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)


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