A Tikhonov-type theorem for abstract parabolic differential inclusions in Banach spaces

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 ε ≥ 0, where ε is the perturbation parameter. Specifically, assuming the existence of a Lipschitz selector of the involved multivalued maps we can define a nonempty subset ${\displaystyle Z_L(\epsilon )}$ of the solution set of the singularly perturbed system. This subset is the set of the Hölder continuous solutions defined in [0,d], d > 0 with prescribed exponent and constant L. We show that ${\displaystyle Z_L(\epsilon )}$ is uppersemicontinuous at ε = 0 in the C[0,d]×C[δ,d] topology for any δ ∈ (0,d].