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Integral Equations, Dichotomy of Evolution Families on the Half-Line and Applications

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Abstract

The purpose of this paper is to provide a new, unified and complete study for uniform dichotomy and exponential dichotomy on the half-line. First we deduce conditions for the existence of uniform dichotomy, using classes of function spaces over \({\mathbb {R}_+}\) which are invariant under translations. After that, we obtain a classification of the main classes of function spaces over \({\mathbb {R}_+}\), in order to deduce necessary and sufficient conditions for the existence of exponential dichotomy, emphasizing on the main technical qualitative properties of the underlying spaces. We motivate our approach by illustrative examples and show that the main hypotheses cannot be dropped. We provide optimal methods regarding the input space in the study of dichotomy and deduce as particular cases some interesting situations as well as several dichotomy results published in the past few years.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, Pârvan Blvd. No. 4, 300223, Timişoara, Romania

    Adina Luminiţa Sasu & Bogdan Sasu

Authors
  1. Adina Luminiţa Sasu
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  2. Bogdan Sasu
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Correspondence to Adina Luminiţa Sasu.

Additional information

The work is supported by the Exploratory Research Grant PN II ID 1080 code 508/2009, director Professor Mihail Megan.

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Sasu, A.L., Sasu, B. Integral Equations, Dichotomy of Evolution Families on the Half-Line and Applications. Integr. Equ. Oper. Theory 66, 113–140 (2010). https://doi.org/10.1007/s00020-009-1735-5

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  • DOI: https://doi.org/10.1007/s00020-009-1735-5

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