Existence and uniqueness of a periodic solution to an indefinite attractive singular equation

Abstract

We study the existence of periodic solutions to differential equations with a singularity at the state variable, extending the recently found existing relation to differential equations with attractive singularity, where the order of the singularity and the regularity of the input functions are closely connected. The results related to the classic open problems in the field of singular equations. The question about uniqueness is also discussed.

1 Introduction

For a long time, the differential equations allow to study systematically natural processes. Together with the development of the general theory of infinitesimal calculus in the last seventieth century, several physical phenomena were understood. In the beginning, the scientific community paid more attention to physical processes which could be mathematically modelled by regular differential equations, understanding that they do not have a discontinuity in the nonlinear term (harmonic oscillators, physical pendulums, ecological and economical models, etc.). However, important problems arising in celestial mechanics, engineering, cellular systems and many others subjects have motivated the researchers to focus the attention on singular differential equations where the nonlinear term presents some types of discontinuity (see Refs. [1, 6, 12, 17, 18, 21, 22]).

Although the historical significance of the singular differential equations is evident, as a discipline the analytical theory is relatively new. To the best to our knowledge, [5, 13] are the first papers where the periodic problem to these types of equations were treated. However, [14] was the key to impulse the development of the recent theory of the periodic problems to singular differential equations.

In that paper, the authors studied the existence of a periodic solution to the following family of singular differential equations

$$\begin{aligned} u''+\frac{\nu }{u^{\lambda }}=h(t), \end{aligned}$$

(1)

where $h\in L_{\omega }$, i.e., $h$ is an $\omega$-periodic function which is Lebesgue integrable on $[0,\omega ]$, $\nu \in \mathbb {R}{\setminus }\{0\}$, and $\lambda >0$. According to the related literature, the singularity in Eq. (1) represents a repulsive singularity if $\nu <0$, and an attractive singularity in the case when $\nu >0$.

The study of the singular differential equations has been focused on the previously mentioned types of singularities (attractive and repulsive), at least in the field of research dealing with the existence of periodic solutions. At the original paper of Lazer and Solimini, the authors present two main results, one of them related to the attractive case and the other one to the repulsive case. In the first case, they have proven that (1) admits an $\omega$-periodic solution if and only if $\overline{h}\mathop {=}\limits ^{def}(1/\omega ) \int _0^{\omega }h>0$, provided that $h$ is a continuous function (condition on the regularity of the external force). However, the similar result is not valid for the repulsive case, unless an additional condition on the order of the singularity of (1), $\lambda$, is taken into account. Precisely, they proved that (1) has at least one $\omega$-periodic solution if and only if $\overline{h}<0$, provided that $\lambda \ge 1$ (in this case, it is not necessary to assume that $h$ is continuous, the relation $h\in L_{\omega }$ is sufficient). However, the condition $\lambda \ge 1$ is not necessary for the existence of an $\omega$-periodic solution to (1) with $\nu <0$, it can be replaced by different kind of conditions (see, e.g., Refs. [8, 9, 19]), and nowadays there is a quite wide theory on the differential equations with repulsive singularities.

The attractive case did not drive such an attention. In spite of that, we can cite some works on the existence of an $\omega$-periodic solution to differential equations with attractive singularities (see Refs. [2, 7, 9–11, 14–16, 20]). In most of them, the equations were studied assuming certain regularity of the function $h$ (for example $h\in L_{\omega }^{\infty }$). On the other hand, in Refs. [8, 9, 19] we can find results on the existence of an $\omega$-periodic solution to the differential equations with attractive singularity whenever $h\in L_{\omega }$ provided that the oscillation of $h$ is sufficiently small. Recently, in Ref. [11], we have proven that the boundedness of the function $h$ in the result of Lazer and Solimini is essential and it cannot be extended to the general case $h\in L_{\omega }^p$ for all $\lambda >0$. More precisely, we have established a relation between the regularity of the external force $h$ and the order of the singularity $\lambda$. The main results of [11] can be formulated in the following way:

• If $h\in L_{\omega }^p$, $\lambda \ge 1/(2p-1)$, then (1) with $\nu >0$ has an $\omega$-periodic solution if and only if $\overline{h}>0$. Moreover, this solution is unique.

• If $\lambda \in (0,1/(2p-1))$ then there exists $h\in L_{\omega }^p$ with $\overline{h}>0$ such that (1) (with $\nu >0$) has no $\omega$-periodic solution.

Thus, the results obtained show that it is not possible to reduce the order of the singularity in (1) without increasing the regularity of $h$.

Another type of equations with attractive singularity was considered in [10], in particular

$$\begin{aligned} u''+\frac{1}{u^{\lambda }}-\frac{1}{u^{\mu }}=h(t)u^{\delta }, \end{aligned}$$

(2)

where $\lambda >\mu \ge 0$ and $\delta \in [0,1)$. Equation (2) is called the Rayleigh–Plesset-type equation, and the results established in [10] show that the order of the singularity $\lambda$ can be decreased by increasing the order of the nonlinearity $\delta$, without increasing the regularity of $h$. More precisely, the following result was established:

• If $h\in L_{\omega }$, $\lambda +2\delta \ge 1$, then (2) has at least one $\omega$-periodic solution if and only if $\overline{h}>0$.

However, the mentioned paper does not deal with the uniqueness of a solution nor the relation between the order of the singularity and the regularity of the external force, as it was done in the Lazer–Solimini-type equation with attractive singularity.

Recently, a lot of attention has been paid to study singular differential equations with indefinite singularity of the type

$$\begin{aligned} u''+\frac{g(t)}{u^{\lambda }}=h(t), \end{aligned}$$

(3)

where $g,h\in L_{\omega }$ and the function $g$ may have zeroes, even it may change sign (see, e.g., [3, 8, 23]). Observe that in such a case, the equation may possess both attractive and repulsive singularities depending on the variable $t$, even the singularity can disappear if $g$ is equal to zero for some subinterval. Despite the fact that the equations of this type arise in modelling of important problems appearing in applied sciences, their qualitative analysis seems to have received little attention in the mathematical literature, if it is compared with the regular cases.

The aim of this work is to answer the open problem proposed in [8], Open problem 4.2]. In that paper, the authors studied Eq. (3) and proved that if $\overline{h}>0$, $g(t)\ge 0$ for a. e. $t\in \mathbb {R}$, and

$$\begin{aligned} \left( \frac{\omega }{4}\int _0^{\omega }[h]_+(s)\hbox {d}s\right) ^{\lambda }\int _0^{\omega }h(s)\hbox {d}s\le \int _0^{\omega }g(s)\hbox {d}s \end{aligned}$$

(4)

holds, then (3) has at least one $\omega$-periodic solution. In spite of the fact that the condition (4) is unimprovable, in general, as shown in [8], Counter-example 4.1], for some particular cases it can be weakened (see Corollary 4 below). However, the main importance of the present paper lies in the answer to the cases when the function $g$ is positive almost everywhere in $\mathbb {R}$ but it is not uniformly essentially bounded from below by some positive constant, in particular, to that cases when $g$ can be estimated by some polynomial with isolated zeroes (see Corollary 5).

2 Statement of the problem and main result

Our results deal with the following family of equations

$$\begin{aligned} u''+\frac{g(t)}{u^{\lambda }}=h(t)u^{\delta }, \end{aligned}$$

(5)

where $\lambda >0$, $\delta \in [0,1)$, $g,h\in L_{\omega }$, and $g$ is a nonnegative function not necessarily essentially bounded from below by some positive constant. We establish a relation between the order of the singularity $\lambda$, the order of the nonlinearity $\delta$, and the regularity $p$ of the function $[h]_+$ guaranteeing the existence of an $\omega$-periodic solution to (5). Furthermore, one can easily check that the obtained results establish a link between the results obtained both in Refs. [10] and [11].

For convenience, we are going to introduce a list of notation which is used throughout the paper:

$\mathbb {N}$, $\mathbb {Z}$, and $\mathbb {R}$ are the sets of all natural, integer, and real numbers, respectively;

$C_{\omega }$ is the Banach space of $\omega$-periodic continuous functions $u:\mathbb {R}\rightarrow \mathbb {R}$, endowed with the norm

$$\begin{aligned} \Vert u\Vert _C=\max \big \{|u(t)|:t\in [0,\omega ]\big \}; \end{aligned}$$

$L_{\omega }^p$, where $1\le p<+\infty$, is the Banach space of $\omega$-periodic functions $h:\mathbb {R}\rightarrow \mathbb {R}$ which are Lebesgue integrable on $[0,\omega ]$ in the $p$-th power, endowed with the norm

$$\begin{aligned} \Vert h\Vert _p=\left( \int _0^{\omega } |h(t)|^p \hbox {d}t\right) ^{1/p}; \end{aligned}$$

$L_{\omega }=L_{\omega }^1$;

$L_{\omega }^{\infty }$ is the Banach space of $\omega$-periodic essentially bounded functions $h:\mathbb {R}\rightarrow \mathbb {R}$, endowed with the norm

$$\begin{aligned} \Vert h\Vert _{\infty }=\hbox {ess sup}\big \{|h(t)|:t\in [0,\omega ]\big \}; \end{aligned}$$

if $h\in L_{\omega }$ then

$$\begin{aligned} \overline{h}=\frac{1}{\omega }\int _0^{\omega }h(s)\hbox {d}s; \end{aligned}$$

if $s,t\in \mathbb {R}$ then $I(s,t)=(\min \{s,t\},\max \{s,t\})$;

if $h\in L_{\omega }$ then

$$\begin{aligned}{}[h]_+(t)=\frac{|h(t)|+h(t)}{2},\qquad [h]_-(t)=\frac{|h(t)|-h(t)}{2}\qquad \hbox {for a. e. }\,t\in \mathbb {R}. \end{aligned}$$

By an $\omega$-periodic solution to Eq. (5), we understand a positive function $u\in C_{\omega }$ which is absolutely continuous together with its first derivative on every compact interval of $\mathbb {R}$ and satisfies (5) almost everywhere on $\mathbb {R}$. To formulate our main result, we need the following notation

Notation 1

Let $g,h\in L_{\omega }$, $g(t)\ge 0$ for a. e. $t\in \mathbb {R}$, $\sigma \ge 0$. Then, for every $t\in \mathbb {R}$, we define

$$\begin{aligned} G(t,\sigma )=\lim _{x\rightarrow t_+}\int _x^{t+\frac{\omega }{2}}\frac{g(s)\hbox {d}s}{(s-t)^{\sigma }}+\lim _{x\rightarrow t_-}\int _{t+\frac{\omega }{2}}^{x+\omega }\frac{g(s)\hbox {d}s}{(t+\omega -s)^{\sigma }},\\ H_-(t,\sigma )=\lim _{x\rightarrow t_+}\int _x^{t+\frac{\omega }{2}}\frac{[h]_-(s)\hbox {d}s}{(s-t)^{\sigma }}+\lim _{x\rightarrow t_-}\int _{t+\frac{\omega }{2}}^{x+\omega }\frac{[h]_-(s)\hbox {d}s}{(t+\omega -s)^{\sigma }}. \end{aligned}$$

Note that, for every fixed $t\in \mathbb {R}$, the limits in Notation 1 exist and each of them is either finite or equal to $+\infty$.

Theorem 1

Let $[h]_+\in L_{\omega }^p$, $p\in [1,+\infty )$,

$$\begin{aligned} g(t)\ge 0\qquad \hbox {for a. e. }\,t\in \mathbb {R},\qquad \overline{g}>0, \end{aligned}$$

(6)

and let there exist a function $\varphi \in L_{\omega }^q$, $q\in [1,+\infty )$, such that¹

$$\begin{aligned}{}[h]_+(t)\le \varphi (t)g^{\frac{q-1}{q}}(t)\qquad \qquad \hbox {for a. e. }\,t\in \mathbb {R}. \end{aligned}$$

(7)

Let, moreover,

$$\begin{aligned} G\left( t,\frac{(2p-1)(\lambda +\delta )q}{(1-\delta )p}\right) + H_+H_-\left( t,\frac{(2p-1)(\lambda +\delta )(q-1)}{(1-\delta )p}\right) >H_+^q\Vert \varphi \Vert _q^q\quad \hbox {for }\,t\in \mathbb {R}\end{aligned}$$

(8)

where

$$\begin{aligned} H_+= & {} \left[ z(p)\left( \frac{(1-\delta )p}{2p-1}\right) ^{\frac{p-1}{p}}(1-\delta )\left\| [h]_+\right\| _p\right] ^{\frac{\lambda +\delta }{1-\delta }}, \end{aligned}$$

(9)

$$\begin{aligned} z(p)= & {} {\left\{ \begin{array}{ll} \left( \frac{p-1}{(1+\delta )p-1}\right) ^{\frac{p-1}{p}}&{}\text{ if } \, p>1,\\ 1&{}\text{ if } \, p=1. \end{array}\right. } \end{aligned}$$

(10)

Then there exists a unique $\omega$-periodic solution to (5) if and only if $\overline{h}>0$.

It can be easily verified that there is no $\omega$-periodic solution to (5) provided (6) is fulfilled and $\overline{h}\le 0$ (see also Sect. 4).

Further, we would like to emphasize that Eq. (5) has at most one $\omega$-periodic solution provided $g,h\in L_{\omega }$ are such that (6) is satisfied, $\delta \in [0,1)$, and $\lambda >0$ (see Sect. 4.2). As far as we know, such a result is new for this type of equations.

In conclusion, our results can be represented as a genuine relation between the order of the singularity, the order of the nonlinear term, and the regularity of the input functions involved, existing in the class of differential equations with attractive singularity.

The structure of the paper is as follows: After the introduction, statement of the problem, and formulation of the main result, we present a section dealing with finding a priori bounds for the $\omega$-periodic solutions to a certain family of equations depending on a parameter $\mu \in (0,1)$ [(homotopic family of equations of (5)]. In Sect. 4, we use the degree theory to prove the existence of an $\omega$-periodic solution (see Sect. 4.1), after this we prove a result on the uniqueness using a suitable change of variable (see Sect. 4.2). Finally, in Sect. 5, the consequences of Theorem 1 are formulated to illustrate the result obtained.

3 A priori estimates

Throughout this section, we establish some a priori bounds of a possible $\omega$-periodic solution to a certain family of second-order differential equations depending on a parameter $\mu \in (0,1]$. It means, we determine some positive constants $M_0$ and $\varepsilon _0$ (which do not depend on $\mu$) such that $\varepsilon _0<u(t)<M_0$ for $t\in \mathbb {R}$, where $u$ is an $\omega$-periodic solution to

$$\begin{aligned} u''+\mu \frac{g(t)}{u^{\lambda }}=\mu h(t)u^{\delta },\qquad \mu \in (0,1]. \end{aligned}$$

(11)

First we introduce a lemma which is proven in [9].

Lemma 1

(see [9], Lemma 2.4]) Let $u\in C_{\omega }$ be a locally absolutely continuous function. Then the inequality

$$\begin{aligned} (M-m)^2\le \frac{\omega }{4}\int _0^{\omega }u'^2(s)\hbox {d}s \end{aligned}$$

holds, where

$$\begin{aligned} M=\max \big \{u(t):t\in \mathbb {R}\big \},\qquad m=\min \big \{u(t):t\in \mathbb {R}\big \}. \end{aligned}$$

(12)

We continue by finding an upper bound for the possible $\omega$-periodic solution to (11).

Lemma 2

Let $\overline{h}>0$. Then there exists $M_0>0$ (which is independent of $\mu$) such that every $\omega$-periodic solution $u$ to (11) admits the estimate

$$\begin{aligned} u(t)<M_0 \qquad \hbox {for }\,t\in \mathbb {R}. \end{aligned}$$

Proof

Assume on the contrary that there exist sequences $(u_n)_{n=1}^{+\infty }$ and $(\mu _n)_{n=1}^{+\infty }\subset (0,1)$, where $u_n$ are $\omega$-periodic solutions to (11) with $\mu =\mu _n$ such that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\Vert u_n\Vert _C=+\infty . \end{aligned}$$

Then, multiplying both sides of (11) by $u_n^{\lambda }$ and integrating over $[0,\omega ]$, we obtain

$$\begin{aligned} \int _0^{\omega } v_n'^2(t)\hbox {d}t=\frac{(1+\lambda )^2\mu _n}{4\lambda }\left[ \int _0^{\omega }g(t)\hbox {d}t-\int _0^{\omega }h(t) u_n^{\lambda +\delta }(t)\hbox {d}t\right] , \end{aligned}$$

(13)

where $v_n(t)=u_n^{\frac{\lambda +1}{2}}(t)$ for $t\in \mathbb {R}$. Since $v_n$ are positive $\omega$-periodic functions, the equality (13), according to Lemma 1, results in

$$\begin{aligned} \left( 1-\left( \frac{m_n}{M_n}\right) ^{\frac{\lambda +1}{2}}\right) ^2\le \frac{\omega (1+\lambda )^2}{16\lambda } \left[ \frac{1}{M_n^{1+\lambda }}\int _0^{\omega }g(t)\hbox {d}t+ \frac{1}{M_n^{1-\delta }}\int _0^{\omega }[h]_-(t)\hbox {d}t\right] , \end{aligned}$$

(14)

where

$$\begin{aligned} m_n=\min \big \{u_n(t):t\in \mathbb {R}\big \},\qquad M_n=\max \big \{u_n(t):t\in \mathbb {R}\big \}. \end{aligned}$$

Since $\delta \in [0,1)$, from (14) it follows that $\frac{m_n}{M_n}\rightarrow 1$ as $n\rightarrow +\infty$. However, the integration of (11) from $0$ to $\omega$ yields

$$\begin{aligned} \frac{1}{m_n^{\lambda }}\int _0^{\omega }g(t)\hbox {d}t\ge m_n^{\delta }\int _0^{\omega }[h]_+(t)\hbox {d}t-M_n^{\delta }\int _0^{\omega }[h]_-(t)\hbox {d}t, \end{aligned}$$

i.e.,

$$\begin{aligned} \frac{1}{m_n^{\lambda +\delta }}\int _0^{\omega }g(t)\hbox {d}t\ge \int _0^{\omega }[h]_+(t)\hbox {d}t-\left( \frac{M_n}{m_n}\right) ^{\delta }\int _0^{\omega }[h]_-(t)\hbox {d}t. \end{aligned}$$

Consequently, passing to the limit as $n\rightarrow +\infty$ in the latter inequality, we obtain a contradiction to $\overline{h}>0$. $\square$

The following assertion shows that the velocity of every $\omega$-periodic solution to (11) is bounded by its position. As a consequence, we can conclude that the velocity of solutions is uniformly bounded.

Lemma 3

Let $[h]_+\in L_{\omega }^p$, $p\in [1,+\infty )$, and let (6) be fulfilled. Then every $\omega$-periodic solution $u$ to (11) admits the estimates

$$\begin{aligned} |u'(t)|\le & {} C u^{\frac{p(1+\delta )-1}{2p-1}}(t)\qquad \hbox {for }\,t\in \mathbb {R}, \end{aligned}$$

(15)

$$\begin{aligned} u(t)\le & {} \left[ u^{\frac{p(1-\delta )}{2p-1}}(t_0)+\widetilde{C}\left( \frac{\omega }{2}-\left| t_0+\frac{\omega }{2}-t\right| \right) \right] ^{\frac{2p-1}{p(1-\delta )}} \qquad \hbox {for }\,t\in [t_0,t_0+\omega ], \end{aligned}$$

(16)

where $t_0\in \mathbb {R}$ is arbitrary,

$$\begin{aligned} C=\left( \frac{2p-1}{p}z(p)\Vert [h]_+\Vert _p\right) ^{\frac{p}{2p-1}},\qquad \widetilde{C}=\frac{p(1-\delta )}{2p-1}C, \end{aligned}$$

(17)

and $z$ is given by (10).

Proof

Let $u$ be an $\omega$-periodic solution to (11). Then

$$\begin{aligned} u''(t)\le [h]_+(t)u^{\delta }(t)\qquad \hbox {for }\,t\in \mathbb {R}. \end{aligned}$$

(18)

Let $\tau \in \mathbb {R}$ be arbitrary but fixed such that $u'(\tau )\not =0$, and let $\sigma ={\text {sgn}}u'(\tau )$. Then there exists $s_0\in I(\tau ,\tau -\sigma \omega )$ such that

$$\begin{aligned} \sigma u'(t)>0\qquad \hbox {for }\,t\in I(s_0,\tau ),\qquad u'(s_0)=0. \end{aligned}$$

If $p=1$, then the integration of (18) over $I(s_0,\tau )$ yields

$$\begin{aligned} |u'(\tau )|\le \Vert [h]_+\Vert _1u^{\delta }(\tau ). \end{aligned}$$

(19)

If $p>1$, then multiplying both sides of (18) by $|u'(t)|^{(p-1)/p}$ and integrating over $I(s_0,\tau )$ we get

$$\begin{aligned} \frac{p}{2p-1}|u'(\tau )|^{\frac{2p-1}{p}}= & {} \sigma \int _{s_0}^{\tau }u''(t)|u'(t)|^{\frac{p-1}{p}}\hbox {d}t\nonumber \\\le & {} \sigma \int _{s_0}^{\tau }[h]_+(t)u^{\delta }(t)|u'(t)|^{\frac{p-1}{p}}\hbox {d}t\!\le \! \Vert [h]_+\Vert _p\left( \sigma \int _{s_0}^{\tau }u^{\frac{p\delta }{p-1}}|u'(t)|\hbox {d}t\right) ^{\frac{p-1}{p}}\nonumber \\= & {} z(p)\Vert [h]_+\Vert _p\left( u^{\frac{p(1+\delta )-1}{p-1}}(\tau )-u^{\frac{p(1+\delta )-1}{p-1}}(s_0)\right) ^{\frac{p-1}{p}}\nonumber \\\le & {} z(p)\Vert [h]_+\Vert _pu^{\frac{p(1+\delta )-1}{p}}(\tau ). \end{aligned}$$

(20)

Thus, (15) follows from (19), resp. (20).

Furthermore, let $t_0\in \mathbb {R}$ be arbitrary but fixed. Then from (15), it follows that

$$\begin{aligned} \frac{u'(t)}{u^{\frac{p(1+\delta )-1}{2p-1}}(t)}\le C,\qquad -\frac{u'(t)}{u^{\frac{p(1+\delta )-1}{2p-1}}(t)}\le C \qquad \hbox {for }\,t\in \mathbb {R}. \end{aligned}$$

(21)

The integration of both inequalities in (21) from $t_0$ to $t$ and from $t$ to $t_0+\omega$, respectively, yields

$$\begin{aligned}&u^{\frac{p(1-\delta )}{2p-1}}(t)\le u^{\frac{p(1-\delta )}{2p-1}}(t_0)+\widetilde{C}(t-t_0)\qquad \hbox {for }\,t\in \left[ t_0,t_0+\frac{\omega }{2}\right] ,\\&u^{\frac{p(1-\delta )}{2p-1}}(t)\le u^{\frac{p(1-\delta )}{2p-1}}(t_0+\omega )+\widetilde{C}(t_0+\omega -t)\qquad \hbox {for }\,t\in \left[ t_0+\frac{\omega }{2},t_0+\omega \right] . \end{aligned}$$

However, the two latter inequalities, with respect to the $\omega$-periodicity of $u$, results in the estimate (16). $\square$

Next, we are going to assume as in Theorem 1 the existence of a function $\varphi \in L_{\omega }^{q}$, where $q\in [1,+\infty )$, such that (7) holds. Thus, our aim is to extend the result proven in Ref. [11] and subsequently to obtain some a priori bound from below of $\omega$-periodic solutions to (11).

Lemma 4

Let (6) be satisfied, and let there exist a function $\varphi \in L_{\omega }^q$, $q\in [1,+\infty )$, such that (7) is fulfilled. Then every $\omega$-periodic solution to (11) admits the estimate

$$\begin{aligned} \int _{t}^{t+\omega }\frac{g(s)\mathrm{d}s}{u^{(\lambda +\delta )q}(s)}+ \int _{t}^{t+\omega }\frac{[h]_-(s)\mathrm{d}s}{u^{(\lambda +\delta )(q-1)}(s)}\le \Vert \varphi \Vert _q^q\qquad \hbox {for }\,t\in \mathbb {R}. \end{aligned}$$

Proof

Put $\gamma =\lambda -(\lambda +\delta )q\le 0$. Then multiplying both sides of (11) by $u^{\gamma }$ and integrating it over $[t,t+\omega ]$, we find

$$\begin{aligned} -\gamma \int _{t}^{t+\omega }u^{\gamma -1}(s)u'^2(s)\hbox {d}s+\mu \int _{t}^{t+\omega }\frac{g(s)\hbox {d}s}{u^{\lambda -\gamma }(s)}= \mu \int _{t}^{t+\omega }h(s)u^{\gamma +\delta }(s)\hbox {d}s. \end{aligned}$$

Since the first term on the left-hand side of the latter equality is positive and $\mu \in (0,1]$, by virtue of (7) and the Hölder’s inequality we get

$$\begin{aligned}&\int _{t}^{t+\omega }\frac{g(s)\hbox {d}s}{u^{\lambda -\gamma }(s)}+ \int _{t}^{t+\omega }[h]_-(s)u^{\gamma +\delta }(s)\hbox {d}s\le \Vert \varphi \Vert _q\left( \int _{t}^{t+\omega }\frac{g(s)\hbox {d}s}{u^{\lambda -\gamma }(s)}\right) ^{\frac{q-1}{q}}\\&\quad \le \Vert \varphi \Vert _q\left( \int _{t}^{t+\omega }\frac{g(s)\hbox {d}s}{u^{\lambda -\gamma }(s)}+ \int _{t}^{t+\omega }[h]_-(s)u^{\gamma +\delta }(s)\hbox {d}s\right) ^{\frac{q-1}{q}}. \end{aligned}$$

Consequently, the result immediately follows from the latter inequality. $\square$

Finally, we are going to obtain an a priori estimate from below for the $\omega$-periodic solutions to (11). To do that, we will need to assume that the singularity of the equation is sufficiently large. It is worth mentioning here that the condition (8) defines the close relation among the singularity $\lambda$, regularity $p$ of the function $[h]_+$, and the order of the nonlinearity $\delta$.

Lemma 5

Let $[h]_+\in L_{\omega }^p$, $p\in [1,+\infty )$, (6) be satisfied, and let there exist a function $\varphi \in L_{\omega }^q$, $q\in [1,+\infty )$, such that (7) holds. Let, moreover, (8) be fulfilled. Then there exists $\varepsilon _0>0$ such that every $\omega$-periodic solution to (11) admits the estimate

$$\begin{aligned} \varepsilon _0<u(t)\qquad \hbox {for }\,t\in \mathbb {R}. \end{aligned}$$

(22)

Proof

Assume on the contrary that for every $n\in \mathbb {N}$ there exists an $\omega$-periodic solution $u_n$ to (11) and $t_n\in [0,\omega ]$ such that

$$\begin{aligned} u_n(t_n)\le \frac{1}{n}. \end{aligned}$$

(23)

Obviously, without loss of generality we can assume that there exists $t_0\in [0,\omega ]$ such that

$$\begin{aligned} \lim _{n\rightarrow +\infty }t_n=t_0. \end{aligned}$$

(24)

According to (8), there exists $\varepsilon >0$ such that

$$\begin{aligned}&\int _{t_0}^{t_0+\omega } g(s) \left[ \varepsilon +\widetilde{C}\left( \frac{\omega }{2}-\left| t_0+\frac{\omega }{2}-s\right| \right) \right] ^{-\frac{(2p-1) (\lambda +\delta )q}{(1-\delta )p}}\hbox {d}s\nonumber \\&\qquad + \int _{t_0}^{t_0+\omega }[h]_-(s) \left[ \varepsilon +\widetilde{C}\left( \frac{\omega }{2}-\left| t_0+\frac{\omega }{2}-s\right| \right) \right] ^{-\frac{(2p-1) (\lambda +\delta )(q-1)}{(1-\delta )p}}\hbox {d}s>\Vert \varphi \Vert _q^q \end{aligned}$$

(25)

where $\widetilde{C}$ is given by (17). Moreover, in view of (24) there exists $n_0\in \mathbb {N}$ such that

$$\begin{aligned}&\left( \frac{1}{n}\right) ^{\frac{p(1-\delta )}{2p-1}}+\widetilde{C}\left( \frac{\omega }{2}-\left| t_n+\frac{\omega }{2}-t\right| \right) \le \varepsilon +\widetilde{C}\left( \frac{\omega }{2}-\left| t_0+\frac{\omega }{2}-t\right| \right) \nonumber \\&\quad \quad \hbox {for }\,t\in [t_n,t_n+\omega ],\quad n\ge n_0, \end{aligned}$$

and consequently, according to Lemmas 3 and (23), we have

$$\begin{aligned} u_n(t)\le \left[ \varepsilon +\widetilde{C}\left( \frac{\omega }{2}-\left| t_0+\frac{\omega }{2}-t\right| \right) \right] ^{\frac{2p-1}{(1-\delta )p}}\qquad \hbox {for }\,t\in [t_n,t_n+\omega ],\quad n\ge n_0. \end{aligned}$$

(26)

On the other hand, in view of Lemma 4, we find

$$\begin{aligned} \int _{t_n}^{t_n+\omega }\frac{g(s)\hbox {d}s}{u_n^{(\lambda +\delta )q}(s)}+ \int _{t_n}^{t_n+\omega }\frac{[h]_-(s)\hbox {d}s}{u_n^{(\lambda +\delta )(q-1)}(s)}\le \Vert \varphi \Vert _q^q\qquad \hbox {for }\,n\in \mathbb {N}. \end{aligned}$$

(27)

Consequently, using (26) in (27), we obtain

$$\begin{aligned}&\int _{t_n}^{t_n+\omega } g(s) \left[ \varepsilon +\widetilde{C}\left( \frac{\omega }{2}-\left| t_0+\frac{\omega }{2}-s\right| \right) \right] ^{-\frac{(2p-1) (\lambda +\delta )q}{(1-\delta )p}}\hbox {d}s\nonumber \\&\qquad + \int _{t_n}^{t_n+\omega }[h]_-(s) \left[ \varepsilon +\widetilde{C}\left( \frac{\omega }{2}-\left| t_0+\frac{\omega }{2}-s\right| \right) \right] ^{-\frac{(2p-1) (\lambda +\delta )(q-1)}{(1-\delta )p}}\hbox {d}s\le \Vert \varphi \Vert _q^q \quad \hbox {for }\,n\ge n_0. \end{aligned}$$

However, the latter inequality, in view of (24), contradicts (25). $\square$

4 Proof of the main result

In this section, we present the proof of Theorem 1 by using the lemmas proven in the previous section. Thus, henceforth we assume the framework of our result.

The proof of Theorem 1 is divided into two parts: In the first one, we deal with the existence of an $\omega$-periodic solution to (5), and the second part is devoted to the proof of the uniqueness.

But at first let us mention that the existence of an $\omega$-periodic solution $u$ to (5) implies $\overline{h}>0$ provided (6) is fulfilled. Indeed, dividing the Eq. (5) by $u^{\delta }$ and integrating it from $0$ to $\omega$ we arrive at

$$\begin{aligned} \int _0^{\omega }\frac{g(s)\hbox {d}s}{u^{\lambda +\delta }}\le \int _0^{\omega }h(s)\hbox {d}s. \end{aligned}$$

However, the latter inequality with respect to (6) results in $\overline{h}>0$.

4.1 Existence of an $\omega$-periodic solution

To start with, we rewrite our problem as an operator equation. This is a standard procedure, so that we describe it briefly. We will work in the Banach space $X=C_{\omega }$ of continuous $\omega$-periodic functions on the real line. The one-dimensional subspace of $X$ formed by constant functions is naturally identified with $\mathbb {R}$. We use this relation to construct two linear projections $\Pi , Q:X\rightarrow X$ on this subspace:

$$\begin{aligned} \Pi [u]\mathop {=}\limits ^{def}u(0),\qquad Q[u]\mathop {=}\limits ^{def}\frac{1}{\omega }\int _0^{\omega }u(s)\hbox {d}s. \end{aligned}$$

For any $u\in {\text {Ker}}Q$, we denote $K[u]$ the primitive of $u$ vanishing at the point $t=0$. The linear operator $K:{\text {Ker}}Q\rightarrow {\text {Ker}}\Pi$ defined in this way is compact. Our Nemytskii operator $N:\Omega \rightarrow L_{\omega }$ is defined by $N[u](t)\mathop {=}\limits ^{def}-g(t)/u^{\lambda }(t)+h(t)u^{\delta }(t)$, where

$$\begin{aligned} \Omega =\left\{ u\in X:u(t)>0\, \text{ for } \, t\in \mathbb {R}\right\} . \end{aligned}$$

The periodic problem associated with (5) can be now rewritten as an existence of a fixed point to an operator equation in $\Omega$:

$$\begin{aligned} u=F[u],\qquad u\in \Omega , \end{aligned}$$

(28)

where the (nonlinear) operator $F:\Omega \rightarrow X$ is given by

$$\begin{aligned} F[u]\mathop {=}\limits ^{def}QN[u]+\Pi [u]+K(I-Q)\circ K(I-Q)N[u] \end{aligned}$$

(we denote by $I$ the identity map on $X$). We point out that $F$ is completely continuous, that is, it is continuous and maps bounded sets of $X$ whose closure is contained in $\Omega$ into relatively compact subsets of $X$.

The objective here is to use the previous functional framework and by using standard arguments to show that the Leray–Schauder fixed point degree of $I-F$ is well defined on a certain open subset $U$ with $\overline{U}\subset \Omega$ and it is nonzero.

Proof of the existence

According to Lemmas 2 and 5, there exist $M_0>0$ and $\varepsilon _0>0$ such that

$$\begin{aligned} \varepsilon _0<\left( \frac{\overline{g}}{\overline{h}}\right) ^{\frac{1}{\lambda +\delta }}<M_0 \end{aligned}$$

(29)

and any (possible) $\omega$-periodic solution $u$ to (11) with $\mu \in (0,1]$ satisfies the inequalities

$$\begin{aligned} \varepsilon _0<u(t)<M_0\qquad \hbox {for }\,t\in \mathbb {R}. \end{aligned}$$

(30)

We will show that the Leray–Schauder degree of $I-F$ on the open set

$$\begin{aligned} U=\left\{ u\in \Omega :\varepsilon _0<u(t)<M_0\, \text{ for } \, t\in \mathbb {R}\right\} \end{aligned}$$

is nonzero. Observe that $\overline{U}\subset \Omega$. In order to check the statement on the degree, we consider the homotopy to the averaged nonlinearity:

$$\begin{aligned} H:[0,1]\times \overline{U}\rightarrow X,\qquad H(\mu ,u)=QN[u]+\Pi [u]+\mu K(I-Q)\circ K(I-Q)N[u]. \end{aligned}$$

Observe that $H$ is completely continuous $H(1,u)=F[u]$ and the $\omega$-periodic solutions to (11) coincide with the fixed points of $H(\mu ,\cdot )$ when $\mu \in (0,1]$. Now our aim is to show that $H(\mu ,u)\ne u$ for any $\mu \in [0,1]$ and $u\in \partial U$.

If $\mu =0$ then ${\text {Im}}H(0,\cdot )\subseteq \mathbb {R}$ and the equality $H(0,u)=u$ implies

$$\begin{aligned} \overline{h} u^{\delta }=\frac{\overline{g}}{u^{\lambda }}\qquad \Longleftrightarrow \qquad u=\left( \frac{\overline{g}}{\overline{h}}\right) ^{\frac{1}{\lambda +\delta }}. \end{aligned}$$

However, the latter in view of (29) results in $u\not \in \partial U$.

Further, we observe that if $H(\mu ,u)=u$ for some $\mu \in (0,1]$ and $u\in \partial U$ then either $u(t_0)=M_0$ or $u(t_0)=\varepsilon _0$ for some $t_0\in \mathbb {R}$ which contradicts to (30).

Thus, $H$ is an admissible homotopy and

$$\begin{aligned} d_{\mathrm{LS}}(I-F,U,0)=d_{\mathrm{LS}}(I-H(1,\cdot ),U,0)=d_{\mathrm{LS}}(I-H(0,\cdot ),U,0) \end{aligned}$$

(31)

($d_{\mathrm{LS}}$ denotes the Leray–Schauder degree). Since $H(0,\cdot )$ is contained in $\mathbb {R}$, combining (31) with [4], Theorem 8.7] we find that

$$\begin{aligned} d_{\mathrm{LS}}(I-F,U,0)=d_{\mathrm{B}}((I-H(0,\cdot ))|_{\mathbb {R}},U\cap \mathbb {R},0)=-1 \end{aligned}$$

($d_\mathrm{B}$ denotes the Brouwer degree). $\square$

4.2 Uniqueness

Our strategy to prove the result on the uniqueness of $\omega$-periodic solution to (5) will consist in considering the change of variable $v(t)=(1-\delta )^{-1}u^{1-\delta }(t)$ for $t\in \mathbb {R}$. In this way, Eq. (5) can be rewritten in the form

$$\begin{aligned} v''=-\left( \frac{\delta }{1-\delta }\right) \frac{v'^2}{v}-\frac{\widetilde{g}(t)}{v^{\frac{\lambda +\delta }{1-\delta }}}+h(t), \end{aligned}$$

(32)

where $\widetilde{g}(t)=(1-\delta )^{-\frac{\lambda +\delta }{1-\delta }}g(t)$. The proof presented below is based on the argument done in [11] but adapted to Eq. (32).

Proof of the uniqueness

Assume that there exist two $\omega$-periodic solutions to (5), $u_1$ and $u_2$. Then, we define the $\omega$-periodic functions $v_1(t)=(1-\delta )^{-1}u_1^{1-\delta }(t)$ and $v_2(t)=(1-\delta )^{-1}u_2^{1-\delta }(t)$ for $t\in \mathbb {R}$. The functions $v_1$ and $v_2$ are $\omega$-periodic solutions to (32). Obviously, it is sufficient to consider only the following two possibilities:

• (i) $v_1(t)>v_2(t)$ for every $t\in \mathbb {R}$;

• (ii) the images of the functions $v_1$ and $v_2$ have a nonempty intersection.

Let us start by showing that the case (i) cannot occured provided (6) is fuflilled. Indeed, put $w(t)=v_1(t)-v_2(t)$ for $t\in \mathbb {R}$. Then

$$\begin{aligned} w''(t)\ge & {} -\left( \frac{\delta }{1-\delta }\right) \left( \frac{v_1'^2(t)}{v_1(t)}-\frac{v_2'^2(t)}{v_2(t)}\right) \ge -\left( \frac{\delta }{1-\delta }\right) \left( \frac{v_1'^2(t)-v_2'^2(t)}{v_2(t)}\right) \\= & {} -\left( \frac{\delta }{1-\delta }\right) \left( \frac{v_1'(t)+v_2'(t)}{v_2(t)}\right) w'(t)\qquad \hbox {for a. e. }\,t\in \mathbb {R}. \end{aligned}$$

However, since $w$ and $w'$ are continuous $\omega$-periodic functions, there exists $t_0\in \mathbb {R}$ such that $w'(t_0)=w'(t_0+\omega )=0$. Consequently, by virtue of the Gronwall–Bellman lemma, we conclude that $w'\equiv 0$. This implies that $v_1'\equiv v_2'$ and therefore

$$\begin{aligned} w''(t)=-\frac{\delta }{1-\delta }v_1'^2(t)\left( \frac{1}{v_1(t)}-\frac{1}{v_2(t)}\right) -\widetilde{g}(t)\left( v_1^{-\frac{\lambda +\delta }{1-\delta }}(t)-v_2^{-\frac{\lambda +\delta }{1-\delta }}(t)\right) \qquad \hbox {for a. e. }\,t\in \mathbb {R}. \end{aligned}$$

Since $\overline{g}>0$, the latter equality contradicts (i).

Concerning the case (ii), there exist $t_*\in [0,\omega )$, $t^*\in (t_*,t_*+\omega ]$, and $t_0\in (t_*,t^*)$ such that

$$\begin{aligned} w(t)>0\qquad \hbox {for }\,t\in (t_*,t^*),\qquad w(t_*)=w(t^*)=0,\qquad w'(t_0)=0. \end{aligned}$$

(33)

where $w(t)=v_1(t)-v_2(t)$ for $t\in \mathbb {R}$. In the same manner as in the case (i), we obtain

$$\begin{aligned} w''(t)\ge -\left( \frac{\delta }{1-\delta }\right) \left( \frac{v_1'(t)+v_2'(t)}{v_2(t)}\right) w'(t)\qquad \hbox {for a. e. }\,t\in [t_*,t^*]. \end{aligned}$$

Cosequently, according to Gronwall–Bellman lemma, we obtain $w'(t)\ge 0$ for $t\in [t_0,t^*]$ and $w'(t)\le 0$ for $t\in [t_*,t_0]$, which contradicts (33). $\square$

Remark 1

Note that from the proof of the uniqueness, it follows that (5) has at most one solution provided $\delta \in [0,1)$, $\lambda >0$, and $g,h\in L_{\omega }$ fulfill (6). That means that if there exists an $\omega$-periodic solution to (5), then it is unique.

5 Applications and examples

Below we discuss some particular cases illustrating the result obtained. The first assertion shows that Theorem 1, being applied when the function $g$ is uniformly bounded from below by a positive constant, yields as a corollary the results proven in Refs. [10, 11].

Corollary 1

Let $[h]_+\in L_{\omega }^p$, $p\in [1,+\infty )$, $g(t)\ge \delta _0>0$ for a. e. $t\in \mathbb {R}$, and let

$$\begin{aligned} \frac{\lambda +\delta }{1-\delta }\ge \frac{1}{2p-1}. \end{aligned}$$

(34)

Then there exists a unique $\omega$-periodic solution to (5) if and only if $\overline{h}>0$.

Proof

It is sufficient to put $\varphi \equiv \delta _0^{(1-p)/p}[h]_+$, $q=p$, and notice that $G\left( t,\frac{(2p-1)(\lambda +\delta )}{(1-\delta )}\right) =+\infty$ for every $t\in \mathbb {R}$ provided (34) is fulfilled. $\square$

In the case when $[h]_+\in L^{\infty }_{\omega }$, then $[h]_+\in L^p_{\omega }$ for every $p\in [1,+\infty )$. Therefore, from Corollary 1, we obtain the following assertion:

Corollary 2

Let $[h]_+\in L_{\omega }^{\infty }$ and let $g(t)\ge \delta _0>0$ for a. e. $t\in \mathbb {R}$. Then there exists a unique $\omega$-periodic solution to (5) if and only if $\overline{h}>0$.

Example 1

In Ref. [11], it was proven that the equation

$$\begin{aligned} u''+\frac{1}{u^{\lambda }}=h(t) \end{aligned}$$

(35)

with $[h]_+\in L^p_{\omega }$ and $\overline{h}>0$ and has a unique $\omega$-periodic solution if $\lambda \ge 1/(2p-1)$. Moreover, there is also established an example showing that for every $\lambda \in (0,1/(2p-1))$, there exists $h\in L^p_{\omega }$ with $\overline{h}>0$ such that (35) has no $\omega$-periodic solution.

Corollary 1 says that if a sub-linear term is added to (35), the condition $\lambda \ge 1/(2p-1)$ can be weakened. In particular,

$$\begin{aligned} u''+\frac{1}{u^{\lambda }}=h(t)u^{\delta } \end{aligned}$$

(36)

has a unique $\omega$-periodic solution for any $\lambda >0$ if $\delta \in [1/(2p),1)$, provided $[h]_+\in L^p_{\omega }$, $\overline{h}>0$. On the other hand, a relation of this type was proven in [10] by using the method of lower and upper functions in the case when $p=1$. It is worth mentioning here that Corollary 1 joins both results of [10] and [11] and establishes a relation between the orders of the singularity and nonlinearity, and the regularity of the input function, guaranteeing the existence of a unique $\omega$-periodic solution to (36).

As it was mentioned in Example 1, in the case when $\lambda <1/(2p-1)$, an additional condition on $h$ is required in order to guarantee the existence of an $\omega$-periodic solution to (35). One of such a condition can be obtained from Theorem 1 immediately by putting $\varphi \equiv [h]_+$ and $q=p$. Then we have the following assertion:

Corollary 3

Let $[h]_+\in L_{\omega }^p$, $p\in [1,+\infty )$, and let

$$\begin{aligned} \frac{\lambda +\delta }{1-\delta }<\frac{1}{2p-1}. \end{aligned}$$

Let, moreover,

$$\begin{aligned} \frac{2(1-\delta )}{1+\lambda -2p(\lambda \!+\!\delta )}\left( \frac{\omega }{2}\right) ^{\frac{1+\lambda -2p(\lambda +\delta )}{1-\delta }} \!+\!H_+\left( \frac{2}{\omega }\right) ^{\frac{(2p-1)(\lambda +\delta )(p-1)}{(1-\delta )p}}\int _0^{\omega }[h]_-(s)\hbox {d}s>H_+^p\Vert [h]_+\Vert _p^p\nonumber \\ \end{aligned}$$

(37)

where $H_+$ is given by (9). Then there exists a unique $\omega$-periodic solution to (36) if and only if $\overline{h}>0$.

If $p=1$ and $\delta =0$, then Corollary 3 results in the following consequence:

Corollary 4

Let $[h]_+\in L_{\omega }$ and let $\lambda <1$. Let, moreover,

$$\begin{aligned} \left( \frac{\omega }{4}\int _0^{\omega }[h]_+(s)\hbox {d}s\right) ^{\lambda }\int _0^{\omega }h(s)\hbox {d}s<\frac{\omega }{2^{\lambda }(1-\lambda )}. \end{aligned}$$

(38)

Then there exists a unique $\omega$-periodic solution to (35) if and only if $\overline{h}>0$.

It is worth mentioning here that the condition (38) improves the condition (4), which was obtained in [8], for Eq. (35).

However, as it was mentioned in the introduction, the main importance of Theorem 1 can be distinguished for the cases when $g$ has zeroes at isolated points. Thus, the essence of the condition (8) is different to the one established in [8], Corollary 4.2]. That gives also an answer to the open problem [8], Open problem 4.2].

Corollary 5

Let $h\in L_{\omega }^p$, $p\in [1,+\infty )$, and let there exist $c>0$, $\alpha _i,\beta _i\ge 0$, $t_i\in \mathbb {R}$ $(i=1,\dots ,n)$ such that $t_1<t_2<\dots <t_n<t_1+\omega$ and

$$\begin{aligned} g(t)\ge & {} c(t_{i+1}-t)^{\alpha _{i+1}}(t-t_i)^{\beta _i}\qquad \hbox {for a. e. }\,t\in (t_i,t_{i+1}),\quad i=1,\dots ,n-1,\\ g(t)\ge & {} c(t_1+\omega -t)^{\alpha _1}(t-t_n)^{\beta _n}\qquad \hbox {for a. e. }\,t\in (t_n,t_1+\omega ). \end{aligned}$$

Let, moreover,

$$\begin{aligned} \frac{\lambda +\delta }{1-\delta }>\frac{(1+\gamma _0)(1+\gamma p)}{(1+\gamma )(2p-1)}\quad \text{ if } \, p>1,\qquad \frac{\lambda +\delta }{1-\delta }\ge 1+\gamma _0\quad \text{ if } \, p=1, \end{aligned}$$

(39)

where

$$\begin{aligned} \gamma _0=\max \big \{\min \big \{\alpha _i,\beta _i\big \}: i=1,\dots ,n\big \},\qquad \gamma =\max \big \{\alpha _i,\beta _i:i=1,\dots ,n\big \}. \end{aligned}$$

Then there exists a unique $\omega$-periodic solution to (5) if and only if $\overline{h}>0$.

Proof

First suppose that $p>1$. Put

$$\begin{aligned} g_0(t)&\mathop {=}\limits ^{def}&{\left\{ \begin{array}{ll} c(t_{i+1}-t)^{\alpha _{i+1}}(t-t_i)^{\beta _i}\qquad \hbox {for }\,t\in [t_i,t_{i+1}),\quad i=1,\dots ,n-1,\\ c(t_1+\omega -t)^{\alpha _1}(t-t_n)^{\beta _n}\qquad \hbox {for }\,t\in [t_n,t_1+\omega ), \end{array}\right. } \\ g_0(t)&\mathop {=}\limits ^{def}&g_0(t-k\omega )\qquad \hbox {for }\,t\in [t_1+k\omega ,t_1+(k+1)\omega ),\quad k\in \mathbb {Z}\setminus \{0\}, \end{aligned}$$

and

$$\begin{aligned} \varphi (t)\mathop {=}\limits ^{def}[h]_+(t)g_0^{\frac{1-q}{q}}(t)\qquad \hbox {for a. e. }\,t\in \mathbb {R}, \end{aligned}$$

where $q<(1+\gamma )p/(1+\gamma p)$ is such that

$$\begin{aligned} \frac{\lambda +\delta }{1-\delta }\ge \frac{(1+\gamma _0)p}{(2p-1)q}. \end{aligned}$$

(40)

Then using the Hölder’s inequality, we obtain

$$\begin{aligned} \int _0^{\omega }\varphi ^q(s)\hbox {d}s=\int _0^{\omega }\frac{[h]^q_+(s)\hbox {d}s}{g_0^{q-1}(s)}\le \Vert [h]_+\Vert _p^q\left( \int _0^{\omega }g_0^{\frac{p(1-q)}{p-q}}(s)\hbox {d}s\right) ^{\frac{p-q}{p}}<+\infty , \end{aligned}$$

and consequently, $\varphi \in L^q_{\omega }$,

$$\begin{aligned}{}[h]_+(t)\le \varphi (t)g^{\frac{q-1}{q}}(t)\qquad \hbox {for a. e. }\,t\in \mathbb {R}. \end{aligned}$$

Furthermore, from (40) it follows that

$$\begin{aligned} G\left( t,\frac{(2p-1)(\lambda +\delta )q}{(1-\delta )p}\right) =+\infty \qquad \hbox {for every }\,t\in \mathbb {R}. \end{aligned}$$

(41)

If $p=1$, then we put $q=1$, $\varphi \equiv [h]_+$, and it can be easily verified that, in view of (39), the relation (41) holds again. $\square$

Now, we are going to present an example of a differential equation to which our result can be efficiently applied. To illustrate the result, we have selected a particular case of a physical model studied in [17], Section 5] submitted to the action of an external force. The dynamic of a trapless 3D Bose–Einstein condensate with variable scattering length can be ruled by Eq. (5), where $g$ models the S-wave scattering length, which is assumed to vary $\omega$-periodically in time. In our case, a nonnegative $g$ corresponds to attractive interactions between the elementary particles and $h$ is an external force (usually $\delta =0$). Then the existence of an $\omega$-periodic solution to (5) is interpreted as a bound state of the condensate with external trap. To simplify the model, the function $g$ can be considered as a polynomial function, which may have several zeroes. However, we also can investigate the problem in the case when $g$ is trigonometric. Obviously, the latter case seems to be more useful in applications but also more complicated to study analytically. Nevertheless, according to Corollary 5, it is sufficient to check the condition (8) in a neighborhood of each zero of $g$. Thus, having approximated $g$ by a Taylor polynomial, the problem is reduced to the much simpler one—the polynomial case. This makes our result efficiently applicable in several cases.

Example 2

Consider the equation

$$\begin{aligned} u''+\frac{g(t)}{u^{\lambda }}=h(t), \end{aligned}$$

(42)

where

$$\begin{aligned} g(t)= & {} ct^{\alpha }(\omega -t)^{\alpha }\qquad \hbox {for }\,t\in [0,\omega ),\\ g(t)= & {} g(t-k\omega ) \qquad \hbox {for }\,t\in [k\omega ,(k+1)\omega ),\quad k\in \mathbb {Z}\setminus \{0\}, \end{aligned}$$

$c$, $\alpha$, and $\lambda$ are positive numbers, $\overline{h}>0$. According to Corollary 5, one can conclude that (42) has a unique $\omega$-periodic solution if one of the following conditions is fulfilled:

• $[h]_+\in L_{\omega }^{\infty }$ and $\lambda >\alpha /2$;

• $[h]_+\in L_{\omega }^p$, $p\in (1,+\infty )$, and $\lambda >\frac{1+\alpha p}{2p-1}$;

• $[h]_+\in L_{\omega }$ and $\lambda \ge 1+\alpha$.

Note that the result for the case when $[h]_+\in L_{\omega }^{\infty }$ is obtained applying Corollary 5 for $p\in (1,+\infty )$ sufficiently large.