Some boundary value problems for nonlinear functional-differential equations (Q1603157)

The author deals with boundary value problems of the form \[ u^{(n)}(t)=f(u)(t),\quad \ell_i(u)=0,\;i=1,\dots,n-1,\quad u^{(n-1)}(t_0)=\alpha(u), \] where \(n\geq 2\), \(f\) is an operator acting from some subspace of the space of \(n-1\) times continuously differentiable functions on the interval \([a,b]\) into the space of integrable functions on the same interval, \(t_0\in[a,b]\), and \(\ell_i\), \(i=1,\dots,n-1\), and \(\alpha\) are some functionals. Effective sufficient conditions for the solvability and unique solvability of the considered problem are established. Note that the obtained results are sharp and that they cannot be improved in a sense, as shown in examples constructed at the end of the paper.