Perturbing the boundary conditions of the generator of a cosine family

Abstract

Let A be a densely defined, closed linear operator (which we shall call maximal operator) with domain D(A) on a Banach space X and consider closed linear operators L:D(A)→∂X and Φ:D(A)→∂X (where ∂X is another Banach space called boundary space). Putting conditions on L and Φ, we show that the second order abstract Cauchy problem for the operator A Φ with A Φu=Au and domain D(A Φ):={u∈D(A):Lu=Φu} is well-posed and thus it generates a cosine operator function on the Banach space X. © 2011 Springer Science+Business Media, LLC.