Asymptotic boundary value problems in Banach spaces

A continuation principle is given for solving boundary value problems on arbitrary (possibly infinite) intervals to Carathéodory differential inclusions in Banach spaces. For this aim, the appropriate fixed point index is defined to condensing decomposable multivalued operators in Fréchet spaces. This index extends and unifies the one for compact maps in Andres et al. [Trans. Amer. Math. Soc. 351 (1999) 4861–4903] as well as the one for operators in Banach spaces in Bader [Ph.D. Thesis, University of Munich, 1995]. As an application, we prove the existence of an entirely bounded solution of a semilinear evolution inclusion.  2002 Elsevier Science (USA). All rights reserved