TY - JOUR

T1 - A basis theorem for a class of max-plus eigenproblems

AU - Mallet-Paret, John

AU - Nussbaum, Roger D.

N1 - Funding Information:
This paper was supported in part by National Science Foundation Grants DMS-9970319 (JM-P) and DMS-0070829 (RDN).

PY - 2003/4/10

Y1 - 2003/4/10

N2 - 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.

AB - 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.

KW - Additive eigenvalue

KW - Differential-delay equation

KW - Max-plus operator

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U2 - 10.1016/S0022-0396(02)00087-6

DO - 10.1016/S0022-0396(02)00087-6

M3 - Article

AN - SCOPUS:0037431214

SN - 0022-0396

VL - 189

SP - 616

EP - 639

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -