On Ulam Stability of a Partial Differential Operator in Banach Spaces

Abstract

In this paper, we prove that, if infx∈A|f(x)|=m>0, then the partial differential operator D defined by D(u)=∑k=1nfk∂u∂xk−fu, where f,fi∈C(A,R),u∈C1(A,X),i=1,…,n,I⊂R is an interval, A=I×Rn−1 and X is a Banach space, is Ulam stable with the Ulam constant K=1m. Moreover, if infx∈A|f(x)|=0, we prove that D is not generally Ulam stable.