Oscillatory solutions of boundary value problems

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Abstract

We consider boundary value problems of the form xʺ = f (t, x, xʹ), x(a) = A, x(b) = B, assuming that f is continuous together with fx and fxʹ. We study also equations in a quasi-linear form xʺ + p(t)xʹ + q(t)x = F(t, x, xʹ). Introducing types of solutions of boundary value problems as an oscillatory type of the respective equation of variations, we show that for a solution of definite type, the problem can be reformulated in a quasi-linear form. Resonant problems are considered separately. Any resonant problem that has no solutions of indefinite type is in fact nonresonant. The ways of how to detect solutions of definite types are discussed.

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Sadyrbaev, F. (2016). Oscillatory solutions of boundary value problems. In Springer Proceedings in Mathematics and Statistics (Vol. 164, pp. 109–117). Springer New York LLC. https://doi.org/10.1007/978-3-319-32857-7_11

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