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

doi:10.12677/AAM.2017.69140

Advances in Applied Mathematics
Vol.06 No.09(2017), Article ID:23136,12 pages
10.12677/AAM.2017.69140

Existence of Positive Solutions for Singular Boundary Value Problems on the Infinite Interval

Ke Wang, Ying Wang^*

●How to Cite this Article

School of Mathematics and Statistics, Linyi University, Linyi Shandong

Received: Nov. 30^th, 2017; accepted: Dec. 15^th, 2017; published: Dec. 22^nd, 2017

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.

Keywords:Positive Solutions, Cone, Fixed Points, Infinite Interval

无穷区间上奇异边值问题正解的存在性

王克，王颖^*

临沂大学数学与统计学院，山东临沂

收稿日期：2017年11月30日；录用日期：2017年12月15日；发布日期：2017年12月22日

摘要

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。

关键词 :正解，锥，不动点，无穷区间

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

本文，我们主要讨论无穷区间上奇异边值问题(BVP)正解的存在性：

${\begin{cases} {(p (t) x^{'} (t) )}^{'} - k^{2} x (t) + f (t, x (t), (x^{'} (t))) = 0, t \in (0, + \infty), \\ α_{1} x (0) - β_{1} \lim_{t \to 0^{+}} p (t) x^{'} (t) = γ_{1}, \\ α_{2} \lim_{t \to + \infty} x (t) - β_{2} \lim_{t \to + \infty} p (t) x^{'} (t) = γ_{2}, \end{cases}$ (1.1)

其中 $k \in (- \infty, + \infty)$ ， $f : (0, + \infty) \times (- \infty, + \infty) \to [0, + \infty)$ 是连续函数并且允许在 $t = 0$ 的点奇异； $p \in C [0, + \infty) \cap C^{1} (0, + \infty)$ 且 $p$ 在 $(0, + \infty)$ 上大于0， $\int_{0}^{+ \infty} \frac{1}{p (s)} d s < + \infty$ ； $α_{i}, β_{i}, γ_{i} (i = 1, 2)$ ，

$ρ = α_{2} β_{1} + α_{1} β_{2} + α_{1} α_{2} B (0, + \infty) > 0, B (t, s) = \int_{0}^{+ \infty} \frac{1}{p (v)} d v$

由于其强大的应用背景，无穷区间上的边值问题已经引起人们越来越多的兴趣 [1] [2] [3] [4] . Zima [5] 研究了没有奇异项的下述方程：

${\begin{cases} x^{″} (t) - k^{2} x (t) + f (t, x (t)) = 0, t \in (0, + \infty), \\ x (0) = 0, \lim_{t \to \infty} x (t) = 0, \end{cases}$

其中 $k > 0$ ， $f : [0, + \infty) \times [0, + \infty) \to [0, + \infty)$ 是非负连续函数且 $f (t, s) \leq a (t) + b (t) x$ ，

$(t, x) \in [0, + \infty) \times [0, + \infty)$ ， $a, b : [0, + \infty) \to [0, + \infty)$ 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：

$x^{″} (t) - k^{2} x (t) + m f (t, x (t)) = 0,$ (1.2)

其中 $f : [0, + \infty) \times [0, + \infty) \to [0, + \infty)$ 是连续函数且 $\sup {f (t, x) : (t, x) \in [0, + \infty) \times [0, + \infty)} < + \infty$ ， $m : (0, + \infty) \to [0, + \infty)$ 连续并且允许在 $t = 0$ 点奇异。文 [7] 中的作者研究的是方程

${(p (t) x^{'} (t) )}^{'} + λ f (t, x (t)) - k^{2} x (t) = 0,$

在BVP(1.1)中，当 $γ_{1} = γ_{2} = 0$ 时的情况，应用Krasnosel-skii不动点理论，文 [8] 获得了 $k = 0$ 时方程(1.2)正解的存在性。

受以上文章的启发，本文讨论奇异BVP(1.1)的正解。与文 [5] [6] [7] [8] 相比，一方面，本文研究的是无穷区间上具有一般边值条件的奇异微分方程；另一方面，本文中，我们使用逼近的方法，通过建立特殊的空间和特殊的锥，利用锥上不动点定理，来解决奇异性与无穷区间所带来的困难。

本文的主要目的是获得BVP(1.1)正解的存在性。一般来说，我们通过求解一个积分算子 $T$ 的不动点来解决，本文中，我们获得的是无穷区间 $[0, + \infty)$ 上BVP(1.1)正解的存在性，即 $t$ 的定义域从有限区间推广到无穷区间。其中的主要困难是证明 $T$ 是一个全连续算子，由于在无穷区间 $[0, + \infty)$ 上，Ascoli-Arzela失效，关于无穷区间上的算子紧性的判断准则(引理2.2)可以帮助我们解决这个问题。

2. 预备知识

记

$a (t) = β_{1} + α_{1} B (0, t), b (t) = β_{2} + α_{2} B (t, \infty)$

$a (\infty) = \lim_{t \to \infty} a (t) = β_{1} + α_{1} B (0, \infty) < + \infty, a (0) = \lim_{t \to \infty} a (t) = β_{1}$

$b (\infty) = \lim_{t \to \infty} b (t) = β_{2}, b (0) = \lim_{t \to \infty} b (t) = β_{2} + α_{1} B (0, \infty) < + \infty$

$x (t) = ρ^{- 1} [1 + a (t) b (t) y_{1} (t)], x^{'} (t) = y_{2} (t), f (t, u, u^{'}) = g (t, v_{1}, v_{1})$

引理2.1 [8] ：假设条件 $\int_{0}^{+ \infty} \frac{1}{p (s)} d s < + \infty$ 成立且 $ρ < 0$ ，则

${\begin{cases} {(p (t) u^{'} (t) )}^{'} + v (t) = 0, t \in (0, + \infty), \\ α_{1} u (0) - β_{1} \lim_{t \to 0^{+}} p (t) u^{'} (t) = 0, \\ α_{2} \lim_{t \to + \infty} u (t) - β_{2} \lim_{t \to + \infty} p (t) u^{'} (t) = 0, \end{cases}$ (2.1)

对任意的 $v \in L (0, \infty)$ ，BVP(2.1)有唯一解且此唯一解可以表示为

$u (t) = \int_{0}^{+ \infty} G (t, s) v (s) d s .$

其中

$G (t, s) = \frac{1}{ρ} {\begin{cases} (β_{1} + α_{1} B (0, s)) (β_{2} + α_{2} B (t, \infty)), 0 \leq s < t \leq + \infty, \\ (β_{1} + α_{1} B (0, t)) (β_{2} + α_{2} B (s, \infty)), 0 \leq t < s \leq + \infty, \end{cases}$ (2.2)

注2.1：由(2.2)定义的 $G (t, s)$ 有下列性质：

(1) $G (t, s)$ 在 $[0, + \infty) \times [0, + \infty)$ 上连续；

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

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

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

(5) $G (t, s) \leq G (s, s) \leq \frac{1}{ρ} (β_{1} + α_{1} B (0, s)) (β_{2} + α_{2} B (s, \infty)) < + \infty$ ；

(6) $\bar{G} (s) = \lim_{t \to + \infty} G (t, s) = \frac{1}{ρ} β_{2} (β_{1} + α_{1} B (0, s)) \leq G (s, s) < + \infty$ ；

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

$c^{*} \frac{| x (t) |}{ρ^{- 1} [1 + a (u) b (u)]} \leq \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} \leq 1$ ， $\frac{a (t) + b (t)}{[1 + a (t) b (t)]} \geq c^{*} \frac{a (u) + b (u)}{[1 + a (u) b (u)]}$ 。

本文，我们在空间

X

中研究BVP (1.1)

$X = {x \in C^{1} [0, + \infty) : \sup_{t \in [0, + \infty)} \frac{| x (t) |}{ρ^{- 1} [1 + a (t) b (t)]} < + \infty, \sup_{t \in [0, + \infty)} | x^{'} (t) | < + \infty}$ (2.3)

显然， $(X, ‖ \cdot ‖)$ 是Banach空间， $‖ x ‖ = \max {{‖ x ‖}_{0}, ‖ x^{'} ‖}$ ， $‖ x_{0} ‖ = \sup_{t \in [0, + \infty)} \frac{| x (t) |}{ρ^{- 1} [1 + a (t) b (t)]}$ ， $‖ x^{'} ‖ = \sup_{t \in [0, + \infty)} | x^{'} (t) |$ (见 [9] )令

$K = {x \in X : x (t) \geq 0, \min_{t \in [a^{*}, b^{*}]} \frac{| x (t) |}{ρ^{- 1} [1 + a (t) b (t)]} \geq c^{*} \frac{| x (u) |}{ρ^{- 1} [1 + a (u) b (u)]}, t, u \in [0, + \infty)}$

易知 $K$ 是 $X$ 中的一个锥。

类似于文 [9] [10] ，下面的引理2.2成立：

引理2.2：假设 $M \subseteq X$ ，若 $M$ 满足下列条件：

(1) $M$ 在 $X$ 中一致有界；

(2) 函数 ${y_{1}, y_{2} | y_{1} (t) = \frac{| x (t) |}{ρ^{- 1} [1 + a (t) b (t)]}, y_{2} (t) = x^{'} (t) : x, x^{'} \in M}$ 在 $[0, + \infty)$ 上局部等度连续；

(3) 函数 ${y_{1}, y_{2} | y_{1} (t) = \frac{| x (t) |}{ρ^{- 1} [1 + a (t) b (t)]}, y_{2} (t) = x^{'} (t) : x, x^{'} \in M}$ 在 $+ \infty$ 处一致收敛，则称 $M$ 在 $X$ 中是相对紧的。

引理2.3 [11] [12] ：假设 $P$ 是Banach空间 $E$ 中的正锥，定义 $P_{γ} = {x \in P : ‖ x ‖ < γ}$ ， ${\bar{P}}_{γ, R} = {x \in P : γ \leq ‖ x ‖ < R}$ ， $0 < γ < R < + \infty$ 令 $A : {\bar{P}}_{γ, R} \to P$ 是全连续算子，如果下列条件成立：

(1) $‖ A x ‖ \leq ‖ x ‖, \forall x \in \partial P$ ；

(2) 存在 $x_{0} \in \partial P_{1}$ ，使得 $x \neq A x + m x_{0}, \forall x \in \partial P_{γ}, m > 0$ ，则 $A$ 在 ${\bar{P}}_{γ, R}$ 存在不动点；

注2.2：假设条件(1)在 $x \in \partial P_{γ}$ 且条件(2)在 $x \in \partial P_{R}$ 上分别成立，则引理2.3的结论仍然成立。

3. 主要结果

(H₁) $f, g : (0, + \infty) \times [0, + \infty) \times (- \infty, + \infty) \to [0, + \infty)$ 是连续函数并且满足 $k^{2} u \leq f (t, u, u^{'}) = g (t, v_{1}, v_{2}) \leq ϕ (t) q (t, v_{1}, v_{2})$ ，其中 $ϕ : (0, + \infty) \to [0, + \infty)$ 连续且在 $t = 0$ 点奇异， $ϕ (t)$ 在 $[0, + \infty)$ 上不恒为0， $q : [0, + \infty) \times [0, + \infty) \times (- \infty, + \infty) \to [0, + \infty)$ 是连续函数，对任意的 $0 \leq t \leq + \infty$ ， $v_{1}, | v_{2} |$ 属于 $[0, + \infty)$ 上的有界集， $q (t, v_{1}, v_{2})$ 有界。

(H₂) $\int_{0}^{+ \infty} ϕ (s) d s < + \infty .$

根据以上假设，定义积分算子 $T : K \to X$ ：

$(T x) (t) = \frac{γ_{1} b (t)}{ρ} + \frac{γ_{2} a (t)}{ρ} + \int_{0}^{+ \infty} G (t, s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s, t \in [0, + \infty),$ (3.1)

显然BVP(1.1)有解 $x$ 当且仅当 $x \in K$ 是由(3.1)定义的算子 $T$ 的不动点。

引理3.1：假设条件(H₁) (H₂)成立。则 $T : K \to K$ 是全连续算子。

证明：首先，我们证明 $T : K \to X$ 是有定义的且 $T (K) \subseteq K$ 。对任意固定的 $x \in K$ ，存在 $γ^{*} > 0$ 有 $‖ x ‖ \leq γ^{*}$ ，即 $y_{1} (t) = \frac{| x (t) |}{ρ^{- 1} [1 + a (t) b (t)]} \leq γ^{*}, y_{2} (t) = x^{'} (t) \leq γ^{*}, t \in [0, + \infty)$ 。由条件(H₁)易知 $T_{x} \geq 0$ ， $S_{γ^{*}} : = \sup {q (t, v_{1}, v_{2}) : 0 \leq t \leq + \infty, 0 \leq v_{1}, | v_{2} | \leq γ^{*}} < + \infty$ 。因而，由条件(H₁) (H₂)，对任意的 $t \in [0, + \infty)$ ，有

$\begin{array}{l} \frac{(T x) (t)}{ρ^{- 1} [1 + a (t) b (t)]} = ρ^{- 1} [1 + a (t) b (t)] (\frac{γ_{1} b (t)}{ρ} + \frac{γ_{2} a (t)}{ρ}) \\ + ρ^{- 1} [1 + a (t) b (t)] \int_{0}^{+ \infty} G (t, s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s \\ \leq \frac{γ_{1} b (t) + γ_{2} a (t)}{[1 + a (t) b (t)]} + \int_{0}^{+ \infty} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s \\ \leq \frac{γ_{1} b (0) + γ_{2} a (\infty)}{[1 + β_{1} β_{2}]} + \int_{0}^{+ \infty} ϕ (s) q (s, y_{1} (s), y_{2} (s)) d s \\ \leq \frac{γ_{1} b (0) + γ_{2} a (\infty)}{[1 + β_{1} β_{2}]} + S_{γ^{*}} \int_{0}^{+ \infty} ϕ (s) d s < + \infty, \end{array}$ (3.2)

\begin{array}{l} | T {(x)}^{'} (t) | = | {(\frac{γ_{1} b (t)}{ρ} + \frac{γ_{2} a (t)}{ρ} + \int_{0}^{+ \infty} G (t, s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s)}^{'} | \\ = | \frac{α_{1} γ_{2}}{ρ p (t)} - \frac{α_{2} γ_{1}}{ρ p (t)} - \frac{α_{2}}{ρ p (t)} \int_{0}^{t} a (s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s + \frac{α_{1}}{ρ p (t)} \int_{t}^{+ \infty} b (s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s | \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ} + \int_{0}^{t} \frac{α_{2} a (s)}{ρ} (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s + \int_{t}^{+ \infty} \frac{α_{1} b (s)}{ρ} (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s) \end{array}

$\begin{array}{l} \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ} + \int_{0}^{+ \infty} f (s, x (s), x^{'} (s)) d s) \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ} + \int_{0}^{+ \infty} ϕ (s) q (s, y_{1} (s), y_{2} (s)) d s) \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ} + S_{γ^{*}} \int_{0}^{+ \infty} ϕ (s) d s) < + \infty . \end{array}$ (3.3)

由(3.2)和(3.3)可得 $T (x) (t) \in X$ 。由 $G (t, s)$ 的性质对任意的 $t \in [a^{*}, b^{*}]$ ，有

$\begin{array}{l} \frac{(T x) (t)}{ρ^{- 1} [1 + a (t) b (t)]} \\ = \frac{γ_{1} B (t)}{ρ ρ^{- 1} [1 + a (t) b (t)]} + \frac{γ_{2} a (t)}{ρ ρ^{- 1} [1 + a (t) b (t)]} + \int_{0}^{+ \infty} G (t, s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s \\ \geq c^{*} \frac{γ_{1} b (u) + γ_{2} a (u)}{ρ ρ^{- 1} [1 + a (u) b (u)]} + \int_{0}^{+ \infty} c^{*} \frac{G (u, s)}{ρ^{- 1} [1 + a (u) b (u)]} (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s \\ = c^{*} (\frac{γ_{1} b (u) + γ_{2} a (u)}{ρ ρ^{- 1} [1 + a (u) b (u)]} + \int_{0}^{+ \infty} \frac{G (u, s)}{ρ^{- 1} [1 + a (u) b (u)]} (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s) \\ = \frac{(T x) (u)}{ρ^{- 1} [1 + a (u) b (u)]}, \forall u \in [0, + \infty) . \end{array}$ (3.4)

由(3.4)可知， $\min_{t \in [a^{*}, b^{*}]} \frac{(T x) (t)}{ρ^{- 1} [1 + a (t) b (t)]} \geq c^{*} \frac{(T x) (u)}{ρ^{- 1} [1 + a (u) b (u)]}, \forall u \in [0, + \infty)$ 。因此， $T (K) \subseteq K$ 。

其次，对任意的自然数 $m$ ，定义算子 $T_{m} : K \to K$ ：

$(T_{m} x) (t) = \frac{γ_{1} b (t)}{ρ} + \frac{γ_{2} a (t)}{ρ} + \int_{\frac{1}{m}}^{\infty} G (t, s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s, t \in [0, + \infty) .$ (3.5)

下面，对 $m \geq 1$ ，我们将证明 $T_{m} : K \to K$ 是全连续的。首先证明 $T_{m} : K \to K$ 是连续的。假设 $x_{n}, x \in K$ 且 $‖ x_{n} - x ‖ \to 0$ 。由(3.5)和条件(H₂)可知

$\begin{array}{l} \frac{| (T_{m} x_{n}) (t) - (T_{m} x) (t) |}{ρ^{- 1} [1 + a (t) b (t)]} = | \int_{\frac{1}{m}}^{\infty} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} (f (s, x_{n} (s), {x^{'}}_{n} (s)) - k^{2} x_{n} (s)) d s \\ - \int_{\frac{1}{m}}^{\infty} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s | \\ \leq \int_{\frac{1}{m}}^{\infty} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} | f (s, x_{n} (s), {x^{'}}_{n} (s)) + f (s, x (s), x^{'} (s)) | d s \\ = \int_{\frac{1}{m}}^{\infty} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} | g (s, y_{1 n} (s), y_{2 n} (s)) + g (s, y_{1} (s), y_{2} (s)) | d s \end{array}$

$\begin{array}{l} \leq \int_{\frac{1}{m}}^{\infty} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} ϕ (s) (q (s, y_{1 n} (s), y_{2 n} (s)) + q (s, y_{1} (s), y_{2} (s))) d s \\ \leq \int_{\frac{1}{m}}^{\infty} ϕ (s) (q (s, y_{1 n} (s), y_{2 n} (s)) + q (s, y_{1} (s), y_{2} (s))) d s \leq 2 S_{γ^{*}} \int_{0}^{\infty} ϕ (s) d s < + \infty, \end{array}$

其中 $S_{γ^{*}} : = \sup {q (t, v_{1}, v_{2}) : 0 \leq t \leq + \infty, 0 \leq v_{1}, | v_{2} | \leq γ^{*}} < + \infty$ (由条件(H₁)可得)， $γ^{*}$ 是一个实数并且 $γ^{*} \geq \max_{n \in N} {‖ x ‖, ‖ x_{n} ‖}$ ， $N$ 表示自然数集。

\begin{array}{l} | {(T_{m} x_{n})}^{'} (t) - {(T_{m} x)}^{'} (t) | \\ = | - \frac{α_{2}}{ρ p (t)} \int_{\frac{1}{m}}^{t} a (s) (f (s, x_{n} (s), {x^{'}}_{n} (s)) - k^{2} x_{n} (s)) d s + \frac{α_{1}}{ρ p (t)} \int_{t}^{\infty} b (s) (f (s, x_{n} (s), {x^{'}}_{n} (s)) - k^{2} x_{n} (s)) d s \\ - (- \frac{α_{2}}{ρ p (t)} \int_{\frac{1}{m}}^{t} a (s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s + \frac{α_{1}}{ρ p (t)} \int_{t}^{\infty} b (s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s) | \\ \leq \frac{α_{2}}{ρ p (t)} \int_{\frac{1}{m}}^{t} a (s) | f (s, x_{n} (s), {x^{'}}_{n} (s)) + f (s, x (s), x^{'} (s)) | d s \\ + \frac{α_{1}}{ρ p (t)} \int_{t}^{\infty} b (s) | f (s, x_{n} (s), {x^{'}}_{n} (s)) + f (s, x (s), x^{'} (s)) | d s \end{array}

$\begin{array}{l} \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\int_{\frac{1}{m}}^{t} \frac{α_{2} a (s)}{ρ} | f (s, x_{n} (s), {x^{'}}_{n} (s)) + f (s, x (s), x^{'} (s)) | d s \\ + \int_{t}^{\infty} \frac{α_{1} b (s)}{ρ} | f (s, x_{n} (s), {x^{'}}_{n} (s)) + f (s, x (s), x^{'} (s)) | d s) \\ = \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\int_{\frac{1}{m}}^{t} \frac{α_{2} a (s)}{ρ} | g (s, y_{1 n} (s), y_{2 n} (s)) + g (s, y_{1} (s), y_{2} (s)) | d s \end{array}$

$\begin{array}{l} + \int_{t}^{\infty} \frac{α_{1} b (s)}{ρ} | g (s, y_{1 n} (s), y_{2 n} (s)) + g (s, y_{1} (s), y_{2} (s)) | d s) \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} \int_{\frac{1}{m}}^{t} 2 (g (s, y_{1 n} (s), y_{2 n} (s)) + g (s, y_{1} (s), y_{2} (s))) d s \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} \int_{\frac{1}{m}}^{t} 2 ϕ (s) (q (s, y_{1 n} (s), y_{2 n} (s)) + q (s, y_{1} (s), y_{2} (s))) d s \\ \leq 4 S_{γ^{*}} \sup_{t \in [0, + \infty)} \frac{1}{p (t)} \int_{0}^{\infty} ϕ (s) d s < + \infty . \end{array}$

对任意的 $ε > 0$ ，由条件(H₂)，存在充分大的 $A_{0}, {A^{'}}_{0} (A_{0}, {A^{'}}_{0} > \frac{1}{m})$ 满足

$2 λ S_{γ^{*}} \int_{A_{0}}^{\infty} ϕ (s) d s < \frac{ε}{3},$ (3.6)

$4 S_{γ^{*}} \sup_{t \in [0, + \infty)} \frac{1}{p (t)} \int_{{A^{'}}_{0}}^{\infty} ϕ (s) d s < \frac{ε}{3} .$ (3.7)

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

$\frac{| x_{n} (s) - x (s) |}{ρ^{- 1} [1 + a (s) b (s)]} \leq ‖ x_{n} - x ‖ < \frac{ε}{3} {(k^{2} \int_{\frac{1}{m}}^{A_{0}} \frac{G (s, s) (1 + a (s) b (s))}{1 + β_{1} β_{2}} d s)}^{- 1} .$ (3.8)

由 $g (t, v_{1}, v_{2})$ 在 $[1 / m, A_{0}] * [0, γ^{*}] * [0, γ^{*}]$ 上的连续性，对上述的 $ε > 0$ ，存在 $δ > 0$ ，对任意的 $s \in [1 / m, A_{0}]$ ， $v_{1}, v_{2}, {v^{'}}_{1}, {v^{'}}_{2} \in [0, γ^{*}] * [0, γ^{*}]$ ，当 $| v_{1} - {v^{'}}_{1} | < δ, | v_{2} - {v^{'}}_{2} | < δ$ 时，有

$| g (s, v_{1}, v_{2}) - g (s, {v^{'}}_{1}, {v^{'}}_{2}) | < \frac{ε}{3} {(A_{0} - 1 / m)}^{- 1} .$ (3.9)

由 $‖ x_{n} - x ‖ \to 0, n \to + \infty$ ，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 $s \in [1 / m, A_{0}]$ ，有

$\frac{| x_{n} (s) - x (s) |}{ρ^{- 1} [1 + a (s) b (s)]} \leq ‖ x_{n} - x ‖ < δ, | {x^{'}}_{n} (s) - x^{'} (s) | \leq ‖ x_{n} - x ‖ < δ .$

所以，由(3.9)，当 $n > N_{1}$ 时，对任意的 $s \in [1 / m, A_{0}]$ ，可得

$| f (s, x_{n} (s), {x^{'}}_{n} (s)) - f (s, x (s), x^{'} (s)) | = | g (s, y_{1 n} (s), y_{2 n} (s)) - g (s, y_{1} (s), y_{2} (s)) | < \frac{ε}{3} {(A_{0} - 1 / m)}^{- 1} .$ (3.10)

因此，由(3.6)，(3.8)~(3.10)，当 $n > N_{1}$ 时，对任意的 $t \in [0, + \infty)$ ，有

$\begin{array}{l} \frac{| | (T_{m} x_{n}) (t) - (T_{m} x) (t) | |}{ρ^{- 1} [1 + a (t) b (t)]} = | \int_{\frac{1}{m}}^{\infty} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} (f (s, x_{n} (s), {x^{'}}_{n} (s)) - k^{2} x_{n} (s)) d s \\ - \int_{\frac{1}{m}}^{\infty} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s | \\ \leq \int_{\frac{1}{m}}^{A_{0}} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} (| f (s, x_{n} (s), {x^{'}}_{n} (s)) - f (s, x (s), x^{'} (s)) | + k^{2} | x_{n} (s) - x (s) |) d s \\ + \int_{A_{0}}^{\infty} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} (f (s, x_{n} (s), {x^{'}}_{n} (s)) + f (s, x (s), x^{'} (s))) d s \end{array}$

$\begin{array}{l} \leq \int_{\frac{1}{m}}^{A_{0}} | f (s, x_{n} (s), {x^{'}}_{n} (s)) - f (s, x (s), x^{'} (s)) | \\ + \int_{\frac{1}{m}}^{A_{0}} \frac{k^{2} G (s, s) (1 + a (s) b (s))}{[1 + β_{1} β_{2}]} \frac{| x_{n} (s) - x (s) |}{ρ^{- 1} [1 + a (s) b (s)]} d s \\ + \int_{A_{0}}^{\infty} (f (s, x_{n} (s), {x^{'}}_{n} (s)) + f (s, x (s), x^{'} (s))) d s \\ \leq \frac{2 ε}{3} + \int_{A_{0}}^{\infty} ϕ (s) (q (s, y_{1 n} (s), y_{2 n} (s)) + q (s, y_{1} (s), y_{2} (s))) d s \\ \leq \frac{2 ε}{3} + 2 S_{γ^{*}} \int_{A_{0}}^{\infty} ϕ (s) d s < ε . \end{array}$ (3.11)

另一方面，类似于(3.11)的证明，结合(3.7)，可得 $| {(T_{m} x_{n})}^{'} (t) - {(T_{m} x)}^{'} (t) | < ε$ 。即对每一个自然数 $m$ ， $T_{m} : K \to K$ 是连续的。

下面证明，对每一个自然数 $m$ ， $T_{m} : K \to K$ 是紧的。假设 $M$ 是 $K$ 中的任意有界子集，则存在常数 $r^{0} > 0$ ，使得 $‖ x ‖ \leq r^{0},$ $x \in M$ 。由(3.5)，条件(H₁)和(H₂)，对任意的 $x \in M$ ，有

$\begin{array}{l} \frac{(T_{m} x) (t)}{ρ^{- 1} [1 + a (t) b (t)]} = ρ^{- 1} [1 + a (t) b (t)] (\frac{γ_{1} b (t)}{ρ} + \frac{γ_{2} a (t)}{ρ}) \\ + ρ^{- 1} [1 + a (t) b (t)] \int_{\frac{1}{m}}^{\infty} G (t, s) (f (s, x_{n} (s), {x^{'}}_{n} (s)) - k^{2} x_{n} (s)) d s \\ \leq \frac{γ_{1} b (0) + γ_{2} a (\infty)}{1 + β_{1} β_{2}} + \int_{\frac{1}{m}}^{\infty} ϕ (s) q (s, y_{1} (s), y_{2} (s)) d s \leq \frac{γ_{1} b (0) + γ_{2} a (\infty)}{1 + β_{1} β_{2}} + S_{γ^{0}} \int_{0}^{\infty} ϕ (s) d s < + \infty, \end{array}$

\begin{array}{l} | {(T_{m} x)}^{'} (t) | = | {(\frac{γ_{1} b (t)}{ρ} + \frac{γ_{2} a (t)}{ρ} + \int_{\frac{1}{m}}^{\infty} G (t, s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s)}^{'} | \\ = | \frac{α_{1} γ_{2}}{ρ p (t)} - \frac{α_{2} γ_{1}}{ρ p (t)} - \frac{α_{2}}{ρ p (t)} \int_{\frac{1}{m}}^{t} a (s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s + \frac{α_{1}}{ρ p (t)} \int_{t}^{\infty} b (s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s | \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ} + \int_{\frac{1}{m}}^{t} ϕ (s) q (s, y_{1} (s), y_{2} (s)) d s) \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ} + S_{γ^{*}} \int_{0}^{\infty} ϕ (s) d s) < + \infty, \end{array}

其中 $S_{γ^{0}} : = \sup {q (t, v_{1}, v_{2}) : 0 \leq t \leq + \infty, 0 \leq v_{1}, | v_{2} | \leq γ^{0}}$ 。因此， $T_{m} M$ 一致有界。对任意的 $t, t^{'} \in [0, + \infty)$ ，可得

\begin{array}{l} | \frac{(T_{m} x) (t)}{ρ^{- 1} [1 + a (t) b (t)]} - \frac{(T_{m} x) (t^{'})}{ρ^{- 1} [1 + a (t) b (t)]} | \\ = | \frac{γ_{1} b (t) + γ_{2} a (t)}{1 + a (t) b (t)} - \frac{γ_{1} b (t^{'}) + γ_{2} a (t^{'})}{1 + a (t^{'}) b (t^{'})} + \int_{\frac{1}{m}}^{\infty} (\frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} - \frac{G (t^{'}, s)}{ρ^{- 1} [1 + a (t^{'}) b (t^{'})]}) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s | \\ \leq \frac{γ_{1} | b (t) - b (t^{'}) | + γ_{2} | a (t) - a (t^{'}) |}{1 + β_{1} β_{2}} + \int_{0}^{\infty} \frac{| G (t, s) - G (t^{'}, s) |}{ρ^{- 1} (1 + β_{1} β_{2})} ϕ (s) (q (s, y_{1 n} (s), y_{2 n} (s)) + q (s, y_{1} (s), y_{2} (s))) d s \\ \leq \frac{γ_{1} | b (t) - b (t^{'}) | + γ_{2} | a (t) - a (t^{'}) |}{1 + β_{1} β_{2}} + \frac{2 S_{γ^{0}}}{ρ^{- 1} (1 + β_{1} β_{2})} \int_{0}^{\infty} | G (t, s) - G (t^{'}, s) | ϕ (s) d s \to 0, t \to t^{'}, \end{array}

\begin{array}{l} | (T_{m} x) (t) - {(T_{m} x)}^{'} (t) | \\ \leq | \frac{α_{1} γ_{2} - α_{2} γ_{1}}{ρ} | \frac{| p (t) - p (t^{'}) |}{p (t) p (t^{'})} + \sup_{t, t^{'} \in [0, + \infty)} {\frac{1}{p (t)}, \frac{1}{p (t^{'})}} (\int_{t}^{t^{'}} \frac{α_{1} b (s) + α_{2} a (s)}{ρ} (f (s, x_{n} (s), {x^{'}}_{n} (s)) - k^{2} x_{n} (s)) d s) \\ \leq | \frac{α_{1} γ_{2} - α_{2} γ_{1}}{ρ} | \frac{| | p (t) - p (t^{'}) | |}{p (t) p (t^{'})} + \sup_{t, t^{'} \in [0, + \infty)} {\frac{1}{p (t)}, \frac{1}{p (t^{'})}} (S_{γ^{0}} \int_{t}^{t^{'}} \frac{α_{1} b (s) + α_{2} a (s)}{ρ} ϕ (s) d s) \to 0, t \to t^{'} . \end{array}

即 $T_{m} M$ 在 $[0, + \infty)$ 上等度连续。由于

\begin{array}{l} | \frac{(T_{m} x) (t)}{ρ^{- 1} [1 + a (t) b (t)]} - \frac{(T_{m} x) (\infty)}{ρ^{- 1} [1 + a (\infty) b (\infty)]} | \\ \leq \frac{γ_{1} | b (t) - b (\infty) | + γ_{2} | a (t) - a (\infty) |}{1 + β_{1} β_{2}} + \int_{0}^{\infty} \frac{| G (t, s) - \bar{G} (s) |}{ρ^{- 1} (1 + β_{1} β_{2})} ϕ (s) (q (s, y_{1 n} (s), y_{2 n} (s)) + q (s, y_{1} (s), y_{2} (s))) d s \\ \leq \frac{γ_{1} | b (t) - b (\infty) | + γ_{2} | a (t) - a (\infty) |}{1 + β_{1} β_{2}} + \frac{2 S_{r^{0}}}{ρ^{- 1} (1 + β_{1} β_{2})} \int_{0}^{\infty} | G (t, s) - \bar{G} (s) | ϕ (s) d s \to 0, t \to \infty . \end{array}

并且可以证明 $| {(T_{m} x)}^{'} (t) - {(T_{m} x)}^{'} (\infty) | \to 0, t \to \infty$ ，所以 $T_{m} M$ 一致收敛。由引理2.2，对每一个自然数 $m$ ， $T_{m} : K \to K$ 是全连续算子。

最后证明， $T : K \to K$ 是全连续算子。对任意的 $t \in [0, + \infty), x \in K, ‖ x ‖ \leq 1$ ，即

$y_{1} (t) = \frac{x (t)}{ρ^{- 1} [1 + a (t) b (t)]} \leq 1$ ， $y_{2} (t) = x^{'} (t) \leq | x^{'} (t) | \leq 1$ ， $t \in [0, + \infty)$ 。

$\begin{matrix} \frac{| (T x) (t) - (T_{m} x) (t) |}{ρ^{- 1} [1 + a (t) b (t)]} = \int_{0}^{\frac{1}{m}} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s \\ \leq \int_{0}^{\frac{1}{m}} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} g (s, y_{1} (s), y_{2} (s)) d s \\ \leq \int_{0}^{\frac{1}{m}} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} ϕ (s) q (s, y_{1} (s), y_{2} (s)) d s \\ \leq S_{1} \int_{0}^{\frac{1}{m}} ϕ (s) d s \to 0, m \to + \infty . \end{matrix}$

其中 $S_{1} : = \sup {q (t, v_{1}, v_{2}) : 0 \leq v_{1}, | v_{2} | \leq 1} < + \infty .$

$\begin{matrix} | {(T x)}^{'} (t) - {(T_{m} x)}^{'} (t) | = | - \frac{α_{2}}{ρ p (t)} \int_{0}^{\frac{1}{m}} a (s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s | \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} \int_{0}^{\frac{1}{m}} g (s, y_{1} (s), y_{2} (s)) d s \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} \int_{0}^{\frac{1}{m}} ϕ (s) q (s, y_{1} (s), y_{2} (s)) d s \\ \leq S_{1} \sup_{t \in [0, + \infty)} \frac{1}{p (t)} \int_{0}^{\frac{1}{m}} ϕ (s) d s \to 0, m \to + \infty . \end{matrix}$

由此可知 $‖ T x - T_{m} ‖ \to 0, m \to + \infty,$ 即 $T : K \to K$ 是全连续算子。

定理3.1：假设条件(H₁) (H₂)成立，并且 $q, g$ 满足下列条件

(H₃) $0 \leq \lim_{v_{1,}, v_{2} \to 0^{+}} \min_{t \in [0, + \infty)} \frac{q (t, v_{1}, v_{2})}{\max {v_{1}, | v_{2} |}} < L, k^{2} < l < \lim_{v_{1} + | v_{2} | \to + \infty} \min_{t \in [a^{*}, b^{*}]} \frac{g (t, v_{1}, v_{2})}{v_{1} + | v_{2} |} \leq + \infty,$

$L = \max {(1 - \frac{γ_{1} b (0) + γ_{2} a (0)}{r_{1} (1 + β_{1} β_{2})}) {(\int_{0}^{\infty} ϕ (s) d s)}^{- 1}, (\frac{1}{\sup_{t \in [0, + \infty)} \frac{1}{p (t)}} - \frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ r_{1}}) {(\int_{0}^{\infty} ϕ (s) d s)}^{- 1}},$

$l = \max {k^{2} + 1, ({{(\int_{a^{*}}^{b^{*}} G (t, s) d s)}^{- 1} + k^{2})}^{- 1} \frac{1 + a (\infty) b (0)}{ρ}}, r_{1} > 0.$

则BVP(1.1)至少有一个解。

证明：由条件(H₃)的第一个不等式，存在 $ε_{0} > 0, r > 0$ 满足

$q (t, v_{1}, v_{2}) \leq (L - ε_{0}) \max {v_{1}, | v_{2} |}, 0 \leq r, t \in [0, + \infty) .$ (3.12)

$r > r_{1}$ 时，对任意的 $0 \leq v_{1}, | v_{2} | \leq r_{1}, t \in [0, + \infty)$ ，(3.12)同样成立。记 $K_{r_{1}} = {x \in K : ‖ x ‖ < r_{1}}$ ，对任意的 $x \in \partial K_{r_{1}}$ ，有 $y_{1} (t) = \frac{x (t)}{ρ^{- 1} [1 + a (t) b (t)]} \leq r_{1}$ ， $y_{2} (t) = | x^{'} (t) | \leq r_{1}$ ，所以

$q (t, y_{1} (t), y_{2} (t)) \leq (L + ε) {y_{1} (t), | y_{2} (t) |} .$

对任意的 $t \in [0, + \infty)$ ，

$\begin{array}{l} \frac{| (T x) (t) |}{ρ^{- 1} [1 + a (t) b (t)]} = \frac{γ_{1} b (t)}{ρ ρ^{- 1} [1 + a (t) b (t)]} + \frac{γ_{2} a (t)}{ρ ρ^{- 1} [1 + a (t) b (t)]} \\ + \int_{0}^{\infty} G (t, s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s \\ \leq \frac{γ_{1} b (t) + γ_{2} a (t)}{[1 + a (t) b (t)]} + \int_{0}^{\infty} \frac{G (t, s)}{ρ^{- 1} [1 + a (t) b (t)]} g (s, y_{1} (s), y_{2} (s)) d s \\ \leq \frac{γ_{1} b (0) + γ_{2} a (\infty)}{1 + β_{1} β_{2}} + \int_{0}^{\infty} ϕ (s) q (s, y_{1} (s), y_{2} (s)) d s \\ \leq \frac{γ_{1} b (0) + γ_{2} a (\infty)}{1 + β_{1} β_{2}} + (L - ε_{0}) \max {y_{1} (s), | y_{2} (s) |} \int_{0}^{\infty} ϕ (s) d s \leq ‖ x ‖ = r_{1} . \end{array}$

\begin{array}{l} | {(T x)}^{'} (t) | \\ = | {(\frac{γ_{1} b (t)}{ρ} + \frac{γ_{2} a (t)}{ρ} + \int_{0}^{\infty} G (t, s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s)}^{'} | \\ = | \frac{α_{1} γ_{2}}{ρ p (t)} - \frac{α_{2} γ_{1}}{ρ p (t)} - \frac{α_{2}}{ρ p (t)} \int_{0}^{t} a (s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s + \frac{α_{1}}{ρ p (t)} \int_{t}^{\infty} b (s) (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s | \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ} + \int_{0}^{t} \frac{α_{2} a (s)}{ρ} (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s + \int_{t}^{\infty} \frac{α_{1} b (s)}{ρ} (f (s, x (s), x^{'} (s)) - k^{2} x (s)) d s) \end{array}

$\begin{array}{l} \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ} + \int_{0}^{\infty} f (s, x (s), x^{'} (s)) d s) \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ} + \int_{0}^{\infty} ϕ (s) q (s, y_{1} (s), y_{2} (s)) d s) \\ \leq \sup_{t \in [0, + \infty)} \frac{1}{p (t)} (\frac{α_{1} γ_{2} + α_{2} γ_{1}}{ρ} + (L - ε_{0}) \max {y_{1} (s), | y_{2} (s) |} \int_{0}^{\infty} ϕ (s) d s) \leq ‖ x ‖ = r_{1} . \end{array}$

另一方面，由条件(H₃)的第二个不等式，存在 $r_{0} > c^{*} r_{1} > 0$ 满足

$g (t, v_{1}, v_{2}) \geq (l + ε_{0}) (v_{1} + | v_{2} |), v_{1}, | v_{2} | \geq r_{0}, t \in [a^{*}, b^{*}] .$ (3.13)

取 $r_{2} = r_{1} + c^{*} r_{1} > r_{1}$ ， $K_{r 2} = {x \in K, ‖ x ‖ < r_{2}}$ ， $x_{0} = 1 \in \partial K_{1}$ 下面证明

$x \neq T x + μ x_{0}, \forall x \in \partial K_{r 2}, \forall μ > 0.$ (3.14)

否则，存在 $x \in \partial K_{r 2}$ ， $μ_{*} > 0$ ，有 $x_{*} = T x_{*} + μ_{*}$ ，由(3.13)和

$\begin{matrix} y_{1 *} (t) + y_{2 *} = \frac{x_{*} (t)}{ρ^{- 1} (1 + a (t) b (t))} + | {x^{'}}_{*} (t) | \geq c^{*} \frac{x_{*} (u)}{ρ^{- 1} (1 + a (u) b (u))} + | {x^{'}}_{*} (t) | \\ \geq c^{*} (‖ x_{0} ‖ + ‖ x^{'} ‖) > c^{*} r_{2} > r_{0}, t \in [a^{*}, b^{*}], u \in [0, + \infty) \end{matrix}$

可知

$g (t, y_{1 *} (t), y_{2 *} (t)) \geq (l + ε_{0}) (y_{1 *} (t) + | y_{2 *} (t) |) .$ (3.15)

假设 $ξ = \min {x_{*} (t) : t \in [a^{*}, b^{*}]}$ ，则

$\begin{matrix} x_{*} (t) = \frac{γ_{1} b (t)}{ρ} + \frac{γ_{2} a (t)}{ρ} + \int_{0}^{\infty} G (t, s) (f (s, x_{*} (s), {x^{'}}_{*} (s)) - k^{2} x_{*} (s)) d s \\ \geq \int_{a^{*}}^{b^{*}} G (t, s) (g (s, y_{1 *} (s), y_{2 *} (s)) - k^{2} x_{*} (s)) d s + μ_{2} \\ \geq \int_{a^{*}}^{b^{*}} G (t, s) (l + ε_{0}) (y_{1 *} (s) + | y_{2 *} (s) | - k^{2} x_{*} (s)) d s + μ_{2} \\ = \int_{a^{*}}^{b^{*}} G (t, s) (l + ε_{0}) (\frac{x_{*} (s)}{ρ^{- 1} (1 + a (s) b (s))} + | {x^{'}}_{*} (s) | - k^{2} x_{*} (s)) d s + μ_{2} \end{matrix}$

$\begin{array}{l} > \min_{s \in [a^{*}, b^{*}]} x_{*} (s) \int_{a^{*}}^{b^{*}} G (t, s) (\frac{(l + ε_{0})}{ρ^{- 1} (1 + a (s) b (s))} - k^{2}) d s + μ_{2} \\ > \min_{s \in [a^{*}, b^{*}]} x_{*} (s) \int_{a^{*}}^{b^{*}} G (t, s) (\frac{(l + ε_{0})}{ρ^{- 1} (1 + a (\infty) b (0))} - k^{2}) d s + μ_{2} \\ > ξ + μ_{2} > ξ . \end{array}$

即 $ξ < ξ$ ，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 $0 < r_{1} < ‖ x ‖ \leq r_{2}$ 。易知 $x$ 是BVP(1.1)的正解。

类似于定理3.1的证明可得下面的定理3.2成立。

定理3.2：假设条件(H₁) (H₂)成立，并且 $q, g$ 满足下列条件

(H₄) $0 \leq \lim_{v_{1}, | v_{2} | \to \infty} \max_{t \in [0, + \infty)} \frac{q (t, v_{1}, v_{2})}{\max {v_{1}, | v_{2} |}} < L, k^{2} < l < \lim_{v_{1}, | v_{2} | \to 0^{+}} \max_{t \in [a^{*}, b^{*}]} \frac{g (t, v_{1}, v_{2})}{\max {v_{1}, | v_{2} |}} \leq + \infty .$

$L, l$ 的定义同定理3.1。则BVP(1.1)至少有一个正解。

4. 结论

本文，我们主要讨论的是无穷区间上一类微分方程边值问题正解的存在性，其中BVP(1.1)中的非线性项 $f$ 在 $t = 0$ 点是奇异的，文 [5] [6] 中的非线性项 $f$ 限制了连续的条件下，而且我们所研究的方程中， $f$ 增加了导数项，这是在文 [5] [6] [7] [8] 中都没有涉及的，就要求在更为复杂的空间中，构造特殊的函数来讨论BVP(1.1)，同时，相较于文 [5] [6] [7] [8] ，我们研究的边值条件更具有一般性，所以说，本文的结果，在一定程度上，改进和推广了许多已知结果。

基金项目

本文受到临沂大学大学生创新创业训练计划项目(201610452168)部分资助。

文章引用

王克,王颖. 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[J]. 应用数学进展, 2017, 06(09): 1151-1162. http://dx.doi.org/10.12677/AAM.2017.69140

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