A study on the mild solution of impulsive fractional evolution equations. (English) Zbl 1410.34031
Summary: This paper is concerned with the formula of mild solutions to impulsive fractional evolution equation. For linear fractional impulsive evolution equations in many papers, described mild solution as integrals over \((t_k, t_{k + 1}]\) \((k = 1, 2, \dots, m)\) and \([0, t_{1}]\). On the other hand, in [J. Wang et al., Comput. Math. Appl. 64, No. 10, 3389–3405 (2012; Zbl 1268.34033); J. Optim. Theory Appl. 156, No. 1, 13–32 (2013; Zbl 1263.49038); Z. Liu and X. Li, J. Optim. Theory Appl. 156, No. 1, 167–182 (2013; Zbl 1263.93035)], their solutions were expressed as integrals over \([0,t]\). However, it is still not clear what are the correct expressions of solutions to the fractional order impulsive evolution equations. In this paper, firstly, we prove that the solutions obtained in many papers are not correct; secondly, we present the right form of the solutions to linear fractional impulsive evolution equations with order \(0 < \alpha < 1\) and \(1 < \alpha < 2\), respectively; finally, we show that the reason that the solutions to an impulsive ordinary evolution equation are not distinct.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
| 34G20 | Nonlinear differential equations in abstract spaces |
Keywords:
impulsive differential equations; Caputo fractional derivative; mild solutions; fractional partial differential equationsReferences:
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