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Multivalued differential equations on graphs and applications. (English) Zbl 0789.34013
The author has gathered in this thesis his results concerning the problem of existence of solutions of differential inclusions $$u' \in F(t,u)$$, $$u(0)=x_ 0$$, with an additional constraint $$u(t) \in D(t)$$ for $$t \in[0,a]$$. The multifunction $$F$$ is defined only on the graph of the multifunction $$D$$. Most of the results are given for $$u \in X$$, $$X$$ a Banach space. Apart from some standard regularity of $$F$$ a tangential condition of $$F$$ with respect to $$D$$ is needed. The author uses one of the form $$[\{1\} \times F(t,x)] \cap T_ G(t,x) \neq \emptyset$$ for $$t \in[0,a] \backslash N$$, $$x \in D(t)$$ $$[\{1\} \times X] \cap T_ G(t,x) \neq \emptyset$$ for $$t \in N$$, $$x \in D(t)$$. Here $$T_ G(t,x)$$ denotes the tangent cone to the graph of $$D$$ at the point $$(t,x)$$ and $$N$$ is some set of measure 0. The second chapter is devoted to the existence of solutions for upper semicontinuous and almost upper semicontinuous $$F$$. The lower semicontinuous case is also treated for $$F$$ with closed values – the intersection in the first part of the tangential condition must then be replaced with the inclusion $$[\{1\} \times F(t,x)] \subset T_ G (t,x)$$. The third chapter contains the proof showing that the set of all solutions issued from one initial point is compact $$R_ \delta$$ – the limit of decreasing compact contractible sets. Chapter 4 is devoted to the inclusions $$u' \in Au+F(t,u)$$, where $$A$$ is a linear unbounded operator generating a $$C_ 0$$ semigroup; the PDE describing the motion of a flexible string attached to a mass-bob which is guided by a vertical track is discussed. Chapter 5 contains some further applications to the comparison of solutions and weak stability.