EXISTENCE RESULT FOR INITIAL VALUE PROBLEM WITH NONLINEAR FUNCTIONAL RANDOM FRACTIONAL DIFFERENTIAL

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Proceedings of 1st Shri Chhatrapati Shivaji Maharaj QIP Conference on Engineering Innovations Organized by Shri. Chhatrapati Shivaji Maharaj College of Engineering, Nepti, Ahmednagar In Association with JournalNX - A Multidisciplinary Peer Reviewed Journal, ISSN No: 2581-4230 21st - 22nd February, 2018

EXISTENCE RESULT FOR INITIAL VALUE PROBLEM WITH NONLINEAR FUNCTIONAL RANDOM FRACTIONAL DIFFERENTIAL EQUATION Dr. M. K. Bhosale1, Mr. S.S.Parande2 1(Dept. of Mathematics,Shri. ChhatrapatiShivajiMaharaj College of Engineering Dist. Ahmednagar/Pune University, Maharashtra. India, email: mrs.megha_shinde@rediffmail.com) 2(Dept. of Mathematics,Shri. ChhatrapatiShivajiMaharaj College of Engineering Dist. Ahmednagar/Pune University, Maharashtra. India) ABSTRACT:- In this paper we prove the existence result for initial value problems with nonlinear functional random fractional differential equations under Caratheodory condition .

a mapping x: â„Śâ&#x;ś E is measurable if for any B ∈ βE one has

KEY WORDS: - Random fractional differential equation, fractional integral, Caputo fractional derivative etc. 2000 MATHEMATIC SUBJECT CLASSIFICATION:- 26A33, 47H10. 1. INTRODUCTION:The linear as well as nonlinear initial value problem of random differential equations have been studied in the literature by the authors since long time refer a Dhage[4-6]. Similarly the fraction differential equation are frequently use in many branches of engineering and science It has been mentioned first by Liouville in a paper from 1832. There are real world phenomena with anomalous dynamics such as signals transmission, network traffic and so on. In this case the theory of fractional differential equation is a good tool for modeling such as phenomena. For some fundamental result in the theory of fractional differential equations. We refer paper of Lakshmikantham [9, 10, 11] and [13,14]. Let â„? denote the real line and Let I0 = [−r, 0] and I = [0, T] be two closed and bounded interval in â„? for some r> 0 and T > 0. Let đ??˝ = I0 UI. Let C(I0 , â„?) denote the space of continuous â„? valued function I0 . We equip the space C = C(I0 , â„?) with a supremum norm âˆĽ.âˆĽc defined by âˆĽxâˆĽc = sup |x(t)|

Where đ?’œ Ă— βE is the direct product of the Ďƒ- algebras A and βE those defined in â„Ś and E respectively.

tĎľI0

Clearly C is a Banach Space which is also a Banach Space with respect to this norm. For a given t ĎľI define a continuous R-valued function. xt : I0 → â„? by xt (θ) = (t + θ) , θ ĎľI0 Let (Ί, A) be a measurable space i.e. a set đ?›ş with a đ?œŽalgebra of subset of đ?›ş and for given a measurable function xâˆś Ί → C(J, â„?). Consider nonlinear functional random fractional differential equations of the form (in short RFDE) cDÎą x( t , ω )= f( t, x (ω), ω) a.e t∈ J , 0<Îą<1 t x(0, ω) = x0 (ω) (1.1) Where x is a random function; x0 is random, DÎą x is the Caputo fractional derivative of x with respect to the variable t ∈ J and f: J Ă— â„? Ă— Ί → â„? is given function. 2.

EXISTENCE RESULT:Let E denote a Banach space with the norm ||.|| and let Q: Eâ&#x;śE. We further assume that the Banach space E is separable i.e. E has countable dense subset and let βE be the Ďƒ − algebra of Borel subset of E. We say

x-1(B) = {(ω, x) ∈ â„Ś Ă— E : x (ω, x) ∈ B } ∈ đ?’œ Ă— βE

Let Q: â„Ś Ă— Eâ&#x;ś E be a mapping. Then Q is called a random operator if Q (ω, x) is measurable in ω for all x∈E and it is expressed as Q(ω) x = Q(ω, x). A random operator Q (ω) on E is called continuous (resp. compact, totally bounded and completely continuous) If Q(ω, x) is continuous (resp. compact, totally bounded and completely continuous) in x for all ω Ďľ â„Ś. Ě…Ě…Ě…Ě… Lemma 2.1[12]: Let BR (0) and B R (0) be the open and closed ball centered at origin of radius R in the separable Banach space E and let Q: â„ŚĂ— Ě…Ě…Ě…Ě… BR (0)â&#x;ś E be a compact and continuous random operator. Further suppose that there does not exists an uĎľ E with ||u||=R such that Q (ω) u = Îť u for all Îť Ďľ â„Ś where Îť > 1. Then the random equation Q(ω)x = x has a random solution, i.e. there is a Ě…Ě…Ě…Ě… measurable function đ?œ‰:â„Śâ&#x;śB such that R (0) Q(ω)đ?œ‰(ω)=đ?œ‰(ω) for all ωϾ â„Ś . Lemma 2.2[12]: (CarathĂŠodory) Let Q: â„Ś Ă— E â&#x;ś E be a mapping such that Q(.,x) is measurable for all xĎľ E and Q(ω,.) is continuous for all ω Ďľâ„Ś Then the map (ω, x) â&#x;śQ(ω, x) is jointly measurable. We seek random solution of (1.1) in Banach space C (J, â„?) of continuous real valued function defined on J. We equip the space C ( J, â„?) with the supremum norm||.|| defined by ||x|| = suptĎľâ„?|x(t)| It is known that the Banach space C (â„?, â„?) is separable. By L1 (â„?, â„?) we denote the space of Lebesgue measurable real-valued function defined on â„?. By ||.||L1 we denote the usual norm in L1 (â„?, â„?) defined by 1 ||x||L1= âˆŤ0 |x(t)|dt. We need the following definition in the sequel. Definition 2.1: A CarathĂŠodory function f :â„? Ă— â„? Ă— â„Ś â&#x;śâ„? is called random L1- CarathĂŠodory if for each real number r>0 there is a measurable and bounded function hr : â„Ś â&#x;ś L1(â„?,â„?) such that |f (t, x, ω)| ≤ hr (t, ω) a. e.t Ďľ â„? . Where |x|≤ r and for all ω Ďľ â„Ś We consider the following set of hypothesis H1) The function (t, x) → f(t, x, ω) is continuous for a.e. Ď‰âˆˆđ?›ş.

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