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An application of Schauder's fixed point theorem with respect to higher order BVPs

Author: Fu-Hsiang Wong
Journal: Proc. Amer. Math. Soc. 126 (1998), 2389-2397
MSC (1991): Primary 34B15
MathSciNet review: 1476399
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Abstract: We shall provide conditions on the function $f(t,u_{1},\cdots , u_{n-1})$. The higher order boundary value problem

\begin{equation*}\begin{cases}(E)~~ u^{(n)}(t)+ f(t, u(t),u^{(1)}(t),\cdots ,u^{(n-2)}(t))=0~~~~~\hbox {~for~}~~~~~t\in (0,1)~~~~\hbox {and}~~~~~~n\ge 2,\\ (BC)~~ \begin{cases}u^{(i)}(0)=0,~~~~~0\le i \le n-3,\\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\\ \gamma u^{(n-2)}(1)+\delta u^{(n-1)}(1)=0\end{cases} \end{cases} \tag{{BVP}}\end{equation*}

has at least one solution.

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Additional Information

Fu-Hsiang Wong
Affiliation: Department of Mathematics and Science, National Taipei Teacher’s College, 134, Ho-Ping E. Rd. Sec. 2, Taipei 10659, Taiwan, Republic of China

Keywords: Higher order boundary value problems, solution, operator equation, Green's function, Schauder's fixed point theorem, upper-lower-solutions
Received by editor(s): January 22, 1997
Communicated by: Hal L. Smith
Article copyright: © Copyright 1998 American Mathematical Society