Variation of constants formulae for difference equations with advanced arguments (Q5926159)

A vector-valued difference equation with advanced arguments NEWLINE\[NEWLINEy(n+1)= A(n)y(n)+ B(n)y(g(n))+ f(n), \quad g(n)\geq n+1,NEWLINE\]NEWLINE is considered. Under certain assumptions, variation of constants formulas in the form \(y(n)= \Phi(n) \sum_{k=0}^\infty C_k(f)(n)\) are constructed for different sequential spaces where \(\Phi(n)\) denotes the fundamental matrix of the equation \(y(n+1)= A(n)y(n)\) and \(\{C_k\}\) is a sequence of bounded functionals.