Continuous families of exponential attractors for singularly perturbed equations with memory

We establish, for a family of semigroups S_(\u3b5:) H_\u3b5\u2192H_\u3b5 depending on aperturbation parameter \u3b5 08[0,1], where the perturbation is allowed tobecome singular at \u3b5=0, a general theorem on the existenceof exponential attractors E_\u3b5 satisfying a suitableH\uf6lder continuity property with respect tothe symmetric Hausdorff distanceat every \u3b5 08[0,1]. The result is appliedto the abstract evolution equations with memoryx_t (t)+ 2b_0^ 1e\u2592\u2588(k_\u3b5 (s) B_0 (x(t-s) )ds+B_1 (x(t) )=0, \u3b5 08(0,1] )where k_\u3b5 (s)=1/\u3b5 k(s/\u3b5)is the rescaling of a convex summable kernel k with unit mass.Such a family can be viewed as a memory perturbationof the equation x_t (t)+B_0 (x(t) )+B_1 (x(t) )=0formally obtained in the singular limit \u3b5\u21920