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A third-order \(m\)-point boundary-value problem of Dirichlet type involving a \(p\)-Laplacian type operator. (English) Zbl 1172.34008

Summary: Let \(\varphi\) be an odd increasing homeomorphisms from \(\mathbb R\) onto \(\mathbb R\) satisfying \(\varphi (0)=0\), and let \(f:[0,1]\times \mathbb R\times\mathbb R\times\mathbb R\mapsto\mathbb R\) be a function satisfying Carathéodory’s conditions. Let \(\alpha_i\in\mathbb R\), \(\xi_i\in (0,1)\), \(i=1,\dots ,m-2\), \(0<\xi _1<\xi_2<\cdots <\xi _{m-2}<1\) be given. We are interested in the existence of solutions for the \(m\)-point boundary-value problem:
\[ (\varphi (u''))'=f(t,u,u',u''), \quad t\in (0,1), \]
\[ u(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha_iu(\xi_i), \quad u''(0)=0, \]
in the resonance and non-resonance cases. We say that this problem is at resonance if the associated problem
\[ \varphi (u''))'=0, \quad t\in (0,1), \]
with the above boundary conditions has a non-trivial solution. This is the case if and only if \(\sum_{i=1}^{m-2}\alpha_i\xi_i=1\). Our results use topological degree methods. In the non-resonance case; i.e., when \(\sum_{i=1}^{m-2}\alpha_i\xi_i\neq 1\) we note that the sign of degree for the relevant operator depends on the sign of \(\sum_{i=1}^{m-2}\alpha_i\xi_i-1\).

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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