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On a two-point boundary value problem for second-order linear differential equations with coefficients that take values on a given set. (Russian) Zbl 1005.34008

The author studies a two-point boundary value problem for a second-order linear differential equation of the type: \[ x'' + u_{1}(t) x + u_{2}x = f(t), \tag{1} \]
\[ x(a)\sin \alpha x(a)\cos \alpha = A,\qquad x(b)\sin \beta x(b)\cos \beta = B, \tag{2} \] where \(u_{1}, u_{2}, f(t)\) are summable functions on the segment \([a, b],\) \(-\infty < a < b < \infty\), \(A, B, \alpha\) and \(\beta\) are given numbers. Moreover, \[ -\frac{\pi}{2}\leq \beta < \frac{\pi}{2},\qquad -\frac{\pi}{2}< \alpha \leq \frac{\pi}{2}, \qquad \beta < \alpha. \tag{3} \]
Using methods of convex analysis, the length of the maximum segment is estimated, in which any two-point boundary value problem for the linear differential equation of second order with bounded coefficients has a unique solution.

MSC:

34B05 Linear boundary value problems for ordinary differential equations
49J15 Existence theories for optimal control problems involving ordinary differential equations
34H05 Control problems involving ordinary differential equations
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