#### 期刊菜单

The Existence of Zero-Points of Solution for a Class of Second Order Nonlinear Differential Equation
DOI: 10.12677/AAM.2019.87146, PDF, HTML, XML, 下载: 1,015  浏览: 1,277  国家自然科学基金支持

Abstract: In the present paper, we investigate a class of second order differential equation. By establishing several inequalities, some new conditions on the existence of zero-points of solution for the equation are obtained. Several examples are given to illustrate the effectiveness of the obtained conditions.

1. 引言

${\left(p\left(t\right)\Psi \left(y\right)k\left({y}^{\prime }\right)\right)}^{\prime }+r\left(t\right)k\left({y}^{\prime }\right)+q\left(t\right)f\left(y\right)=0$ (1.1)

${\left(p\left({y}^{\prime }\right)\right)}^{\prime }+r\left(t\right){y}^{\prime }+q\left(t\right)y=f\left(t,y,{y}^{\prime }\right)$ (1.2)

${\left(p\left(t\right)\varphi \left(y\right)k\left({y}^{\prime }\right)\right)}^{\prime }+r\left(t\right)k\left({y}^{\prime }\right)+q\left(t\right)y=f\left(t,y,{y}^{\prime }\right)，$ (1.3)

2. 几个不等式

(A1) $\text{0}<{C}_{1}\le \varphi \left(y\right)\le {C}_{2},y\in R$

(A2) ${k}^{2}\left(y\right)\le {C}_{3}yk\left(y\right),y\in R$

$\begin{array}{c}{\left(-\frac{{x}^{2}p\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right)}^{\prime }\ge {\left(\sqrt{\frac{p\varphi \left(y\right)}{{C}_{3}}}\frac{xk\left({y}^{\prime }\right)}{y}+\sqrt{\frac{{C}_{3}}{p\varphi \left(y\right)}}\frac{r\left(t\right)x}{2}-\sqrt{{C}_{3}p\varphi \left(y\right)}{x}^{\prime }\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{C}_{3}}{4{C}_{2}p}{\left[2{C}_{2}p{x}^{\prime }-r\left(t\right)x\right]}^{2}\\ +\frac{1}{4{C}_{2}{C}_{1}p}\left(4{C}_{2}{C}_{1}pq+{C}_{3}\left({C}_{1}-{C}_{2}\right){r}^{2}\right){x}^{2}-\frac{f\left(t,y,{y}^{\prime }\right){x}^{2}}{y}.\end{array}$ (2.1)

${\left(-\frac{{x}^{2}p\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right)}^{\prime }\ge q{x}^{2}+\frac{p\varphi \left(y\right)}{{C}_{3}}{\left({C}_{3}{x}^{\prime }-\frac{xk\left({y}^{\prime }\right)}{y}\right)}^{2}-{C}_{2}{C}_{3}p{{x}^{\prime }}^{2}+\frac{rk\left({y}^{\prime }\right)-f\left(t,y,{y}^{\prime }\right)}{y}{x}^{2}.$ (2.2)

$\begin{array}{l}{\left(-\frac{{x}^{2}p\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right)}^{\prime }\\ \ge q{x}^{2}+\frac{rk\left({y}^{\prime }\right){x}^{2}}{y}-\frac{2x{x}^{\prime }p\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}+\frac{{x}^{2}p\varphi \left(y\right){k}^{2}\left({y}^{\prime }\right)}{{C}_{3}{y}^{2}}-\frac{f\left(t,y,{y}^{\prime }\right){x}^{2}}{y}\\ =q{x}^{2}+\frac{p\varphi \left(y\right)}{{C}_{3}}{\left({C}_{3}{x}^{\prime }-\frac{xk\left({y}^{\prime }\right)}{y}\right)}^{2}+\frac{rk\left({y}^{\prime }\right){x}^{2}}{y}-{C}_{3}p\varphi \left(y\right){{x}^{\prime }}^{2}-\frac{f\left(t,y,{y}^{\prime }\right){x}^{2}}{y}\\ \ge q{x}^{2}+\frac{p\varphi \left(y\right)}{{C}_{3}}{\left({C}_{3}{x}^{\prime }-\frac{xk\left({y}^{\prime }\right)}{y}\right)}^{2}-{C}_{2}{C}_{3}p{{x}^{\prime }}^{2}+\frac{rk\left({y}^{\prime }\right)-f\left(t,y,{y}^{\prime }\right)}{y}{x}^{2}.\end{array}$

$\begin{array}{c}{\left(-{x}^{2}W\left(t\right)\right)}^{\prime }\ge {C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right){\left(\frac{W\left(t\right)}{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}x-{x}^{\prime }-\frac{1}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right){\left({x}^{\prime }+\frac{1}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}+Q\left(t\right){x}^{2}-\frac{f\left(t,y,{y}^{\prime }\right)}{y}\Phi \left(t\right){x}^{2}.\end{array}$ (2.3)

$\begin{array}{l}{W}^{\prime }\left(t\right)=\frac{{\Phi }^{\prime }\left(t\right)}{\Phi \left(t\right)}W\left(t\right)+\Phi \left(t\right)\left[{\left(\frac{p\left(t\right)\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right)}^{\prime }+{\left(p\left(t\right)R\left(t\right)\right)}^{\prime }\right]\\ =\frac{{\Phi }^{\prime }\left(t\right)}{\Phi \left(t\right)}W\left(t\right)+\Phi \left(t\right)\left[\frac{f\left(t,y,{y}^{\prime }\right)-r\left(t\right)k\left({y}^{\prime }\right)-q\left(t\right)y}{y}-\frac{p\left(t\right)\varphi \left(y\right)k\left({y}^{\prime }\right){y}^{\prime }}{{y}^{2}}+{\left(p\left(t\right)R\left(t\right)\right)}^{\prime }\right]\\ \le \frac{{\Phi }^{\prime }\left(t\right)}{\Phi \left(t\right)}W\left(t\right)+\Phi \left(t\right)\left[\frac{f\left(t,y,{y}^{\prime }\right)}{y}-\frac{r\left(t\right)k\left({y}^{\prime }\right)}{y}-q\left(t\right)-\frac{p\left(t\right)\varphi \left(y\right){k}^{2}\left({y}^{\prime }\right)}{{C}_{3}{y}^{2}}+{\left(p\left(t\right)R\left(t\right)\right)}^{\prime }\right]\\ =\frac{{\Phi }^{\prime }\left(t\right)}{\Phi \left(t\right)}W\left(t\right)+\Phi \left(t\right)\left\{-q\left(t\right)+{\left(p\left(t\right)R\left(t\right)\right)}^{\prime }+\frac{f\left(t,y,{y}^{\prime }\right)}{y}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{1}{\varphi \left(y\right)}\left[\frac{r\left(t\right)\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}+\frac{p\left(t\right)}{{C}_{3}}{\left(\frac{\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right)}^{2}\right]\right\}\end{array}$

$\begin{array}{l}=\frac{{\Phi }^{\prime }\left(t\right)}{\Phi \left(t\right)}W\left(t\right)+\Phi \left(t\right)\left\{-q\left(t\right)+\frac{{C}_{3}\left({C}_{2}-{C}_{1}\right)}{4{C}_{1}{C}_{2}}\frac{{r}^{2}\left(t\right)}{p\left(t\right)}+\frac{r\left(t\right)}{{C}_{2}}R\left(t\right)-\frac{p\left(t\right)}{{C}_{2}{C}_{3}}{R}^{2}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left(p\left(t\right)R\left(t\right)\right)}^{\prime }+\left(\frac{2R\left(t\right)}{{C}_{2}{C}_{3}\Phi \left(t\right)}-\frac{r\left(t\right)}{{C}_{2}\Phi \left(t\right)p\left(t\right)}\right)W\left(t\right)-\frac{{W}^{2}\left(t\right)}{{C}_{2}{C}_{3}{\Phi }^{2}\left(t\right)p\left(t\right)}+\frac{f\left(t,y,{y}^{\prime }\right)}{y}\right\}\\ =\frac{{\Phi }^{\prime }\left(t\right)}{\Phi \left(t\right)}W\left(t\right)-\Phi \left(t\right)\left[q\left(t\right)-\frac{{C}_{3}\left({C}_{2}-{C}_{1}\right)}{4{C}_{1}{C}_{2}}\frac{{r}^{2}\left(t\right)}{p\left(t\right)}-\frac{r\left(t\right)}{{C}_{2}}R\left(t\right)+\frac{p\left(t\right)}{{C}_{2}{C}_{3}}{R}^{2}\left(t\right)-{\left(p\left(t\right)R\left(t\right)\right)}^{\prime }\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(\frac{2R\left(t\right)}{{C}_{2}{C}_{3}}-\frac{r\left(t\right)}{{C}_{2}p\left(t\right)}\right)W\left(t\right)-\frac{{W}^{2}\left(t\right)}{{C}_{2}{C}_{3}\Phi \left(t\right)p\left(t\right)}+\frac{f\left(t,y,{y}^{\prime }\right)}{y}\Phi \left(t\right)\\ =-\Phi \left(t\right)\left[q\left(t\right)-\frac{{C}_{3}\left({C}_{2}-{C}_{1}\right)}{4{C}_{1}{C}_{2}}\frac{{r}^{2}\left(t\right)}{p\left(t\right)}-\frac{r\left(t\right)}{{C}_{2}}R\left(t\right)+\frac{p\left(t\right)}{{C}_{2}{C}_{3}}{R}^{2}\left(t\right)-{\left(p\left(t\right)R\left(t\right)\right)}^{\prime }\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(\frac{2R\left(t\right)}{{C}_{2}{C}_{3}}+\frac{{\Phi }^{\prime }\left(t\right)}{\Phi \left(t\right)}-\frac{r\left(t\right)}{{C}_{2}p\left(t\right)}\right)W\left(t\right)-\frac{{W}^{2}\left(t\right)}{{C}_{2}{C}_{3}\Phi \left(t\right)p\left(t\right)}+\frac{f\left(t,y,{y}^{\prime }\right)}{y}\Phi \left(t\right)\\ =-Q\left(t\right)+\stackrel{¯}{R}\left(t\right)W\left(t\right)-\frac{{W}^{2}\left(t\right)}{{C}_{2}{C}_{3}\Phi \left(t\right)p\left(t\right)}+\frac{f\left(t,y,{y}^{\prime }\right)}{y}\Phi \left(t\right).\end{array}$

$\begin{array}{l}{\left[-{x}^{2}W\left(t\right)\right]}^{\prime }\\ \ge -2x{x}^{\prime }W\left(t\right)-{x}^{2}\left(-Q\left(t\right)+\stackrel{¯}{R}\left(t\right)W\left(t\right)-\frac{{W}^{2}\left(t\right)}{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}+\frac{f\left(t,y,{y}^{\prime }\right)}{y}\Phi \left(t\right)\right)\\ ={\left(\frac{W\left(t\right)}{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}x\right)}^{2}-2\frac{W\left(t\right)x}{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}{x}^{\prime }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\frac{W\left(t\right)x}{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}\frac{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}{2}\stackrel{¯}{R}\left(t\right)x+{\left(\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}{x}^{\prime }\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\left(\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}{x}^{\prime }\right)}^{2}+{\left(\frac{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}-{\left(\frac{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\left(\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}{x}^{\prime }\right)\left(\frac{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}{2}\stackrel{¯}{R}\left(t\right)x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\left(\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}{x}^{\prime }\right)\left(\frac{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}{2}\stackrel{¯}{R}\left(t\right)x\right)+Q\left(t\right){x}^{2}-\frac{f\left(t,y,{y}^{\prime }\right)}{y}\Phi \left(t\right){x}^{2}\\ ={\left(\frac{W\left(t\right)}{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}x-\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}{x}^{\prime }-\frac{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\left(\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}{x}^{\prime }+\frac{\sqrt{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+Q\left(t\right){x}^{2}-\frac{f\left(t,y,{y}^{\prime }\right)}{y}\Phi \left(t\right){x}^{2}\end{array}$

$\begin{array}{l}={C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right){\left(\frac{W\left(t\right)}{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}x-{x}^{\prime }-\frac{1}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right){\left({x}^{\prime }+\frac{1}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}+Q\left(t\right){x}^{2}-\frac{f\left(t,y,{y}^{\prime }\right)}{y}\Phi \left(t\right){x}^{2}.\end{array}$

3. 主要结果

${\int }_{0}^{1}\left\{\left[4{C}_{2}{C}_{1}p\left(t\right)q\left(t\right)+{C}_{3}\left({C}_{1}-{C}_{2}\right){r}^{2}\left(t\right)\right]{x}^{2}-{C}_{3}{C}_{1}{\left[2{C}_{2}p\left(t\right){x}^{\prime }-r\left(t\right)x\right]}^{2}\right\}\text{d}t>0,$

$y\left(t\right)$$\left[0,1\right]$ 上至少有一个零点。

$\begin{array}{l}{\int }_{0}^{1}{\left[-\frac{{x}^{2}p\left(t\right)\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right]}^{\prime }\text{d}t\\ \ge {\int }_{0}^{1}\left\{{\left[\sqrt{\frac{p\left(t\right)\varphi \left(y\right)}{{C}_{3}}}\frac{k\left({y}^{\prime }\right)x}{y}+\sqrt{\frac{{C}_{3}}{p\left(t\right)\varphi \left(y\right)}}\frac{r\left(t\right)x}{2}-\sqrt{{C}_{3}p\left(t\right)\varphi \left(y\right)}{x}^{\prime }\right]}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-\frac{{C}_{3}}{4{C}_{2}p\left(t\right)}\left[2{C}_{2}p\left(t\right){x}^{\prime }-r\left(t\right)x\right]+\frac{1}{4{C}_{2}{C}_{1}p\left(t\right)}\left[4{C}_{2}{C}_{1}p\left(t\right)q\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{C}_{3}\left({C}_{1}-{C}_{2}\right){r}^{2}\left(t\right)\right]{x}^{2}-\frac{f\left(t,y,{y}^{\prime }\right){x}^{2}}{y}\right\}\text{d}t.\end{array}$

$x\left(0\right)=x\left(1\right)=0$ ，有

${\int }_{0}^{1}{\left[-\frac{{x}^{2}p\left(t\right)\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right]}^{\prime }\text{d}t={\left[-\frac{{x}^{2}p\left(t\right)\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right]}_{0}^{1}=0.$

${\int }_{0}^{1}\left\{\left[4{C}_{2}{C}_{1}p\left(t\right)q\left(t\right)+{C}_{3}\left({C}_{1}-{C}_{2}\right){r}^{2}\left(t\right)\right]{x}^{2}-{C}_{3}{C}_{1}{\left[2{C}_{2}p\left(t\right){x}^{\prime }-r\left(t\right)x\right]}^{2}\right\}\text{d}t\le 0,$

$\left[4{C}_{2}{C}_{1}p\left(t\right)q\left(t\right)+{C}_{3}\left({C}_{1}-{C}_{2}\right){r}^{2}\left(t\right)\right]{x}^{2}\ge {C}_{3}{C}_{1}{\left[2{C}_{2}p\left(t\right){x}^{\prime }-r\left(t\right)x\right]}^{2}$$t\in \left[0,1\right]$

$\frac{r\left(t\right)k\left(v\right)-f\left(t,u,v\right)}{u}\ge 0\text{}\left(u\ne 0\right)$ ；且

${\int }_{0}^{1}\left[q\left(t\right){x}^{2}-{C}_{2}{C}_{3}p\left(t\right){{x}^{\prime }}^{2}\right]\text{d}t>0.$

$y\left(t\right)$$\left[0,1\right]$ 上至少有一个零点。

$\begin{array}{c}{\int }_{0}^{1}{\left(-\frac{{x}^{2}p\left(t\right)\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right)}^{\prime }\text{d}t\ge {\int }_{0}^{1}\frac{p\left(t\right)\varphi \left(y\right)}{{C}_{3}}{\left({C}_{3}{x}^{\prime }-\frac{k\left({y}^{\prime }\right)}{y}x\right)}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{1}\left[q\left(t\right){x}^{2}-{C}_{2}{C}_{3}p\left(t\right){{x}^{\prime }}^{2}\right]\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{1}\frac{r\left(t\right)k\left({y}^{\prime }\right)-f\left(t,y,{y}^{\prime }\right)}{y}{x}^{2}\text{d}t.\end{array}$

$x\left(0\right)=x\left(1\right)=0$ ，有

${\int }_{0}^{1}{\left(-\frac{{x}^{2}p\left(t\right)\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right)}^{\prime }\text{d}t={\left(-\frac{{x}^{2}p\left(t\right)\varphi \left(y\right)k\left({y}^{\prime }\right)}{y}\right)}_{0}^{1}=0.$

${\int }_{0}^{1}\left[q\left(t\right){x}^{2}-{C}_{2}{C}_{3}p\left(t\right){{x}^{\prime }}^{2}\right]\text{d}t\le 0,$

$\frac{f\left(t,u,v\right)}{u}\le 0\text{}\left(u\ne 0\right)$ ，且

${\int }_{0}^{1}\left[Q\left(t\right){x}^{2}-{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right){\left({x}^{\prime }+\frac{1}{2}x\stackrel{¯}{R}\left(t\right)\right)}^{2}\right]\text{d}t>0,$

$y\left(t\right)$$\left[0,1\right]$ 上至少有一个零点。

$\begin{array}{c}{\int }_{0}^{1}{\left[-{x}^{2}W\left(t\right)\right]}^{\prime }\text{d}t\ge {\int }_{0}^{1}{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right){\left(\frac{W\left(t\right)}{{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right)}x-{x}^{\prime }-\frac{1}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{1}\left[Q\left(t\right){x}^{2}-{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right){\left({x}^{\prime }+\frac{1}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}\right]\text{d}t-{\int }_{0}^{1}\frac{f\left(t,y,{y}^{\prime }\right)}{y}\Phi \left(t\right){x}^{2}\text{d}t.\end{array}$

$x\left(0\right)=x\left(1\right)=0$ ，有

${\int }_{0}^{1}{\left[-{x}^{2}W\left(t\right)\right]}^{\prime }={\left[-{x}^{2}W\left(t\right)\right]}_{0}^{1}=0.$

${\int }_{0}^{1}\left[Q\left(t\right){x}^{2}-{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right){\left({x}^{\prime }+\frac{1}{2}\stackrel{¯}{R}\left(t\right)x\right)}^{2}\right]\text{d}t\le 0,$

${Q}_{2}\left(t\right){x}^{2}\ge {C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right){\left({x}^{\prime }+\frac{1}{2}x{r}_{2}\left(t\right)\right)}^{2},$

4. 实例

${\left[\left(\text{1}+{\mathrm{sin}}^{2}\text{π}t\right)\frac{{y}^{\prime }}{1+\alpha {{y}^{\prime }}^{2}}\right]}^{\prime }+4\text{π}\mathrm{sin}\text{π}t\mathrm{cos}\text{π}t\frac{{y}^{\prime }}{1+\alpha {{y}^{\prime }}^{2}}+q\left(t\right)y+{t}^{2}y{{y}^{\prime }}^{2}=0$ (4.1)

$p\left(t\right)=1+{\mathrm{sin}}^{2}\text{π}t$$\varphi \left(y\right)=1$$k\left(v\right)=\frac{v}{1+\alpha {v}^{2}}$$r\left(t\right)=4\text{π}\mathrm{sin}\text{π}t\mathrm{cos}\text{π}t$$f\left(t,u,v\right)=-{t}^{2}u{v}^{2}$ ，则 $f\in C\left(\left[0,1\right]×R×R;R\right)$$p\in {C}^{1}\left(\left[0,1\right];{R}_{+}\right)$$r\in C\left(\left[0,1\right];R\right)$$k,\varphi \in {C}^{1}\left(R;R\right)$ ，且

$\frac{f\left(t,u,v\right)}{u}=-{t}^{2}{v}^{2}\le 0\text{}\left(u\ne 0\right).$

${C}_{1}={C}_{2}={C}_{3}=1$ ，则条件(A1)、(A2)满足。取 $x\left(t\right)={\mathrm{sin}}^{2}\text{π}t$ ，则 $x\left(0\right)=x\left(1\right)=0$

$t\in \left[0,1\right]$ ，有

$\begin{array}{l}\left[4{C}_{2}{C}_{1}p\left(t\right)q\left(t\right)+{C}_{3}\left({C}_{1}-{C}_{2}\right){r}^{2}\left(t\right)\right]{x}^{2}-{C}_{3}{C}_{1}{\left[2{C}_{2}p\left(t\right){x}^{\prime }-r\left(t\right)x\right]}^{2}\\ =4q\left(t\right)\left(1+{\mathrm{sin}}^{2}\text{π}t\right){\mathrm{sin}}^{4}\text{π}t-{\left[4\text{π}\left(1+{\mathrm{sin}}^{2}\text{π}t\right)\mathrm{sin}\text{π}t\mathrm{cos}\text{π}t-4\text{π}\mathrm{sin}\text{π}t\mathrm{cos}\text{π}t{\mathrm{sin}}^{2}\text{π}t\right]}^{2}\\ =4{\mathrm{sin}}^{2}\text{π}t\left[q\left(t\right)\left(1+{\mathrm{sin}}^{2}\text{π}t\right){\mathrm{sin}}^{2}\text{π}t-4{\text{π}}^{2}{\mathrm{cos}}^{2}\text{π}t\right]\\ \ge 4{\mathrm{sin}}^{2}\text{π}t\left[q\left(t\right){\mathrm{sin}}^{2}\text{π}t-4{\text{π}}^{2}\right]\ge 0,\end{array}$

${\left(\frac{\text{2}+{\mathrm{cos}}^{2}\text{π}t}{1+{\mathrm{cos}}^{2}\text{π}t}\cdot \frac{1+{y}^{2}}{2+{y}^{2}}\cdot \frac{{{y}^{\prime }}^{\text{2}}}{1+\alpha {{y}^{\prime }}^{2}}\right)}^{\prime }+\frac{{t}^{2}{{y}^{\prime }}^{\text{2}}}{1+\alpha {{y}^{\prime }}^{2}}+q\left(t\right)y+\frac{{y}^{3}-{t}^{2}{{y}^{\prime }}^{2}}{1+\alpha {{y}^{\prime }}^{2}}=0$ (4.2)

$p\left(t\right)=\frac{\text{2}+{\mathrm{cos}}^{2}\text{π}t}{1+{\mathrm{cos}}^{2}\text{π}t}$$\varphi \left(y\right)=\frac{1+{y}^{2}}{2+{y}^{2}}$$k\left(v\right)=\frac{{v}^{\text{2}}}{1+\alpha {v}^{2}}$$r\left(t\right)={t}^{2}$$f\left(t,u,v\right)=-\frac{{u}^{3}-{t}^{2}{v}^{2}}{1+a{v}^{2}}$ ，则 $f\in C\left(\left[0,1\right]×R×R;R\right)$$p\in {C}^{1}\left(\left[0,1\right];{R}_{+}\right)$$r\in C\left(\left[0,1\right];R\right)$$k,\varphi \in {C}^{1}\left(R;R\right)$ ，且 $\frac{1}{2}\le \varphi \left(y\right)=\frac{1+{y}^{2}}{2+{y}^{2}}\le 1$$\frac{r\left(t\right)k\left(v\right)-f\left(t,u,v\right)}{u}=\frac{{u}^{2}}{1+\alpha {v}^{2}}\ge 0\text{\hspace{0.17em}}\left(u\ne 0\right)$

${C}_{1}=\frac{1}{2},{C}_{2}={C}_{3}=1$ ，则条件(A1)、(A2)满足。取 $x\left(t\right)=\mathrm{sin}\text{π}t$ ，则 $x\left(0\right)=x\left(1\right)=0$ ，且

$\begin{array}{l}q\left(t\right){x}^{2}-p\left(t\right){C}_{2}{C}_{3}{{x}^{\prime }}^{2}\\ =q\left(t\right){\mathrm{sin}}^{2}\text{π}t-\left(\frac{\text{2}+{\mathrm{cos}}^{2}\text{π}t}{1+{\mathrm{cos}}^{2}\text{π}t}\right){\text{π}}^{2}{\mathrm{cos}}^{2}\text{π}t\\ \ge q\left(t\right){\mathrm{sin}}^{2}\text{π}t-3{\text{π}}^{2}\ge \text{0}\end{array}$

${\left[\left(\text{1}+{\mathrm{sin}}^{2}\text{π}t\right)\frac{{y}^{\prime }}{1+\alpha {{y}^{\prime }}^{2}}\right]}^{\prime }-\text{2π}\mathrm{sin}\text{π}t\mathrm{cos}\text{π}t\frac{{y}^{\prime }}{1+\alpha {{y}^{\prime }}^{2}}+q\left(t\right)y+{t}^{2}y{{y}^{\prime }}^{2}=0,$ (4.3)

$q\left(t\right)\ge {\text{π}}^{2}\left(4+{\mathrm{cot}}^{2}\text{π}t\right),\text{\hspace{0.17em}}t\in \left(0,1\right).$

$p\left(t\right)=1+{\mathrm{sin}}^{2}\text{π}t$$\varphi \left(y\right)=1$$k\left(v\right)=\frac{v}{1+\alpha {v}^{2}}$$r\left(t\right)=-2\text{π}\mathrm{sin}\text{π}t\mathrm{cos}\text{π}t$$f\left(t,u,v\right)=-{t}^{2}uv$ ，则 $f\in C\left(\left[0,1\right]×R×R;R\right)$$p\in {C}^{1}\left(\left[0,1\right];{R}_{+}\right)$$r\in C\left(\left[0,1\right];R\right)$$k,\varphi \in {C}^{1}\left(R;R\right)$ ，且

$\frac{f\left(t,u,v\right)}{u}=-{t}^{2}{v}^{2}\le 0,\text{}u\ne 0.$

${C}_{1}={C}_{2}={C}_{3}=1$ ，则条件(A1)、(A2)成立。取 $x\left(t\right)=\mathrm{sin}\text{π}t$ ，则 $x\left(0\right)=x\left(1\right)=0$

$R\left(t\right)=-\frac{1}{t},\Phi \left(t\right)={t}^{2}$ ，则

$\begin{array}{c}Q\left(t\right)=\Phi \left(t\right)\left[q\left(t\right)-\frac{{C}_{3}\left({C}_{2}-{C}_{1}\right)}{4{C}_{1}{C}_{2}}\frac{{r}^{2}\left(t\right)}{p\left(t\right)}-\frac{r\left(t\right)}{{C}_{2}}R\left(t\right)+\frac{p\left(t\right)}{{C}_{2}{C}_{3}}{R}^{2}\left(t\right)-{\left(p\left(t\right)R\left(t\right)\right)}^{\prime }\right]\\ \ge {t}^{2}\left[{\text{π}}^{2}\left(4+{\mathrm{cot}}^{2}\text{π}t\right)-\frac{\text{2π}\mathrm{sin}\text{π}t\mathrm{cos}\text{π}t}{t}+\frac{1+{\mathrm{sin}}^{2}\text{π}t}{{t}^{2}}-{\left(\frac{1+{\mathrm{sin}}^{2}\text{π}t}{-t}\right)}^{\prime }\right]\\ ={t}^{2}{\text{π}}^{2}\left(4+{\mathrm{cot}}^{2}\text{π}t\right)\end{array}$

$\stackrel{¯}{R}\left(t\right)=\frac{2R\left(t\right)}{{C}_{2}{C}_{3}}+\frac{{\Phi }^{\prime }\left(t\right)}{\Phi \left(t\right)}-\frac{r\left(t\right)}{{C}_{2}p\left(t\right)}=-\frac{2}{t}+\frac{2t}{{t}^{2}}-\frac{-2\text{π}\mathrm{sin}\text{π}t\mathrm{cos}\text{π}t}{1+{\mathrm{sin}}^{2}\text{π}t}=\frac{2\text{π}\mathrm{sin}\text{π}t\mathrm{cos}\text{π}t}{1+{\mathrm{sin}}^{2}\text{π}t}$

$\begin{array}{l}Q\left(t\right){x}^{2}-{C}_{2}{C}_{3}p\left(t\right)\Phi \left(t\right){\left[{x}^{\prime }+\frac{1}{2}x\stackrel{¯}{R}\left(t\right)\right]}^{2}\\ \ge {t}^{2}{\text{π}}^{2}\left(4+{\mathrm{cot}}^{2}\text{π}t\right){\mathrm{sin}}^{2}\text{π}t-\left(1+{\mathrm{sin}}^{2}\text{π}t\right){t}^{2}{\left(\text{π}\mathrm{cos}\text{π}t+\frac{\mathrm{sin}\text{π}t}{2}\cdot \frac{2\pi \mathrm{sin}\text{π}t\mathrm{cos}\text{π}t}{1+{\mathrm{sin}}^{2}\text{π}t}\right)}^{2}\\ ={t}^{2}{\text{π}}^{2}\left(4+{\mathrm{cot}}^{2}\text{π}t\right){\mathrm{sin}}^{2}\text{π}t-\left(1+{\mathrm{sin}}^{2}\text{π}t\right){t}^{2}\left[{\text{π}}^{2}{\mathrm{cos}}^{2}\text{π}t+\frac{2{\pi }^{2}{\mathrm{sin}}^{2}\text{π}t{\mathrm{cos}}^{2}\text{π}t}{1+{\mathrm{sin}}^{2}\text{π}t}+\frac{{\text{π}}^{2}{\mathrm{sin}}^{4}\text{π}t{\mathrm{cos}}^{2}\text{π}t}{{\left(1+{\mathrm{sin}}^{2}\text{π}t\right)}^{2}}\right]\\ ={t}^{2}{\text{π}}^{2}{\mathrm{sin}}^{2}\text{π}t\left\{\left(4+{\mathrm{cot}}^{2}\text{π}t\right)-\left(1+{\mathrm{sin}}^{2}\text{π}t\right)\left[{\mathrm{cot}}^{2}\text{π}t+\frac{2{\mathrm{cos}}^{2}\text{π}t}{1+{\mathrm{sin}}^{2}\text{π}t}+\frac{{\mathrm{sin}}^{2}\text{π}t{\mathrm{cos}}^{2}\text{π}t}{{\left(1+{\mathrm{sin}}^{2}\text{π}t\right)}^{2}}\right]\right\}\\ \ge 2{t}^{2}{\text{π}}^{2}{\mathrm{sin}}^{2}\text{π}t\left\{3+{\mathrm{cot}}^{2}\text{π}t-\left[{\mathrm{cot}}^{2}\text{π}t+3\right]\right\}=0\end{array}$

NOTES

*通讯作者。

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