众所周知,积分方程、 泛函积分方程在物理学、工程学、经济学、生物学等学科中有着广泛的应用^{[1, 2]}. 最近,泛函积分方程理论吸引了广大学者浓厚的兴趣,不少学者对其进行了深入地研究^{[3, 4, 5, 6]},并逐渐形成了非线性分析中一个重要的研究方向.

利用Stieltjes积分理论,可将经典的积分方程推广为Stieltjes型积分方程, 关于这种推广的方法及Stieltjes型积分方程,可参看文献^{[7, 8, 9, 10, 11, 12]}. 利用泛函分析中的工具,拓扑度理论和不动点定理,这类方程的研究得到了快速的发展,然而大部分文献都是在有界区间上探讨方程的可解性^{[5, 13, 14, 15, 16, 17]},对于无界区间,这类方程的研究还是比较少见,如Bana\'s^[18]在无界区间上研究了一类Volterra-Stieltjes积分方程解的存在性.

本文研究如下非线性Volterra-Stieltjes型泛函积分方程 \begin{equation} x(t)=h(t,x(\gamma (t)))+f(t,x(\sigma (t)))\psi (\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s))){{\text{d}}_{s}}g(t,s)),~~~~t\in {{\mathbb{R}}^{+}}. \tag{1.1} \end{equation} 方程$(1.1)$是一类比较广泛的方程,包含了许多非线性分析中的泛函方程、积分方程、泛函积分方程,如经典的Volterra 型积分方程,著名的Chandrasekhar型积分方程都是方程$(1.1)$的特例. 本文在有界连续函数空间内,选取合适的非紧性测度,利用Schauder不动点定理, 在比较弱的条件下,研究方程$(1.1)$解的存在性及渐近行为, 得到的结论具有一般性,推广了有关文献的结果^{[4, 19, 20, 21]}.

首先,我们回顾一下有关有界变差和Stieltjes积分的内容. 假设$x$是定义在 $[a,b]$上的实函数,符号$\bigvee\limits^{b}_{a}x$表示$x$在$[a,b]$上的变差,如果$\bigvee\limits^{b}_{a}x$有限,则称为有界变差. 令函数$g(t,s):[a,b]\times [c,d]\to \mathbb{R}$,则符号$\bigvee\limits^{q}_{t=p}g(t,s)$表示函数$t\rightarrow g(t,s)$在区间$[p,q]\subset [a,b]$上的变差,其中$s\in [c,d]$. 类似可以定义$\bigvee\limits^{q}_{s=p}g(t,s)$. 有关函数的有界变差,可参看文献^[11].

如果$x,\varphi$是定义在$[a,b]$上的实函数,在满足一定的条件下^[11], Stieltjes积分$\int^{b}_{a}x(t){\rm d}\varphi(t)$存在, 则称$x(t)$关于$\varphi(t)$在$[a,b]$上Stieltjes可积. 在文献中, 有很多条件保证Stieltjes积分的存在,比较常用的是 $x$为连续函数,$\varphi$在$[a,b]$上是有界变差函数^{[10, 11]}.

下面,我们介绍几个关于Stieltjes积分常用的性质^{[10, 11]}.

引理2.1 如果函数$x$关于有界变差函数$\varphi$在$[a,b]$是Stieltjes可积,则有 $$\bigg|\int^{b}_{a}x(t){\rm d}\varphi(t)\bigg|\leq \int^{b}_{a}|x(t)| {\rm d}\bigg(\bigvee\limits_{a}^{t}\varphi\bigg).$$

引理2.2 如果函数$x_{1},x_{2}$关于非减函数$\varphi$在$[a,b]$是Stieltjes可积且有$x_{1}(t)\leq x_{2}(t),$ $ t\in[a,b]$,则 $$\int^{b}_{a}x_{1}(t){\rm d}\varphi(t)\leq\int^{b}_{a}x_{2}(t){\rm d}\varphi(t).$$

推论2.1 如果函数$x$关于非减函数$\varphi$在$[a,b]$是Stieltjes可积且有$x(t)\geq 0,t\in[a,b]$,则 $$\int^{b}_{a}x(t){\rm d}\varphi(t)\geq 0.$$

类似,我们可以定义Stieltjes积分$\int^{b}_{a}x(s){\rm d}_{s}g(t,s),$ 其中$g:[a,b]\times [c,d]\to \mathbb{R}$,符号 ${\rm d}_{s}$表示关于变量$s$积分.

假设$x$是定义在$[a,b]$上的实函数,定义$\omega(x,\varepsilon)$如下 $$\omega(x,\varepsilon)=\sup\{ |x(t)-x(s)|:~~ t,s\in[a,b],~~|t-s|\leq \varepsilon \}.$$ 设$p(t,s)=p:[a,b]\times [c,d]\to \mathbb{R}$,则定义 $$\omega(p(t,\cdot),\varepsilon)=\sup\{ |p(t,u)-p(t,v)|:~~ u,v\in[c,d], ~~|u-v|\leq \varepsilon \}, $$ 其中$t\in[a,b].$ 类似可以定义$\omega(p(\cdot,s),\varepsilon)$,$s\in[c,d].$

下面我们介绍有关非紧性测度理论方面的内容. 设$(E,\|\cdot\|)$是实Banach空间且零元$\theta\in E.$$B(x,r)$ 表示球心在$x$,半径为$r$的球,$B_{r}$表示球$B(\theta,r)$. $X$是$E$的非空子集,$\overline{X}$,Conv$X$分别表示$X$的闭包,闭凸子集. $E$的所有非空有界子集记为${\mathfrak{M}}_E$,${\mathfrak{M}}_E$的所有相对紧子集记为${\mathfrak{N}}_E$.

定义2.1^[22] 函数 $\mu :{{\mathfrak{M}}_{E}}\to {{\mathbb{R}}^{+}}:=[0,\infty )$称为$E$上的非紧性测度,如果满足下面的条件

(1) $\ker \mu = \{X \in {\mathfrak{M}}_{E}:~~ \mu(X)=0\}$ 非空且 $\ker \mu\subset {\mathfrak{N}}_{E}$;

(2) $X \subset Y \Rightarrow \mu(X) \leq \mu (Y)$;

(3) $\mu ({\overline{X}}) = \mu({\rm Conv} X)=\mu (X)$;

(4) $\mu(\lambda X+(1-\lambda)Y)\leq \lambda \mu(X)+(1-\lambda)\mu(Y)$, $\lambda\in[0,1]$;

(5) 如果${\mathfrak{M}}_{E}$闭子集中的序列$\{X_n\}_n$满足 $X_{n+1}\subset X_n$ ($n=1,2,\cdots$),且$\lim\limits_{n\to \infty}\mu (X_n)=0$,则$X_{\infty}=\bigcap\limits_{n=1}^{\infty}X_n$是非空集.

定义 2.1 中的$\ker \mu$称为非紧性测度$\mu$的核且$X_\infty$属于$\mu$的核. 事实上,根据不等式$\mu(X_\infty)\leq \mu(X_n)$,$n = 1,2,\cdots$,可知$\mu(X_\infty)$ = 0,所以$X_\infty \in$ $\ker \mu$. 有关非紧性测度理论及其性质,可参看文献^[22].

记Banach空间$BC({{\mathbb{R}}^{+}})$为${{\mathbb{R}}^{+}}$所有实值有界连续函数,且赋予最大值范数 $$\|x\|=\sup\{|x(t)|: t\geq 0\}.$$ 下面介绍定义在$BC({{\mathbb{R}}^{+}})$上的非紧性测度^[22]. $X$为${{\mathbb{R}}^{+}}$上的非空有界闭子集,$T>0$为常数,对于$x \in X$,$\varepsilon > 0$,定义$\omega^T(x,\varepsilon)$如下 $$ \omega^T(x,\varepsilon) = \sup\{|x(t) - x(s)| : t,s \in [0,T], |t - s|\leq \varepsilon\}.$$ 令 $$ \omega^T(X,\varepsilon)= \sup\{\omega^T(x,\varepsilon): x\in X \}, $$ $$ \omega_0^T(X)= \lim_{\varepsilon \to 0} \omega^T(X,\varepsilon), $$ $$ \omega_0(X) = \lim_{T \to \infty} \omega_0^T(X).$$ $$ \beta(X) = \lim_{T \to \infty}\left\{ \sup_{x \in X}\left\{\sup\left[|x(t)| : t \geq T \right]\right\}\right\}.$$ 在${{\mathfrak{M}}_{BC({{\mathbb{R}}^{+}})}}$上定义函数$\mu$ 如下 \begin{equation} \mu (X)={{\omega }_{0}}(X)+\beta (X).\text{ } \tag{2.1} \end{equation}

由文献^[22]可知$\mu $ 是$BC({{\mathbb{R}}^{+}})$上的非紧性测度. $\ker \mu $ 为非空有界子集$X$且$X$上的函数是局部等度连续的,一致关于$X$在无穷远处趋向于$0$,即对于任意的$\varepsilon >0$,存在$T>0$,使得对于$t\ge T$,$x\in X$,有 $|x(t)|<\varepsilon$.

在本节中,我们在空间$BC({{\mathbb{R}}^{+}})$中研究方程$(1.1)$解的存在性及其渐近行为. 首先假设如下条件成立

$(H_{1})$ $\psi :\mathbb{R}\to \mathbb{R}$ 是连续函数,$\psi(0)=0$ 且存在常数 $\lambda >0$ 使得

$$|\psi (x)-\psi (y)|\le \lambda |x-y|,~~~~~~x,~y\in \mathbb{R}.$$

$(H_{2})$ $\gamma ,\sigma ,\eta :{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$ 均为连续函数, $\beta :{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$ 为连续非减函数.

$(H_{3})$ $g:{{\mathbb{R}}^{+}}\times {{\mathbb{R}}^{+}}\to \mathbb{R}$ 满足如下条件

$(H_{3.a})$ 对于所有 ${{t}_{1}},{{t}_{2}}\in {{\mathbb{R}}^{+}}$,$t_{1}<t_{2}$,函数$s\rightarrow g(t_{2},s)-g(t_{1},s)$ 在${{\mathbb{R}}^{+}}$上是非减函数;

$(H_{3.b})$ $s\rightarrow g(0,s)$在${{\mathbb{R}}^{+}}$上是非减函数;

$(H_{3.c})$ 对固定的$t\in {{\mathbb{R}}^{+}}$ 或$s\in {{\mathbb{R}}^{+}}$, 函数$s\rightarrow g(t,s)$,$t\rightarrow g(t,s)$均为${{\mathbb{R}}^{+}}$上的连续函数.

$(H_{4})$ $h:{{\mathbb{R}}^{+}}\times \mathbb{R}\to \mathbb{R}$ 是连续函数且存在函数$l(t)\in BC({{\mathbb{R}}^{+}})$ 使得 $$|h(t,x)-h(t,y)|\le l(t)|x-y|,~~~~~~x,~y\in \mathbb{R},~~t\in {{\mathbb{R}}^{+}}.$$ 而且$\lim\limits_{t\rightarrow\infty}h(t,0)=0$.

$(H_{5})$ $f:{{\mathbb{R}}^{+}}\times \mathbb{R}\to \mathbb{R}$ 是连续函数且存在连续函数$m:{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$ 使得 $$|f(t,x)-f(t,y)|\le m(t)|x-y|,~~~~~~x,~y\in \mathbb{R},~~t\in {{\mathbb{R}}^{+}}.$$ 而且$\lim\limits_{t\rightarrow\infty}f(t,0)=0$.

$(H_{6})$ $u:{{\mathbb{R}}^{+}}\times {{\mathbb{R}}^{+}}\times \mathbb{R}\to \mathbb{R}$是连续函数且存在连续函数$k(t,s):{{\mathbb{R}}^{+}}\times {{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$和连续非减函数$\varphi :{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$使得 $$|u(t,s,x)|\le k(t,s)\varphi (|x|),~~~~~~t,~s\in {{\mathbb{R}}^{+}},~~x\in \mathbb{R}.$$

$(H_{7})$ 函数 $t\rightarrow \int^{\beta(t)}_{0}k(t,s){\rm d}_{s}g(t,s)$ 和 $t\rightarrow m(t)\int^{\beta(t)}_{0}k(t,s){\rm d}_{s}g(t,s)$是${{\mathbb{R}}^{+}}$上的有界函数.

$(H_{8})$ 存在常数 $r_{0}>0$满足如下不等式 \begin{equation} Lr+H +\lambda Mr \varphi(r) +\lambda FK\varphi(r)\leq r % \eqno(3.1)$$ \tag{3.1} \end{equation} 且有 $L+\lambda M\varphi(r_{0})<1$,其中 $$ L=\sup\{l(t): t\geq0\},~~~~ H=\sup\{|h(t,0)|: t\geq0 \},~~~~ F=\sup\{|f(t,0)|: t\geq0 \}, $$ $$ M=\sup\bigg\{m(t)\int^{\beta(t)}_{0}k(t,s){\rm d}_{s} g(t,s): t\geq 0 \bigg\},~~ K=\sup\bigg\{\int^{\beta(t)}_{0}k(t,s){\rm d}_{s} g(t,s): t\geq 0 \bigg\}. $$

注3.1  满足条件$(H_{3})$的函数$g(t,s)$有很多,例如

(1) $g(t,s)=s$.

(2) $g(t,s)=s\ln(t+1)$.

(3) $g(t,s)=\left\{ \begin{array}{*{35}{l}} t\ln \frac{t+s}{t}, & ~~~~t>0,~~s\ge 0; \\ 0,\quad & ~~~~t=0,~~s\ge 0. \\ \end{array} \right.$

(4) $g(t,s)=\int^{t}_{0}(\int^{s}_{0}p(z,y){\rm d}y){\rm d}z$,其中 $p:{{\mathbb{R}}^{+}}\times {{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$是有界可积函数. 更多的例子可见文献^[18].

注3.2  注意到如果$r_{0}>0$满足不等式$(3.1)$,则有 $$L+\lambda M\varphi(r_{0})\leq 1-\frac{H}{r_{0}}-\frac{\lambda FK\varphi(r_{0})}{r_{0}}.$$ 因此,如果$H$,$\lambda FK$有一项不等于$0$,条件$L+\lambda M\varphi(r_{0})<1$即满足.

下面我们给出函数$g(t,s)$的性质,其证明可参看文献^[9].

引理3.1 在满足条件$(H_{3.a})$,$(H_{3.b})$下,对于固定的$t\in {{\mathbb{R}}^{+}}$,$s\rightarrow g(t,s)$在 ${{\mathbb{R}}^{+}}$上是非减函数.

引理3.2  函数$g$满足条件$(H_{3.a})$,则对任意固定的${{s}_{1}},{{s}_{2}}\in {{\mathbb{R}}^{+}}$,$s_{1}<s_{2}$,有函数$t\rightarrow g(t,s_{2})-g(t,s_{1})$在${{\mathbb{R}}^{+}}$是非减的.

本节的主要结论是如下的定理.

定理3.1 在满足条件$(H_{1})$--$(H_{8})$下,方程$(1.1)$至少存在一个解$x\in BC({{\mathbb{R}}^{+}})$且有$\lim\limits_{t\rightarrow\infty}x(t)=0$.

证  在$BC({{\mathbb{R}}^{+}})$上定义算子$\mathcal{F}$如下 $$(\mathcal{F}x)(t)=h(t,x(\gamma (t)))+f(t,x(\sigma (t)))\psi (\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s))){{\text{d}}_{s}}g(t,s)),~~~~t\in \mathbb{R}.$$ 根据假设$(H_{3})$,$(H_{6})$和引理$3.1$,容易证明$\mathcal{F}$的定义是合理的.

定义 $$(Hx)(t)=h(t,x(\gamma(t))),~~~~(Fx)(t)=f(t,x(\sigma(t))), $$ $$ (Ux)(t)=\psi\bigg( \int^{\beta(t)}_{0}u(t,s,x(\eta(s))){\rm d}_{s}g(t,s)\bigg),$$ 则有 $$({\mathcal{F}}x)(t)=(Hx)(t)+(Fx)(t)\times (Ux)(t).$$

对于任意 $T>0$,$\varepsilon>0$,选取 $t_{1},t_{2}\in [0,T]$ 使得 $t_{1}<t_{2}$,$t_{2}-t_{1}\leq\varepsilon$,则对 $x\in BC({{\mathbb{R}}^{+}})$,由引理 $2.1$ 和引理 $3.1$ \begin{eqnarray*} &&|(Ux)(t_{2})-(Ux)(t_{1})| \\ &=&\left|\psi\bigg(\int^{\beta(t_{2})}_{0}u(t_{2},s,x(\eta(s))){\rm d}_{s}g(t_{2},s)\bigg)-\psi\bigg(\int^{\beta(t_{1})}_{0}u(t_{1},s,x(\eta(s))){\rm d}_{s}g(t_{1},s)\bigg)\right| \\ &\leq &\lambda \left|\int^{\beta(t_{2})}_{0}u(t_{2},s,x(\eta(s))){\rm d}_{s}g(t_{2},s)-\int^{\beta(t_{1})}_{0}u(t_{1},s,x(\eta(s))){\rm d}_{s}g(t_{1},s)\right| \\ &\leq& \lambda \left|\int^{\beta(t_{2})}_{0}u(t_{2},s,x(\eta(s))){\rm d}_{s}g(t_{2},s)-\int^{\beta(t_{1})}_{0}u(t_{2},s,x(\eta(s))){\rm d}_{s}g(t_{2},s)\right| \\ & & + \lambda \left|\int^{\beta(t_{1})}_{0}u(t_{2},s,x(\eta(s))){\rm d}_{s}g(t_{2},s)-\int^{\beta(t_{1})}_{0}u(t_{1},s,x(\eta(s))){\rm d}_{s}g(t_{2},s)\right| \\ & & + \lambda \left|\int^{\beta(t_{1})}_{0}u(t_{1},s,x(\eta(s))){\rm d}_{s}g(t_{2},s)-\int^{\beta(t_{1})}_{0}u(t_{1},s,x(\eta(s))){\rm d}_{s}g(t_{1},s)\right| \\ &\leq &\lambda \int^{\beta(t_{2})}_{\beta(t_{1})}\left|u(t_{2},s,x(\eta(s)))\right|{\rm d}_{s}\bigg(\bigvee\limits^{s}_{p=\beta(t_{1})}g(t_{2},p)\bigg) \\ & & + \lambda \int^{\beta(t_{1})}_{0}\left|u(t_{2},s,x(\eta(s)))-u(t_{1},s,x(\eta(s)))\right|{\rm d}_{s}\bigg(\bigvee\limits^{s}_{p=0}g(t_{2},p)\bigg) \\ & & + \lambda \int^{\beta(t_{1})}_{0}|u(t_{1},s,x(\eta(s)))|{\rm d}_{s}\bigg(\bigvee\limits^{s}_{p=0}[g(t_{2},p)-g(t_{1},p)]\bigg) \\ &\leq & \lambda \int^{\beta(t_{2})}_{\beta(t_{1})}\left|u(t_{2},s,x(\eta(s)))\right|{\rm d}_{s}g(t_{2},s) \\ && + \lambda \int^{\beta(t_{1})}_{0}\left|u(t_{2},s,x(\eta(s)))-u(t_{1},s,x(\eta(s)))\right|{\rm d}_{s}g(t_{2},s) \\ & & + \lambda \int^{\beta(t_{1})}_{0}|u(t_{1},s,x(\eta(s)))|{\rm d}_{s}(g(t_{2},s)-g(t_{1},s)) \\ &\leq & \lambda \int^{\beta(t_{2})}_{\beta(t_{1})}k(t_{2},s)\varphi(|x(\eta(s))|){\rm d}_{s}g(t_{2},s) + \lambda \int^{\beta(t_{1})}_{0}\omega^{T}_{u}(\varepsilon){\rm d}_{s}g(t_{2},s)\\ & & + \lambda \int^{\beta(t_{1})}_{0}k(t_{1},s)\varphi(|x(\eta(s))|){\rm d}_{s}(g(t_{2},s)-g(t_{1},s)) \\ &\leq &\lambda k_{T}\varphi(\|x\|)\int^{\beta(t_{2})}_{\beta(t_{1})}{\rm d}_{s}g(t_{2},s)+ \lambda \omega^{T}_{u}(\varepsilon)\int^{\beta(t_{1})}_{0}{\rm d}_{s}g(t_{2},s)\\ & & + \lambda k_{T}\varphi(\|x\|)\int^{\beta(t_{1})}_{0}{\rm d}_{s}(g(t_{2},s)-g(t_{1},s)) \\ &=& \lambda k_{T}\varphi(\|x\|)[g(t_{2},\beta(t_{2}))-g(t_{2},\beta(t_{1}))] + \lambda \omega^{T}_{u}(\varepsilon)[g(t_{2},\beta(t_{1}))-g(t_{2},0)] \\ & & + \lambda k_{T}\varphi(\|x\|)([g(t_{2},\beta(t_{1}))-g(t_{1},\beta(t_{1}))]-[g(t_{2},0)-g(t_{1},0)]) \\ &\leq &\lambda k_{T}\varphi(\|x\|)[g(T,\beta(t_{2}))-g(T,\beta(t_{1}))] + \lambda \omega^{T}_{u}(\varepsilon)[g(T,\beta(t_{1}))-g(T,0)] \\ & & + \lambda k_{T}\varphi(\|x\|)([g(t_{2},\beta(T))-g(t_{1},\beta(T))]+|g(t_{2},0)-g(t_{1},0)|) \\ &\leq &\lambda k_{T}\varphi(\|x\|)\omega^{T}(g(T,\cdot),\omega^{T}(\beta,\varepsilon))+ \lambda \omega^{T}_{u}(\varepsilon)[g(T,\beta(T))-g(T,0)] \\ && + \lambda k_{T}\varphi(\|x\|)[\omega^{T}(g(\cdot,\beta(T)),\varepsilon) + \omega^{T}(g(\cdot,0),\varepsilon)], \end{eqnarray*} 其中 $$ k_{T}=\sup\big\{k(t,s): t\in[0,T],s\in[0,\beta(T)] \big \}, $$ $$ \omega^{T}(g(T,\cdot),\omega^{T}(\beta,\varepsilon))=\sup \big\{ |g(T,\theta_{2})-g(T,\theta_{1})|: \theta_{1},\theta_{2}\in [0,\beta(T)],|\theta_{2}-\theta_{1}|< \omega^{T}(\beta,\varepsilon) \big\}, $$ \begin{eqnarray*} \omega^{T}_{u}(\varepsilon)&=&\sup\big\{|u(t_{2},s,x)-u(t_{1},s,x)|: t_{1},t_{2},\in[0,T],s\in[0,\beta(T)], \\ && |t_{2}-t_{1}|\leq\varepsilon,x\in[-\|x\|,\|x\|] \big \}.\end{eqnarray*} 因为$u$在$[0,T]\times[0,\beta(T)]\times[-\|x\|,\|x\|]$是一致连续的及$\beta$在$[0,T]$的一致连续性,所以在$\varepsilon\rightarrow 0$时,有$\omega^{T}_{u}(\varepsilon)\rightarrow 0$,$\omega^{T}(\beta,\varepsilon)\rightarrow 0$. 从条件$(H_{3.c})$可推得$Ux$在$[0,T]$是连续函数,根据$T$的任意性可知,$Ux$在${{\mathbb{R}}^{+}}$上是连续的. 从条件$(H_{4})$,$(H_{5})$,可知$Fx$,$Hx$均为${{\mathbb{R}}^{+}}$上的连续函数.所以${\mathcal{F}}x$是${{\mathbb{R}}^{+}}$上的连续函数.

下面证明${\mathcal{F}}x$是${{\mathbb{R}}^{+}}$上的有界函数. 对于$x\in BC({{\mathbb{R}}^{+}})$,$t\in {{\mathbb{R}}^{+}}$, \begin{eqnarray} & |(\mathcal{F}x)(t)|\le |h(t,x(\gamma (t)))|+|f(t,x(\sigma (t)))|\cdot \left| \varphi (\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s))){{\text{d}}_{s}}g(t,s)) \right| \\ & \le (|h(t,x(\gamma (t)))-h(t,0)|+|h(t,0)|)+\lambda (|f(t,x(\sigma (t)))-f(t,0)|+|f(t,0)|) \\ & \times \int_{0}^{\beta (t)}{\left| u(t,s,x(\eta (s))) \right|}{{\text{d}}_{s}}(\underset{p=0}{\overset{s}{\mathop{V}}}\,g(t,p)) \\ & \le (l(t)|x(\gamma (t))|+|h(t,0)|)+\lambda (m(t)|x(\sigma (t))|+|f(t,0)|) \\ \tag{3.2}\end{eqnarray} 因此 $$|({\mathcal{F}}x)(t)|\leq L\|x\|+H + \lambda M\varphi(\|x\|)\|x\| + \lambda FK\varphi(\|x\|).$$ 从而 ${\mathcal{F}}x$是${{\mathbb{R}}^{+}}$上的有界函数,即$\mathcal{F}$ 将 $BC({{\mathbb{R}}^{+}})$映射到自身.

根据$(3.2)$式,可知 $$\|{\mathcal{F}}x\|\leq L\|x\|+H + \lambda M\|x\| \varphi(\|x\|) +\lambda FK\varphi(\|x\|).$$ 利用$(H_{8})$,可推得存在常数$r_{0}>0$使得$L+ \lambda M\varphi(r_{0})<1$, 因此 $\mathcal{F}$将球$B_{r_{0}}$映射到自身.

对任意$T>0$,定义 $$m_{T}=\sup\{m(t): t\in[0,T]\},~~\sigma_{T}=\sup\{\sigma(t): t\in[0,T]\}, ~~\gamma_{T}=\sup\{\gamma(t): t\in[0,T]\}.$$ 选取非空子集$X\subseteq B_{r_{0}}$,$x\in X$,对任意$T>0$,$\varepsilon>0$,令$t_{1},t_{2}\in [0,T]$使得$t_{1}<t_{2}$,$t_{2}-t_{1}\leq \varepsilon$,则有 \begin{eqnarray*} &&|(Fx)(t_{2})(Ux)(t_{2})-(Fx)(t_{1})(Ux)(t_{1})|\\ & \leq& |f(t_{2},x(\sigma(t_{2})))-f(t_{2},x(\sigma(t_{1})))| \cdot \left|\psi\bigg(\int^{\beta(t_{2})}_{0}u(t_{2},s,x(\eta(s))){\rm d}_{s}g(t_{2},s)\bigg)\right| \\ &&+ \left|f(t_{2},x(\sigma(t_{1})))- f(t_{1},x(\sigma(t_{1})))\right| \cdot \left|\psi\bigg(\int^{\beta(t_{2})}_{0}u(t_{2},s,x(\eta(s))){\rm d}_{s}g(t_{2},s)\bigg)\right| \\ &&+ |f(t_{1},x(\sigma(t_{1})))| \\ &&\cdot \left|\psi\bigg(\int^{\beta(t_{2})}_{0}u(t_{2},s,x(\eta(s))){\rm d}_{s}g(t_{2},s)\bigg) - \psi\bigg(\int^{\beta(t_{2})}_{0}u(t_{2},s,x(\eta(s))){\rm d}_{s}g(t_{1},s)\bigg)\right| \\ &&+ |f(t_{1},x(\sigma(t_{1})))|\\ && \cdot \left|\psi\bigg(\int^{\beta(t_{2})}_{0}u(t_{2},s,x(\eta(s))){\rm d}_{s}g(t_{1},s)\bigg) -\psi\bigg(\int^{\beta(t_{2})}_{0}u(t_{1},s,x(\eta(s))){\rm d}_{s}g(t_{1},s)\bigg)\right| \\ &&+ |f(t_{1},x(\sigma(t_{1})))| \\ &&\cdot \left|\psi\bigg(\int^{\beta(t_{2})}_{0}u(t_{1},s,x(\eta(s))){\rm d}_{s}g(t_{1},s)\bigg) -\psi\bigg(\int^{\beta(t_{1})}_{0}u(t_{1},s,x(\eta(s))){\rm d}_{s}g(t_{1},s)\bigg)\right| \\ & \leq & \lambda m(t_{2})|x(\sigma(t_{2}))-x(\sigma(t_{1}))|\int^{\beta(t_{2})}_{0}|u(t_{2},s,x(\eta(s)))|{\rm d}_{s}\bigg(\bigvee\limits_{p=0}^{s}g(t_{2},p)\bigg) \\ &&+ \lambda \left|f(t_{2},x(\sigma(t_{1})))-f(t_{1},x(\sigma(t_{1})))\right|\int^{\beta(t_{2})}_{0}|u(t_{2},s,x(\eta(s)))| {\rm d}_{s}\bigg(\bigvee\limits_{p=0}^{s}g(t_{2},p)\bigg) \\ & &+ \lambda \left|f(t_{1},x(\sigma(t_{1})))\right|\int^{\beta(t_{2})}_{0}|u(t_{2},s,x(\eta(s)))|{\rm d}_{s}\bigg(\bigvee\limits_{p=0}^{s}[g(t_{2},p)-g(t_{1},p)]\bigg) \\ & &+ \lambda \left|f(t_{1},x(\sigma(t_{1})))\right|\int^{\beta(t_{2})}_{0}|u(t_{2},s,x(\eta(s)))-u(t_{1},s,x(\eta(s)))| {\rm d}_{s}\bigg(\bigvee\limits_{p=0}^{s}g(t_{1},p)\bigg) \\ & &+ \lambda \left|f(t_{1},x(\sigma(t_{1})))\right|\int^{\beta(t_{2})}_{\beta(t_{1})}|u(t_{1},s,x(\eta(s)))| {\rm d}_{s}\bigg(\bigvee\limits_{p=\beta(t_{1})}^{s}g(t_{1},p)\bigg) \\ & \leq & \lambda m(t_{2})\omega^{T}(x,\omega^{T}(\sigma,\varepsilon))\int^{\beta(t_{2})}_{0}k(t_{2},s)\varphi(|x(\eta(s))|){\rm d}_{s}g(t_{2},s) \\ & &+ \lambda \omega^{T}_{r_{0}}(f,\varepsilon)\int^{\beta(t_{2})}_{0}k(t_{2},s)\varphi(|x(\eta(s))|){\rm d}_{s}g(t_{2},s)\\ & &+ \lambda [m(t_{1})|x(\sigma(t_{1}))|+|f(t_{1},0)|]\int^{\beta(t_{2})}_{0}k(t_{2},s)\varphi(|x(\eta(s))|){\rm d}_{s}[g(t_{2},s)-g(t_{1},s)] \\ & &+ \lambda [m(t_{1})|x(\sigma(t_{1}))|+|f(t_{1},0)|]\int^{\beta(t_{2})}_{0}\omega^{T}_{r_{0}}(u,\varepsilon){\rm d}_{s}g(t_{1},s) \\ & &+ \lambda [m(t_{1})|x(\sigma(t_{1}))|+|f(t_{1},0)|]\int^{\beta(t_{2})}_{\beta(t_{1})}k(t_{1},s)\varphi(|x(\eta(s))|){\rm d}_{s}g(t_{1},s)\\ & \leq& \lambda M\varphi(\|x\|)\omega^{T}(x, \omega^{T}(\sigma,\varepsilon)) + \lambda K\varphi(\|x\|)\omega^{T}_{r_{0}}(f,\varepsilon)+ \lambda (m_{T}\|x\|+F) \\ && \times \bigg\{k_{T}\varphi(\|x\|) \int^{\beta(t_{2})}_{0}{\rm d}_{s}[g(t_{2},s)-g(t_{1},s)] + \omega^{T}_{r_{0}}(u,\varepsilon)\int^{\beta(t_{2})}_{0}{\rm d}_{s}g(t_{1},s)\\ && + k_{T}\varphi(\|x\|)\int^{\beta(t_{2})}_{\beta(t_{1})}{\rm d}_{s}g(t_{1},s) \bigg\}\\ & \leq &\lambda M\varphi(r_{0})\omega^{T}(x,\omega^{T}(\sigma,\varepsilon))+\lambda K\varphi(r_{0})\omega^{T}_{r_{0}}(f,\varepsilon) \\ &&+ \lambda (m_{T}r_{0}+F)\times \left\{k_{T}\varphi(r_{0}) ([g(t_{2},\beta(t_{2}))-g(t_{1},\beta(t_{2}))]-[g(t_{2},0)-g(t_{1},0)] )\right. \\ &&+ \omega^{T}_{r_{0}}(u,\varepsilon)[g(t_{1},\beta(t_{2}))-g(t_{1},0)]+k_{T}\varphi(r_{0})[g(t_{1},\beta(t_{2}))-g(t_{1},\beta(t_{1}))] \}\\ & \leq & \lambda M\varphi(r_{0})\omega^{T}(x,\omega^{T}(\sigma,\varepsilon))+ \lambda K\varphi(r_{0})\omega^{T}_{r_{0}}(f,\varepsilon) \\ && + \lambda (m_{T}r_{0}+F)\times \left\{k_{T}\varphi(r_{0}) ([g(t_{2},\beta(T))-g(t_{1},\beta(T))]+|g(t_{2},0)-g(t_{1},0)| )\right. \\ && +\omega^{T}_{r_{0}}(u,\varepsilon)[g(T,\beta(t_{2}))-g(T,0)]+k_{T}\varphi(r_{0})[g(T,\beta(t_{2}))-g(T,\beta(t_{1}))] \}\\ & \leq& \lambda M\varphi(r_{0})\omega^{T}(x,\omega^{T}(\sigma,\varepsilon))+ \lambda K\varphi(r_{0})\omega^{T}_{r_{0}}(f,\varepsilon) \\ && + \lambda (m_{T}r_{0}+F)\times \left\{k_{T}\varphi(r_{0}) [\omega^{T}(g(\cdot,\beta(T)),\varepsilon)+\omega^{T}(g(\cdot,0),\varepsilon)]\right. \\ && +\omega^{T}_{r_{0}}(u,\varepsilon)[g(T,\beta(T))-g(T,0)]+k_{T}\varphi(r_{0})\omega^{T}(g(T,\cdot),\omega^{T}(\beta,\varepsilon)) \}, \end{eqnarray*} 其中 $$ \omega^{T}(x,\omega^{T}(\sigma,\varepsilon))=\sup\big\{ |x(\theta_{2})-x(\theta_{1})|: \theta_{1},\theta_{2}\in [0,\sigma_{T}],|\theta_{2}-\theta_{1}|< \omega^{T}(\sigma,\varepsilon) \big \}, $$ $$ \omega^{T}_{r_{0}}(f,\varepsilon)=\sup\big\{|f(t_{2},x)-f(t_{1},x)|: t_{1},t_{2}\in[0,T], |t_{2}-t_{1}|\leq\varepsilon,x\in[-r_{0},r_{0}] \big \}, $$ \begin{eqnarray*} \omega^{T}_{r_{0}}(u,\varepsilon)&=&\sup\big\{|u(t_{2},s,x)-u(t_{1},s,x)|: t_{1},t_{2},\in[0,T],s\in[0,\beta(T)], \\ && |t_{2}-t_{1}|\leq\varepsilon,x\in[-r_{0},r_{0}]\big\}, \end{eqnarray*} 且有 \begin{eqnarray*} &&|(Hx)(t_{2})-(Hx)(t_{1})|\\ &=&|h(t_{2},x(\gamma(t_{2})))-h(t_{1},x(\gamma(t_{1})))| \\ &\leq &|h(t_{2},x(\gamma(t_{2})))-h(t_{2},x(\gamma(t_{1})))|+|h(t_{2},x(\gamma(t_{1})))-h(t_{1},x(\gamma(t_{1})))| \\ &\leq &L|x(\gamma(t_{2}))-x(\gamma(t_{1}))|+|h(t_{2},x(\gamma(t_{1})))-h(t_{1},x(\gamma(t_{1})))| \\ &\leq &L\omega^{T}(x,\omega^{T}(\gamma,\varepsilon)) +\omega^{T}_{r_{0}}(h,\varepsilon), \end{eqnarray*} 其中 $$ \omega^{T}(x,\omega^{T}(\gamma,\varepsilon))=\sup\{ |x(\theta_{2})-x(\theta_{1})|: \theta_{1},\theta_{2}\in [0,\gamma_{T}],|\theta_{2}-\theta_{1}|< \omega^{T}(\gamma,\varepsilon) \}, $$ $$ \omega^{T}_{r_{0}}(h,\varepsilon)=\sup\{|h(t_{2},x)-h(t_{1},x)|: t_{1},t_{2}\in[0,T], |t_{2}-t_{1}|\leq\varepsilon,x\in[-r_{0},r_{0}] \}.$$ 从而 \begin{eqnarray*} &&|({\mathcal{F}}x)(t_{2})-({\mathcal{F}}x)(t_{1})| \\ &\leq& |(Hx)(t_{2})-(Hx)(t_{1})| +|(Fx)(t_{2})(Ux)(t_{2})-(Fx)(t_{1})(Ux)(t_{1})| \\ &\leq &L\omega^{T}(x,\omega^{T}(\gamma,\varepsilon)) +\omega^{T}_{r_{0}}(h,\varepsilon) + \lambda M\varphi(r_{0})\omega^{T}(x,\omega^{T}(\sigma,\varepsilon)) +\lambda K\varphi(r_{0})\omega^{T}_{r_{0}}(f,\varepsilon) \\ &&+\lambda (m_{T}r_{0}+F) \times \left\{k_{T}\varphi(r_{0}) [\omega^{T}(g(\cdot,\beta(T)),\varepsilon)+\omega^{T}(g(\cdot,0),\varepsilon)]\right. \\ &&+\omega^{T}_{r_{0}}(u,\varepsilon)[g(T,\beta(T))-g(T,0)]+k_{T}\varphi(r_{0})\omega^{T}(g(T,\cdot), \omega^{T}(\beta,\varepsilon)) \}.\end{eqnarray*} 因此 \begin{eqnarray*} \omega^{T}({\mathcal{F}}x,\varepsilon)&\leq& L\omega^{T}(x,\omega^{T}(\gamma,\varepsilon)) +\omega^{T}_{r_{0}}(h,\varepsilon) +\lambda M\varphi(r_{0})\omega^{T}(x,\omega^{T}(\sigma,\varepsilon)) +\lambda K\varphi(r_{0})\omega^{T}_{r_{0}}(f,\varepsilon) \\ && +\lambda (m_{T}r_{0}+F) \times \{k_{T}\varphi(r_{0}) [\omega^{T}(g(\cdot,\beta(T)),\varepsilon)+\omega^{T}(g(\cdot,0),\varepsilon)] \\ &&+\omega^{T}_{r_{0}}(u,\varepsilon)[g(T,\beta(T))-g(T,0)]+k_{T}\varphi(r_{0})\omega^{T}(g(T,\cdot), \omega^{T}(\beta,\varepsilon)) \}.\end{eqnarray*} 从而 \begin{eqnarray*} \omega^{T}({\mathcal{F}}X,\varepsilon)&\leq &L\omega^{T}(X,\omega^{T}(\gamma,\varepsilon)) +\omega^{T}_{r_{0}}(h,\varepsilon) +\lambda M\varphi(r_{0})\omega^{T}(X,\omega^{T}(\sigma,\varepsilon)) +\lambda K\varphi(r_{0})\omega^{T}_{r_{0}}(f,\varepsilon) \\ && +\lambda (m_{T}r_{0}+F) \times \{k_{T}\varphi(r_{0}) [\omega^{T}(g(\cdot,\beta(T)),\varepsilon)+\omega^{T}(g(\cdot,0),\varepsilon)] \\ &&+\omega^{T}_{r_{0}}(u,\varepsilon)[g(T,\beta(T))-g(T,0)] +k_{T}\varphi(r_{0})\omega^{T}(g(T,\cdot), \omega^{T}(\beta,\varepsilon)) \}.\end{eqnarray*} 所以 $$\omega^{T}_{0}({\mathcal{F}}X)\leq (L+\lambda M\varphi(r_{0}))\omega^{T}_{0}(X). $$ 从而可得到 \begin{equation} \omega_{0}({\mathcal{F}}X)\leq (L+\lambda M\varphi(r_{0}))\omega_{0}(X).%\eqno(3.3)$$ \tag{3.3} \end{equation}

另一方面,对$T>0$和任意$x\in X$,由$(3.2)$式 $$|({\mathcal{F}}x)(t)|\leq L|x(\gamma(t))|+|h(t,0)|+\lambda M\varphi(r_{0})|x(\sigma(t))|+\lambda K\varphi(r_{0})|f(t,0)|,~~~~t\geq T.$$ 根据$(H_{4})$,$(H_{5})$,有 \begin{equation} \beta({\mathcal{F}}X)\leq(L+ \lambda M\varphi(r_{0}))\beta(X). % \eqno(3.4)$$ \tag{3.4} \end{equation} 从而由$(3.3)$,$(3.4)$式可推得 \begin{equation} \mu({\mathcal{F}}X)\leq (L+ \lambda M\varphi(r_{0}))\mu(X),%\eqno(3.5)$$ \tag{3.5} \end{equation} 其中$\mu$ 是 $(2.1)$式中定义的非紧性测度.

考虑序列集$(B^{n}_{r_{0}})$,其中$B^{1}_{r_{0}}={\rm Conv} {\mathcal{F}}(B_{r_{0}})$, $B^{2}_{r_{0}}={\rm Conv}{\mathcal{F}}(B^{1}_{r_{0}}),\cdots$. 易知这些集合均为闭凸集,且$(B^{n}_{r_{0}})$是递减的,即$B^{n+1}_{r_{0}}\subset B^{n}_{r_{0}}$,$n=1,2,\cdots .$ 根据$(3.5)$式可得 $$\mu(B^{n}_{r_{0}})\leq q^{n}\mu(B_{r_{0}}),$$ 其中 $q=L+ \lambda M\varphi(r_{0})$,而且容易证明$\mu(B_{r_{0}})=3r_{0}$.由 $q<1$,可知 $$\lim\limits_{n\rightarrow\infty}\mu(B^{n}_{r_{0}})=0.$$ 因此,根据非紧性测度的定义,可推得$Y=\bigcap\limits^{\infty}_{n=1}B^{n}_{r_{0}}$ 是非空有界闭凸集. 不难证明,$Y\in \ker \mu$且 $\mathcal{F}$将$Y$映入$Y$.

下面我们将证明$\mathcal{F}$是$Y$上的连续函数. 对任意$\varepsilon>0$,选取$x,y\in Y$使得 $\|x-y\|\leq\varepsilon$. 因为 $y\in Y$,$Y\in \ker \mu$且 $\lim\limits_{t\rightarrow\infty}f(t,0)=0$,所以存在$T>0$,使得对于$t>T$,有 $|y(\sigma(t))|<\varepsilon$,$|f(t,0)|<\varepsilon$. 利用引理$2.1$,引理$2.2$,对$t>T$,有 \begin{eqnarray} & |(\mathcal{F}x)(t)-(\mathcal{F}y)(t)| \\ & \le |(Hx)(t)-(Hy)(t)|+|(Fx)(t)\cdot (Ux)(t)-(Fy)(t)\cdot (Uy)(t)| \\ & \le l(t)\|x-y\|+|f(t,x(\sigma (t)))-f(t,y(\sigma (t)))|\cdot \left| \psi (\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s))){{\text{d}}_{s}}g(t,s)) \right| \\ & +|f(t,y(\sigma (t)))|\cdot \left| \psi (\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s))){{\text{d}}_{s}}g(t,s))-\psi (\int_{0}^{\beta (t)}{u}(t,s,y(\eta (s))){{\text{d}}_{s}}g(t,s)) \right| \\ & \le L\varepsilon +\lambda m(t)\|x-y\|\int_{0}^{\beta (t)}{|}u(t,s,x(\eta (s)))|{{\text{d}}_{s}}(\underset{p=0}{\overset{s}{\mathop{V}}}\,g(t,p)) \\ & +\lambda (m(t)|y(\sigma (t))|+|f(t,0)|)\int_{0}^{\beta (t)}{|}u(t,s,x(\eta (s)))-u(t,s,y(\eta (s)))|{{\text{d}}_{s}}(\underset{p=0}{\overset{s}{\mathop{V}}}\,g(t,p)) \\ & \le L\varepsilon +\lambda \varepsilon m(t)\int_{0}^{\beta (t)}{k}(t,s)\varphi (|x(\eta (s))|){{\text{d}}_{s}}g(t,s)+\lambda (m(t)|y(\sigma (t))|+|f(t,0)|) \\ & \times \int_{0}^{\beta (t)}{(|u(}t,s,x(\eta (s)))|+|u(t,s,y(\eta (s)))|){{\text{d}}_{s}}g(t,s) \\ & \le L\varepsilon +\lambda \varepsilon \varphi ({{r}_{0}})m(t)\int_{0}^{\beta (t)}{k}(t,s){{\text{d}}_{s}}g(t,s)+\lambda (m(t)|y(\sigma (t))|+|f(t,0)|) \\ & \times 2\varphi ({{r}_{0}})\int_{0}^{\beta (t)}{k}(t,s){{\text{d}}_{s}}g(t,s) \\ & \le L\varepsilon +\lambda M\varphi ({{r}_{0}})\varepsilon +2\lambda \varphi ({{r}_{0}})(m(t)|y(\sigma (t))|+|f(t,0)|)\int_{0}^{\beta (t)}{k}(t,s){{\text{d}}_{s}}g(t,s) \\ & \le (L+3\lambda M\varphi ({{r}_{0}})+2\lambda K\varphi ({{r}_{0}}))\varepsilon , \\ \tag{3.6}\end{eqnarray} 另一方面,$t\in[0,T]$, \begin{eqnarray} & |(\mathcal{F}x)(t)-(\mathcal{F}y)(t)| \\ & \le l(t)\|x-y\|+|f(t,x(\sigma (t)))-f(t,y(\sigma (t)))|\cdot \left| \psi (\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s))){{\text{d}}_{s}}g(t,s)) \right| \\ & +|f(t,y(\sigma (t)))|\cdot \left| \psi (\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s))){{\text{d}}_{s}}g(t,s))-\psi (\int_{0}^{\beta (t)}{u}(t,s,y(\eta (s))){{\text{d}}_{s}}g(t,s)) \right| \\ & \le L\varepsilon +\lambda m(t)\|x-y\|\int_{0}^{\beta (t)}{k}(t,s)\varphi (|x(\eta (s))|){{\text{d}}_{s}}g(t,s) \\ & +\lambda (m(t)|y(\sigma (t))|+|f(t,0)|)\int_{0}^{\beta (t)}{|}u(t,s,x(\eta (s)))-u(t,s,y(\eta (s)))|{{\text{d}}_{s}}g(t,s) \\ & \le L\varepsilon +\lambda M\varphi ({{r}_{0}})\varepsilon +\lambda ({{m}_{T}}{{r}_{0}}+F){{{\bar{\omega }}}_{{{r}_{0}}}}(u,\varepsilon )[g(t,\beta (t))-g(t,0)] \\ & \le L\varepsilon +\lambda M\varphi ({{r}_{0}})\varepsilon +\lambda ({{m}_{T}}{{r}_{0}}+F){{{\bar{\omega }}}_{{{r}_{0}}}}(u,\varepsilon )[g(T,\beta (T))-g(T,0)], \\ \tag{3.7}\end{eqnarray} 其中 $$\overline{\omega}_{r_{0}}(u,\varepsilon)=\sup\{|u(t,s,x)-u(t,s,y)|: t\in[0,T], s\in[0,\beta(T)],x,y\in[-r_{0},r_{0}],|x-y|\leq\varepsilon\}.$$ 由$u$在$[0,T]\times[0,\beta(T)]\times[-r_{0},r_{0}]$的一致连续性,可推得$\overline{\omega}_{r_{0}}(u,\varepsilon)\rightarrow 0$,$\varepsilon\rightarrow 0$. 利用$(3.6)$,$(3.7)$式可知$\mathcal{F}$是$Y$上的连续函数. 根据Schauder不动点定理,$\mathcal{F}$在$Y$上至少存在一解$x$. 不难证明$x=x(t)$是方程$(1.1)$的解. 因为 $Y\in \ker \mu$,从而 $\lim\limits_{t\rightarrow\infty}x(t)=0$. 定理 $3.1$ 证毕.

如果 $\psi(x)=x$,此时$(H_{1})$ 成立,方程$(1.1)$即为 \begin{equation} x(t)=h(t,x(\gamma (t)))+f(t,x(\sigma (t)))\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s))){{\text{d}}_{s}}g(t,s),~~t\in {{\mathbb{R}}^{+}}. \tag{3.8} \end{equation} 根据定理3.1,如下结论成立.

推论3.1 如果$(H_{2})$--$(H_{8})$成立,则方程$(3.8)$至少存在一解$x\in BC({{\mathbb{R}}^{+}})$且$\lim\limits_{t\rightarrow\infty}x(t)=0$.

注3.3 当$f(t,x)=1$ 或者$f(t,x)=a(t)$,$a(t)$在无穷远处不趋于零,此时条件$(H_{5})$不满足,这时候就不能应用定理3.1,推论3.1. 针对这种情况,我们下面给出如下的结论.

假设如下条件成立

$(H'_{5})$ $f:{{\mathbb{R}}^{+}}\times \mathbb{R}\to \mathbb{R}$ 是连续函数且存在连续函数 $m:{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$ 使得 $$|f(t,x)-f(t,y)|\le m(t)|x-y|,~~~~x,y\in \mathbb{R},~~t\in {{\mathbb{R}}^{+}}.$$ 且有$f(t,0)\in BC({{\mathbb{R}}^{+}})$.

$(H_{9})$ $\lim\limits_{t\rightarrow\infty}\int^{\beta(t)}_{0}k(t,s){\rm d}_{s}g(t,s)=0.$

类似于定理3.1,推论3.1的证明,有

定理3.2 假设$(H_{1})$--$(H_{4})$,$(H'_{5})$,$(H_{6})$--$(H_{9})$成立,则方程$(1.1)$至少存在一解$x\in BC({{\mathbb{R}}^{+}})$且有$\lim\limits_{t\rightarrow\infty}x(t)=0$.

推论3.2 假设$(H_{2})$--$(H_{4})$,$(H'_{5})$,$(H_{6})$--$(H_{9})$成立, 则方程$(3.8)$至少存在一解$x\in BC({{\mathbb{R}}^{+}})$ 且有$\lim\limits_{t\rightarrow\infty}x(t)=0$.

在本节中,我们举一些例子来验证本文的主要结论.

例4.1 如果$g(t,s)=s$,容易证明 $(H_{3})$成立,此时方程$(1.1)$即为 \begin{equation} x(t)=h(t,x(\gamma (t)))+f(t,x(\sigma (t)))\psi (\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s)))\text{d}s),~~~~t\in {{\mathbb{R}}^{+}}. \tag{4.1}\end{equation} 由定理$3.1$,定理$3.2$,可知

定理4.1 假设$(H_{1})$,$(H_{2})$,$(H_{4})$--$(H_{8})$成立, 则方程$(4.1)$至少存在一解$x\in BC({{\mathbb{R}}^{+}})$且有$\lim\limits_{t\rightarrow\infty}x(t)=0$.

定理4.2 假设$(H_{1})$,$(H_{2})$,$(H_{4})$,$(H'_{5})$,$(H_{6})$--$(H_{9})$ 成立, 则方程$(4.1)$至少存在一解$x\in BC({{\mathbb{R}}^{+}})$且有$\lim\limits_{t\rightarrow\infty}x(t)=0$.

注意到参考文献中广泛研究的积分方程、泛函积分方程,很多都是方程$(4.1)$的特殊情况, 定理4.1,定理4.2也推广了相关的结论,例如

$\bullet$  如果 $\psi(x)=x$,$f(t,x)=1$,方程 $(4.1)$即为如下形式^[4] \begin{equation} x(t)=h(t,x(\gamma (t)))+\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s)))\text{d}s,~~~~t\in {{\mathbb{R}}^{+}}. \tag{4.2}\end{equation} 在方程$(4.2)$中,如果$\gamma(t)=\beta(t)=\eta(t)=t$,即为文献^[19] 研究的方程. 如果$h(t,x)=q(t)$,$\beta(t)=\eta(t)=t$, 方程$(4.2)$变成经典的非线性Volterra积分方程^{[1, 2]}.

$\bullet$  如果$\psi(x)=x$,$\gamma(t)=\sigma(t)=\beta(t)=\eta(t)=t$,方程$(4.1)$变为如下的非线性二次Volterra型积分方程^[21] \begin{equation} x(t)=h(t,x(t))+f(t,x(t))\int_{0}^{t}{u}(t,s,x(s))\text{d}s,~~~~t\in {{\mathbb{R}}^{+}}. \tag{4.3}\end{equation} 在方程$(4.3)$中,如果$h(t,x)=0$,文献^[23]研究了此类方程; 如果$f(t,x)=x$, 文献^[24]对这类方程进行了探讨.

$\bullet$  如果$h(t,x)=q(t)$,则方程为如下形式^[25] \begin{equation} x(t)=q(t)+f(t,x(\sigma (t)))\psi (\int_{0}^{\beta (t)}{u}(t,s,x(\eta (s)))\text{d}s),~~~~t\in {{\mathbb{R}}^{+}}. \tag{4.4}\end{equation} 在方程$(4.4)$中,如果$\psi(x)=x$,文献^[20]对这类方程进行了研究.

例4.2 考虑如下二次Volterra型积分方程 \begin{equation} x(t)=h(t,x(t))+x(t)\int_{0}^{t}{\frac{t}{t+s}}u(t,s,x(s))\text{d}s,~~~~~t\in {{\mathbb{R}}^{+}}. \tag{4.5}\end{equation} 这类方程包含了经典的二次Chandrasekhar型积分方程^{[14, 18, 26, 27]}.

如果 $$g(t,s)=\left\{ \begin{array}{*{35}{l}} t\ln \frac{t+s}{t}, & ~~~~t>0,~~s\ge 0; \\ 0,\quad & ~~~~t=0,~~s\ge 0. \\ \end{array} \right.$$ 则此时方程$(4.5)$为方程$(1.1)$的特殊情况,其中$\psi(x)=x$,$\gamma(t)=\sigma(t)=\eta(t)=\beta(t)=t$,且 $f(t,x)=x$,从而条件$(H_{1})$--$(H_{3})$成立 $(\lambda=1)$. 另外一方面,不难验证$(H_{5})$也满足,此时$m(t)=1,$ $f(t,0)=0,F=0$. 根据定理$3.1$,有

定理4.3 假设$(H_{4})$,$(H_{6})$满足且如下条件成立:

$(H'_{7})$ 函数$t\rightarrow \int^{t}_{0}\frac{t}{t+s}k(t,s){\rm d}s$在${{\mathbb{R}}^{+}}$上有界;

$(H'_{8})$ 存在常数$r_{0}>0$满足不等式 $Lr+H + \widetilde{K}r \varphi(r) \leq r$,且有$L+\widetilde{K}\varphi(r_{0})<1$,其中 $$\widetilde{K}= \sup\left\{\int^{t}_{0}\frac{t}{t+s} k(t,s){\rm d}s: t\geq 0 \right\}. $$ 则方程$(4.5)$至少存在一解$x\in BC({{\mathbb{R}}^{+}})$且有$\lim\limits_{t\rightarrow\infty}x(t)=0.$

最后,我们给出满足定理3.1所有条件的一个具体例子.

例4.3 考虑如下二次Volterra型泛函积分方程 \begin{equation} x(t)=\frac{1}{32}{{\text{e}}^{-{{t}^{2}}}}+\frac{{{t}^{2}}+2}{32({{t}^{2}}+1)}x(2t)+({{\text{e}}^{-2t}}+{{t}^{2}}x(\sqrt{t}))\int_{0}^{{{t}^{2}}}{\frac{{{x}^{2}}(s)}{1+{{t}^{4}}+{{s}^{2}}}}\text{d}s,~~~~t\in {{\mathbb{R}}^{+}}. \tag{4.6}\end{equation} 易知方程$(4.6)$是方程$(1.1)$的特殊情形,其中$\psi(x)=x,g(t,s)=s,\gamma(t)=2t, \sigma(t)=\sqrt{t},\beta(t)=t^{2},$ $ \eta(t)=t$,且 $$h(t,x(\gamma(t)))=\frac{1}{32}{\rm e}^{-t^{2}}+\frac{t^{2}+2}{32(t^{2}+1)}x(2t), $$ $$ f(t,x(\sigma(t)))={\rm e}^{-2t}+t^{2}x(\sqrt{t}),~~u(t,s,x(\eta(s)))=\frac{x^{2}(s)}{1+t^{4}+s^{2}}.$$ 从而$\lambda=1,l(t)=\frac{t^{2}+2}{32(t^{2}+1)},h(t,0)=\frac{1}{32}{\rm e}^{-t^{2}}$, 即有 $L=\frac{1}{16},H=\frac{1}{32}$,所以$(H_{1})$--$(H_{4})$成立.另一方面,我们有 $m(t)=t^{2}$,$k(t,s)=\frac{1}{1+t^{4}+s^{2}}$,$f(t,0)={\rm e}^{-2t}$, $\varphi(x)=x^{2}$,因此$F=1$,$M=K=\frac{\pi}{4}$.最后,注意到$(H_{8})$的不等式有如下形式 $$\frac{1}{16}r+\frac{1}{32}+\frac{\pi}{4}r^{3}+\frac{\pi}{4}r^{2}\leq r.$$ 不难证明,存在正常数$r_{0}$满足此不等式且有$L+\lambda M\varphi(r_{0})<1$ (如$r_{0}=\frac{2}{3}$). 从而$(H_{5})$--$(H_{8})$成立. 根据定理3.1,方程 $(4.6)$至少存在一解$x\in BC({{\mathbb{R}}^{+}})$且有$\lim\limits_{t\rightarrow\infty}x(t)=0.$