Abstract
We prove an existence result for systems of differential inclusions driven by multivalued mappings which need not assume closed or convex values everywhere, and need not be semicontinuous everywhere. Moreover, we consider differentiation with respect to a nondecreasing function, thus covering discrete, continuous and impulsive problems under a unique formulation. We emphasize that our existence result appears to be new even when the derivator is the identity, i.e. when derivatives are considered in the usual sense. We also apply our existence theorem for inclusions to derive a new existence result for discontinuous Stieltjes differential equations. Examples are given to illustrate the main results.
Highlights
Stieltjes differential equations are equations of the form xg(t) = f t, x(t), t ∈ [t0, t0 + T), (t0 ∈ R, T > 0), (1.1)where, roughly speaking, the Stieltjes derivative of x with respect to g is x(s) – x(t) xg (t) =lim s→t g(s) g(t), where the derivator function g : R −→ R is nondecreasing
The definition of Stieltjes derivative is consistent with Stieltjes integration, in the sense that every bonna fide function can be recovered as the indefinite Lebesgue–Stieltjes integral of its g-derivative
We shall constantly refer to results in [8], we remark that there exist much older, similar notions of derivatives with respect to functions and fundamental theorems for Stieltjes integrals, see for example [3]
Summary
(2) The function f fulfills the following properties: (a) There exists fg (t) for g-almost all t ∈ [a, b) (i.e., for all t ∈ [a, b) except on a set of μg -measure zero). We have the following sufficient condition for relative compactness of sets of g-absolutely continuous functions in BCg([a, b])
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