On solvability conditions for nonlinear operator equations

New sufficient conditions are found for solvability and unique solvability of nonlinear operator equations in the Banach space. In particular, abstract analogues of the Conti–Opial-type theorems are established, which concern the solvability of nonlinear boundary value problems. On the basis of these results, new sufficient conditions are obtained for the solvability of a periodic problem at resonance for nonlinear higher-order functional differential equations.