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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 ${u}^{\text{'}}\in F\left(t,u\right)$, $u\left(0\right)={x}_{0}$, with an additional constraint $u\left(t\right)\in D\left(t\right)$ for $t\in \left[0,a\right]$. 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 $\left[\left\{1\right\}×F\left(t,x\right)\right]\cap {T}_{G}\left(t,x\right)\ne \varnothing$ for $t\in \left[0,a\right]\setminus N$, $x\in D\left(t\right)$ $\left[\left\{1\right\}×X\right]\cap {T}_{G}\left(t,x\right)\ne \varnothing$ for $t\in N$, $x\in D\left(t\right)$. Here ${T}_{G}\left(t,x\right)$ denotes the tangent cone to the graph of $D$ at the point $\left(t,x\right)$ 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 $\left[\left\{1\right\}×F\left(t,x\right)\right]\subset {T}_{G}\left(t,x\right)$. 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 ${u}^{\text{'}}\in Au+F\left(t,u\right)$, 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.