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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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