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Multivalued differential equations on graphs and applications. (English) Zbl 0789.34013
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 \(u' \in F(t,u)\), \(u(0)=x_ 0\), 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.

34A60 Ordinary differential inclusions