Abstract
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.
Highlights
Introduction and PreliminariesThe goal of this paper is to prove the existence of limit and integral solutions for a class of nonlinear evolution inclusions with nonlocal initial conditions of the form:ẋ (t) ∈ Ax (t) + f (t), t ∈ (t0, T ) (1)f (t) ∈ F (t, x (t)), t ∈ (t0, T )x (t0 ) = g( x (·)) ∈ D ( A).Here, A : D ( A) ⊂ E ⇒ E is an m–dissipative operator, E is a real Banach space with norm | · |, F : I × E ⇒ E a multifunction with nonempty, closed convex and bounded values, I = [t0, T ], and g : C ( I; E) → D ( A) is a given function
We prove the existence of limit and integral solutions for (1) in general Banach spaces
We prove here the existence of limit solutions in general Banach spaces under much weaker assumptions
Summary
The goal of this paper is to prove the existence of limit and integral solutions for a class of nonlinear evolution inclusions with nonlocal initial conditions of the form:. We prove the existence of limit and integral solutions for (1) in general Banach spaces. The multifunction F : I × E ⇒ E is said to be one-sided Lipschitz (OSL) if there exists a Lebesgue integrable L(·) (maybe negative) and a null set I ⊂ I such that for every u, v ∈ E, every t ∈ I \ I, every f u ∈ F (t, u) and every ε > 0 there exists f v ∈ F (t, v) such that:. The OSL condition in the case of multifunctions was first introduced (in a more general form and with different name) in [17]. Let L1 (Ω) be the set of all real valued Lebesgue integrable functions on the bounded domain Ω ⊂ R. It is easy to see that f (·) is OSL with a negative constant
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