On measure of noncompactness in Lebesgue and Sobolev spaces with an application to the functional integro-differential equation

In this paper we attempt to define axiomatic measures of non-compactness for Sobolev spaces of integer order W n , p ( Ω ) , where Ω ⊂ R d (which is equivalent to Ω being any set of infinite measure). We consider two cases, one with Ω being an open subset with finite measure, and another when Ω = R d , and discuss basic features of the measure of non-compactness in each case. Next we define a partial measure of non-compactness on space L p ( Ω ; B ) , where B is a Banach space. Furthermore, we give some application to solve the functional integro-differential equation in W 1 , p ( Ω ) .