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

${u}^{\text{'}}\in F(t,u)$,

$u\left(0\right)={x}_{0}$, with an additional constraint

$u\left(t\right)\in D\left(t\right)$ 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

$[\left\{1\right\}\times F(t,x)]\cap {T}_{G}(t,x)\ne \varnothing $ for

$t\in [0,a]\setminus N$,

$x\in D\left(t\right)$ $[\left\{1\right\}\times X]\cap {T}_{G}(t,x)\ne \varnothing $ for

$t\in N$,

$x\in D\left(t\right)$. 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

$[\left\{1\right\}\times F(t,x)]\subset {T}_{G}(t,x)$. 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

${u}^{\text{'}}\in Au+F(t,u)$, 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.