\((N, \lambda)\)-periodic solutions to abstract difference equations of convolution type (Q6612217)

The authors study the existence of a class of solutions, defined \((N,\lambda)\)-periodic in [\textit{E. Alvarez} et al., Mediterr. J. Math. 19, No. 1, Paper No. 47, 15 p. (2022; Zbl 1484.39010); Adv. Difference Equ. 2019, Paper No. 105, 12 p. (2019; Zbl 1459.39027)], to equations of the form \N\[\Nw(n+1,x)= A\sum_{r=-\infty}^n a(n-r)w(r+1,x)+\sum_{r=-\infty}^n b(n-r)h(r,w(r,x)), \N\]\Nwhere \(n\) is an integer, \(x\in X\), \(X\) is a Banach space, \(A\) is a closed linear operator in \(X\), \(a,b\) are sequences of positive reals and \(h\) is a function of \(n\) and \(x\). Sufficient conditions on the existence and uniqueness of solutions of the above equations are found in the linear and semilinear cases. The conditions are stated in terms of the resolvent family generated by the linear operator \(A\) and some properties related to the \((N,\lambda)\)-periodicity of the resolvent family.