Some surjectivity results for a class of multivalued maps and applications

Abstract

Let X be a Banach space over k (R or $ℂ$) and let $F:X→ X$ be a multivalued upper semicontinuous (u.s.c.) map with acyclic values. In [MV] Martelli and Vignoli extended to multivalued maps F the definition of a quasinorm of F (notation $|F|$) given by Granas in [Gr] for singlevalued ones.Using this definition they gave some surjectivity results in the context of ?-nonexpansive and condensing maps with $|F|<1$. In the present paper we improve these results in two ways. Firstly, we use the numerical radius of F (notation $n(F)$) instead of the quasinorm of F (we have $n(F)≤ |F|$ and there exist examples showing that this inequality can be strict). Secondly, we consider the class of admissible maps, which contains the u.s.c. acyclic valued ones.As a consequence we obtain, in particular, surjectivity results for the sum of two singlevalued maps not necessarily one-to-one.We will see that such results could not be obtained by using u.s.c. acyclic valued maps instead of admissible maps.It seems that only Webb [W] has obtained some surjectivity results of this kind.