One Sided Lipschitz Evolution Inclusions in Banach Spaces

Using the notion of limit solution, we study multivalued perturbations of m-dissipative differential inclusions with nonlocal initial conditions. These solutions enable us to work in general Banach spaces, in particular L1. The commonly used Lipschitz condition on the right-hand side is weakened to a one-sided Lipschitz one. No compactness assumptions are required. We consider the cases of an arbitrary one-sided Lipschitz condition and the case of a negative one-sided Lipschitz constant. Illustrative examples, which can be modifications of real models, are provided.

Relaxation of nonlocal m-dissipative differential inclusions

We show here that the set of the integral solutions of a nonlocal differential inclusion is dense in the set of the solution set of the corresponding relaxed differential inclusion. We further define a notion of limit solution and show that the set of limit solutions is closed and is the closure of the set of integral solutions. An illustrative example is provided.