## C. Chilarescu – A. Pogan – C. Preda ∗ A CHARACTERIZATION OF THE EXPONENTIAL STABILITY OF EVOLUTIONARY PROCESSES IN TERMS OF THE

### BibTeX

@MISC{Of_c.chilarescu,

author = {Admissibilty Of and Orlicz Spaces},

title = {C. Chilarescu – A. Pogan – C. Preda ∗ A CHARACTERIZATION OF THE EXPONENTIAL STABILITY OF EVOLUTIONARY PROCESSES IN TERMS OF THE},

year = {}

}

### OpenURL

### Abstract

Abstract. A characterization of the exponential stability of evolutionary rocesses in terms of the admissibility of some pairs of spaces, is given. The method of ”test functions ” from a very large class of spaces is used. Thus are obtained generalizations of some results given by N. van Minh, F. Rabiger and R. Schnaubelt. 1.

### Citations

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Citation Context ...nential stability and exponential dichotomy for the case of exponentially bounded and strongly continuous evolution families were obtained by N. van Minh ([11], [12]), F. Räbiger [11] , Y. Latushkin (=-=[2]-=-, [6], [7], [8]), M. Megan ([10], [15]), P. Preda ([15], [16], [17]), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], [18]). In the spirit of Perron’s idea, the aim of this paper is to propose a ver... |

25 | Evolutionary semigroups and Lyapunov theorems in Banach spaces
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Citation Context ...al stability and exponential dichotomy for the case of exponentially bounded and strongly continuous evolution families were obtained by N. van Minh ([11], [12]), F. Räbiger [11] , Y. Latushkin ([2], =-=[6]-=-, [7], [8]), M. Megan ([10], [15]), P. Preda ([15], [16], [17]), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], [18]). In the spirit of Perron’s idea, the aim of this paper is to propose a very gen... |

16 | The asymptotic behaviour of semigroups of linear operators, volume 88 of Operator Theory: Advances and Applications
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Citation Context ...volutionary processes and the admissibility of certain function spaces, using a well-known class of function spaces which are translation invariant and also rearrangement invariant( see for instance (=-=[13]-=-). Theses type of spaces are called in the literature Orlicz spaces. Related to the subject above, we note that until now the most common classes of spaces used as ”input” and ”output” spaces were the... |

15 |
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Citation Context ...rical Engineering of UCLA, U.S.A. Research suported in part under NSF Grant no. ECS-0400730. 169170 C. Chilarescu – A. Pogan – C. Preda to the paper ”Die stabilitätsfrage bei differentialgeighungen” =-=[14]-=-,where Perron gave a characterization of exponential stability of the solutions to the linear differential equations dx = A(t)x, t ∈ [0,+∞), x ∈ Rn dt where A(t) is a matrix bounded continuous functio... |

13 | Sufficient conditions for exponential stability and dichotomy of evolution equations
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Citation Context ...s were obtained by N. van Minh ([11], [12]), F. Räbiger [11] , Y. Latushkin ([2], [6], [7], [8]), M. Megan ([10], [15]), P. Preda ([15], [16], [17]), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], =-=[18]-=-). In the spirit of Perron’s idea, the aim of this paper is to propose a very general and useful approach for the subject concerning the connection between the exponential stability of evolutionary pr... |

12 | Dichotomy of differential equations on Banach spaces and an algebra of weighted translation operators
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Citation Context ...ability and exponential dichotomy for the case of exponentially bounded and strongly continuous evolution families were obtained by N. van Minh ([11], [12]), F. Räbiger [11] , Y. Latushkin ([2], [6], =-=[7]-=-, [8]), M. Megan ([10], [15]), P. Preda ([15], [16], [17]), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], [18]). In the spirit of Perron’s idea, the aim of this paper is to propose a very general ... |

12 | Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line, Int. Eqns. Oper. Theory 32
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Citation Context ...In last few years, several results about exponential stability and exponential dichotomy for the case of exponentially bounded and strongly continuous evolution families were obtained by N. van Minh (=-=[11]-=-, [12]), F. Räbiger [11] , Y. Latushkin ([2], [6], [7], [8]), M. Megan ([10], [15]), P. Preda ([15], [16], [17]), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], [18]). In the spirit of Perron’s ide... |

9 |
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Citation Context ...ty and exponential dichotomy for the case of exponentially bounded and strongly continuous evolution families were obtained by N. van Minh ([11], [12]), F. Räbiger [11] , Y. Latushkin ([2], [6], [7], =-=[8]-=-), M. Megan ([10], [15]), P. Preda ([15], [16], [17]), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], [18]). In the spirit of Perron’s idea, the aim of this paper is to propose a very general and u... |

3 | On an application of a Banach-Steinhaus theorem to the study of the boundedness of solutions of nonlinear differential equations - Bellman |

3 |
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Citation Context ...al dichotomy for the case of exponentially bounded and strongly continuous evolution families were obtained by N. van Minh ([11], [12]), F. Räbiger [11] , Y. Latushkin ([2], [6], [7], [8]), M. Megan (=-=[10]-=-, [15]), P. Preda ([15], [16], [17]), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], [18]). In the spirit of Perron’s idea, the aim of this paper is to propose a very general and useful approach fo... |

3 |
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Citation Context ...t few years, several results about exponential stability and exponential dichotomy for the case of exponentially bounded and strongly continuous evolution families were obtained by N. van Minh ([11], =-=[12]-=-), F. Räbiger [11] , Y. Latushkin ([2], [6], [7], [8]), M. Megan ([10], [15]), P. Preda ([15], [16], [17]), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], [18]). In the spirit of Perron’s idea, the... |

3 |
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Citation Context ...hotomy for the case of exponentially bounded and strongly continuous evolution families were obtained by N. van Minh ([11], [12]), F. Räbiger [11] , Y. Latushkin ([2], [6], [7], [8]), M. Megan ([10], =-=[15]-=-), P. Preda ([15], [16], [17]), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], [18]). In the spirit of Perron’s idea, the aim of this paper is to propose a very general and useful approach for the ... |

3 |
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Citation Context ...exponentially bounded and strongly continuous evolution families were obtained by N. van Minh ([11], [12]), F. Räbiger [11] , Y. Latushkin ([2], [6], [7], [8]), M. Megan ([10], [15]), P. Preda ([15], =-=[16]-=-, [17]), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], [18]). In the spirit of Perron’s idea, the aim of this paper is to propose a very general and useful approach for the subject concerning the ... |

2 |
AND SCHÄFFER J.J., Linear differential equations and function spaces
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Citation Context ...der the inhomogeneous Cauchy problem: (3) x ′ (t) = A(t)x(t) + f (t), x(0) = 0, t ≥ 0 The function u f given by 1 is called a mild solution of 3. In some well-known particular cases (see for instance =-=[9]-=-) u f is the unique classical solution of 3. This type of characterization of exponential stability for 2 has a fairly long history that goes back ∗ This research was done when the last author was Pos... |

2 | Exponential dichotomy of evolutionary processes in Banach spaces
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- 1985
(Show Context)
Citation Context ...ntially bounded and strongly continuous evolution families were obtained by N. van Minh ([11], [12]), F. Räbiger [11] , Y. Latushkin ([2], [6], [7], [8]), M. Megan ([10], [15]), P. Preda ([15], [16], =-=[17]-=-), T. Randolph ([7], [8]), R. Schnaubelt ([8], [11], [18]). In the spirit of Perron’s idea, the aim of this paper is to propose a very general and useful approach for the subject concerning the connec... |

1 | Dichotomies in stability theory, Lect - COPPEL - 1978 |

1 |
AMS Subject Classification: 34D05,47D06, 93D20. Constantin CHILARESCU, Department of Mathematics, West University of Timisoara, 4 Parvan Bulevard, 300223 Timisoara, ROMANIA e-mail: cchilarescu@rectorat.uvt.ro Alin POGAN
- ZAANEN, Integration, et al.
- 1967
(Show Context)
Citation Context ... ‖ f ‖E(X) = ‖ ‖ f (·)‖ ‖E REMARK 4. If { fn}n∈N ⊂ E(X), f ∈ E(X), and fn → f in E(X) when n → ∞, then there exists { fnk } k∈N a subsequence of { fn} n∈N such that fnk → f a.e. For proof of this see =-=[19]-=-. PROPOSITION 1. If � is a Young function of the Orlicz space L� then i) The map a� : R∗ + → R∗ + given by a�(t) = t�−1 ( ) 1 t is nondecreasing. ii) ∫t 0 | f (s)|ds ≤ a�(t)‖ f ‖�, for all t > 0, f ∈ ... |