Multiplicative perturbations of incomplete second order abstract differential equations

Abstract

PurposeTo study the multiplicative perturbation of local C‐regularized cosine functions associated with the following incomplete second order abstract differential equations in a Banach space X u″(t)=A(I+B)u(t), u(0)=x, u′(0)=y,(*) where A is a closed linear operator on X and B is a bounded linear operator on X.Design/methodology/approachThe multiplicative perturbation of exponentially bounded regularized C‐cosine functions is generally studied by the Laplace transformation. However, C‐cosine functions might not be exponentially bounded, so that the new method for the multiplicative perturbation of the nonexponentially bounded regularized C‐cosine functions should be applied. In this paper, the property of regularized C‐cosine functions is directly used to obtain the desired results.FindingsThe new results of the multiplicative perturbations of the nonexponentially bounded C‐cosine functions are obtained.Originality/valueThe new techniques differing from those given previously in the literature are employed to deduce the desired conclusions. The results can be applied to deal with incomplete second order abstract differential equations which stem from cybernetics, engineering, physics, etc.