Multivalued differential equations on closed sets. (English) Zbl 0715.34114

Summary: Let X be a Banach space, \(D\subset X\) closed, \(J=[0,a]\subset R\) and F: \(J\times D\to 2^ X\setminus \{\emptyset \}\) a multivalued map. We consider the initial value problem \[ (1)\quad u'\in F(t,u)\quad a.e.\text{ on } J,\quad u(0)=x_ 0\in D. \] Solutions of (1) are understood to be a.e. differentiable with \(u'\in L^ 1_ X(J)\) such that \(u(t)=x_ 0+\int^{t}_{0}u'(s)ds\) on J and (1) is satisfied. We first give sufficient conditions for existence of solutions in the autonomous case \(F(t,x)=F(x)\) and indicate relations to the fixed point problem \(x\in F(x)\). We concentrate on upper semicontinuous F with compact convex images and give counter-examples if F is lower semicontinuous or the F(x) are not convex. Then the time-dependent problem (1) is considered by reduction to the autonomous case. Finally we prove existence of solutions which are monotone with respect to a preorder or Lyapunov-like functions. Since we allow dim X\(=\infty\), we find it necessary to exploit Zorn’s Lemma. As a by-product of this approach we achieve considerable simplification of proofs or improvement of some results known for dim X\(<\infty\).


34G20 Nonlinear differential equations in abstract spaces
34A60 Ordinary differential inclusions
47H10 Fixed-point theorems