Multiple Bounded Solutions of Differential Inclusions: The Nielsen Theory Approach

The generalized Nielsen number is defined for self–maps, which are composed by operators with Rδ-values, on compact connected ANRs. Then it is applied to Carathéodory differential inclusions with constraints for obtaining the multiplicity criteria. More precisely, such problems are transformed to those for the lower estimate of fixed points of the related operators with the given properties on bounded, compact, connected neighbourhood retracts of Fréchet spaces. In this way, multiple solutions can be proved e.g. for the multivalued initial value problem on the half–line or, in the single–valued case, for boundary value problems to ordinary differential equations.