A basis theorem for a class of max-plus eigenproblems

We study the max-plus equation P þ cðxÞ 1⁄4 max xpspM ðHðsÞ þ cðgðsÞÞÞ; xA1⁄20;M ; ð*Þ where H : 1⁄20;M -ð N;NÞ and g : 1⁄20;M -1⁄20;M are given functions. The function c : 1⁄20;M -1⁄2 N;NÞ and the quantity P are unknown, and are, respectively, an eigenfunction and additive eigenvalue. Eigensolutions c 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 j; for 1pipq; with the same eigenvalue P; such that the general solution c to ð*Þ is given by cðxÞ 1⁄4 ðc þ jðxÞÞ3ðc þ jðxÞÞ3?3ðc þ jðxÞÞ: Here cA1⁄2 N;NÞ are arbitrary quantities and 3 denotes the maximum operator. In many cases q 1⁄4 1 so the solution c is unique up to an additive constant. r 2002 Elsevier Science (USA). All rights reserved.