Evolution Inclusions in Banach Spaces under Dissipative Conditions

Abstract

We develop a new concept of a solution, called the limit solution, to fully nonlinear differential inclusions in Banach spaces. That enables us to study such kind of inclusions under relatively weak conditions. Namely we prove the existence of this type of solutions and some qualitative properties, replacing the commonly used compact or Lipschitz conditions by a dissipative one, i.e., one-sided Perron condition. Under some natural assumptions we prove that the set of limit solutions is the closure of the set of integral solutions.

One Sided Lipschitz Evolution Inclusions in Banach Spaces

“…We present an example, which is a modification of the one from [12], to illustrate the applicability of the abstract results obtained.…”

“…Notice that, in this case, the right-hand side is a one-sided Perron. The results of [12] are not applicable, because here we study the nonlocal problem.…”

“…In [12], a new concept of solution for the local problem (4) was introduced, called the limit solution.…”

“…First, we present the following result regarding the limit solutions of the local problem (4), which comes directly from Theorem 2 in [12]. Proposition 4.…”

“…First, we consider F to be a one-sided Lipschitz with respect to a positive Lipschitz function. For the function g(•) we assume the same growth condition as in [7,12], which is weaker than the one from [6]. Then, we consider the case when F is a one-sided Lipschitz with respect to a negative constant.…”