The study of differential inclusions may be described as the study of dynamical systems having velocities not uniquely determined by the state of the system, but depending loosely upon it. Hence, the replacement of differential equations

by differential inclusions

where F is a set-valued map that associates to (t,x) the set of feasible velocities.

Besides mathematical motivations for studying inclusions, social and biological sciences provide many instances of differential inclusions. Inclusions allow taking into account uncertainty and the variety of available dynamics, in describing the evolution of states of systems derived from economics, social and biological sciences.

In studying differential equations and inclusions, one is invariably led to the mathematical concept of Integration. Of recent vintage is Kurzweil-Henstock integration, which is said to provide the most general framework into which most integration processes can be fitted. We study differential inclusions using the Kurzweil-Henstock integral, and show that under suitable growth and semicontinuity conditions, the initial value problem has a nonempty solution set, and furthermore, Kneser's and Hukuhara's theorems hold. We also look into functional differential inclusions (delay differential inclusions) using the Kurzweil integral. Finally, we extend the definition of the integral of a multifunction with values in $[/cal P](/Re/sp[n]$) to the integral of a multifunction with values in the power set of a separable Hilbert space.