Existence Results for a System of Fractional Differential Equations with Fractional Order Random Time Scale

INTRODUCTION

Consider the fractional differential equations with initial conditions as given below where SasD is the S time scale Riemann Liouville fractional derivative of order a. SsI the fractional integral of Riemann-Liouville and 00,Sss is an arbitrary interval on S. As a part of theoretical & potential applications, theory of time lamina calculus is involved to concern together difference and differential equation [1]. suppose that hs is a continuous function with right dense. Many authors have tried and proved the life and one-of-a-kindness of the first order differential equations with initial and boundary time scales conditions by various methods and criteria. The idea of this content arises from reference papers [10-24] in which Krasnoselskii-Krein and Nagumo conditions on non-linear term h, excluding Lipschitz assumption are exposed to derive the main results. Consider the first of the following orders ordinary differential equation with two classes and fractional differential equation with fractional order In section 2, few definitions and fundamental statements are added in such a way to prove main results. In section 3, the main theorem is illustrated. We first set up unique solution for first order problem under Krasnoselskii-Krein conditions. Then we extend the proof to successive approximation, which converge to unique solution.

We recollect basic consequence and definition from time lamina calculus. A chronometer Let card (S)>=2 is a non-empty closed subset of S. The forward and backward jump operators ,:SS are respectively defined by & The point sS is defined as follows

 

;leftdense;leftscattered ;rightdense;rightscattered

  

 Let

\max;whenadmitsaleftscatteredmaximum S;otherwise

 



Denote .sAAS SI is interval of ,S where I is an interval of .R Definition 2.1. Delta Derivative [1] Assume :hSR and let .sS Define provided the limit exist. Here hs is called delta derivative of h at.s Also, h is referred as delta differentiable on S provided h exists for all .ss The function :hsR is called the delta derivative of h on .s Definition 2.2. [6] A function :hSR Only if it is rd-continuous is it considered rd-continuous right dense point continuous in S and its left sided limits exists at left dense points in .S rdC denotes a Banach space with norm and a set of rd-continuous functions. Similarly, a function :hSR is called ld-continuous only if it is continuous at left dense point in S. The set of ld- continuous function :hSR is represented by ldC. For ,define .rdrdhChSuphs Definition 2.3. Delta antiderivative [6] A function :,SHR A function's delta antiderivative is referred to as a function's delta antiderivative.:,SHR provided H is continuous on ,,S delta-differentiable on ,S and Hshs for all ,.Ss Then we define the  integral oh h from to by

Suppose is a time scale, is an interval of 1 & f is an integrable function on ,. Let 01.a Then the left fractional integral of order a of f is defined by where  is a gamma function Definition 2.5. [Fractional Riemann Liouville Derivative on time Scale] Let S be a time scale, ,01,sSa and :.fSR And there was the left. Fractional derivative of order Riemann-Liouville a of f is defined by We can use SasI instead of 0SassI and SasI instead of 0SassDwhen 0.s Lemma 2.1. Let h be a non-decreasing continuous function on the ,.S We define extension h of h to the non-imaginary interval , by

,hshs for every ,.Ss Lemma 2.2. [5] Let be continuous. Then the general solution of the differential equation is given by (2.11)

Lemma 2.3. [6] For any function h integrable on 00,Sss, we have the following

Lemma 2.5. [6] Let 01a and 00:,.ShssRR The function v is a solution of problem (1.2) if and only if it is a solution of the following integral equation

Lemma 2.6. [22] The of the equation is given by where 01111and &1aRLsLaD is the fractional Riemann-Liouville derivative of order 0,1a on the interval 00,.ss

To prove the main result, define 000,:,,,,.TsysssyR

Theorem 3.1.1. (Conditions of Krasnoselskii-Krein) Let ,hsy be non-discontinuous in 0T and for all 0,,,sysyT satisfying for some positive constants c and k; also the non- imaginary number  which lies between 0 and 1 such that 11.k Then, the first order initial value problem (1.2) has only one solution on 00,.Sss

Suppose p and q are two solutions of (1.2) in 00,.Sss We have to prove that .pq Let us define s and Qs by Such that  is the extension of  to the real interval 00,.ss From condition (A2) that Consequently, since 000,0for ,and =c,QsQsssQss for every 00,.Ssss It is concluded from (3.18) and (3.19) that ,QscQs for every That is 1111QsQsdscQQsds. It is reduced to Hence That is the exponent of s in the above constraint is non-negative, since 111k. Hence 0lim0.sss Therefore if we define 00,s then the function is rd-continuous in 00,.Sss To prove 0 on 00,.Sss Assume that  does not disappear at some points s; that is 0s on 00,.ss Then there arise a maximum 0,n when s equals to some 1010:ssss such that 101,for ,.Stnstss From condition (A1), we have which is a contradiction. Hence, there exist unique solution. Theorem 3.1.2. Kooi‘s Condition Let ,hsy be non-discontinuous in 0T and for all 0,,,sysyT satisfying (B2) 0,,,bsshsyhsycyy for some positive constants c and k from real line. Also, real numbers ,b are defined as 01,and11.bkb Then the first order initial value problem (1.2) has only one solution on 00,.Sss Proof. Similar procedure from theorem 3.1.1 is followed here to prove the given statement.

Theorem 3.2.3. Assume that conditions (A1) and (A2) are satisfied, then the consecutive estimations given by Converge uniformly to the unique solution p of (1.2) on 00,,ss where min,/,N and N is the bound for h on 0.T Proof: Since we proved uniqueness in theorem 3.1.1, it is enough to prove existence of solution by Arzela-Ascoli theorem. Step:1 The consecutive approximations 1,0,1,2,...mpm given by (3.25) are well defined and continuous. This gives the following result for By induction, the sequence 1jps is well defined and uniformly bounded on 00,.Sss Step: 2 To prove x is continuous function in 00,,Sss where x is defined by For we have In (3.29), the right side expression in inequality is at most 2212XsNss for large m if0 provided that 21.2ssN For some arbitrary  and interchangeable 12,ss we get Hence X is continuous on 00,.Sss By condition (A2) and definition of successive approximations, we get The sequence mp is equicontinuous: that is 1200,,Sssss for each function mp and some positive . If there exist N such that 21,ss then The family jp fulfills all conditions of Arzela Ascoli theorem in 00,.rdScss Hence there exists a subsequence jkp converging uniformly on 00,.kSssasj Let us assume If 10,jjppasj then the limiting case of any subsequence is the only one solution [unique solution] p of (3.25). It follows that the entire sequence jp converges uniformly to .p To show that *0.()Xiens is null. Set and by denoting *.kQssXs To show that *0lim0.ss Hence *0 by absurdity. Assume that *0s for 00,;Ssss then there exists 1s such that

By condition (A1), Which is contradiction. So *0. Hence (3.25) converge uniformly to a unique solution  of (1.2) on 00,Sss by successive approximation.

Theorem 3.3.1. [Conditions of Krasnoselskii-Krein] Denote 0000,,|,,uSSCssRppCssR and 1000,,.vSsspCssR Let ,hsy be continuous in 0T and satisfying for all 0,,,sysyT (C1) 00,,,ahsyhsyklassyyss (C2) ,,hsyhsycyy where ,,clk are negative constants such that 1,kkla and 111k, and all real numbers  lies between 0 and 1. Then the fractional order initial value problem (1.3) has only one solution on 00,.ss Proof: Suppose p and q are two solutions of (1.3) in 00,.Sss To show that .pq To prove the result, define andsQs by Such that ˆ is the extension of  to the real interval 00,.ss From condition (B2), it follows Also 00ˆ,forevery,.SasSDQssQssss for every 00,.Ssss By (3.37) and (3.38) and using lemma 2.6, we get for every where andL are defined in lemma 2.6.



0.ksss We get for every 00,.sss Hence 0lim0.sss Therefore, if we define 00,s then the function is rd-continuous in 00,.Sss Next to show that 0. Assume contrarily  does not disappear at few points ;s that is 0s on 00,.Sss Then there exists a maximum 0n attained when s is equal to some 1010:ssss such that 101,for,.Stnstss By hypothesis (B1), we have 101knssss which is contradiction. Hence the solution is unique. Theorem 3.3.2. Conditions of Kooi‘s Let ,hsy be non-discontinuous in 0T and for all 0,,,sysyT satisfying (D1) 00,,,ahsyhsyklassyyss (D2) 0,,bsshsyhsycyy for some non-negative constants ,cl and ;k also the non- imaginary positive numbers ,,,bkl are such that 01b and 11&kb.kla Then, the first order initial value problem of first order FDE (1.3) has at most one solution on 00,.Sss

Assume that (C1) and (C2) are satisfied; then the consecutive approximation towards solution is given by tends to a finite limit uniformly to the unique solution p of (1.3) on 00,,ss where 11min,aaN and N is the bound for h on 0.T (3.43) Proof: Since uniqueness of the solution have been proved by theorem 3.3.1, we have to prove the existence of solution by Arzela Ascoli theorem. The successive approximation 1,0,1,2,...mpm given in (3.42) are properly defined and continuous. By mathematical induction, the flow of sequence 1jps is properly defined and uniformly bounded on 00,.Sss Step: 2 To prove X is continuous function in 00,,Sss where X is defined by

For 1200,,,Sssss we have That is

211if0giventhat.4

aamssN Since  is arbitrary and 12,ss can be interchangeable, then That is X continuous on 00,.Sss By condition (C2) and the definition consecutive approximations, we get therefore the sequence mp is equicontinuous. For each function mp and 0, 1200,,.Sssss If there exists 211;aassN then 11121221ammNpspsssa. Let us denote *1lim.jkjkknspsps Further, if 10as,jjppj then the limiting case of any subsequence is the unique solution p of (3.42). Let 01saascQsstXtdta and define *kssXs and then using lemma 2.6, we get that 10jkasLss which gives that *0lim0.ss And also proved that *0 by absurdity. presume that *0s at any point in 00,;Sss then there exist 1s such that 100**1,0max.Ssssnss For condition (C1), we obtain this is an inconsistency. (i.e.) *0. Hence Picard‘s successive approximation (3.42) tends to finite limit (uniform convergence) to unique solution p of (1.2) on 00,.Sss

Hence, we can establish the solution of non-linear FDE with order 0,1a by few basic named conditions.

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Assistant Professor, PG Department of Mathematics, Vellalar College for Women (Autonomous)