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 Open Access
Study on variable coefficients singular differential equation via constant coefficients differential equation
 Shaowen Yao^{1} and
 Jie Liu^{1}Email authorView ORCID ID profile
 Received: 14 June 2018
 Accepted: 16 December 2018
 Published: 1 February 2019
Abstract
Keywords
 Singular differential equations
 Positive periodic solutions
 Variable coefficients
 Constant coefficients
 Green’s function
1 Introduction
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].
2 Preliminary
2.1 Constant coefficients differential equation
Lemma 2.1
(see [14])
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 \((t,s)\in [0,\omega ]\times [0, \omega ]\).
Lemma 2.3
(see [15])
Lemma 2.4
Assume that \(M<\frac{8\pi ^{3}}{3\sqrt{3} \omega ^{3}}\) holds, then \(G^{*}(t,s)\geq 0\) for all \((t,s)\in [0,\omega ]\times [0,\omega ]\).
2.2 Variable coefficients differential equation
Lemma 2.5
Proof
Remark 1
Q is completely continuous in X.
3 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].
3.1 Case (I) \(\gamma _{*}=0\)
Theorem 3.1
 \((H_{1})\) :

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

There exists a positive constant \(R>0\) such that \(R>(\varPhi _{ R})_{*}\) andwhere$$ R\geq \frac{M}{m} \biggl( \bigl(g\bigl((\varPhi _{R})_{*} \bigr)\bigr) \biggl(1+ \frac{h(R)}{g(R)} \biggr) \biggr)\varLambda ^{*}+ \Vert \gamma \Vert ), $$$$ \varPhi _{R}(t)= \int ^{\omega }_{0}G(t,s) (\phi _{R}) (s)\,ds, \qquad \varLambda (t)= \int ^{\omega }_{0}G(t,s)\zeta (s)\,ds, \qquad \Vert \gamma \Vert =\max _{t\in [0,\omega ]} \bigl\vert \gamma (t) \bigr\vert . $$
Proof
Corollary 3.2
 \((F_{1})\) :

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
3.2 Case (II) \(\gamma _{*}>0\)
Theorem 3.3
 \((H_{4})\) :

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 \(Q(\varOmega )\subset \varOmega \).
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
 \((F_{2})\) :

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
Corollary 3.5
Proof
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
 \((H_{2}')\) :

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

Let us defineand assume that \(f(t,x)\geq 0\) for all \((t,x)\in [0,\omega ]\times (0,R]\).$$ R:=\frac{M}{m} \bigl(g(\gamma _{*})\varLambda ^{*}+ \gamma ^{*} \bigr), $$
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:
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
Proof
3.3 Case (III) \(\gamma ^{*}<0\)
Theorem 3.9
 \((H_{5})\) :

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 \(Q(\varOmega )\subset \varOmega \).
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
Proof
4 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])
 (I)
F has a fixed point in Ū; or
 (II)
there is a \(x\in \partial U\) and \(\lambda \in (0,1)\) with \(x=\lambda Fx\).
4.1 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:
Proof
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.
Combining the above three steps, the proof is completed. □
Corollary 4.3
 \((F_{5})\) :

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
From Theorems 3.6 and 4.2, we obtain the following conclusion.
Theorem 4.4
 \((H_{6}')\) :

there exists a positive constant \(R>0\) such thatand we assume that \(f(t,x)\geq 0\) for all \((t,x)\in [t,\omega ]\times (0,R]\).$$ \frac{M}{m}g \biggl(\frac{\sigma m}{M}R+\gamma _{*} \biggr) \varLambda ^{*}< R, $$
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
Proof
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.
Theorem 4.6
 \((H_{6}^{*})\) :

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
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
 \((F_{6})\) :

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
4.2 Case (II) \(\gamma ^{*}\leq 0\)
Theorem 4.8
 \((H_{6}^{**})\) :

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
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.
5 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].
Declarations
Acknowledgements
SWY and JL are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.
Availability of data and materials
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
Funding
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].
Authors’ contributions
SWY and JL contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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