Skip to main content

Nonconstant periodic solutions created by impulses for singular differential equations


In this work we discuss the existence of nonconstant periodic solutions for nonautonomous singular second order differential equations in the presence of impulses. Our approach is variational.

1 Introduction

This paper is devoted to the study of the existence of nonconstant periodic solutions for non-autonomous singular second order differential equations,

$$ u^{\prime\prime}(t)+f(t,u)=e(t),\quad\mbox{for a.e. }t\in [ 0,T ] , T>0, $$

under impulse conditions

$$ \Delta u^{\prime}(t_{j})=I_{j}\bigl(u(t_{j}) \bigr),\quad j=1,2,\ldots,p-1, $$

where f is a singular negative function and \(\Delta u^{\prime }(t_{j})=u^{\prime}(t_{j}^{+})-u^{\prime}(t_{j}^{-})\), with \(u^{\prime }(t_{j}^{\pm})= \lim_{t\rightarrow t_{j}^{\pm}} u^{\prime }(t)\); \(t_{j}\) for \(j=1,2,\ldots,p-1\), are the instants where the impulses occur with \(0=t_{0}< t_{1}<\cdots<t_{p-1}<t_{p}=T\), \(t_{j+p}=t_{j}+T\). The functions \(I_{j}:\mathbb{R} \rightarrow \mathbb{R} \); \(j=1,2,\ldots,p-1\), are continuous and represent the jump discontinuities of \(u^{\prime}\) at the impulse moments, and \(I_{j+p}\equiv I_{j}\). Applications of impulsive differential equations with or without delays occur in medicine, population dynamics, and chaos theory; see [1, 2]. For the general aspects of impulsive differential equations, we refer the reader to the classical monographs [3, 4]. Due to its significance, a great deal of work has been done in the theory of impulsive differential equations; see for example [57]. It was pointed out in [8] that singular differential equations of the form (1.1) appear in the description of many phenomena in the applied sciences, such as nonlinear elasticity. Singular problems without impulse effects have been investigated extensively in the literature (see [2, 912] and the references therein). Some classical tools have been used to study such problems. These classical techniques include the coincidence degree theory of Mawhin and Willem [13], the method of upper and lower solutions [14], some fixed point theorems [15], and variational methods [16, 17]. For example, the authors in [18] obtained multiple periodic solutions for second-order perturbed Hamiltonian systems with impulse effects via variational methods. We believe that singular problems with impulsive effects have not been sufficiently studied; for some work on the subject, see [19, 20]. Inspired by the above facts, and the following important result (see [12]): if e is an integrable T-periodic function, then (1.1) has a positive T-periodic weak solution if and only if \(\int_{0}^{T}e(t)\,dt<0\), the aim of this paper is to prove a new existence result on a weak nonconstant T-periodic solutions generated by impulses (1.2) for the singular equation (1.1). Here, we say that a solution is generated by impulses if this solution exists when \(I_{j}\neq 0\), for some \(1< j< p-1\), and if it disappears when \(I_{j}\equiv 0\) for all \(1< j< p-1\).

The paper is organized as follows. Section 2 contains the basic preliminaries. An existence result of periodic solutions is given in Section 3. We conclude with an example.

2 Preliminaries

In this section we introduce some basic notions that will be used in the rest of the paper. \(P_{T}\) denotes the set of T-periodic functions \(u: \mathbb{R} \rightarrow \mathbb{R} \) satisfying \(u(t+T)=u(t)\) for all \(t\in \mathbb{R} \); \(C_{T}=\{u\in P_{T}; u\mbox{ is continuous}\}\). For \(u\in C_{T}\) we denote its norm by \(\Vert u\Vert _{\infty}=\sup \{ \vert u(t)\vert ; t\in [ 0,T ] \} \). Then \(( C_{T}, \Vert \cdot \Vert _{\infty} ) \) is a Banach space. For \(p\geq1\), \(L^{p}:=L^{p}(0,T;\mathbb{R} )\) is the classical Lebesgue space of measurable functions \(u:[0;T]\rightarrow \mathbb{R} \) such that \(\vert u(\cdot)\vert ^{p}\) is integrable, and for \(u\in L^{p}\) we define its norm by

$$ \Vert u\Vert _{L^{p}}= \biggl( \int_{0}^{T} \bigl\vert u(t)\bigr\vert ^{p}\,dt \biggr) ^{\frac{1}{p}}. $$

We consider the Sobolev space \(H_{T}^{1}=\{u:\mathbb{R} \rightarrow \mathbb{R} ; u \mbox{ is absolutely continuous}, u^{\prime}\in L^{2}, \mbox{ and}\mbox{ }u(t)=u(t+T) \mbox{ for }t\in \mathbb{R} \}\). \(H_{T}^{1}\), equipped with the inner product

$$ (u,v)=\int_{0}^{T}u^{\prime}(t)v^{\prime }(t)\,dt+ \int_{0}^{T}u(t)v(t)\,dt $$

and the norm

$$ \Vert u\Vert _{H_{T}^{1}}:= \bigl( \Vert u\Vert _{L^{2}}^{2}+ \bigl\Vert u^{\prime}\bigr\Vert _{L^{2}}^{2} \bigr) ^{\frac{1}{2}} $$

is a reflexive Banach space. Also, \(H_{T}^{1}\) admits the orthogonal decomposition, \(H_{T}^{1}=E+F\), where F is the subspace of constant functions in \(H_{T}^{1}\) and E denotes the subspace of functions in \(H_{T}^{1}\) with zero mean value. E is a weakly closed subspace of \(H_{T}^{1}\). If \(u\in E\), then the Wirtinger inequality

$$ \int_{0}^{T}\bigl\vert u(t)\bigr\vert ^{2}\,dt\leq\frac {T^{2}}{4\pi ^{2}}\int_{0}^{T} \bigl\vert u^{\prime}(t)\bigr\vert ^{2}\,dt $$

implies that, on E, we can obtain the equivalent norm

$$ \| u\|:=\bigl\Vert u^{\prime}\bigr\Vert _{L^{2}}. $$

Also, for \(u\in E\) we have

$$ \Vert u\Vert _{\infty}\leq\sqrt{T}\Vert u\Vert . $$

It is easy to see that a T-periodic solution of (1.1), (1.2) with zero mean value must be a nonconstant T-periodic solution of (1.1), (1.2).

Definition 1

\(u\in H_{T}^{1}\) is solution of (1.1), (1.2) if \(u\in C_{T}\) such that for every j, \(u_{j}=u|_{[t_{j},t_{j+1}]} \in H^{2}(t_{j},t_{j+1})\), and it satisfies (1.1) for a.e. \(t\in[0,T]\), \(t\neq t_{j}\), the limits \(u^{\prime}(t_{j}^{-})\), \(u^{\prime}(t_{j}^{+})\) exist and the impulsive conditions (1.2) are satisfied.

3 Main result

We consider the impulsive second-order periodic boundary value problem,

$$ \left \{ \textstyle\begin{array}{@{}l} u^{\prime\prime}(t)+f(t,u)=e(t),\quad\mbox{for }t\in(0,T), t\neq t_{j}, \\ \Delta u^{\prime}(t_{j})=I_{j}(u(t_{j})),\quad j=1,2,\ldots,p-1, \\ u(0)-u(T)=0, \qquad u^{\prime}(0)-u^{\prime}(T)=0,\end{array}\displaystyle \right . $$

under the following assumptions:

  1. (H1)
    1. (i)

      \(f: \mathbb{R} \times(0,+\infty)\rightarrow \mathbb{R} \), is a negative Carathéodory function which is T-periodic in its first argument,

    2. (ii)

      \(\lim_{u\rightarrow0^{+}}f(t,u)=-\infty\), for a.e. \(t \in [0,T]\),

    3. (iii)

      \(\lim_{u\rightarrow +\infty}f(t,u)=0\), for a.e. \(t \in[0,T]\),

  2. (H2)
    1. (i)

      e is a locally integrable T-periodic function and \(\bar{e}:=\frac{1}{T}\int_{0}^{T}e(t)\,dt>0\),

    2. (ii)

      \(I_{j}:\mathbb{R} \rightarrow \mathbb{R} \), is a continuous bounded function for all \(j=1,\ldots,p-1\), such that \(m=\inf I_{j}<\sup I_{j}=M<-\frac{T}{p-1}\bar{e}\).

Remark 1

(H1)(iii) implies that \(\lim_{u\rightarrow+\infty}\frac {F(t,u)}{u^{2}}=0\), for a.e. \(t \in [0,T]\) where \(F(t,u):=\int_{1}^{u}f(t,s)\,ds\).

Remark 2

Consider (1.1) and suppose that \(I_{j}\equiv0\) for all \(1< j< p-1\). In this case f verifies the conditions of the second result in [12]. Then (1.1) has a positive T-periodic solution if and only if \(\bar{e}<0\). This means that (1.1) under (H1) and (H2)(i) does not have a T-periodic weak solution. However, if the impulses happen, i.e. if (H2)(ii) is fulfilled for this singular equation (1.1), there may exist a positive T-periodic weak solution. Such a solution is called a periodic solution generated by impulses as pointed out in [20].

Theorem 1

Suppose (H1) and (H2) hold. Then (3.1) admits at least one weak nonconstant T-periodic solution.


To prove this result, we rely on a variational method. In order to study problem (3.1), we consider the following modified problem:

$$ \left \{ \textstyle\begin{array}{@{}l} u^{\prime\prime}(t)+f_{r}(t,u(t))=e(t),\quad \mbox{for a.e. } t\in (0,T), t\neq t_{j}, \\ \Delta u^{\prime}(t_{j})=I_{j}(u(t_{j})),\quad j=1,2,\ldots,p-1, \\ u(0)-u(T)=0, \qquad u^{\prime}(0)-u^{\prime}(T)=0, \end{array}\displaystyle \right . $$

where \(f_{r}:[0,T]\times \mathbb{R} \rightarrow \mathbb{R} \) is the truncation function defined for \(r\in(0,1]\) by

$$ f_{r}(t,u)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} f(t,r), & u\leq r, \\ f(t,u), & u>r. \end{array}\displaystyle \right . $$

\(f_{r}\) is a negative, continuous, and T-periodic function in t, which satisfies (H1)(iii). Let \(F_{r}(t,u):=\int_{1}^{u}f_{r}(t,s)\,ds\).

Take \(v\in H_{T}^{1}\) and multiply the two sides of the equality \(-u^{\prime \prime}-f_{r}(t;u)+e(t)=0\) by v and integrate from 0 to T

$$ \int_{0}^{T}\bigl[-u^{\prime\prime}(t)-f_{r} \bigl(t;u(t)\bigr)+e(t)\bigr]v(t)\,dt=0. $$

Due to the jump discontinuities of \(u^{\prime}\) at each \(t_{j}\), \(j=1,2,\ldots,p-1\), and since v is T-periodic and \(u^{\prime }(0)-u^{\prime }(T)=0\), the first term of (3.3) becomes

$$\begin{aligned} \int_{0}^{T}u^{\prime\prime}(t)v(t)\,dt =&\sum _{j=0}^{p}\int_{t_{j}}^{t_{j+1}}u^{\prime\prime }(t)v(t)\,dt \\ =&u^{\prime}(T)v(T)-u^{\prime}(0)v(0)-\sum _{j=1}^{p-1}\Delta u^{\prime}(t_{j})v(t_{j})- \int_{0}^{T}u^{\prime}(t)v^{\prime}(t)\,dt \\ =&-\sum_{j=1}^{p-1}\Delta u^{\prime }(t_{j})v(t_{j})-\int_{0}^{T}u^{\prime}(t)v^{\prime}(t)\,dt. \end{aligned}$$

Combining the above with (3.3) we obtain

$$ \sum_{j=1}^{p-1}\Delta u^{\prime }(t_{j})v(t_{j})+ \int_{0}^{T}u^{\prime}(t)v^{\prime }(t)\,dt- \int_{0}^{T}f_{r}\bigl(t;u(t)\bigr)\,dt+ \int_{0}^{T}e(t)]v(t)\,dt=0. $$

As a result, we introduce the concept of a weak solution for problem (3.2). We say that a function u is a weak solution of problem (3.2) if (3.4) holds for any \(v\in H_{T}^{1}\).

Hence, we define the energy functional \(\Phi_{r}\): \(H_{T}^{1}\rightarrow \mathbb{R} \), associated to (3.2) by

$$ \Phi_{r}(u):=\frac{1}{2}\int_{0}^{T}\bigl|u^{\prime }(t)\bigr|^{2}\,dt+ \sum_{j=1}^{p-1}\int_{0}^{u(t_{j})}I_{j}(s)\,ds-\int_{0}^{T}F_{r}\bigl(t,u(t)\bigr)\,dt+ \int_{0}^{T}e(t)u(t)\,dt. $$

Clearly, \(\Phi_{r}\) is well defined on \(H_{T}^{1}\). Combining the weak lower semicontinuity of the \(L^{2}\)-norm and Fatou’s lemma we infer that \(\Phi_{r}\) is weakly lower semi continuous, by means of the assumptions (H1)(i), (H2). Also, it is a differentiable functional whose derivative is the functional \(\Phi _{r}^{\prime}(u)\), given by

$$ \Phi_{r}^{\prime}(u)v=\int_{0}^{T}u^{\prime}(t)v^{\prime }(t)\,dt+ \sum_{j=1}^{p-1}I_{j} \bigl(u(t_{j})\bigr)v(t_{j})-\int _{0}^{T}f_{r}\bigl(t,u(t)\bigr)v(t)\,dt+\int _{0}^{T}e(t)v(t)\,dt. $$

Obviously, from (3.4), if \(u\in H_{T}^{1}\) is a critical point of the functional \(\Phi_{r}\), then u is a weak solution of problem (3.2). So, to obtain nonconstant weak solutions, it is sufficient to prove the existence of critical points of \(\Phi_{r}\), on the weakly closed subspace E of \(H_{T}^{1}\).

Now, we claim that \(\Phi_{r}\) is coercive on E. Indeed, the assumption (H1)(iii), implies that, for all \(\varepsilon\in(0,\min(1,\frac{\pi^{2}}{T^{2}}))\), there exists \(\delta _{\varepsilon}>0\) such that, for almost every \(t\in[0,T]\), we have

$$ \bigl\vert f_{r}(t,u)\bigr\vert \leq2\varepsilon u $$

whenever \(\vert u\vert >\delta _{\varepsilon }\). Using (3.2), (3.6), and (H1)(iii) we obtain for all \(u\in \mathbb{R} \) and a.e \(t\in[0,T]\)

$$ \bigl\vert F_{r}(t,u)\bigr\vert \leq\varepsilon u^{2}+ \max_{\vert u\vert \leq\delta_{\varepsilon}} \bigl\vert F_{r}(t;u)\bigr\vert - \varepsilon, $$

so that

$$ \int_{0}^{T}F_{r}\bigl(t,u(t)\bigr)\,dt \leq\varepsilon \int_{0}^{T}\bigl\vert u(t)\bigr\vert ^{2}\,dt+C_{\varepsilon}, $$

where \(C_{\varepsilon}=\int_{0}^{T}\max_{\vert u\vert \leq\delta} \vert F_{r}(t;u)\vert \,dt-T\varepsilon<+\infty\). Also, one can easily see that

$$ \int_{0}^{T}e(t)u(t)\,dt+\sum _{j=1}^{p-1}\int _{0}^{u(t_{j})}I_{j}(s)\,ds \geq \bigl( m(p-1)-\Vert e\Vert _{L^{1}} \bigr) \Vert u\Vert _{\infty}. $$

Thus, for \(u\in E\), by (H2) and the previous inequalities we obtain

$$\begin{aligned} \Phi_{r}(u) =&\frac{1}{2}\int_{0}^{T}\bigl|u^{\prime }(t)\bigr|^{2}\,dt- \int_{0}^{T}F_{r}\bigl(t,u(t)\bigr)\,dt+ \int _{0}^{T}e(t)u(t)\,dt+\sum _{j=1}^{p-1}\int _{0}^{u(t_{j})}I_{j}(s)\,ds \\ \geq&\frac{1}{2}\int_{0}^{T}\bigl|u^{\prime}(t)\bigr|^{2}\,dt- \varepsilon \int_{0}^{T}u(t)^{2}\,dt-C_{\varepsilon}T+ \bigl( m(p-1)-\Vert e\Vert _{L^{1}} \bigr) \Vert u\Vert _{\infty} \\ \geq&\frac{1}{2}\biggl(1-\frac{T^{2}}{2\pi^{2}}\varepsilon\biggr)\Vert u \Vert ^{2}+\sqrt{T} \bigl( m(p-1)-\Vert e\Vert _{L^{1}} \bigr) \Vert u\Vert -C\varepsilon T. \end{aligned}$$

So, \(\Phi_{r}(u)\rightarrow+\infty\) as \(\Vert u\Vert \rightarrow+\infty\), which shows that \(\Phi_{r}\) is coercive on E. Since E is a weakly closed subspace of \(H_{T}^{1}\), using the direct method of the calculus of variations, we see that there exists \(u^{\ast}\in E\) such that

$$ \Phi_{r}\bigl(u^{\ast}\bigr)=\inf_{E} \Phi_{r}. $$

Notice that by (H1)(i) and (H2)(i), we have \(\int _{0}^{T}f_{r}(t,0)\,dt=\int_{0}^{T}f(t,r)\,dt\leq 0<\int_{0}^{T}e(t)\,dt\), so that the function \(u\equiv0\) cannot be a solution of (3.2). Hence \(u^{\ast}\) is a nontrivial solution of (3.2). □

In the following, we shall show that \(u^{\ast}\) is a solution of (3.1). For this purpose we introduce the following auxiliary result.

Lemma 1

There exist \(r_{0}\in(0,1)\) and a constant \(\beta_{0}>0\) such that each solution u of (3.2) satisfies \(r_{0}\leq u(t)\leq\beta_{0}\), for all t. In particular, any T-periodic solution of (3.2) is a solution of (3.1).


Here, we shall use some ideas from [10].

We proceed by contradiction. Suppose, on the contrary, that, for each \(r\in (0,1)\) and for each constant \(\beta>0\), there exists a T-periodic solution u of (3.2) which satisfies

$$ u(t)< r\quad \mbox{or}\quad u(t)>\beta\quad\mbox{for some }t\in [ 0,T ] . $$

In particular, if for each integer \(n>1\) we consider \(r_{n}=\frac{1}{n}\) and \(\beta=n\), the above assumption implies that there exists a solution \(u_{n}\) of (3.2) for \(r=r_{n}\) such that

$$ \bigl\{ u_{n}(t);t\in \mathbb{R} \bigr\} \nsubseteqq [ r_{n},n ] . $$

We will show that this assumption leads to a contradiction.

First, we claim that for every \(n>1\) there must exist \(\tau _{n}\in[ 0,T ] \) such that

$$ u_{n}(\tau_{n})\in\biggl[\frac{1}{n},n\biggr]. $$

Indeed, suppose on the contrary that there exists a subsequence of \((u_{n})_{n}\), which we label the same, for which \(\min u_{n}(t)>n\). It follows from (H1)(iii) and the Fatou lemma that

$$\begin{aligned} (p-1)M \geq&\liminf_{n\rightarrow+\infty}\sum _{j=1}^{p-1} I_{j}\bigl(u_{n}(t_{j})\bigr) =\liminf _{n\rightarrow +\infty}\int_{0}^{T}\bigl(f_{r_{n}} \bigl(t,u_{n}(t)\bigr)-e(t)\bigr)\,dt\\ \geq&\int_{0}^{T}\liminf_{n\rightarrow+\infty} \bigl(f_{r_{n}}\bigl(t,u_{n}(t)\bigr)-e(t)\bigr)\,dt \end{aligned}$$


$$\begin{aligned} \int_{0}^{T}\liminf_{n\rightarrow+\infty} \bigl(f_{r_{n}}\bigl(t,u_{n}(t)\bigr)-e(t)\bigr)\,dt =&\int _{0}^{T} \liminf_{x\rightarrow+\infty} \bigl(f(t,x)-e(t)\bigr)\,dt\\ =&\int_{0}^{T}\lim_{x\rightarrow+\infty} \bigl(f(t,x)-e(t)\bigr)\,dt=-T\bar{e}, \end{aligned}$$

which leads to

$$ (p-1)M\geq-T\bar{e}. $$

This is a contradiction to (H2)(ii). Similarly, we will arrive at a contradiction with (H2), if we assume that \(\max u_{n}<\frac{1}{n}\). In fact, by the Fatou lemma we have

$$\begin{aligned} \limsup_{n\rightarrow+\infty}\sum_{j=1}^{p-1} I_{j}\bigl(u_{n}(t_{j})\bigr) =&\limsup _{n\rightarrow+\infty} \int_{0}^{T} \bigl( f_{r_{n}}\bigl(t,u_{n}(t) \bigr) -e(t)\bigr)\,dt\\ \leq&\int_{0}^{T}\limsup_{n\rightarrow+\infty} \bigl(f_{r_{n}}\bigl(t,u_{n}(t)\bigr)-e(t)\bigr)\,dt\\ \leq&\int_{0}^{T}\limsup_{x\rightarrow0^{+}} \bigl[ f(t,x)-e(t) \bigr]\,dt. \end{aligned}$$


$$ \limsup_{n\rightarrow+\infty}\sum _{j=1}^{p-1}I_{j} \bigl(u_{n}(t_{j})\bigr)\leq\int_{0}^{T} \lim_{x\rightarrow0^{+}} \bigl[ f(t,x)-e(t) \bigr]\,dt=-\infty. $$

This contradicts the assumption that \(I_{j}\) is bounded.

Next, we show that \(u_{n}\) is bounded from above. Since for all \(n>1\), \(u_{n} \) is a T-periodic solution of (3.2), \(\Phi_{r}^{\prime }(u_{n})=0\). Hence for all \(v\in H_{T}^{1}\) and for all \(n>1\) we have, for all ε, \(0<\varepsilon<1\),

$$ \Biggl\vert \int_{0}^{T} \bigl[ u_{n}^{{\prime }}(t)v^{{\prime}}(t)-f_{r_{n}} \bigl(t,u_{n}(t)\bigr)v(t)+e(t)v(t) \bigr] +\sum _{j=1}^{p-1}I_{j}\bigl(u_{n}(t_{j+sp}) \bigr)v(t_{j+sp})\Biggr\vert \leq \varepsilon \Vert v\Vert . $$

Taking \(v(t)\equiv-1\) in the above inequality, we obtain

$$\begin{aligned} &\biggl\vert \int_{0}^{T} \bigl[ f_{r_{n}}\bigl(t,u_{n}(t)\bigr)-e(t) \bigr]\,dt\biggr\vert - \Biggl\vert \sum_{j=1}^{p-1}I_{j}(u_{n}(t_{j}) \Biggr\vert \\ &\quad\leq\biggl\vert \int_{0}^{T} \bigl[ f_{r_{n}}\bigl(t,u_{n}(t)\bigr)-e(t) \bigr]\,dt-\sum _{j=1}^{p-1}I_{j}(u_{n}(t_{j}) \biggr\vert \leq\varepsilon\sqrt{T}. \end{aligned}$$


$$\begin{aligned} \biggl\vert \int_{0}^{T} \bigl[ f_{r_{n}}\bigl(t,u_{n}(t)\bigr)-e(t) \bigr]\,dt\biggr\vert \leq&\Biggl\vert \sum_{j=1}^{p-1}I_{j}(u_{n}(t_{j}) \Biggr\vert +\varepsilon\sqrt {T}\ \\ \leq&(p-1)\vert m\vert +\varepsilon\sqrt{T}. \end{aligned}$$

Now, from the above inequality, we get for all \(n>1\)

$$\begin{aligned} \int_{0}^{T}\bigl\vert f_{r_{n}}\bigl(t,u_{n}(t)\bigr)\bigr\vert \,dt \leq&\int _{0}^{T}\bigl\vert f_{r_{n}} \bigl(t,u_{n}(t)\bigr)-e(t)\bigr\vert \,dt+\int_{0}^{T} e(t)\,dt \\ =& \biggl\vert \int_{0}^{T} \bigl[ f_{r_{n}}\bigl(t,u_{n}(t)\bigr)-e(t) \bigr]\,dt\biggr\vert + \bar{e}T \\ \leq&(p-1)\vert m\vert +\varepsilon\sqrt{T}+T\bar{e}. \end{aligned}$$

Also, taking \(v=u_{n}\) in (3.11), we obtain

$$ \varepsilon \Vert u_{n}\Vert \geq\bigl\Vert u^{\prime }\bigr\Vert _{{L_{2}}}^{2}-\int_{0}^{T} \bigl[ f_{r_{n}}\bigl(t,u_{n}(t)\bigr)-e(t) \bigr] u_{n}(t)\,dt+\sum _{j=1}^{p-1}I_{j} \bigl(u_{n}(t_{j})\bigr)u_{n}(t_{j}). $$

Using (3.12) we get for all \(n >1\)

$$\begin{aligned} \biggl\vert \int_{0}^{T} \bigl[ f_{r_{n}}\bigl(t,u_{n}(t)\bigr)-e(t) \bigr] u_{n}(t)\,dt\biggr\vert \leq&\Vert u_{n}\Vert _{\infty } \biggl( \int_{0}^{T}\bigl\vert f_{r_{n}}\bigl(t,u_{n}(t)\bigr)\bigr\vert \,dt+\int _{0}^{T}\bigl\vert e(t)\bigr\vert \,dt \biggr) \\ \leq&\Vert u_{n}\Vert _{\infty} \bigl( \varepsilon\sqrt {T}+T\bar{e}+(p-1)|m|+\Vert e\Vert _{L^{1}} \bigr) . \end{aligned}$$

Thus (3.13) implies that

$$ \varepsilon \Vert u_{n}\Vert \geq \Vert u_{n}\Vert ^{2} -\Vert u_{n}\Vert _{\infty} \bigl( \varepsilon \sqrt{T}+T\bar{e}+\Vert e\Vert _{L^{1}}+(p-1)|m| \bigr) . $$

Wirtinger’s inequality (2.1) combined with (3.14) gives, for all \(n>1\),

$$ \varepsilon \Vert u_{n}\Vert \geq \Vert u_{n}\Vert ^{2}-\sqrt{\frac{T}{12}}\Vert u_{n}\Vert \bigl( \varepsilon \sqrt{T}+T\bar{e}+\Vert e\Vert _{L^{1}}+(p-1)|m| \bigr) . $$

We deduce that, for \(n >1\),

$$ \Vert u_{n}\Vert \leq\beta_{1}, $$


$$ \beta_{1}= \biggl( 1+\frac{T}{2\sqrt{3}} \biggr) +\sqrt{ \frac {T}{12}} \bigl( \Vert e\Vert _{L^{1}}+T\bar{e}+(p-1)|m| \bigr) . $$

Notice that \(\beta_{1}\) is independent of n. Hence \((u_{n})_{n}\) is bounded in \(H_{T}^{1}\). Since \(\Vert u_{n}\Vert _{\infty }\leq \sqrt{T}\Vert u_{n}\Vert \), we deduce that

$$ u_{n}(t)\leq\beta:=\beta_{1}\sqrt{T}. $$

Consequently, for n sufficiently large (\(n>\beta\)), for all \(t\in [ 0,T ] \), we have \(u_{n}(t)\leq n\). Furthermore, we cannot have \(u_{n}(t)\geq\frac{1}{n}\) for all \(t\in [ 0,T ] \); otherwise we would get \(\frac{1}{n}\leq u_{n}(t)\leq n\) for all \(t\in [ 0,T ] \) and this contradicts the assumption (3.10). Therefore, for n sufficiently large (\(n>\beta\)), there must exist a \(t_{n}^{\ast}\in [ 0,T ] \) such that \(u_{n}(t_{n}^{\ast})<\frac{1}{n}\). This means that \(t_{n}^{\ast}\in I_{\frac{1}{n}}\), where \(I_{\frac{1}{n}}\) is the set defined by

$$ I_{\frac{1}{n}}=\bigl\{ t\in [ 0,T ] ; u_{n}(t)< r_{n} \bigr\} . $$

Hence the set \(I_{\frac{1}{n}}\) is not empty. The continuity of the solution \(u_{n}\) at \(t=t_{n}^{\ast}\) implies that \(\operatorname {meas}(I_{\frac {1}{n}})>0\), which implies

$$ \int_{I_{\frac{1}{n}}}\bigl[f_{{r_{n}}}\bigl(t,u_{n}(t) \bigr)-e(t)\bigr]\,dt\neq0. $$

Now, consider the sets

$$\begin{aligned}& I_{1,\beta}= \bigl\{ t\in [ 0,T ] ; 1\leq u_{n}(t)\leq \beta \bigr\} , \end{aligned}$$
$$\begin{aligned}& I_{\frac{1}{n},1}= \bigl\{ t\in [ 0,T ] ; r_{n}\leq u_{n}(t)< 1 \bigr\} , \end{aligned}$$

so that we can write

$$ [ 0,T ] =I_{\frac{1}{n}}\cup I_{\frac{1}{n},1}\cup I_{1,\beta}. $$

Then integrating the differential equation in (3.2) from 0 to T we obtain

$$\begin{aligned} \Upsilon_{n} :=&\int_{0}^{T}-u_{n}^{\prime\prime }(t)\,dt= \int_{0}^{T}\bigl(f_{r_{n}} \bigl(t,u_{n}(t)\bigr)-e(t)\bigr)\,dt \\ =&\int_{I_{\frac{1}{n}}}\bigl[f_{{r_{n}}}\bigl(t,u_{n}(t) \bigr)-e(t)\bigr]\,dt \\ &{}+\int_{I_{\frac{1}{n},1}}\bigl[f_{{r_{n}}}\bigl(t,u_{n}(t) \bigr)-e(t)\bigr]\,dt+\int_{I_{1,\beta}}\bigl[f_{{r_{n}}} \bigl(t,u_{n}(t)\bigr)-e(t)\bigr]\,dt. \end{aligned}$$

(1) Assume we are integrating positively on all subintervals of \([ 0,T ] \).

If \(t\in I_{\frac{1}{n}}\) then \(u_{n}(t)< r_{n}\). It follows from (3.15) and (H1)(ii) that

$$ \int_{I_{\frac{1}{n}}}\bigl[f_{{r_{n}}}\bigl(t,u_{n}(t) \bigr)-e(t)\bigr]\,dt= \int_{I_{\frac{1}{n}}}\bigl[f(t,r_{n})-e(t) \bigr]\,dt< 0, $$

which yields

$$ \Upsilon_{n}< \int_{I_{\frac{1}{n},1}} \bigl[f_{{r_{n}}} \bigl(t,u_{n}(t)\bigr)-e(t)\bigr]\,dt+\int_{I_{1,\beta}} \bigl[f_{{r_{n}}}\bigl(t,u_{n}(t)\bigr)-e(t)\bigr]\,dt. $$

If \(t\in I_{1,\beta}\) then \(u_{n}(t)\in [1,\beta]\). This means that \(u_{n}(t)\) is bounded on \(I_{1,\beta}\), since \(f_{{r_{n}}}\) is continuous in u, then \(f_{{r_{n}}}\) is bounded almost everywhere in \(I_{1,\beta}\). Let

$$ C=C(\beta)=\max \bigl\{ \bigl\vert f_{{r_{n}}}(t,x)\bigr\vert ; t\in [ 0,T ] ,1\leq x\leq\beta \bigr\} . $$


$$ \biggl\vert \int_{I_{1,\beta}} \bigl[ f_{{r_{n}}} \bigl(t,u_{n}(t)\bigr)-e(t) \bigr]\,dt\biggr\vert \leq\int _{I_{1,\beta}}\bigl\vert f_{{r_{n}}}\bigl(t,u_{n}(t) \bigr)\bigr\vert +\bigl\vert e(t)\bigr\vert \,dt\leq T\bigl(C+\Vert e\Vert _{L^{1}}\bigr), $$

and (3.19) leads to

$$ \Upsilon_{n}\leq\int_{I_{\frac{1}{n},1}} \bigl[ f_{{r_{n}}} \bigl(t,u_{n}(t)\bigr)-\bar{e} \bigr]\,dt+T(C+\bar{e}). $$

By (H1)(ii), we see that, for every \(\sigma>0\), there exists \(\gamma_{\sigma}>0\) such that \(f(t,x)-\bar{e} <-\sigma\), for all \(x\in I_{\gamma_{\sigma}}:= ( 0,\gamma _{\sigma } ) \) and for every \(t\in [ 0,T ] \). Then, for n large enough (\(n>\beta\)), we have \(J:=I_{\frac{1}{n},1}\cap I_{\gamma _{\sigma }}\neq\emptyset\). Hence, (H1)(i) implies

$$ \int_{I_{\frac{1}{n},1}} \bigl[ f_{{r_{n}}}\bigl(t,u_{n}(t) \bigr)-\bar {e} \bigr]\,dt< \int_{J} \bigl[ f_{{r_{n}}}\bigl(t,u_{n}(t)\bigr)-\bar {e} \bigr]\,dt< -\sigma \operatorname {meas}(J). $$

Thus, for \(\sigma=\frac{1}{\operatorname {meas}(J)}n^{2}T(C+\bar{e})\), we obtain

$$\begin{aligned} \Upsilon_{n} < &\int_{I_{{\frac{1}{n},1}}} \bigl[ f_{{r_{n}}}\bigl(t,u_{n}(t)\bigr)-\bar{e} \bigr]\,dt+T(C+\bar{e}) \\ < &T(C+\bar{e}) \bigl(1-n^{2}\bigr)\mathop{\rightarrow}\limits_{n\rightarrow+\infty} - \infty. \end{aligned}$$

Then \(\Upsilon_{n}\) is not bounded.

(2) If we integrate negatively on all subintervals of \([ 0,T ] \) then, instead of (3.23), we get

$$ \int_{I_{{\frac{1}{n},1}}} \biggl[ f\biggl(t,u_{0}+ \frac {1}{n}\biggr)-\bar{e} \biggr]\,dt>\sigma \operatorname {meas}(I_{J}). $$

This, together with (3.20), leads to

$$ \Upsilon_{n}\rightarrow+\infty,\quad\mbox{as }n \rightarrow+\infty. $$

On the other hand, integrating the differential equation in (3.2) from 0 to T and using T-periodicity of \(u_{n}^{\prime}\), we obtain

$$\begin{aligned} \Upsilon_{n} =&-\int_{0}^{T}u_{n}^{\prime\prime }(t)\,dt=- \sum_{j=0}^{p}\int_{t_{j}^{+}}^{t_{j+1}^{-}}u_{n}^{\prime\prime}(t)\,dt \\ =&\sum_{j=1}^{p-1}\Delta u_{n}^{\prime }(t_{j})=\sum _{j=1}^{p-1}I_{j}\bigl(u_{n}(t_{j}) \bigr)\leq(p-1)M. \end{aligned}$$

Thus by (H2)

$$ \mbox{for each }n\in \mathbb{N} ^{\ast}, \Upsilon_{n}< 0 \mbox{ and }\Upsilon_{n}\mbox{ is bounded}. $$

We see that (3.26) contradicts (3.24) and (3.25). This contradiction shows that Lemma 1 is proved. In particular, Lemma 1 shows that there exists \(r\in(0,1)\) such that every T-periodic solution u of (3.2) is a solution of (3.1), since it satisfies \(u(t)\geq r\) for all \(t\in \mathbb{R} \) and \(f_{r}(t,u(t))=f(t,u(t))\), if \(u(t)\geq r\). Therefore \(u^{\ast}\) is a nonconstant T-periodic solution of (1.1), (1.2). This completes the proof of our main result. □

4 Example

Consider the impulsive singular problem

$$ \left \{ \textstyle\begin{array}{@{}l} u^{\prime\prime}(t)-\frac{e^{t}}{u^{\alpha}}=e(t),\quad \mbox{for } t\in(0,T), t\neq t_{j}, \\ \Delta u^{\prime}(t_{j})=I_{j}(u(t_{j})),\quad j=1, \\ u(0)-u(T)=0, \end{array}\displaystyle \right . $$

where \(\alpha>1\) and \(T>0\). Take \(I_{j}(s)=\cos s-2\), and \(e\in L^{2}([0;T],\mathbb{R} )\) such that \(\bar{e}<\frac{1}{T}\). In this case \(m=-3\) and \(M=-1\). Then (H1)-(H2) hold. Therefore, by Theorem 1, problem (4.1) has at least one nonconstant T-periodic solution.


  1. Nenov, SI: Impulsive controllability and optimization problems in population dynamics. Nonlinear Anal. 36, 881-890 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Gao, S, Chen, L, Nieto, JJ, Torres, A: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24, 6037-6045 (2006)

    Article  Google Scholar 

  3. Agarwal, RP, Franco, D, O’Regan, D: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math. 69, 83-96 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Book  MATH  Google Scholar 

  5. Li, W, Chang, Y, Nieto, JJ: Solvability of impulsive neutral evolution differential inclusions with state-dependent delay. Math. Comput. Model. 49, 1920-1927 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Li, J, Nieto, JJ: Existence of positive solutions for multipoint boundary value problem on the half-line with impulses. Bound. Value Probl. 2009, Article ID 834158 (2009)

    MathSciNet  Google Scholar 

  7. Ahmad, B, Nieto, JJ: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal. 69, 3291-3298 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chu, J, Nieto, JJ: Recent existence results for second order singular periodic differential equations. Bound. Value Probl. 2009, Article ID 540863 (2009)

    Article  MathSciNet  Google Scholar 

  9. Agarwal, RP, Perera, K, O’Regan, D: Multiple positive solutions of singular problems by variational methods. Proc. Am. Math. Soc. 134, 817-824 (2005)

    Article  MathSciNet  Google Scholar 

  10. Boucherif, A, Daoudi-Merzagui, N: Periodic solutions of singular nonautonomous second order differential equations. Nonlinear Differ. Equ. Appl. 15, 147-158 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Daoudi-Merzagui, N: Periodic solutions of nonautonomous second order differential equations with a singularity. Appl. Anal. 73, 449-462 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lazer, AC, Solimini, S: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 99, 109-114 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  14. Chen, L, Sun, J: Nonlinear boundary value problem for first-order impulsive functional differential equations. J. Math. Anal. Appl. 318, 726-741 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chen, L, Tisdell, CC, Yuan, R: On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl. 331, 233-244 (2007)

    MathSciNet  Google Scholar 

  16. Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10, 680-690 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Tian, Y, Ge, W: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 51, 509-527 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Sun, J, Chen, H, Nieto, JJ, Otero-Novoa, M: Multiplicity of solutions for perturbed second order Hamiltonian systems with impulsive effects. Nonlinear Anal. 72, 4575-4586 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sun, J, O’Regan, D: Impulsive periodic solutions for singular problems via variational methods. Bull. Aust. Math. Soc. 86, 193-204 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhang, H, Li, Z: Periodic and homoclinic solutions generated by impulses. Nonlinear Anal., Real World Appl. 12, 39-51 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references


The authors would like to thank an anonymous referee and Professor Ravi Agarwal for comments that led to the improvement of the presentation of the manuscript. Also, A Boucherif is grateful to King Fahd University of Petroleum and Minerals for its constant support.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Abdelkader Boucherif.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Daoudi-Merzagui, N., Boucherif, A. Nonconstant periodic solutions created by impulses for singular differential equations. Bound Value Probl 2015, 193 (2015).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: