Semilinear evolution equations on discrete time and maximal regularity (Q1036174)

In the present paper, the authors study the existence of bounded solutions and stability for semilinear equations \[ \Delta x_n-Ax_n=f(n,x_n),\;n\in {\mathbb Z}_+, \] by means of the maximal regularity properties for the vector-valued discrete time evolution equation \[ \Delta x_n-Ax_n=f_n,\;n\in {\mathbb Z}_+, \] with initial condition \(x_0=0\). Here, \(A\) is a bounded linear operator defined on a complex Banach space \(X\). First of all, the authors prove a very general theorem on the existence of bounded solutions for the semilinear problem, whose first discrete derivatives are in \(l_p({\mathbb Z}_+;X)\) spaces. Then, they give a general stability criterion. Finally, as an application, they examine the asymptotic behavior of discrete control systems.