A Landau–Kolmogorov inequality for generators of families of bounded operators

Abstract

differential equations A Landau-Kolmogorov type inequality for generators of a wide class of strongly continuous families of bounded and linear operators defined on a Banach space is shown. Our approach allows us to recover (in a unified way) known results about uniformly bounded C 0 -semigroups and cosine functions as well as to prove new results for other families of operators. In particular, if A is the generator of an α-times integrated family of bounded and linear operators arising from the well-posedness of fractional differential equations of order β + 1 then, we prove that the inequalityholds for all x ∈ D( A 2 ).

On Landau-Kolmogorov inequalities for dissipative operators

“…There are many papers on the inequalities of the form (1.1). We refer the reader to [1][2][3][4][5][6][7][8][9][10][11][12][13][14], for instance.…”