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Nonconstant periodic solutions created by impulses for singular differential equations
Boundary Value Problems volume 2015, Article number: 193 (2015)
Abstract
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,
under impulse conditions
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 [5–7]. 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, 9–12] 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
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
and the norm
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
implies that, on E, we can obtain the equivalent norm
Also, for \(u\in E\) we have
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,
under the following assumptions:
-
(H1)
-
(i)
\(f: \mathbb{R} \times(0,+\infty)\rightarrow \mathbb{R} \), is a negative Carathéodory function which is T-periodic in its first argument,
-
(ii)
\(\lim_{u\rightarrow0^{+}}f(t,u)=-\infty\), for a.e. \(t \in [0,T]\),
-
(iii)
\(\lim_{u\rightarrow +\infty}f(t,u)=0\), for a.e. \(t \in[0,T]\),
-
(i)
-
(H2)
-
(i)
e is a locally integrable T-periodic function and \(\bar{e}:=\frac{1}{T}\int_{0}^{T}e(t)\,dt>0\),
-
(ii)
\(I_{j}:\mathbb{R} \rightarrow \mathbb{R} \), is a continuous bounded function for all \(j=1,\ldots,p-1\), such that \(m=\inf I_{j}<\sup I_{j}=M<-\frac{T}{p-1}\bar{e}\).
-
(i)
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.
Proof
To prove this result, we rely on a variational method. In order to study problem (3.1), we consider the following modified problem:
where \(f_{r}:[0,T]\times \mathbb{R} \rightarrow \mathbb{R} \) is the truncation function defined for \(r\in(0,1]\) by
\(f_{r}\) 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
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
Combining the above with (3.3) we obtain
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
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
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
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 \(t\in[0,T]\)
so that
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
Thus, for \(u\in E\), by (H2) and the previous inequalities we obtain
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
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).
Proof
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
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
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
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
and
which leads to
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
Hence
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, \(0<\varepsilon<1\),
Taking \(v(t)\equiv-1\) in the above inequality, we obtain
Then
Now, from the above inequality, we get for all \(n>1\)
Also, taking \(v=u_{n}\) in (3.11), we obtain
Using (3.12) we get for all \(n >1\)
Thus (3.13) implies that
Wirtinger’s inequality (2.1) combined with (3.14) gives, for all \(n>1\),
We deduce that, for \(n >1\),
where
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
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
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
Now, consider the sets
so that we can write
Then integrating the differential equation in (3.2) from 0 to T we obtain
(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
which yields
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
Then
and (3.19) leads to
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
Thus, for \(\sigma=\frac{1}{\operatorname {meas}(J)}n^{2}T(C+\bar{e})\), we obtain
Then \(\Upsilon_{n}\) is not bounded.
(2) If we integrate negatively on all subintervals of \([ 0,T ] \) then, instead of (3.23), we get
This, together with (3.20), leads to
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
Thus by (H2)
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
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.
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Acknowledgements
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.
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Daoudi-Merzagui, N., Boucherif, A. Nonconstant periodic solutions created by impulses for singular differential equations. Bound Value Probl 2015, 193 (2015). https://doi.org/10.1186/s13661-015-0458-6
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DOI: https://doi.org/10.1186/s13661-015-0458-6