Integral and derivative in the banach space c0

Source abstract

Let A be the generator of a strongly continuous periodic one-parameter group of bounded linear operators in a Banach space H, and let Rλ be the resolvent of A at the point λ and Φ(A) is a polynomial of A with complex coefficients. We have obtained conditions for compactness be the operator Rλ. Using this result, conditions of the existence of a solution to the nonlinear equation Φ(A)x = f(x) are obtained, where f is a continuous mapping of H into itself.