Multiple bounded solutions of differential inclusions: The Nielsen theory approach. (English) Zbl 0940.34008

Andres, Jan

J. Differ. Equations 155, No. 2, 285-310 (1999).

Summary: The generalized Nielsen number is defined for self-maps, which are composed by operators with \(R_\delta\)-values, on compact connected ANRs. Then it is applied to Carathéodory differential inclusions with constraints for obtaining 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.

MSC:

34A60	Ordinary differential inclusions
34G20	Nonlinear differential equations in abstract spaces
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

generalized Nielsen number; Carathéodory differential inclusions; retracts of Fréchet spaces; multiple solutions; boundary value problems

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