Existence, uniqueness and limiting behavior of solutions of a class of differential equations in Banach space
John E. Lagnese · Pacific Journal of Mathematics · 1974
Let X be a Banach space (real or complex) and A n and B be linear operators in X with D{B) £ D(A n ), n~l, 2, •••.The following note is concerned with existence and uniqueness of solutions of the problem (1.1) -|-[(I -A n )u(t)] -Bu(t) = 0, (t > 0), u(0) = u 0 , at and the limiting behavior of these solutions as the operators A n tend to zero in a sense to be specified.We will show that for a large class of operators the problem (1.1) is well posed and that its solutions tend to the solution of the problem (1.2) -^j&--Bu(t) = 0, (t > 0), u(Q) = u 0 .