On Regulated Solutions of Impulsive Differential Equations with Variable Times

Caponetti, Diana; Cichoń, Mieczysław; Marraffa, Valeria

doi:10.3390/math8122164

Cited by 2 publications

(1 citation statement)

References 29 publications

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“…They prove that even though the dynamics of the system and the delay have ideal continuity properties, the right side may not even have limits at some points due to the impact of past impulses in the present. In [6], the authors investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. They give sufficient conditions to ensure that a regulated solution of an impulsive problem is globally defined.…”

Section: Introductionmentioning

confidence: 99%

An Existence Result in <i>α</i>-norm for Impulsive Functional Differential Equations with Variable Times

Zabsonre¹

2022

AJAM

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The dynamics of evolving processes is often subjected to abrupt changes such as shocks, harvesting, and natural disasters. Often these short-term perturbations are treated as having acted instantaneously or in the form of "impulses." In fact, there are many processes and phenomena in the real world, which are subjected during their development to the short-term external influences. Their duration is negligible compared with the total duration of the studied phenomena and processes. Impulsive differential equations take an important place in some area such that physics, chemical technology, population dynamics, biotechnology, and economics. The study of such equations is relatively less developed due to the difficulties created by the state-dependent impulses. In the case of impulses at variable times, a "beating phenomenon" may occur, that is to say, a solution of the differential equation may hit a given barrier several times (including infinitely many times). In this work, we study the existence of solutions for some partial impulsive functional differential equations with variable times in Banach spaces by using the fractional power of closed operators theory. We suppose that the undelayed part admits an analytic semigroup. The delayed part is assumed to be Lipschitz. We use Schaefer fixed-point Theorem to prove the existence of solutions for this first order equation with impulse in α-norm.

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Section: Introductionmentioning

confidence: 99%

An Existence Result in <i>α</i>-norm for Impulsive Functional Differential Equations with Variable Times

Zabsonre¹

2022

AJAM

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Fixed-time and state-dependent time discontinuities in the theory of Stieltjes differential equations

Satco

2022

Electron. J. Qual. Theory Differ. Equ.

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In the present paper, we are concerned with a very general problem, namely the Stieltjes differential Cauchy problem involving state-dependent discontinuities. Given that the theory of Stieltjes differential equations covers the framework of impulsive problems with fixed-time impulses, in the present work we generalize this setting by allowing the occurrence of fixed-time impulses, as well as the occurrence of state-dependent impulses. Along with an existence result obtained under an overarching set of assumptions involving Stieltjes integrals, it is showed that a least and a greatest solution can be found.

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On Regulated Solutions of Impulsive Differential Equations with Variable Times

Cited by 2 publications

References 29 publications

An Existence Result in <i>α</i>-norm for Impulsive Functional Differential Equations with Variable Times

An Existence Result in <i>α</i>-norm for Impulsive Functional Differential Equations with Variable Times

Fixed-time and state-dependent time discontinuities in the theory of Stieltjes differential equations

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On Regulated Solutions of Impulsive Differential Equations with Variable Times

Cited by 2 publications

References 29 publications

An Existence Result in &lt;i&gt;α&lt;/i&gt;-norm for Impulsive Functional Differential Equations with Variable Times

An Existence Result in &lt;i&gt;α&lt;/i&gt;-norm for Impulsive Functional Differential Equations with Variable Times

Fixed-time and state-dependent time discontinuities in the theory of Stieltjes differential equations

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An Existence Result in <i>α</i>-norm for Impulsive Functional Differential Equations with Variable Times

An Existence Result in <i>α</i>-norm for Impulsive Functional Differential Equations with Variable Times