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Solvability for two classes of higher-order multi-point boundary value problems at resonance. (English) Zbl 1149.34008

Summary: Using the theory of coincidence degree, we establish existence results of positive solutions for higher-order multi-point boundary value problems at resonance for ordinary differential equation
\[ u(n)(t)=f(t,u(t),u'(t),\dots,u^{(n-1)}(t))+e(t),\quad t\in (0,1), \]
with one of the following boundary conditions:
\[ u^{(i)}(0)=0,\quad i=1,2,\dots, n-2, u^{(n-1)}(0)=u^{(n-1)}(\xi), u^{(n-2)}(1)=\sum_{j=1}^{m-2}\beta_ju^{(n-2)}(\eta_j), \] and
\[ u(i)(0)=0,\quad i=1,2,\dots, n-1,\quad u^{(n-2)}(1)=\sum_{j=1}^{m-2}\beta_ju^{(n-2)}(\eta_j), \]
where \(f:[0,1]\times\mathbb R^n\to \mathbb =(-\infty,+\infty)\) is a continuous function, \(e(t)\in L^1[0,1]\beta_j\in\mathbb R\) \((1\leq j\leq m-2, m\geq 4)\), \(0<\eta 1<\eta^2<\cdots <\eta_{m-2}<1\), \(0<\xi<1,\) all the \(\beta_{-j}^{-s}\) have not the same sign. We also give some examples to demonstrate our results.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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