## 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.

### MSC:

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