A boundary value problem for implicit vector differential inclusions without assumptions of lower semicontinuity

摘要

Given a multifunction F : [ a , b ] × R n × R n → 2 R $F:[a,b]imesmathbf{R}^{n}imesmathbf{R}^{n}o2^{mathbf{R}}$ and a function h : X → R $h:Xomathbf{R}$ (with X ⊆ R n $Xsubseteq mathbf{R}^{n}$ ), we consider the following implicit two-point problem: find u ∈ W 2 , p ( [ a , b ] , R n ) $uin W^{2,p}([a,b], mathbf{R}^{n})$ such that { h ( u ″ ( t ) ) ∈ F ( t , u ( t ) , u ′ ( t ) ) a.e. in  [ a , b ] , u