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Mixed boundary-value problems for singular second-order ordinary differential equations

Authors:

M N Yakovlev

Institution:

(1) St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia

Abstract:

It is proved that the boundary-value problem

$\begin{gathered} - u' + p_0 (t)u(t) + \sum\limits_{k = 2}^m {q_k (t)u^{2k + 1} (t) + f_0 (t)_\varphi (u(t)) = f(t), 0 < t < 1,} \hfill \\ u(a) = 0, u'(b) = 0 \hfill \\ \end{gathered}$

, has a solution, provided that the following conditions are fulfilled:

$\begin{gathered} \left| {p_0 (t)} \right|(t - a) \in L(a,b), f(t)\sqrt {t - a} \in L(a,b), 0 \leqslant f_0 (t)\sqrt {t - a} \in L(a,b), 0 \leqslant q_k (t)(t - a)^{k + 1} \in L(a,b), \hfill \\ - c\left| u \right| \leqslant \varphi (u)u, c > 0, 1 - \int\limits_a^b {p_0^ - (t)(t - a)dt > 0} \hfill \\ \end{gathered}$

, and, for ϕ(u) ≡ 0, the Galerkin method converges in the norm of the space H¹(a, b; a). Several theorems of a similar kind are presented. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 246–266.

Keywords:

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