1 Introduction
In recent years, there has been a fairly great amount of work on periodic solutions for Duffing equations with a singularity (see [1‐5, 7‐11] and the references cited therein), in account of its applications in applied sciences. For example, the Brillouin electron beam focusing problem [2, 8] is to prove the existence of a positive π-periodic solutions of
where a is a positive constant.
$$ u''+a(1+\cos 2t)u=\frac{1}{u}, $$
In 2012, Cheng and Ren [1] discussed the existence and multiplicity of positive periodic solutions for the following Duffing equation:
where the nonlinear term g has a strong singularity of repulsive type at \(u=0\) and satisfies super-linearity condition at \(u=+\infty \). It is concluded that there exist infinitely many positive periodic solutions for Eq. (1.1) by applications of the generalized Poincaré–Birkhoff twist theorem. Afterwards, Wang and Ma [11] investigated Eq. (1.1) with g has a strong singularity of repulsive type at \(u = 0\) and a semi-linearity condition at \(x =+\infty \), using the Poincaré–Birkhoff theorem, the authors proved the existence of infinitely many 2π-periodic solutions.
$$ u''(t)+g\bigl(u(t)\bigr)=p(t), $$
(1.1)
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Proceeding from [1, 2, 8, 11], in this paper, we further consider the following p-Laplacian neutral Duffing equation with singularity:
where \(p\geq 2\), \(\varphi _{p}(u)=|u|^{p-2}u\) for \(u\neq 0\) and \(\varphi _{p}(0)=0\), c is a constant and \(|c|\neq 1\), P, τ are constants and \(0\leq \tau < T\); \(e\in L^{2}(\mathbb{R})\) is a T-periodic function; \(g:(0,+\infty )\to \mathbb{R}\) is a \(L^{2}\)-Carathéodory function, the nonlinear term g of (1.2) has a singularity at \(u=0\), i.e., we have a strong singularity of repulsive type (\(g_{1}\)):
or a strong singularity of attractive type (\(g_{2}\)):
$$ \bigl(\phi _{p}\bigl(u(t)-cu(t-\tau ) \bigr)'\bigr)'+Pu'(t)+g\bigl(u(t) \bigr)=e(t), $$
(1.2)
$$ \lim_{u\to 0^{+}}g(u)=-\infty , \quad \mbox{and} \quad \lim _{u\to 0^{+}} \int ^{u}_{1}g(s)\,ds=+\infty ; $$
$$ \lim_{u\to 0^{+}}g(u)=+\infty , \quad \mbox{and} \quad \lim _{u\to 0^{+}} \int ^{u}_{1}g(s)\,ds=-\infty . $$
Applying topological degree theory [6], we obtain the following conclusions.
Theorem 1.1
Assume that conditions
\(|c|<1\)
and (\(g_{1}\)) hold. Suppose the following conditions hold:
Then Eq. (1.2) has at least one positive
T-periodic solution.
(\(\mathrm{H}_{1}\))
There exist two positive constants
\(d_{1}\), \(d_{2}\)
with
\(d_{1}< d_{2}\)
such that
\(g(u)-e(t)<0\)
for
\((t,u)\in [0,T]\times (0,d _{1})\)
and
\(g(u)-e(t)>0\)
for
\((t,u)\in [0,T]\times (d_{2},+\infty )\).
Theorem 1.2
Assume that the conditions
\(|c|>1\), (\(g_{1}\)) and (\(\mathrm{H}_{1}\)) hold. Suppose the following condition holds:
Then Eq. (1.2) has at least one positive
T-periodic solution if
\(\alpha ^{\frac{1}{p}}(1+2|c|)^{\frac{1}{p}}T<|c|-1\).
(\(\mathrm{H}_{2}\))
There exist two positive constants
α, β
such that
$$ g(u)\leq \alpha u^{p-1}+\beta , \quad \textit{for all } u>0. $$
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Remark 1.3
It is worth mentioning that the condition of the nonlinear term g is relatively weak in the case that \(|c|<1\), i.e., the nonlinear term g may satisfy the sub-linearity, semi-linearity or super-linearity conditions at \(x=\infty \). The nonlinear term g in the case that \(|c|>1\) only satisfies the semi-linearity condition at \(u=\infty \). Obviously, our result can be more general.
Remark 1.4
The condition (\(g_{2}\)) contradicts the condition (\(g_{1}\)). Therefore, the above methods of [1, 11] and condition of Theorems 1.1–1.2 are no long applicable to prove the existence of a positive periodic solutions for Eq. (1.2) with strong singularity of attractive type. We need to give another method and conditions to get over this problem.
Theorem 1.5
Assume that conditions
\(|c|<1\)
and (\(g_{2}\)) hold. Furthermore, suppose the following condition holds:
Then Eq. (1.2) has at least one positive
T-periodic solution.
(\(\mathrm{H}_{3}\))
There exist two positive constants
\(d_{3}\), \(d_{4}\)
with
\(d_{3}< d_{4}\)
such that
\(g(u)-e(t)>0\)
for
\((t,u)\in [0,T]\times (0,d _{3})\)
and
\(g(u)-e(t)<0\)
for
\((t,u)\in [0,T]\times (d_{4},+\infty )\).
Theorem 1.6
Assume that conditions
\(|c|>1\), (\(g_{2}\)) and (\(\mathrm{H}_{3}\)) hold. Suppose the following condition holds:
Then Eq. (1.2) has at least one positive
T-periodic solution if
\(m^{\frac{1}{p}}(1+2|c|)^{\frac{1}{p}}T<|c|-1\).
(\(\mathrm{H}_{4}\))
There exist two positive constants
m, n
such that
$$ -g(u)\leq m u^{p-1}+n, \quad \textit{for all } u>0. $$
2 Periodic solution for Eq. (1.2) with strong singularity of repulsive type
Firstly, let \(A: C_{T}\rightarrow C_{T}\) be the operator on \(C_{T}:=\{u\in C(\mathbb{R}, \mathbb{R}): u(t+T)\equiv u(t)\ \forall t\in \mathbb{R}\}\) given by
$$ (Au) (t):=u(t)-cu(t-\tau ) \quad \forall u\in C_{T} , t\in \mathbb{R} . $$
Lemma 2.1
(see [12])
The operator
A
has a continuous inverse
\(A^{-1}\)
on
\(C_{T}\), satisfying
(1)
\(\vert [A^{-1}f ](t) \vert \leq \frac{\|f\|}{|1-|c||}\), \(\forall f\in C_{T}\), where
\(\|f\|:= \max_{t\in \mathbb{R}}|f(t)| \).
(2)
\(\int ^{T}_{0} \vert [A^{-1}f ](t) \vert \,dt \leq \frac{1}{|1-|c||}\int ^{T}_{0}|f(t)|\,dt\), \(\forall f\in C_{T}\).
Secondly, we embed Eq. (1.2) into the following equation family with a parameter \(\lambda \in (0,1]\):
The following lemma is a consequence of Theorem 3.1 of [6].
$$ \bigl(\phi _{p}(Au)'(t) \bigr)'+\lambda Pu'(t)+\lambda g\bigl(u(t)\bigr)=\lambda e(t). $$
(2.1)
Lemma 2.2
Assume that there exist positive constants
\(E_{1}\), \(E_{2}\), \(E_{3}\)
and
\(E_{1}< E_{2}\)
such that the following conditions hold:
Then Eq. (1.2) has at least one
T-periodic solution.
(1)
Each possible periodic solution
u
to Eq. (2.1) such that
\(E_{1}< u(t)< E_{2}\), for all
\(t\in [0,T]\)
and
\(\|u'\|< E_{3}\).
(2)
Each possible solution
C
to the equation
satisfies
\(C\in (E_{1},E_{2})\).
$$ g(C)-\frac{1}{T} \int ^{T}_{0} e(t)\,dt=0 $$
(3)
We have
$$ \biggl(g(E_{1})-\frac{1}{T} \int ^{T}_{0} e(t)\,dt \biggr) \biggl(g(E_{2})- \frac{1}{T} \int ^{T}_{0} e(t)\,dt \biggr)< 0. $$
2.1 Proof of Theorem 1.1
Proof of Theorem 1.1
Firstly, integrating both sides of Eq. (2.1) over \([0,T]\), we get
In view of the mean value theorem of integrals, there exists a point \(\xi \in (0,T)\) such that
From condition (\(\mathrm{H}_{1}\)) and \(u(t)\) being continuous, we have
$$ \int ^{T}_{0}\bigl[g\bigl(u(t)\bigr)-e(t)\bigr] \,dt=0. $$
(2.2)
$$ g\bigl(u(\xi )\bigr)-e(\xi )=0. $$
$$ d_{1}\leq u(\xi )\leq d_{2}. $$
(2.3)
Multiplying both sides of Eq. (2.1) by \((Au)'(t)\) and integrating from 0 to T, we deduce
Moreover,
and
since \((Au)'(t)=(Au')(t)\) and \(du(t)=\frac{du(t-\tau )}{d(t-\tau )}\,dt=du(t- \tau )\).
$$ \begin{aligned}[b] & \int ^{T}_{0}\bigl(\phi _{p} \bigl((Au)'(t)\bigr)\bigr)'(Au)'(t)\,dt+ \lambda P \int ^{T}_{0}u'(t) (Au)'(t) \,dt + \lambda \int ^{T}_{0}g\bigl(u(t)\bigr) (Au)'(t) \,dt \\ &\quad =\lambda \int ^{T}_{0}e(t) (Au)'(t)\,dt. \end{aligned} $$
(2.4)
$$ \int ^{T}_{0}\bigl(\phi _{p}(Au)'(t) \bigr)'(Au)'(t)\,dt= \int ^{T}_{0}(Au)'(t)\,d\phi _{p}(Au)'(t)=0 $$
(2.5)
$$ \begin{aligned}[b] \int ^{T}_{0}g\bigl(u(t)\bigr) (Au)'(t) \,dt&= \int ^{T}_{0}g\bigl(u(t)\bigr) \bigl(Au' \bigr) (t)\,dt \\ &= \int ^{T}_{0}g\bigl(u(t)\bigr) \bigl(u'(t)-cu'(t- \tau )\bigr)\,dt \\ &= \int ^{T}_{0}g\bigl(u(t)\bigr)\,du(t)-c \int ^{T}_{0}g\bigl(u(t)\bigr)\,du(t-\tau ) \\ &=-c \int ^{T}_{0}g\bigl(u(t)\bigr)\,du(t) \\ &=0, \end{aligned} $$
(2.6)
Substituting Eqs. (2.5) and (2.6) into (2.4), we obtain
From Eq. (2.7), we arrive at
From the Hölder inequality and \(\int ^{T}_{0}|u'(t-\tau )|^{2}\,dt= \int ^{T}_{0}|u'(t)|^{2}\,dt\), we see that
where \(\|e\|_{2}:= (\int ^{T}_{0}|e(t)|^{2}\,dt )^{ \frac{1}{2}}\). Obviously, \(|P|-|P||c|>0\), since \(|c|<1\). Hence, we deduce
From Eqs. (2.3), (2.8) and the Hölder inequality, we see that
$$ P \int ^{T}_{0}u'(t) \bigl(Au'\bigr) (t)\,dt= \int ^{T}_{0}e(t) \bigl(Au'\bigr) (t) \,dt. $$
(2.7)
$$ \biggl\vert P \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert ^{2}\,dt \biggr\vert = \biggl\vert Pc \int ^{T}_{0}u'(t)u'(t- \tau )\,dt+ \int ^{T}_{0}e(t) \bigl(Au'\bigr) (t) \,dt \biggr\vert . $$
$$\begin{aligned} \vert P \vert \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert ^{2}\,dt \leq & \vert P \vert \vert c \vert \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \bigl\vert u'(t- \tau ) \bigr\vert \,dt+\bigl(1+ \vert c \vert \bigr) \int ^{T}_{0} \bigl\vert e(t) \bigr\vert \bigl\vert u'(t) \bigr\vert \,dt \\ \leq & \vert P \vert \vert c \vert \biggl( \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \biggl( \int ^{T}_{0} \bigl\vert u'(t-\tau ) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\ &{}+\bigl(1+ \vert c \vert \bigr) \biggl( \int ^{T}_{0} \bigl\vert e(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \biggl( \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr)^{\frac{1}{2}} \\ = & \vert P \vert \vert c \vert \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert ^{2}\,dt+\bigl(1+ \vert c \vert \bigr)\|e\|_{2} \biggl( \int ^{T} _{0} \bigl\vert u'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}, \end{aligned}$$
$$ \biggl( \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}\leq \frac{(1+|c|) \|e\|_{2}}{|P|-|P||c|}:=M_{1}'. $$
(2.8)
$$ \begin{aligned}[b] u(t)&=u(\xi )+ \int ^{t}_{\xi }u'(s)\,ds\leq d_{2}+\sqrt{T} \biggl( \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\ &\leq d_{2}+\sqrt{T} \frac{(1+|c|) \|e\|_{2}}{|P|-|P||c|}:=M_{1}. \end{aligned} $$
(2.9)
On the other hand, from Eq. (2.2), it is clear that
Case (I). If \(\overline{e}:=\frac{1}{T}\int ^{T}_{0}e(t)\,dt\leq 0\), from Eq. (2.10), we have
where \(g^{+}(u):=\max \{g(u),0\}\). Since \(g^{+}(u(t))-e(t)\geq 0\), from condition (\(\mathrm{H}_{1}\)), we know \(u(t)\geq d_{2}\). Then we deduce
where \(\|g^{+}_{M_{1}}\|:=\max_{d_{2}\leq u\leq M_{1}}g^{+}(u)\). As \((Au)(0)=(Au)(T)\), there exists a point \(t_{1}\in [0,T]\) such that \((Au)'(t_{1})=0\), from Eqs. (2.2), (2.8) and (2.11), we have
$$ \begin{aligned}[b] \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt&= \int _{g(u(t))\geq 0}g\bigl(u(t)\bigr)\,dt- \int _{g(u(t))\leq 0}g\bigl(u(t)\bigr)\,dt \\ &=2 \int _{g(u(t))\geq 0}g\bigl(u(t)\bigr)\,dt- \int ^{T}_{0}e(t)\,dt. \end{aligned} $$
(2.10)
$$ \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt \leq 2 \int ^{T}_{0}\bigl(g^{+}\bigl(u(t) \bigr)-e(t)\bigr)\,dt, $$
$$ \begin{aligned}[b] \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt&\leq 2 \int ^{T}_{0}g^{+}\bigl(u(t)\bigr)\,dt+ \int ^{T}_{0} \bigl\vert e(t) \bigr\vert \,dt \\ &\leq 2T \bigl\Vert g^{+}_{M_{1}} \bigr\Vert +T^{\frac{1}{2}}\|e\|_{2}, \end{aligned} $$
(2.11)
$$ \begin{aligned}[b] \bigl\Vert \phi _{p}\bigl((Au)'\bigr) \bigr\Vert &=\max _{t\in [t_{1},t_{1}+T]} \biggl\{ \biggl\vert \int ^{t}_{t_{1}}\bigl(\phi _{p} \bigl((Au)'(s)\bigr)\bigr)'\,ds \biggr\vert \biggr\} \\ &\leq \vert P \vert \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt+ \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt+ \int ^{T}_{0} \bigl\vert e(t) \bigr\vert \,dt \\ &\leq \vert P \vert T^{\frac{1}{2}} \biggl( \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert ^{2}\,dt \biggr)^{ \frac{1}{2}}+ \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt+ \int ^{T}_{0} \bigl\vert e(t) \bigr\vert \,dt \\ &\leq |P|T^{\frac{1}{2}} M_{1}^{\prime \frac{1}{2}}+2T \bigl\Vert g^{+}_{M_{1}} \bigr\Vert +2T ^{\frac{1}{2}}\|e \|_{2}:=M_{2}'. \end{aligned} $$
(2.12)
We claim that there exists a positive constant \(M_{2}^{**}>M_{2}'+1\) such that, for all \(t\in \mathbb{R,}\)
In fact, if \(x'\) is not bounded, there exists a positive constant \(M_{2}''\) such that \(\|u'\|>M_{2}''\) for some \(u'\in \mathbb{R}\). Therefor, it is clear that
Then we have a contradiction. So, Eq. (2.13) holds. By Lemma 2.1 and Eq. (2.13), we get
since \(|c|<1\).
$$ \bigl\Vert (Au)' \bigr\Vert \leq M_{2}^{**}. $$
(2.13)
$$ \bigl\Vert \phi _{p}(Au)' \bigr\Vert = \bigl\Vert \phi _{p}\bigl(Au'\bigr) \bigr\Vert = \bigl\Vert Au' \bigr\Vert ^{p-1}=\bigl(1+|c|\bigr)^{p-1} \bigl\Vert u' \bigr\Vert ^{p-1}\geq \bigl(1+ \vert c \vert \bigr)^{p-1}M_{2}^{\prime \prime p-1}:=M_{2}^{*}. $$
$$ \begin{aligned}[b] \bigl\Vert u' \bigr\Vert &= \bigl\Vert A^{-1}Au' \bigr\Vert = \bigl\Vert A^{-1}(Au)' \bigr\Vert \\ &\leq \frac{ \Vert (Au)' \Vert }{1-|c|} \\ &\leq \frac{M_{2}^{**}}{1-|c|}:=M_{2}, \end{aligned} $$
(2.14)
Case (II). If \(\overline{e}>0\), from Eq. (2.10), we obtain
Since \(g^{+}(u(t))\geq 0\), from condition (\(\mathrm{H}_{1}\)), we know that there exists a positive constant \(d_{2}^{*}\) such that \(u(t)\geq d_{2}^{*}\). Therefore, we see that
where \(\|g^{+}_{M}\|:=\max_{d_{2}^{*}\leq u\leq M_{1}}g^{+}(x)\). Similarly, we deduce \(\|u'\|\leq M_{2}\).
$$ \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt\leq 2 \int ^{T}_{0}g^{+}\bigl(u(t)\bigr)\,dt. $$
$$\begin{aligned} \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt&\leq 2 \int ^{T}_{0}g^{+}\bigl(u(t)\bigr)\,dt \\ &\leq 2T \bigl\Vert g^{+}_{M} \bigr\Vert , \end{aligned}$$
Multiplying both sides of Eq. (2.1) by \(u'(t)\) and integrating on the interval \([\xi ,t]\), where \(\xi \in [0,T]\) is defined in Eq. (2.3), we get
Furthermore, from Eqs. (2.12), (2.8) and (2.14), we get
From the repulsive condition (\(g_{1}\)), we know that there exists a constant \(M_{3}>0\) such that
Similarly, we can discuss \(t\in [0,\xi ]\).
$$ \begin{aligned}[b] \lambda \int ^{u(t)}_{u(\xi )}g(v)\,dv&=\lambda \int ^{t}_{\xi }g\bigl(u(s)\bigr)u'(s) \,ds \\ &=- \int ^{t}_{\xi }\bigl(\phi _{p} \bigl((Au)'(s)\bigr)\bigr)'u'(s)\,ds- \lambda P \int ^{t}_{ \xi }|u'(s)|^{2}\,ds \\ &\quad {}+\lambda \int ^{t}_{\xi }e(s)u'(s)\,ds. \end{aligned} $$
(2.15)
$$\begin{aligned} \lambda \biggl\vert \int ^{u(t)}_{u(\xi )}g(v)\,dv \biggr\vert \leq & \int ^{t}_{ \xi } \bigl\vert \bigl(\phi _{p}\bigl((Au)'(s)\bigr)\bigr)' \bigr\vert \bigl\vert u'(s) \bigr\vert \,ds+\lambda P \int ^{t}_{\xi } \bigl\vert u'(s) \bigr\vert ^{2}\,ds \\ &{}+\lambda \int ^{t}_{\xi } \bigl\vert e(s) \bigr\vert \bigl\vert u'(s) \bigr\vert \,ds \\ \leq &\lambda M_{2}M_{2}'+\lambda \vert P \vert \bigl(M_{1}'\bigr)^{2}+\lambda M_{2}T^{ \frac{1}{2}}\|e\|_{2}:=\lambda M_{3}'. \end{aligned}$$
$$ u(t)\geq M_{3}, \quad \forall t\in [\xi ,T]. $$
(2.16)
From Eqs. (2.3), (2.9), (2.14) and (2.16), it is obvious that a periodic solution u to Eq. (2.1) satisfies
Then the condition (1) of Lemma 2.2 is satisfied. For a possible solution C to the equation
we have \(C\in (E_{1},E_{2})\). Hence, the condition (2) of Lemma 2.2 holds. Finally, it is clear that the condition (3) of Lemma 2.2 is also satisfied. In fact, from condition (\(\mathrm{H}_{1}\)), we can get
and
Using Lemma 2.2, it is concluded that Eq. (1.2) has at least one positive periodic solution. □
$$ E_{1}< u(t)< E_{2}, \qquad \bigl\Vert u' \bigr\Vert < E_{3}. $$
$$ g(C)-\frac{1}{T} \int ^{T}_{0}e(t)\,dt=0, $$
$$ g(E_{1})-\frac{1}{T} \int ^{T}_{0}e(t)\,dt< 0 $$
$$ g(E_{2})-\frac{1}{T} \int ^{T}_{0}e(t)\,dt>0. $$
2.2 Proof of Theorem 1.2
Proof of Theorem 1.2
The same strategy and notation are followed as in the proof of Theorem 1.1. Then we see that
$$ u(t)\leq d_{2}+ \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt. $$
(2.17)
Multiplying both sides of Eq. (2.1) by \((Au)(t)\) and integrating on the interval \([0,T]\), it is clear that
Substituting \(\int ^{T}_{0}(\phi _{p}(Au)'(t))'(Au)(t)\,dt=-\int ^{T}_{0}|(Au)'(t)|^{p}\,dt\) and \(P\int ^{T}_{0}u'(t)u(t)\,dt=0\) into Eq. (2.18), we have
Furthermore, we deduce
From condition (\(\mathrm{H}_{2}\)) and \(u(t)>0\), we see that
Substituting Eqs. (2.20) and (2.21) into (2.19), applying the Hölder inequality, we obtain
From Eq. (2.10) and condition (\(\mathrm{H}_{2}\)), we get
Substituting Eqs. (2.17) and (2.23) into (2.22), we see that
where \(N_{1}:=(1+2|c|)(\beta T+T^{\frac{1}{2}}\|e\|_{2})\). Applying Lemma 2.1, we have
since \(\frac{1}{p}+\frac{1}{q}=1\) and \(|c|>1\). We apply the inequality
Substituting Eqs. (2.24) into (2.25), we have
Since \(\alpha ^{\frac{1}{p}}(1+2|c|)^{\frac{1}{p}}T<|c|-1\), we know that there exists a positive constant \(M_{1}'\) such that
From Eqs. (2.17) and (2.26), we have
The proof is left for the reader, being the same as that of Theorem 1.1. □
$$ \begin{aligned}[b] & \int ^{T}_{0}\bigl(\phi _{p} \bigl((Au)'(t)\bigr)\bigr)'(Au) (t)\,dt+\lambda P \int ^{T}_{0}u'(t) (Au) (t)\,dt+ \lambda \int ^{T}_{0}g\bigl(u(t)\bigr) (Au) (t)\,dt \\ &\quad =\lambda \int ^{T}_{0}e(t) (Au) (t)\,dt. \end{aligned} $$
(2.18)
$$\begin{aligned} \int ^{T}_{0} \bigl\vert (Au)'(t) \bigr\vert ^{p}\,dt&=-\lambda Pc \int ^{T}_{0}u'(t)u(t-\tau )\,dt+ \lambda \int ^{T}_{0}g\bigl(u(t)\bigr) \bigl(u(t)-cu(t-\tau )\bigr)\,dt \\ &\quad {}-\lambda \int ^{T}_{0}e(t) \bigl(u(t)-cu(t-\tau )\bigr)\,dt. \end{aligned}$$
(2.19)
$$ \int ^{T}_{0}u'(t)u(t-\tau )\,dt= \int ^{T}_{0}u(t-\tau )\,du(t)= \int ^{T} _{0}u(t-\tau )\,du(t-\tau )=0. $$
(2.20)
$$ \int ^{T}_{0}g\bigl(u(t)\bigr)u(t)\,dt\leq \alpha \int ^{T}_{0}\bigl(u(t)\bigr)^{p}\,dt+ \beta \int ^{T}_{0}u(t)\,dt. $$
(2.21)
$$ \begin{aligned}[b] \int ^{T}_{0} \bigl\vert (Au)'(t) \bigr\vert ^{p}\,dt &\leq \alpha \int ^{T}_{0}\bigl(u(t)\bigr)^{p}\,dt+ \beta \int ^{T}_{0}u(t)\,dt+|c| \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \bigl\vert u(t-\tau ) \bigr\vert \,dt \\ &\quad {}+\bigl(1+ \vert c \vert \bigr)\|u\| \int ^{T}_{0}\bigl|e(t)\bigr|\,dt \\ &\leq \alpha T\|u\|^{p}+\beta T\|u\|+|c|\|u\| \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt \\ &\quad {}+\bigl(1+ \vert c \vert \bigr) \|u\|T^{\frac{1}{2}}\|e \|_{2}. \end{aligned} $$
(2.22)
$$ \begin{aligned}[b] \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt&=2 \int _{g(u(t))\geq 0}g\bigl(u(t)\bigr)\,dt- \int ^{T}_{0}e(t)\,dt \\ &\leq 2\alpha \|u\|^{p-1}T+2\beta T+\|e\|_{2}T^{\frac{1}{2}}. \end{aligned} $$
(2.23)
$$ \begin{aligned}[b] &\int ^{T}_{0} \bigl\vert (Au)'(t) \bigr\vert ^{p}\,dt \\ &\quad \leq \alpha \bigl(1+2 \vert c \vert \bigr)T \|u\|^{p}+\bigl(1+2 \vert c \vert \bigr) \bigl( \beta T+T^{\frac{1}{2}}\|e\|_{2}\bigr)\|u\| \\ &\quad \leq \alpha \bigl(1+2 \vert c \vert \bigr)T \biggl(d_{2}+ \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr)^{p}+N _{1} \biggl(d_{2}+ \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr) \\ &\quad = \alpha \bigl(1+2 \vert c \vert \bigr)T \biggl( \biggl( \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr)^{p}+pd _{2} \biggl( \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr)^{p-1} +\cdots +d_{2}^{p} \biggr) \\ &\quad \quad {}+N_{1} \biggl(d_{2}+ \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr), \end{aligned} $$
(2.24)
$$ \begin{aligned}[b] \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt&= \int ^{T}_{0} \bigl\vert \bigl(A^{-1}Au' \bigr) (t) \bigr\vert \,dt \\ &\leq \frac{\int ^{T}_{0}|(Au)'(t)|\,dt}{|c|-1} \\ &\leq \frac{T^{\frac{1}{q}} (\int ^{T}_{0}|(Au)'(t)|^{p}\,dt ) ^{\frac{1}{p}}}{|c|-1}, \end{aligned} $$
(2.25)
$$ (x+y)^{k}\leq x^{k}+ y^{k}, \quad \mbox{for } x, y>0, 0< k< 1. $$
$$\begin{aligned} \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt \leq &\frac{\alpha ^{\frac{1}{p}}(1+2|c|)^{ \frac{1}{p}}T\int ^{T}_{0}|u'(t)|\,dt}{|c|-1} \\ &{}+\frac{\alpha ^{\frac{1}{p}}(1+2|c|)^{\frac{1}{p}}(pd_{2})^{ \frac{1}{p}}T (\int ^{T}_{0}|u'(t)|\,dt )^{\frac{p-1}{p}}}{|c|-1} \\ &{}+\cdots +\frac{T^{\frac{1}{q}} (N_{1}^{\frac{1}{p}}+(pd_{2} ^{p-1})^{\frac{1}{p}} ) (\int ^{T}_{0}|u'(t)|\,dt )^{ \frac{1}{p}}+ T^{\frac{1}{p}}(d_{2}+ (N_{1}d_{2})^{\frac{1}{p}})}{|c|-1}. \end{aligned}$$
$$ \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt\leq M_{1}'. $$
(2.26)
$$ \|u\|\leq d_{2}+\frac{1}{2} \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt\leq d_{2}+\frac{M _{1}'}{2}:=M_{1}. $$
3 Periodic solution for Eq. (1.2) with strong singularity of attractive type
3.1 Proof of Theorem 1.5
Proof of Theorem 1.5
We follow the same strategy and notation as in the proof of Theorem 1.1. We can get
From Eq. (2.10), we have
where \(g^{-}:=\min \{g(u),0\}\).
$$ u(t)\leq M_{1}. $$
$$ \begin{aligned}[b] \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt&= \int _{g(u(t))\geq 0}g\bigl(u(t)\bigr)\,dt- \int _{g(u(t))\leq 0}g\bigl(u(t)\bigr)\,dt \\ &=-2 \int _{g(u(t))\leq 0}g^{-}\bigl(u(t)\bigr)\,dt+ \int ^{T}_{0}e(t)\,dt, \end{aligned} $$
(3.1)
Case (I). If \(\overline{e}\geq 0\), from Eq. (3.1), we obtain
Since \(g^{-}(u(t))-e(t)\leq 0\), from condition (\(\mathrm{H}_{3}\)), we know that \(u(t)\geq d_{4}\). Then we deduce
where \(\|g^{-}_{M_{1}}\|:=\max_{d_{4}\leq u\leq M_{1}}(-g^{-}(u))\). From Eqs. (2.12) and (3.2), we see that
$$ \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt \leq -2 \int ^{T}_{0}\bigl(g^{-}\bigl(u(t) \bigr)-e(t)\bigr)\,dt. $$
$$ \begin{aligned}[b] \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt&\leq -2 \int ^{T}_{0}g^{-}\bigl(u(t)\bigr)\,dt+ \int ^{T}_{0} \bigl\vert e(t) \bigr\vert \,dt \\ &\leq 2T \bigl\Vert g^{-}_{M_{1}} \bigr\Vert +T^{\frac{1}{2}}\|e\|_{2}, \end{aligned} $$
(3.2)
$$ \begin{aligned}[b] \bigl\Vert \phi _{p}\bigl((Au)'\bigr) \bigr\Vert &\leq \vert p \vert \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt+ \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt+ \int ^{T}_{0} \bigl\vert e(t) \bigr\vert \,dt \\ &\leq |p|T^{\frac{1}{2}}M_{1}^{\prime \frac{1}{2}}+2T \bigl\Vert g^{-}_{M_{1}} \bigr\Vert +2T ^{\frac{1}{2}}\|e \|_{2}:=M_{2}'. \end{aligned} $$
(3.3)
Case (II). If \(\overline{e}<0\), from Eq. (3.1), we arrive at
Since \(g^{-}(u(t))\leq 0\), from condition (\(\mathrm{H}_{3}\)), we know that there exists a positive constant \(d_{4}^{*}\) such that \(u(t)\geq d_{4}^{*}\). Therefore, we have
where \(\|g^{-}_{M}\|:=\max_{d_{4}^{*}\leq u\leq M_{1}}(-g^{-}(u))\). Similarly, we can get \(|\phi _{p}((Au)'(t))|\leq M_{2}'\).
$$ \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt\leq -2 \int ^{T}_{0}g^{-}\bigl(u(t)\bigr)\,dt. $$
$$\begin{aligned} \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt&\leq -2 \int ^{T}_{0}g^{-}\bigl(u(t)\bigr)\,dt \\ &\leq 2T \bigl\Vert g^{-}_{M} \bigr\Vert , \end{aligned}$$
The proof left for the reader, being the same as that of Theorem 1.1. □
3.2 Proof of Theorem 1.6
Proof of Theorem 1.6
The same strategy and notation are followed as in the proof of Theorem 1.2. We only consider \(\int ^{T}_{0}|g(u(t))|\,dt\). From Eqs. (2.23), (3.1) and condition (\(\mathrm{H}_{4}\)), we deduce
From Eqs. (2.24) and (3.1), we obtain
$$ \begin{aligned}[b] \int ^{T}_{0} \bigl\vert g\bigl(u(t)\bigr) \bigr\vert \,dt&= -2 \int _{g(u)\leq 0}g^{-1}\bigl(u(t)\bigr)\,dt+ \int ^{T} _{0}e(t)\,dt \\ &\leq 2m\|u\|^{p-1}T+2nT+T^{\frac{1}{2}}\|e\|_{2}. \end{aligned} $$
(3.4)
$$\begin{aligned} \int ^{T}_{0} \bigl\vert (Au)'(t) \bigr\vert ^{p}\,dt \leq& m\bigl(1+2 \vert c \vert \bigr)T \biggl( \biggl( \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr) ^{p}+\cdots +d_{4}^{p} \biggr) \\ &{}+N_{1} \biggl(d_{4}+ \int ^{T}_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr). \end{aligned}$$
The proof left for the reader, being the same as that of Theorem 1.1. □
4 Examples
Example 4.1
Consider the following neutral Duffing equation with strong singularity of repulsive type:
where τ is a positive constant and \(0<\tau <T\), P and μ are constants and \(\mu \geq 1\), n is an integer.
$$ \biggl(\phi _{p} \biggl(u(t)- \frac{1}{2}u(t-\tau ) \biggr)' \biggr)'+Pu'(t)+ \sum_{i=1}^{n}u^{2i}(t)- \frac{1}{u^{\mu }}=e^{\cos t}, $$
(4.1)
Comparing Eqs. (4.1) to (1.2), we know that \(g(u)=\sum_{i=1}^{n}u^{2i}(t)-\frac{1}{u^{\mu }}\), \(T=2\pi \), \(c= \frac{1}{2}<1\). Obviously, there exist constants \(d_{1}=0.1\) and \(d_{2}=1\) such that condition (\(\mathrm{H}_{1}\)) holds. In fact, \(\lim_{u\to 0^{+}}\int ^{u}_{1}g(s)\,ds=\lim_{u\to 0^{+}}\int ^{u}_{1} (\sum_{i=1}^{n}u^{2i}(t)-\frac{1}{u^{\mu }} )\,ds=+ \infty \), thus, the condition (\(\mathrm{H}_{2}\)) holds. Therefore, applying Theorem 1.1, we know that Eq. (4.1) has at least one positive 2π-periodic solution.
Example 4.2
Consider the following neutral Duffing equation with strong singularity of attractive type:
where \(p=4\), τ is a constant and \(0\leq \tau < T\).
$$ \bigl(\phi _{p}\bigl(u(t)-40u(t-\tau ) \bigr)'\bigr)'+8u'(t)-16u^{3}+ \frac{1}{u}=e^{ \sin 2t}, $$
(4.2)
It is clear that \(T=\pi \), \(P=8\), \(c=40>1\), \(g(u)=16u^{3}-\frac{1}{u}\), \(e(t)=e^{\sin 2t}\). Obviously, it is easy to see that there exist constants \(d_{3}=\frac{1}{4}\) and \(d_{4}=\frac{3}{5}\) such that condition (\(\mathrm{H}_{3}\)) holds. \(\lim_{u\to 0^{+}}\int ^{u}_{1}g(s)\,ds= \lim_{u\to 0^{+}}\int ^{u}_{1}(-16s^{3}+\frac{1}{s})\,ds=-\infty \), thus, the condition (\(\mathrm{g}_{2}\)) holds. Consider \(-g(u)\leq 16u^{3}+1\), where \(m=16\), \(n=1\). So, condition (\(\mathrm{H}_{4}\)) is satisfied. Next, it is verified that
Therefore, using Theorem 1.6, it is concluded that Eq. (4.2) has at least one positive π-periodic solution.
$$ \frac{m^{\frac{1}{p}}(1+2|c|)^{\frac{1}{p}}T}{|c|-1} = \frac{2\times 3\times \pi }{39}< 1. $$
5 Conclusion
In this paper, using the Manásevich–Mawhin theorem on the continuous case of topological degree, we discuss the existence of a positive periodic solution for the p-Laplacian neutral Duffing equation (1.2). The nonlinear term g satisfies the strong singularity of attractive and repulsive type at \(u=0\) and may obey the sub-linearity, semi-linearity or super-linearity conditions at \(u=\infty \). First, we obtain the existence of a positive periodic solution for Eq. (1.2) with a strong singularity of repulsive type. Afterwards, we prove the existence of a positive periodic solution for Eq. (1.2) with a strong singularity of attractive type. Our results improve and extend the results in [1, 2, 8, 11].
Acknowledgements
SWY and XZZ are grateful to anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.
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