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Periodic solutions for prescribed mean curvature p-Laplacian equations with a singularity of repulsive type and a time-varying delay
Advances in Difference Equations volume 2016, Article number: 178 (2016)
Abstract
In this article, the authors study the existence of positive periodic solutions for a prescribed mean curvature p-Laplacian equation with a singularity of repulsive type and a time-varying delay
where \(g\rightarrow-\infty\) when \(x\rightarrow0^{+}\). The existence of positive periodic solutions conditions is devised by using the coincidence degree theory and some analysis methods. A numerical example demonstrates the validity of the main results.
1 Introduction
The problems of periodic solution have been studied widely for some types of differential equations with a singularity (see [1–8] and the references therein). For example, in [2], Zhang studied periodic solutions for the following Liénard equation with a singularity:
where \(f: {\mathbb {R}}\rightarrow{ \mathbb {R}}\), \(g: {\mathbb {R}}\times(0,+\infty )\rightarrow{ \mathbb {R}}\) is an \(L^{2}\)-Carathéodory function, \(g(t,x)\) is a T-periodic function in the first argument and can be singular at \(x=0\), i.e., \(g(t,x)\) can be unbounded as \(x\rightarrow0^{+}\).
On the basis of work of Zhang, Wang in [8] further studied periodic solutions for the Liénard equation with a singularity and a deviating argument, which is different from the literature [2],
where \(0\leq\sigma< T\) is a constant, \(f: {\mathbb {R}}\rightarrow{ \mathbb {R}}\), \(g: {\mathbb {R}}\times(0,+\infty)\rightarrow{ \mathbb {R}}\) is an \(L^{2}\)-Carathéodory function, \(g(t,x)\) is a T-periodic function in the first argument and can be singular at \(x=0\), i.e., \(g(t,x)\) can be unbounded as \(x\rightarrow0^{+}\).
Nowadays, the prescribed mean curvature equation and its modified forms, which arise from some problems associated with differential geometry and physics such as combustible gas dynamics, have been studied widely (see [9–12] and the references therein). Moreover, we note that the existence of periodic solutions for the prescribed curvature mean equations has attracted much attention from researchers. In [13], Feng considered a kind of prescribed mean curvature Liénard equation
where \(\tau,e \in C({\mathbb {R}},{\mathbb {R}}) \) are T-periodic, and \(g\in C({\mathbb {R}}\times{ \mathbb {R}},{\mathbb {R}})\) is T-periodic in the first argument, \(T>0\) is a constant. By applying Mawhin’s continuation theorem and given some sufficient conditions, the author showed that equation (1.1) has at least one periodic solution.
On the basis of work of Feng, various types of prescribed curvature mean equations have been studied (see [14–17] and the references therein). But, to the best of our knowledge, the study of positive periodic solutions for the prescribed mean curvature equation with a singularity is relatively infrequent. This is due to the fact that the mechanism on which how the solution is influenced by the singularity and the nonlinear term \((\frac{u'(t)}{\sqrt{1+(u'(t))^{2}}})'\) associated to prescribed mean curvature equation is far away from clear.
To address this issue, recently, Lu and Kong in [18] studied periodic solutions for a kind of prescribed mean curvature Liénard equation with a singularity and a deviating argument:
where \(0 \leq\sigma< T\), \(g:(0,+\infty)\rightarrow \mathbb {R}\) is a continuous function and can be singular at \(u=0\). However, \(\sigma =kT\), k is an integer. If \(\sigma\neq kT\), it is difficult to estimate a priori bounds of periodic solutions by using method in [18]. Therefore, it is significant to consider time-varying delay for the prescribed mean curvature equations.
Inspired by the above facts, in this paper, we consider the following prescribed mean curvature Duffing-type equation with a singularity of repulsive type and a time-varying delay:
where \(g: [0,T]\times(0,+\infty)\times(0,+\infty)\rightarrow{ \mathbb {R}}\) is a continuous function. g can be singular at \(x=0\), i.e., g can be unbounded as \(x\rightarrow0^{+}\). \(\tau, p\in(\mathbb {R}, \mathbb {R})\) are T-periodic with \(\int_{0}^{T} p(t)\,dt=0\), β is a constant. By applying Mawhin’s continuation theorem, we prove that equation (1.3) has at least one positive T-periodic solution. So, our research is meaningful and feasible.
The rest of the paper is organized as follows. In Section 2, some necessary definitions and lemmas are introduced. The existence of periodic solutions conditions is presented in Section 3. A numerical example is illustrated to show the validity of the proposed method in Section 4.
2 Preliminary
First, we recall the following definition and lemmas.
Definition 2.1
Let X and Y be two Banach spaces with norms \(\Vert \cdot \Vert _{X}\), \(\Vert \cdot \Vert _{Y}\), respectively. A linear operator
is said to be a Fredholm operator of index zero provided that
-
(i)
ImL is a closed subset of Y,
-
(ii)
\(\dim \ker L=\operatorname {codim}\operatorname {Im}L<\infty\).
Definition 2.2
Let X and Y be two Banach spaces with norms \(\Vert \cdot \Vert _{X}\), \(\Vert \cdot \Vert _{Y}\), respectively, \(\Omega\subset X\) be an open and bounded set;
is a Fredholm operator of index zero, and we have a continuous operator
being L-compact in Ω̄ provided that
-
(I)
\(K_{p}(I-Q)N(\bar{\Omega})\) is a relative compact set of X,
-
(II)
\(QN (\bar{\Omega})\) is a bounded set of Y,
where we denote \(X_{1}=\ker L\), \(Y_{2}=\operatorname {Im}L\). Then we have the decompositions \(X=X_{1}\oplus X_{2}\), \(Y=Y_{1}\oplus Y_{2}\), and we let
be continuous linear projectors (meaning \(P^{2}=P\) and \(Q^{2}=Q\)), and \(K_{p}=L\vert^{-1}_{\ker P\cap D(L)}\).
Lemma 2.1
[19]
Let X and Y be two real Banach spaces, \(L: D(L)\subset X \rightarrow Y\) be a Fredholm operator with index zero, \(\Omega\subset X\) be an open bounded set, and \(N:\bar{\Omega}\subset X\rightarrow Y\) be L-compact on Ω̄. Suppose that all of the following conditions hold:
-
(1)
\(Lx\neq\lambda Nx\), \(\forall x \in\partial\Omega\cap D(L)\), \(\forall \lambda\in(0,1)\);
-
(2)
\(QNx\neq0\), \(\forall x\in\partial\Omega\cap\ker L\);
-
(3)
\(\deg\{JQN,\Omega\cap\ker L,0\}\neq0\), where \(J: \operatorname {Im}Q\rightarrow\ker L\) is a homeomorphism map.
Then the equation \(Lx=Nx\) has at least one solution on \(D(L)\cap\bar {\Omega}\).
In order to use Lemma 2.1, let us consider the following system:
where \(\varphi_{q}(s)=\vert s^{q-2}\vert s\), \(\frac{1}{p}+\frac{1}{q}=1 \), \(x_{2}(t)=\varphi_{p} (\frac{x'_{1}(t)}{\sqrt {1+(x'_{1}(t))^{2}}} )=\phi^{-1}(x'_{1}(t))\). Obviously, if \(x(t)=(x_{1}(t), x_{2}(t))^{\top}\) is a solution of (2.1), then \(x_{1}(t)\) is a solution of (1.3).
Let
where the normal \(\Vert x\Vert =\max\{\vert x_{1}\vert _{0},\vert x_{2}\vert _{0}\}\), and \(\vert x_{1}\vert _{0}=\max_{t\in[0,T]}\vert x_{1}(t)\vert \), \(\vert x_{2}\vert _{0}= \max_{t\in[0,T]}\vert x_{2}(t)\vert \). It is obvious that X and Y are Banach spaces. Furthermore, for \(\varphi\in C_{T}\), \(\Vert \varphi \Vert _{r}= (\int^{T}_{0}\vert \varphi (t)\vert ^{r}\,dt )^{\frac{1}{r}}\), \(r>1\).
Now we define the operator L
where \(D(L)=\{x\mid x=(x_{1}(t),x_{2}(t))^{\top}\in C^{1}({\mathbb {R}}, {\mathbb {R}}^{2}), x(t)=x(t+T)\}\).
Define a nonlinear operator \(N:\bar{\Omega}\subset X\rightarrow Y\)
Then problem (2.1) can be written as \(Lx=Nx\) in Ω̄.
We know
then \(x'_{1}(t)=0\), \(x'_{2}(t)=0\), obviously \(x_{1}\in \mathbb {R}\), \(x_{2}\in {\mathbb {R}}\), thus \(\ker L={\mathbb {R}}^{2}\), and it is also easy to prove that \(\operatorname {Im}L= \{y\in Y, \int_{0}^{T}y(s)\,ds=0 \}\), so L is a Fredholm operator of index zero.
Let
and
Let \(K_{p}=L\vert^{-1}_{\ker p\cap D(L)}\), then it is easy to see that
where
It implies that \(\forall\Omega\subset X\) is an open and bounded set with \(\bar{\Omega} \subset X\), \(K_{p}(I-Q)N(\bar{\Omega})\) is a relative compact set of X, \(QN (\bar{\Omega})\) is a bounded set of Y, so the operator N is L-compact in Ω̄.
3 Main results
Theorem 3.1
For problem (1.3), assume the following conditions hold:
- \((g_{1})\) :
-
(Balance condition) There exist positive constants \(A_{1}\) and \(A_{2}\) with \(A_{1}< A_{2}\) such that if x is a positive continuous T-periodic function satisfying \(\int_{0}^{T} g(t,x(t),x(t-\tau(t)))\,dt=0\), then
$$A_{1}\leq x(\varepsilon)\leq A_{2} $$for some \(\varepsilon\in[0,T]\).
- \((g_{2})\) :
-
(Degree condition) \(\overline{g}(x)<0\) for all \(x\in(0,A_{1})\) and \(\overline{g}(x)>0\) for all \(x>A_{2}\), where \(\overline{g}(x)=\frac{1}{T} \int_{0}^{T} g(t,x(t),x(t-\tau(t)))\,dt\), \(x>0\).
- \((g_{3})\) :
-
(Decomposition condition) \(g(t,x(t),x(t-\tau (t)))=g_{1}(t,x(t-\tau(t)))+g_{0}(x(t))\), where \(g_{0}\) is a continuous function and there exist positive constants \(a_{i}\), \(c_{i}\), \(i=1, 2\), and b such that
$$g \bigl(t,x(t),x \bigl(t-\tau(t) \bigr) \bigr)\leq a_{1}x(t)+a_{2}x \bigl(t-\tau(t) \bigr)+b, \quad (t,x)\in [0,T]\times(0,+\infty). $$Meanwhile, \(\vert g_{1}(t,x)\vert \leq c_{1}x+c_{2}\).
- \((g_{4})\) :
-
(Strong force condition at \(x=0\)) \(\int_{0}^{1} g_{0}(x)\,dx=-\infty\).
- \((g_{5})\) :
-
\(B:= ( \int_{0}^{T}\vert p(t)\vert ^{2}\,dt )^{\frac {1}{2}}+\sup_{t\in[0,T]}\vert p(t)\vert <+\infty\), \(\vert \beta \vert >c_{1}T\), and
$$\vert \beta \vert M_{1}+T \bigl[2(a_{1}+a_{2})M_{1}+2b+B \bigr]< 1, $$where \(M_{1}= A_{2}+ \frac{c_{1}A_{2}T+c_{2}T+B\sqrt{T}}{ \vert \beta \vert -c_{1}T}\).
Then equation (1.3) has at least one positive T-periodic solution.
Proof
Let \(\Omega_{1}=\{x\in\bar{\Omega},Lx=\lambda Nx,\forall\lambda \in(0,1)\}\). If, \(\forall x\in\Omega_{1}\), we have
where \(v(t)=\phi^{-1} (\frac{u'(t)}{\lambda} )=\varphi _{p} (\frac{\frac{1}{\lambda}u'(t)}{\sqrt{1+\frac {(u'(t))^{2}}{\lambda^{2}}}} )\).
Integrating the second equation of (3.1) from 0 to T, we have
Combining with \((g_{1})\), we can see that there exist positive constants \(A_{1}\), \(A_{2}\), and \(\varepsilon\in[0,T]\) such that
Therefore, we have
Multiplying the second equation of (3.1) by \(u'(t)\) and integrating on the interval \([0,T]\), we can get
It follows from \((g_{3})\) that
i.e.,
which together with (3.4) gives
It follows from \(\vert \beta \vert >aT\) that
Substituting (3.6) into (3.4), we obtain
Furthermore, from the second equation of (3.1), we can get
Write
Then we can get from (3.2) and \((g_{3})\)
Substituting (3.9) into (3.8) and combining with (3.6) and (3.7), we obtain
Integrating the first equation of (3.1) on the interval \([0,T]\), we have
Then we can see that there exists \(\eta\in[0,T]\) such that \(v(\eta )=0\). It implies that
which combining with (3.10) gives
Since \(\vert \beta \vert M_{1}+T(2(a_{1}+a_{2})M_{1}+2b+B)<1\), we have
From (3.1), we can also have
On the other hand, from the second equation of (3.1) and by \((g_{3})\), we can see that
Take \(\omega\in[0,T]\), then
Multiplying both sides of equation (3.13) by \(u'(\omega)\), we have
Let \(\varepsilon\in[0,T]\) be as in (3.3). For any \(\omega\in [\varepsilon,T]\), integrating equation (3.14) on the interval \([\varepsilon,T]\), we have
where \(G_{M_{1}}=\max_{\vert u\vert \leq M_{1}}g_{1}(t,u)\). It follows from (3.16) that
According to \((g_{4})\), we can see that there exists a constant \(M_{2}>0\) such that, for \(\omega\in[\varepsilon,T]\),
For the case \(\omega\in[0,\varepsilon]\), we can handle it similarly. Thus, we have
Let us define
Then by (3.3), (3.7), and (3.18) we can obtain
Set
Then the condition (1) of Lemma 2.1 is satisfied. Suppose that there exists \(x\in\partial\Omega\cap\ker L\) such that \(QN x=\frac{1}{T}\int_{0}^{T}{N}x(s)\,ds=(0,0)^{\top}\), i.e.,
Since \(\ker L=\mathbb {R}^{2}\), and \(u\in{ \mathbb {R}}\), \(v\in{ \mathbb {R}}\) are constant, combining with the first equation of (3.20), we obtain
From the second equation of (3.20), we have
From \((g_{1})\) we can see that
which contradicts the assumption \(x\in\partial\Omega\). So, for all \(x\in\ker L\cap\partial\Omega\), we have \(QN x\neq 0\). Then the condition (2) of Lemma 2.1 is satisfied.
In the following, we prove that the condition (3) of Lemma 2.1 is also satisfied.
Let
then we have
Define \(J:\operatorname {Im}Q\rightarrow\ker L\) to be a linear isomorphism with
and define
Then
Now we claim that \(H(\mu,x)\) is a homotopic mapping. Assume, by way of contradiction, that there exist \(\mu_{0}\in[0,1]\) and \(x_{0}=\bigl ({\scriptsize\begin{matrix}{}u_{0}\cr v_{0}\end{matrix}} \bigr ) \in\partial\Omega\) such that \(H(\mu_{0},x_{0})=0\).
Substituting \(\mu_{0}\) and \(x_{0}\) into (3.21), we have
Since \(H(\mu_{0},x_{0})=0\), we can see that
Combining with \(\mu_{0}\in[0,1]\), we obtain \(v_{0}=0\). Thus \(u_{0}=A_{1}\) or \(A_{2}\).
If \(u_{0}=A_{1}\), it follows from \((g_{2})\) that \(g (u_{0})<0\), then substituting \(v_{0}=0\) into (3.22), we have
If \(u_{0}=A_{2}\), it follows from \((g_{2})\) that \(g (u_{0})>0\), then substituting \(v_{0}=0\) into (3.22), we have
Combining with (3.23) and (3.24), we can see that \(H(\mu _{0},x_{0})\neq0\), which contradicts the assumption. Therefore \(H(\mu,x)\) is a homotopic mapping and \(x^{\top}H(\mu,x)\neq0\), \(\forall(x,\mu )\in(\partial\Omega\cap\ker L)\times[0,1]\). Then
Thus, the condition (3) of Lemma 2.1 is also satisfied. Therefore, by applying Lemma 2.1, we can conclude that equation (1.3) has at least one positive T-periodic solution. □
4 Numerical example
In this section, we provide an example to illustrate results from the previous sections.
Example 4.1
As an application, we consider the following example:
Conclusion
Problem (4.1) has at least one positive \(\frac{\pi}{50}\)-periodic solution.
Proof
Corresponding to Theorem 3.1 and (1.3), we have
then we can have and choose
and \(B:= (\int_{0}^{T}\vert p(t)\vert ^{2}\,dt )^{\frac{1}{2}}+\sup_{t\in[0,T]} \vert p(t)\vert <\frac{1}{36} <+\infty\). Then we can see that \((g_{1})\)-\((g_{4})\) hold. Moreover, \(\vert \beta \vert >c_{1}T\) and
Hence, by applying Theorem 3.1, we can see that equation (4.1) has at least one positive T-periodic solution. □
Remark 4.1
Since all the results in [1–19] and the references therein cannot be applicable to equation (4.1) for solving positive periodic solutions, Theorem 3.1 in this paper is essentially new.
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Acknowledgements
The authors express their thanks to the referee for his (or her) valuable suggestions. This work was supported by Natural Science Foundation of Wu Yi University (No. XQ201305) and young-middle-aged teachers education scientific research project in Fujian province (No. JA15524).
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Chen, W., Kong, F. Periodic solutions for prescribed mean curvature p-Laplacian equations with a singularity of repulsive type and a time-varying delay. Adv Differ Equ 2016, 178 (2016). https://doi.org/10.1186/s13662-016-0879-2
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DOI: https://doi.org/10.1186/s13662-016-0879-2
Keywords
- prescribed mean curvature equation
- coincidence degree theory
- periodic solutions
- singularity
- delay