Solvability of nonlinear equations in a cone of a Banach space

The solvability conditions for the equation $Tu+F(u)=0$ are found in the case where the operator $[T+F'(u)]^{-1}$ exists only for $u\in K$, where $K$ is a cone in the Banach space $X$. An application concerning the solvability of boundary-value problems for a system of second-order differential equations is provided.