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In this paper we provide some results on Ulam stability for the linear difference equation of order one in Banach spaces and we determine its best Ulam constant. The main result is applied to a process of loan amortization.

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The linear differential operator with constant coefficients D(y)=y(n)+a1y(n−1)+…+any,y∈Cn(R,X) acting in a Banach space X is Ulam stable if and only if its characteristic equation has no roots on the imaginary axis. We prove that if the characteristic equation of D has distinct roots rk satisfying Rerk>0,1≤k≤n, then the best Ulam constant of D is KD=1|V|∫0∞|∑k=1n(−1)kVke−rkx|dx, where V=V(r1,r2,…,rn) and Vk=V(r1,…,rk−1,rk+1,…,rn),1≤k≤n, are Vandermonde determinants.