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.
Highlights
Introduction and PreliminariesLet X be a real Banach space with the norm | · |, A : D ( A) ⊂ X ⇒ X an m–dissipative operator generating the semigroup {S(t) : D ( A) → D ( A); t ≥ 0} and F : I × X ⇒ X a multifunction with nonempty, closed and bounded values, where I = [t0, T ].In this paper, we study evolution inclusions of the form ẋ (t) ∈ Ax (t) + F (t, x (t)), x (t0 ) = x0 ∈ D ( A). (1)Notice that many parabolic systems can be written in the form (1)
(I) We prove that the set of limit solutions of (1) is nonempty and closed in C ( I, X ) when X is a general Banach space and F (·, ·) is almost continuous and satisfies a one-sided Perron condition
(II) We prove that in the case when A generates a compact semigroup, the closure of the set of integral solutions of (1) is exactly the set of limit solutions, which in general does not coincide with the set of integral solutions of the relaxed system
Summary
Our definition given above is more convenient for the study of the qualitative properties of the set of integral solutions of (1) in the case when X is an arbitrary Banach space. (I) We prove that the set of limit solutions of (1) is nonempty and closed in C ( I, X ) when X is a general Banach space and F (·, ·) is almost continuous and satisfies a one-sided Perron condition. We recall that the Carathéodory function w : I × R+ → R+ is said to be Perron function if it is integrally bounded on bounded sets, w(t, 0) ≡ 0, w(t, ·) is nondecreasing for every t ∈ I and the zero function is the only solution of the scalar differential equation r 0 (t) = w(t, r (t)), r (t0 ) = 0, on I. For every solution x (·) of (5), in particular for every ε–solution x (·) of (1), with the pseudoderivative f x (·), we have that dist( f x (t), F (t, x (t))) ≤ 2N on I, since | f x (t)| ≤ N and k F (t, x (t))k ≤ N for every t ∈ I
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