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Zbl 0789.34013
Bothe, Dieter
Multivalued differential equations on graphs and applications.
(English)
[B D] Paderborn: Gesamthochschule Paderborn, Fachbereich Mathematik-Informatik, 102 p. (1992).

The author has gathered in this thesis his results concerning the problem of existence of solutions of differential inclusions $u' \in F(t,u)$, $u(0)=x\sb 0$, 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 $[\{1\} \times F(t,x)] \cap T\sb G(t,x) \ne \emptyset$ for $t \in[0,a] \backslash N$, $x \in D(t)$ $[\{1\} \times X] \cap T\sb G(t,x) \ne \emptyset$ for $t \in N$, $x \in D(t)$. Here $T\sb G(t,x)$ 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 $[\{1\} \times F(t,x)] \subset T\sb G (t,x)$. The third chapter contains the proof showing that the set of all solutions issued from one initial point is compact $R\sb \delta$ -- 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 $C\sb 0$ 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.
[T.Rzezuchowski (Warszawa)]
MSC 2000:
*34A60 ODE with multivalued right-hand sides

Keywords: existence of solutions of differential inclusions; constraint; Banach space; tangential condition; motion of a flexible string attached to a mass-bob; comparison of solutions; weak stability

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