The aim of the paper is to give a theorem about the existence and uniqueness of a solution of the following nonlocal abstract Cauchy problem in a Banach space:
where , , , , , , , and , where are Banach spaces with norms . The Banach theorem about the fixed point is used to prove the existence and uniqueness of a solution of the problem considered. The results obtained can be applied among other things to the description of motion phenomena with better effect than the classical Cauchy problem. They are a continuation of those given by the first author [Z. Angew. Math. Mech. 70, 3, 202-206 (1990); J. Appl. Math. Stochastic Anal. 3, No.3, 65-79 (1990); J. Math. Anal. Appl. (to appear) (1990); J. Appl. Math. Stochastic Anal. (to appear) (1990); Appl. Anal. (to appear) (1990)] and generalize the known theorem about the existence and uniqueness of the solution considered by the second author and S. Leela [Nonlinear Differential Equations in Abstract Spaces (1981; Zbl 0456.34002)] and by W. Kołodziej [Mathematical Analysis (1970; Zbl 0209.362)].