### 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 |
---|---|

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 |

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### Keywords

- C-semigroup
- Perturbation
- Relatively bounded

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Far East Journal of Mathematical Sciences*,

*29*(3), 555-575.

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

Research output: Contribution to journal › Article

*Far East Journal of Mathematical Sciences*, vol. 29, no. 3, pp. 555-575.

}

TY - JOUR

T1 - Relative bounded perturbation of abstract cauchy problem

AU - Chang, Yu Hsien

AU - Hong, Cheng Hong

PY - 2008/6/1

Y1 - 2008/6/1

N2 - 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.

AB - 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.

KW - C-semigroup

KW - Perturbation

KW - Relatively bounded

UR - http://www.scopus.com/inward/record.url?scp=84891651729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84891651729&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84891651729

VL - 29

SP - 555

EP - 575

JO - Far East Journal of Mathematical Sciences

JF - Far East Journal of Mathematical Sciences

SN - 0972-0871

IS - 3

ER -