Relative bounded perturbation of abstract cauchy problem

Yu Hsien Chang, Cheng Hong Hong

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In this paper, we study the perturbed abstract Cauchy equation du(t) dt = (A + B)u(t) with initial condition u(0) = x, where A is a generator of a C-semigroup on a Banach space X and B is a relatively bounded linear operator on X. We show that if the perturbation operator B is an A-bounded linear operator which commutates with C and its Abound is sufficiently small, then (A + B) generates a C-semigroup {V(t)}t≥0 on X, and hence the perturbed abstract Cauchy problem has a unique mild solution as long as the initial data x is in the subspace [Im(C)]. It is remarkable that we can directly apply these results to some differential equations.

Original languageEnglish
Pages (from-to)555-575
Number of pages21
JournalFar East Journal of Mathematical Sciences
Volume29
Issue number3
Publication statusPublished - 2008 Jun 1

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Abstract Cauchy Problem
Bounded Linear Operator
Semigroup
Cauchy Equation
Perturbation
Mild Solution
Initial conditions
Subspace
Banach space
Generator
Differential equation
Operator

Keywords

  • C-semigroup
  • Perturbation
  • Relatively bounded

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Relative bounded perturbation of abstract cauchy problem. / Chang, Yu Hsien; Hong, Cheng Hong.

In: Far East Journal of Mathematical Sciences, Vol. 29, No. 3, 01.06.2008, p. 555-575.

Research output: Contribution to journalArticle

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