On first and second order linear Stieltjes differential equations

Fernández, Francisco Javier; Albés, Ignacio Márquez; Tojo, F. Adrián F.

doi:10.1016/j.jmaa.2022.126010

Cited by 10 publications

(14 citation statements)

References 18 publications

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“…Let us recall the following definition of Stieltjes derivative in [4,Definition 3.7]. To that end, we consider a, b ∈ , a < b, such that a ∈ N − g and b ∈ D g ∪ C g ∪ N + g .…”

Section: Preliminariesmentioning

confidence: 99%

“…To that end, we consider a, b ∈ , a < b, such that a ∈ N − g and b ∈ D g ∪ C g ∪ N + g . A careful reader might observe that throughout the entirety of [4] it is also required that g(a) = 0 and a ∈ D g . The first of these conditions can easily be avoided by redefining the map g if necessary; whereas the condition a ∈ D g can be imposed without loss of generality, as pointed out by [4,5] and [15,Proposition 4.28], when the focus of the study is the existence and uniqueness of solution of differential problems, which is not our case.…”

Section: Preliminariesmentioning

confidence: 99%

“…Instead, they provided a definition at almost every point for a certain measure, being thus comparable to the Radon-Nikodym derivative. Recently, in [4], the definition was extended in order to include all points of the domain, which also had an important repercussion in previously known results such as the product rule, which, in this more general context, reads…”

Section: Introductionmentioning

confidence: 99%

“…It would seem that, at least, ∆g ought to be always g-differentiable as its regularity is intrinsically defined by the same function g with respect to which we are differentiating, but this is far from reality, as previously known examples show -cf. [4,Remark 3.16]. Hence, we need to ask ourselves, when is ∆g * g-differentiable?…”

Section: Introductionmentioning

confidence: 99%

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Consequences of the product rule in Stieltjes differentiability

Fernandez¹,

Albés²,

Tojo³

2022

Preprint

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This work revolves around the study of differentiability in the Stieltjes sense of a product of functions. A formula for the first order derivative has been obtained in the past, which is similar to the usual one with some extra terms in its expression. The aim of this paper is to take this behavior into account to study under which conditions we can guarantee the existence of higher order derivatives, while obtaining some other interesting results for the Stieltjes derivative along the way. We also investigate the regularity of the product of two functions.

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“…Let us recall the following definition of Stieltjes derivative in [4,Definition 3.7]. To that end, we consider a, b ∈ , a < b, such that a ∈ N − g and b ∈ D g ∪ C g ∪ N + g .…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Consequences of the product rule in Stieltjes differentiability

Fernandez¹,

Albés²,

Tojo³

2022

Preprint

Self Cite

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“…In the literature related to Stieltjes differential equations we can find from foundational works regarding the Stieltjes derivative [13], to the study of the first order linear problem along with Picard and Peano type existence results [7] as well as existence results in another settings [9,10]. In [4], for the first time, the authors were able to Stieltjes-differentiate functions at every point of their domain. This allows to successfully define the notion of higher order Stieltjes derivatives and thus to consider higher order differential problems [4,6].…”

Section: Introductionmentioning

confidence: 99%

Stieltjes analytical functions and higher order linear differential equations

Cora¹,

Fernández²,

Tojo³

2023

Preprint

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In this work we develop a theory of Stieltjes-analytic functions. We first define the Stieltjes monomials and polynomials and we study them exhaustively. Then, we introduce the ganalytic functions locally, as an infinite series of these Stieltjes monomials and we study their properties in depth and how they relate to higher order Stieltjes differentiation. We define the exponential series and prove that it solves the first order linear problem. Finally, we apply the theory to solve higher order linear homogeneous Stieltjes differential equations with constant coefficients.

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Relaxation Theorem for Stieltjes Differential Inclusions on Infinite Intervals

Marraffa,

Satco

2023

Mediterr. J. Math.

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For a very general first-order differential problem on an infinite-time horizon involving the Stieltjes derivative with respect to a left-continuous non-decreasing function $$g:[0,\infty )\rightarrow \mathbb {R}$$ g : [ 0 , ∞ ) → R $$\begin{aligned} \left\{ \begin{array}{l} x'_g(t) \in F(t,x(t)),\; t\in [0,\infty )\\ x(0)=x_0, \end{array} \right. \end{aligned}$$ x g ′ ( t ) ∈ F ( t , x ( t ) ) , t ∈ [ 0 , ∞ ) x ( 0 ) = x 0 , we study the possibility to approximate the solutions of the convexified inclusion by the solutions of the non-convexified problem. Via a generalization to this framework of a classical result concerning continuous selection of trajectory, we thus present a relaxation theorem which states that, similarly to the setting of usual differential inclusions, the approximation can be achieved once we allow to the initial value to differ (but remaining close to) from the initial value of the considered solution of the relaxed inclusion.

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On first and second order linear Stieltjes differential equations

Cited by 10 publications

References 18 publications

Consequences of the product rule in Stieltjes differentiability

Consequences of the product rule in Stieltjes differentiability

Stieltjes analytical functions and higher order linear differential equations

Relaxation Theorem for Stieltjes Differential Inclusions on Infinite Intervals

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