Solvability of an \(m\)-point boundary value problem for second order ordinary differential equations. (English) Zbl 0819.34012

The paper deals with the \(m\)-point boundary value problem (1) \(x''(t) = f(t,x(t), x'(t)) + e(t)\), \(t \in (0,1)\), \(x(1) = \sum^{m-2}_{i=1} a_ i x(\xi_ i)\), \(x'(0) = 0\), where \(f\) is a Carathéodory function, \(e \in L(0,1)\), \(a_ i \in \mathbb{R}\) have the same sign, \(0 < \xi_ 1 < \cdots < \xi_{m-2} < 1\). The authors transform the above problem onto the auxiliary three-point problem with the same equation and the boundary conditions (2) \(x'(0) = 0\), \(x(1) = \alpha x (\eta)\), where \(\alpha = \sum^{m-2}_{i=1} a_ i\), \(\eta \in [\xi_ 1, \xi_{m-2}]\). Then they find a common a priori bound for all solutions of \(x''(t) = \lambda f(t,x(t), x'(t)) + \lambda e(t)\), (2), which is independent on \(\eta\) and \(\lambda \in [0,1]\). Finally, using the Mawhin’s continuation theorem, they prove the existence of solutions of the auxiliary three-point problem which implies the existence of a solution of problem (1). The existence results are proved provided \(f\) satisfies a linear growth conditions with respect to the phase variables with sufficiently small coefficients.


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