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

It is proved that the boundary-value problem -u″ + p 0(t)u(t) + ∑k=2m qk (t)u 2k+1(t) + f0(t)φ(u(t)) = f(t), 0 < t < 1, u(a) = 0, u′(b) = 0, has a solution, provided that the following conditions are fulfilled: |p0(t)|(t -a) ∈ L(a, b), f(t)√t - a ∈ L(a, b), 0 ≤ f0(t)√t -a ∈ L(a, b), 0 ≤ q k(t)(t - a)k+1 ∈ L(a, b), -c|u| ≤ φ(u)u, c > 0, 1 - ∫ab p0-(t)(t - a) dt > 0, 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. © 2007 Springer Science+Business Media, Inc.