## 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$$<\infty$$.

### MSC:

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

### Keywords:

Banach space; Lyapunov-like functions