﻿ 无穷区间上奇异边值问题正解的存在性

# 无穷区间上奇异边值问题正解的存在性Existence of Positive Solutions for Singular Boundary Value Problems on the Infinite Interval

Banach空间中的微分方程理论是非线性泛函分析的重要分支，奇异方程边值问题是微分方程学科的组成部分，处于微分方程理论和线性及非线性泛函分析的交叉结合点上，广泛存在于弹簧的振动、梁的非弹性振动、种群生态系统等自然界的各种数学模型中，本文主要应用锥上的不动点理论，通过建立特殊的空间和范数，在非线性项f奇异的条件下，讨论了无穷区间上一类微分方程边值问题解的存在性，获得了方程至少存在一个正解的结论。本文的结果在一定程度上推广了奇异和非奇异条件下的许多已知结果。

Abstract: The theory of differential equations in Banach spaces is an important branch of nonlinear analysis. The boundary value problem of differential equation is the component of the differential equation subject, which is in the intersection of differential equation theory and linear and nonlinear functional analysis. It exists widely in various mathematical models of nature, such as spring vibration, inelastic vibration of beams and population ecosystem. In this paper, by using the fixed point theory in the cone with a special norm and space, the authors discuss the existence of positive solutions for a class of boundary value problems of differential equation on the infinite interval and obtain that the equation has at least one positive solution. The results improve many known results including singular and non-singular cases to a certain extent.

1. 引言

$\left\{\begin{array}{l}{\left(p\left(t\right){x}^{\prime }\left(t\right)\text{​}\right)}^{\prime }-{k}^{2}x\left(t\right)+f\left(t,x\left(t\right),\left({x}^{\prime }\left(t\right)\right)\right)=0,\text{\hspace{0.17em}}t\in \left(0,+\infty \right),\\ {\alpha }_{1}x\left(0\right)-{\beta }_{1}\underset{t\to {0}^{+}}{\mathrm{lim}}p\left(t\right){x}^{\prime }\left(t\right)={\gamma }_{1},\\ {\alpha }_{2}\underset{t\to +\infty }{\mathrm{lim}}x\left(t\right)-{\beta }_{2}\underset{t\to +\infty }{\mathrm{lim}}p\left(t\right){x}^{\prime }\left(t\right)={\gamma }_{2},\end{array}$ (1.1)

$\rho ={\alpha }_{2}{\beta }_{1}+{\alpha }_{1}{\beta }_{2}+{\alpha }_{1}{\alpha }_{2}B\left(0,+\infty \right)>0,B\left(t,s\right)={\int }_{0}^{+\infty }\frac{1}{p\left(v\right)}\text{d}v$

$\left\{\begin{array}{l}{x}^{″}\left(t\right)-{k}^{2}x\left(t\right)+f\left(t,x\left(t\right)\right)=0,\text{\hspace{0.17em}}t\in \left(0,+\infty \right),\\ x\left(0\right)=0,\text{\hspace{0.17em}}\underset{t\to \infty }{\mathrm{lim}}x\left(t\right)=0,\end{array}$

$\left(t,x\right)\in \left[0,+\infty \right)×\left[0,+\infty \right)$$a,\text{\hspace{0.17em}}b:\left[0,+\infty \right)\to \left[0,+\infty \right)$ 是连续函数。类似于文 [5] 中的边值条件，Hao等 [6] 建立了下面微分方程正解的存在性理论：

${x}^{″}\left(t\right)-{k}^{2}x\left(t\right)+mf\left(t,x\left(t\right)\right)=0,$ (1.2)

${\left(p\left(t\right){x}^{\prime }\left(t\right)\text{​}\text{​}\right)}^{\prime }+\lambda f\left(t,x\left(t\right)\right)-{k}^{2}x\left(t\right)=0,$

2. 预备知识

$a\left(t\right)={\beta }_{1}+{\alpha }_{1}B\left(0,t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}b\left(t\right)={\beta }_{2}+{\alpha }_{2}B\left(t,\infty \right)$

$a\left(\infty \right)=\underset{t\to \infty }{\mathrm{lim}}a\left(t\right)={\beta }_{1}+{\alpha }_{1}B\left(0,\infty \right)<+\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}a\left(0\right)=\underset{t\to \infty }{\mathrm{lim}}a\left(t\right)={\beta }_{1}$

$b\left(\infty \right)=\underset{t\to \infty }{\mathrm{lim}}b\left(t\right)={\beta }_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}b\left(0\right)=\underset{t\to \infty }{\mathrm{lim}}b\left(t\right)={\beta }_{2}+{\alpha }_{1}B\left(0,\infty \right)<+\infty$

$x\left(t\right)={\rho }^{-1}\left[1+a\left(t\right)b\left(t\right){y}_{1}\left(t\right)\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}^{\prime }\left(t\right)={y}_{2}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(t,u,{u}^{\prime }\right)=g\left(t,{v}_{1},{v}_{1}\right)$

$\left\{\begin{array}{l}{\left(p\left(t\right){u}^{\prime }\left(t\right)\text{​}\right)}^{\prime }+v\left(t\right)=0,\text{\hspace{0.17em}}t\in \left(0,+\infty \right),\\ {\alpha }_{1}u\left(0\right)-{\beta }_{1}\underset{t\to {0}^{+}}{\mathrm{lim}}p\left(t\right){u}^{\prime }\left(t\right)=0,\\ {\alpha }_{2}\underset{t\to +\infty }{\mathrm{lim}}u\left(t\right)-{\beta }_{2}\underset{t\to +\infty }{\mathrm{lim}}p\left(t\right){u}^{\prime }\left(t\right)=0,\end{array}$ (2.1)

$u\left(t\right)={\int }_{0}^{+\infty }G\left(t,s\right)v\left(s\right)\text{d}s.$

$G\left(t,s\right)=\frac{1}{\rho }\left\{\begin{array}{l}\left({\beta }_{1}+{\alpha }_{1}B\left(0,s\right)\right)\left({\beta }_{2}+{\alpha }_{2}B\left(t,\infty \right)\right),\text{\hspace{0.17em}}0\le s (2.2)

(1) $G\left(t,s\right)$$\left[0,+\infty \right)×\left[0,+\infty \right)$ 上连续；

(2) 对任意的 $s\in \left[0,+\infty \right)$$G\left(t,s\right)$$\left[0,+\infty \right)$ 上连续可微( $t=s$ 点除外)；

(3) ${\frac{\partial G\left(t,s\right)}{\partial t}|}_{t={s}^{+}}=-{\frac{\partial G\left(t,s\right)}{\partial t}|}_{t={s}^{-}}=-\frac{1}{p\left(s\right)}$

(4) 对任意的 $s\in \left[0,+\infty \right)$$G\left(t,s\right)$$\left[0,+\infty \right)$ 上除 $t=s$ 点外满足与其对应的齐次方程(BVP(2.1) $v\left(t\right)\equiv 0$ )即 $G\left(t,s\right)$ 是BVP(2.1)的Green’s函数；

(5) $G\left(t,s\right)\le G\left(s,s\right)\le \frac{1}{\rho }\left({\beta }_{1}+{\alpha }_{1}B\left(0,s\right)\right)\left({\beta }_{2}+{\alpha }_{2}B\left(s,\infty \right)\right)<+\infty$

(6) $\stackrel{¯}{G}\left(s\right)=\underset{t\to +\infty }{\mathrm{lim}}G\left(t,s\right)=\frac{1}{\rho }{\beta }_{2}\left({\beta }_{1}+{\alpha }_{1}B\left(0,s\right)\right)\le G\left(s,s\right)<+\infty$

(7) 对任意的 $t\in \left[{a}^{*},{b}^{*}\right]\subset \left(0,+\infty \right)$$s,u\in \left[0,+\infty \right)$ ，存在 ${c}^{*}\left(0<{c}^{*}<1\right)$ ，满足

${c}^{*}\frac{|x\left(t\right)|}{{\rho }^{-1}\left[1+a\left(u\right)b\left(u\right)\right]}\le \frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\le 1$$\frac{a\left(t\right)+b\left(t\right)}{\left[1+a\left(t\right)b\left(t\right)\right]}\ge {c}^{*}\frac{a\left(u\right)+b\left(u\right)}{\left[1+a\left(u\right)b\left(u\right)\right]}$

$X=\left\{x\in {C}^{1}\left[0,+\infty \right):\underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{|x\left(t\right)|}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}<+\infty ,\underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}|{x}^{\prime }\left(t\right)|<+\infty \right\}$ (2.3)

$K=\left\{x\in X:x\left(t\right)\ge 0,\underset{t\in \left[{a}^{*},{b}^{*}\right]}{\mathrm{min}}\frac{|x\left(t\right)|}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\ge {c}^{*}\frac{|x\left(u\right)|}{{\rho }^{-1}\left[1+a\left(u\right)b\left(u\right)\right]},t,u\in \left[0,+\infty \right)\right\}$

(1) $M$$X$ 中一致有界；

(2) 函数 $\left\{{y}_{1},{y}_{2}|{y}_{1}\left(t\right)=\frac{|x\left(t\right)|}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]},{y}_{2}\left(t\right)={x}^{\prime }\left(t\right):x,{x}^{\prime }\in M\right\}$$\left[0,+\infty \right)$ 上局部等度连续；

(3) 函数 $\left\{{y}_{1},{y}_{2}|{y}_{1}\left(t\right)=\frac{|x\left(t\right)|}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]},{y}_{2}\left(t\right)={x}^{\prime }\left(t\right):x,{x}^{\prime }\in M\right\}$$+\infty$ 处一致收敛，则称 $M$$X$ 中是相对紧的。

(1) $‖Ax‖\le ‖x‖,\forall x\in \partial P$

(2) 存在 ${x}_{0}\in \partial {P}_{1}$ ，使得 $x\ne Ax+m{x}_{0},\forall x\in \partial {P}_{\gamma },\text{\hspace{0.17em}}m>0$ ，则 $A$${\stackrel{¯}{P}}_{\gamma ,R}$ 存在不动点；

3. 主要结果

(H1) $f,g:\left(0,+\infty \right)×\left[0,+\infty \right)×\left(-\infty ,+\infty \right)\to \left[0,+\infty \right)$ 是连续函数并且满足 ${k}^{2}u\le f\left(t,u,{u}^{\prime }\right)=g\left(t,{v}_{1},{v}_{2}\right)\le \varphi \left(t\right)q\left(t,{v}_{1},{v}_{2}\right)$ ，其中 $\varphi :\left(0,+\infty \right)\to \left[0,+\infty \right)$ 连续且在 $t=0$ 点奇异， $\varphi \left(t\right)$$\left[0,+\infty \right)$ 上不恒为0， $q:\left[0,+\infty \right)×\left[0,+\infty \right)×\left(-\infty ,+\infty \right)\to \left[0,+\infty \right)$ 是连续函数，对任意的 $0\le t\le +\infty$${v}_{1},|{v}_{2}|$ 属于 $\left[0,+\infty \right)$ 上的有界集， $q\left(t,{v}_{1},{v}_{2}\right)$ 有界。

(H2) ${\int }_{0}^{+\infty }\varphi \left(s\right)\text{d}s<+\infty .$

$\left(Tx\right)\left(t\right)=\frac{{\gamma }_{1}b\left(t\right)}{\rho }+\frac{{\gamma }_{2}a\left(t\right)}{\rho }+{\int }_{0}^{+\infty }G\left(t,s\right)\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s,\text{\hspace{0.17em}}t\in \left[0,+\infty \right),$ (3.1)

$\begin{array}{l}\frac{\left(Tx\right)\left(t\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}={\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]\left(\frac{{\gamma }_{1}b\left(t\right)}{\rho }+\frac{{\gamma }_{2}a\left(t\right)}{\rho }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]{\int }_{0}^{+\infty }G\left(t,s\right)\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s\\ \le \frac{{\gamma }_{1}b\left(t\right)+{\gamma }_{2}a\left(t\right)}{\left[1+a\left(t\right)b\left(t\right)\right]}+{\int }_{0}^{+\infty }\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s\\ \le \frac{{\gamma }_{1}b\left(0\right)+{\gamma }_{2}a\left(\infty \right)}{\left[1+{\beta }_{1}{\beta }_{2}\right]}+{\int }_{0}^{+\infty }\varphi \left(s\right)q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\text{d}s\\ \le \frac{{\gamma }_{1}b\left(0\right)+{\gamma }_{2}a\left(\infty \right)}{\left[1+{\beta }_{1}{\beta }_{2}\right]}+{S}_{{\gamma }^{*}}{\int }_{0}^{+\infty }\varphi \left(s\right)\text{d}s<+\infty ,\end{array}$ (3.2)

$\begin{array}{l}\le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}\left(\frac{{\alpha }_{1}{\gamma }_{2}+{\alpha }_{2}{\gamma }_{1}}{\rho }+{\int }_{0}^{+\infty }f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)\text{d}s\right)\\ \le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}\left(\frac{{\alpha }_{1}{\gamma }_{2}+{\alpha }_{2}{\gamma }_{1}}{\rho }+{\int }_{0}^{+\infty }\varphi \left(s\right)q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\text{d}s\right)\\ \le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}\left(\frac{{\alpha }_{1}{\gamma }_{2}+{\alpha }_{2}{\gamma }_{1}}{\rho }+{S}_{{\gamma }^{*}}{\int }_{0}^{+\infty }\varphi \left(s\right)\text{d}s\right)<+\infty .\end{array}$ (3.3)

$\begin{array}{l}\frac{\left(Tx\right)\left(t\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\\ =\frac{{\gamma }_{1}B\left(t\right)}{\rho {\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}+\frac{{\gamma }_{2}a\left(t\right)}{\rho {\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}+{\int }_{0}^{+\infty }G\left(t,s\right)\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s\\ \ge {c}^{*}\frac{{\gamma }_{1}b\left(u\right)+{\gamma }_{2}a\left(u\right)}{\rho {\rho }^{-1}\left[1+a\left(u\right)b\left(u\right)\right]}+{\int }_{0}^{+\infty }{c}^{*}\frac{G\left(u,s\right)}{{\rho }^{-1}\left[1+a\left(u\right)b\left(u\right)\right]}\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s\\ ={c}^{*}\left(\frac{{\gamma }_{1}b\left(u\right)+{\gamma }_{2}a\left(u\right)}{\rho {\rho }^{-1}\left[1+a\left(u\right)b\left(u\right)\right]}+{\int }_{0}^{+\infty }\frac{G\left(u,s\right)}{{\rho }^{-1}\left[1+a\left(u\right)b\left(u\right)\right]}\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s\right)\\ =\frac{\left(Tx\right)\left(u\right)}{{\rho }^{-1}\left[1+a\left(u\right)b\left(u\right)\right]},\forall u\in \left[0,+\infty \right).\end{array}$ (3.4)

$\left({T}_{m}x\right)\left(t\right)=\frac{{\gamma }_{1}b\left(t\right)}{\rho }+\frac{{\gamma }_{2}a\left(t\right)}{\rho }+{\int }_{\frac{1}{m}}^{\infty }G\left(t,s\right)\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s,t\in \left[0,+\infty \right).$ (3.5)

$\begin{array}{l}\frac{|\left({T}_{m}{x}_{n}\right)\left(t\right)-\left({T}_{m}x\right)\left(t\right)|}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\text{=}|{\int }_{\frac{\text{1}}{m}}^{\infty }\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\left(f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)-{k}^{2}{x}_{n}\left(s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{\frac{1}{m}}^{\infty }\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s|\\ \le {\int }_{\frac{1}{m}}^{\infty }\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}|f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)+f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)|\text{d}s\\ ={\int }_{\frac{1}{m}}^{\infty }\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}|g\left(s,{y}_{1n}\left(s\right),{y}_{2n}\left(s\right)\right)+g\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)|\text{d}s\end{array}$

$\begin{array}{l}\le {\int }_{\frac{1}{m}}^{\infty }\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\varphi \left(s\right)\left(q\left(s,{y}_{1n}\left(s\right),{y}_{2n}\left(s\right)\right)+q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\right)\text{d}s\\ \le {\int }_{\frac{1}{m}}^{\infty }\varphi \left(s\right)\left(q\left(s,{y}_{1n}\left(s\right),{y}_{2n}\left(s\right)\right)+q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\right)\text{d}s\le 2{S}_{{\gamma }^{*}}{\int }_{0}^{\infty }\varphi \left(s\right)\text{d}s<+\infty ,\end{array}$

$\begin{array}{l}\le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}\left({\int }_{\frac{1}{m}}^{t}\frac{{\alpha }_{2}a\left(s\right)}{\rho }|f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)+f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)|\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{t}^{\infty }\frac{{\alpha }_{1}b\left(s\right)}{\rho }|f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)+f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)|\text{d}s\right)\\ =\underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}\left({\int }_{\frac{1}{m}}^{t}\frac{{\alpha }_{2}a\left(s\right)}{\rho }|g\left(s,{y}_{1n}\left(s\right),{y}_{2n}\left(s\right)\right)+g\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)|\text{d}s\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{t}^{\infty }\frac{{\alpha }_{1}b\left(s\right)}{\rho }|g\left(s,{y}_{1n}\left(s\right),{y}_{2n}\left(s\right)\right)+g\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)|\text{d}s\right)\\ \le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}{\int }_{\frac{1}{m}}^{t}2\left(g\left(s,{y}_{1n}\left(s\right),{y}_{2n}\left(s\right)\right)+g\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\right)\text{d}s\\ \le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}{\int }_{\frac{1}{m}}^{t}2\varphi \left(s\right)\left(q\left(s,{y}_{1n}\left(s\right),{y}_{2n}\left(s\right)\right)+q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\right)\text{d}s\\ \le 4{S}_{{\gamma }^{*}}\underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}{\int }_{0}^{\infty }\varphi \left(s\right)\text{d}s<+\infty .\end{array}$

$2\lambda {S}_{{\gamma }^{*}}{\int }_{{A}_{0}}^{\infty }\varphi \left(s\right)\text{d}s<\frac{\epsilon }{3},$ (3.6)

$4{S}_{{\gamma }^{*}}\underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}{\int }_{{{A}^{\prime }}_{0}}^{\infty }\varphi \left(s\right)\text{d}s<\frac{\epsilon }{3}.$ (3.7)

$‖{x}_{n}-x‖\to 0,n\to +\infty$ ，存在一个充分大的自然数 ${N}_{0}$ 使得 $n>{N}_{0}$ 时，对任意的 $s\in \left[0,+\infty \right)$ ，有

$\frac{|{x}_{n}\left(s\right)-x\left(s\right)|}{{\rho }^{-1}\left[1+a\left(s\right)b\left(s\right)\right]}\le ‖{x}_{n}-x‖<\frac{\epsilon }{3}{\left({k}^{2}{\int }_{\frac{1}{m}}^{{A}_{0}}\frac{G\left(s,s\right)\left(1+a\left(s\right)b\left(s\right)\right)}{1+{\beta }_{1}{\beta }_{2}}\text{d}s\right)}^{-1}.$ (3.8)

$g\left(t,{v}_{1},{v}_{2}\right)$$\left[1/m,{A}_{0}\right]*\left[0,{\gamma }^{*}\right]*\left[0,{\gamma }^{*}\right]$ 上的连续性，对上述的 $\epsilon >0$ ，存在 $\delta >0$ ，对任意的 $s\in \left[1/m,{A}_{0}\right]$${v}_{1},{v}_{2},{{v}^{\prime }}_{1},{{v}^{\prime }}_{2}\in \left[0,{\gamma }^{*}\right]*\left[0,{\gamma }^{*}\right]$ ，当 $|{v}_{1}-{{v}^{\prime }}_{1}|<\delta ,|{v}_{2}-{{v}^{\prime }}_{2}|<\delta$ 时，有

$|g\left(s,{v}_{1},{v}_{2}\right)-g\left(s,{{v}^{\prime }}_{1},{{v}^{\prime }}_{2}\right)|<\frac{\epsilon }{3}{\left({A}_{0}-1/m\right)}^{-1}.$ (3.9)

$‖{x}_{n}-x‖\to 0,n\to +\infty$ ，存在一个充分大的自然数 ${N}_{1}>{N}_{0}$ ，使得 $n>{N}_{1}$ 时，对任意的 $s\in \left[1/m,{A}_{0}\right]$ ，有

$\frac{|{x}_{n}\left(s\right)-x\left(s\right)|}{{\rho }^{-1}\left[1+a\left(s\right)b\left(s\right)\right]}\le ‖{x}_{n}-x‖<\delta ,\text{\hspace{0.17em}}|{{x}^{\prime }}_{n}\left(s\right)-{x}^{\prime }\left(s\right)|\le ‖{x}_{n}-x‖<\delta .$

$|f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)-f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)|=|g\left(s,{y}_{1n}\left(s\right),{y}_{2n}\left(s\right)\right)-g\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)|<\frac{\epsilon }{3}{\left({A}_{0}-1/m\right)}^{-1}.$ (3.10)

$\begin{array}{l}\frac{||\left({T}_{m}{x}_{n}\right)\left(t\right)-\left({T}_{m}x\right)\left(t\right)||}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\text{=}|{\int }_{\frac{\text{1}}{m}}^{\infty }\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\left(f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)-{k}^{2}{x}_{n}\left(s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{\frac{1}{m}}^{\infty }\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s|\\ \le {\int }_{\frac{1}{m}}^{{A}_{0}}\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\left(|f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)-f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)|+{k}^{2}|{x}_{n}\left(s\right)-x\left(s\right)|\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{{A}_{0}}^{\infty }\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\left(f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)+f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)\right)\text{d}s\end{array}$

$\begin{array}{l}\le {\int }_{\frac{1}{m}}^{{A}_{0}}|f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)-f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{\frac{1}{m}}^{{A}_{0}}\frac{{k}^{2}G\left(s,s\right)\left(1+a\left(s\right)b\left(s\right)\right)}{\left[1+{\beta }_{1}{\beta }_{2}\right]}\frac{|{x}_{n}\left(s\right)-x\left(s\right)|}{{\rho }^{-1}\left[1+a\left(s\right)b\left(s\right)\right]}\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{{A}_{0}}^{\infty }\left(f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)+f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)\right)\text{d}s\\ \le \frac{2\epsilon }{3}+{\int }_{{A}_{0}}^{\infty }\varphi \left(s\right)\left(q\left(s,{y}_{1n}\left(s\right),{y}_{2n}\left(s\right)\right)+q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\right)\text{d}s\\ \le \frac{2\epsilon }{3}+2{S}_{{\gamma }^{*}}{\int }_{{A}_{0}}^{\infty }\varphi \left(s\right)\text{d}s<\epsilon .\end{array}$ (3.11)

$\begin{array}{l}\frac{\left({T}_{m}x\right)\left(t\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}={\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]\left(\frac{{\gamma }_{1}b\left(t\right)}{\rho }+\frac{{\gamma }_{2}a\left(t\right)}{\rho }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]{\int }_{\frac{1}{m}}^{\infty }G\left(t,s\right)\left(f\left(s,{x}_{n}\left(s\right),{{x}^{\prime }}_{n}\left(s\right)\right)-{k}^{2}{x}_{n}\left(s\right)\right)\text{d}s\\ \le \frac{{\gamma }_{1}b\left(0\right)+{\gamma }_{2}a\left(\infty \right)}{1+{\beta }_{1}{\beta }_{2}}+{\int }_{\frac{1}{m}}^{\infty }\varphi \left(s\right)q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\text{d}s\le \frac{{\gamma }_{1}b\left(0\right)+{\gamma }_{2}a\left(\infty \right)}{1+{\beta }_{1}{\beta }_{2}}+{S}_{{\gamma }^{0}}{\int }_{0}^{\infty }\varphi \left(s\right)\text{d}s<+\infty ,\end{array}$

${T}_{m}M$$\left[0,+\infty \right)$ 上等度连续。由于

${y}_{1}\left(t\right)=\frac{x\left(t\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\le 1$${y}_{2}\left(t\right)={x}^{\prime }\left(t\right)\le |{x}^{\prime }\left(t\right)|\le 1$$t\in \left[0,+\infty \right)$

$\begin{array}{c}\frac{|\left(Tx\right)\left(t\right)-\left({T}_{m}x\right)\left(t\right)|}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}={\int }_{0}^{\frac{1}{m}}\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s\\ \le {\int }_{0}^{\frac{1}{m}}\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}g\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\text{d}s\\ \le {\int }_{0}^{\frac{1}{m}}\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\varphi \left(s\right)q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\text{d}s\\ \le {S}_{1}{\int }_{0}^{\frac{1}{m}}\varphi \left(s\right)\text{d}s\to 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}m\to +\infty .\end{array}$

$\begin{array}{c}|{\left(Tx\right)}^{\prime }\left(t\right)-{\left({T}_{m}x\right)}^{\prime }\left(t\right)|=|-\frac{{\alpha }_{2}}{\rho p\left(t\right)}{\int }_{0}^{\frac{1}{m}}a\left(s\right)\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s|\\ \le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}{\int }_{0}^{\frac{1}{m}}g\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\text{d}s\\ \le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}{\int }_{0}^{\frac{1}{m}}\varphi \left(s\right)q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\text{d}s\\ \le {S}_{1}\underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}{\int }_{0}^{\frac{1}{m}}\varphi \left(s\right)\text{d}s\to 0,m\to +\infty .\end{array}$

(H3) $0\le \underset{{v}_{1,},{v}_{2}\to {0}^{+}}{\mathrm{lim}}\underset{t\in \left[0,+\infty \right)}{\mathrm{min}}\frac{q\left(t,{v}_{1},{v}_{2}\right)}{\mathrm{max}\left\{{v}_{1},|{v}_{2}|\right\}}

$L=\mathrm{max}\left\{\left(1-\frac{{\gamma }_{1}b\left(0\right)+{\gamma }_{2}a\left(0\right)}{{r}_{1}\left(1+{\beta }_{1}{\beta }_{2}\right)}\right){\left({\int }_{0}^{\infty }\varphi \left(s\right)\text{d}s\right)}^{-1},\text{\hspace{0.17em}}\left(\frac{1}{{\mathrm{sup}}_{t\in \left[0,+\infty \right)}\frac{1}{p\left(t\right)}}-\frac{{\alpha }_{1}{\gamma }_{2}+{\alpha }_{2}{\gamma }_{1}}{\rho {r}_{1}}\right){\left({\int }_{0}^{\infty }\varphi \left(s\right)\text{d}s\right)}^{-1}\right\},$

$l=\mathrm{max}\left\{{k}^{2}+1,\text{\hspace{0.17em}}\left({{\left({\int }_{{a}^{*}}^{{b}^{*}}G\left(t,s\right)\text{d}s\right)}^{-1}+{k}^{2}\right)}^{-1}\frac{1+a\left(\infty \right)b\left(0\right)}{\rho }\right\},\text{\hspace{0.17em}}{r}_{1}>0.$

$q\left(t,{v}_{1},{v}_{2}\right)\le \left(L-{\epsilon }_{0}\right)\mathrm{max}\left\{{v}_{1},|{v}_{2}|\right\},\text{\hspace{0.17em}}0\le r,\text{\hspace{0.17em}}t\in \left[0,+\infty \right).$ (3.12)

$r>{r}_{1}$ 时，对任意的 $0\le {v}_{1},|{v}_{2}|\le {r}_{1},\text{\hspace{0.17em}}t\in \left[0,+\infty \right)$ ，(3.12)同样成立。记 ${K}_{{r}_{1}}=\left\{x\in K:‖x‖<{r}_{1}\right\}$ ，对任意的 $x\in \partial {K}_{{r}_{1}}$ ，有 ${y}_{1}\left(t\right)=\frac{x\left(t\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\le {r}_{1}$${y}_{2}\left(t\right)=|{x}^{\prime }\left(t\right)|\le {r}_{1}$ ，所以

$q\left(t,{y}_{1}\left(t\right),{y}_{2}\left(t\right)\right)\le \left(L+\epsilon \right)\left\{{y}_{1}\left(t\right),|{y}_{2}\left(t\right)|\right\}.$

$\begin{array}{l}\frac{|\left(Tx\right)\left(t\right)|}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}=\frac{{\gamma }_{1}b\left(t\right)}{\rho {\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}+\frac{{\gamma }_{2}a\left(t\right)}{\rho {\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{\infty }G\left(t,s\right)\left(f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)-{k}^{2}x\left(s\right)\right)\text{d}s\\ \le \frac{{\gamma }_{1}b\left(t\right)+{\gamma }_{2}a\left(t\right)}{\left[1+a\left(t\right)b\left(t\right)\right]}+{\int }_{0}^{\infty }\frac{G\left(t,s\right)}{{\rho }^{-1}\left[1+a\left(t\right)b\left(t\right)\right]}g\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\text{d}s\\ \le \frac{{\gamma }_{1}b\left(0\right)+{\gamma }_{2}a\left(\infty \right)}{1+{\beta }_{1}{\beta }_{2}}+{\int }_{0}^{\infty }\varphi \left(s\right)q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\text{d}s\\ \le \frac{{\gamma }_{1}b\left(0\right)+{\gamma }_{2}a\left(\infty \right)}{1+{\beta }_{1}{\beta }_{2}}+\left(L-{\epsilon }_{0}\right)\mathrm{max}\left\{{y}_{1}\left(s\right),|{y}_{2}\left(s\right)|\right\}{\int }_{0}^{\infty }\varphi \left(s\right)\text{d}s\le ‖x‖={r}_{1}.\end{array}$

$\begin{array}{l}\le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}\left(\frac{{\alpha }_{1}{\gamma }_{2}+{\alpha }_{2}{\gamma }_{1}}{\rho }+{\int }_{0}^{\infty }f\left(s,x\left(s\right),{x}^{\prime }\left(s\right)\right)\text{d}s\right)\\ \le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}\left(\frac{{\alpha }_{1}{\gamma }_{2}+{\alpha }_{2}{\gamma }_{1}}{\rho }+{\int }_{0}^{\infty }\varphi \left(s\right)q\left(s,{y}_{1}\left(s\right),{y}_{2}\left(s\right)\right)\text{d}s\right)\\ \le \underset{t\in \left[0,+\infty \right)}{\mathrm{sup}}\frac{1}{p\left(t\right)}\left(\frac{{\alpha }_{1}{\gamma }_{2}+{\alpha }_{2}{\gamma }_{1}}{\rho }+\left(L-{\epsilon }_{0}\right)\mathrm{max}\left\{{y}_{1}\left(s\right),|{y}_{2}\left(s\right)|\right\}{\int }_{0}^{\infty }\varphi \left(s\right)\text{d}s\right)\le ‖x‖={r}_{1}.\end{array}$

$g\left(t,{v}_{1},{v}_{2}\right)\ge \left(l+{\epsilon }_{0}\right)\left({v}_{1}+|{v}_{2}|\right),\text{\hspace{0.17em}}{v}_{1},|{v}_{2}|\ge {r}_{0},\text{\hspace{0.17em}}t\in \left[{a}^{*},{b}^{*}\right].$ (3.13)

${r}_{2}={r}_{1}+{c}^{*}{r}_{1}>{r}_{1}$${K}_{r2}=\left\{x\in K,‖x‖<{r}_{2}\right\}$${x}_{0}=1\in \partial {K}_{1}$ 下面证明

$x\ne Tx+\mu {x}_{0},\text{\hspace{0.17em}}\forall x\in \partial {K}_{r2},\text{\hspace{0.17em}}\forall \mu >0.$ (3.14)

$\begin{array}{c}{y}_{1*}\left(t\right)+{y}_{2*}=\frac{{x}_{*}\left(t\right)}{{\rho }^{-1}\left(1+a\left(t\right)b\left(t\right)\right)}+|{{x}^{\prime }}_{*}\left(t\right)|\ge {c}^{*}\frac{{x}_{*}\left(u\right)}{{\rho }^{-1}\left(1+a\left(u\right)b\left(u\right)\right)}+|{{x}^{\prime }}_{*}\left(t\right)|\\ \ge {c}^{*}\left(‖{x}_{0}‖+‖{x}^{\prime }‖\right)>{c}^{\text{*}}{r}_{2}>{r}_{0},\text{ }t\in \left[{a}^{*},{b}^{*}\right],\text{\hspace{0.17em}}u\in \left[0,+\infty \right)\end{array}$

$g\left(t,{y}_{1*}\left(t\right),{y}_{2*}\left(t\right)\right)\ge \left(l+{\epsilon }_{0}\right)\left({y}_{1*}\left(t\right)+|{y}_{2*}\left(t\right)|\right).$ (3.15)

$\begin{array}{c}{x}_{*}\left(t\right)=\frac{{\gamma }_{1}b\left(t\right)}{\rho }+\frac{{\gamma }_{2}a\left(t\right)}{\rho }+{\int }_{0}^{\infty }G\left(t,s\right)\left(f\left(s,{x}_{*}\left(s\right),{{x}^{\prime }}_{*}\left(s\right)\right)-{k}^{2}{x}_{*}\left(s\right)\right)\text{d}s\\ \ge {\int }_{{a}^{*}}^{{b}^{*}}G\left(t,s\right)\left(g\left(s,{y}_{1*}\left(s\right),{y}_{2*}\left(s\right)\right)-{k}^{2}{x}_{*}\left(s\right)\right)\text{d}s+{\mu }_{2}\\ \ge {\int }_{{a}^{*}}^{{b}^{*}}G\left(t,s\right)\left(l+{\epsilon }_{0}\right)\left({y}_{1*}\left(s\right)+|{y}_{2*}\left(s\right)|-{k}^{2}{x}_{*}\left(s\right)\right)\text{d}s+{\mu }_{2}\\ ={\int }_{{a}^{*}}^{{b}^{*}}G\left(t,s\right)\left(l+{\epsilon }_{0}\right)\left(\frac{{x}_{*}\left(s\right)}{{\rho }^{-1}\left(1+a\left(s\right)b\left(s\right)\right)}+|{{x}^{\prime }}_{*}\left(s\right)|-{k}^{2}{x}_{*}\left(s\right)\right)\text{d}s+{\mu }_{2}\end{array}$

$\begin{array}{l}>\underset{s\in \left[{a}^{*},{b}^{*}\right]}{\mathrm{min}}{x}_{*}\left(s\right){\int }_{{a}^{*}}^{{b}^{*}}G\left(t,s\right)\left(\frac{\left(l+{\epsilon }_{0}\right)}{{\rho }^{-1}\left(1+a\left(s\right)b\left(s\right)\right)}-{k}^{2}\right)\text{d}s+{\mu }_{2}\\ >\underset{s\in \left[{a}^{*},{b}^{*}\right]}{\mathrm{min}}{x}_{*}\left(s\right){\int }_{{a}^{*}}^{{b}^{*}}G\left(t,s\right)\left(\frac{\left(l+{\epsilon }_{0}\right)}{{\rho }^{-1}\left(1+a\left(\infty \right)b\left(0\right)\right)}-{k}^{2}\right)\text{d}s+{\mu }_{2}\\ >\xi +{\mu }_{2}>\xi .\end{array}$

$\xi <\xi$ ，所以(3.14)成立。根据以上的讨论，引理3.1和2.3， $T$ 有不动点 $x$ 满足 $0<{r}_{1}<‖x‖\le {r}_{2}$ 。易知 $x$ 是BVP(1.1)的正解。

(H4) $0\le \underset{{v}_{1},|{v}_{2}|\to \infty }{\mathrm{lim}}\underset{t\in \left[0,+\infty \right)}{\mathrm{max}}\frac{q\left(t,{v}_{1},{v}_{2}\right)}{\mathrm{max}\left\{{v}_{1},|{v}_{2}|\right\}}

$L,\text{\hspace{0.17em}}l$ 的定义同定理3.1。则BVP(1.1)至少有一个正解。

4. 结论

[1] Liu, B.G., Liu, L.S. and Wu, Y.H. (2010) Unbounded Solutions for Three-Point Boundary Value Problems with Non-linear Boundary Conditions on . Nonlinear Analysis, 73, 2923-2932.
https://doi.org/10.1016/j.na.2010.06.052

[2] Smail, D., Quiza, S. and Yan, B.Q. (2012) Positive Solutions for Singular BVPs on the Positive Half-Line Arising from Epidemiology and Combustion Theory. Acta Mathematica Sci-entia, 32, 672-694.
https://doi.org/10.1016/S0252-9602(12)60048-4

[3] Liu, L.S., Wang, Z.G. and Wu, Y.H. (2009) Multiple Pos-itive Solutions of the Singular Boundary Value Problems for Secong-Order Differential Equations on the Haly-Line. Nonlinear Analysis, 71, 2564-2575.
https://doi.org/10.1016/j.na.2009.01.092

[4] Ma, R.Y. and Zhu, B. (2009) Existence of Positive Solutions for a Semipositone Boundary Value Problem on the Half-Line. Computers & Mathematics with Applications, 58, 1672-1686.
https://doi.org/10.1016/j.camwa.2009.07.005

[5] Zimbabwe, M. (2001) On Solutions of Boundary Value Problems on the Half-Line. Journal of Mathematical Analysis and Applications, 259, 127-136.
https://doi.org/10.1006/jmaa.2000.7399

[6] Hao, Z.C., Laing, J. and Xiamen, T.J. (2006) Positive Solutions of Operator Equations on Halt-Line. Journal of Mathematical Analysis and Applications, 314, 423-435.
https://doi.org/10.1016/j.jmaa.2005.04.004

[7] Wang, Y., Liu, L.S. and Wu, Y.H. (2008) Positive Solutions of Singular Boundary Value Problems on the Half-Line. Applied Mathematics and Computation, 197, 789-796.

[8] Liam, H.R. and Ge, W.G. (2006) Existence of Positive for Sturm-Liouville Boundary Value Problems on the Half-Line. Journal of Mathematical Analysis and Applications, 321, 781-792.
https://doi.org/10.1016/j.jmaa.2005.09.001

[9] Xing, M.H., Zhang, K.M. and Gao, H.L. (2009) Existence of Positive Solutions for General Storm-Liouville Boundary Value Problems. Acta Mathematica Scientia, 29A, 929-939.

[10] Yan, B.Q., Q’Regan, D. and Agartala, R.P. (2006) Unbounded Solutions for Singular Boundary Value Problems on the Semi-Infinite Interval: Upper and Lower Solutions and Multiplicity. Journal of Computational and Applied Mathematics, 1997, 365-386.

[11] Guo, D.J. and Lakshmikantham, V. (1998) Nonlinear Problems in Abstract Cones. Academic Press, New York.

[12] Amani, H. (1976) Fixed Point Equations and Nonlinear Eigenvalue Problems in Erdered Banach Space. SIAM Review, 18, 620-709.
https://doi.org/10.1137/1018114

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