Abstract
Abstract A semilinear multivalued evolution equation is considered in a reflexive Banach space. The nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. No strong compactness is assumed, neither on the evolution operator generated by the linear part, or on the nonlinear term. A wide family of nonlocal associated boundary value problems is investigated by means of a fixed point technique. Applications are given to an optimal feedback control problem, to a nonlinear hyperbolic integro-differential equation arising in age-structure population models, and to a multipoint boundary value problem associated to a parabolic partial differential equation. MSC:34G25, 34B10, 34B15, 47H04, 28B20, 34H05.
Highlights
The paper deals with the following semilinear evolution inclusion in a reflexive Banach space E:
In [ ] and [ ] we proved the existence of classical solutions for the inclusion ( . ) associated with a two-point boundary condition by means of weak topology, avoiding hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term F
As far as we know, all the results showed are new even for the case of a single-valued nonlinear term F
Summary
For every t ∈ [a, b], xn(t) x(t); Denoting = {(t, s) ∈ [a, b] × [a, b] : a ≤ s ≤ t ≤ b}, we recall that a two-parameter family {U(t, s)}(t,s)∈ , where U(t, s) : E → E is a bounded linear operator and (t, s) ∈ , is called an evolution system if the following conditions are satisfied: . We observe that since the evolution operator U is strongly continuous on the compact set , by the uniform boundedness theorem, there exists a constant D = D > such that
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