Applications of Stieltjes Derivatives to Periodic Boundary Value Inclusions

Abstract

In the present paper, we are interested in studying first-order Stieltjes differential inclusions with periodic boundary conditions. Relying on recent results obtained by the authors in the single-valued case, the existence of regulated solutions is obtained via the multivalued Bohnenblust–Karlin fixed-point theorem and a result concerning the dependence on the data of the solution set is provided.

Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity

“…Applying now an idea presented (and used in the study of initial value problems) in [4], we prove that the existence results in [17] can be obtained in the less restrictive case where F is convex, compact-valued, and upper semi-continuous with respect to its second variable, except for a set (which can be dense in R d , e.g., Example 3.11 in [4]), where a condition involving the notion of contingent g-derivative must be imposed. We also obtain the compactness (in the topology of uniform convergence) of the solution set.…”

“…The concept of solution we are searching for is given in [17] (Definition 4): u : [0, T] → R d is a solution of problem (1) if it is g-absolutely continuous and satisfies…”

“…In [17] (Theorem 4) the existence of solutions of (1) is proved under the hypotheses that F is convex, compact-valued and upper semicontinuous with respect to its second argument.…”

“…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.…”

Existence and Uniqueness of Solution for Stieltjes Differential Equations with Several Derivators

“…The usual setting for Stieltjes differential equations in the literature involves a single derivator either in its theoretical -see for example [4,6,9,11,15,16,22]-or numerical studies [4,5]. This is also the case for other differential problems involving Stieltjes derivatives such as in [13,17,21], or even the corresponding integral counterparts. Nevertheless, it is the new setting of differential problems with Stieltjes derivatives that offers the possibility of a new type of problems: systems of differential equations in which each of the components is differentiated with respect to a different nondecreasing and left-continuous function.…”