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Second-order boundary value problems with nonhomogeneous boundary conditions. II. (English) Zbl 1119.34009

Summary: Sufficient conditions are given for the existence of solutions of the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition \[ u''+f(t,u,u')=0,\quad t\in(0,1), \]
\[ u(0)-\sum^m_{i=1}a_iu(t_i)= \lambda_1,\quad u(1)-\sum^m_{i=1}b_iu(t_i) =\lambda_2. \] We prove that the whole plane \(\mathbb{R}^2\) is divided by a “continuous decreasing curve” \(\Gamma\) into two disjoint connected regions \(\Lambda^E\) and \(\Lambda^N\) such that the above problem has at least one solution for \((\lambda_1,\lambda_2)\in\Gamma\), has at least two solutions for \((\lambda_1,\lambda_2)\in\Lambda^E\setminus \Gamma\), and has no solution for \((\lambda_1,\lambda_2)\in \Lambda^N\). We also find explicit sub-regions of \(\Lambda^E\) where the above problem has at least two solutions and two positive solutions, respectively.
Part I, cf. Math. Nachr. 278, No. 1–2, 173–193 (2005; Zbl 1060.34005).

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 1060.34005
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References:

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