For a family of semigroups Sε(t) : ℌε → ℌε depending on a perturbation parameter ε ∈ [0, 1], where the perturbation is allowed to become singular at ε = 0, we establish a general theorem on the existence of exponential attractors εε satisfying a suitable Hölder continuity property with respect to the symmetric Hausdorff distance at every ε ∈ [0, 1]. The result is applied to the abstract evolution equations with memory
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0308210509000365/resource/name/S0308210509000365_inline1.gif?pub-status=live)
where kε(s) = (1/ε)k(s/ε) is the rescaling of a convex summable kernel k with unit mass. Such a family can be viewed as a memory perturbation of the equation
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0308210509000365/resource/name/S0308210509000365_inline2.gif?pub-status=live)
formally obtained in the singular limit ε → 0.