THE EXISTENCE OF A WEAK SOLUTION OF THE SEMILINEAR FIRST-ORDER DIFFERENTIAL EQUATION IN A BANACH SPACE

This paper is devoted to the investigation of the existence and uniqueness of a suitably defined weak solution of the abstract semilinear value problem u_ (t) = Au(t) + f(t; u(t)); u(0) = x with x 2 X; where X is a Banach space. We are concerned with two types of solutions: weak and mild. Under the assumption that A is the generator of a strongly continuous semigroup of linear, bounded operators, we also establish sufficient conditions such that if u is a weak (mild) solution of the initial value problem, then u is a mild (weak) solution of that problem