On the solvability of some nonlinear differential equations

The author proves some existence results for nonlinear operator equations in so-called quasisemimetric spaces and applies them to the solvability of some semilinear equations in function spaces. As examples of applicability of these results, he considers the solvability of some semilinear partial differential equations where the linear part is either an elliptic second-order operator, or the wave or the heat operator.