On a method of constructing a solution of a nonlinear resonance boundary problem (Q1177766)

The paper deals with solvability of the nonlinear BVP (*) \(x'=A(t)x+f(t,x,x')\), \(\ell(x)=0\), where \(\ell: C([0,\omega],\mathbb{R}^ n)\to\mathbb{R}^ n\) is a linear continuous operator. Using some results concerning the operator equation \({\mathcal A}x=Fx\), where \({\mathcal A}\) is a linear Fredholm operator, the conditions on the nonlinearity \(f\) and the operator \(\ell\) are given which guarantee that a certain iterative sequence \(x_ m(t)\in C^ 1([0,\omega],\mathbb{R}^ n)\) converges to a solution \(x(t)\) of (*). The estimate for the speed of this convergence in the \(C^ 1[0,\omega]\)-norm is given.