Solvability of semilinear differential equations with singularity

We prove local theorems on the existence of solutions of the Cauchy problem for singular equations of the form {fx262-01} in Banach spaces. Solvability conditions depend on the type of the singularity of the pencil λA + B of closed linear operators A and B . Examples and applications to finite-dimensional differential-algebraic equations, infinite systems of differential equations, and partial differential equations of the non-Kovalevskaya type are presented.