Given a multifunction $F:[a,b] imes r o 2^{f R}$, we consider the implicit multivalued boundary value problem $$cases{h(u^{primeprime}(t))in F(t,u(t))& a.e. in $[a,b]$cr cr u(a)=u(b)=0.cr}$$ We prove an existence theorem for solutions $uin W^{2,p}([a,b])$, where for each $tin [a,b]$ the multifunction $F(t,cdot)$ can fail to be lower semicontinuous even at all points $xin{f R}$. In particular, our assumptions are satisfied, for instance, if there exist a neglegible set $Esub{f R}$ and a multifunction $G:[a,b] imes {f R} o 2^{f R}$ such that for a.a. $tin[a,b]$ one has $$ig{xin{f R}:G(t,cdot,)hbox{ is not l.s.c. at }xig}cup ig{xin{f R}:G(t,x) e F(t,x)ig}sub E.$$ No monotonicity assumption is required for $h$ or $F$. Our result extends Theorem 3 of [5], in which the explicit case is considered.
Second-order implicit differential inclusions with discontinuous right-hand side
CUBIOTTI, Paolo;
2014-01-01
Abstract
Given a multifunction $F:[a,b] imes r o 2^{f R}$, we consider the implicit multivalued boundary value problem $$cases{h(u^{primeprime}(t))in F(t,u(t))& a.e. in $[a,b]$cr cr u(a)=u(b)=0.cr}$$ We prove an existence theorem for solutions $uin W^{2,p}([a,b])$, where for each $tin [a,b]$ the multifunction $F(t,cdot)$ can fail to be lower semicontinuous even at all points $xin{f R}$. In particular, our assumptions are satisfied, for instance, if there exist a neglegible set $Esub{f R}$ and a multifunction $G:[a,b] imes {f R} o 2^{f R}$ such that for a.a. $tin[a,b]$ one has $$ig{xin{f R}:G(t,cdot,)hbox{ is not l.s.c. at }xig}cup ig{xin{f R}:G(t,x) e F(t,x)ig}sub E.$$ No monotonicity assumption is required for $h$ or $F$. Our result extends Theorem 3 of [5], in which the explicit case is considered.File | Dimensione | Formato | |
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