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

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.