Abstract
The paper deals with a semilinear evolution equation in a reflexive and separable Banach space. The non-linear term is multivalued, upper Caratheodory and it depends on a retarded argument. The existence of global almost exact, i.e. classical, solutions is investigated. The results are based on a continuation principle for condensing multifields and the required transversalities derive from the introduction of suitable generalized guiding functions. As a consequence, the equation has a bounded globally viable set. The results are new also in the lack of retard and in the single valued case. A brief discussion of a non-local diffusion model completes this investigation.
Highlights
The paper deals with the multivalued evolution equation x′(t) + A(t)x(t) ∈ F (t, xt), t ∈ [a, b]depending on a retarded argument and satisfying the initial condition (1.1)x(t) = φ(t − a), t ∈ [a − h, a]. (1.2)We assume that the state space E is a reflexive Banach space; when it is needed we take E separable
We say that a multivalued function F : I × A ⊸ E2, where I is a closed real interval, A ⊆ E1 and E1, E2 are Banach spaces, is integrably bounded on every bounded set if, for every bounded subset Ω ⊂ A there exists μFΩ ∈ L1+(I) such that y E2 ≤ μFΩ(t), for a.a. t ∈ I, all x ∈ Ω and y ∈ F(t, x)
In this paper we show that a continuation principle for condensing multifields can be used, in alternative, and we introduce suitable generalized guiding functions in order to prove its transversality condition, i.e. condition (d) in Theorem 3.1
Summary
The paper deals with the multivalued evolution equation x′(t) + A(t)x(t) ∈ F (t, xt), t ∈ [a, b]. The theory of guiding and bounding functions was generalized in [2, 3, 4, 5] for multivalued equations of first and second order in infinite dimensional Banach spaces. Assume the existence of an open, bounded, convex subset K ⊂ E and of a generalized guiding function on ∂K, V : E → R, satisfying (V); let κ > 0 be such that VxG is Lipschitzian on ∂K + κB. Thanks to a ScorzaDragoni type result discussed, we introduce a sequence of related initial value problems (Pm) Their solvability depends on some preliminary existence results, the continuation principle in Section 3 combined with the generalized guiding function V. Whenever F satisfies (F1-2), it is superpositionally measurable (see e.g. [19, Theorem 1.3.5])
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