Existence of solutions and periodicity of differential inclusions (Q1201354)

The differential inclusion (1) \(x'(t)\in A(t,x(t))x(t)+F(t,x(t))\) together with the condition \(Lx=r\) is considered in \(R^ n\). Here \(A\) is an \(n\times n\) continuous matrix map, \(F\) is measurable in \(t\) and upper semicontinuous in \(x\) with nonempty convex compact values, satisfying \(\| F(t,x)\|\leq\alpha(t)\| x\|+\beta(t)\). \(L\) is a linear continuous operator such that the problem \(x'(t)-A(t,v(t))x(t)=f(t)\), \(Lx=r\) admits solutions for all \(f\) belonging to a linear manifold \(\Phi\) in \(C[J\times R^ n,R^ n\times R^ n]\) for each fixed \(v\). Moreover, it is assumed that the set of all measurable selections from \(F(t,v(t))\) for each fixed \(v\) is closed convex subset of \(\Phi\). Under these assumptions the existence of a solution for (1) is proved, provided the integral of \(\alpha\) is sufficiently small. Under some additional assumptions an existence theorem for periodic solutions of (1) is proved, provided the dimension of the space \(R^ n\) is odd. Some examples are given.