Upper semicontinuity of attractors for lattice systems under singular perturbations

Consider the following first order lattice system (u) over dot(m) + (2u(m) - u(m-1) - u(m+1)) + lambda(m)u(m) + f(m)(u(m)) = g(m), m is an element of Z, which is perturbed by the epsilon-small two order term epsilon(u) double over dot(m) + (u) over dot(m) + (2u(m) - u(m-1) - u(m+1)) + lambda(m)u(m) + f(m)(u(m)) = g(m), m is an element of Z. Under certain conditions on f(m), lambda(m) and g(m), the original systems and the epsilon-small perturbed systems have global attractors A in l(2) and A(epsilon) in l(2) x l(2), respectively, and A can be naturally embedded into a compact set A(0) in l(2) x l(2). We prove the upper semicontinuity of A(0) with respect to the attractors A(epsilon) at zero by showing that for any neighborhood O(A(0)) of A(0), A(epsilon) enters O(A(0)) if epsilon is small enough. (C) 2009 Published by Elsevier Ltd