Asymptotic behavior of mild solutions for a class of abstract nonlinear difference equations of convolution type

Abstract

Abstract We prove the existence and uniqueness of a weighted pseudo asymptotically mild solution to the following class of abstract semilinear difference equations: u(n+1)=A∑k=−∞na(n−k)u(k+1)+∑k=−∞nb(n−k)f(k,u(k)),n∈Z, $$ u(n+1)= A \sum_{k=-\infty }^{n} a(n-k)u(k+1)+ \sum _{k=-\infty }^{n} b(n-k)f\bigl(k,u(k)\bigr),\quad n\in \mathbb{Z}, $$ where A is the generator of a resolvent sequence {S(n)}n∈N0 $\{S(n)\}_{n\in \mathbb{N}_{0}}$ of bounded and linear operators defined in a Banach space X, the sequences a,b $a, b$ are complex-valued, and f∈l1(Z×X,X) $f\in l^{1}( \mathbb{Z}\times X, X)$.

Keywords

• Weighted pseudo asymptotically mild solutions
• Abstract difference equations
• Resolvent sequences of operators