Initial value problems of fractional order with fractional impulsive conditions
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 \[ \mathcal{D}_{0+}^{\gamma}u(t_k+) - \mathcal{D}_{0+}^{\gamma}u(t_k-) = J_k(u(t_k)), \quad k = 1, 2, \dots, m \] where \(\delta \in (1, 2)\), \(0 < \gamma < 1\) and \(D_{0+}^\delta\) is the Caputo derivative operator of order \(\delta\) defined by \[ \begin{aligned} D_{0+}^\delta u(t) = & \left ( \mathcal{D}_{0+}^\delta \left [ u(s) - \sum_{k=0}^{n-1} \frac{u^(k)(0)}{k!} s^k \right ] \right ) (t)\\ = & \mathcal{D}_{0+}^\delta u(t) - \sum_{k=0}^{n-1} \frac{u^{(k)}(0)}{\Gamma(k-\delta + 1)} t^{k-\delta}\\ = & \frac{1}{\Gamma(n-\delta)} \int_0^t \! (t-s)^{n-\delta-1}u^{(n)}(s) \, ds, \end{aligned} \] where \(n = [\delta] + 1\) and the last inequality holds if \(u \in AC^n[0, 1]\) is studied. This paper is self-contained and overall well written. In the second section the author states the problems as well as the definitions of both fractional differential operators and the Riemann-Liouville integral operators. In Section 3, the author gives a counterexample showing that certain ``inversion'' of impulsive fractional differential equations are not correct. In the fourth section, the author states technical results, including the definitions of what is meant by a solution to the impulsive initial value problems, and demonstrates the proper way to invert the problems. The author also includes the statements of the fixed points theorems that are employed to show the existence of solutions. Finally, in Sections 5 and 6, the author states and proves existence theorems for the initial value problems. This manuscript accurately describes a method to invert a fractional order differential equation with impulse conditions. It points out errors in previous results. This paper is an essential manuscript for anyone studying fractional order equations with impulsive conditions.
scientific article; zbMATH DE number 6186674
scientific article