Periodic boundary value problems involving Stieltjes derivatives

Applications of Stieltjes Derivatives to Periodic Boundary Value Inclusions

“…In this section, relying on the theory in [4], we present an existence result for the linear Stieltjes differential equation with periodic boundary conditions…”

“…Allowing the study in a unique framework of many classical problems: ordinary differential or difference equations (in the case of an absolutely continuous measure-with respect to the Lebesgue measure-respectively of a discrete measure), impulsive differential problems (for a sum of Lebesgue measure with a discrete one), dynamic equations on time scales (see [1]) and generalized differential equations (e.g., [2,3]), it is clear why the theory of differential equations driven by measures has seen a significant growth (e.g., [1,4]).…”

“…Based on the results obtained in [4] for measure-driven differential equations with periodic boundary conditions, in the present paper we focus on nonlinear differential inclusions of the form:…”

“…As far as the authors know, periodic differential problems driven by a non-decreasing left-continuous function g have been studied only in the single-valued case in [4].…”

Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity

“…Aware of the significance of periodic boundary conditions when studying real life processes, the present work focuses on the analysis of a first-order differential inclusion with periodic boundary value conditions u g (t) + p(t)u(t) ∈ F(t, u(t)), µ g −a.e. t ∈ [0, T] u(0) = u(T) (1) involving the Stieltjes derivative with respect to a left-continuous non-decreasing function g : [0, T] → R and a real function p, Lebesgue-Stieltjes integrable with respect to g. This problem was investigated in the single-valued framework in [16] and then generalized to the multivalued setting in [17] under the assumptions that F : [0, T] × R d → P (R d ) has convex, compact values and it is a Carathéodory multifunction.…”