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Existence of solutions for a higher order multi-point boundary value problem. (English) Zbl 1173.34008
The authors study the $n$-th order multi-point boundary value problem $$u^{(n)}(t)+f(t,u(t),u'(t),\dots,u^{(n-1)}(t))=\lambda p(t),\quad t\in(0,1)$$ with the following multi-point boundary conditions: $$\align &u^{(i)}(0)=A_i,i=0,1,\dots,n-3,\\ &u^{(n-2)}(0)-\sum_{j=1}^m a_ju^{(n-2)}(t_j)=A_{n-2},\\ &u^{(n-2)}(1)-\sum_{j=1}^m b_ju^{(n-2)}(t_j)=A_{n-1}, \endalign$$ where $n\geq 3$ and $m\geq 1$ are integers, $\lambda\in \mathbb{R}$ is a parameter, $f\in C([0,1]\times \mathbb{R}^n,\Bbb R)$, $p\in\Bbb C([0,1],\Bbb R)$ with $p(t)>0$ on $[0,1]$, $A_i\in \mathbb{R}$ for $i=0,1,\dots,n-1$,$a_j,b_j\in \mathbb{R}^+:=[0,\infty)$ for $j=1,\dots,m$. By using the lower and upper solution method and topological degree theory, sufficient conditions are obtained for the existence of one and two solutions of the problem for different values of $\lambda$. The result obtained extend and improve some recent work in the literature.

34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
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