%0 Journal Article
%T THE EXISTENCE OF A WEAK SOLUTION OF THE SEMILINEAR
	FIRST-ORDER DIFFERENTIAL EQUATION IN A BANACH
	SPACE
%A Jużyniec, Mariusz
%J Czasopismo Techniczne
%V 2014
%R 10.4467/2353737XCT.14.300.3388
%N Nauki Podstawowe Zeszyt 2 NP (16) 2014
%P 59-62
%K operator, semigroup, weak solution
%@ 0011-4561
%D 2015
%U https://ejournals.eu/czasopismo/czasopismo-techniczne/artykul/the-existence-of-a-weak-solution-of-the-semilinear-first-order-differential-equation-in-a-banach-space
%X 
	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.