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Properties of the solution of evolution inclusions driven by time dependent subdifferentials. (English) Zbl 0757.34055
An evolution problem of the type (1) \(- x'(t)\in\partial\phi(t,x(t))+F(t,x(t))\) and the convexified problem \(- x'(t)\in\partial\phi(t,x(t))+\overline{\text{conv}} F(t,x(t))\) are considered in a Hilbert space \(H\). It is assumed that (1) for every \(t\in T\), \(\phi(t,\cdot)\) is convex, lower semicontinuous with compact level sets \(\{x\in H\mid\;\phi(t,x)+\| x\|^ 2\leq\lambda\}\); (2) for every positive integer \(b\), there exists a constant \(K_ b>0\), an absolutely continuous function \(g_ b: T\mapsto\mathbb{R}\) with \(\dot g_ b\in L^ \beta(T)\) and a function of bounded variation \(h_ b: T\mapsto\mathbb{R}\) s.t. if \(t\in T\), \(x\in\text{dom} \phi(t,\cdot)\) with \(\| x\|\leq b\) and \(s\in[t,r]\), then there exists \(\hat x\in\text{dom} \phi(s,\cdot)\) satisfying \(\| \hat x-x\|\leq| g_ b(s)-g_ b(t)|(\phi(t,x)+K_ b)^ \alpha\) and \(\phi(s,\hat x)\leq\phi(t,x)+| h_ b(s)-h_ b(t)|(\phi(t,s)+K_ b)\), where \(\alpha\in[0,1]\), and \(\beta=2\) if \(\alpha\in[0,1/2]\) or \(\beta=1-\alpha\) if \(\alpha\in[1/2,1]\); (3) \(F: T\times H\rightsquigarrow H\) is a set- valued map with non-empty, closed values, which is measurable in \(t\); (4) \(h(F(t,x),F(t,y))\leq\theta(t)\| x-y\|\) with \(\theta(\cdot)\in L_ +^ 1\), \(x,y\in H\); (5) \(| F(t,x)|=\sup\{\| y\|\): \(y\in F(t,x)\}\leq a(t)+b(t)\| x\|\) a.e. with \(a,b\in L_ +^ 2\). Under these assumptions a relaxation theorem is proved: the closure of the solution set of (1) starting from \(x_ 0\in\text{dom} \phi(0,\cdot)\) equals the solution set of the convexified problem. Moreover, using this relaxation theorem it is proved that the closedness of the solution of (1) in \(C(T,H)\) implies that \(F(t,x)\) is convex valued for almost all \(t\in T\).
MSC:
34G20 Nonlinear differential equations in abstract spaces
34A60 Ordinary differential inclusions
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