Existence of mild solutions for the semilinear nonlocal problem in Banach spaces

In this paper we study the existence of mild solutions of the semilinear nonlocal initial value problem on a bounded and on an unbounded interval. In our considerations we apply a new calculation method based on the technique of modulus of continuity in conjunction with idea of the family of measures of noncompactness in Fréchet space. The results obtained in this paper generalize several ones obtained earlier by many authors.