Solvability of an \(m\)-point boundary value problem for second order ordinary differential equations
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.
scientific article; zbMATH DE number 743918