Application of the infinite matrix theory to the solvability of a system of differential equations

In this paper we deal with the solvability of the infinite system of differential equations x′(t)=Δ(λ)x(t)+b with x(0)=a, where Δ(λ) is the triangle defined by the infinite matrix whose the nonzero entries are [Δ(λ)]nn=λn and [Δ(λ)]n,n−1=λn−1 for all n∈N, for a given sequence λ and a,b are two given infinite column matrices. We use a new method based on Laplace transformations to solve this system.