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multivalued boundary value problem; differential inclusion in Banach space; compact operator; fixed point theorem
The paper deals with the multivalued boundary value problem $x'\in A(t,x)x+F(t,x)$ for a.a.\ $t \in [a,b]$, $Mx(a)+Nx(b) =0$, in a separable, reflexive Banach space $E$. The nonlinearity $F$ is weakly upper semicontinuous in $x$. We prove the existence of global solutions in the Sobolev space $W^{1,p}([a,b], E)$ with $1<p<\infty $ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.
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