Linearized stability analysis of discrete Volterra equations

AbstractVarious different types of stability are defined, in a unified framework, for discrete Volterra equations of the type x(n)=f(n)+∑nj=0K(n,j,x(n)) (n⩾0). Under appropriate assumptions, stability results are obtainable from those valid in the linear case (K(n,j,x(n))=B(n,j)x(j)), and a linearized stability theory is studied here by using the fundamental and resolvent matrices. Several necessary and sufficient conditions for stability are obtained for solutions of the linear equation by considering the equations in various choices of Banach space B, the elements of which are sequences of vectors (x(n),f(n)∈Ed, B(n,j):Ed→Ed, n,j⩾0, etc.). We show that the theory, including a number of new results as well as results already known, can be presented in a systematic framework, in which results parallel corresponding results for classical Volterra integral equations of the second kind