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Asymptotic stability and integral inequalities for solutions of linear systems on Radon-Nikodým spaces. (English) Zbl 1077.35024
The paper deals with the homogeneous Cauchy problem $$u'(t)=A(t)u(t) +f(t)$$ for $$t\geq 0$$, $$u(0)=0$$ on a Radon-Nikodým space $$X$$, where $$A(t)$$ is a linear operator-valued function and $$f$$ is an $$X$$-valued, locally Bochner integrable function. Let $$u_f$$ be the mild solution of this problem. Suppose that the homogeneous system $$u'(t)=A(t)u(t)$$ is exponentially stable. Under certain additional conditions it is proved that for each function $$f$$ belonging to the Sobolev space $$W_p^1(\mathbb R_+,X)$$, $$1\leq p<\infty$$, the solution $$u_f$$ lies in the same space.
##### MSC:
 35B35 Stability in context of PDEs 46N20 Applications of functional analysis to differential and integral equations 26D10 Inequalities involving derivatives and differential and integral operators 34G10 Linear differential equations in abstract spaces
##### Keywords:
evolution family; mild solution