Solvability of Fractional Order Semi-linear Stochastic Impulsive Differential Equation with State-Dependent Delay
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RESEARCH ARTICLE
Solvability of Fractional Order Semi-linear Stochastic Impulsive Differential Equation with State-Dependent Delay Mohd Nadeem1 • Jaydev Dabas1
Received: 7 August 2017 / Revised: 4 March 2019 / Accepted: 15 March 2019 Ó The National Academy of Sciences, India 2019
Abstract In this paper, we present a mathematical model of fractional order semi-linear functional stochastic differential equations with impulsive effects in a Hilbert space. The sufficient conditions of the existence and uniqueness of mild solutions for the considered problem are proved by using the classical fixed point theorems. Finally, we presented two examples to verify the results. Keywords Fractional order differential equation Stochastic functional differential equations Existence results Impulsive conditions Mathematics Subject Classification 26A33 34K50 34A12 34A37
1 Introduction In the last two decades, fractional differential equations have been proven to be a valuable tool in the modeling of various phenomena in the fields of science and engineering such as physics, electrochemistry, fluid dynamics, control theory, porous media, viscoelasticity and so forth (see [1–6], and references therein). On the other hand, differential equations of fractional order have made notable contributions for the description of memory and
& Mohd Nadeem [email protected] Jaydev Dabas [email protected] 1
Department of Applied Science and Engineering, IIT Roorkee, Saharanpur Campus, Saharanpur, U.P. 247001, India
hereditary properties of various materials and processes (see [7–14] and references therein). The stochastic problems incorporate the randomness into mathematical equations so they are better than deterministic ones as they describe the problems more accurately as in [15–17]. Hence, stochastic differential equations appear as a natural description of several realworld phenomena. Nowadays, stochastic differential equations have been considered to attention in both aspects applied as well as theoretical disciplines of science and engineering [18–21]. The impulsive effects also exist widely in stochastic differential systems in various fields, such as mechanical systems, engineering, biology, economics, finance and population dynamics, when abrupt changes arise in their state at certain moments of time between intervals of continuous evolution [19, 20, 22–24]. The delay differential equations arise in various disciplines such as biological and physical sciences, and they often force us to consider variable delay or state-dependent delays. The stochastic model with such delay is important from the applications point of view, where numeric real-world problems are converted into this type of mathematical models. A very few works are reported with fractional model of stochastic problem with variable delay [25, 26] and that motivated us to study such problems. Recently, many workers report the modeling of fractional (order a 2 ð0; 1Þ) stochastic differential equations in all possible aspects such as controllability, st
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