Nonlincnr Andysir. Theory, Merhodr & Applications. Vol. 9. No. 8, pp. 775-786, 1985. Printed m Great Britarn. 0362-546X,& $3.00 + .OO 0 1985 Pergamon Press Ltd. zyxwvutsrqpon EXISTENCE OF BOUNDED SOLUTIONS FOR MULTIVALUED DIFFERENTIAL SYSTEMS M. CECCHI Istituto di Matematica Applicata, Facolta di Ingegneria, Universita di Firenze, Via S. Marta 3, 50139 Firenze, Italy M. MARINI Dipartimento di Ingegneria Elettronica, Universita di Firenze, Via S. Marta 3, 50139 Firenze, Italy and P. ZECCA Dipartimento di Sistemi e Informatica, Universita di Firenze, Via S. Marta 3, 50139 Firenze, Italy zyxwvutsrqponmlkj (Received 27 February 1984; received for publication 9 October 1984) Key words and phrases: Multivalued systems. boundary value problems, existence theorems. INTRODUCTION MANY papers deal with the problem of the existence of bounded solutions on an interval ZC R for the multivalued differential system (0.1) where F is a convex valued upper-semicontinuous map, A is an n x n matrix with Caratheodory conditions, Y is a subset of a Banach space (see e.g. [8,9] and references therein). The technique for solving such problems is essentially that of reducing (0.1) to the abstract form XEMX, (0.2) and to apply some suitable fixed point theorem to the operator M. Lasota and Opial, [6], solved (0.1) when the interval I is compact, and A independent of X. For such systems, they developed a general theory which, under many aspects, is similar to the classical theory of single valued differential equations. It is well known that if the interval I is not compact, the operator M in (0.2) is often not upper semicontinuous, and this makes many fixed point theorems difficult to be applied. In the case when A is independent of x, such a difficulty has been overcome by Anichini and Zecca and Zecca and Zezza in [l, 93, first solving the problem in the compact case, and then using a diagonal extension technique. Nevertheless, this procedure requires strong hypotheses to guarantee the convergence of the diagonal methods. In this paper, our aim is to prove the existence of bounded solutions for (0.1) when I is a possibly unbounded real interval. The method is similar to the one used in [2] for differential equations with single valued right-hand sides. Such an approach reduces the problem of proving the existence of solutions for (0.1) or (0.2) to the one of finding suitable a priori 775