Existence and Ulam stability for random fractional integro-differential equation
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Existence and Ulam stability for random fractional integro-differential equation Le Si Dong1 · Ngo Van Hoa2,3 · Ho Vu1 Received: 3 August 2019 / Accepted: 2 May 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this paper, we prove the existence and uniqueness of solution for random fractional integrodifferential equation. The existence of at least mean square continuous solution for this problem is discussed. Using the fixed point theorem, Ulam–Hyers stability and Ulam–Hyers– Rassias stability of random fractional integro-differential equation are studied. Finally, we give an example to illustrate our results. Keywords Random fractional differential equation · Second-order stochastic processes · Mean square continuous solution Mathematics Subject Classification 34A12 · 34A30 · 34D20
1 Introduction Fractional differential equations (FDEs) can serve as useful tools for description of mathematical modelling of systems and processes in the fields of mechanics, physics, economics and control theory, etc. For more details, the reader can find in the books (see e.g. Kilbas et al. [11], Podlubny [13]) and the references therein. Hafiz et al. [8] have constructed the concept of stochastic fractional calculus in the mean square sense, for which some properties similar to that of the deterministic fractional calculus are considered. Hafiz [7] has studied the mean square Caputo ( or Riemann–Liouville) fractional integration for mean square integrable stochastic processes. Author has discussed the mean square fractional derivative in the sense of Caputo (or Riemann–Liouville). El-Sayed et al. [5] have discussed the local existence of a unique mean square continuous solution for delay stochastic fractional differential equation. The continuous dependence of the solution
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Ngo Van Hoa [email protected]
1
Faculty of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam
2
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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L. S. Dong et al.
on the initial-valued problem is studied. In [6], authors have proved the existence of solution of random fractional differential equation with nonlocal condition. Vu et al. [15] proved the existence and uniqueness of solution for random fractional differential equation with impulses via Banach fixed point technique and Schauder fixed point technique. Moreover, authors also presented for continuous dependence of the solution of initial data. The construction of numerical solutions of random fractional differential equations in mean square sense has been interested in recent years by many mathematicians. We refer to the readers [1–3,10]. In [1,3], authors have solved the general random fractional linear differential equation under general assumptions on random input data. A mean square rand
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