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We study the solvability of a two-point nonlocal boundary-value problem for an operatordifferential equation with weakly nonlinear right-hand side. The proof of theorems is carried out within the Nash-Moser iterative scheme. In this scheme, the important point is a construction of estimates to norms of inverse linearized operators arising at each step of this scheme as well as the related problem of small denominators. The inverse operator for linearized operator is obtained by the method developed in the work of M. Berti, P. Bolle (Duke Math. J., 134 (2), 359–419 (2006)). The problem of small denominators is solved by using the metric approach.

Two-point nonlocal problem for a weak nonlinear differential-operator equation