| Summary: | 碩士 === 國立中央大學 === 數學系 === 85 === \magnification=1200\baselineskip24pt\font\mm=tcl
scaled\magstep3\font\tm=tcr scaled\magstep2\font\tr=tcb
scaled\magstep1\font\sm=tcl scaled\magstep2\font\tl=tcl
scaled\magstep0\pageno=-1\centerline {\tm 論 文 摘 要 } \vskip1
cm自然界中,有很都現象$\circ $ 這些現象常被科學家模擬成數學模型$
\circ $而最常用的數學模型,乃是"微分方程式"$\circ $ 藉由微分方程
式解的存在性,可以說明自然界問題的現象和預測未來的結果$\circ $ 因
此有很都數學家對於探討微分方程式解之存在性有相當濃厚的興趣,並以
用各種不同的數學方法來求出解,而順利解決問題,使得此數學模型能得到
一些合理的解釋和預測,對人類盡一些微薄之力$\circ$ \par在此篇論文
之中,我們所關心的問題是有關於高階非線性常微分方程式解之存在性$
\circ$ $$\left\{\eqalign{ (E)~~&u^{(n+2)
}(t)+f(t,u)=0,\quad 0<t<1,\cr (BC)~~
&\left\{\eqalign{ & u^{(r_i-1)
}(0)=0,~~~~1\le i \le k, \cr & u^{(
s_j-1)}(1)=0,~~~~ 1\le j \le n-k, \cr
&\alpha u^{(n)}(0)-\beta u^{(n+1)}(0)=0, \cr
&\gamma u^{(n)}(1)+\delta u^{(n+1)}(1)=0, \cr}\right.
\cr } \right.\leqno
(BVP)$$其中 $\{r_1,\dots ,r_k\}\cup \{s_1,\dots ,s_{n-k}\}$ 是
$\{1,\dots ,n\}$的分割$\circ$ \par 事實上,有關此問題之研究,早已
是目前數學家所關心的,而且也是數學界的主流$\circ$ 而我們這篇文章
的主要目的,乃在於推廣 Agarwal 和 Henderson 等數學家的結果, 他們
的結果只現至於 $$\left\{\eqalign{
&u^{(n+2)}(t)+a(t)g(u)=0,~~ 0<t<1,\cr
&\eqalign{ & u^{(r_i-1)}(0)=0,1
\le i \le k, \cr & u^{(s_j-1)
}(1)=0, 1\le j \le n-k, \cr
&u^{(n)}(0)=0, u^{(n+1)}(1)=0,\hbox{~or~} \cr
&u^{(n)}(1)=0, u^{(n+1)}(0)=0, \cr}
\cr } \right.\leqno (BVP)_1$$因
此我們便藉由他們所作的論文中得到了靈感,而推廣了他們的結果,在推廣
的過程中,亦用到葉哲志老師,王富祥及連偉成學長他們論文中之技巧,才
能得以完成此文章$\circ$\end
\font\msbmten=msbm10 \font\msbmseven=msbm7 \font\msbmfive=
msbm5\newfam\msbmfam \textfont\msbmfam=
\msbmten\scriptfont\msbmfam=\msbmseven
\scriptscriptfont\msbmfam=
\msbmfive\def\Bbb{\fam\msbmfam}\magnification=1440%
\documentstyle{amsppt}%\voffset 0.75 true in%\hoffset 0.75 true
in\centerline {\bf 1. Introduction }\bigskip\bigskipLet
$\{r_1, r_2, \dots ,r_k \}$ and $\{s_1, s_2, \dots , s_{n-k}\}$
be a disjoint partition of $\{1, 2, \dots ,n\}$ such that
$r_1<r_2\dots <r_k$ and $s_1<s_2<\dots <s_{n-k}$. For each
$0\le i\le n$, set $$\sigma _i=\hbox{card}\{j|s_j>i\}+1.$$ Let
the interval $J\subseteq R$ be defined by $$J=
\left\{\eqalign{ & [0,\infty ) \hbox{~~if~~}
(-1)^{\sigma _0}=1, \cr & (-\infty ,0]
\hbox{~~if~~} (-1)^{\sigma _0}=-1.\cr}\right.
$$ In this paper, we consider the existence ofpositive
solutions of the following two-point boundary value problem.$$
\left\{\eqalign{ (1.1)~~~~~&u^{(n+2)}(t)+ f(t,u)=0,
\quad 0<t<1,\cr (1.2)~~~~~ &\left\{\eqalign{
& u^{(r_i-1)}(0)=0,~~~~1\le i \le k \cr
& u^{(s_j-1)}(1)=0,~~~~ 1\le j \le n-k \cr
&\alpha u^{(n)}(0)-\beta u^{(n+1)}(0)=0, \cr
&\gamma u^{(n)}(1)+\delta u^{(n+1)}(1)=0, \cr}
\right. \cr
} \right.\leqno (BVP)$$ where\item {(i)}
$n$ is a nonnegative integer,\item {(ii)} $\alpha, \beta,
\gamma, \delta$ are nonnegative real constants with
$\rho=\gamma \beta+\alpha \gamma+\alpha \delta>0$,\item {(iii)}
$f\in C([0,1]\times J,[0,\infty))$.We define$$max f_0:=
\lim\limits_{{(-1)^{\sigma _0}}u \to 0^+} max_{t \in [0,1]}{f(t,
u)\over u},$$$$min f_0:=\lim\limits_{{(-1)^{\sigma _0}}u \to 0^+
} min_{t \in [0,1]}{f(t,u)\over u},$$$$max f_\infty:=
\lim\limits_{{(-1)^{\sigma _0}}u \to \infty} max_{t \in [0,1
]}{f(t,u)\over u}$$\noindent and$$min f_\infty:=
\lim\limits_{{(-1)^{\sigma _0}}u \to \infty} min_{t \in [0,1
]}{f(t,u)\over u}.$$\smallskipIn 1996, Agarwal and Henderson [1]
showed the following veryexcellent result.\par\bigskip{\bf
Theorem A.} ( Agarwal and Henderson [1])
Consider the following nonlinear boundary value problem
$$\left\{\eqalign{ &u^{(n+2)}(t)+a(t)g(
u)=0,~~ 0<t<1,\cr &\eqalign{
& u^{(r_i-1)}(0)=0,1\le i \le k, \cr
& u^{(s_j-1)}(1)=0, 1\le j \le n-k, \cr
&u^{(n)}(0)=0, u^{(n+1)}(1)=0,\hbox{~or~} \cr
&u^{(n)}(1)=0, u^{(n+1)}(0)=0, \cr}
\cr } \right.\leqno (BVP)_1$$
where\item{(a)} $\alpha,\beta,\gamma,\delta $ are nonnegative
real constants,\item{(b)} $g\in C(J,[0,\infty))$,\item{(c)}
$a\in C([0,1],[0,\infty))$ and does not vanish identically on
any subinterval.If $~~g_0:=\lim\limits_{(-1)^{\sigma
_0} u \to 0^+}{g(u)\over u}=0 \hbox {~~and~~} {(-1)^{\sigma _0}}
g_\infty:=(-1)^{\sigma _0}\lim\limits_{(-1)^ {\sigma _0}u \to
0^+}{g(u)\over u} \break = \infty ,\hbox{~~or~~}
(-1)^{\sigma _0}g_0=\infty \hbox{~~and~~}g_\infty =0,$then ${(
BVP)_1}$ has at least one positive solution.\bigskipThe purpose
of this note is to generalize Theorem A to the nonlinear
boundaryvalue problem $(BVP)$ even if $(-1)^{\sigma _0
}minf_0,(-1)^{\sigma _0}maxf_0, (-1)^{\sigma _0}$\break $
minf_\infty, (-1)^{\sigma _0}maxf_\infty \not \in \{0,\infty\}$
by combining the techniques of Agarwal andHenderson [1] and
Lian-Wong-Yeh [11].For other related results, we refer to [2],
[3], [4], [6], [7], [8], [9], [12], [13].\bigskip\end
|
|---|
