QUALITATIVE RESULTS FOR FINITE SYSTEM OF R-L FRACTIONAL DIFFERENTIAL EQUATIONS WITH INITIAL TIME DIFFERENCE

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Qualitative results such as existence and uniqueness of finite system of Riemann-Liouville (R-L) fractional differential equations with initial time difference are obtained. Monotone technique coupled with method of lower and upper solutions is developed to obtain existence and uniqueness of solutions of finite system of R-L fractional differential equations with initial time difference.


INTRODUCTION
Due to wide applications of fractional calculus in sciences, engineering, nature and social sciences numerous methods of solving fractional differential equations are developed [1,8,9]. V.Lakshmikantham et.al [6] obtained local and global existence results for solutions of Riemann-Liouville fractional differential equations.The Caputo fractional differential equation with periodic boundary conditions have been studied in [3,4] and developed monotone method.
Existence and uniqueness of solution of Riemann-Liouville fractional differential equation with integral boundary conditions is proved in [11,12]. Monotone method for Riemann-Liouville fractional differential equations with initial conditions is established by McRae [7]. Vasundhara Devi considered [2] the general monotone method for periodic boundary value problem of Caputo fractional differential equation. Recently, initial value problems involving Riemann-Liouville fractional derivative was studied by authors [5,13]. Yaker et.al. proved existence and uniqueness of solutions of fractional differential equations with initial time difference for locally Holder continuous functions [14]. Authors have generalized these results for the class of continuous functions [10] and extended for system. In this paper, we consider the finite system of Riemann-Liouville fractional differential equations with initial time difference when the function on the right hand side is mixed quasi-monotone and construct two monotone convergent sequences to obtain existence and uniqueness of solution for the finite system.
The paper is organized as follows: In section 2, basic definitions and results are given. Section 3 is devoted to obtain main results .

BASIC RESULTS
Some basic definitions and results used for the development of monotone technique for the problem are given in this section.
The Riemann-Liouville fractional derivative of order q(0 < q < 1) [?] is defined as Let m ∈ C p (J, R) and for any t 1 ∈ (t 0 , T ] we have m(t 1 ) = 0 and m(t) < 0 for t 0 ≤ t ≤ t 1 . Then it follows that D q m(t 1 ) ≥ 0. and (ii) D q w(t) ≥ f (t, w(t)), t 0 < t ≤ T. Assume f (t, u) satisfy one sided Lipschitz condition The results proved by Yakar et.al. for the following problem , are generalized by authors [10] for class of continuous functions u(t).
The corresponding Volterra fractional integral is given by and that every solution of (2.2) is a solution of (2.1).
In this paper, we develop monotone technique coupled with lower and upper solutions for the class of continuous functions for the following finite system of Riemann-Liouville fractional differential equations with initial time difference and obtain existence and uniqueness of solution for the problem.
1 − q are said to be ordered lower and upper solutions When either r i or s i is equal to zero a special case of the mixed quasimonotone property is defined as follows:

QUALITATIVE RESULTS
This section is devoted to develop monotone method for system of Riemann-Liouville fractional differential equations with initial time difference and obtain existence and uniqueness of solution of the problem (2.3).
Then there exists monotone sequences {v n (t)} and {w n (t)} such that where v(t) and w(t) are minimal and maximal solutions of the problem (2.3) respectively.
Since f (t, u 1 , u 2 , . . . , u N ) is quasimonotone nondecreasing in t for each u i we have and and which proves that v i0 and w i0 are lower and upper solutions of IVP (2.3) respectively.
For each η(t) and µ(t) in C p (J, R) such that v 0 is the unique solution of the problem (3.1). Firstly, we prove that A possesses the monotone property on the segment is the unique solution of system (3.1) with and p i (t) ≤ 0.
Applying Corollary 2.1, we get p i (t) ≤ 0 on 0 ≤ t ≤ T and hence v 0 and p i (t) ≤ 0.
Thus it follows that the mapping A possesses monotone property on the segment [v 0 , w 0 ].
It follows from Corollary 2.1 that p i (t) ≤ 0, which gives v k Thus the sequences {v n (t)} and {w n (t)} are bounded from below and bounded from above respectively and monotonically nondecreasing and monotonically non-increasing on J. Hence point-wise limit exist and are given by Using corresponding Volterra fractional integral equations It follows that v(t) and w(t) are solutions of system (2.3). Lastly, we prove that v(t) and w(t) are the minimal and maximal solution of the problem (2.3).
Let u(t) = (u 1 , . . . , u N ) be any solution of (2.3) other than v(t) and w(t), so that there exists k and p i (t) ≥ 0.
Thus v k+1 i (t) ≤ u i (t) on J. Since v 0 i (t) ≤ u i (t) on J, by induction it follows that v k i (t) ≤ u i (t) for all k. Similarly, we can prove u i ≤ w k i for all k on J. Hence v k i (t) ≤ u i (t) ≤ w k i (t) on J. Taking limit as n → ∞, it follows that v i (t) ≤ u i (t) ≤ w i (t) on J. Now, we obtain the uniqueness of solution of the problem (2.3) in the following