### 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 language | English |
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Pages (from-to) | 555-575 |

Number of pages | 21 |

Journal | Far East Journal of Mathematical Sciences |

Volume | 29 |

Issue number | 3 |

Publication status | Published - 2008 Jun 1 |

### Keywords

- C-semigroup
- Perturbation
- Relatively bounded

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Chang, Y. H., & Hong, C. H. (2008). Relative bounded perturbation of abstract cauchy problem.

*Far East Journal of Mathematical Sciences*,*29*(3), 555-575.