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Study on variable coefficients singular differential equation via constant coefficients differential equation
Boundary Value Problems volume 2019, Article number: 3 (2019)
Abstract
In this paper, we consider the following thirdorder singular differential equation with variable coefficients:
By using the Green’s function of the linear differential equation with constant coefficients and some fixed point theorems, i.e., Leray–Schauder alternative principle and Schauder’s fixed point theorem, we prove the existence of positive periodic solutions of the above equation.
Introduction
Since the 1980s, many researchers began to discuss the singular differential equations. More concretely, in 1987, Lazer and Solimini [11] investigated the singular equations
and
where \(\lambda >0\), g, h and f are periodic functions with period ω. In [11], the authors said that the (1.1) has an attractive singularity, whereas (1.2) has a repulsive singularity. For positive constant functions h, g and a continuous forcing term f, the authors gave the necessary and sufficient conditions for the existence of periodic solutions of (1.1) and (1.2). One of the common conditions to guarantee the existence of a positive periodic solution is a socalled strong force condition (corresponding to the case \(\lambda \geq 1\) in (1.1)) or (1.2); see [3, 9, 10, 19, 20, 22] and the references therein. On the other hand, the existence of positive periodic solutions of the singular differential equations has been established with a weak force condition (corresponding to the case \(0<\lambda <1\) in (1.1) or (1.2)); see [2, 4,5,6,7, 12, 13, 16, 18].
From then on, the study of existence of positive periodic solutions for secondorder singular differential equations has attracted many researchers’ attention (see [2, 3, 6, 7, 9, 10, 12, 16, 18, 20]). For example, in 2007 Torres [16] investigated a kind of secondorder nonautonomous singular differential equation,
By Schauder’s fixed point theorem, the author showed that the additional assumption of a weak singularity is enough to guarantee the existence of periodic solutions. Afterwards, Chu and Torres [6] improved the above results and presented a new assumption weaker than the singular condition in [16]. In 2010, Wang [18] investigated the existence and multiplicity of positive periodic solutions of the singular systems (1.3) with superlinearity or sublinearity assumptions at infinity for some \(e(t)\). The proof of their results was based on the Krasnoselskii fixed point theorem.
At the same time, some authors began to consider thirdorder singular differential equation. For example, in 2006, Chu and Zhou [8] discussed the thirdorder singular differential equation
with periodic boundary conditions \(x^{(i)}(0)=x^{(i)}(2\pi )\), \(i=0,1,2\). Here κ is a positive constant and nonlinearity \(f(t,x)\) is singular at \(x=0\). By the Green’s function and a fixed point theorem in cones, they obtained the existence of periodic solutions for (1.4) in a small range of \(\kappa _{*}\), and to be concrete \(\kappa _{*}\in (0,\frac{1}{\sqrt{3}})\).
All the aforementioned results concern secondorder singular differential equations with variable coefficients or singular thirdorder differential equations with constant coefficients. There are few results on the singular thirdorder equation with variable coefficients. Motivated by [6, 8, 16, 18], in this paper, we discuss the existence of positive periodic solutions for the following singular differential equation with variable coefficients:
where \(e(t)\in C(\mathbb{R}, \mathbb{R})\) is an ωperiodic function, \(a(t)\in C(\mathbb{R},\mathbb{R}^{+})\) is an ωperiodic function and the nonlinear term \(f(t,x)\in C( \mathbb{R}\times \mathbb{R}^{+},\mathbb{R})\). Moreover, f is an ωperiodic function on t and is of singularity at origin, i.e.,
We say (1.5) is of repulsive type (resp. attractive type) if \(f(t,x)\rightarrow +\infty \) (resp. \(f(t,x)\rightarrow \infty \)) as \(x\rightarrow 0^{+}\).
As is well known, it is very complicated to calculate the Green’s function of a thirdorder linear differential equation with variable coefficients
where \(h\in C(\mathbb{R},\mathbb{R}^{+})\) is an ωperiodic function. In this paper, we first discuss the Green’s function of the thirdorder linear differential equation with constant coefficients
where \(M:=\max_{t\in [0, \omega ]}a(t)\). By application of the Green’s function of (1.6) and some fixed point theorems, i.e. Schauder’s fixed point theorem and a nonlinear alternative principle of Leray–Schauder, we obtain the existence of a positive periodic solution for (1.5).
The paper is organized as follows: In Sect. 2, the Green’s function for constant coefficients differential equation (1.6) will be given. Some useful properties for the Green’s function are shown also. In Sect. 3, we will prove that a weak singularity enables the achievement of new existence criteria by means of Schauder’s fixed point theorem. Moreover, we consider the periodic solution of (1.5) with attractiverepulsive singularities. In Sect. 4, by employing Green’s function of (1.6) and the nonlinear alternative principle of Leray–Schauder, we prove the existence results of positive periodic solutions of (1.5), which are applicable to the case of a strong singularity as well as to the case of a weak singularity. Our new results generalize some recent results contained in [8].
Preliminary
Constant coefficients differential equation
In this section, we discuss the Green’s function of the thirdorder differential equations
where \(M:=\max_{t\in [0,\omega ]}a(t)\). We will use it to investigate the existence of a positive periodic solution for (1.5). In the following, we introduce Green’s functions of (2.1) and some properties, which can be found in [14].
Lemma 2.1
(see [14])
The equation (2.1) has an unique ωperiodic solution
where
Now we present some properties of the Green’s functions for (2.1). Denote
Lemma 2.2
(see [14])
\(\int ^{\omega }_{0}G(t, s) \,ds= \frac{1}{M}\) and if \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds, then \(0< A\leq G(t, s)\leq B\) for all \((t,s)\in [0,\omega ]\times [0, \omega ]\).
On the other hand, let \(M=\varrho ^{3}\), then (2.1) is transformed into
and
Moreover, the solutions of (2.4) can be written as
where
Lemma 2.3
(see [15])
The boundary problem (2.5) is equivalent to the integral equation
where
Moreover, for \(G_{2}(t,s)\), if \(\varrho <\frac{2\pi }{\sqrt{3} \omega }\), then we have the estimates
According to the above lemmas, the solution of (2.1) can be written as
Thus, let
we can get
Lemma 2.4
Assume that \(M<\frac{8\pi ^{3}}{3\sqrt{3} \omega ^{3}}\) holds, then \(G^{*}(t,s)\geq 0\) for all \((t,s)\in [0,\omega ]\times [0,\omega ]\).
Proof
From Lemma 2.3, we know \(G_{2}(t,s)\geq 0\). Since \(G_{1}(t,s)>0\), from (2.7) we can see that \(G^{*}(t,s)\geq 0\) for all \((t,s)\in [0,\omega ]\times [0,\omega ]\). □
Variable coefficients differential equation
In this section, we consider the variable coefficients differential equations
where \(h \in C(\mathbb{R},\mathbb{R}^{+}) \) is an ωperiodic function and \(a(t)\in C(\mathbb{R},\mathbb{R}^{+})\) is the same ωperiodic functions in (1.5). Obviously, the calculation of the Green’s function of (2.9) is very complicated. To overcome this difficulty, we will make a shift on the linear term.
Let \(X=\{\phi \in C(\mathbb{R},\mathbb{R}):\phi (t+\omega )=\phi (t) \}\) with the maximum norm \(\\phi \=\max_{t\in [0, \omega ]} \phi (t)\). Obviously, X is a Banach space. Denote
then (2.9) can be rewritten as
Define the operators \(T, H: X\to X\) by
Clearly, T, H are completely continuous. Furthermore, we see that \((Th)(t)>0\) if \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\). At the same time we have \(\H\\leq Mm\). By Lemma 2.1, the solution of (2.10) can be written in the form
Therefore, we have
using the fact \(\int ^{\omega }_{0}G(t,s)\,ds=\frac{1}{M}\). Hence the solution of (2.10) can be written as
Define an operator \(P:X\to X\),
it is obvious that \(x(t)=(Ph)(t)\) is the unique periodic solution of (2.9) for any \(h(t)\). Moveover, we have the following.
Lemma 2.5
P is completely continuous and
Proof
By the Neumann expansion of P, we have
Since T and H are completely continuous, P is completely continuous. Since \(Th(t)>0\) for any t, we get
Noting that \(\TH\<1\), we get
□
For the singular differential equation with variable coefficients considered in this paper,
where \(e(t)\in C(\mathbb{R}, \mathbb{R})\) is an ωperiodic function, \(a(t)\in C(\mathbb{R},\mathbb{R}^{+})\) is an ωperiodic function and the nonlinear term \(f(t,x)\in C( \mathbb{R}\times \mathbb{R}^{+},\mathbb{R})\), we define an operator \(Q: X\to X\),
According to Lemma 2.5, it is easy to verify the following remark.
Remark 1
Q is completely continuous in X.
Weak singularity
In this section, we establish the existence of positive periodic solutions for thirdorder differential equation (1.5) by using Schauder’s fixed point theorem [21].
We write \(d(t)\succ 0\) if \(d(t)\geq 0\) for a.e. \(t\in [0,\omega ]\) and \(d(t)>0\) on a set of positive measure. For a given function \(e(t)\in C[0,\omega ]\), we denote the essential supremum and infimum by \(e^{*}\) and \(e_{*}\) if they exist. Define the function \(\gamma : \mathbb{R}\rightarrow \mathbb{R,}\)
which is the unique ωperiodic solution of
Case (I) \(\gamma _{*}=0\)
Theorem 3.1
Suppose that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds. Furthermore, assume that the following conditions hold:
 \((H_{1})\) :

For each \(L>0\), there exists a continuous function \(\phi _{L}\succ 0\) such that \(f(t,x)\geq \phi _{L}(t)\) for all \((t,x)\in [0,\omega ]\times (0,L]\).
 \((H_{2})\) :

There exist continuous, nonnegative functions \(g(x)\), \(h(x)\) and \(\zeta (t)\) such that
$$ 0\leq f(t,x)\leq \zeta (t) \bigl(g(x)+h(x)\bigr) \quad \textit{for all }(t,x)\in [0,\omega ]\times (0,\infty ), $$and \(g(x)>0\) is nonincreasing and \(h(x)\) is nondecreasing in \(x\in (0,\infty )\).
 \((H_{3})\) :

There exists a positive constant \(R>0\) such that \(R>(\varPhi _{ R})_{*}\) and
$$ R\geq \frac{M}{m} \biggl( \bigl(g\bigl((\varPhi _{R})_{*} \bigr)\bigr) \biggl(1+ \frac{h(R)}{g(R)} \biggr) \biggr)\varLambda ^{*}+ \Vert \gamma \Vert ), $$where
$$ \varPhi _{R}(t)= \int ^{\omega }_{0}G(t,s) (\phi _{R}) (s)\,ds, \qquad \varLambda (t)= \int ^{\omega }_{0}G(t,s)\zeta (s)\,ds, \qquad \Vert \gamma \Vert =\max _{t\in [0,\omega ]} \bigl\vert \gamma (t) \bigr\vert . $$
Proof
It is obvious that an ωperiodic solution of (1.5) is just a fixed point of the operator equation
Let R be the positive constant satisfying \((H_{3})\) and denote
then we have \(R>r>0\). Now define
Obviously, Ω is a closed convex set. Next we will prove \(Q(\varOmega )\subset \varOmega \).
In fact, for each \(x\in \varOmega \) and for all \(t\in [0,\omega ]\), using the fact that \(G(t,s)>0\) for all \((t,s)\in [0,\omega ]\times [0, \omega ]\), together with condition \((H_{1})\) and Lemma 2.5, we have
On the other hand, by conditions \((H_{2})\), \((H_{3})\) and Lemma 2.5, we have
In conclusion, \(Q(\varOmega )\subset \varOmega \). From Remark 1, we know that Q is compact in Ω. Therefore, the proof is completed by Schauder’s fixed point theorem. □
Corollary 3.2
Suppose that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds. Assume the following condition holds:
 \((F_{1})\) :

There exist continuous functions \(d(t)\), \(\hat{d}(t)\succ 0\) and \(0<\rho <1\) such that satisfy
$$ 0\leq \frac{\hat{d}(t)}{x^{\rho }}\leq f(t,x)\leq \frac{d(t)}{x^{ \rho }}, \quad \textit{for all }x>0, \textit{ and a.e. }t. $$
Proof
Take
then conditions \((H_{1})\) and \((H_{2})\) are satisfied and the existence condition \((H_{3})\) becomes
where \(\hat{\varPsi }=\int ^{\omega }_{0}G(t,s)\hat{d}(s)\,ds\), \(\varPsi = \int ^{\omega }_{0}G(t,s)d(s)\,ds\), for some \(R>0\). Note that \(\hat{\varPsi }_{*}>0\), since \(0<\rho <1\), we can choose \(R>0\) large enough so that (3.3) is satisfied. By Theorem 3.1, we can complete the proof. □
Case (II) \(\gamma _{*}>0\)
Theorem 3.3
Suppose that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds. And \(f(t,x)\) satisfies \((H_{2})\). Furthermore, assume that the following condition holds:
 \((H_{4})\) :

There exists \(R>0\) such that
$$ \frac{M}{m} \biggl(g(\gamma _{*}) \biggl(1+\frac{h(R)}{g(R)} \biggr) \varLambda ^{*}+\gamma ^{*} \biggr) \leq R. $$
Proof
We shall adopt the same strategy and notation as in the proof of Theorem 3.1. Let R be the positive constant satisfying \((H_{4})\) and \(r=\gamma _{*}\), then \(R>r>0\) since \(R>\gamma ^{*}\). Next we prove that \(Q(\varOmega )\subset \varOmega \).
For each \(x\in \varOmega \) and for all \(t\in [0,\omega ]\), by the nonnegative sign of \(G(t,s)\) and \(f(t,x)\) we have
On the other hand, from Lemma 2.5, we have
since \(\gamma _{*}>0\), we know \(\gamma (t)>0\), then \(\\gamma \=\gamma ^{*}\). Therefore, by \((H_{2})\) and \((H_{4})\), we have
In conclusion, \(Q(\varOmega )\subset \varOmega \). From Remark 1, it is easy to show that Q is compact in Ω. Therefore, by Schauder’s fixed point theorem, our result is proven. □
Corollary 3.4
Suppose that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds. Assume the following condition holds:
 \((F_{2})\) :

There exist a continuous function \(d(t)\succ 0\) and a constant \(\rho >0\) such that satisfy
$$ 0\leq f(t,x)\leq \frac{d(t)}{x^{\rho }}, \quad \textit{for all }x>0, \textit{ and a.e. }t. $$
Proof
Take
then condition \((H_{2})\) is satisfied and the existence condition \((H_{4})\) is also satisfied if we take \(R>0\) with
By Theorem 3.3, the result is obvious. □
Corollary 3.5
Suppose that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds. Assume the following condition holds: \((F_{3})\) There exist a continuous function \(d(t)\succ 0\) and constants \(\rho >0\), \(0\leq \eta <1\) that satisfy
If \(\gamma _{*}>0\), then (1.5) has at least one positive periodic solution.
Proof
Take
then condition \((H_{2})\) is satisfied and the existence condition \((H_{4})\) is also satisfied if we take \(R>0\) with
Using Theorem 3.3, the proof is complete. □
On the other hand, condition \((H_{2})\) implies that the nonlinearity \(f(t,x)\) is nonnegative for all values \((t,x)\), which is quite a hard restriction. In the following, we will show how to avoid this restriction for \(\gamma _{*}>0\).
Theorem 3.6
Assume that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds. Furthermore, assume that the following conditions hold:
 \((H_{2}')\) :

There exist continuous, nonnegative functions \(g(x)\) and \(\zeta (t)\), such that
$$ f(t,x)\leq \zeta (t)g(x) \quad \textit{for all }(t,x)\in [0,\omega ]\times (0, \infty ), $$and \(g(x)>0\) is nonincreasing in \(x\in (0,\infty )\).
 \((H_{3}')\) :

Let us define
$$ R:=\frac{M}{m} \bigl(g(\gamma _{*})\varLambda ^{*}+ \gamma ^{*} \bigr), $$and assume that \(f(t,x)\geq 0\) for all \((t,x)\in [0,\omega ]\times (0,R]\).
If \(\gamma _{*}>0\), then (1.5) has at least one positive periodic solution.
Proof
Let R be the positive constant satisfying \((H_{3}')\) and \(r=\gamma _{*}\), then \(R>r>0\) since \(R>\gamma ^{*}\). Using the same method in the proof of Theorem 3.3, it is easy to prove that \(T(\varOmega ) \subset \varOmega \). We omit it. Then, by Schauder’s fixed point theorem, we complete the proof. □
Corollary 3.7
Assume \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds. Assume the following condition hold:
\((F_{4})\) there exist constants \(\rho , \eta , \mu >0\) such that satisfy
If \(\gamma _{*}>0\), then (1.5) has at least one positive periodic solution for each \(0\leq \mu <\mu '\), where \(\mu '\) is some positive constant.
Proof
The nonlinearity is \(f(t,x)=\frac{1}{x^{\rho }}\mu x^{\eta }\), and therefore \((H_{2}')\) holds with \(\zeta (t)=1\), \(g(x)= \frac{1}{x^{\rho }}\). Define \(R=\frac{M}{m} (\frac{\varUpsilon ^{*}}{( \gamma _{*})^{\rho }}+\gamma ^{*} )\). Note that \(f(t,x)\geq 0\) if and only if \(x^{\rho +\eta }\leq 1/ \mu \). Therefore, \((H_{3}')\) is verified for any \(\mu <(R)^{(\rho +\eta )}\). As a consequence, the result holds for \(\mu '= (\frac{M}{m} (\frac{\varUpsilon ^{*}}{( \gamma _{*})^{\rho }} +\gamma ^{*} ) )^{(\rho +\eta )}\). □
In the following, we will investigate (1.5) with attractive–repulsive singularities.
Corollary 3.8
Assume that \(\frac{3\sqrt{3}\omega M^{\frac{1}{3}}}{4\pi }<1\) holds and the nonlinearity in (1.5) is
where \(\alpha >\beta >0\) and \(\kappa >0\) is a positive parameter. If \(\gamma _{*}>0\) and \(\kappa \in [0,\kappa _{1}]\), where
then (1.5) has at least one positive periodic solution.
Proof
To apply Theorem 3.6, we take
It is obvious that the condition \((H_{2})'\) holds. Take \(R>0\) with
where \(\frac{1}{M}=\int ^{\omega }_{0}G(t,s)\,ds\).
Next, we prove \(f(t,x)\geq 0\) for all \((t,x)\in [0,\omega ]\times (0,R]\). In fact, since \(\alpha >\beta >0\) and \(\kappa \in [0,\kappa _{1}]\), we have
Therefore, we get \(f(t,x)\geq 0\) for all \((t,x)\in [0,\omega ]\times (0,R]\). Then condition \((H_{3}')\) is satisfied. □
Case (III) \(\gamma ^{*}<0\)
Theorem 3.9
Assume that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds and \(f(t,x)\) satisfies \((H_{1})\) and \((H_{2})\). Furthermore, assume that the following condition holds:
 \((H_{5})\) :

There exists \(R>0\) such that \(R>(\varPhi _{R})_{*}+\gamma _{*}>0\) and
$$ \frac{M}{m}g\bigl((\varPhi _{R})_{*}+\gamma _{*}\bigr) \biggl(1+\frac{h(R)}{g(R)} \biggr) \varLambda ^{*}\leq R. $$
Proof
This theorem can be proved in the same way as Theorem 3.1.
Let R be a positive constant satisfying \((H_{5})\) and \(r=(\varPhi _{R})_{*}+ \gamma _{*}\), then \(R>r>0\) since \(R>(\varPhi _{R})_{*}+\gamma _{*}\). Next we will prove that \(Q(\varOmega )\subset \varOmega \).
For each \(x\in \varOmega \) and for all \(t\in [0,\omega ]\), from \((H_{1})\), \((H_{5})\) and Lemma 2.5, we have
On the other hand, from Lemma 2.5, we have
since \(\gamma ^{*}\leq 0\), \(G(t,s)\) and \(f(t,x(t))\) are nonnegative, \((\varPhi _{R})_{*}+\gamma _{*}>0\), then we know
Therefore, by \((H_{2})\) and \((H_{5})\), we have
In conclusion, \(Q(\varOmega )\subset \varOmega \). From Remark 1, it is easy to verify that Q is compact in Ω. Therefore, by Schauder’s fixed point theorem, our result is proven. □
Corollary 3.10
Assume \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) and \((F_{1})\) hold. If \(\gamma ^{*}<0\) and
where \(0<\rho <1\) appears in \((F_{1})\), then there exists a positive periodic solution of (1.5).
Proof
Take
Then conditions \((H_{1})\) and \((H_{2})\) are satisfied. Next, we consider the condition \((H_{5})\) also to be satisfied. Taking \(R:=\frac{M\varPsi ^{*}}{m(r)^{\rho }}\), then \((\varPhi _{R})_{*}+\gamma _{*}>0\) holds if r satisfies
or, equivalently,
The function \(f(r)\) possesses a minimum at \(r_{0}:= [\frac{ \hat{\varPsi }_{*}m^{\rho }}{(M\varPsi ^{*})^{\rho }}\rho ^{2} ]^{\frac{1}{1 \rho ^{2}}}\). Let \(r=r_{0}\), then \((\varPhi _{R})_{*}+\gamma _{*}>0\) holds in \((H_{5})\) if \(\gamma _{*}\geq f(r_{0})\), which is just the condition
The \((H_{5})\) holds directly by the choice of R, and it remain to prove that \(R=\frac{M\varPsi ^{*}}{m(r_{0})^{\rho }}>r_{0}\). This is easily verified through elementary computations. Using Theorem 3.9, we complete the proof. □
Strong and weak singularities
In the section, we state and prove the existence results which are applicable to the case of a strong singularity as well as to the case of a weak singularity. The proof is based on the following nonlinear alternative of Leray–Schauder, which can be found in [1].
Lemma 4.1
([1])
Let C be a convex subset of a normed linear space E, and let U be an open subset of C with \(0\in U\). Then every compact, continuous map \(F:\bar{U}\rightarrow C\) has at least one of the following properties,

(I)
F has a fixed point in Ū; or

(II)
there is a \(x\in \partial U\) and \(\lambda \in (0,1)\) with \(x=\lambda Fx\).
Let
By Lemma 2.2, we have \(B>A>0\) and \(0<\sigma <1\).
Define the operator \(Q^{*}:X\rightarrow X\),
and a cone K in X,
Case (I) \(\gamma _{*}\geq 0\)
Theorem 4.2
Assume that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\), \((H_{1})\) and \((H_{2})\) hold. Suppose the following conditions are satisfied:
\((H_{6})\) there exists a positive constant \(R>0\) such that
If \(\gamma _{*}\geq 0\), then (1.5) has at least one positive ωperiodic solution x with \(x(t)>\gamma (t)\) for all t and \(0<\x\gamma \<R\).
Proof
Step 1. Consider the equation
It is easy to see that if (4.2) has a positive ωperiodic solution x satisfying \(x(t)+\gamma (t)>0\) for \(t\in [0,\omega ]\) and \(0<\x\<R\), then \(u(t)=x(t)+\gamma (t)\) is a positive ωperiodic solution of (1.5) with \(0<\u \gamma \<R\). So we only need to consider (4.2).
Since \((H_{6})\) holds, we can choose \(n_{0}\in \{1,2,\ldots \}\) such that \(\frac{1}{n_{0}}<\frac{\sigma m}{M} R+\gamma _{*}\) and
Let \(N_{0}=\{n_{0},n_{0}+1,\ldots \}\) and fix \(n\in N_{0}\), we consider the family of equations
where \(\mu \in [0,1]\) and
Now we prove that (4.3) has a periodic solution for each n. If x is a periodic solution of problem (4.3), define an operator \(Q_{n}^{*}: K\rightarrow X\) by
So, solving (4.3) is equivalent to solving the following operator equation:
Let \(\varOmega =\{x\in K \x \< R\}\). We claim \(Q_{n}^{*}(\varOmega ) \subset K\). In fact, \(\forall x\in K\), from Lemma 2.5, we have
since \(\int ^{\omega }_{0}G(t,s)=\frac{1}{M}\). Therefore, from Lemma 2.2 and Lemma 2.5, we have
Since \((Th)(t)\leq (Ph)(t)\leq \frac{M}{m}\Th\\), we can get \(\Th\\geq \frac{m}{M}\Ph\\). This implies that \(Q_{n}^{*}(\varOmega ) \subset K\). Besides, from Lemma 2.5, we know that \(Q_{n}^{*}: \varOmega \rightarrow K\) is completely continuous.
Step 2. We claim that any fixed point x of (4.4) for all \(\mu \in [0,1]\) must satisfy \(\x\\neq R\). Otherwise, assume that x is a fixed point of (4.4) for some \(\mu \in [0,1]\) such that \(\x\=R\). Thus, from Lemma 2.5, we have
Therefore, we have
since \(\frac{1}{n}\leq \frac{1}{n_{0}}<\sigma \frac{m}{M} R+\gamma _{*}\). Thus, from \((H_{2})\) we have
Therefore, we have
This is a contradiction to the choice of \(n_{0}\) and the claim is proved.
From this claim, Lemma 4.1 guarantees that
has a fixed point, denoted by \(x_{n}\), in Ω, i.e.,
has a ωperiodic solution \(x_{n}\) with \(\x_{n}\< R\). Since \(x_{n}(t)\geq \frac{1}{n}>0\) for all \(t\in [0,\omega ]\), \(x_{n}\) is actually a positive ωperiodic solution of (4.6).
Now we show that \(x_{n}(t)+\gamma (t)\) have an uniform positive lower bound, i.e., there exists a constant \(\vartheta >0\) independent of \(n\in N_{0}\) such that
for all \(n\in N_{0}\). To see this, we know from \((H_{1})\) that there exists a continuous function \(\phi _{R+\gamma ^{*}}(t)>0\) such that \(f(t,x)\geq \phi _{R+\gamma ^{*}}(t)\) for a.e. t and \(0< x\leq R+\gamma ^{*}\). Let \(x_{R+\gamma ^{*}}(t)\) be the unique ωperiodic solution to
according to Lemma 2.5 and \(\gamma _{*}\geq 0\), we have
where \(\varPhi (t)=\int ^{\omega }_{0}G(t,s)\phi _{r+\gamma ^{*}}(s)\,ds\). Since \(x_{n}(t)+\gamma (t)\leq R+\gamma ^{*}\) and \(x_{n}+\gamma _{*}\geq \frac{1}{n}\), we have
So we have \(x_{n}(t)+\gamma (t)\geq \vartheta \) for all n.
Step 3. In order to pass from the solution \(x_{n}\) of (4.6) to that of the original problem (4.2), we need to show \(\{x_{n}\}_{n\in N_{0}}\) is compact.
First we claim
for some constant \(D_{1}>0\) (independent of \(n\in N_{0}\)) and for all \(n\in N_{0}\).
In fact, since \(x_{n}\) is an ωperiodic solution of (4.6) we have
Multiplying both sides of (4.9) by \(x_{n}'(t)\) and integrating from 0 to ω, we have
Substituting \(\int ^{\omega }_{0}x_{n}'''(t)x_{n}'(t)\,dt=\int ^{\omega }_{0}x_{n}''(t)^{2}\,dt\) into (4.10),
where
Using the Writinger inequality (see [17], Lemma 2.4), we have
It is easy to see that there exists a constant \(D_{1}'>0\) such that
From \(x_{n}(0)=x_{n}(\omega )\), we know that there exists a point \(t_{0}\in [0,\omega ]\) such that \(x_{n}'(t_{0})=0\). Therefore, we have
The fact that \(\x_{n}\< R\) and (4.8) show that \(\{x_{n}\} _{n\in N_{0}}\) is bounded and an equicontinuous family on R. Now the Arzelà–Ascoli theorem guarantees that \(\{x_{n}\}_{n\in N_{0}}\) has a subsequence \(\{x_{n_{k}}\}_{k\in N}\), converging uniformly on R to a function \(x\in X\). From the fact that \(\x_{n}\< R\) and \(\vartheta \leq x_{n}+\gamma \), we know that x satisfies \(\vartheta \leq x(t)+ \gamma (t)\leq R+\gamma ^{*}\) for all t. Moreover, \(x_{n_{k}}\) satisfies the integral equation
Let \(k\rightarrow \infty \), we get
Therefore, x is a positive periodic solution of (4.2) and satisfies \(0<\ x\\leq R\). Besides, it is not difficult to show that \(\x\< R\), by noting that if \(\x\=R\), the argument similar to the proof of the first claim will yield a contradiction.
Combining the above three steps, the proof is completed. □
Corollary 4.3
Assume that \(\frac{3\sqrt{3}\omega M^{\frac{1}{3}}}{4\pi }<1\) holds. Suppose the following condition is satisfied:
 \((F_{5})\) :

there exist continuous functions \(d(t)\), \(\hat{d}(t)\succ 0\) and \(\rho >0\), \(0\leq \eta <1\) such that
$$ 0\leq \frac{\hat{d}(t)}{x^{\rho }}\leq f(t,x)\leq \frac{d(t)}{x^{ \rho }}+d(t)x^{\eta } \quad \textit{for all } x>0 \textit{ and a.e. } t. $$
Proof
We will apply Theorem 4.2. We take
Then \((H_{1})\) and \((H_{2})\) are satisfied and the existence condition \((H_{6})\) becomes
where \(\varPsi (t)=\int ^{\omega }_{0}G(t,s)d(s)\,ds\), for some \(r>0\). Since \(\rho >0\), \(0\leq \eta <1\) and \(\gamma _{*}\geq 0\), we can choose \(R>0\) large enough that (4.11) is satisfied. □
From Theorems 3.6 and 4.2, we obtain the following conclusion.
Theorem 4.4
Assume that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\), \((H_{1})\) and \((H_{2}')\) hold. Suppose the following conditions are satisfied:
 \((H_{6}')\) :

there exists a positive constant \(R>0\) such that
$$ \frac{M}{m}g \biggl(\frac{\sigma m}{M}R+\gamma _{*} \biggr) \varLambda ^{*}< R, $$and we assume that \(f(t,x)\geq 0\) for all \((t,x)\in [t,\omega ]\times (0,R]\).
If \(\gamma _{*}\geq 0\), then (1.5) has at least one positive ωperiodic solution x with \(x(t)>\gamma (t)\) for all t and \(0<\x\gamma \<R\).
Corollary 4.5
Assume that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds and the nonlinearity in (1.5) is
where \(\alpha >\beta >0\) and \(\mu >0\) is a positive parameter. If \(\gamma _{*}\geq 0\), then there exists a positive constant \(\kappa _{2}\) such that (1.5) has at least one positive periodic solution for each \(0\leq \kappa \leq \kappa _{2}\).
Proof
We will apply Theorem 4.4. Take
It is easy to see that the condition \((H_{2}')\) holds. The existence condition \((H_{6}')\) becomes
where \(\frac{1}{M}=\int ^{\omega }_{0} G(t,s)\,ds\). we can choose \(R>0\) appropriately large such that (4.12) is satisfied.
Next, we show that the nonlinear term \(f(t,x)\geq 0\), for all \((t,x)\in [0,\omega ]\times (0,R]\). In fact, \(f(t,x)\geq 0\) if and only if \(\kappa \leq x^{\beta \alpha }\). There exists a positive constant \(\kappa _{2}\) such that \(\kappa _{2}< R^{\beta \alpha }\). In view of \(\kappa <\kappa _{2}\) and \(\beta <\alpha \), we get \(\kappa < R^{\beta  \alpha }< x^{\beta \alpha }\) for all \(x\in (0,R]\). Therefore, the condition \((H_{6}')\) holds.
Finally, we show that \((H_{1})\) is satisfied. Let
and
Since \(\alpha >\beta \), one can easily verify that \(s_{1}< s_{2}\) and \(P(s_{1})=0\), \(P'(s_{2})=0\), \(P'(s)<0\), \(s\in (0,s_{2})\). Therefore, \(P(s)\) is decreasing in \((0,s_{1})\subset (0,s_{2})\). On the other hand, we can choose \(\kappa >0\) small enough such that \(R\in (0,s_{1})\). Thus,
This implies that the condition \((H_{1})\) is satisfied if we take
□
Theorem 4.6
Assume that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) and \((H_{2})\) hold. Assume that the following condition holds:
 \((H_{6}^{*})\) :

There exists a positive constant \(R>0\) such that
$$ \frac{M}{m}g \biggl(\frac{\sigma m}{M} R \biggr) \biggl(1+ \frac{h(R+ \gamma ^{*})}{g(R+\gamma ^{*})} \biggr)\varLambda ^{*}< R. $$
Proof
The proof of this theorem can be completed by the method analogous to that in Theorem 4.2. Step 1 and Step 2 are the same as in the proof of Theorem 4.2. Now, we consider that \(x_{n}(t)+\gamma (t)\) have a uniform positive lower bound, i.e., there exists a constant \(\vartheta _{1}>0\), independent of \(n\in N_{0}\), such that
for all \(n\in N_{0}\).
Since \(x_{n}+\gamma >\frac{1}{n}\) and \(\gamma _{*}>0\), from Lemma 2.2, we know that G and f are of a nonnegative sign. Thus we have \(x_{n}(t)+\gamma (t)\geq \gamma (t)\geq \gamma _{*}\geq \vartheta _{1}\). The rest of the proof is the same as Theorem 4.2. □
Corollary 4.7
Assume that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) holds. Suppose the following condition is satisfied:
 \((F_{6})\) :

there exist continuous function \(d(t)\geq 0\) for a.e. \(t\in [0,\omega ]\) and \(\rho >0\), \(0\leq \eta <1\) such that
$$ 0\leq f(t,x)\leq \frac{d(t)}{x^{\rho }}+d(t)x^{\eta }, \quad \textit{for all } x>0, \textit{ for a.e. } t. $$
Proof
We will apply Theorem 4.6. Take
then \((H_{2})\) is satisfied and the existence condition \((H_{6}^{*})\) becomes
Since \(\tau >0\), \(0\leq \eta <1\) and \(\gamma _{*}>0\), we can choose \(R>0\) large enough such that (4.13) is satisfied. □
Case (II) \(\gamma ^{*}\leq 0\)
Theorem 4.8
Assume that \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\), \((H_{1})\) and \((H_{2})\) hold, and the following condition is satisfied:
 \((H_{6}^{**})\) :

There exists a positive constant \(R>0\) such that \(\frac{\sigma m}{M}R+\gamma _{*}>0\) and
$$ \frac{M}{m}g \biggl(\frac{\sigma m}{M}R+\gamma _{*} \biggr) \biggl(1+ \frac{h(R)}{g(R)} \biggr)\varLambda ^{*}< R. $$  \((H_{7})\) :

\(\gamma _{*}+\varPhi '_{*}>0\), here \(\varPhi '(t)=\int ^{\omega } _{0}G(t,s)\phi _{R}(s)\,ds\).
If \(\gamma ^{*}\leq 0\), then (1.5) has at least one positive ωperiodic solution x with \(x(t)>\gamma (t)\) for all t and \(0<\x\gamma \<R\).
Proof
The proof of this theorem can be completed by the method analogous to that in Theorem 4.2. Step 1 and Step 2 are the same as in the proof of Theorem 4.2. Now, it remains to consider that \(x_{n}(t)+\gamma (t)\) has a uniform positive lower bound, i.e., there exists a constant \(\vartheta _{2}>0\), independent of \(n\in N_{0}\), such that
for all \(n\in N_{0}\).
Since \((H_{1})\) and \(\gamma ^{*}\leq 0\), we know that there exists a continuous function \(\phi _{R}(t)>0\) such that \(f(t,x)\geq \phi _{R}(t)\) for a.e. t and \(0< x\leq R+\gamma ^{*}\leq R\). Let \(x_{R}(t)\) be the unique ωperiodic solution to
from \((H_{6}^{*})\), then we have
Since \(x_{n}(t)+\gamma (t)\leq R+\gamma ^{*}< R\) and \(x_{n}+\gamma _{*} \geq \frac{1}{n}\), we have
So we have \(x_{n}(t)+\gamma (t)\geq \vartheta _{2}\). The rest of the proof is the same as Theorem 4.2. □
Remark 2
Replacing above assumptions \(M<\frac{64\pi ^{3}}{81\sqrt{3}\omega ^{3}}\) by assumption \(M<\frac{8\pi }{3\sqrt{3}\omega ^{3}}\), we can get similar existence results, which we omit here.
Conclusions
The paper is devoted to the existence of a positive periodic solution for Eq. (1.5). As is well known, it is very complicated to calculate the Green’s function of the thirdorder linear differential equation with variable coefficients. In this paper, we first discuss the Green’s function of the thirdorder linear differential equation with constant coefficients (1.6). By application of the Green’s function of (1.6) and some fixed point theorems, i.e. Schauder’s fixed point theorem and a nonlinear alternative principle of Leray–Schauder, we obtain the existence of a positive periodic solution for (1.5). Our results are applicable to the case of a strong singularity as well as to the case of a weak singularity; these new results generalize some recent results obtained in [8].
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Acknowledgements
SWY and JL are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.
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This work was supported by National Natural Science Foundation of China under Grand [11626087, 11601048]; and Fundamental Research Funds for the Universities of Henan Province under Grand [NSFRF170302]; and Henan Polytechnic University Doctor Fund under Grand [B2016058].
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Yao, S., Liu, J. Study on variable coefficients singular differential equation via constant coefficients differential equation. Bound Value Probl 2019, 3 (2019). https://doi.org/10.1186/s136610181112x
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Keywords
 Singular differential equations
 Positive periodic solutions
 Variable coefficients
 Constant coefficients
 Green’s function