Existence, uniqueness and approximation of classical solutions to nonlinear two-point boundary value problems (Q1612616)

Sufficient conditions are obtained for the existence and uniqueness of classical solutions to the Dirichlet boundary value problem \[ u''(x)= \lambda u(x)+ f(u)+ g(x),\quad u(0)= u(\pi)= 0. \] Here, the function \(f\) may be unbounded. Besides that, solutions to the problem \[ -u''(x)= b(u,x),\quad u(0)= u(\pi)= 0, \] where the right-hand side \(b\) is bounded and smooth, are approximated by solutions to an initial boundary value problem for the parabolic equation \(v_t- v_{xx}= b(v,x)\). Examples are given.