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 Open Access
Some results for fourthorder nonlinear differential equation with singularity
 Yun Xin^{1}Email author and
 Zhibo Cheng^{2}
 Received: 28 May 2015
 Accepted: 24 October 2015
 Published: 30 October 2015
Abstract
By application of Green’s function and some fixed point theorems, i.e., LeraySchauder alternative principle and Schauder’s fixed point theorem, we establish two new existence results of positive periodic solutions for a nonlinear fourthorder singular differential equation, which extend and improve significantly existing results in the literature.
Keywords
 fourthorder differential equation
 Green’s function
 singularity
 LeraySchauder alternative principle
 Schauder’s fixed point theorem
MSC
 34B16
 34C25
 34B18
1 Introduction
Ding and Taliaferro’s work has attracted the attention of many specialists in differential equations. More recently, the method of lower and upper solutions [3, 4], the PoincaréBirkhoff twist theorem [5–7], topological degree theory [8, 9], Schauder’s fixed point theorem [10], the Krasnoselskii fixed point theorem in a cone [11, 12] and fixed point index theory [13] have been employed to investigate the existence of positive periodic solutions of singular secondorder and thirdorder differential equations.
To conclude this introduction, some notations are presented as follows: for a.e. means for almost every. We write \(d(t)\succ0\) if \(d(t)\geq0\) for a.e. \(t\in[0,\omega]\) and it is positive in a set of positive measure. The set of positive real numbers is denoted by \(R^{+}\). For a given function \(e\in L^{1}[0,\omega]\), we denote the essential supremum and infimum by \(e^{*}\) and \(e_{*}\) if they exist. Define \(C_{\omega}=\{x(t)\in C(R,R) : x(t+\omega)=x(t)\}\). Let \(X=\{\phi\in C(\mathbb{R},\mathbb{R}):\phi (t+\omega)=\phi(t)\}\) with the maximum norm \(\\phi\=\max_{0\leq t\leq\omega}\phi(t)\). Obviously, X is a Banach space.
2 Green’s function of fourthorder differential equation
Lemma 2.1
Proof
By a direct calculation, we get that the solution u satisfies the periodic boundary value condition of problem (2.1). □
Now we present the properties of Green’s functions for (2.1).
Lemma 2.2
\(\int^{\omega}_{0}G(t, s)\, ds=\frac{1}{\rho^{4}}\) and if \(\rho<\frac{\pi}{\omega}\) holds, then \(0< l< G(t, s)\leq L\) for all \(t\in[0,\omega]\) and \(s\in[0,\omega]\).
Proof
From (2.3) we can get \(\int^{\omega}_{0}G(t, s)\, ds=\frac{1}{\rho^{4}}\). If \(\rho<\frac{\pi}{\omega}\), we get \(G(t,s)>0\) for all \(t\in[0,\omega]\) and \(s\in[0,\omega]\).
3 Existence results (I)
In the section, we state and prove the first existence results. The proof is based on the following nonlinear alternative of LeraySchauder, which can be found in [21].
Lemma 3.1
 (I)
F has a fixed point in Ū or
 (II)
there is a \(u\in\partial U\) and \(\lambda\in(0,1)\) with \(x=\lambda Fx\).
Theorem 3.1
 (H_{1}):

For each constant \(L>0\), there exists a continuous function \(\phi_{L}\succ0\) such that \(f(t,x)\geq\phi_{L}(t)\) for a.e. t and \(x\in(0,L]\).
 (H_{2}):

There exist continuous, nonnegative functions \(g(x)\), \(h(x)\) and \(k(t)\) such thatand \(g(x)>0\) is nonincreasing, \(h(x)/g(x)\) is nondecreasing in x.$$ 0\leq f(t,x)\leq k(t) \bigl(g(x)+h(x)\bigr) \quad \textit{for all } x\in(0, \infty) \textit{ and a.e. } t, $$
 (H_{3}):

There exists a positive number \(r>0\) such thatwhere \(K(t)=\int^{\omega}_{0}G(t,s)k(s)\,ds\).$$ \frac{r}{g(\sigma r+\gamma_{*}) (1+\frac{h(r+\gamma^{*})}{g(r+\gamma^{*})} )}>K^{*}, $$
Proof
Step 2. In order to study (3.2), we first consider a family of equations.
Now we prove that (3.3) has a periodic solution for each n.
Step 3. In order to pass the solutions \(x_{n}\) of (3.6) to that of the original problem (3.2), we need to show that \(\{x_{n}\}_{n\in N_{0}}\) is compact.
Combining the three steps, the proof is completed. □
Corollary 3.1
 (F_{1}):

There exist continuous functions \(d(t), \hat{d}(t)\succ0\) and \(\tau>0\), \(0\leq\eta<1\), such that$$ 0\leq\frac{\hat{d}(t)}{x^{\tau}}\leq f(t,x)\leq\frac{d(t)}{x^{\tau}}+d(t)x^{\eta}\quad \textit{for all } x>0 \textit{ and a.e. } t. $$
Proof
Since \(\tau>0\), \(0\leq\eta<1\) and \(\gamma_{*}\geq0\), we can choose \(r>0\) large enough such that (3.11) is satisfied. □
Theorem 3.2
Assume that \(\rho<\frac{\pi}{\omega}\) and (H_{2})(H_{3}) hold. If \(\gamma_{*}>0\), then (1.3) has at least one positive ωperiodic solution x with \(x(t)>\gamma(t)\) for all t and \(0<\x\gamma\<r\).
Proof
The proof left is the same as in Theorem 3.1. □
Corollary 3.2
 (F_{2}):

There exists a continuous function \(d(t)\geq0\) for a.e. \(t\in[0,\omega]\) and \(\tau>0\), \(0\leq\eta<1\), such that$$0\leq f(t,x)\leq\frac{d(t)}{x^{\tau}}+d(t)x^{\eta}\quad \textit{for all } x>0, \textit{for a.e. } t. $$
Proof
Since \(\tau>0\), \(0\leq\eta<1\) and \(\gamma_{*}>0\), we can choose \(r>0\) large enough such that (3.12) is satisfied. □
Theorem 3.3
 (H_{4}):

\(\gamma_{*}+\Phi_{*}>0\), here \(\Phi(t)=\int^{\omega}_{0}G(t,s)\phi_{r+\gamma^{*}}(s)\,ds\).
Proof
The proof left is the same as in Theorem 3.1. □
As an application of Theorem 3.3, we consider the case \(\gamma_{*}=0\). The following Corollary 3.3 is a direct result of Theorem 3.3.
Corollary 3.3
Assume that \(\rho<\frac{\pi}{\omega }\) and (H_{1})(H_{3}) hold. If \(\gamma_{*}=0\), then (1.3) has at least one positive ωperiodic solution.
Theorem 3.4
 (H_{5}):

\(\gamma_{*}+\Phi'_{*}>0\), here \(\Phi'(t)=\int^{\omega}_{0}G(t,s)\phi_{r}(s)\,ds\).
Proof
The proof left is the same as in Theorem 3.1. □
Corollary 3.4
Assume that \(\rho<\frac{\pi}{\omega }\) and (F_{1}) hold. If \(\gamma^{*}\leq0\) and \(\gamma_{*}>\frac{\hat{\Psi}_{*}}{r^{\tau}}\), here \(\hat{\Psi}=\int^{\omega}_{0}G(t,s)\hat{d}(s)\,ds\), then (1.3) has at least one positive ωperiodic solution.
Proof
4 Existence results (II)
In this section, we establish the existence of positive periodic solutions for the fourthorder differential equation (1.3) by using Schauder’s fixed point theorem, which can be found in [22] (see p.61).
Lemma 4.1
([22], see p.61)
A compact operator \(A:M\rightarrow M\) has a fixed point provided M is a bounded, closed, convex, nonempty subset of a Banach space X over \(\mathbb{R}\).
Theorem 4.1
 (\(\mathrm{H}_{3}'\)):

There exists a positive constant \(R>0\) such that \(R>(\Phi_{R})_{*}+\gamma_{*}>0\) andwhere \(\Phi_{R}(t)=\int^{\omega}_{0}G(t,s)\phi_{R}(s)\,ds\).$$R\geq g\bigl((\Phi_{R})_{*}+\gamma_{*}\bigr) \biggl(1+\frac{h(R)}{g(R)} \biggr)K^{*}+\gamma^{*}, $$
Proof
Therefore, using the ArzelaAscoli theorem, it is easy to show that \(T^{*}\) is compact in Ω. Hence, the proof is finished by Schauder’s fixed point theorem. □
As an application of Theorem 4.1, we consider the case \(\gamma_{*}=0\). The following theorem is a direct result of Theorem 4.1.
Theorem 4.2
 (\(\mathrm{H}_{3}''\)):

There exists a positive constant \(R>0\) such that \(R>(\Phi_{R})_{*}\) and$$g\bigl((\Phi_{R})_{*}\bigr) \biggl(1+\frac{h(R)}{g(R)} \biggr)K^{*}+ \gamma^{*}\leq R. $$
Corollary 4.1
 (F_{3}):

There exist continuous functions \(d(t), \hat{d}(t)\succ0\) and \(0<\tau<1\) satisfying$$0\leq\frac{\hat{d}(t)}{x^{\tau}}\leq f(t,x)\leq\frac{d(t)}{x^{\tau}} \quad \textit{for all } x>0 \textit{ and a.e. } t. $$
Proof
Corollary 4.2
 (F_{4}):

There exist continuous functions \(d(t), \hat{d}(t)\succ0\) and \(0<\tau<1\), \(0\leq\eta<1\) satisfying$$0\leq\frac{\hat{d}(t)}{x^{\tau}}\leq f(t,x)\leq\frac{d(t)}{x^{\tau}}+d(t)x^{\eta}\quad \textit{for all } x>0 \textit { and a.e. } t. $$
Proof
The next results explore the case when \(\gamma_{*}>0\).
Theorem 4.3
 (\(\mathrm{H}_{3}'''\)):

There exists \(R>0\) such that$$g(\gamma_{*}) \biggl(1+\frac{h(R)}{g(R)} \biggr)K^{*}+\gamma^{*}\leq R. $$
Proof
We follow the same strategy and notation as in the proof of Theorem 4.1. Let R be the positive constant satisfying (\(\mathrm{H}_{3}'''\)) and let \(r=\gamma_{*}\); then \(R>r>0\) since \(R>\gamma^{*}\). Next we prove that \(T^{*}(\Omega)\subset\Omega\).
Corollary 4.3
 (F_{5}):

There exist a continuous function \(d(t)\succ0\) and a constant \(\tau>0\) satisfying$$0\leq f(t,x)\leq\frac{d(t)}{x^{\tau}} \quad \textit{for all } x>0 \textit{ and a.e. } t. $$
Proof
Corollary 4.4
Assume that \(\rho<\frac{\pi}{\omega}\) and (F_{2}) hold. If \(\gamma_{*}>0\), then (1.3) has at least one positive periodic solution.
Proof
On the other hand, condition (H_{2}) implies in particular that the nonlinearity \(f(t,x)\) is nonnegative for all values \((t,x)\), which is quite a hard restriction. In the following, we show how to avoid this restriction for \(\gamma_{*}>0\).
Theorem 4.4
 (\(\mathrm{H}_{2}^{*}\)):

There exist continuous, nonnegative functions \(g(x)\) and \(k(t)\) such thatand \(g(x)>0\) is nonincreasing in \(x\in(0,\infty)\).$$f(t,x)\leq k(t)g(x) \quad \textit{for all } (t,x)\in[0,\omega]\times(0,\infty), $$
 (\(\mathrm{H}_{3}^{*}\)):

Let us defineand assume that \(f(t,x)\geq0\) for all \((t,x)\in[0,\omega]\times(0,R]\).$$R:=g(\gamma_{*})K^{*}+\gamma^{*}, $$
Proof
We again use Schauder’s fixed point theorem. Let R be the positive constant satisfying (\(\mathrm{H}_{3}^{*}\)) and \(r=\gamma_{*}\); then \(R>r>0\) since \(R>\gamma^{*}\). By again using the method of Theorem 4.3, it is easy to prove that \(T^{*}(\Omega)\subset\Omega\). We omit the details. □
Corollary 4.5
 (F_{6}):

There exist constants \(\tau, \eta, \mu>0\) satisfying$$f(t,x)=\frac{1}{x^{\tau}}\mu x^{\eta}\quad \textit{for all } x>0 \textit{ and a.e. } t. $$
Proof
Finally, we consider \(\gamma^{*}\leq0\).
Theorem 4.5
Proof
Declarations
Acknowledgements
YX and ZC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (Nos. 11501170, 11271339) and the Fundamental Research Funds for the Universities of Henan Province (NSFRF140142).
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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