Unstable periodic solutions of a parabolic problem in the presence of non-well-ordered lower and upper solutions

We prove the solvability of the Dirichlet-periodic problem for a semilinear parabolic equation, assuming the existence of a lower solution alpha and an upper solution beta, which do not satisfy the ordering condition alpha less than or equal to beta. Our results yield the existence of unstable solutions, localized by means of alpha and beta, requiring weak regularity of the coefficients of the equation. (C) 2000 Academic Press.