# Second-order differential equation with indefinite and repulsive
singularities 111 Yonghui Xia acknowledges the support from National Natural
Science Foundation of China under Grant (No. 11671176, 11931016).

Xiaoxiao Cuia Yonghui Xiaa222Corresponding author. Yonghui Xia,
xiadoc<EMAIL_ADDRESS>
a College of Mathematics and Computer Science, Zhejiang Normal University,
321004, Jinhua, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract—This paper concerns a second-order differential equation with
indefinite and repulsive singularities. It is the first time to study
differential equation containing both indefinite and repulsive singularities
simultaneously. A set of sufficient conditions are obtained for the existence
of positive periodic solutions. The theoretical underpinnings of this paper
are the positivity of Green’s function and fixed point theorem in cones. Our
results improve and extend the results in previous literatures. Finally, three
examples and their numerical simulations (phase diagrams and time diagrams of
periodic solutions) are given to show the effectiveness of our conclusions.

Keyword—indefinite and repulsive singularities; fixed point theorem in cones;
positive periodic solution.

MSC— 34B16; 34B18; 34C25.

##  1 Introduction

There are many works on the problem of periodic solutions for the nonlinear
ordinary differential equations with deviating argument, $p$-Laplacian,
neutral type operator and singularity (see e.g. [1, 2, 3, 4, 5, 6, 7, 9, 8,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52]). In this paper, we pay particular attention to the
differential equations with singularity. In fact, the differential equations
with singularity has attracted many mathematicians’ attention (see e.g.[3, 6,
8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30,
31, 32, 33, 35, 36, 37, 38, 43, 44, 45, 46, 47, 49, 51, 52]). By the method of
upper and lower solution, Lazer and Solimini [31] initially studied the
periodic solutions of the following singular equations

$x^{\prime\prime}(t)+\frac{1}{x^{\mu}(t)}=m(t),$ (1.1)

and

$x^{\prime\prime}(t)-\frac{1}{x^{\mu}(t)}=m(t),$ (1.2)

where $m(t)\in C(\mathbb{R},\mathbb{R})$ is a periodic function and $\mu>0$ is
a constant. It is said that equation (1.1) has an attractive singularity and
equation (1.2) has a repulsive singularity. Pino and Manásevich [43] studied
the periodic solutions for a problem arising in nonlinear elasticity. The
equations is formulated as

$x^{\prime\prime}(t)+f(t,x(t))=0.$ (1.3)

Further, Pino, Manásevich and Montero,[43, 42] considered the equation (1.3)
with singularities. Hakl and Torres [22] discussed equation (1.3) with
attractive-repulsive singularities:

$x^{\prime\prime}(t)=\frac{b(t)}{x^{\nu_{1}}(t)}-\frac{c(t)}{x^{\nu_{2}}(t)}+e(t),$
(1.4)

where $b(t),~{}c(t)\in C(\mathbb{R},(0,+\infty))$ are periodic functions,
$\nu_{1}>\nu_{2}>0$ are constants. The positivity of $b(t),~{}c(t)$ states the
equation has both attractive and repulsive singularities. Bravo and Torres [6]
investigated equation (1.3) with an indefinite singularity as follows:

$x^{\prime\prime}(t)=\frac{b(t)}{x^{3}(t)},$ (1.5)

where $b(t)$ is a piecewise-constant sign-changing function. Motivated by the
two works [6, 22], the mathematicians paid their attention to the singular
equations containing both attractive and repulsive singularities
simultaneously or indefinite singularity (see [8, 10, 21, 24, 35, 36, 37, 47,
52]). Thereinto, Hakl and Zamora [24], Godoy and Zamora [21] generalized
equation (1.5) in a wider application. In the recent years, Chu et. al. [16,
17, 28], Li and Zhang [33], Torres [45, 46] obtained some existence results
for positive periodic solutions of the following equation by some fixed point
theorems

${}x^{\prime\prime}(t)+a(t)x(t)=f(t,x)+e(t),$ (1.6)

where $f(t,x)$ may be singular at $x=0$. Subsequently, Cheng and Cui [8]
considered equation (1.6) with $f(t,x)$ an indefinite singularity, i.e.,

${}x^{\prime\prime}(t)+q(t)x(t)=\frac{b(t)}{x^{\rho}(t)}+e(t),$ (1.7)

where $\rho>0$ is a constant and $b(t)\in C(\mathbb{R},\mathbb{R})$ is a
periodic function.Lu et. al. [35, 36, 37] by means of coincidence degree
theory generalized indefinite singularity to Liénard equation.

In this paper, we study the following differential equation with indefinite
and repulsive singularities simultaneously:

$x^{\prime\prime}(t)+p(t)x^{\prime}(t)+q(t)x(t)=\frac{b(t)}{x^{\rho_{1}}(t)}+\frac{c(t)}{x^{\rho_{2}}(t)}+e(t),$
(1.8)

where $p(t),~{}q(t),~{}b(t)\in C(\mathbb{R},\mathbb{R})$ and $c(t),~{}e(t)\in
C(\mathbb{R},(0,+\infty))$ are $\omega$-periodic functions,
$\rho_{1},~{}\rho_{2}$ are positive constants. The term
$\frac{b(t)}{x^{\rho_{1}}(t)}$ has an indefinite singularity in view of the
uncertainty of the sign of weight function $b(t)$, even the singularity will
be disappeared if $b(t)=0$ in some subintervals. In addition to this, the
equation has a repulsive singularity because of the term
$\frac{c(t)}{x^{\rho_{2}}(t)}$ and $c(t)>0$. In particular, if equation (1.8)
has no damping term (i.e., $p(t)\equiv 0$), it reduces to the following
differential equation:

$x^{\prime\prime}(t)+q(t)x(t)=\frac{b(t)}{x^{\rho_{1}}(t)}+\frac{c(t)}{x^{\rho_{2}}(t)}+e(t).$
(1.9)

We generalize and improve the previous works in three aspects. Firstly,
equation (1.8) is investigated in a very general form containing both
indefinite singularity and repulsive singularity simultaneously. Obviously, it
is more general and complex than equations (1.3)-(1.7). In fact, equations
(1.3)-(1.7) either do not contain indefinite singularity or single
singularity. This means that the previous methods can not be directly applied
to equation (1.8) any more. Secondly, noticing the term
$\frac{b_{1}(t)}{x^{\nu_{1}}}-\frac{c_{1}(t)}{x^{\nu_{2}}}$, Chu et al. [12],
Cheng and Ren [10], Hakl and Torres [22] required $\nu_{1}>\nu_{2}$. In fact,
$\nu_{1}>\nu_{2}$ implies that equation (1.4) tends to be more repulsive
singularity. In this paper, we consider all the three cases,
$\rho_{1}>\rho_{2}$, $\rho_{1}=\rho_{2}$ and $\rho_{1}<\rho_{2}$.
Consequently, our results generalize and improve the existing literatures.
Thirdly, our results can be applied to both strong singularity and weak
singularity. In fact, following Lazer and Solimini’s work, there are many good
papers appearing in this field by different kinds of methods such as the fixed
point theorems [8, 9, 27], lower and upper solution [1, 22, 23], Leray-
Schauder alternative principle [12, 32, 28] and coincidence degree theory [5,
37, 29], and so on. Different from above methods, in this paper, we use the
theory of the fixed point in cones to study the periodic solution.

The rest structure of this paper is as follows. In section 2, the sufficient
conditions for the positivity of Green’s function of the second-order
differential equation are given. Our main results are stated in Section 3. In
section 4, three examples and their simulations are given to confirm our
conclusions.

##  2 Positivity of Green’s function

In order to make good use of fixed point theorem to get the existence of
positive periodic solution for equation (3.1), first of all we need to
guarantee the invariance of the sign of Green’s function of the nonhomogeneous
linear equation corresponding to equation (3.1). According to the specific
situation of this paper, we consider the positivity of Green’s function.

Let $C_{\omega}^{1}:=\\{\phi(t),~{}\phi^{\prime}(t)\in
C(\mathbb{R},\mathbb{R}):~{}\phi(t+\omega)=\phi(t),~{}\phi^{\prime}(t+\omega)=\phi^{\prime}(t),~{}t\in\mathbb{R}\\}$,
equipped with the norm
$\|\phi\|=\|\phi\|_{\infty}+\|\phi^{\prime}\|_{\infty}$, and $C_{\omega}^{1}$
in the norm $\|\cdot\|$ sense is a Banach space, here
$\|\phi\|_{\infty}=\max\limits_{t\in[0,\omega]}|\phi(t)|$. For a continuous
function $\phi(t)$, we define $\phi^{*}:=\max\limits_{0\leq
t\leq\omega}\phi(t)$ and $\phi_{*}:=\min\limits_{0\leq t\leq\omega}\phi(t)$.

Considering the following second-order nonhomogeneous linear differential
equation,

$\begin{cases}u^{\prime\prime}(t)+p(t)u^{\prime}(t)+l(t)u(t)=h(t),\\\
u(0)=u(\omega),~{}u^{\prime}(0)=u^{\prime}(\omega),\end{cases}$ (2.1)

where $l(t)=\frac{q(t)}{\alpha}$, $h(t)\in C(\mathbb{R},(0,+\infty))$ is a
continuous $\omega$-periodic function. The unique $\omega$-periodic solution
of equation (2.1) is expressed as

$u(t)=\int^{\omega}_{0}G(t,s)h(s)ds,$

where $G(t,s)$ is Green’s function of equation (2.1). Now we talk about
sufficient conditions to make $G(t,s)>0$ such that $\omega$-periodic solution
of equation (2.1) is positive. To describe the problem conveniently, we assume
the following condition holds:

$(A)$ The Green’s function $G(t,s)>0$ for all
$(t,s)\in[0,\omega]\times[0,\omega]$.

In 2005, Wang et al.[48, Lemma 2.4] proved condition $(A)$ can be satisfied
under the following two conditions:

$(A_{1})$ There are continuous $\omega$-periodic functions $a_{1}(t)$ and
$a_{2}(t)$ such that
$\int^{\omega}_{0}a_{1}(t)dt>0,~{}~{}\int^{\omega}_{0}a_{2}(t)dt>0$ and

$a_{1}(t)+a_{2}(t)=p(t),~{}~{}~{}~{}a_{1}^{\prime}(t)+a_{1}(t)a_{2}(t)=l(t),~{}~{}~{}~{}\mbox{for}~{}~{}t\in\mathbb{R}.$

$(A_{2})$ $\left(\int^{\omega}_{0}p(t)dt\right)^{2}\geq
4\omega^{2}\exp\left(\frac{1}{\omega}\int^{\omega}_{0}\ln l(s)ds\right).$

On the basis of Wang’s work, Cheng and Ren gave the following conclusion:

######  Lemma 2.1.

[10, Lemma 2.2] Assume that condition $(A_{1})$ holds, then condition $(A)$
holds.

In 2012, Chu et al. [12] also proved condition $(A)$ by using antimaximum
principle. First, authors defined two functions

$\varsigma(p)(t)=exp\left(\int^{t}_{0}p(s)ds\right)$

and

$\varsigma_{1}(p)(t)=\varsigma(p)(\omega)\int^{t}_{0}p(s)ds+\int^{\omega}_{t}\varsigma(p)(s)ds,$

and then gave the following theorem:

######  Lemma 2.2.

[12, Corollary 2.6] Assume that

$\int^{\omega}_{0}l(s)\varsigma(p)(s)\varsigma_{1}(-p)(s)ds\geq 0$

and

$\sup\limits_{0\leq
t\leq\omega}\left\\{\int^{t+\omega}_{t}\varsigma(-p)(s)ds\int^{t+\omega}_{t}\max\\{l(s),0\\}\varsigma(p)(s)ds\right\\}\leq
4$

hold. If $l(t)\not\equiv 0$, then condition $(A)$ holds.

In particular, as the damping term of equation (2.1) is absent (i.e.,
$p(t)\equiv 0$), a specific value about Green’s function can be obtained, and
condition $(A)$ is ensured.

######  Lemma 2.3.

[25, Lemma 2.5] In the case $l(t)=\xi^{2}$ with $\xi>0$, the expression of the
Green’s function as follows,

$G(t,s)=\left\\{\begin{aligned}
&\frac{\cos\xi(t-s-\frac{\omega}{2})}{2\xi\sin\frac{\xi\omega}{2}},~{}~{}~{}~{}0\leq
s\leq t\leq\omega,\\\
&\frac{\cos\xi(t-s+\frac{\omega}{2})}{2\xi\sin\frac{\xi\omega}{2}},~{}~{}~{}~{}0\leq
t<s\leq\omega.\end{aligned}\right.$ (2.2)

If

$\xi<\frac{\pi}{\omega},$ (2.3)

then condition $(A)$ holds.

Under the circumstances that condition $(A)$ holds, one can say

$G^{*}:=\max\limits_{0\leq s,t\leq\omega}G(t,s)>G_{*}:=\min\limits_{0\leq
s,t\leq\omega}G(t,s)>0,$

and we define

$\sigma:=\frac{G_{*}}{G^{*}+\max\limits_{0\leq
s,t\leq\omega}\left|\frac{\partial G(t,s)}{\partial
t}\right|},~{}~{}\delta:=\frac{\max\limits_{0\leq
s,t\leq\omega}\left|\frac{\partial G(t,s)}{\partial t}\right|}{G_{*}}.$

##  3 Existence of positive periodic solution for equation (1.8)

Our main results and their proofs are presented in this section. To study
eqution (1.8), we make a transformation of variable $x=y^{\alpha}$ and
$\alpha=\frac{1}{\rho_{1}+1}$. Then equation (1.8) is rewritten as follows:

$y^{\prime\prime}(t)+p(t)y^{\prime}(t)+\frac{q(t)}{\alpha}y(t)=\frac{c(t)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(t)+\frac{e(t)}{\alpha}y^{1-\alpha}(t)+(1-\alpha)\frac{|y^{\prime}(t)|^{2}}{y(t)}+\frac{b(t)}{\alpha}.$
(3.1)

Obviously, by this transformation, it no longer contains indefinite
singularity. We see that the term $(1-\alpha)\frac{|y^{\prime}(t)|^{2}}{y(t)}$
has a repulsive singularity. Now let’s consider the term
$\frac{c(t)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(t)$:

$(i)$ If $\rho_{1}>\rho_{2}$ (i.e. $0<1-\alpha-\alpha\rho_{2}<1$), the term
has no singularity, in this case, equation (3.1) has just a repulsive
singularity;

$(ii)$ if $\rho_{1}<\rho_{2}$ (i.e. $1-\alpha-\alpha\rho_{2}<0$), the term has
a repulsive singularity, in this case, equation (3.1) has two repulsive
singularities;

$(iii)$ if $\rho_{1}=\rho_{2}$ (i.e. $1-\alpha-\alpha\rho_{2}=0$), same
situation as $(i)$.

Consequently, the existence of positive periodic solution for equation (1.8)
is equivalent to that of equation (3.1). To prove the existence of positive
periodic solutions, we need a fixed point theorem in cones (see [40]).

######  Lemma 3.1.

Let $X$ be a Banach space and $K$ is a cone in $X$. Assume that
$\Lambda_{1},~{}\Lambda_{2}$ are open subsets of $X$ with
$0\in\Lambda_{1},~{}\overline{\Lambda}_{1}\subset\Lambda_{2}$. Let

$\mathcal{T}:K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1})\rightarrow K$

be a continuous and completely continuous operator such that

$(i)$ $\|\mathcal{T}y\|\leq\|y\|$ for $y\in K\cap\partial\Lambda_{2}$;

$(ii)$ there exists $y_{0}\in K\backslash\\{0\\}$ such that
$y\neq\mathcal{T}y+\lambda y_{0}$ for $y\in K\cap\partial\Lambda_{1}$ and
$\lambda>0$.
Then $\mathcal{T}$ has a fixed point in
$K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1})$.

All the discussion throughout this paper takes place in Banach space
$C^{1}_{\omega}$, define a cone $K$ in space $C^{1}_{\omega}$ as follows:

$K=\\{y\in C^{1}_{\omega}:~{}\min
y(t)\geq\sigma\|y\|,~{}|y^{\prime}(t)|\leq\delta y(t),~{}0\leq
t\leq\omega\\}.$

In what follows, we discuss the periodic solutions with three cases:
$\rho_{1}>\rho_{2}$, $\rho_{1}<\rho_{2}$ and $\rho_{1}=\rho_{2}$.

######  Theorem 3.1.

Assume that condition $(A)$ and $\rho_{1}>\rho_{2}$ hold. Furthermore, suppose
there exist $R>r>0$ such that:

$(H_{1})$
$\frac{1}{\sigma}\left(-\frac{b_{*}}{e_{*}}\right)^{\frac{1}{1-\alpha}}\leq
r\leq\left(\frac{G_{*}c_{*}\omega\sigma^{1-\alpha-\alpha\rho_{2}}}{\alpha}\right)^{\frac{1}{\alpha+\alpha\rho_{2}}}$.

$(H_{2})$ $\frac{(1-\alpha)(G^{*}+\delta G_{*})\delta^{2}\omega}{\sigma}<1$.
Then there is at least one positive $\omega$-periodic solution of equation
(3.1), saying $y(t)$ satisfying $r\leq\|y\|\leq R$. Consequently, it is a
positive periodic solution of equation (1.8).

###### Proof.

To use the fixed point in cones, take

$\Lambda_{1}:=\\{y\in C_{\omega}^{1}:\|y\|<r\\},~{}~{}~{}\Lambda_{2}:=\\{y\in
C_{\omega}^{1}:\|y\|<R\\},$

and define an operator $\mathcal{T}$ as

$(\mathcal{T}y)(t)=\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds,$

then a fixed point of operator equation $y=\mathcal{T}y$ is an
$\omega$-periodic solution of equation (3.1). To this end, we divide it into
four steps.

Step 1. We prove
$\mathcal{T}(K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1}))\subset K$. In
fact, for $\forall~{}y\in K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1})$,
by definitions of $K,~{}\Lambda_{1}$ and $\Lambda_{2}$, one can say

$\sigma r\leq\sigma\|y\|\leq y(t)\leq\|y\|_{\infty}\leq\|y\|\leq
R~{}\mbox{and}~{}|y^{\prime}(t)|\leq\delta y(t)\leq\delta R.$

It follows from the first half of condition $(H_{1})$ that

$\begin{split}&\frac{c(t)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(t)+\frac{e(t)}{\alpha}y^{1-\alpha}(t)+(1-\alpha)\frac{|y^{\prime}(t)|^{2}}{y(t)}+\frac{b(t)}{\alpha}\\\
\geq&\frac{c(t)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(t)+\frac{e(t)}{\alpha}y^{1-\alpha}(t)+\frac{b(t)}{\alpha}\\\
\geq&\frac{c_{*}\sigma^{1-\alpha-\alpha\rho_{2}}}{\alpha}r^{1-\alpha-\alpha\rho_{2}}+\frac{e_{*}\sigma^{1-\alpha}}{\alpha}r^{1-\alpha}+\frac{b_{*}}{\alpha}\\\
>&\frac{e_{*}\sigma^{1-\alpha}}{\alpha}r^{1-\alpha}+\frac{b_{*}}{\alpha}\\\
\geq&0.\end{split}$ (3.2)

On the basis of condition $(A)$ and (3.2), we have the following two
inequalities,

$\begin{split}&\min\limits_{0\leq t\leq\omega}(\mathcal{T}y)(t)\\\
=&\min\limits_{0\leq
t\leq\omega}\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
\geq&G_{*}\int^{\omega}_{0}\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
=&\sigma\left(G^{*}+\max\limits_{0\leq s,t\leq\omega}\left|\frac{\partial
G(t,s)}{\partial
t}\right|\right)\int^{\omega}_{0}\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
=&\sigma
G^{*}\int^{\omega}_{0}\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
&+\sigma\max\limits_{0\leq s,t\leq\omega}\left|\frac{\partial G(t,s)}{\partial
t}\right|\int^{\omega}_{0}\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
\geq&\sigma\max\limits_{0\leq
t\leq\omega}\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
&+\sigma\max\limits_{0\leq t\leq\omega}\int^{\omega}_{0}\left|\frac{\partial
G(t,s)}{\partial
t}\right|\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
=&\sigma\|\mathcal{T}y\|_{\infty}+\sigma\|(\mathcal{T}y)^{\prime}\|_{\infty}=\sigma\|\mathcal{T}y\|,\end{split}$

and

$\begin{split}|(\mathcal{T}y)^{\prime}(t)|=&\left|\int^{\omega}_{0}\frac{\partial
G(t,s)}{\partial
t}\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\right|\\\
\leq&\int^{\omega}_{0}\left|\frac{\partial G(t,s)}{\partial
t}\right|\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
\leq&\max\limits_{0\leq t\leq\omega}\left|\frac{\partial G(t,s)}{\partial
t}\right|\int^{\omega}_{0}\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
=&\delta
G_{*}\int^{\omega}_{0}\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
\leq&\delta\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
=&\delta(\mathcal{T}y)(t).\end{split}$

This shows that
$\mathcal{T}(K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1}))\subset K$.

Step 2. We prove that $\mathcal{T}$ is a completely continuous operator. For
any $y\in K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1})$, we have

$\begin{split}|\mathcal{T}y|=&\left|\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\right|\\\
=&\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
\leq&G^{*}\left(\frac{c^{*}}{\alpha}R^{1-\alpha-\alpha\rho_{2}}+\frac{e^{*}}{\alpha}R^{1-\alpha}+(1-\alpha)\frac{(\delta
R)^{2}}{\sigma r}+\frac{b^{*}}{\alpha}\right)\omega:=N_{1},\end{split}$

and

$\begin{split}|(\mathcal{T}y)^{\prime}(t)|=&\left|\int^{\omega}_{0}\frac{\partial
G(t,s)}{\partial
t}\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\right|\\\
\leq&\delta
G_{*}\int^{\omega}_{0}\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
\leq&\delta
G_{*}\left(\frac{c^{*}}{\alpha}R^{1-\alpha-\alpha\rho_{2}}+\frac{e^{*}}{\alpha}R^{1-\alpha}+(1-\alpha)\frac{(\delta
R)^{2}}{\sigma r}+\frac{b^{*}}{\alpha}\right)\omega:=N_{2}.\end{split}$

Furthermore, for any $y_{1},~{}y_{2}\in
K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1})$ with $y_{1}\neq y_{2}$,

$|\mathcal{T}y_{1}-\mathcal{T}y_{2}|=\left|\frac{d\mathcal{T}(\theta
y_{1}+(1-\theta)y_{2})}{dt}(y_{1}-y_{2})\right|\leq N_{2}|y_{1}-y_{2}|,$

where $0<\theta<1$. That is to say, $\mathcal{T}$ is a uniformly bounded and
equicontinuity operator, it follows the Arzela-Ascoli theorem that
$\mathcal{T}:K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1})\rightarrow K$
is a continuous and completely continuous operator.

Step 3. We prove $(i)$ of Lemma 3.1, i.e.,

$\|\mathcal{T}y\|\leq\|y\|,~{}~{}\mbox{for}~{}~{}y\in
K\cap\partial\Lambda_{2}.$ (3.3)

In view of $y\in K\cap\partial\Lambda_{2}$, there is

$\sigma R\leq
y(t)\leq\|y\|_{\infty}\leq\|y\|=R~{}\mbox{and}~{}|y^{\prime}|\leq\delta
y(t)\leq\delta R.$

Therefore, we arrive at

$\begin{split}\|\mathcal{T}y\|=&\|\mathcal{T}y\|_{\infty}+\|(\mathcal{T}y)^{\prime}\|_{\infty}\\\
=&\max\limits_{0\leq
t\leq\omega}\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
&+\max\limits_{0\leq t\leq\omega}\int^{\omega}_{0}\left|\frac{\partial
G(t,s)}{\partial
t}\right|\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
\leq&(G^{*}+\delta
G_{*})\left(\frac{c^{*}}{\alpha}R^{1-\alpha-\alpha\rho_{2}}+\frac{e^{*}}{\alpha}R^{1-\alpha}+\frac{(1-\alpha)\delta^{2}}{\sigma}R+\frac{b^{*}}{\alpha}\right)\omega.\end{split}$
(3.4)

In view of $1-\alpha-\alpha\rho_{2}<1$ and $1-\alpha<1$, condition $(H_{2})$
implies that there is a sufficiently large $R$ such that

$(G^{*}+\delta
G_{*})\left(\frac{c^{*}}{\alpha}R^{1-\alpha-\alpha\rho_{2}}+\frac{e^{*}}{\alpha}R^{1-\alpha}+\frac{(1-\alpha)\delta^{2}}{\sigma}R+\frac{b^{*}}{\alpha}\right)\omega\leq
R=\|y\|.$

Therefore, inequality (3.3) holds.

Step 4. We prove $(ii)$ of Lemma 3.1. To do so, let $y_{0}=1$, then $y_{0}\in
K\backslash\\{0\\}$. Now we prove that

$y\neq\mathcal{T}y+\lambda y_{0},~{}~{}\mbox{for}~{}~{}y\in
K\cap\partial\Lambda_{1},~{}~{}\lambda>0$ (3.5)

by way of contradiction. If inequality (3.5) does not hold, we can assume that
there exist $y_{1}\in K\cap\partial\Lambda_{1}$ and $\lambda_{0}>0$ such that

$y_{1}=\mathcal{T}y_{1}+\lambda_{0}y_{0}.$

To be sure, for $y_{1}\in K\cap\partial\Lambda_{1}$, it holds that

$\sigma r=\sigma\|y_{1}\|\leq
y_{1}(t)\leq\|y_{1}\|_{\infty}\leq\|y_{1}\|=r~{}\mbox{and}~{}|y^{\prime}_{1}(t)|\leq\delta
y_{1}(t)\leq\delta r.$

Therefore, from condition $(H_{1})$, we obtain

$\begin{split}y_{1}=&\mathcal{T}y_{1}+\lambda_{0}y_{0}\\\
=&\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y_{1}^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y_{1}^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}_{1}(s)|^{2}}{y_{1}(s)}+\frac{b(s)}{\alpha}\right)ds+\lambda_{0}\\\
>&\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y_{1}^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y_{1}^{1-\alpha}(s)+\frac{b(s)}{\alpha}\right)ds\\\
\geq&G_{*}\left(\frac{c_{*}}{\alpha}(\sigma
r)^{1-\alpha-\alpha\rho_{2}}+\frac{e_{*}}{\alpha}(\sigma
r)^{1-\alpha}+\frac{b_{*}}{\alpha}\right)\omega\\\
\geq&\frac{G_{*}c_{*}\sigma^{1-\alpha-\alpha\rho_{2}}\omega}{\alpha}r^{1-\alpha-\alpha\rho_{2}}\\\
\geq&r=\|y_{1}\|,\end{split}$ (3.6)

which contradicts with $y_{1}\in K\cap\partial\Lambda_{1}$. Hence, inequality
(3.5) holds.

It follows from above four steps that all the conditions of Lemma 3.1 are
satisfied. Therefore, it follows that $\mathcal{T}$ has a fixed point $y\in
K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1})$. Obviously, the fixed
point is an $\omega$-periodic solution of equation (3.1). That is, equation
(3.1) has at least one positive $\omega$-periodic solution $y(t)$ satisfying
$r\leq\|y\|\leq R$. ∎

######  Corollary 3.1.

Assume that inequality (2.3), conditions $(H_{1}),~{}(H_{2})$ and
$\rho_{1}>\rho_{2}$ hold. Then equation (1.9) has at least one positive
$\omega$-periodic solution.

######  Theorem 3.2.

Assume that conditions $(A),~{}(H_{2})$ and $\rho_{1}<\rho_{2}$ hold.
Furthermore, suppose there exist $R>r>0$ such that:

$(H_{3})$
$\frac{1}{\sigma}\left(-\frac{b_{*}}{e_{*}}\right)^{\frac{1}{1-\alpha}}\leq
r\leq\left(\frac{G_{*}c_{*}\omega}{\alpha}\right)^{\frac{1}{\alpha+\alpha\rho_{2}}}$.
Then there is at least one positive $\omega$-periodic solution of equation
(3.1), saying $y(t)$ satisfying $r\leq\|y\|\leq R$. Consequently, it is a
positive periodic solution of equation (1.8).

###### Proof.

We follow the same strategy and notations as in the proof of Theorem 3.1. Now
we prove (3.3), similar to (3.4), by using condition $(H_{2})$ and
$\rho_{1}<\rho_{2}$, one can prove that

$\begin{split}\|\mathcal{T}y\|=&\|\mathcal{T}y\|_{\infty}+\|(\mathcal{T}y)^{\prime}\|_{\infty}\\\
=&\max\limits_{0\leq
t\leq\omega}\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
&+\max\limits_{0\leq t\leq\omega}\int^{\omega}_{0}\left|\frac{\partial
G(t,s)}{\partial
t}\right|\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
\leq&(G^{*}+\delta
G_{*})\left(\frac{c^{*}\sigma^{1-\alpha-\alpha\rho_{2}}}{\alpha}R^{1-\alpha-\alpha\rho_{2}}+\frac{e^{*}}{\alpha}R^{1-\alpha}+\frac{(1-\alpha)\delta^{2}}{\sigma}R+\frac{b^{*}}{\alpha}\right)\omega\\\
\leq&R=\|y\|\end{split}$

holds if $R$ is large enough. This means that inequality (3.3) holds.

In what follows, we prove that inequality (3.5), from condition $(H_{3})$ and
$\rho_{1}<\rho_{2}$, (3.6) gives

$\begin{split}y_{1}=&\mathcal{T}y_{1}+\lambda_{0}y_{0}\\\
=&\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y_{1}^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y_{1}^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}_{1}(s)|^{2}}{y_{1}(s)}+\frac{b(s)}{\alpha}\right)ds+\lambda_{0}\\\
>&G_{*}\left(\frac{c_{*}}{\alpha}r^{1-\alpha-\alpha\rho_{2}}+\frac{e_{*}}{\alpha}(\sigma
r)^{1-\alpha}+\frac{b_{*}}{\alpha}\right)\omega\\\
\geq&\frac{G_{*}c_{*}\omega}{\alpha}r^{1-\alpha-\alpha\rho_{2}}\\\
\geq&r=\|y_{1}\|,\end{split}$

which contradicts with $y_{1}\in K\cap\partial\Lambda_{1}$. Then inequality
(3.5) follows. Therefore, the conclusion follows from Lemma 3.1 immediately. ∎

######  Corollary 3.2.

Assume that inequality (2.3), conditions $(H_{2}),~{}(H_{3})$ and
$\rho_{1}<\rho_{2}$ hold. Then there is at least one positive
$\omega$-periodic solution of equation (1.9).

######  Theorem 3.3.

Assume that conditions $(A),~{}(H_{2})$ and $\rho_{1}=\rho_{2}$ hold.
Furthermore, suppose there exist $R>r>0$ such that one of the following two
cases holds:

Case I: $b_{*}+c_{*}\geq 0$;

Case II: $b_{*}+c_{*}<0$ and condition $(H_{4})$ hold:

$(H_{4})$
$\frac{1}{\sigma}\left(-\frac{b_{*}+c_{*}}{e_{*}}\right)^{\frac{1}{1-\alpha}}<r\leq\frac{G_{*}\bar{b}^{+}\omega}{\alpha}$,
where
$b^{+}(t):=\max\\{0,b(t)\\},~{}\bar{b}^{+}:=\frac{1}{\omega}\int_{0}^{\omega}b^{+}(t)dt$.
Then there is at least one positive $\omega$-periodic solution of equation
(3.1), saying $y(t)$ satisfying $r\leq\|y\|\leq R$. Consequently, it is a
positive periodic solution of equation (1.8).

###### Proof.

If $\rho_{1}=\rho_{2}$, equation (3.1) can be rewritten as

$y^{\prime\prime}(t)+p(t)y^{\prime}(t)+\frac{q(t)}{\alpha}y(t)=\frac{e(t)}{\alpha}y^{1-\alpha}(t)+(1-\alpha)\frac{|y^{\prime}(t)|^{2}}{y(t)}+\frac{b(t)+c(t)}{\alpha},$

and operator $\mathcal{T}$ as

$(\mathcal{T}y)(t)=\int^{\omega}_{0}G(t,s)\left(\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)+c(s)}{\alpha}\right)ds.$

Similar to the proof of Theorem 3.1. For $\forall~{}y\in
K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1})$, we have

$\begin{split}&\frac{e(t)}{\alpha}y^{1-\alpha}(t)+(1-\alpha)\frac{|y^{\prime}(t)|^{2}}{y(t)}+\frac{b(t)+c(t)}{\alpha}\\\
\geq&\frac{e(t)}{\alpha}y^{1-\alpha}(t)+\frac{b(t)+c(t)}{\alpha}\\\
\geq&\frac{e_{*}\sigma^{1-\alpha}}{\alpha}r^{1-\alpha}+\frac{b_{*}+c_{*}}{\alpha}\\\
>&0.\end{split}$

Case I: if $b_{*}+c_{*}\geq 0$, it is obvious that the above inequality holds;

Case II: if $b_{*}+c_{*}<0$, from the first half part of condition $(H_{4})$,
we can get the above inequality holds as well.

Furthermore, one can prove that
$\mathcal{T}(K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1}))\subset K$ and
$\mathcal{T}:K\cap(\overline{\Lambda}_{2}\backslash\Lambda_{1})\rightarrow K$
is a continuous and completely continuous operator.

Now we prove inequality (3.3), similar to (3.4), by using condition $(H_{2})$
and $\rho_{1}=\rho_{2}$, one can prove that

$\begin{split}\|\mathcal{T}y\|=&\|\mathcal{T}y\|_{\infty}+\|(\mathcal{T}y)^{\prime}\|_{\infty}\\\
=&\max\limits_{0\leq
t\leq\omega}\int^{\omega}_{0}G(t,s)\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
&+\max\limits_{0\leq t\leq\omega}\int^{\omega}_{0}\left|\frac{\partial
G(t,s)}{\partial
t}\right|\left(\frac{c(s)}{\alpha}y^{1-\alpha-\alpha\rho_{2}}(s)+\frac{e(s)}{\alpha}y^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}(s)|^{2}}{y(s)}+\frac{b(s)}{\alpha}\right)ds\\\
\leq&(G^{*}+\delta
G_{*})\left(\frac{e^{*}}{\alpha}R^{1-\alpha}+\frac{(1-\alpha)\delta^{2}}{\sigma}R+\frac{b^{*}+c^{*}}{\alpha}\right)\omega\\\
\leq&R=\|y\|\end{split}$

holds if $R$ is large enough. This means that inequality (3.3) holds.

In the following, we prove that inequality (3.5).

Case I: if $b_{*}+c_{*}\geq 0$, the inequality

$\begin{split}y_{1}=&\mathcal{T}y_{1}+\lambda_{0}y_{0}\\\
=&\int^{\omega}_{0}G(t,s)\left(\frac{e(s)}{\alpha}y_{1}^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}_{1}(s)|^{2}}{y_{1}(s)}+\frac{b(s)+c(s)}{\alpha}\right)ds+\lambda_{0}\\\
>&G_{*}\left(\frac{e_{*}}{\alpha}(\sigma
r)^{1-\alpha}+\frac{b_{*}+c_{*}}{\alpha}\right)\omega\\\
\geq&\frac{G_{*}e_{*}\sigma^{1-\alpha}\omega}{\alpha}r^{1-\alpha}\\\
\geq&r=\|y_{1}\|\end{split}$ (3.7)

can be satisfied if $r>0$ is small enough.

Case II: if $b_{*}+c_{*}<0$, from condition $(H_{4})$, we have

$\begin{split}y_{1}=&\int^{\omega}_{0}G(t,s)\left(\frac{e(s)}{\alpha}y_{1}^{1-\alpha}(s)+(1-\alpha)\frac{|y^{\prime}_{1}(s)|^{2}}{y_{1}(s)}+\frac{b(s)+c(s)}{\alpha}\right)ds+\lambda_{0}\\\
>&G_{*}\int^{\omega}_{0}\left(\frac{e_{*}}{\alpha}(\sigma
r)^{1-\alpha}+\frac{b^{+}(s)}{\alpha}+\frac{b^{-}(s)}{\alpha}+\frac{c_{*}}{\alpha}\right)ds\\\
\geq&G_{*}\int^{\omega}_{0}\left(\frac{e_{*}\sigma^{1-\alpha}}{\alpha}r^{1-\alpha}+\frac{b^{+}(s)}{\alpha}+\frac{b_{*}}{\alpha}+\frac{c_{*}}{\alpha}\right)ds\\\
>&G_{*}\int^{\omega}_{0}\frac{b^{+}(s)}{\alpha}ds\\\
=&\frac{G_{*}\bar{b}^{+}\omega}{\alpha}\\\ \geq&r=\|y_{1}\|,\end{split}$ (3.8)

where $b^{-}(t):=\min\\{0,b(t)\\}$. Both cases I and II contradict with
$y_{1}\in K\cap\partial\Lambda_{1}$, so inequality (3.5) follows.

Therefore, Lemma 3.1 shows equation (3.1) has a positive $\omega$-periodic
solution $y(t)$ satisfying $r\leq\|y\|\leq R$. ∎

######  Corollary 3.3.

Assume that equation (2.3), condition $(H_{2})$ and $\rho_{1}=\rho_{2}$ hold.
Then equation (1.9) has at least one positive $\omega$-periodic solution
$y(t)$ with $r\leq\|y\|\leq R$ if one of the following two cases holds:

Case I: $b_{*}+c_{*}\geq 0$;

Case II: $b_{*}+c_{*}<0$ and condition $(H_{4})$ hold.

######  Remark 3.1.

Corollary 3.3 generalizes the results in literature [8].

##  4 Example

In this section, we give three examples and the simulations to illustrate our
results.

######  Example 4.1.

Consider differential equation with indefinite and repulsive singularities:

${}x^{\prime\prime}+\frac{1}{40}x=\frac{1+2\cos(3t)}{x^{\frac{3}{2}}}+\frac{e^{2\sin(3t)}}{x^{\frac{13}{10}}}+10+\cos(3t).$
(4.1)

Comparing equation (4.1) with equation (1.9), we see that $q(t)=\frac{1}{40}$,
$b(t)=1+2\cos(3t)$, $b_{*}=-1$, $c(t)=e^{2\sin(3t)}$, $c_{*}=e^{-2}$,
$e(t)=10+\cos(3t)$, $e_{*}=9$, $\rho_{1}=\frac{3}{2}>\rho_{2}=\frac{13}{10}$,
so we can apply Corollary 3.1.

In view of $\alpha=\frac{1}{1+\rho_{1}}=\frac{2}{5}$,
$\frac{q(t)}{\alpha}=\frac{1}{16}$,
$\xi=\sqrt{\frac{q(t)}{\alpha}}=\frac{1}{4}$, $\omega=\frac{2\pi}{3}$, we know
$\xi<\frac{\pi}{\omega}$, equation (2.3) hold. From equation (2.2), we can
obtain $G_{*}=\frac{2\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}$,
$G^{*}=\frac{4}{\sqrt{2-\sqrt{3}}}$, $\max\limits_{0\leq
s,t\leq\omega}\left|\frac{\partial G(t,s)}{\partial
t}\right|=\frac{\sqrt{2-\sqrt{3}}}{2}$,
$\sigma=\frac{G_{*}}{G^{*}+\max\limits_{0\leq
s,t\leq\omega}\left|\frac{\partial G(t,s)}{\partial
t}\right|}=\frac{4\sqrt{2+\sqrt{3}}}{10-\sqrt{3}}$,
$\delta=\frac{\max\limits_{0\leq s,t\leq\omega}\left|\frac{\partial
G(t,s)}{\partial t}\right|}{G_{*}}=\frac{2-\sqrt{3}}{4\sqrt{2+\sqrt{3}}}$.

Conditions $(H_{1})$ and $(H_{2})$ can be satisfied by calculation. So by
using Corollary 3.1, equation (4.1) has at least one positive
$\frac{2\pi}{3}$-periodic solution.

Figure 1: Phase diagram of the $\frac{2\pi}{3}$-periodic solution and its time
series diagram. (a) Phase diagram of the periodic solution with initial value
$(399.93015,0)$. (b) Time series diagram of the periodic solution.

######  Example 4.2.

Consider differential equation with indefinite and repulsive singularities:

${}x^{\prime\prime}+\frac{1}{40}x=\frac{1+2\cos(3t)}{x^{\frac{3}{2}}}+\frac{e^{2\sin(3t)}}{x^{2}}+10+\cos(3t).$
(4.2)

Since $\rho_{1}=\frac{3}{2}<\rho_{2}=2$, we can apply Corollary 3.2. Similar
to Example 4.1, we can get equation (2.3), conditions $(H_{2})$ and $(H_{3})$
hold. So by using Corollary 3.2, equation (4.2) has at least one positive
$\frac{2\pi}{3}$-periodic solution.

Figure 2: Phase diagram of the $\frac{2\pi}{3}$-periodic solution and its time
series diagram. (a) Phase diagram of the periodic solution with initial value
$(399.8941,0)$. (b) Time series diagram of the periodic solution.

######  Example 4.3.

Consider differential equation with indefinite and repulsive singularities:

${}x^{\prime\prime}+\frac{1}{40}x=\frac{1+2\cos(3t)}{x^{\frac{3}{2}}}+\frac{e^{2\sin(3t)}}{x^{\frac{3}{2}}}+10+\cos(3t).$
(4.3)

Obviously, $\rho_{1}=\frac{3}{2}=\rho_{2}$ and $b_{*}+c_{*}<0$. Moreover,

$\bar{b}^{+}=\frac{1}{\omega}\int_{0}^{\omega}b^{+}(t)dt=\frac{3}{2\pi}\int_{-\frac{2\pi}{9}}^{\frac{2\pi}{9}}(1+2\cos(3t))dt=\frac{2}{3}+\frac{\sqrt{3}}{\pi}.$

By Example 4.1, we can get equation (2.3), conditions $(H_{2})$ and $(H_{4})$
hold. So by using Case II of Corollary 3.3, equation (4.3) has at least one
positive $\frac{2\pi}{3}$-periodic solution.

Figure 3: Phase diagram of the $\frac{2\pi}{3}$-periodic solution and its time
series diagram. (a) Phase diagram of the periodic solution with initial value
$(399.9045,0)$. (b) Time series diagram of the periodic solution.

## Data Availability Statement

My manuscript has no associated data. It is pure mathematics.

## Conflict of interest statement

The authors declare that there is no conflict of interests regarding the
publication of this article.

## Contributions

We declare that all the authors have same contributions to this paper.

## References

  * [1] P. Amster, M. Mariani, Oscillating solutions of a nonlinear fourth order ordinary differential equation, J. Math. Anal. Appl., 325 (2007) 1133-1141.
  * [2] C. Bereanu, Periodic solutions of some fourth-order nonlinear differential equations, Nonlinear Anal., 71 (2009) 53-57.
  * [3] C. Bereanu, J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular $\varphi$-Laplacian, J. Differential Equations, 243 (2007) 536-557.
  * [4] A. Boscaggin, G. Feltrin, Pairs of positive radial solutions for a Minkowski-curvature Neumann problem with indefinite weight, Nonlinear Anal., 196 (2020) 14 pp.
  * [5] A. Boscaggin, G. Feltrin, Positive periodic solutions to an indefinite Minkowski-curvature equation, J. Differential Equations, 269 (2020) 5595-5645.
  * [6] J. Bravo, P. Torres, Periodic solutions of a singular equation with indefinite weight, Adv. Nonlinear Stud., 10 (2010) 927-938.
  * [7] A. Cabada, S. Lois, Maximum principles for fourth and sixth order periodic boundary value problems, Nonlinear Anal., 29 (1997) 1161-1171.
  * [8] Z. Cheng, X. Cui, Positive periodic solution to an indefinite singular equation, Appl. Math. Lett., 112 (2021) 106740.
  * [9] Z. Cheng, X. Cui, Positive periodic solution for a second-order damped singular equation via fixed point theorem in cones, Bull. Malays. Math. Sci. Soc., 44 (2021) 2675-2691.
  * [10] Z. Cheng, J. Ren, Periodic solution for second order damped differential equations with attractive-repulsive singularities, Rocky Mountain J. Math., 48 (2018) 753-768.
  * [11] W. Cheung, J. Ren, Periodic solutions for $p$-Laplacian Rayleigh equations, Nonlinear Anal., 65 (2006) 2003-2012.
  * [12] J. Chu, N. Fan, P. Torres, Periodic solutions for second order singular damped differential equations, J. Math. Anal. Appl., 388 (2012) 665-675.
  * [13] J. Chu, M. Li, Positive periodic solutions of Hill’s equations with singular nonlinear perturbations, Nonlinear Anal., 69 (2008) 276-286.
  * [14] J. Chu, M. Li, Twist periodic solutions of second order singular differential equations, J. Math. Anal. Appl., 355 (2009) 830-838.
  * [15] J. Chu, J. Nieto, Impulsive periodic solutions of first-order singular differential equations, Bull. Lond. Math. Soc., 40 (2008) 143-150.
  * [16] J. Chu, P. Torres, Applications of Schauder’s fixed point theorem to singular differential equations, Bull. Lond. Math. Soc., 39 (2007) 653-660.
  * [17] J. Chu, P. Torres, M. Zhang, Periodic solutions of second order non-autonomous singular dynamical systems, J. Differential Equations, 239 (2007) 196-212.
  * [18] C. Fabry, J. Mawhin, M. Nkashma, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull. Lond. Math. Soc., 18 (1986) 173-180.
  * [19] A. Fonda, A. Sfecci, On a singular periodic Ambrosetti-Prodi problem, Nonlinear Anal., 149 (2017) 146-155.
  * [20] J. Godoy, R. Hakl, X. Yu, Existence and multiplicity of periodic solutions to differential equations with attractive singularities, Proc. R. Soc. Edinb., Sect. A, Math., (2021) 1-26.
  * [21] J. Godoy, M. Zamora, Periodic solutions for a second-order differential equation with indefinite weak singularity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019) 1135-1152.
  * [22] R. Hakl, P. Torres, On periodic solutions of second-order differential equations with attractive-repulsive singularities, J. Differential Equations, 248 (2010) 111-126.
  * [23] R. Hakl, P. Torres, M. Zamora, Periodic solutions of singular second order differential equations: Upper and lower functions, Nonlinear Anal., 74 (2011) 7078-7093.
  * [24] R. Hakl, M. Zamora, Periodic solutions to second-order indefinite singular equations, J. Differential Equations, 263 (2017) 451-469.
  * [25] W. Han, J. Ren, Some results on second-order neutral functional differential equations with infinite distributed delay, Nonlinear Anal., 70 (2009) 1393-1406.
  * [26] P. Jebelean, J. Mawhin, Periodic solutions of singular nonlinear perturbations of the ordinary $p$-Laplacian, Adv. Nonlinear Stud., 2 (2002) 299-312.
  * [27] D. Jiang, J. Chu, D. O’Regan, R. Agarwal, Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces, J. Math. Anal. Appl., 286 (2003) 563-576.
  * [28] D. Jiang, J. Chu, M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. Differential Equations, 211 (2005) 282-302.
  * [29] F. Kong, S. Lu, Existence of positive periodic solutions of fourth-order singular $p$-Laplacian neutral functional differential equations, Filomat, 31 (2017) 5855-5868.
  * [30] F. Kong, S. Lu, Z. Liang, Existence of positive periodic solutions for neutral Liénard differential equations with a singularity, Electron. J. Differential Equations, 242 (2015) 1-12.
  * [31] A. Lazer, S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc., 99 (1987) 109-114.
  * [32] S. Li, Y. Wang, Multiplicity of positive periodic solutions to second order singular dynamical systems, Mediterr. J. Math., 14 (2017) 13 pp.
  * [33] X. Li, Z. Zhang, Periodic solutions for second order differential equations with a singular nonlinearity, Nonlinear Anal., 69 (2008) 3866-3876.
  * [34] S. Lu, W. Ge, Z. Zhen, Periodic solutions for a kind of Rayleigh equation with a deviating argument, Appl. Math. Lett., 17 (2004) 443-449.
  * [35] S. Lu, Y. Guo, L. Chen, Periodic solutions for Liénard equation with an indefinite singularity, Nonlinear Anal. Real World Appl., 45 (2019) 542-556.
  * [36] S. Lu, R. Xue, Periodic solutions for a singular Liénard equation with indefinite weight, Topol. Methods Nonlinear Anal., 54 (2019) 203-218.
  * [37] S. Lu, X. Yu, Existence of positive periodic solutions for Liénard equations with an indefinite singularity of attractive type, Bound. Value Probl., 101 (2018) 19 pp.
  * [38] R. Martins, Existence of periodic solutions for second-order differential equations with singularities and the strong force condition, J. Math. Anal. Appl., 317 (2006) 1-13.
  * [39] P. Omari, F. Zanolin, On forced nonlinear oscillations in nth order differential systems with geometric conditions, Nonlinear Anal., 8 (1984) 723-748.
  * [40] D. O’Regan, Existence Theory for Nonlinear Ordinary Differential Equations, Kluwer Academic Publishers Group, Dordrecht, (1997).
  * [41] S. Peng, S. Zhu, Periodic solutions for $p$-Laplacian Rayleigh equations with a deviating argument, Nonlinear Anal., 67 (2007) 138-146.
  * [42] M. Pino, R. Manásevich, Infinitely many $T$-periodic solutions for a problem arising in nonlinear elasticity, J. Differential Equations, 103 (1993) 260-277.
  * [43] M. Pino, R. Manásevich, A. Montero, $T$-periodic solutions for some second order differential equations with singularities, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992) 231-243.
  * [44] P. Torres, Mathematical models with singularities-A zoo of singular creatures. Atlantis Briefs in Differential Equations, Atlantis Press, Paris, (2015).
  * [45] P. Torres, Bounded solutions in singular equations of repulsive type, Nonlinear Anal., 32 (1998) 117-125.
  * [46] P. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations, 232 (2007) 277-284.
  * [47] A. Ureña, Periodic solutions of singular equations, Topol. Methods Nonlinear Anal., 47 (2016) 55-72.
  * [48] Y. Wang, H. Lian, W. Ge, Periodic solutions for a second order nonlinear functional differential equation, Appl. Math. Lett., 20 (2007) 110-115.
  * [49] Z. Wang, Periodic solutions of Liénard equation with a singularity and a deviating argument, Nonlinear Anal. Real World Appl., 16 (2014) 227-234.
  * [50] Z. Wang, L. Qian, S. Lu, On the existence of periodic solutions to a fourth-order $p$-Laplacian differential equation with a deviating argument, Nonlinear Anal. Real World Appl., 11 (2010) 1660-1669.
  * [51] X. Yu, S. Lu, A multiplicity result for periodic solutions of Liénard equations with an attractive singularity, Appl. Math. Comput., 346 (2019) 183-192.
  * [52] X. Yu, S. Lu, F. Kong, Existence and multiplicity of positive periodic solutions to Minkowski-curvature equations without coercivity condition, J. Math. Anal. Appl., 507 (2022) 15 pp.


W_1^*  &=   \sum_{j=1}^m  \big (  \mathbb{E}^*_j
- \mathbb{E}^*_{j-1} \big)
r_j{^*}^\top D_j^* (z_1)^{-2} D_j^* (z_2)^{-1} r_j^* ~,
\\
W_2^* &=
\sum_{j=1}^m  \big (  \mathbb{E}^*_j
- \mathbb{E}^*_{j-1} \big)  \Big [
\beta_j^*(z_1)
r_j{^*}^\top D_j^* (z_1)^{-2} D_j^* (z_2)^{-1} r_j^*  \gamma_j^*(z_1)
\Big   ] ~.
\end{align*}

Case (a): ${m^{3+\varepsilon} \over n} =o(1)$, $\mathbb{E}[ |X_1|^{4 + \varepsilon}]$
Note that

\begin{align}
\label{det41}
| H_{2n} |^{1+r} \leq  c \big (
| b_n^*(z_1)W_1^*  |^{1+r}  + |  b_n^*(z_1)W_2^* |^{1+r}\big ) ~,
\end{align}

and that
Hölder's inequality gives for both terms on the right hand side

\begin{align}
\label{det42}
\mathbb{E} \big [ | b_n^*(z_1)  W_j^* |^{1+r}
\big ] \leq \big (  \mathbb{E} \big [ | b_n^*(z_1) |^{2(1+r)/(1-r)} \big ]  \big )^{(1-r)/2} \big (
\E \big [ | W_j^* |^{2}
\big ] \big )^{(1+r)/2} ~~~~~~~~~(j=1,2).
\end{align}

We now show that both factors terms on the right hand side are bounded, that is

\begin{align}
\label{det43a}
\mathbb{E} \big [ | b_n^*(z) |^{q(r)}\big ]  &\leq c
\\
\E | W_j |^{2}
&\leq c  ~~~~~~~~~(j=1,2)
\label{det43b}
\end{align}

for $z \in  \mathcal{C}_n$ and $q(r)=q = 2(1+r)/(1-r)$ with a
sufficiently. small number $r> 0$.

Proof of (<ref>)
We recall the definition of the set $\mathcal{A}_n$ in (<ref>), the estimate
in (<ref>) and use the decomposition

| b_n^*(z) | = | b_n^*(z) | ( \mathds 1_{\mathcal{A}_n}+\mathds 1_{\mathcal{A}_n^c}).

For the second  term we have

\begin{align*}
\E \big [ \mathds 1_{\mathcal{A}_n^c} | b_n^*(z) |^q \big ] \leq \Big  | {z \over \Im (z)} \Big |^q
\mathbb{P} ( \mathcal{A}_n^c )
\leq m^{q} \varepsilon^{-q}
\leq m^{q(1+\alpha )} \mathbb{P} ( \mathcal{A}_n^c )  = o(1),
\end{align*}

by (<ref>). For the first term we note that on the set
$\mathcal{A}_n$ we have $ \|  D^*(z)^{-1} \|_{S_\infty} < c$
for some constant  by the arguments given  in the derivation of (<ref>). Moreover, we have

\begin{align}
\|r_1^* \|_2^2  & =  \|  r_1^* {r_1^*}^\top  \|_{S_\infty}
\leq  \Big \|  \sum_{j=1}^m r_j^* {r_j^*}^\top  \Big \|_{S_\infty}  +\Big \|  \sum_{j=2}^m r_j^* {r_j^*}^\top  \Big \|_{S_\infty}  \leq 2 K_l,
\end{align}

and,  recalling the definition of $D^*$ in (<ref>) yields

\begin{equation}
\label{det62}
|  \beta_1^*(z) | = | 1 -{r_j^*}^\top  D^*(z)^{-1}{r_j^*} |  \leq 1  + \|  D^*(z)^{-1} \|_{S_\infty}
\| {r_j^*} \|^2 < c
\end{equation}

on $\mathcal{A}_n$ for some positive constant $c$.
This  gives
\begin{align}
\label{det44}
\E \big [ \mathds 1_{\mathcal{A}_n} | b_n^*(z) |^q \big ]  \leq c \Big (  1    +
\E \big [
\mathds 1_{\mathcal{A}_n}
| b_n^*(z) |^q
|   \gamma_1^*(z) |^q
\big ]
\Big ).
\end{align}

Next, we show $ \E \big [ \mathds 1_{\mathcal{A}_n}  |  \gamma_1^*(z) |^q \big ]$ converges to $0$.
For this purpose we recall the definition of $\varepsilon_1^* (z) $ in (<ref>) and obtain, using the same arguments as on page 580 in  baisil2004, the bound

\begin{align}
\nonumber
\E \big [ \mathds 1_{\mathcal{A}_n}   |  \gamma_1^*(z)  - \varepsilon_1^* (z)   |^q \big ]
& \leq {c \over m^{q/2} }
\E \big [ \mathds 1_{\mathcal{A}_n} \big |
\beta_{12}^*(z) {r_2^*}^\top  D_{12}^*(z)^{-1}L_nL_n^\top
\big  |^q \big ]
\end{align}

where $\beta_{12}^*(z) $ is defined in (<ref>)
and satisfies $  |
\beta_{12}^*(z) | \leq c $ on ${\mathcal{A}_n} $
which follows by similar arguments as in ??.
Similarly, w$ \|  D^*_{12} (z)^{-1} \|_{S_\infty} < c$
on the set
$\mathcal{A}_n$, which  gives

\begin{align}
\E \big [ \mathds 1_{\mathcal{A}_n}   |  \gamma_1^*(z)  - \varepsilon_1^* (z)   |^q \big ]
& \leq {c \over m^{q/2} } \mathbb{P} ( \mathcal{A}_n )
\leq  {c \over m^{q/2} } ~.
\label{det50a}
\end{align}

An application of Proposition <ref>
with the matrix (<ref>) then gives

\begin{align}
\label{det50}
\E \big [ \mathds 1_{\mathcal{A}_n}   |  \gamma_1^*(z)  |^q \big ]
& \leq  c \Big (  \E \big [ \mathds 1_{\mathcal{A}_n}   |  \varepsilon_1^* (z)   |^q \big ] +
{1 \over m^{q/2} }  \Big )  \leq c \Big (
{\delta_m^{2q-4} \over m } +
{m \over n } +  {1 \over m^{q/2} }
\Big )  .
\end{align}

We now continue with  (<ref>) using Hölder's inequality  (with $q=1/a$ and $q'=1/(1-a)$ for any $a \in (0,1)$)

\begin{align}
\label{det51}
\E \big [ \mathds 1_{\mathcal{A}_n} | b_n^*(z) |^q \big ]  & \leq c \Big  \{
1 +
\Big | { z \over \Im (z) } \Big |^{q(1-a)}
\E \big [
\mathds 1_{\mathcal{A}_n}
| b_n^*(z) |^{qa}
|   \gamma_1^*(z) |^{q} \big ]
\Big \}  \\
\nonumber
& \leq
c \Big \{
1 +
m^{q(1-a)} \varepsilon_n^{-q(1-a)}
\big (   \E \big [ \mathds 1_{\mathcal{A}_n} | b_n^*(z) |^q \big ]  \big )^{a}
( \E \big [ \mathds 1_{\mathcal{A}_n}   |  \gamma_1^*(z)  |^{q/(1-a)} )^{1-a}
\Big \}
\\
\nonumber
& \leq
c \Big \{
1 +  \big (   \E \big [ \mathds 1_{\mathcal{A}_n} | b_n^*(z) |^q \big ]  \big )^{a} \kappa_n
\Big \}  ~,
\end{align}

where the sequence $\kappa_n$ is defined by

\begin{align*}
\kappa_n & =
m^{q(1-a) }  \varepsilon_n^{-q(1-a)}
\Big (  {\delta_n^{2q-4(1-a)} \over m^{1-a}} + \Big ( {m \over n } \Big )^{1-a}
+{1 \over m^{q/2}}
\Big )
\end{align*}

Note that this sequence converges to $0$ if we choose
$n=m^{3 + \epsilon}$, $\delta_2$ and $\alpha$  sufficiently small This is wrong !

Proof of (<ref>)

Note that

W_1^*  =   \sum_{j=1}^m
\big (  \mathbb{E}^*_j
- \mathbb{E}^*_{j-1} \big)
\Big \{
r_j{^*}^\top D_j^* (z_1)^{-2} D_j^* (z_2)^{-1} r_j^*
-  {1 \over m} \tr  \big (
L_n L_n^\top  D_j^* (z_1)^{-2}
D_j^* (z_2)^{-1} \big )
\Big
\}

and Proposition <ref> gives

\begin{align}
\nonumber
\mathbb{E} \big [  | W_1 |^2
\big ]
&= \mathbb{E} \big [  \mathbb{E}^* [
| W_1 |^2] \big ] \\
&= \sum_{j=1}^m\mathbb{E}
\Big [
\Big |
\big (  \mathbb{E}^*_j
- \mathbb{E}^*_{j-1} \big) \Big \{ r_j{^*}^\top D_j^* (z_1)^{-2} D_j^* (z_2)^{-1} r_j^*
- {1 \over m } {\rm tr}  \big (
L_n^\top D_j^* (z_1)^{-2} D_j^* (z_2)^{-1}L_n
\big ) \Big ) \Big |^2
\Big ]
\nonumber \\
\nonumber
% \label{det2}
\leq  4 m
\mathbb{E}
\Big [ \Big |
r_1{^*}^\top D_1^* (z_1)^{-2} D_1^* (z_2)^{-1} r_1^*
- {1 \over m } {\rm tr}  \big (
L_n^\top D_1^* (z_1)^{-2} D_1^* (z_2)^{-1}L_n
\big ) \Big |^2
\Big ]   \\
& \leq 4 m K_2 \Big ( {1\over m}  + {m^2 \over m } \big )
\leq 4 K_p (1 + o(1))
\nonumber
\end{align}

We now consider the term $W_2$, that is

\begin{align*}
\mathbb{E}[ | W_2^*||^2 ]  &=     \mathbb{E}\big [ \mathbb{E} [| W_2|^*|^2 ]  \big ] \\
&=   \sum_{j=1}^m \mathbb{E} \Big [ \Big |
\big (  \mathbb{E}^*_j
- \mathbb{E}^*_{j-1} \big)  \big [
\beta_j^*(z_1)
r_j{^*}^\top D_j^* (z_1)^{-2} D_j^* (z_2)^{-1} r_j^*  \gamma_j^*(z_1)
\big   ] \Big |^2 \Big ] \\
& \leq  4m \mathbb{E} \big |
\beta_1^*(z_1)
r_1{^*}^\top D_1^* (z_1)^{-2} D_1^* (z_2)^{-1} r_1^*  \gamma_1^*(z_1)
\big |^2
\end{align*}

We now use the decomposition $\mathds{1} = \mathds{1}_{\mathcal{A}_n} +  \mathds{1}_{\mathcal{A}_n^c}$,   (<ref>) and (<ref>)
to obtain

\begin{align*}
\mathbb{E}[ | W_2^*||^2 ]  & \leq  4m \mathbb{E} \big |
\mathds{1}_{\mathcal{A}_n}
\gamma_1^*(z_1)
\big |^2   + o(1)  =O(1),
\end{align*}

where the last equality follows from (<ref>).

The  term $H_{3n}$ in the decomposition ??
can be bounded in the same way.

For the first term $H_{1n}$ we use the same arguments as in baisil2004 and obtain the representation

\begin{align*}
b_n^*(z_1)b_n^*(z_2)Y_1^* -b_n^*(z_2)Y_2^* -b_n^*(z_1)b_n^*(z_2)Y_3^*
\end{align*}

\begin{align*}
Y_1^* &= \sum_{j=1}^m
\big ( \mathbb{E}_{j}^* -\mathbb{E}_{j-1}^* \big )
\big [ \big (r_j{^*}^\top (D_j^* (z_1))^{-1} (D_j^* (z_2))^{-1} r_j^* \big ) ^2   - \big ( {1 \over m }  {\rm tr } \big ( L_n (D_j^* (z_1))^ {-1} (D_j^* (z_2) )^{-1}
L_n^\top  \big ) \big )^2 \big ]
\\
Y_2^* &=   \sum_{j=1}^m
\big ( \mathbb{E}_{j}^* -\mathbb{E}_{j-1}^* \big ) \beta_j^* (z_1) \beta_j^* (z_2) \big (r_j{^*}^\top (D_j^* (z_1))^{-1} (D_j^* (z_2))^{-1} r_j^* \big ) ^2 \gamma_j^* (z_2)  \\
Y_3^* &=   \sum_{j=1}^m
\big ( \mathbb{E}_{j}^* -\mathbb{E}_{j-1}^* \big ) \beta_j^* (z_1) \beta_j^* (z_2) \big (r_j{^*}^\top (D_j^* (z_1))^{-1} (D_j^* (z_2))^{-1} r_j^* \big ) ^2 \gamma_j^* (z_1) \\
\end{align*}

This yields

\begin{align*}
\mathbb{E} [ |H _{1n} |^{1+r} ]
& \leq(\big (  \mathbb{E} \big [ | b_n^*(z_1) b_n^*(z_2) |^{2(1+r)/(1-r)} \big ]  \big )^{(1-r)/2} \big (
\E \big [ | Y_1^* |^{2}
\big ] \big )^{(1+r)/2}   \\
& + (\big (  \mathbb{E} \big [ |  b_n^*(z_2) |^{2(1+r)/(1-r)} \big ]  \big )^{(1-r)/2} \big (
\E \big [ | Y_2^* |^{2}
\big ] \big )^{(1+r)/2}  \\
& +
(\big (  \mathbb{E} \big [ | b_n^*(z_1) b_n^*(z_2) |^{2(1+r)/(1-r)} \big ]  \big )^{(1-r)/2} \big (
\E \big [ | Y_3^* |^{2}
\big ] \big )^{(1+r)/2}
\end{align*}

In the following we will show
that all three terms in this expression. We restrict ourselves to the first term, the other terms can be estimated similarly.

Todo: checken und gfs einfügen ! By similar arguments as given in the derivation of ??  we can show that

\begin{align*}
\mathbb{E} \big [ |  b_n^*(z_1)b_n^*(z_2) |^q \big ]  \leq c
\end{align*}

where $q=q(r)= {2(1+r)/(1-r)}$, and therefore it remains to derive a bound for  $\mathbb{E} |Y_1^*|^2$.
To estimate this moment

\begin{align*}
\mathbb{E} |Y_1^*|^2 & =
\sum_{j=1}^{m} \E
\Big |  \big ( \mathbb{E}_{j}^* -\mathbb{E}_{j-1}^* \big )
\big [ \big (r_j{^*}^\top (D_j^* (z_1))^{-1} (D_j^* (z_2))^{-1} r_j^* \big ) ^2   - \big ( {1 \over m }  {\rm tr } \big ( L_n (D_j^* (z_1))^ {-1} (D_j^* (z_2) )^{-1}
L_n^\top  \big ) \big )^2 \big ]
\Big |^2 \\
&  \leq
4 m \E
\Big |
\big (r_1{^*}^\top (D_1^* (z_1))^{-1} (D_1^* (z_2))^{-1} r_1^* \big ) ^2   - \big ( {1 \over m }  {\rm tr } \big ( L_n (D_1^* (z_1))^ {-1} (D_1^* (z_2) )^{-1}
L_n^\top  \big ) \big )^2
\Big |^2
\end{align*}

We now use  the decomposition
$\mathds{1} = \mathds{1}_{\mathcal{A}_n} +  \mathds{1}_{\mathcal{A}_n^c}$ and the identity
check $|a^2-b^2|^2 = |a-b|^4 + 2( \bar a b + a\bar  b)
|a-b|^2$ and obtain

\begin{align*}
&  \E  \mathds{1}_{\mathcal{A}_n}
\Big |
\big (r_1{^*}^\top (D_1^* (z_1))^{-1} (D_1^* (z_2))^{-1} r_1^* \big ) ^2   - \big ( {1 \over m }  {\rm tr } \big ( L_n (D_1^* (z_1))^ {-1} (D_1^* (z_2) )^{-1}
L_n^\top  \big ) \big )^2  \Big |^2 \\
& \leq    \mathbb{E}  \Big |
r_1{^*}^\top (D_1^* (z_1))^{-1} (D_1^* (z_2))^{-1} r_1^*    -  {1 \over m }  {\rm tr } \big ( L_n (D_1^* (z_1))^ {-1} (D_1^* (z_2) )^{-1}
L_n^\top  \big ) \Big |^4 \\
& + 4  \mathbb{E}  \Big [ \Big |
r_1{^*}^\top (D_1^* (z_1))^{-1} (D_1^* (z_2))^{-1} r_1^*    -  {1 \over m }  {\rm tr } \big ( L_n (D_1^* (z_1))^ {-1} (D_1^* (z_2) )^{-1}
L_n^\top  \big ) \Big |^2
\\
& \times
\mathds{1}_{\mathcal{A}_n}
\| (D_1^* (z_1))^ {-1} (D_1^* (z_2) )^{-1} \|_{{S}_\infty}^2  {q \over m} \|r_1^* \|^2
\Big ]  \\
&  = O \Big (  {1 \over m} \Big )
\end{align*}

where we have used Proposition <ref> for the matrix   (<ref>).  The bound on $\E |Y_1^*|^2$ now follows observing (<ref>).

Case (b): $f$ is analytic on  a sufficiently large interval If

x_l < K_l -  \limsup_{n \to \infty}  \| \Sigma_n \|_{S_\infty} ~~\text{ and }~x_r < K_r +  \limsup_{n \to \infty}  \| \Sigma_n \|_{S_\infty}

it follows that

\| D_1^*(z)^{-1}\|_{S_\infty}
\leq  \max \Big \{
{1 \over | x_l -K_l| },  {1 \over | x_r -K_r| }
\Big \}  <  {1  \over \limsup_{n \to \infty}  \| \Sigma_n \|_{S_\infty} }

on the set $\mathcal{A}_n$. This implies that

\limsup_{n \to \infty} {1 \over m } \Big |   \mathbb{E}^* \tr \Big ( L_nL_n^\top D_1^*(z)^{-1} \Big) \Big |
\leq
\limsup_{n \to \infty}  \|  L_nL_n^\top \|_{S_\infty}
\| D_1^*(z)^{-1}\|_{S_\infty} < 1 .

Consequently, we obtain the on the set $\mathcal{A}_n$ that

\limsup_{n \to \infty} | b_n(z) |
\leq
\limsup_{n \to \infty} {1 \over 1 -  \|  L_nL_n^\top \|_{S_\infty}
\| D_1^*(z)^{-1}\|_{S_\infty} } < \infty .

Wir müssen jetzt noch die Arugmentation aus dem ersten Teil aufgreifen  !!

ich bin nicht sicher, ob das wirklich gut in die Einleitung poasst. Eigentlich ziehen wir mit zuücklegen - oder ?Eventuell ans Ende von Kapitel 4! Es wäre schön, noch an einem Bsp. zu demonstrieren, dass unsere Vorausetzungen an $m$ scharf sind!
The challenges  in the construction of a consistent  ”$m$ out of $n$” bootstrap can easily be visualized in the example $T_n(f)={\rm tr} (\widehat \Sigma_n^2)$. If the population matrix is given by $\Sigma_n = I_p$, a simple calculation gives for the expectation of this statistic

\begin{equation}\label{ex4a}
\mathbb{E} \Big[{\rm tr} \big(\widehat \Sigma_n^2 \big)\Big] = \frac {p}{n} (p+1+n )= n (1+c) c \cdot (1+o(1))
\end{equation}

as $n,p \to \infty$ and $p/n \to c$. Now
let $\widehat \Sigma^*_n=m^{-1}q\sum_{i=1}^mZ_i^*(Z_i^*)^\top$  denote the corresponding matrix which is obtained by first sampling $m$ times with replacement from $Y_1, \ldots, Y_n$,
and then  resampling $q$ coordinates   as in Example <ref>  (i.e.
then drawing $q$ components randomly). Then it is easy to see

\widehat \Sigma^*_n  ~{\stackrel {\mathcal{D}}{=}} ~\frac {1}{m} \sum^n_{i=1} W_i \Pi_n Y_i (\Pi_n Y_i)^\top  ,

where $W=(W_1,\ldots,W_n)$ has a multinomial  $(m, \frac {1}{n}, \ldots, \frac {1}{n})$ distribution independent of ${\cal Y} =(Y_{1}, \ldots , Y_{n})$
and $\Pi_n$ denotes a random variable independent of  ${\cal Y} $ and $W$ and uniformly distributed on the set of all matrices $A \in \{ 0,1 \}^{q \times p}$ with exactly one entry equal to $1$ in each row and at most one entry equal to $1$ in each column korrekt?  Da müssen wir ganz genau aufpassen - ziehen mit oder ohne zurücklegen !  Vorne schrieben wir ohne - wie argumentieren wir hier ??. A straightforward calculation finally gives

\begin{eqnarray*}
\mathbb{E} \Big [ \mbox{tr} \big (  ( \widehat \Sigma^*_n )^2  \big ) \big |  {\cal Y} \Big ]
&=& \frac {1}{n} \big(1 - \frac {1}{n} + \frac {m}{n}\big) \frac{1}{m} \sum^n_{i=1} \| \Pi_n Y_i \|^4 + \frac {1}{n^2} \frac {m-1}{m} \sum_{i_1 \neq i_2} (\Pi_n Y_{i_1})^\top   (\Pi_n Y_{i_2} ) \\
& {\stackrel {\mathcal{D}}{=}}  & \frac {1}{nm} \big ( 1- \frac {1}{n} + \frac {m}{n}\big) \sum^n_{i=1} \| Y_i^{(1)} \|^4  +  \frac {1}{n^2} \frac {m-1}{m}  \sum_{i_1 \neq i_2} Y^{(1)^\top  }_{i_1} Y^{(1)}_{i_2}
\end{eqnarray*}

where $Y^{(1)}_1 , \ldots , Y^{(1)}_n  \in \mathbb{R}^{q}$ denote the vectors containing the first $q$ components of the vectors $Y_1 , \ldots , Y_{n} \in \mathbb{R}^p$, respectively  (note  that $\Sigma_{n} =I_{p}$, and by (<ref>)
the components of $Y_{i}$ are independent identically distributed).  Consequently, we obtain

\begin{eqnarray}
\label{h4}
\mathbb{E} \Big [ \mbox{tr} \big (  ( \widehat \Sigma^*_n )^2  \big )  ] &=& \frac {1}{m} \big ( 1 - \frac {1}{n} + \frac {m}{n} \big) (2 q + q^2) + \frac {n-1}{n} \frac {m-1}{m} q \\
&=& \frac {q}{m}\Big \{ \big( 1 - \frac {1}{n} + \frac {m}{n} \big) (2+q) + \frac {n-1}{n} (m-1) \Big \}~.
\nonumber
\end{eqnarray}

From this expression we already see that the classical  bootstrap  ”sampling with replacement”, which corresponds to $m=n$, $q=p$ yields to a bias,
that is $ \mathbb{E}  [  \mbox{tr} ( ( \widehat \Sigma^*_n  )^2)  ] =  nc (2c+1) \cdot (1+o(1))$.
On the other hand, if $m=o(1)$ and $q/m \to c$  the expectation in  (<ref>)
is of the order $mc (1+c) \cdot (1+o(1))$ indicating consistency of our approach.

On the other hand, for the variance additional conditions on $m \to \infty$ are required. It is well known (and relatively easy to verify) that

{\rm Var} \big( {\rm tr} \big ( \widehat \Sigma_n ^2\big)  \big) = (8c + 20c^2 +8c^3) \cdot  (1+o(1))

as $p/n \to \infty$. However, a similar statement holds for the bootstrap version only if $m$ is substantially smaller than $n$. More precisely, it can be shown

\mathbb{E} \Big [ {\rm Var} \Big ( {\rm tr} \big ( ( \widehat \Sigma^*_n)^2  \big) \mid {\cal Y} \Big) \Big ] = (8c + 20c^2 + 8c^3) (1+o(1))

if $m = o (\sqrt{n})$.