Semilinear evolution equations of second order via maximal regularity (Q937544)

Let \(X\) be a complex Banach space and assume that \(A:X\rightarrow X\) is a bounded linear operator. This paper is concerned with existence and stability results for the semilinear discrete problem \[ \Delta^2x_n-Ax_n=f(n,x_n,\Delta x_n),\quad n\geq 1, \] where \(x_0=x_1=0\) and \(f:{\mathbb Z}_+\times X\times X\rightarrow X\) is a Lipschitz function such that \(f(\cdot ,0,0)\in l_1({\mathbb Z}_+,X)\). The main result of the present paper establishes the existence of bounded solutions whose second discrete derivative is in \(l_p\) (\(1<p<\infty\)). The proof combines the Banach contraction principle with maximal regularity arguments. By means of the discrete Gronwall inequality, the authors also deduce \textit{a priori} estimates for the solutions \(x_n\) and their discrete derivatives \(\Delta x_n\) and \(\Delta^2x_n\).