Infinite delay fractional stochastic integro-differential equations with Poisson jumps of neutral type
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ORIGINAL RESEARCH PAPER
Infinite delay fractional stochastic integro-differential equations with Poisson jumps of neutral type R. Jahir Hussain1 • S. Satham Hussain1 Received: 22 July 2020 / Accepted: 20 October 2020 Forum D’Analystes, Chennai 2020
Abstract This paper addresses the study of nonlinear fractional stochastic neutral integrodifferential equations with Poisson jumps and infinite delay in Hilbert space. Initially, the existence and uniqueness of mild solution for the proposed system are verified by applying the suitable conditions of sectorial operators, fixed point technique and stochastic analysis with non-Lipschitz condition and Lipschitz condition as a special case. Also, continuous dependence of mild solutions for proposed model is determined. Finally an example is provided to illustrate the developed theory. Keywords Existence of mild solutions Fractional differential equations Infinite delay Poisson jumps Sectorial operators
Mathematics Subject Classification 34A08 93E03 34A12 34K40 60G57
1 Introduction In the last two decades, fractional calculus is one of the best tool to characterize long memory processes and materials, anomalous diffusion, long range interactions, powerlaws, economics, chemical technology, medicine and many other related fields and the corresponding mathematical models are fractional differential equations (FDEs) which involves fractional derivative. FDEs arises frequently in a variety of branches of science and engineering problems (see [11–13]). One of the & S. Satham Hussain [email protected] R. Jahir Hussain [email protected] 1
PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Trichy, Tamil Nadu 620 020, India
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R. J. Hussain, S. S. Hussain
emerging problem in the field is the theory of fractional evolution equations and these equations are motivated by the applications to some problems in fluid dynamics traffic model, heat conduction in materials with memory, etc (see [11, 17]). Every one knows that the deterministic system often fluctuate due to noise. Hence it is mandatory to develop the system with stochastic ones. Familiar that arbitrary fluctuations are regular in the real world, scientific (mathematical) models for complex systems are frequently subject to instabilities. Stochastic differential equations (SDEs) are important to modelling the real life phenomena where there is a need for an aspect of randomness. SDEs in infinite dimensional spaces are motivated by the random phenomena studied in the natural sciences like physics, chemistry, mathematical finance and in control theory. To model these stochastic phenomena, researchers used Wiener process and Poisson jumps. SDEs with Poisson jumps become very popular in modelling and widely used to describe the asset and commodity price dynamics (see [3]). The existence, uniqueness, stability and qualitative analysis of the mild solutions of SDEs are studied by many authors (see [4–6, 14–17]). Integro-different
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