t\in [0,T]}\gamma (t). \end{aligned}$$

Define

$$\begin{aligned} K:=\{u\in X: u(t)\ge 0~~\text{ for } \text{ all }~~t~\in {\mathbb {R}}~~\text{ and }~~\min _{t\in {\mathbb {R}}}u(t)\ge \sigma \Vert u\Vert \}. \end{aligned}$$

It is easy to verify that K is a cone in \(X:=\{u\in C({\mathbb {R}},{\mathbb {R}}):~u(t+T)-u(t)\equiv 0\}\) with norm \(\Vert u\Vert :=\max \limits _{t\in {\mathbb {R}}}|u(t)|.\) Our proof is based on the following fixed point theorem in cones, which can be found in [9].

Lemma 2.3

(See [9]) Let X be a Banach space and K is a cone in X. Assume that \(\Omega _1,~\Omega _2\) are open subsets of X with \(0\in \Omega _1,~{\bar{\Omega }}_1\subset \Omega _2.\) Let

$$\begin{aligned} \Psi :~K\cap ({\bar{\Omega }}_2\setminus \Omega _1)\rightarrow K \end{aligned}$$

be a continuous and completely continuous operator such that

  1. (i)

    \(\Vert \Psi u\Vert \le \Vert u\Vert \) for \(u\in K\cap \partial \Omega _2\); and

  2. (ii)

    there exists \(u_0\in K\setminus \{0\}\) such that \(u\ne \Psi u+\lambda u_0\) for \(u\in K\cap \partial \Omega _1\) and \(\lambda >0\). Then, \(\Psi \) has a fixed point in \(K\cap ({\bar{\Omega }}_2\setminus \Omega _1).\)

3 Main Results

Theorem 3.1

Assume that the condition (A) holds. Furthermore, suppose the following conditions are satisfied:

\((H_1)\):

\(f(t,u)+e(t)\ge 0\), for all \((t,u)\in [0,T]\times (0,+\infty );\)

\((H_2)\):

there exist continuous nonnegative functions \(\varphi ,~\kappa \) and positive function g such that

$$\begin{aligned} 0\le f(t,u)\le \kappa (t)(g(u)+\varphi (u))~~~~~~\text{ for } \text{ all } ~~(t,u)\in [0,T]\times (0,+\infty ), \end{aligned}$$

and g(u) is non-increasing, \(\varphi (u)/g(u)\) is non-decreasing in u;

\((H_3)\):

there exists a positive number R such that

$$\begin{aligned} g(\sigma R)\left( 1+\frac{\varphi (R)}{g(R)}\right) \Gamma ^*+\gamma ^*\le R, \end{aligned}$$

where \(\Gamma (t)=\int ^T_0G(t,s)\kappa (s)\hbox {d}s.\)

\((H_4)\):

there exists a positive number r with \(r<R\) such that

$$\begin{aligned} {} \gamma _*\ge r. \end{aligned}$$
(3.1)

Then, Eq. (1.3) has at least one positive T-periodic solution u with \(u\in [r,R].\)

Proof

It is easy to see that a T-periodic solution of Eq. (1.3) is just a fixed point of the operator equation \(u=\Psi u\), where the operator \(\Psi \) is defined as

$$\begin{aligned} (\Psi u)(t)=\int ^T_0G(t,s)(f(s,u(s))+e(s))\hbox {d}s. \end{aligned}$$

Now we define two open sets

$$\begin{aligned} \Omega _1:=\{u\in X:~\Vert u\Vert<r\},~~~\Omega _2:=\{u\in X:~\Vert u\Vert <R\}. \end{aligned}$$

We claim that \(\Psi (K\cap ({\bar{\Omega }}_2\setminus \Omega _1))\subset K\). For \(\forall ~u\in K\cap ({\bar{\Omega }}_2\setminus \Omega _1)\), from condition \((H_1)\), we deduce

$$\begin{aligned} \begin{aligned} \min _{t\in {\mathbb {R}}}(\Psi u)(t)=&\min _{t\in {\mathbb {R}}} \int ^T_0G(t,s)(f(s,u(s))+e(s))\hbox {d}s\\ \ge&A\int ^T_0(f(s,u(s))+e(s))\hbox {d}s\\ \ge&\sigma B\int ^T_0(f(s,u(s))+e(s))\hbox {d}s\\ \ge&\sigma \max _{t\in {\mathbb {R}}}\int ^T_0G(t,s)(f(s,u(s))+e(s))\hbox {d}s\\ \ge&\sigma \Vert \Psi u\Vert . \end{aligned} \end{aligned}$$

This implies that \(\Psi (K\cap ({\bar{\Omega }}_2\setminus \Omega _1))\subset K.\) Let W be an any bounded subset in \(K\cap ({\bar{\Omega }}_2\backslash \Omega _1)\), then for \(\forall ~u\in W\), we have

$$\begin{aligned} \begin{aligned} \Vert \Psi u\Vert =&\max \limits _{t\in {\mathbb {R}}}\left| \int ^T_0G(t,s)(f(s,u(s))+e(s))\hbox {d}s\right| \\ \le&B\int ^T_0\max \limits _{t\in {\mathbb {R}}}|f(s,u(s))+e(s)|\hbox {d}s\\ \le&B\int ^T_0\max \limits _{t\in {\mathbb {R}}}(|f(t,u(s))|+|e(s)|)\hbox {d}s\\ \le&BT(\Vert f_R\Vert +\Vert e\Vert ):=N_1, \end{aligned} \end{aligned}$$
(3.2)

where \(\Vert f_R\Vert :=\max \limits _{r<u\le R}|f(t,u)|\) from \(u\in K\cap ({\bar{\Omega }}_2\setminus \Omega _1)\).

On the another hand, we arrive at

$$\begin{aligned} \begin{aligned} \left| \frac{d\Psi u}{dt}\right| =&\left| \int ^T_0\frac{\partial G(t,s)}{\partial t}(f(s,u(s))+e(s))\hbox {d}s\right| \\ \le&\int ^T_0\left| \frac{\partial G(t,s)}{\partial t}\right| (|f(s,u(s))|+|e(s)|)\hbox {d}s\\ \le&N_2' T(\Vert f_R\Vert +\Vert e\Vert ):=N_2, \end{aligned} \end{aligned}$$
(3.3)

where \(N_2':=\max \limits _{t\in {\mathbb {R}}}\left| \frac{\partial G(t,s)}{\partial t}\right| \).

Therefore, using the Arzela–Ascoli theorem, it is easy to see that \(\Psi : K\cap ({\bar{\Omega }}_2\backslash \Omega _1)\rightarrow K\) is a continuous and completely continuous operator.

Let \(u_0\equiv 1,\) then \(u_0\in K\). Now we prove that

$$\begin{aligned} {} u\ne \Psi u+\lambda u_0,~~\forall ~u\in K\cap \partial \Omega _1,~\text{ and }~\lambda >0. \end{aligned}$$
(3.4)

If not, there would exists \(u_1\in K\cap \partial \Omega _1\) (i.e., \(\Vert u_1\Vert =r\)), and \(\lambda _0>0\) such that

$$\begin{aligned} u_1=\Psi u_1+\lambda _0u_0. \end{aligned}$$

From conditions (A), \((H_2)\) and \((H_4)\), we know that the nonlinear term f and the Green’s function G are nonnegative functions. Therefore, we obtain

$$\begin{aligned} \begin{aligned} u_1(t)&=(\Psi u_1)(t)+\lambda _0 u_0\\&=\int ^T_0G(t,s)(f(s,u_1(s))+e(s))\hbox {d}s+\lambda _0\\&\ge \gamma (t)+\lambda _0\\&\ge \gamma _*+\lambda _0\\&> r. \end{aligned} \end{aligned}$$

This is a contradictory to \(u_1\in K\cap \partial \Omega _1\). Then, (3.4) holds.

On the other hand, we claim

$$\begin{aligned} {} \Vert \Psi u\Vert \le \Vert u\Vert ,~~~\forall ~u\in K\cap \partial \Omega _2. \end{aligned}$$
(3.5)

In fact, for any \(u\in K\cap \partial {\Omega _2}\), we can get \(\sigma R\le u(t)\le R\). Hence, from conditions \((H_2)\) and \((H_3)\), it is clear that

$$\begin{aligned} \begin{aligned} (\Psi u)(t)=&\int ^T_0G(t,s)(f(s,u(s))+e(s))\hbox {d}s\\ \le&\int ^T_0G(t,s)\kappa (s)g(u(s))\left( 1+\frac{\varphi (u(s))}{g(u(s))}\right) \hbox {d}s+\gamma (t)\\ \le&g(\sigma R)\left( 1+\frac{\varphi (R)}{g(R)}\right) \Gamma ^*+\gamma ^*\\ \le&R=\Vert u\Vert . \end{aligned} \end{aligned}$$

Therefore, \(\Vert \Psi u\Vert \le \Vert u\Vert \), i.e., (3.5) holds.

By Lemma 2.3, we obtain that \(\Psi \) has a fixed point \(u\in K\cap ({\bar{\Omega }}_2\setminus \Omega _1)\). Obviously, this fixed point is a positive T-periodic solution of Eq. (1.3) and satisfies \(u\in [r,R]\). \(\square \)

Corollary 3.1

Assume that the condition (A) holds. Furthermore, suppose the following condition is satisfied:

\((F_1)\):

there exist continuous nonnegative function l and constants \(\varepsilon >0,~~0\le \eta <1\) such that

$$\begin{aligned} 0\le f(t,u)\le \frac{l(t)}{u^\varepsilon }+l(t)u^\eta ,~~~~\text{ for } \text{ all }~~ (t,u)\in [0,T]\times (0,+\infty ). \end{aligned}$$

If the external force \(e(t)>0\), then Eq. (1.3) has at least one positive T-periodic solution.

Proof

Apply Theorem 3.1, take

$$\begin{aligned} \kappa (t)=l(t),~~~~g(u)=\frac{1}{u^\varepsilon },~~~~\varphi (u)=u^\eta , \end{aligned}$$

then conditions \((H_1)\) and \((H_2)\) are satisfied and the existence condition \((H_3)\) becomes

$$\begin{aligned} {} L^*<\frac{(R-\gamma ^*)(\sigma R)^\varepsilon }{1+R^{\eta +\varepsilon }}, \end{aligned}$$
(3.6)

where \(L(t)=\int ^\omega _0G(t,s)l(s)\hbox {d}s, \) for some \(R>0\). Since \(\varepsilon >0,~~0\le \eta <1\) and \(e(t)>0\), we can choose \(R>0\) large appropriate such that (3.6) is satisfied.

On the other hand, we consider condition \((H_4)\). In fact, we may choose \(r>0\) small appropriate such that (3.1) holds. \(\square \)

From condition \((F_1)\), it is easy to see that the nonlinear term f only satisfies sub-linearity condition. Next, we consider that the nonlinear term f satisfies semi-linearity and super-linearity conditions.

Corollary 3.2

Assume that the condition (A) holds. Furthermore, suppose the following condition is satisfied:

\((F_2)\):

there exist continuous nonnegative functions l,  c and constants \(\varepsilon>0,~~\eta \ge 1,~~\mu >0\) such that

$$\begin{aligned} f(t,u)=\frac{l(t)}{u^\varepsilon }+\mu c(t)u^\eta ,~~~~\text{ for } \text{ all }~~ (t,u)\in [0,T]\times (0,+\infty ). \end{aligned}$$

If the external force \(e(t)>0\), then Eq. (1.3) has at least one positive T-periodic solution for each \(0<\mu <\mu ^*\), where \(\mu ^*\) is some positive constant.

Proof

We follow the same strategy and notation as in the proof of Corollary 3.1. Take

$$\begin{aligned} \kappa (t)=1,~~~~g(u)=\frac{l^*}{u^\varepsilon },~~~~\varphi (u)=\mu c^*u^\eta , \end{aligned}$$

then conditions \((H_1)\) and \((H_2)\) are satisfied and the existence condition \((H_3)\) becomes

$$\begin{aligned} \mu <\frac{(R-\gamma ^*)\sigma ^\varepsilon R^\varepsilon -l^*}{c^*\sigma ^\varepsilon R^{\eta +\varepsilon }}. \end{aligned}$$

Take \(\mu ^*:=\sup \limits _{R>0}\frac{(R-\gamma ^*)\sigma ^\varepsilon R^\varepsilon -l^*}{c^*\sigma ^\varepsilon R^{\eta +\varepsilon }}.\) Since \(0<\mu <\mu ^*\), then condition \((H_3)\) holds. The remaining part of the proof is the same as that of Corollary 3.1. \(\square \)

Remark 3.1

Theorem 3.1 applies to the example

$$\begin{aligned} u''+p(t)u'+q(t)u=\frac{d(t)}{u^\varepsilon }+\mu c(t)e^u+e^{\sin t}. \end{aligned}$$

A result similar to Corollary 3.2 holds for this example.

Remark 3.2

It is easy to verify that condition \((H_4)\) and \(\gamma _*=0\) are contradictory. Therefore, the condition \((H_4)\) is no longer applicable to prove the existence of a positive T-periodic solution for Eq. (1.3) in the case that \(\gamma _*=0\). Next, we need to find other conditions to get over this problem.

Theorem 3.2

Assume that conditions (A), \((H_1){-}(H_3)\) hold. Furthermore, suppose the following conditions are satisfied:

\((H_5)\):

There exists a continuous nonnegative function \(\phi _R\) such that \(f(t,u)\ge \phi _R(t)\) for all \((t,u)\in [0,T]\times (0,R]\);

\((H_6)\):

there exists a positive number r with \(r<R\) such that

$$\begin{aligned} \Phi _*+\gamma _*\ge r, \end{aligned}$$

where \(\Phi (t)=\int ^T_0G(t,s)\phi \) Then, Eq. (1.3) has at least one positive T-periodic solution u with \(u\in [r,R]\).

Proof

We will follow the same strategy and notations as the proof of Theorem 3.1. Now, we prove (3.4) holds. From conditions \((H_2)\), \((H_5)\) and \((H_6)\), we see that

$$\begin{aligned} \begin{aligned} u_1(t)&=(\Psi u_1)(t)+\lambda _0 u_0\\&=\int ^T_0G(t,s)(f(s,u_1(s))+e(s))\hbox {d}s+\lambda _0\\&\ge \int ^T_0G(t,s)\phi _r(s)\hbox {d}s+\gamma (t)+\lambda _0\\&\ge \Phi _*+\gamma _*+\lambda _0\\&> r. \end{aligned} \end{aligned}$$

This is a contradictory to \(u_1\in K\cap \partial \Omega _1\). Then, (3.4) holds.

The remaining part of the proof is the same as that of Theorem 3.1.

\(\square \)

Applying Theorem 3.2, we get the following conclusion.

Corollary 3.3

Assume that conditions (A), \((H_1){-}(H_3)\) and \((H_5)\) hold. Furthermore, suppose the following condition is satisfied:

\((H_7)\) there exists a positive number r with \(r<R\) such that

$$\begin{aligned} \Phi _*\ge r. \end{aligned}$$

If \(\gamma _*=0\), then Eq. (1.3) has at least one positive T-periodic solution u with \(u\in [r,R]\).

Corollary 3.4

Assume that condition (A) holds. Furthermore, suppose the following condition holds,

\((F_3)\):

there exist continuous nonnegative functions \(l,~~{\hat{l}}\) and constants \(\varepsilon >0,~~0\le \eta <1\) such that

$$\begin{aligned} 0\le \frac{{\hat{l}}(t)}{u^\varepsilon }\le f(t,u)\le \frac{l(t)}{u^\varepsilon }+l(t)u^\eta ,~~~~\text{ for } \text{ all }~~ (t,u)\in [0,T]\times (0,+\infty ). \end{aligned}$$

If the external force \(e(t)=0\), then Eq. (1.3) has at least one positive T-periodic solution.

Proof

Applying Corollary 3.3, take

$$\begin{aligned} \phi _R(t)=\frac{{\hat{l}}(t)}{R^\varepsilon },~~~~\kappa (t)=l(t),~~~~g(u)=\frac{1}{u^\varepsilon },~~~~\varphi (u)=u^\eta , \end{aligned}$$

then conditions \((H_1)\), \((H_2)\) and \((H_5)\) hold and the existence condition \((H_3)\) becomes

$$\begin{aligned} {} \frac{\sigma ^\varepsilon R^{\varepsilon +1}}{1+R^{\varepsilon +\eta }}>L^*. \end{aligned}$$
(3.7)

Since \(\varepsilon >0,~0\le \eta <1\) and \(e(t)=0\), we can choose \(R>0\) large appropriate such that (3.7) is satisfied.

On the other hand, we consider that condition \((H_7)\) is satisfied. In fact, we discuss the following inequality holds

$$\begin{aligned} {} r<{\hat{L}}^{\frac{1}{\varepsilon +1}}_*, \end{aligned}$$
(3.8)

where \({\hat{L}}(t)=\int ^T_0G(t,s){\hat{l}}(s)\hbox {d}s.\) We can choose \(r>0\) small appropriate such that (3.8) holds. \(\square \)

Corollary 3.5

Assume that conditions (A) and \((F_2)\) hold. If the external force \(e(t)=0\), then Eq. (1.3) has at least one positive T-periodic solution for each \(0<\mu <\mu ^*\), where \(\mu ^*\) is some positive constant.

Proof

We follow the same strategy and notation as in the proof of Corollary 3.4. Take

$$\begin{aligned} \phi _R(t)=\frac{l_*}{R^\varepsilon },~~~~\kappa (t)=1,~~~~g(u)=\frac{l^*}{u^\varepsilon },~~~~\varphi (u)=\mu c^*u^\eta , \end{aligned}$$

then conditions \((H_1)\), \((H_2)\) and \((H_5)\) hold and the existence condition \((H_3)\) becomes

$$\begin{aligned}{} \mu <\frac{\sigma ^\varepsilon R^{\varepsilon +1}-l^*}{c^*\sigma ^\varepsilon R^{\eta +\varepsilon }}. \end{aligned}$$

Take \(\mu ^*:=\sup \limits _{R>0}\frac{\sigma ^{\varepsilon +1} R^\varepsilon -l^*}{c^*\sigma ^\varepsilon R^{\eta +\varepsilon }}.\) Since \(0<\mu <\mu ^*\), then condition \((H_3)\) holds. The remaining part of the proof is the same as that of Corollary 3.4. \(\square \)

Remark 3.3

Theorem 3.2 applies to the example

$$\begin{aligned} u''+p(t)u'+q(t)u=\frac{l(t)}{u^{{\widetilde{\varepsilon }}}}+\mu c(t)u^{{\widetilde{\eta }}}, \end{aligned}$$

where \(\mu \) is a positive constant and \({\tilde{\xi }}>0,~{\widetilde{\eta }}>1\).

4 Applications

In this section, we consider Eq. (1.4) and Ermakov–Pinney equation (1.5). Applying Corollary 3.3, we obtain the following conclusions.

We first consider Eq. (1.4), which is written as the following form

$$\begin{aligned} {} u''+\frac{4C_1}{\rho r_1^2}u=\frac{4C_1}{\rho r_1^2}\frac{1}{u^5}+\frac{3}{\rho r_1^2}h(t)u^2. \end{aligned}$$
(4.1)

Theorem 4.1

Assume that condition \(\frac{4C_1}{\rho r_1^2}<(\frac{\pi }{T})^2\) holds. Furthermore,

  1. (i)

    If \(\Vert h\Vert =0\), then Eq. (1.4) has at least one positive T-periodic solution.

  2. (ii)

    If \(\Vert h\Vert >0\), then Eq. (1.4) has at least one positive T-periodic solution for each

    $$\begin{aligned} 0<\Vert h\Vert <\sup \limits _{R>0}\frac{4C_1(\sigma ^5R^6-1)}{3R^7}, \end{aligned}$$

    where R is a positive constant.

Proof

By Remark 2.2, we have

$$\begin{aligned} a(t)=\frac{4C_1}{\rho r_1^2}=\delta ^2. \end{aligned}$$

Obviously, if \(\frac{4C_1}{\rho r_1^2}<(\frac{\pi }{T})^2\), the Green function \(G_3(t,s)\) is positive.

Comparing Eq. (1.4) with Eq. (1.3), we see that

$$\begin{aligned} f(t,u)=\frac{4C_1}{\rho r_1^2}\frac{1}{u^5}+\frac{3}{\rho r_1^2}h(t)u^2,~~~e(t)=0. \end{aligned}$$

Take

$$\begin{aligned} \phi _R(t)=\frac{4C_1}{\rho r_1^2}\frac{1}{R^5},\quad \kappa (t)=1,\quad g(u)=\frac{4C_1}{\rho r_1^2}\frac{1}{u^5}, \quad \varphi (u)=\frac{3}{\rho r_1^2}\Vert h\Vert u^2, \end{aligned}$$

then conditions \((H_1)\), \((H_2)\) and \((H_5)\) hold and the existence condition \((H_7)\) becomes

$$\begin{aligned} {} \int ^T_0 G_3(t,s)\frac{4C_1}{\rho r_1^2}\frac{1}{r^5}\hbox {d}s\ge r. \end{aligned}$$
(4.2)

The Green’s function (2.7) can be transformed into

$$\begin{aligned} G_3(t,s)=\frac{\cos \delta \left( t-s+\frac{T}{2}\right) }{2\delta \sin \frac{\delta T}{2}}=\frac{\cos \sqrt{a}\left( t-s+\frac{T}{2}\right) }{2\sqrt{a}\sin \frac{\sqrt{a}T}{2}},~~~~t\le s\le t+T. \end{aligned}$$
(4.3)

From Eq. (4.3), we arrive at

$$\begin{aligned} {} \begin{aligned} \int ^T_0 G_3(t,s)\hbox {d}s=&\int ^{t+T}_t G_3(t,s)\hbox {d}s\\ =&\int ^{t+T}_t\frac{\cos \delta \left( t-s+\frac{T}{2}\right) }{2\delta \sin \frac{\delta T}{2}}\hbox {d}s\\ =&\frac{1}{2\delta \sin \frac{\delta T}{2}}\left( -\frac{1}{\delta }\sin \delta \left( t-s+\frac{T}{2}\right) \right) \big {|}^{t+T}_t\\ =&\frac{1}{\delta ^2}=\frac{\rho r_1^2}{4C_1}. \end{aligned} \end{aligned}$$
(4.4)

Substituting (4.4) into (4.2), we get

$$\begin{aligned} {} r^6\le 1, \end{aligned}$$
(4.5)

we can choose \(r<1\) to such that (4.5) holds. Thus, the condition \((H_7)\) holds.

On the other hand, we consider condition \((H_3)\), which becomes

$$\begin{aligned} \frac{4C_1}{\rho r_1^2}\frac{1}{(\sigma R)^5}\left( 1+\frac{\frac{3}{\rho r_1^2}\Vert h\Vert R^2}{\frac{4C_1}{\rho r_1^2}\frac{1}{R^5}}\right) \int ^T_0 G_3(t,s)\hbox {d}s\le R. \end{aligned}$$
(4.6)

Substituting (4.4) into (4.6), we obtain

$$\begin{aligned} {} \frac{1}{(\sigma R)^5}+\frac{3\Vert h\Vert R^7}{4C_1(\sigma R)^5}\le R. \end{aligned}$$
(4.7)

Then, we consider the following two cases.

Case (i). If \(\Vert h\Vert =0\), from (4.7), we can choose \(R\ge \sigma ^{-\frac{5}{6}}\) such that condition \((H_3)\) holds. Applying Corollary 3.3, we obtain that Eq. (1.4) has at least one positive T-periodic solution.

Case (ii). If \(\Vert h\Vert >0\), from (4.7), we get

$$\begin{aligned} \Vert h\Vert \le \frac{4C_1(\sigma ^5R^6-1)}{3R^7}. \end{aligned}$$

Then, for each \(0<\Vert h\Vert <\sup \limits _{R>0}\frac{4C_1(\sigma ^5R^6-1)}{3R^7}\), it is easy to verify that condition \((H_3)\) holds. The proof is completed.

\(\square \)

Next, we consider existence of a positive periodic solution to Ermakov–Pinney equation (1.5).

Theorem 4.2

Assume that (2.6) holds. Then, there exist two positive constants r,  R with \(r<R\) such that Eq. (1.5) has at least one positive T-periodic solution u with \(u\in [r,R]\).

Proof

Comparing Eq. (1.5) with Eq. (1.3), we can see

$$\begin{aligned} f(t,u)=\frac{\nu ^2}{u^3},~~~e(t)=0. \end{aligned}$$

Take

$$\begin{aligned} \phi _R(t)=\frac{\nu ^2}{R^3},~~~~\kappa (t)=\nu ^2,~~~~g(u)=u^3,~~~~\varphi (u)=0, \end{aligned}$$

then conditions \((H_1)\), \((H_2)\) and \((H_5)\) hold and the existence condition \((H_3)\) becomes

$$\begin{aligned} {} \frac{\nu ^2\int ^T_0G(t,s)\hbox {d}s}{\sigma ^3}\le R^4, \end{aligned}$$
(4.8)

we can choose \(R\ge \left( \frac{\nu ^2\int ^T_0G(t,s)\hbox {d}s}{\sigma ^3}\right) ^{\frac{1}{4}}\) such that (4.8) is satisfied.

On the other hand, we consider that condition \((H_7)\) is satisfied. In fact, we discuss the following inequality holds

$$\begin{aligned} {} \nu ^2\int ^T_0G(t,s)\hbox {d}s\ge r^4. \end{aligned}$$
(4.9)

We can choose \(r\le \left( \nu ^2\int ^T_0G(t,s)\hbox {d}s\right) ^{\frac{1}{4}}\) such that Eq. (4.9) holds. It is clear

$$\begin{aligned} r\le \left( \nu ^2\int ^T_0G(t,s)\hbox {d}s\right) ^{\frac{1}{4}}\le \left( \frac{\nu ^2\int ^T_0G(t,s)\hbox {d}s}{\sigma ^3}\right) ^{\frac{1}{4}}\le R, \end{aligned}$$

since \(\sigma \le 1\). Applying Corollary 3.3, we obtain that Eq. (1.5) has at least one positive T-periodic solution u with \(u\in [r,R]\). \(\square \)