Oscillation analysis of solutions of non-zero continuous linear functional equations

Abstract

As a mathematical model of mechanical and electronic oscillation, the study and analysis of the oscillation characteristics of the solution of the non-zero continuous linear functional equation are of great significance in theory and practice. In view of the oscillation characteristics of the solutions of the second and third order non-zero continuous functional equations, this paper puts forward a hypothesis, studies the oscillation and asymptotics of the non-zero continuous linear functional differential equations by using the generalized Riccati transformation and the integral average technique, and establishes some new sufficient conditions for the oscillation or convergence to zero of all solutions of the equations, so as to obtain a new theorem for the solutions of the non-zero continuous linear functional equations.

Keywords

Non-zero continuous linear functional equation oscillation characteristic

1. Introduction

Oscillation phenomena are common in daily production and life, such as mechanical oscillation, vocal cord oscillation, electromagnetic oscillation, thermal motion and atomic motion [1, 2, 3]. Because of the complexity of oscillation, people often simplify the hypothesis, establish the corresponding mathematical model, and use a relatively simple mathematical method to describe the complex oscillation problem, that is, the oscillation theory of the dynamic equation [4, 5, 6]. Vibration theory of dynamic equation is an important branch of qualitative theory of differential equation, which has important application value in control engineering, mechanical oscillation, biopharmaceutical, mechanics and so on [7, 8, 9]. This has attracted the attention and research of scholars in various fields.

In recent years, the oscillation of analytical solutions of non-zero continuous linear functional equations have been widely concerned, e.g., in reference [10, 11]. Because there are mathematical models in biology, medicine and other fields, but there are few studies on the oscillation characteristics of solutions of non-zero continuous linear functional equations. Reference [12] and reference [13] respectively study the number of hematopoietic model and population dynamics model. It provides research support for hematopoietic data and population dynamics analysis in the field of biology. In reference [14], the oscillation of numerical solutions of piecewise continuous delay differential equations with independent variables is studied. This method well reflects the oscillatory characteristics of numerical solutions of piecewise continuous time-delay differential equations, but the equations can only be applied the field of biology. They are all functional differential equations in special cases [15], the oscillation characteristics of nonzero continuous linear functional equations are not studied.

Therefore, based on the above research, analyzing the oscillation characteristics of the solution of non-zero continuous linear functional equation. This paper mainly studies the oscillation characteristics of the solution of non-zero continuous linear functional equation by taking the second-order and third-order non-zero continuous linear functional equation as an example. The oscillation and asymptotic behavior of nonzero continuous linear functional differential equations are established by using generalized Ricardian transform and integral average technique. The generalized Ricardian transform and integral average technique can set up many physical quantities in physics, which reflect the characteristics of self-similarity observed at different scales.

2. Application theory of the algorithm

2.1 Analysis on the oscillation characteristics of the solution of the second-order non-zero continuous linear functional equation

The oscillation characteristics of the solutions of the second-order non-zero continuous linear functional equation are considered as follows:

$\displaystyle(a(t)\psi(x(t)){x}^{\prime}(t){)}^{\prime}+p(t){x}^{\prime}(t)+Q(% t,x(t),x(\sigma(t)))=R(t,x(t),{x}^{\prime}(\delta(t)))$ (1)

2.1.1 Preparatory knowledge

By general definition, let $x(t)$ be the solution of a differential equation, which is not equal to zero on any infinite interval $[T,+\infty)$ . Such a solution is called a regular solution. If $x(t)$ is the regular solution of a differential equation, and there is a sequence of $\{t_{i}\},\mathop{\lim}\limits_{i\to+\infty}\{t_{i}\}=+\infty$ , so that $x(t_{i})=0$ , then $x(t)$ is an oscillatory solution of the differential equation. If ${x}^{\prime}(t)$ can be changed for any large $t$ , then $x(t)$ is a weakly oscillatory solution of the equation [16]. If all regular solutions of a differential equation are oscillatory, the equation is called oscillatory [17].

Agreement in this article:

( $H_{1}$ ) $a,p,\sigma,\delta\in C\left[{\left[{t_{0},+\infty}\right),R}\right](R=(-\infty% ,+\infty)),Q,R\in C\left[{\left[{t_{0},+\infty}\right)\times R^{2},R}\right],% \psi\in C\left[R\right]$ , and $a(t)>0$ , $\psi(x)>0$ , when $t\to+\infty$ , $\sigma(t)\to+\infty$ . ( $H_{2}$ ) has continuous functions $qq(t),r(t):\left[{t_{0},+\infty}\right)\to R$ and f $f_{1}(x),f_{2}(x),g(x):R\to R$ , and $r(t)\geqslant 0$ , $t\geqslant t_{0}$ , $xf_{i}(x)>0$ , $f_{i}^{\prime}\geqslant 0$ , $i=1,2$ , $|f_{2}(x)|\leqslant|f_{1}(x)|$ , $x\neq 0$ , $0<g(x)\leqslant c$ , $c>0$ is a constant. ( $H_{3}$ ) when $uv>0$ , $\frac{Q(t,u,v)}{f_{1}(u)}\geqslant q(t)$ , $\frac{R(t,u,v)}{f_{2}(u)}\leqslant r(t)g(v)$ .

2.1.2 Main results

Theorem 1: let $\psi(x)f_{1}^{\prime}(x)\geqslant k>0$ , $x\neq 0$ and

$\displaystyle 0<\int{{}_{0}^{\varepsilon}\frac{\psi(u)}{f_{1}(u)}}du<+\infty,% \int{{}_{-u}^{0}\frac{\psi(u)}{f_{1}(u)}}du>-\infty$ (2)

And for every constant $M$ , there are:

$\displaystyle\int{{}_{s_{0}}^{+\infty}\frac{M}{a(s)}}ds-\int{{}_{s_{0}}^{+% \infty}\frac{1}{a(s)}}\int{{}_{\tau_{0}}^{s}\left[{q(\tau)-cr(\tau)-\frac{p^{2% }(\tau)}{4ka(\tau)}}\right]d\tau ds=-\infty}$ (3)

Then Eq. (1) is oscillatory.

It is proved that if $x(t)$ is a non oscillatory solution of Eq. (1), then there is $t_{1}$ . Let’s suppose that when $t\geqslant t_{1}\geqslant t_{0}$ , $x(t)>0$ , and $x(\sigma(t))>0$ . Considering the function

$\displaystyle W(t)=\frac{a(t)\psi(x(t)){x}^{\prime}(t)}{f_{1}(x(t))},t% \geqslant t_{1},$

from Eq. (1), it can get:

$\displaystyle{W}^{\prime}(t)=\frac{p(t){x}^{\prime}(t)}{f_{1}(x(t))}-\frac{Q(t% ,x(t),x(\sigma(t)))}{f_{1}(x(t))}+\frac{R(t,x(t),{x}^{\prime}(\delta(t)))}{f_{% 1}(x(t))}-a(t)\psi(x(t))f_{1}^{\prime}(x(t))\frac{{x}^{\prime 2}(t)}{f_{1}^{2}% (x(t))}\leqslant-q(t)+cr(t)+\frac{p^{2}(t)}{4a(t)\psi(x(t))f_{1}^{\prime}(x(t)% )}-\left[{\sqrt{a(t)\psi(x(t))f_{1}^{\prime}(x(t))}\frac{{x}^{\prime}(t)}{f_{1% }(x(t))}+\frac{p(t)}{\sqrt[2]{a(t)\psi(x(t))f_{1}^{\prime}(x(t))}}}\right]^{2}% \leqslant-q(t)+cr(t)+\frac{p^{2}(t)}{4ka(t)}$

The above equation is integrated by $t_{1}$ to $t$ to obtain:

$\displaystyle\frac{a(t)\psi(x(t)){x}^{\prime}(t)}{f_{1}(x(t))}\leqslant M-\int% _{t_{1}}^{t}{\left[{q(t)-cr(t)-\frac{p^{2}(t)}{4ka(t)}}\right]dt}$

where,

$\displaystyle M=\frac{a(t_{1})\psi(x(t_{1})){x}^{\prime}(t_{1})}{f_{1}(x(t_{1}% ))},$

both sides are divided by $a(t)$ and integrated to get:

$\displaystyle\int_{t_{1}}^{t}{\frac{\psi(x(t)){x}^{\prime}(t)}{f(x(t))}dt}% \leqslant\int_{s_{1}}^{s}{\frac{M}{a(s)}ds}-\int_{s_{1}}^{s}{\frac{1}{a(s)}% \int_{\tau_{1}}^{s}{\left[{q(\tau)-cr(\tau)-\frac{p^{2}(\tau)}{4ka(\tau)}}% \right]d\tau ds}}.$

From Eq. (3), when $t\to+\infty$ , we can get:

$\displaystyle I(t)=\int_{s_{1}}^{s}{\frac{\psi(x(s)){x}^{\prime}(s)}{f_{1}(x(s% ))}}ds=\int_{u(t_{1})}^{u(t)}{\frac{\psi(u)}{f_{1}(u)}}du\to-\infty$ (4)

On the other hand, for a sufficiently large $t$ , if $x(t)\geqslant x(t_{1})\geqslant 0$ , then $\frac{\psi(u)}{f_{1}(u)}>0$ based on condition $(H_{2})$ , then $I(t)\geqslant 0$ , which is contradictory to Eq. (4); If $x(t)<x(t_{1})$ , then $I(t)=\int_{0}^{u}{\frac{\psi(u)}{f_{1}(u)}}du$ based on Eq. (2), which is contradictory [18], so Eq. (1) is oscillatory.

Similarly, it can be proved that $x(t)<0$ when $t\geqslant t_{1}$ .

Next, the oscillation properties of the equation are discussed from the asymptotic property of the solution.

In order to discuss the oscillation of Eq. (1) from the asymptotic state of the solution, all the regular solutions of Eq. (1) are divided into the following four categories:

$S^{+}=\left\{{x=x(t):x(t)}\right.$ is the regular solution of Eq. (1), and there is $t_{x}\geqslant t_{0}$ . When $t\geqslant t_{x}$ , there is $x(t){x}^{\prime}(t)>\left.0\right\}$ .

$S^{-}=\left\{{x=x(t):x(t)}\right.$ is the regular solution of Eq. (1), and there is $t_{x}\geqslant t_{0}$ , $x(t){x}^{\prime}(t)\leqslant\left.0\right\}$ .

$S^{0}=\left\{{x=x(t):x(t)}\right.$ is the regular solution of Eq. (1), and there is $\left\{{t_{N}}\right\},t_{N}\to+\infty$ , then $x(t_{N})=\left.0\right\}$ .

$S^{w0}=\left\{{x=x(t):x(t)}\right.$ is the solution of Eq. (1), for a sufficiently large $t$ , there is $x(t)\neq 0$ , and for all $t_{a}>t_{0}$ , there is $t_{a_{1}}>t_{a}$ , $t_{a_{2}}>t_{a}$ and let ${x}^{\prime}(t_{a_{1}}){x}^{\prime}(t_{a_{2}})<\left.0\right\}$ .

It is easy to prove that $S^{+}\cap S^{-}\cap S^{0}\cap S^{wo}=\emptyset$ . According to the above definition, the solution in $S^{+}$ is ultimately a positive non decreasing function or a negative non increasing function, the solution in $S^{-}$ is finally a positive non increasing function or a negative non decreasing function, the solution in $S^{0}$ is oscillatory, and the solution in $S^{wo}$ is finally weakly oscillatory [19].

Lemma 1: Let $\psi(x)f_{1}^{\prime}(x)\geqslant k>0$ , $x\neq 0$ , if:

$\displaystyle\mathop{\lim}\limits_{t\to+\infty}\int_{t_{0}}^{t}{\left[{q(s)-cr% (s)-\frac{p^{2}(s)}{4ka(s)}}\right]ds=+\infty}$ (5)

Then for Eq. (1), there is $S^{+}=\emptyset,S^{wo}=\emptyset$ .

It is proved that: (I) supposing that Eq. (1) has a solution $x\in S^{+}$ , which is not out of generality. If $t_{1}>t_{0}$ , let $t>t_{1}$ , then there are $x(t)>0$ , $x(\sigma(t))>0$ , ${x}^{\prime}(t)\geqslant 0$ . The functions

$\displaystyle W(t)=\rho(t)\frac{a(t)\psi(x(t)){x}^{\prime}(t)}{f_{1}(x(t))},t% \geqslant t_{1},$ $\displaystyle{W}^{\prime}(t)=\frac{p(t){x}^{\prime}(t)}{f_{1}(x(t))}-\frac{Q(t% ,x(t),x(\sigma(t)))}{f_{1}(x(t))}+\frac{R(t,x(t),{x}^{\prime}(\delta(t)))}{f_{% 1}(x(t))}-a(t)\psi(x(t))f_{1}^{\prime}(x(t))\frac{{x}^{\prime 2}(t)}{f_{1}^{2}% (x(t))}\leqslant-q(t)+cr(t)+\frac{p^{2}(t)}{4a(t)\psi(x(t)){f}^{\prime}_{1}(x(% t))}-\left[{\sqrt{a(t)\psi(x(t))f_{1}^{\prime}(x(t))}\frac{{x}^{\prime}(t)}{f_% {1}(x(t))}+\frac{p(t)}{2\sqrt{a(t)\psi(x(t))f_{1}^{\prime}(x(t))}}}\right]% \leqslant-q(t)+cr(t)+\frac{p^{2}(t)}{4ka(t)}$

are considered.

The above equation is integrated by $t_{1}$ to $t=(t\geqslant t_{1})$ to obtain.

$\displaystyle\frac{a(t)\psi(x(t)){x}^{\prime}(t)}{f_{1}(x(t))}-\frac{a(t_{1})% \psi(x(t_{1})){x}^{\prime}(t_{1})}{f_{1}(x(t_{1}))}\leqslant-\int_{t_{1}}^{t}{% \left[{q(s)-cr(s)-\frac{p^{2}(s)}{4ka(s)}}\right]}ds.$

From Eq. (5), it can get $\mathop{\lim}\limits_{t\to+\infty}\frac{a(t)\psi(x(t)){x}^{\prime}(t)}{f_{1}(x% (t))}=-\infty$ , which contradicts with $x(t)>0,{x}^{\prime}(t)\geqslant 0(t\geqslant t_{1})$ .

Similarly, it can be proved that the state of $x(t)<0,{x}^{\prime}(t)<0$ when $t\geqslant t_{1}$ .

(II) If Eq. (1) has solution $x(t)\in S^{wo}$ , then there is $t_{1}\geqslant t_{0}$ . If $x(t)>0,x(\sigma(t))>0$ , then for all $t_{a}\geqslant t_{1}$ , there is $t_{a_{1}}\geqslant t_{a}$ , $t_{a_{2}}\geqslant t_{a}$ , so that ${x}^{\prime}(t_{a_{1}}){x}^{\prime}(t_{a_{2}})<0$ . As shown in (I),

$\displaystyle\frac{a(t)\psi(x(t)){x}^{\prime}(t)}{f_{1}(x(t))}\leqslant\frac{a% (t_{1})\psi(x(t_{1})){x}^{\prime}(t_{1})}{f_{1}(x(t_{1}))}--\int_{t_{1}}^{t}{% \left[{q(s)-cr(s)-\frac{p^{2}(s)}{4ka(s)}}\right]}ds$

is obtained.

According to Eq. (5), for a sufficiently large $t$ , there is always a contradiction between ${x}^{\prime}(t)<0$ and ${x}^{\prime}(t_{a_{1}}){x}^{\prime}(t_{a_{2}})<0$ , and it is proved.

Similarly, it can be proved that the state of $x(t)<0,x(\sigma(t))<0$ when $t\geqslant t_{1}$ .

Lemma 2: let $\psi(x)f_{1}^{\prime}(x)\geqslant k>0$ , $x\neq 0$ , if $\frac{f_{1}(u)}{\psi(u)}$ is strongly linear, that is:

$\displaystyle\int_{0}^{\varepsilon}{\frac{f_{1}(u)}{\psi(u)}}du<+\infty,\int_{% -\varepsilon}^{0}{\frac{\psi(u)}{f_{1}(u)}}du>-\infty$ (6)

And for all $T\geqslant t_{0}$ , there are:

$\displaystyle\mathop{\lim}\limits_{t\to\infty}\sup\int_{T}^{t}{\frac{1}{a(s)}}% \left[{q(\tau)-cr(\tau)-\frac{p^{2}(\tau)}{4ka(\tau)}}\right]d\tau ds=+\infty$ (7)

Then for Eq. (1), there is $S^{-}=\emptyset$ .

It is proved that if Eq. (1) has a solution $x(t)\in S^{-}$ , it is not general. If there is $t_{1}\geqslant t_{0}$ , then when $t\geqslant t_{1}$ , there is $x(t)>0$ , $x(\sigma(t))>0$ , ${x}^{\prime}(t)\leqslant 0$ . Considering the function $W(t)\frac{a(t)\psi(x(t)){x}^{\prime}(t)}{f_{1}(x(t))}$ , $t\geqslant t_{1}$ , from Eq. (1), it can get:

$\displaystyle{W}^{\prime}(t)=-\frac{p(t){x}^{\prime}(t)}{f_{1}(x(t))}-\frac{Q(% t,x(t),x(\sigma(t)))}{f_{1}(x(t))}+\frac{R(t,x(t),x(\delta(t)))}{f_{1}(x(t))}-% a(t)\psi(x(t))f_{1}^{\prime}(x(t))\frac{{x}^{\prime 2}(t)}{f_{1}^{2}(x(t))}% \leqslant-q(t)+cr(t)-\frac{p(t){x}^{\prime}(t)}{f_{1}(x(t))}-a(t)\psi(x(t))f_{% 1}^{\prime}(x(t))\frac{{x}^{\prime 2}(t)}{f_{1}^{2}(x(t))}=-q(t)+cr(t)+\frac{p% ^{2}(t)}{4a(t)\psi(x(t)){f}^{\prime}_{1}(x(t))}-\left[{\sqrt{a(t)\psi(x(t))f_{% 1}^{\prime}(x(t))}\frac{{x}^{\prime}(t)}{f_{1}(x(t))}+\frac{p(t)}{2\sqrt{a(t)% \psi(x(t)){f}^{\prime}_{1}(x(t))}}}\right]\leqslant-q(t)+cr(t)+\frac{p^{2}(t)}% {4ka(t)}.$

From the integral of $t_{1}$ to $t(t\geqslant t_{1})$ , the following equation is obtained:

From the known condition, it can obtain:

$\displaystyle\frac{\psi(x(t)){x}^{\prime}(t)}{f_{1}(x(t))}\leqslant\frac{1}{a(% t)}\int_{t_{1}}^{t}{\left[{q(s)-cr(s)-\frac{p^{2}(s)}{4ka(s)}}\right]}ds.$

Integration from T1 to T2 on both sides can obtain:

$\displaystyle\int_{t_{1}}^{t}{\frac{\psi(x(t)){x}^{\prime}(t)}{f_{1}(x(t))}ds}% \leqslant-\int_{t_{1}}^{t}{\frac{1}{a(s)}\int_{t_{1}}^{s}{\left[{q(\tau)-cr(% \tau)-\frac{p^{2}(\tau)}{4ka(\tau)}}\right]}}d\tau ds,$

that is:

$\displaystyle\int_{x(t)}^{x(t_{1})}{\frac{\psi(u)}{f_{1}(u)}du}\geqslant\int_{% t_{1}}^{t}{\frac{1}{a(s)}\int_{t_{1}}^{s}{\left[{q(\tau)-cr(\tau)-\frac{p^{2}(% \tau)}{4ka(\tau)}}\right]}}d\tau ds.$

From the above equation, it is noted that $\mathop{\lim}\limits_{t\to\infty}\int_{x(t)}^{x(t_{1})}{\frac{\psi(u)}{f_{1}(u% )}du}=+\infty$ in Eq. (7). This is in contradiction with Eq. (6) and it is proved.

Similarly, the state of $x(t)<0,x(\sigma(t))<0$ , ${x}^{\prime}(t)\geqslant 0$ can be proved, when $t\geqslant t_{1}$ .

Theorem 2: if $p(t)\geqslant 0$ , $t\geqslant t_{0}$ , $\psi(x)f_{1}^{\prime}(x)\geqslant k>0$ , $x\neq 0$ , let Eq. (5) be true, and:

$\displaystyle\mathop{\lim}\limits_{t\to\infty}\int_{t_{0}}^{t}{\frac{1}{a(s)}}% ds=+\infty$ (8)

Then Eq. (1) is oscillatory.

Proof: from Lemma 1, for Eq. (1), there is $S^{+}=S^{wo}=\emptyset$ . Therefore, it is only necessary to prove that for Eq. (1), there is $S^{-}=\emptyset$ . Supposing that Eq. (1) has a solution $x(t)\in S^{-}$ , without losing generality. Supposing that there is $t_{1}\geqslant t_{0}$ , so that when $t\geqslant t_{1}$ , there is $x(t)>0$ , ${x}^{\prime}(t)\leqslant 0$ . As the proof process of Lemma 1, it can get:

$\displaystyle\frac{a(t)\psi(x(t)){x}^{\prime}(t)}{f_{1}(x(t))}\leqslant\frac{a% (t_{1})\psi(x(t_{1})){x}^{\prime}(t_{1})}{f_{1}(x(t_{1}))}-\int_{t_{1}}^{t}{% \left[{q(s)-cr(s)-\frac{p^{2}(s)}{4ka(s)}}\right]ds}.$

From the above equation and Eq. (5), there is $t_{2}\geqslant t_{1}$ , so that when $t\geqslant t_{2}$ , ${x}^{\prime}(t)<0$ . Then from the condition, there is $t_{3}\geqslant t_{2}$ , so that $\int_{t_{0}}^{t}{\left[{q(s)-cr(s)-\frac{p^{2}(s)}{4ka(s)}}\right]ds}\geqslant 0$ , $t\geqslant t_{3}$ . From $t_{3}$ to $t$ integral Eq. (1), it can get:

$\displaystyle a(t)\psi(x(t)){x}^{\prime}(t)=a(t_{3})\psi(x(t_{3})){x}^{\prime}% (t_{3})+\int_{t_{3}}^{t}{R(s,x(s),{x}^{\prime}(\delta(s)))ds}-$ $\displaystyle\quad\int_{t_{3}}^{t}{Q(s,x(s),x(\sigma(s)))ds}-\int_{t_{3}}^{t}{% p(s),{x}^{\prime}(s)ds}\leqslant$ $\displaystyle\quad a(t_{3})\psi(x(t_{3})){x}^{\prime}(t_{3})-f_{1}(x(t))\int_{% t_{3}}^{t}{\left[{q(s)-cr(s)-\frac{p^{2}(s)}{4ka(s)}}\right]ds}+$ $\displaystyle\quad\int_{t_{3}}^{t}{f_{1}^{\prime}(x(s)){x}^{\prime}(s)\int_{t_% {3}}^{s}{\left[{q(\tau)-cr(\tau)-\frac{p^{2}(\tau)}{4ka(\tau)}}\right]d\tau ds% }\leqslant}a(t_{3})\psi(x(t_{3})){x}^{\prime}(t_{3})=k(k<0)$

For $t\geqslant t_{3}$ , there is $\int_{x(t_{3})}^{x(t)}{\psi(u)du\leqslant k\int_{t_{3}}^{t}{\frac{1}{a(s)}}}ds$ . From Eq. (8) and $k<0$ , $\mathop{\lim}\limits_{t\to+\infty}\int_{x(t)}^{x(t_{3})}{\psi(u)du=+\infty}$ is obtained, which is in contradiction with the existence and finiteness of $\mathop{\lim}\limits_{t\to+\infty}x(t)$ and the continuity of $\psi$ . similarly, when $t\geqslant t_{1}$ , $x(t)<0$ , ${x}^{\prime}(t)\geqslant 0$ can be proved. It satisfies Theorem 2, and the equation oscillates.

2.2 Analysis on the oscillation characteristics of the solution of the third-order non-zero continuous linear functional equation

In this paper, the oscillation characteristics of the solutions of the following third-order non-zero continuous linear functional differential equations are considered.

$\displaystyle\left[{r(t)\left[{x(t)+p(t)x(\tau(t))}\right]^{\prime\prime}}% \right]^{\prime}+m(t)\left[{x(t)+p(t)x(\tau(t))}\right]^{\prime\prime}+q(t)f(x% (\sigma(t)))g({x}^{\prime}(t))=0,$

(9) $\displaystyle\quad t\geqslant t_{0}$

Supposing that the following conditions are true:

(H1) $r(t),q(t)\in C\left({\left[{t_{0},+\infty}\right),\left({0,+\infty}\right)}% \right),p(t),m(t)\in C\left({\left[{t_{0},+\infty}\right),\left({0,+\infty}% \right)}\right),0\leqslant p(t)\leqslant p<1,$ and $\int_{t_{0}}^{+\infty}{\frac{1}{r(t)}}\exp\left({-\int_{t_{0}}^{t}{\frac{m(s)}% {r(s)}ds}}\right)dt=+\infty$ ; (H2) $\tau(t),\sigma(t)\in C\left({\left[{t_{0},+\infty}\right),\left({0,+\infty}% \right)}\right),\tau(t)\leqslant t,\sigma(t)\leqslant t,\mathop{\lim}\limits_{% t\to\infty}\tau(t)=\mathop{\lim}\limits_{t\to\infty}\sigma(t)=\infty;$ (H3) $f(t)\in C\left({R,R}\right),\frac{f(u)}{u}\geqslant K>0,u\neq 0;$ (H4) $g(t)\in C\left({R,\left[{L,+\infty}\right)}\right),L>0.$

The definition function is shown in Eq. (10):

$\displaystyle y(t)=x(t)+p(t)x(\sigma(t)),t\geqslant t_{0}$ (10)

If the function $x(t)\in C^{1}\left[{T_{x},\infty}\right),T_{x}\geqslant t_{0}$ , let $r(t)\bar{y}(t)\in C^{1}\left[{T_{x},\infty}\right)$ and satisfy Eq. (2.2) on $\left[{T_{x},\infty}\right)$ , then $x(t)$ is called the solution of Eq. (2.2).

In this paper, it considers that Eq. (2.2) satisfies the solution of $\sup\left\{{|x(t)|:t\geqslant T}\right\}>0$ for all $T\geqslant T_{x}$ . If the solution of Eq. (2.2) has any large zero point, it is called oscillatory; If all the solutions of Eq. (2.2) are oscillatory, it is called oscillatory, otherwise Eq. (2.2) is non oscillatory [20].
2.2.1 Some Lemmas

Lemma 3: if $u(t)>0,{u}^{\prime}(t)>0,{u}^{\prime\prime}(t)\leqslant 0,t\geqslant t_{0}$ , then $t_{\alpha}\geqslant t_{0}$ exists for any $\alpha\in(0,1)$ , so $u(h(t))\geqslant\alpha\frac{h(t)}{t}u(t),t\geqslant T_{\alpha}$ .

Lemma 4: if $u(t)>0,{u}^{\prime}(t)>0,{u}^{\prime\prime}(t)>0,{u}^{\prime\prime\prime}(t)% \leqslant 0,t\geqslant T_{\alpha}$ , then $\beta\in(0,1)$ and $t_{\beta}\geqslant t_{0}$ exist, so $u(t)\geqslant\beta tu(t),t\geqslant T_{\beta}$ .

Lemma 5: if $x(t)$ is the final positive solution of Eq, (2.2), then $y(t)$ has only two possibilities, that is, $T\geqslant t_{0}$ , which makes $(A)y(t)>0,{y}^{\prime}(t)>0,{y}^{\prime\prime}(t)>0;(B)y(t)>0,{y}^{\prime}(t)<% 0,{y}^{\prime\prime}(t)>0.$ when $t\geqslant T$ .

Proof: if $x(t)$ is the final positive solution and condition (H3) of Eq. (2.2), there is $t_{1}\geqslant t_{0}$ , when $t\geqslant t_{1}$ , there is $x(t)>0$ , $x(\tau(t))>0$ , $x(\sigma(t))>0$ , and it is easy to know $y(t)>x(t)>0$ and:

$\displaystyle\left[{r(t){y}^{\prime\prime}(t)}\right]^{\prime}+m(t){y}^{\prime% \prime}(t)\leqslant-q(t)f(x(\sigma(t)))g({x}^{\prime}(t))<0,$

Then:

$\displaystyle\frac{d}{dt}\left({\exp\left({\int_{t_{0}}^{t}{\frac{m(s)}{r(s)}% ds}}\right)r(t){y}^{\prime\prime}(t)}\right)<0$

Therefore, $\exp\left({\int_{t_{0}}^{t}{\frac{m(s)}{r(s)}ds}}\right)r(t){y}^{\prime\prime}% (t)$ is a subtractive function, and the number is finally determined, so there are two possibilities, namely ${y}^{\prime\prime}(t)<0$ or ${y}^{\prime\prime}(t)>0,t\leqslant t_{1}$ .

If ${y}^{\prime\prime}(t)<0$ , then there is a constant $M>0$ , making $\exp\left({\int_{t_{0}}^{t}{\frac{m(s)}{r(s)}ds}}\right)r(t){y}^{\prime\prime}% (t)\leqslant-M<0,t\geqslant t_{2}$ .

On $\left[{t_{2},t}\right)$ , the integral of the above equation is ${y}^{\prime}(t)\leqslant{y}^{\prime}(t_{2})-M\exp\int_{t_{2}}^{t}{\frac{1}{r(s% )}\exp}\left({-\int_{t_{2}}^{s}{\frac{m(v)}{r(v)}dv}}\right)ds$ .

In the above equation, let $t\to\infty$ , using $(H1)$ , ${y}^{\prime}(t)\to-\infty$ , therefore, ${y}^{\prime}(t)$ is finally negative. But ${y}^{\prime}(t)$ and ${y}^{\prime\prime}(t)$ are finally negative, so $y(t)$ is finally negative, which contradicts the hypothesis of $y(t)>0$ , so ${y}^{\prime\prime}(t)>0$ . Therefore, $y(t)$ can only have and two types of $(A)$ and $(B)$ , and Lemma 5 has been proved.

Lemma 6: let $x(t)$ be the final positive solution of Eq. (2.2) and the response $y(t)$ be type $(B)$ , if:

$\displaystyle\mathop{\limsup}\limits_{t\to\infty}\int_{t_{0}}^{t}{\int_{v}^{% \infty}{\left({\frac{1}{r(u)}\int_{u}^{\infty}{q(s)ds}}\right)dudv}}=\infty$ (11)

Then $\mathop{\lim}\limits_{t\to\infty}x(t)=\mathop{\lim}\limits_{t\to\infty}y(t)=0$ .

It is proved that if $x(t)$ is the final positive solution of Eq. (2.2), and the corresponding $y(t)$ is of type $(B)$ , there is a constant $l\geqslant 0$ , making $\mathop{\lim}\limits_{t\to\infty}y(t)=l$ . It can be asserted that $l=0$ . In fact, if $l>0$ , then for any $\varepsilon>0$ , there is $t_{2}>t_{1}$ , making $l<y(t)<l+\varepsilon$ , $t\geqslant t_{2}$ . Taking $0<\varepsilon<\frac{l(1-p)}{p}$ , there are.

$\displaystyle x(t)=y(t)-p(t)x(\tau(t))>l-p(l+\varepsilon)$ (12)

Let $M=\frac{l-p(l+\varepsilon)}{l+\varepsilon}$ , then $M>0$ . Thus, Eq. (12) is as:

$\displaystyle x(t)>M(l+\varepsilon)>\textit{My(t)}$ (13)

In combination with (H3) and (H3), Eqs (2.2) and (13), the following results are obtained:

$\displaystyle(r(t){y}^{\prime\prime}(t){)}^{\prime}+m(t){y}^{\prime\prime}(t)+% \textit{KLMq}(t)y(\sigma(t))\leqslant 0,t\geqslant t_{2}$ (14)

Equation (14) can be written as:

$\displaystyle(r(t){y}^{\prime\prime}(t)r(t){)}^{\prime}+\textit{KLMq}(t)y(% \sigma(t))z(t)\leqslant 0,t\geqslant t_{2}$ (15)

where, $z(t)=\exp\left({\int_{t_{0}}^{t}{\frac{m(v)}{r(v)}}dv}\right)$ , the Eq. (15) is made integral from $t$ to $\infty$ , $-r(t){y}^{\prime\prime}(t)z(t)+\textit{KLM}\int_{t}^{\infty}{q(}s)y(\sigma(s))% z(s)ds\leqslant 0$ is obtained. It is noted that $y(\sigma(t))>l,t\geqslant t_{3}\geqslant t_{2}$ has $-{y}^{\prime\prime}(t)+\frac{\textit{KLMl}}{r(t)z(t)}\int_{t}^{\infty}{q(}s)z(% s)ds\leqslant 0,t\geqslant t_{3}$ , and ${z}^{\prime}(t)\geqslant 0$ has:

$\displaystyle-{y}^{\prime\prime}(t)+\frac{\textit{KLMl}}{r(t)}\int_{t}^{\infty% }{q(}s)ds\leqslant 0,t\geqslant t_{3}$ (16)

From $t_{3}$ to $t$ , the integral of Eq. (16) produces

$\displaystyle\int_{t_{3}}^{t}{\int_{v}^{\infty}{\left({\frac{1}{r(u)}\int_{u}^% {\infty}{q(s)ds}}\right)dudv\leqslant\frac{y(t_{3})}{KMLl}}}.$

The above equation conflicts with Eq. (11). Therefore, $l=0$ , and $0<x(t)\leqslant y(t)$ , then $\mathop{\lim}\limits_{t\to\infty}x(t)=0$ . Lemma 6 is proved.

2.2.2 Main results

Theorem 3: let $n>1$ and Eq. (11) be true. If there is a function $\rho\in C^{1}\left({\left[{t_{0},\infty}\right),(0,\infty)}\right)$ , then

$\displaystyle\mathop{\limsup}\limits_{t\to\infty}\frac{1}{t^{n}}\int_{t_{0}}^{% t}{(t-s)^{n}\left[{\rho(s)Q(s)-\frac{\rho(s)r(s)}{4}\left({\frac{{\rho}^{% \prime}(s)}{\rho(s)}-\frac{m(s)}{r(s)}}\right)^{2}}\right]ds}=\infty$ (17)

Then, for Eq. (2.2), every solution $x(t)$ oscillates, or when $t\to\infty$ , $x(t)\to 0$ . Where:

$\displaystyle Q(s)=\alpha\beta KL(1-p)\frac{\sigma^{2}(s)}{s}q(s)$ (18)

Here $\alpha$ and $\beta$ are defined by Lemmas 3 and 4.

It is proved that: let Eq. (2.2) have non oscillatory solution $x(t)$ , and not be generality; Let $x(t)$ finally be positive (it can be similarly proved when $x(t)$ finally is negative), so there is sufficient large $t_{1}\geqslant T$ ( $T$ is mentioned in Lemma) [21]. Let $x(t)>0$ , $x(\tau(t))>0$ , $x(\sigma(t))>0$ when $t\geqslant t_{1}$ . $y(t)$ is defined by Eq. (11), and based on Lemma 5, $y(t)$ may be type $(A)$ or type $(B)$ .

If $y(t)$ is type $(A)$ , then $x(t)=y(t)-p(t)x(\tau(t))\geqslant y(t)-py(\tau(t))\geqslant(1-p)y(t)$ .

Therefore, from $(H3)$ , $(H4)$ and Eq. (2.2), it can get:

$\displaystyle(r(t){y}^{\prime\prime}(t){)}^{\prime}+m(t){y}^{\prime\prime}(t)+% KL(1-p)q(t)y(\sigma(t))$ (19)

Let:

$\displaystyle W(t)=\rho(t)\frac{r(t){y}^{\prime\prime}(t)}{{y}^{\prime}(t)},t% \geqslant t_{1}$ (20)

Then:

$\displaystyle{W}^{\prime}(t)\leqslant\frac{KL(1-p)\rho(t)q(t)y(\sigma(t))}{{y}% ^{\prime}(t)}+\left({\frac{{\rho}^{\prime}(t)}{\rho(t)}-\frac{m(t)}{r(t)}}% \right)W(t)-\frac{W^{2}(t)}{\rho(t)r(t)}$ (21)

Using Lemmas 3 and 4, there are:

$\displaystyle{W}^{\prime}(t)\leqslant-\rho(t)Q(t)-\left[{\frac{W(t)}{\sqrt{% \rho(t)r(t)}}-\frac{\sqrt{\rho(t)r(t)}}{2}\left({\frac{{\rho}^{\prime}(t)}{% \rho(t)}-\frac{m(t)}{r(t)}}\right)}\right]^{2}+\frac{\rho(t)r(t)}{4}\left({% \frac{{\rho}^{\prime}(t)}{\rho(t)}-\frac{m(t)}{r(t)}}\right)^{2}\leqslant-% \left[{\rho(t)Q(t)-\frac{\rho(t)r(t)}{4}\left({\frac{{\rho}^{\prime}(t)}{\rho(% t)}-\frac{m(t)}{r(t)}}\right)^{2}}\right]$ (22)

where $Q(t)$ is defined by Eq. (2.2). Therefore:

$\displaystyle\int_{t_{1}}^{t}{(t-s)^{n}}\left[{\rho(s)Q(s)-\frac{\rho(s)r(s)}{% 4}\left({\frac{{\rho}^{\prime}(s)}{\rho(s)}-\frac{m(s)}{r(s)}}\right)^{2}}% \right]ds\leqslant-\int_{t_{1}}^{t}{(t-s)^{n}}{W}^{\prime}(s)ds,t\geqslant t_{% 1}.$

It is noticed that

$\displaystyle\int_{t_{1}}^{t}{(t-s)^{n}}{W}^{\prime}(s)ds=n\int_{t_{1}}^{t}{(t% -s)^{n-1}}W(s)ds-W(t_{1})(t-t_{1})^{n}$

and to obtain:

$\displaystyle\frac{1}{t^{n}}\int_{t_{1}}^{t}{(t-s)^{n}}G(s)ds\leqslant W(t_{1}% )(1-\frac{t_{1}}{t})^{n}-\frac{n}{t^{n}}\int_{t_{1}}^{t}{(t-s)^{n-1}}W(s)ds,$

where:

$\displaystyle G(s)=\rho(s)Q(s)-\frac{\rho(s)r(s)}{4}\left({\frac{{\rho}^{% \prime}(s)}{\rho(s)}-\frac{m(s)}{r(s)}}\right)^{2},$

so there is $\mathop{\limsup}\limits_{t\to\infty}\frac{1}{t^{n}}\int_{t_{1}}^{t}{(t-s)^{n}G% (s)ds<+\infty}$ , which is contradiction with condition Eq. (17).

Supposing that $y(t)$ is type $(B)$ , then there is $\mathop{\lim}\limits_{t\to\infty}x(t)=\mathop{\lim}\limits_{t\to\infty}y(t)=0$ based on Lemma 6. Theorem 3 is proved.

Following, the Philos type of integral average technique is used to give a new oscillation theorem [22] for Eq. (2.2), for which the following function $\wp$ is introduced, let $D=\left\{{\left({t,s}\right)t\geqslant s\geqslant t_{0}}\right\},D_{0}=\left\{% {\left({t,s}\right)t>s\geqslant t_{0}}\right\}$ .

The function $H\left({t,s}\right)\in C\left({D,R}\right)$ belongs to the type $\wp$ , which is recorded as $H\in\wp$ , if:

(i) $H\left({t,t}\right)=0,t\geqslant t_{0};H\left({t,s}\right)>0,\left({t,s}\right% )\in D_{0}$ ;

(ii) $\frac{\partial H\left({t,t}\right)}{\partial s}\leqslant 0,\left({t,s}\right)% \in D,$ and $-\frac{\partial H\left({t,t}\right)}{\partial s}=h\left({t,s}\right)\sqrt{H% \left({t,s}\right)}$ , $h\in C\left({D,R}\right),\left({t,s}\right)\in D_{0}$ .

Theorem 4: let Eq. (11) hold, and there are functions $H\in\wp$ and $\rho\in C^{1}\left({\left[{t_{0},\infty}\right)(0,\infty)}\right)$ , so that:

$\displaystyle\mathop{\limsup}\limits_{t\to\infty}\frac{1}{H(t,t_{0})}\int_{t_{% 0}}^{t}{\left[{H(t,s)\rho(s)Q(s)-\frac{1}{4}\rho(s)r(s)h_{1}^{2}(t,s)}\right]}% ds=+\infty$ (23)

where $Q(s)$ has a definition of Eq. (14), and:

$\displaystyle h_{1}(t,s)=h(t,s)-\left({\frac{{\rho}^{\prime}(s)}{\rho(s)}-% \frac{m(s)}{r(s)}}\right)\sqrt{H(t,s)}$ (24)

Then every solution $x(t)$ of Eq. (2.2) will vibrate, or $x(t)\to 0$ when $t\to\infty$ .

It is proved that, just like the proof of Theorem 3, let $x(t)$ be a non-oscillatory solution of Eq. (2.2), let $x(t)>0$ , $x(\tau(t))>0$ , $x(\sigma(t))>0$ , $t\geqslant t_{1}$ . $y(t)$ is defined by Eq. (11), and $y(t)$ may be type $(A)$ or type $(B)$ according to Lemma 5.

If $y(t)$ is type $(A)$ , the function $W(t)$ is defined as Eq. (20). Then $W(t)>0$ and Eq. (21) holds. By using Lemmas 3 and 4, it generates by Eq. (20) that:

$\displaystyle{W}^{\prime}(t)\leqslant-\rho(t)Q(t)+\left({\frac{{\rho}^{\prime}% (t)}{\rho(t)}-\frac{m(t)}{r(t)}}\right)W(t)-\frac{1}{\rho(t)r(t)}W^{2}(t)$ (25)

where $Q(t)$ is defined by Eq. (14), let $A(t)=\frac{{\rho}^{\prime}(t)}{\rho(t)}-\frac{m(t)}{r(t)},B(t)=\frac{1}{\rho(t% )r(t)}$ , then there are:

$\displaystyle\int_{t_{1}}^{t}{H(t,s)}\rho(s)Q(s)ds$ $\displaystyle\quad\leqslant\int_{t_{1}}^{t}{H(t,s)}\left[{-{W}^{\prime}(s)+A(s% )W(s)-B(s)W^{2}(s)}\right]ds$ $\displaystyle\quad=-H(t,s)W(s)_{t_{1}}^{t}+\int_{t_{1}}^{t}{\left\{{\frac{% \partial H(t,s)}{\partial s}W(s)+H(t,s)\left[{A(s)W(s)-B(s)W^{2}(s)}\right]}% \right\}}ds$ (26) $\displaystyle\quad=H(t,t_{1})W(t_{1})-\int_{t_{1}}^{t}{\left[{\sqrt{H(t,s)}h_{% 1}(t,s)W(s)+H(t,s)B(s)W^{2}(s)}\right]}ds$ $\displaystyle\quad=H(t,t_{1})W(t_{1})-\int_{t_{1}}^{t}{\left[{\sqrt{H(t,s)B(s)% }W(s)+\frac{h_{1}(t,s)}{2\sqrt{B(s)}}}\right]^{2}ds+}\int_{t_{1}}^{t}{\frac{h_% {1}^{2}(t,s)}{4B(s)}}ds$

where $h_{1}(t,s)$ is defined by Eq. (24), thus, $\frac{1}{H(t,t_{1})}\int_{t_{1}}^{t}{\left[{H(t,s)\rho(s)Q(s)-\frac{h_{1}^{2}(% t,s)}{4B(s)}}\right]}ds\leqslant W(t_{1})$ is contradiction with condition Eq. (23).

If $y(t)$ is type $(B)$ , the proof is the same as the second part of Theorem 3, so it is omitted. Theorem 4 is proved.

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