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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 81, Issue 1-2
  • January 1978, pp. 131-151

Periodic solutions of special differential equations: an example in non-linear functional analysis†

  • Roger D. Nussbaum (a1)
  • DOI: http://dx.doi.org/10.1017/S0308210500010490
  • Published online: 14 November 2011
Abstract
Synopsis

We consider differential-delay equations which can be written in the form

The functions fi and gk are all assumed odd. The equation

is a special case of such equations with q = N + 1 (assuming f is an odd function). We obtain an essentially best possible theorem which ensures the existence of a non-constant periodic solution x(t) with the properties (1) x(t)≧0 for 0≦tq, (2) x(–t) = –x(t) for all t and (3) x(t + q) = –x(t) for all t. We also derive uniqueness and constructibility results for such special periodic solutions. Our theorems answer a conjecture raised in [8].

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

1H. Amann Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620709.

2P. J. Bushell Hilbert's metric and positive contraction mappings in a Banach space. Arch. Rational Mech. Anal. 52 (1973), 330338.

3E. N. Dancer Global solution branches for positive mappings. Arch. Rational Mech. Anal. 52 (1973), 181192.

4E. N. Dancer On the structure of solutions of non-linear eigenvalue problems. Indiana Univ. Math. J. 23 (1974), 10691076.

8J. Kaplan and J. Yorke Ordinary differential equations which yield periodic solutions of differential delay equations. J. Math. Anal. Appl. 48 (1974), 317324.

12R. D. Nussbaum Periodic solutions of some nonlinear, autonomous functional differential equations. II. J. Differential Equations 14 (1973), 360394.

13R. D. Nussbaum A global bifurcation theorem with applications to functional differential equations. J. Functional Analysis 19 (1975), 319339.

15P. Rabinowitz Some global results for nonlinear eigenvalue problems. J. Functional Analysis 7 (1971), 487513.

16H. H. Schaefer Topological Vector Spaces (New York: Springer, 1971).

17A. C. Thompson On certain contraction mappings in a partially ordered vector space. Proc. Amer. Math. Soc. 14 (1963), 438443.

18R. E. L. Turner Transversality and cone maps. Arch. Rational Mech. Anal. 58 (1975), 151179.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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