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| The application of the classical Hartman-type conditions, as presented in [Trans. Amer. Math. Soc. 96 (1960), 493--509] i.e. sign conditions w.r.t. the first state variable and growth conditions (sometimes called as the Bernstein--Nagumo--Hartman conditions) w.r.t. the second state variable, will be discussed, on various levels of abstraction, for a multivalued Dirichlet problem. For instance, in abstract spaces, if the right-hand sides are Marchaud (i.e. globally upper-semicontinuous) and condensing, then the growth conditions can be very liberal. On the other hand, for multivalued upper-Carath\'eodory condensing right-hand sides, the situation is more delicate. Nevertheless, the related obtained criteria can be still not worse than those of Hartman, i.e. as for vector equations in finite-dimensional spaces.
The approach is based on the combination of topological degree arguments, bounding (Liapunov-like) functions and a Scorza--Dragoni approximation technique.
An illustrative application of the main existence and localization results can concern partial integro-differential equations involving discontinuities in state variables. |
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