, Volume 149, Issue 5, pp 1504-1516

On the solvability of a singular boundary-value problem for the equation f(t, x, x′, x″) = 0

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Abstract

In this work, we consider boundary-value problems of the form $$f(t, x, x', x'') = 0, 0 < t < 1, x(0) = 0, x'(1) = b, b > 0$$ , where the scalar function f(t, x, p, q) may be singular at x = 0. As far as we know, the solvability of the singular boundary-value problems of this form has not been treated yet. Here we try to fill in this gap. Examples illustrating our main result are included.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 16, Differential and Functional Differential Equations. Part 2, 2006.