On the peano theorem in infinite-dimensional spaces

New sufficient conditions for a first-order ordinary differential equation in an infinite Banach space to be locally solvable and for the solution to depend continuously on a parameter are obtained. The results are applied to an integrodifferential equation.Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 569–576, May, 1972.The author is grateful to B. N. Sadovskii for posing the problem and for his continuous interest in the work.