Abstract
In this paper, we consider the following p-Laplacian Liénard type differential equation with singularity and deviating argument:
By applications of coincidence degree theory and some analysis techniques, sufficient conditions for the existence of positive periodic solutions are established.
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1 Introduction
In this paper, we consider the following p-Laplacian Liénard type differential equation with singularity and deviating argument:
where \(\varphi_{p}:\mathbb{R}\rightarrow\mathbb{R}\) is given by \(\varphi _{p}(s)=\vert s\vert ^{p-2}s\), here \(p>1\) is a constant, f is continuous function; g is a continuous function defined on \(\mathbb{R}^{2}\) and periodic in t with \(g(t,\cdot)=g(t+T,\cdot)\), g has a singularity at \(x=0\); σ is a constant and \(0\leq \sigma< T\); \(e:\mathbb{R}\rightarrow\mathbb{R}\) are continuous periodic functions with \(e(t+T)\equiv e(t)\) and \(\int^{T}_{0}e(t)\,dt=0\).
As is well known, the existence of periodic solutions for Liénard type differential equations was extensively studied (see [1–10] and the references therein). In recent years, there also appeared some results on a Liénard type differential equation with singularity; see [11, 12]. In 1996, using coincidence degree theory, Zhang considered the existence of T-periodic solutions for the scalar Liénard equation
when g becomes unbounded as \(x\rightarrow0^{+}\). The main emphasis was on the repulsive case, i.e. when \(g(t,x)\rightarrow+\infty \), as \(x\rightarrow0^{+}\). Afterwards, Wang [12] studied the existence of periodic solutions of the Liénard equation with a singularity and a deviating argument,
where σ is a constant. When g has a strong singularity at \(x = 0\) and satisfies a new small force condition at \(x =\infty\), the author proved that the given equation has at least one positive T-periodic solution.
However, the Liénard type differential equation (1.1), in which there is a p-Laplacian Liénard type differential equation, has not attracted much attention in the literature. There are not so many existence results for (1.1) even as regards the p-Laplacian Liénard type differential equation with singularity and deviating argument. In this paper, we try to fill this gap and establish the existence of a positive periodic solution of (1.1) using coincidence degree theory. Our new results generalize in several aspects some recent results contained in [11, 12].
2 Preparation
Let X and Y be real Banach spaces and \(L:D(L)\subset X\rightarrow Y\) be a Fredholm operator with index zero, here \(D(L)\) denotes the domain of L. This means that ImL is closed in Y and \(\dim \operatorname {Ker}L=\dim(Y/\operatorname {Im}L)<+\infty\). Consider supplementary subspaces \(X_{1}\), \(Y_{1}\) of X, Y, respectively, such that \(X=\operatorname {Ker}L \oplus X_{1}\), \(Y=\operatorname {Im}L\oplus Y_{1}\). Let \(P:X\rightarrow \operatorname {Ker}L\) and \(Q:Y\rightarrow Y_{1}\) denote the natural projections. Clearly, \(\operatorname {Ker}L\cap(D(L)\cap X_{1})=\{0\}\) and so the restriction \(L_{P}:=L|_{D(L)\cap X_{1}}\) is invertible. Let K denote the inverse of \(L_{P}\).
Let Ω be an open bounded subset of X with \(D(L)\cap\Omega\neq\emptyset\). A map \(N:\overline{\Omega}\rightarrow Y\) is said to be L-compact in Ω̅ if \(QN(\overline{\Omega})\) is bounded and the operator \(K(I-Q)N:\overline{\Omega}\rightarrow X\) is compact.
Lemma 2.1
(Gaines and Mawhin [13])
Suppose that X and Y are two Banach spaces, and \(L:D(L)\subset X\rightarrow Y\) is a Fredholm operator with index zero. Let \(\Omega\subset X\) be an open bounded set and \(N:\overline{\Omega}\rightarrow Y \) be L-compact on Ω̅. Assume that the following conditions hold:
-
(1)
\(Lx\neq\lambda Nx\), \(\forall x\in\partial\Omega\cap D(L)\), \(\lambda\in(0,1)\);
-
(2)
\(Nx\notin \operatorname {Im}L\), \(\forall x\in\partial\Omega\cap \operatorname {Ker}L\);
-
(3)
\(\deg\{JQN,\Omega\cap \operatorname {Ker}L,0\}\neq0\), where \(J:\operatorname {Im}Q\rightarrow \operatorname {Ker}L\) is an isomorphism.
Then the equation \(Lx=Nx\) has a solution in \(\overline{\Omega}\cap D(L)\).
For the sake of convenience, throughout this paper we will adopt the following notation:
Lemma 2.2
([14])
If \(\omega\in C^{1}(\mathbb{R},\mathbb{R})\) and \(\omega(0)=\omega(T)=0\), then
where \(1\leq p<\infty\), \(\pi_{p}=2\int^{(p-1)/p}_{0}\frac{ds}{(1-\frac{s^{p}}{p-1})^{1/p}}=\frac {2\pi(p-1)^{1/p}}{p\sin(\pi/p)}\).
Lemma 2.3
If \(x\in C^{1}(\mathbb{R},\mathbb{R})\) with \(x(t+T)=x(t)\), and \(t_{0}\in[0,T]\) such that \(\vert x(t_{0})\vert < d\), then
Proof
Let \(\omega(t)=x(t+t_{0})-x(t_{0})\), and then \(\omega(0)=\omega(T)=0\). By Lemma 2.2 and Minkowski’s inequality, we have
This completes the proof of Lemma 2.3. □
In order to apply the topological degree theorem to study the existence of a positive periodic solution for (1.1), we rewrite (1.1) in the form
where \(\frac{1}{p}+\frac{1}{q}=1\). Clearly, if \(x(t)=(x_{1}(t),x_{2}(t))^{\top}\) is an T-periodic solution to (2.1), then \(x_{1}(t)\) must be an T-periodic solution to (1.1). Thus, the problem of finding an T-periodic solution for (1.1) reduces to finding one for (2.1).
Now, set \(X=Y=\{x=(x_{1}(t),x_{2}(t))\in C^{1}(\mathbb{R},\mathbb{R}^{2}): x(t+T)\equiv x(t)\}\) with the norm \(\Vert x\Vert =\max\{\vert x_{1}\vert _{\infty}, \vert x_{2}\vert _{\infty}\}\). Clearly, X and Y are both Banach spaces. Meanwhile, define
by
and \(N: X\rightarrow Y\) by
Then (2.1) can be converted to the abstract equation \(Lx=Nx\). From the definition of L, one can easily see that
So L is a Fredholm operator with index zero. Let \(P:X\rightarrow \operatorname {Ker}L\) and \(Q:Y\rightarrow \operatorname {Im}Q\subset\mathbb {R}^{2}\) be defined by
then \(\operatorname {Im}P=\operatorname {Ker}L\), \(\operatorname {Ker}Q=\operatorname {Im}L\). Let K denote the inverse of \(L|_{\operatorname {Ker}p\cap D(L)}\). It is easy to see that \(\operatorname {Ker}L=\operatorname {Im}Q=\mathbb{R}^{2}\) and
where
From (2.2) and (2.3), it is clear that QN and \(K(I-Q)N\) are continuous, \(QN(\overline{\Omega})\) is bounded and then \(K(I-Q)N(\overline{\Omega})\) is compact for any open bounded \(\Omega\subset X\), which means N is L-compact on Ω̅.
3 Main results
Assume that
exists uniformly a.e. \(t\in[0,T]\), i.e., for any \(\varepsilon>0\) there is \(g_{\varepsilon}\in L^{2}(0,T)\) such that
for all \(x>0\) and a.e. \(t\in[0,T]\). Moreover, \(\psi\in C(\mathbb{R},\mathbb{R})\) and \(\psi(t+T)=\psi(t)\).
For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:
(H1) (Balance condition) There exist constants \(0< D_{1}< D_{2}\) such that if x is a positive continuous T-periodic function satisfying
then
for some \(\tau\in[0,T]\).
(H2) (Degree condition) \(\bar{g}(x)<0\) for all \(x \in(0,D_{1})\), and \(\bar{g}(x)>0\) for all \(x>D_{2}\).
(H3) (Decomposition condition) \(g(t,x)=g_{0}(x)+g_{1}(t,x)\), where \(g_{0}\in C((0,\infty);\mathbb{R}) \) and \(g_{1}:[0,T]\times[0,\infty)\rightarrow\mathbb{R}\) is an \(L^{2}\)-Carathéodory function, i.e. it is measurable in the first variable and continuous in the second variable, and for any \(b>0\) there is \(h_{b}\in L^{2}(0,T;\mathbb{R}_{+})\) such that
(H4) (Strong force condition at \(x=0\)) \(\int^{1}_{0}g_{0}(x)\,dx=-\infty\).
Theorem 3.1
Assume that conditions (H1)-(H4) hold. Suppose the following condition is satisfied:
(H5) \((\frac{T}{\pi_{p}} )^{p}\vert \psi \vert _{\infty}<1\).
Then (1.1) has at least one positive T-periodic solution.
Proof
Consider the equation
Set \(\Omega_{1}=\{x:Lx=\lambda Nx,\lambda\in (0,1)\}\). If \(x(t)=(x_{1}(t),x_{2}(t))^{\top}\in\Omega_{1}\), then
Substituting \(x_{2}(t)=\frac{1}{\lambda^{p-1}}\varphi_{p}(x_{1}'(t))\) into the second equation of (3.3)
Integrating both sides of (3.4) over \([0,T]\), we have
From (H1), there exist positive constants \(D_{1}\), \(D_{2}\), and \(\xi\in[0,T]\) such that
Then we have
and
Combining the above two inequalities, we obtain
Multiplying both sides of (3.4) by \(x_{1}(t)\) and integrating over the interval \([0,T]\), we get
Substituting \(\int^{T}_{0}(\varphi_{p}(x_{1}'(t)))'x_{1}(t)\,dt=-\int^{T}_{0}\vert x_{1}'(t)\vert ^{p}\,dt\), \(\int ^{T}_{0}f(x_{1}(t))x_{1}'(t)x_{1}(t)\,dt=0\) into (3.8), we have
For any \(\varepsilon>0\), there exists a function \(g_{\varepsilon}\in L^{2}(0,T)\) such that (3.2) holds. Since \(x_{1}(t)>0\), \(t\in[0,T]\), it follows from (3.4) that
We infer from (3.9) and (3.10)
From Lemma 2.3 and (3.7), we have
Substituting (3.7), (3.12) into (3.11), we get
where \(\vert g_{\varepsilon} \vert _{2}= (\int^{T}_{0}\vert g_{\varepsilon}(t)\vert ^{2}\,dt )^{\frac{1}{2}}\). Since ε is sufficiently small, from (H5) we know that \((\frac{T}{\pi_{p}} )^{p}\vert \psi \vert _{\infty}<1\). So, it is easy to see that there exists a positive constant \(M'_{1}\) such that
From (3.7), we have
Write
Then we get from (3.2) and (3.6)
By the second equations of (3.3) and (3.15), we obtain
where \(\vert f\vert _{M_{1}}=\max_{0< x_{1}\leq M_{1}}\vert f(x_{1}(t))\vert \). By the first equation of (3.3), we have
which implies that there is a constant \(t_{2}\in[0,T]\) such that \(x_{2}(t_{2})=0\), so
On the other hand, it follows from (3.4) that
Namely,
Multiplying both sides of (3.19) by \(x_{1}'(t)\), we get
Let \(\tau\in[0,T]\), for any \(\tau\leq t\leq T\), we integrate (3.20) on \([\tau, t]\) and get
By (3.14), (3.15), (3.16), (3.17), and (3.18), we have
We have
where \(g_{M_{1}}=\max_{0\leq x\leq M_{1}}\vert g_{1}(t,x)\vert \in L^{2}(0,T)\) is as in (H3). We have
From these inequalities we can derive from (3.21) that
for some constant \(M_{3}'\) which is independent on λ, x, and t. In view of the strong force condition (H4), we know that there exists a constant \(M_{3}>0\) such that
The case \(t\in[0,\tau]\) can be treated similarly.
From (3.14), (3.17), and (3.23), we let
where \(0< E_{1}<\min(M_{3}, D_{1})\), \(E_{2}>\max(M_{1}, D_{2}) \), \(E_{3}>M_{2}\). \(\Omega_{2}=\{x:x\in\partial\Omega\cap \operatorname {Ker}L\}\) then \(\forall x\in \partial\Omega\cap \operatorname {Ker}L\)
If \(QNx=0\), then \(x_{2}(t)=0\), \(x_{1}=E_{2}\) or \(-E_{2}\). But if \(x_{1}(t)=E_{2}\), we know
From assumption (H2), we have \(x_{1}(t)\leq D_{2}\leq E_{2}\), which yields a contradiction. Similarly if \(x_{1}=-E_{2}\). We also have \(QNx\neq0\), i.e., \(\forall x\in\partial\Omega\cap \operatorname {Ker}L\), \(x\notin \operatorname {Im}L\), so conditions (1) and (2) of Lemma 2.1 are both satisfied. Define the isomorphism \(J:\operatorname {Im}Q\rightarrow \operatorname {Ker}L\) as follows:
Let \(H(\mu,x)=-\mu x+(1-\mu)JQNx\), \((\mu,x)\in[0,1]\times\Omega\), then \(\forall (\mu,x)\in(0,1)\times(\partial\Omega\cap \operatorname {Ker}L)\),
We have \(\int^{T}_{0}e(t)\,dt=0\). So, we can get
From (H2), it is obvious that \(x^{\top}H(\mu,x)<0\), \(\forall (\mu,x)\in(0,1)\times(\partial\Omega\cap \operatorname {Ker}L)\). Hence
So condition (3) of Lemma 2.1 is satisfied. By applying Lemma 2.1, we conclude that the equation \(Lx=Nx\) has a solution \(x=(x_{1},x_{2})^{\top}\) on \(\bar{\Omega}\cap D(L)\), i.e., (2.1) has an T-periodic solution \(x_{1}(t)\). □
Finally, we present an example to illustrate our result.
Example 3.1
Consider the p-Laplacian Liénard type differential equation with singularity and deviating argument:
where \(\kappa\geq1\) and \(p=4\), f is a continuous function, σ is a constant, and \(0\leq\sigma< T\).
It is clear that \(T=\pi\), \(g(t,x)=\frac{1}{5}(\cos2t+2)x^{3}(t-\sigma)-\frac{1}{x^{\kappa}(t-\sigma)}\), \(\psi(t)=\frac{1}{5}(\cos2t+2)\). It is obvious that (H1)-(H4) hold. Now we consider the assumption (H5). Since \(\vert \psi \vert _{\infty}\leq\frac{3}{5}\), we have
So by Theorem 3.1, we know (3.24) has at least one positive π-periodic solution.
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Acknowledgements
YX and ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC Project (No. 11501170), Fundamental Research Funds for the Universities of Henan Province (NSFRF140142), Henan Polytechnic University Outstanding Youth Fund (J2015-02) and Henan Polytechnic University Doctor Fund (B2013-055).
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YX and ZBC worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
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Xin, Y., Cheng, Z. Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument. Adv Differ Equ 2016, 41 (2016). https://doi.org/10.1186/s13662-015-0721-2
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DOI: https://doi.org/10.1186/s13662-015-0721-2