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Multi-point boundary value problems on an unbounded domain at resonance. (English) Zbl 1138.34006

Summary: We consider the second-order nonlinear differential equation
\[ (p(t)u'(t))'=f(t,u(t),u'(t))\text{ a.e. in }(0,\infty), \]
satisfying two sets of boundary conditions:
\[ u'(0)=0,\quad\sum^n_{i=1}\kappa_iu(T_i)=\lim_{t\to\infty}u(t) \]
and
\[ u(0)=0,\quad \sum^n_{i=1}\kappa_iu(T_i)=\lim_{t\to\infty} u(t), \]
where \(n\geq1\), \(f:[0,\infty)\times \mathbb R^2\to\mathbb R\) is Carathéodory with respect to \(L_1[0,\infty)\). The parameters in the multi-point boundary conditions are such that the corresponding differential operator is non-invertible but nevertheless is a Fredholm map of index zero. As a result the coincidence degree theory can be applied to establish existence theorems.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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References:

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