Existence of multiple positive solutions of higher order multi-point nonhomogeneous boundary value problem

Dapeng Xie, Hefei Normal University, Hefei, Anhui, P. R. China
Yang Liu, Hefei Normal University, Hefei, Anhui, P. R. China
Chuanzhi Bai, Huaiyin Normal University, Huaian, Jiangsu, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 33. (2010), pp. 1-13.

Abstract: In this paper, by using the Avery and Peterson fixed point theorem, we establish the existence of multiple positive solutions for the following higher order multi-point nonhomogeneous boundary value problem
$ u^{(n)}(t) + f(t,u(t),u'(t),\ldots,u^{(n-2)}(t)) = 0, t\in (0,1)$,
$ u(0)= u'(0)=\cdots=u^{(n-3)}(0)=u^{(n-2)}(0)=0, u^{(n-2)}(1)-\sum_{i=1}^{m} a_i u^{(n-2)}(\xi_i)=\lambda$,
where $n\ge3$ and $m\ge1$ are integers, $0<\xi_1<\xi_2<\cdots<\xi_m<1$ are constants, $\lambda\in [0,\infty)$ is a parameter, $a_i>0$ for $1\le i\le m$ and $\sum_{i=1}^{m} a_i\xi_i<1$, $f(t,u,u',\cdots,u^{(n-2)})\in C([0,1]\times[0,\infty)^{n-1}, [0,\infty))$. We give an example to illustrate our result.

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