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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:
\begin{aligned} &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}, \end{aligned} where $$n\geq 3$$ and $$m\geq 1$$ are integers, $$\lambda\in \mathbb{R}$$ is a parameter, $$f\in C([0,1]\times \mathbb{R}^n,\mathbb R)$$, $$p\in\mathbb C([0,1],\mathbb R)$$ 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.
Reviewer: Minghe Pei (Jilin)

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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