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Hopf bifurcation for non-densely defined Cauchy problems

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Abstract

In this paper, we establish a Hopf bifurcation theorem for abstract Cauchy problems in which the linear operator is not densely defined and is not a Hille–Yosida operator. The theorem is proved using the center manifold theory for non-densely defined Cauchy problems associated with the integrated semigroup theory. As applications, the main theorem is used to obtain a known Hopf bifurcation result for functional differential equations and a general Hopf bifurcation theorem for age-structured models.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Zhihua Liu

  2. UMR CNRS 5251, IMB& INRIA sud-ouest Anubis, Université de Bordeaux, 146 rue Léo Saignat, 33076, Bordeaux, France

    Pierre Magal

  3. Department of Mathematics, University of Miami, Coral Gables, FL, 33124-4250, USA

    Shigui Ruan

Authors
  1. Zhihua Liu
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  2. Pierre Magal
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  3. Shigui Ruan
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Correspondence to Shigui Ruan.

Additional information

Z. Liu, P. Magal—Research was partially supported by the French Ministry of Foreign and European Affairs program EGIDE (20932UL).

S. Ruan—Research was partially supported by the National Science Foundation (DMS-0715772 and DMS-1022728).

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Liu, Z., Magal, P. & Ruan, S. Hopf bifurcation for non-densely defined Cauchy problems. Z. Angew. Math. Phys. 62, 191–222 (2011). https://doi.org/10.1007/s00033-010-0088-x

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