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Some results for fourthorder nonlinear differential equation with singularity
Boundary Value Problems volume 2015, Article number: 200 (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.
1 Introduction
Generally speaking, differential equations with singularities have been considered from the very beginning of the discipline. The main reason is that singular forces are ubiquitous in applications, gravitational and electromagnetic forces being the most obvious examples. In 1965, Ding [1] discussed the Brillouin electron beam focusing system
and obtained the existence of a positive periodic solution for the model if \(0< a<\frac{1}{4}\).
Afterwards, in 1979, Taliaferro [2] discussed the model equation with singularity
subject to
and obtained the existence of a solution for the problem. Here, \(\alpha>0\), \(q\in C(0,1)\) with \(q>0\) on \((0,1)\) and \(\int_{0}^{1} t(1t)q(t)\,dt<\infty\). We call the equation with strong force condition if \(\alpha\geq1\) and we call it with weak force condition if \(0<\alpha<1\).
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.
At the beginning, most of work concentrated on secondorder and thirdorder singular differential equations, as in the references we mentioned above. Recently some results on fourthorder differential equations have been published (see [14–20]). In 2003, Conti et al. [15] studied the fourthorder equation
with periodic boundary conditions, where \(c\geq(\pi/T)^{2}\), \(f:{\mathbb{R}}^{2}\to {\mathbb{R}}\) is continuous, Tperiodic in t and has a superlinear behavior at 0 and at infinity. Under these assumptions, they showed that for each positive integer \(n\geq1\) the problem admits a Tperiodic solution having precisely 2n simple zeroes in \([0,T]\). The proof was inspired by Nehari’s argument of combining variational methods and nodal properties of solutions. However, here a new and subtle minmax procedure is built, allowing one to interpret nodal properties of solutions of the problem as a topological property and to get these solutions by means of a variational principle with two constraints. In 2009, Li and Zhang [17] used some Sobolev constants to explicitly characterize a class of potentials \(q(t)\in L^{p}(0,T)\) for which the periodic beam equation with periodic boundary condition
is nondegenerate. As an application, they obtained the uniqueness of periodic solutions of a certain class of superlinear beam equations. Recently, Mosconi and Santra [19] proved that if \(F\in C(\mathbb{R})\) was coercive and \(\{F'=0\}\) was discrete, then the EFK equation
possesses \(L^{\infty}(\mathbb{R})\) solutions if and only if \(F'\) changed sign at least twice. As a corollary they proved that if \(u_{n}\) solved
then \(\u_{n}\_{\infty}\rightarrow+\infty\) if \(c_{n}\rightarrow0\), provided F has a unique local minimum, its only minimum is nondegenerate and \(\operatorname{int}(\{F'=0\})=\varnothing\). Finally, they gave criteria ensuring existence and nonexistence of Tperiodic solutions to (1.2) when F had multiple well.
In the above papers, the authors investigated fourthorder equations. However, the study on the singular fourthorder equation is relatively infrequent. Motivated by [15, 17, 19], in this paper, we further consider a fourthorder nonlinear differential equation as follows:
where \(\rho\in R^{+}\), \(e(t)\in L^{1}(R)\) is an ωperiodic function, \(f\in \operatorname{Car}(R\times R^{+},R)\) is an \(L^{2}\)Carathéodory function, i.e., it is measurable in the first variable and continuous in the second variable, and for every \(0< r< s\), there exists \(h_{r,s}\in L^{2}[0,\omega]\) such that \(f(t,x(t))\leq h_{r,s}\) for all \(x\in[r,s]\) and a.e. \(t\in[0,\omega]\), f is an ωperiodic function about t. The nonlinear term f of (1.3) can be with a singularity at origin, i.e.,
It is said that (1.3) is of repulsive type (resp. attractive type) if \(f(t,x)\rightarrow+\infty\) (resp. \(f(t,x)\rightarrow\infty\)) as \(x\rightarrow0^{+}\).
The remaining part of the paper is organized as follows. In Section 2, the Green’s function for the fourthorder linear differential equation
will be given. Here, \(h\in C(R, (0, +\infty))\) is an ωperiodic function. Some useful properties for Green’s function are obtained. In Section 3, by employing Green’s function and a nonlinear alternative principle of LeraySchauder, we state and prove the first existence result for (1.3). The result is applicable to the case of a strong singularity as well as the case of a weak singularity. In Section 4, we get a second existence result for (1.3). We prove that a weak singularity enables the achievement of new existence criteria through a basic application of Schauder’s fixed point theorem.
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
For \(\rho>0\) and \(h\in X\), the equation
has a unique solution which is of the form
where
Proof
It is easy to check that the associated homogeneous equation of (2.1) has the solution
Applying the method of variation of parameters, we get
and then
Noting that \(u(0)=u(\omega)\), \(u'(0)=u'(\omega)\), \(u''(0)=u''(\omega )\), \(u'''(0)=u'''(\omega)\), we obtain
Therefore
where \(G(t, s)\) is defined as in (2.3).
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 \(s\in[0,\omega]\).
Next we compute a lower and an upper bound for \(G(t, s)\) for \(s\in[0, \omega]\). We have
and the proof is complete. □
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
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 \(u\in\partial U\) and \(\lambda\in(0,1)\) with \(x=\lambda Fx\).
Define the function \(\gamma:R \rightarrow R\) by
which under the hypothesis in Section 2 is the unique ωperiodic solution of
Under Lemma 2.2, we always denote
Obviously, we have \(0<\sigma<1\).
We take \(X=C_{\omega}\) with \(\x\=\max_{t}x(t)\). Define the operator \(T:X\rightarrow X\),
Define the cone K in X by
Theorem 3.1
Assume that \(\rho<\frac{\pi}{\omega}\) holds. Suppose that the following conditions are satisfied:
 (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 that
$$ 0\leq f(t,x)\leq k(t) \bigl(g(x)+h(x)\bigr) \quad \textit{for all } x\in(0, \infty) \textit{ and a.e. } t, $$and \(g(x)>0\) is nonincreasing, \(h(x)/g(x)\) is nondecreasing in x.
 (H_{3}):

There exists a positive number \(r>0\) such that
$$ \frac{r}{g(\sigma r+\gamma_{*}) (1+\frac{h(r+\gamma^{*})}{g(r+\gamma^{*})} )}>K^{*}, $$where \(K(t)=\int^{\omega}_{0}G(t,s)k(s)\,ds\).
If \(\gamma_{*}\geq0\), 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
Step 1. Consider the equation
It is easy to see that if (3.2) 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 (1.3) with \(0<\u\gamma\<r\). So we only need to consider (3.2).
Step 2. In order to study (3.2), we first consider a family of equations.
Since (H_{3}) holds, we can choose \(n_{0}\in\{1,2,\ldots\}\) such that \(\frac{1}{n_{0}}<\sigma r+\gamma_{*}\) and
Let \(N_{0}=\{n_{0},n_{0}+1,\ldots\}\). Fix \(n\in N_{0}\). Consider the family of equations
where \(\mu\in[0,1]\), and
Now we prove that (3.3) has a periodic solution for each n.
If x is a periodic solution of problem (3.3), we have from Lemma 2.1 that
Define \(T_{n}: K\rightarrow X\) by
So, solving (3.3) is equivalent to the following fixed point problem:
Let \(\Omega=\{x\in K\mid\x \< r\}\). We claim \(T_{n}(\Omega )\subset K\). In fact, \(\forall x\in K\), we have
here \(f_{n}^{+}(t,x)=\max\{0,f_{n}(t,x)\}\). This implies that \(T_{n}(\Omega)\subset K\). Besides, it is easy to see that \(T_{n}:\Omega\rightarrow K\) is completely continuous.
We claim that any fixed point x of (3.4) for all \(\mu\in[0,1]\) must satisfy \(\x\\neq r\). Otherwise, assume that x is a fixed point of (3.4) for some \(\mu\in[0,1]\) such that \(\x\=r\). Thus, we have
Therefore, we have
So, we have
since \(\frac{1}{n}\leq\frac{1}{n_{0}}<\sigma r+\gamma_{*}\). Thus, from (H_{2}) we have
where \(f^{+}(t,x)=\max\{0,f(t,x)\}\).
Therefore,
This is a contradiction to the choice of \(n_{0}\) and the claim is proved.
From this claim, Lemma 3.1 guarantees that
has a fixed point, denoted by \(x_{n}\), in Ω, i.e.,
has an 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 (3.6).
Now we show that \(x_{n}(t)+\gamma(t)\) has a 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
then we have
where \(\Phi(t)=\int^{\omega}_{0}G(t,s)\phi_{r+\gamma^{*}}(s)\,ds\).
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 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.
First we show
for some constant \(M_{1}>0\) (independent of \(n\in N_{0}\)) and for all \(n\in N_{0}\).
In fact, since \(x_{n}\) is an ωperiodic solution of (3.6), we have
Multiplying both sides of (3.9) by \(x_{n}(t)\) and integrating from 0 to ω, we have
Substituting \(\int^{\omega}_{0}x_{n}^{(4)}(t)x_{n}(t)\,dt=\int^{\omega}_{0}x_{n}''(t)^{2}\,dt\) into (3.10), we have
where \(f_{r}=\max_{\vartheta\leq x_{n}(t)+\gamma\leq r+\gamma^{*}}f_{n}(t,x_{n}(t)+\gamma(t))\), \(f_{r}_{2}= (\int^{\omega}_{0}f_{r}^{2}\,dt )^{\frac{1}{2}}\). It is easy to see that there exists a constant \(M_{1}'>0\) 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 \(\x_{n}\< r\) and (3.8) show that \(\{x_{n}\}_{n\in N_{0}}\) is bounded and an equicontinuous family on R. Now the ArzelaAscoli 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 \(\x_{n}\< r\) and \(\vartheta\leq x_{n}+\gamma\), x satisfies \(\vartheta\leq x(t)+\gamma(t)\leq r+\gamma^{*}\) for all t. Moreover, \(x_{n_{k}}\) satisfies the integral equation
Letting \(k\rightarrow\infty\), we get
Therefore, x is a positive periodic solution of (3.2) 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 three steps, the proof is completed. □
Corollary 3.1
Assume that \(\rho<\frac{\pi}{\omega }\) holds. Suppose that the following condition is satisfied:
 (F_{1}):

There exist continuous functions \(d(t), \hat{d}(t)\succ0\) and \(\tau>0\), \(0\leq\eta<1\), such that
$$ 0\leq\frac{\hat{d}(t)}{x^{\tau}}\leq f(t,x)\leq\frac{d(t)}{x^{\tau}}+d(t)x^{\eta}\quad \textit{for all } x>0 \textit{ and a.e. } t. $$
If \(\gamma_{*}\geq0\), then (1.3) has at least one positive ωperiodic solution.
Proof
We will apply Theorem 3.1. We take
Then (H_{1}) and (H_{2}) are satisfied and the existence condition (H_{3}) becomes
where \(\Psi(t)=\int^{\omega}_{0}G(t,s)d(s)\,ds\) for some \(r>0\).
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
We will follow the same strategy and notations as in the proof of Theorem 3.1. Step 1 and Step 2 are the same as in the proof of Theorem 3.1. Now, we consider that \(x_{n}(t)+\gamma(t)\) has 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 nonnegative sign. Thus we have
So we have \(x_{n}(t)+\gamma(t)\geq\vartheta_{1}\).
The proof left is the same as in Theorem 3.1. □
Corollary 3.2
Assume that \(\rho<\frac{\pi}{\omega }\) holds. Suppose that the following condition is satisfied:
 (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
$$0\leq f(t,x)\leq\frac{d(t)}{x^{\tau}}+d(t)x^{\eta}\quad \textit{for all } x>0, \textit{for a.e. } t. $$
If \(\gamma_{*}>0\), then (1.3) has at least one positive ωperiodic solution.
Proof
We will apply Theorem 3.2. Take
then (H_{2}) is satisfied and the existence condition (H_{3}) becomes
where \(\Psi(t)=\int^{\omega}_{0}G(t,s)d(s)\,ds \) for some \(r>0\).
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
Assume that \(\rho<\frac{\pi}{\omega}\) and (H_{1})(H_{3}) hold. And the following condition is satisfied:
 (H_{4}):

\(\gamma_{*}+\Phi_{*}>0\), here \(\Phi(t)=\int^{\omega}_{0}G(t,s)\phi_{r+\gamma^{*}}(s)\,ds\).
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
We will follow the same strategy and notations as in the proof of Theorem 3.1. Step 1 and Step 2 are the same as in the proof of Theorem 3.1. Now, we mainly 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}) holds, we know 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
From (H_{4}) we have
Since \(x_{n}(t)+\gamma(t)\leq r+\gamma^{*}\) and \(x_{n}+\gamma_{*}\geq\frac{1}{n}\), then
i.e., \(x_{n}(t)+\gamma(t)\geq\vartheta_{2}\).
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
Assume that \(\rho<\frac{\pi}{\omega}\) and (H_{1})(H_{3}) hold. And the following condition is satisfied:
 (H_{5}):

\(\gamma_{*}+\Phi'_{*}>0\), here \(\Phi'(t)=\int^{\omega}_{0}G(t,s)\phi_{r}(s)\,ds\).
If \(\gamma^{*}\leq0\), 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
We will follow the same strategy and notations as in the proof of Theorem 3.1. Step 1 and Step 2 are the same as in the proof of Theorem 3.1. Now, we mainly consider that \(x_{n}(t)+\gamma(t)\) has a uniform positive lower bound, i.e., there exists a constant \(\vartheta_{3}>0\), independent of \(n\in N_{0}\), such that
for all \(n\in N_{0}\).
Since (H_{1}) and \(\gamma^{*}\leq0\), 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_{5}) then we have
Since \(x_{n}(t)+\gamma(t)\leq r+\gamma^{*}\leq r\) and \(x_{n}+\gamma_{*}\geq\frac{1}{n}\), we have
So we have \(x_{n}(t)+\gamma(t)\geq\vartheta_{3}\).
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
We will apply Theorem 3.4. Take
then (H_{1}) and (H_{2}) are satisfied. Since \(\gamma^{*}\leq0\) and \(\gamma_{*}>\frac{\hat{\Psi}_{*}}{r^{\tau}}\), we know that condition (H_{5}) holds.
Next, the existence condition (H_{3}) becomes
for some \(r>0\). Since \(\tau>0\), \(0\leq\eta<1\) and \(\gamma^{*}\leq0\), we can choose \(r>0\) large enough such that (3.13) is satisfied. □
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
Assume that \(\rho<\frac{\pi}{\omega}\) and (H_{1})(H_{2}) hold. Furthermore, assume that the following condition holds:
 (\(\mathrm{H}_{3}'\)):

There exists a positive constant \(R>0\) such that \(R>(\Phi_{R})_{*}+\gamma_{*}>0\) and
$$R\geq g\bigl((\Phi_{R})_{*}+\gamma_{*}\bigr) \biggl(1+\frac{h(R)}{g(R)} \biggr)K^{*}+\gamma^{*}, $$where \(\Phi_{R}(t)=\int^{\omega}_{0}G(t,s)\phi_{R}(s)\,ds\).
Then (1.3) has at least one positive periodic solution.
Proof
An ωperiodic solution of (1.3) is just a fixed point of the map \(T^{*}:X\rightarrow X\) defined by
Let R be a positive constant satisfying (\(\mathrm{H}_{3}'\)) and
Then we have \(R>r>0\). Now we define the set
Obviously, Ω is a closed convex set. Next we prove \(T^{*}(\Omega)\subset\Omega\).
In fact, for each \(x\in\Omega\) 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}), we get
where \(f^{+}(t,x)=\max\{0,f(t,x)\}\).
On the other hand, by conditions (H_{2}) and (\(\mathrm{H}_{3}'\)), we have
In conclusion, \(T(\Omega)\subset\Omega\). Say W is any bounded subset in Ω. Then, for \(\forall x\in W\), we have
where \(\gamma^{*}=\max_{t\in[0,\omega]}\gamma(t)\), \(f_{R}=\max_{r\leq x(t)\leq R}f(t,x(t))\), \(f_{R}_{1}=\int^{\omega}_{0}f_{R}\,dt=\omegaf_{R}\).
Due to the continuity of G, we have
where \(B'=\max \vert \frac{\partial G(t,s)}{\partial t}\vert \) for all \((t,s)\in[0,\omega]\times[0,\omega]\), \(e_{1}=\int^{\omega}_{0}e(s)\,ds\).
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
Assume that \(\rho<\frac{\pi}{\omega}\) holds and \(f(t,x)\) satisfies conditions (H_{1}) and (H_{2}). Furthermore, assume that the following condition holds:
 (\(\mathrm{H}_{3}''\)):

There exists a positive constant \(R>0\) such that \(R>(\Phi_{R})_{*}\) and
$$g\bigl((\Phi_{R})_{*}\bigr) \biggl(1+\frac{h(R)}{g(R)} \biggr)K^{*}+ \gamma^{*}\leq R. $$
If \(\gamma_{*}=0\), then (1.3) has at least one positive periodic solution.
Corollary 4.1
Assume that \(\rho<\frac{\pi}{\omega}\) holds. Assume that the following condition holds:
 (F_{3}):

There exist continuous functions \(d(t), \hat{d}(t)\succ0\) and \(0<\tau<1\) satisfying
$$0\leq\frac{\hat{d}(t)}{x^{\tau}}\leq f(t,x)\leq\frac{d(t)}{x^{\tau}} \quad \textit{for all } x>0 \textit{ and a.e. } t. $$
If \(\gamma_{*}=0\), then (1.3) has at least one positive periodic solution.
Proof
We will apply Theorem 4.2. We take
Then conditions (H_{1}) and (H_{2}) are satisfied and the existence condition (\(\mathrm{H}_{3}''\)) becomes
where \(\hat{\Psi}=\int^{\omega}_{0}G(t,s)\hat{d}(t)\,dt\), \(\Psi=\int^{\omega}_{0}G(t,s)d(t)\,dt\) for some \(R>0\). Note that \(\Psi_{*}>0\), since \(0<\tau<1\), we can choose \(R>0\) large enough so that (4.3) is satisfied and the proof is complete. □
Corollary 4.2
Assume that \(\rho<\frac{\pi}{\omega}\) holds. Assume that the following condition holds:
 (F_{4}):

There exist continuous functions \(d(t), \hat{d}(t)\succ0\) and \(0<\tau<1\), \(0\leq\eta<1\) satisfying
$$0\leq\frac{\hat{d}(t)}{x^{\tau}}\leq f(t,x)\leq\frac{d(t)}{x^{\tau}}+d(t)x^{\eta}\quad \textit{for all } x>0 \textit { and a.e. } t. $$
If \(\gamma_{*}=0\), then (1.3) has at least one positive periodic solution.
Proof
We will apply Theorem 4.2. We take
Then conditions (H_{1}) and (H_{2}) are satisfied and the existence condition (\(\mathrm{H}_{3}''\)) becomes
for some \(R>0\). So, we can choose \(R>0\) large enough so that (4.4) is satisfied and the proof is complete. □
The next results explore the case when \(\gamma_{*}>0\).
Theorem 4.3
Assume that \(\rho<\frac{\pi}{\omega}\) holds and \(f(t,x)\) satisfies (H_{2}). Furthermore, assume that the following condition holds:
 (\(\mathrm{H}_{3}'''\)):

There exists \(R>0\) such that
$$g(\gamma_{*}) \biggl(1+\frac{h(R)}{g(R)} \biggr)K^{*}+\gamma^{*}\leq R. $$
If \(\gamma_{*}>0\), then (1.3) has at least one positive periodic solution.
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 \(T^{*}(\Omega)\subset\Omega\).
For each \(x\in\Omega\) and for all \(t\in[0,\omega]\), by the nonnegative sign of \(G^{11}(t,s)\) and \(f(t,x)\), we have
On the other hand, by (H_{2}) and (\(\mathrm{H}_{3}'''\)), we have
In conclusion, \(T^{*}(\Omega)\subset\Omega\). Meanwhile, using the ArzelaAscoli theorem, it is easy to show that \(T^{*}\) is compact in Ω. Therefore, by Schauder’s fixed point theorem, our result is proven. □
Corollary 4.3
Assume that \(\rho<\frac{\pi}{\omega}\) holds. Assume that the following condition holds:
 (F_{5}):

There exist a continuous function \(d(t)\succ0\) and a constant \(\tau>0\) satisfying
$$0\leq f(t,x)\leq\frac{d(t)}{x^{\tau}} \quad \textit{for all } x>0 \textit{ and a.e. } t. $$
If \(\gamma_{*}>0\), then (1.3) has at least one positive periodic solution.
Proof
We apply Theorem 4.3. We take
Then condition (H_{2}) is satisfied and the existence condition (\(\mathrm{H}_{3}'''\)) is also satisfied if we take \(R>0\) with
□
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
We apply Theorem 4.3. We take
Then condition (H_{2}) is satisfied and the existence condition (\(\mathrm{H}_{3}'''\)) is also satisfied if we take \(R>0\) with
□
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
Assume that \(\rho<\frac{\pi}{\omega}\) holds. Furthermore, assume that the following conditions hold:
 (\(\mathrm{H}_{2}^{*}\)):

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

Let us define
$$R:=g(\gamma_{*})K^{*}+\gamma^{*}, $$and assume that \(f(t,x)\geq0\) for all \((t,x)\in[0,\omega]\times(0,R]\).
If \(\gamma_{*}>0\), then (1.3) has at least one positive periodic solution.
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
Assume that \(\rho<\frac{\pi}{\omega}\) holds. Assume that the following condition holds:
 (F_{6}):

There exist constants \(\tau, \eta, \mu>0\) satisfying
$$f(t,x)=\frac{1}{x^{\tau}}\mu x^{\eta}\quad \textit{for all } x>0 \textit{ and a.e. } t. $$
If \(\gamma_{*}>0\), then (1.3) has at least one positive periodic solution for each \(0\leq\mu<\mu'\), where \(\mu'\) is some positive constant.
Proof
The nonlinearity is
and therefore (\(\mathrm{H}_{2}^{*}\)) holds with \(k(t)=1\), \(g(x)=\frac{1}{x^{\tau}}\). Let \(\Upsilon(t)=\int^{\omega}_{0}G(t,s)\,ds\). Then R as defined in (\(\mathrm{H}_{3}^{*}\)) is just \(R=\frac{\Upsilon^{*}}{\gamma_{*}^{\tau}}+\gamma^{*}\). Note that \(f(t,x)\geq0\) if and only if \(x^{\tau+\eta}\leq1/ \mu\). Therefore, (\(\mathrm{H}_{3}^{*}\)) is verified for any \(\mu< R^{(\tau+\eta)}\). As a consequence, the result holds for
□
Finally, we consider \(\gamma^{*}\leq0\).
Theorem 4.5
Assume that \(\rho<\frac{\pi}{\omega}\) and (F_{3}) hold. If \(\gamma^{*}\leq0\) and
then there exists a positive ωperiodic solution of (1.3).
Proof
In this case, to prove that \(T^{*}(\Omega)\subset\Omega\) defined by (4.2), it is sufficient to find \(0< r< R\) such that
Taking \(R=\frac{\Psi^{*}}{r^{\tau}}\), the first inequality holds if r verifies
or equivalently,
The function \(f(r)\) possesses a minimum at \(r_{0}:= [\frac{\hat{\Psi}_{*}}{(\Psi^{*})^{\tau}}\tau^{2} ]^{\frac {1}{1\tau^{2}}}\). Let \(r=r_{0}\). Then the first inequality holds in (4.6) if \(\gamma_{*}\geq f(r_{0})\), which is just condition (4.5). The second inequality holds directly by the choice of R, and it would remain to prove that \(R=\frac{\Psi^{*}}{r_{0}^{\tau}}>r_{0}\). This is easily verified through elementary computations. Using the ArzelaAscoli theorem, it is easy to show that \(T^{*}\) is compact in Ω. Therefore, the proof is completed by Schauder’s fixed point theorem. □
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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).
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YX and ZC 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. Some results for fourthorder nonlinear differential equation with singularity. Bound Value Probl 2015, 200 (2015). https://doi.org/10.1186/s1366101504648
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DOI: https://doi.org/10.1186/s1366101504648