Existence for second order difference inclusions on R+ governed by monotone operators

TitleExistence for second order difference inclusions on R+ governed by monotone operators
Publication TypeJournal Article
AuthorsMorosanu, Gheorghe
Journal titleAdvanced Nonlinear Studies
Year2014
Pages661-670
Volume14
Issue3
Abstract

Consider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u"(t) +
q(t)u'(t) ∈ Au(t) + f (t) for a.a. t ∈ R+ = [0,∞), subject to the condition u(0) = x ∈
Cl {D(A)}, where A: D(A) ⊂ H → H is a (possibly set-valued) maximal monotone operator whose range contains 0; p, q ∈ L∞(R+), with ess inf p > 0 and either ess inf q > 0 or ess sup q < 0. Eq. (E) has been investigated under various assumptions by Barbu [1]-[2], Brezis [4], V´eron [9]-[10] and others. Recently (see Morosanu [8]), we proved existence and uniqueness of a strong solution to Eq. (E) subject to u(0) = x ∈ D(A) in the weighted space X = L^2_b(R+; H), where b(t) = a(t)/p(t), a(t) = exp(\int_0^t q(s)/p(s) ds), under our weak conditions on p and q (see above). Here we extend this result to the general case x ∈ Cl {D(A)}, while keeping the other assumptions unchanged, thus solving a long standing problem. In addition, the proof of our recent result, which is the starting point of the present paper, is considerably simplified. Furthermore, some qualitative properties of solutions are pointed out, an application to a minimization problem is given, and some open problems are formulated.

ISSN1536 - 1365
LanguageEnglish
Publisher linkhttp://www.advancednonlinearstudies.com
Unit: 
Department of Mathematics and its Applications