A Tikhonov type Theorem for abstract parabolic differential inclusions in Banach spaces

Abstract

We consider a class of singularly perturbed systems of semilinear parabolic differential inclusions in infinite dimensional spaces. For such a class we prove a Tikhonov-type theorem for a suitably defined subset of the set of all solutions for e ≥ 0, where e is the perturbation parameter. Specifically, assuming the existence of a Lipschitz selector of the involved multivalued maps we can define a nonempty subset ZL(e) of the solution set of the singularly perturbed system. This subset is the set of the Holder continuous solutions defined in [0, d], d > 0 †Research supported by the project “Qualitative analysis and control of dynamical systems” at the University of Siena and FRBR grant 99-01-00333. 208 A. Gudovich, M. Kamenski and P. Nistri with prescribed exponent and constant L. We show that ZL(e) is uppersemicontinuous at e = 0 in the C[0, d]× C[δ, d] topology for any δ ∈ (0, d].