Initial value problems of fractional order with fractional impulsive conditions
type: Item
instance of: scholarly article
title: Initial value problems of fractional order with fractional impulsive conditions
zbMATH Open document ID: 1286.34011
author: Nickolai Kosmatov
published in: Results in Mathematics
publication date: 2013-07-11T00:00:00Z
review text: Over the past decade there have been numerous papers dedicated to the existence and uniqueness of initial and boundary value problems of fractional order. One of the main issues with fractional differential equations is that inverting the problem is not as straight forward as for ordinary differential equations. The author has two main goals in this paper. The first is to show the correct method of inversion technique for fractional differential equations with fractional impulsive conditions, and the second is to study initial value problems with fractional impulsive conditions involving the Riemann-Liouville derivative of order \(0 < \alpha \leq 1\) and involving the Caputo derivative of order \(1 < \delta < 2\). In particular, the author studies the following problems: The existence of solutions for the Riemann-Liouville integro-differential equation with \(0 < \alpha < 1\), \[ \mathcal{D}_{0^+}^\alpha u(t) = f(t, u(t)), \quad t \in (0, 1] \setminus \{t_1, t_2, \dots, t_m \} \] with solutions satisfying the initial condition \[ \mathcal{I}_0^{1-\alpha}u(0) = u_0, \] and the impulsive conditions \[ \mathcal{D}_{0+}^{\beta}u(t_k+) - \mathcal{D}_{0+}^{\beta}u(t_k-) = J_k(u(t_k)), \quad k = 1, 2, \dots, m \] where \(0 < \beta < \alpha\). The existence of solutions for the first order ordinary differential equation \[ u'(t) = f(t, u(t)), \quad t \in (0, 1] \setminus \{t_1, t_2, \dots, t_m \} \] with solutions satisfying the initial condition \[ u(0) = u_0 \] and the impulsive conditions \[ \mathcal{D}_{0+}^{\beta}u(t_k+) - \mathcal{D}_{0+}^{\beta}u(t_k-) = J_k(u(t_k)), \quad k = 1, 2, \dots, m \] where \(0 < \beta < \alpha\) is recalled. The existence of solutions for the fractional order equation \[ D_{0+}^\delta u(t) = f(t, u(t)), \quad t \in (0, 1] \setminus \{t_1, t_2, \dots, t_m \}, \] with solutions satisfying the initial conditions \[ u(0) = u_0, \, u'(0) = u_1 \] and the impulsive conditions \[ \mathcal{D}_{0+}^{\gamma}u(t_k+) - \mathcal{D}_{0+}^{\gamma}u(t_k-)…
reviewed by: Eric R. Kaufmann
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 6186674
zbMATH Keywords: Caputo derivative
zbMATH Keywords: fixed point impulsive conditions
zbMATH Keywords: Riemann-Liouville derivative
MaRDI profile type: Publication
full work available at URL: s00025-012-0269-3
OpenAlex ID: W1965580586
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DOI: 10.1007/S00025-012-0269-3
description: scientific article; zbMATH DE number 6186674