Multiple positive solutions for a nonlinear 2n-th order m-point boundary value problems

Youyu Wang, Tianjin University of Finance and Economics, Tianjin, P. R. China
Yantao Tang, Tianjin University of Finance and Economics, Tianjin, P. R. China
Meng Zhao, Tianjin University of Finance and Economics, Tianjin, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 39. (2010), pp. 1-13.

Abstract: In this paper, we consider the existence of multiple positive solutions for the 2n-th order $m$-point boundary value problems:
$$\left\{\begin{array}{ll} x^{(2n)}(t)=f(t,x(t),x^{''}(t),\cdots ,x^{(2(n-1))}(t)), 0\leq t\leq 1,\\
x^{(2i+1)}(0)=\sum\limits_{j=1}^{m-2}\alpha_{ij}x^{(2i+1)}(\xi_j),\quad
x^{(2i)}(1)=\sum\limits_{j=1}^{m-2}\beta_{ij}x^{(2i)}(\xi_j), 0\leq i\leq n-1,\\
\end{array}\right.$$
where $\alpha_{ij}, \beta_{ij} \ (0\leq i\leq n-1,1\leq j\leq m-2) \in [0,\infty)$, $\sum\limits_{j=1}^{m-2}\alpha_{ij},\sum\limits_{j=1}^{m-2}\beta_{ij}\in (0,1)$, $0<\xi_1<\xi_2<\ldots<\xi_{m-2}<1$. Using Leggett-Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem.

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