A basis theorem for a class of max-plus eigenproblems

John Mallet-Paret, Roger D. Nussbaum

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


We study the max-plus equation where H:[0, M]→(-∞, ∞) and γ:[0, M]→[0, M] are given functions. The function ψ:[0, M]→[-∞, ∞) and the quantity P are unknown, and are, respectively, an eigenfunction and additive eigenvalue. Eigensolutions ψ are known to describe the asymptotics of certain solutions of singularly perturbed differential equations with state dependent time lags. Under general conditions we prove the existence of a finite set (a basis) of eigensolutions, for 1≤i≤q, with the same eigenvalue P, such that the general solution ψ to (*) is given by Here ci ∈[-∞, ∞) are arbitrary quantities and v denotes the maximum operator. In many cases q = 1 so the solution ψ is unique up to an additive constant.

Original languageEnglish (US)
Pages (from-to)616-639
Number of pages24
JournalJournal of Differential Equations
Issue number2
StatePublished - Apr 10 2003

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics


  • Additive eigenvalue
  • Differential-delay equation
  • Max-plus operator


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