MONOTONE ITERATIVE TECHNIQUE FOR FINITE SYSTEM OF FRACTIONAL DIFFERENCE EQUATIONS

In this paper, we consider non-linear fractional finite difference system and establish the existence of solutions using monotone iterative technique. AMS Subject Classification: 39A10, 39A99


Introduction
Fractional calculus gained importance during the past three decades due to its applicability in diverse fields of science and engineering.The notions of fractional calculus may be traced back to the works of Euler, but the idea of fractional difference is very recent.G.V.S.R. Deekshitulu and J. Jagan Mohan [2] modified the definition of fractional difference given by Atsushi Nagai [11] and discussed some basic inequalities, comparison theorems and qualitative properties of the solutions of fractional difference equations [2,3,4,5,6].
Being an important tool in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of finite systems of differential equations and difference equations, comparison theorems and their applications have attracted great interests of many mathematicians.The investigation of scalar fractional order difference inequalities was initiated by G.V.S.R. Deekshitulu and J. Jagan Mohan [2].
In this paper, we use monotone iterative technique for finite system of fractional difference equations and derive the convergence of monotone sequences to upper and lower solutions.Then using these upper and lower solutions, existence of solutions to finite system of fractional difference inequalities to obtain more general results.

Preliminaries
In this section, we introduce some basic definitions and results concerning nabla discrete fractional calculus.Definition 2.2.For any complex number α and f defined over the integer set {a − p, a − p + 1, ....n}, the α th order difference of f (n) over {a, a + 1, ....n} is defined by Later, Hirota took the first n terms of Taylor series of ∆ α −n = ε −α (1 − B) α and gave the following definition.Definition 2.3.Let α ∈ R. Then difference operator of order α is defined by Here a n , (a ∈ R, n ∈ Z) stands for a binomial coefficient defined by In 2002, Atsushi Nagai [1] introduced another definition of fractional difference which is a slight modification of Hirota's fractional difference operator.
Definition 2.4.Let α ∈ R and m be an integer such that m − 1 < α ≤ m.The difference operator ∆ * ,−n of order α is defined as G.V.S.R.Deekshitulu and J.Jagan Mohan [2] rearranged the terms in Atsushi Nagai's [11] definition for 0 < α < 1 in such a way that the expression for ∇ α does not involve any difference operator and the term (−1) j inside the summation index as follows.
Definition 2.5.The fractional sum operator of order α is defined as (2.5) Definition 2.6.The Caputo type fractional difference operator of order α is defined as Corollary 1.The equivalent form of (2.6) is (2.7) Theorem 2.1.(Discrete Langenhop Inequality) Let y(n), a(n) and b(n) be any three non negative functions defined for n ∈ N + 0 .If for n, k ∈ N + 0 such that k ≤ n the following inequality be satisfied where y(n) is not necessarily non negative.Then, for all n, k ∈

Main Results
In this section, we consider the following finite system of non linear fractional difference equations of order α, 0 < α < 1 where u and f are n-vectors with components u i (n) : We first establish some basic vectorial inequalities.Vectorial inequalities mean that the same inequalities hold between the corresponding components.We need the following properties in this study.
) is said to be quasi monotone non-decreasing (non-increasing) if for any fixed u i (i = 1, 2) f i is non-decreasing (non-increasing) in u j for j = i.
It means that if for all i,f i are quasi monotone non-decreasing( quasi monotone non increasing), then f is said to be quasi monotone non decreasing quasi monotone non increasing).
Similarly a function w(n) : N + 0 → R n is said to be an upper solution of (3.1) if Theorem 3.1.Let the function f (n, v(n)) be quasi monotone non-decreasing and v(n) and w(n) be lower and upper solutions of (3.1) defined on N + 0 such that v(0) ≤ w(0).Then for all n ∈ N + 0 , v(n) ≤ w(n).
Proof.Suppose there exists a k ∈ N + 0 such that v(k) ≤ w(k) and v(k +1) > w(k + 1).Since 0 < α < 1 ≤ j and j−α j > 0, we have . This is a contradiction to the quasi monotone property of f (n, u(n)).Hence the proof.
Further, if u(n) is a solution of (3.1), then the by repeatedly applying the above theorem, we obtain Proof.If it is false, then there exists a k ∈ N + 0 such that m k > 0 and m n ≤ 0 for n < k.Consider

Monotone iterative technique:
Now we shall apply a general theory of monotone iterative technique for finite system of fractional difference equations of order α(0 < α < 1.) We need the following notions.
For each i, 1 ≤ i ≤ n , let p i , q i be two non-negative integers such that p i + q i = n − 1 so that we can split the vector u into u = (u i , [u] pi , [u] qi ).Then the system (3.1)canbe written as ) then v, w are said to be coupled lower and upper quasi solutions of (3.4), if v and w are said to be coupled quasi solutions of (3.4) if ] is said to possess a mixed quasi monotone property (mqmp), if for each i, f i (n, u i , [u] pi , [u] qi ) is monotone non-decreasing in [u] pi components and monotone non-increasing in [u] qi components.
In particular f (n, u) is said to possess non-decreasing property if f i (n, u) is non-decreasing in u 1 , u 2 , ..., u n for all fixed n ∈ N + 0 .Theorem 3.3.Let the function f (n, u) possess mixed monotonic property.Further, let there exists functions v(n) and w(n) defined on N + 0 such that Then for all n Proof.Define a function z(n) as follows: (3.5) By using principle of mathematical induction we shall show that z i (n) ≥ 0 for all n ∈ N + 0 .We have, v p (0) ≤ w p (0), v q (0) ≥ w q (0) i.e.,z p (0) ≥ 0, z q (0) ≥ 0. Let v p (m) ≤ w p (m), v q (m) ≥ w q (m) for some fixed m ∈ N + 0 or z p (m) ≥ 0, z q (m) ≥ 0. Now consider ≤ w p (m + 1).
By principle of mathematical induction, z i (n) ≥ 0 for all n ∈ N + 0 i.e., z(n) ≥ 0 for all n ∈ N + 0 implies v p (n) ≤ w p (n), v q (n) ≥ w q (n).Hence the proof.
Theorem 3.4.Let the assumptions of the Theorem 3.2 hold good and If u is a solution of (3.4), then the result follows from Theorems (3.1) and (3.2).
possess mixed quasi monotone property,and let v 0 , w 0 be coupled lower and upper quasi solutions of system (3.4) on N + 0 .Suppose further that whenever v 0 ≤ u ≤ w 0 and v 0,i ≤ s i ≤ u i ≤ w 0,i and M > 0. Then there exists monotone sequences {v n }, {w n } such that v n → v and w n → w as n → ∞ uniformly and monotonically to coupled minimal and maximal quasi solutions of (3.4)on N + 0 provided v 0 (0) ≤ u 0 ≤ w 0 (0).Further if u is any solution of (3.4) such that v 0 ≤ u ≤ w 0 on N + 0 then v ≤ u ≤ w on N + 0 .
Proof.For any y, z ∈ C[N + 0 , R n ] such that v 0 ≤ y, z ≤ w 0 on N + 0 ,we define Clearly, (3.7) is a non-homogeneous equation in u i and has an unique solution [8].In order to construct and establish the convergence of the monotone sequences {v n }, {w n }, we define a mapping A such that A[y, z] = u, where u is the unique solution of (3.7).This sequences {v n }, are {w n } derived in the following steps. (a).
(b). 'A' possesses the mixed quasi monotone property on the segment [v 0 , w 0 ], where the segment To prove (a): Let v 1 be the unique solution of (3.7) with y = v 0 , z = w 0 .Then A[v 0 , w 0 ] = v 1 .Now let p i = v 0,i − v 1,i and consider

And also p
Hence by Theorem 3.2, we have Similarly we can prove w 0,i ≥ A[w 0,i , v 0,i ].
To prove (b): Let us take y 1 , y 2 , z ∈ [v 0 , w 0 ] be such that Using similar arguments as above, we have p i+1 ≤ 0. Thus u Consequently this implies that A[y, z] ≤ A[z, y] for y ≤ z.Or A satisfies mixed quasi monotone property.
Continuing this way using (a) and (b) ,we can define the sequences Since the sequences are monotonic and hence by Dini's theorem they converge uniformly to coupled quasi solutions of (v, w) of (3.7).Therefore Now we show that (v, w) are coupled minimal and maximal quasi solutions of (3.4) respectively.Let (u 1 , u 2 ) be any coupled quasi solutions of (3.4) such that u 1 , u 2 ∈ [v 0 , w 0 ].Let us suppose that for some integer k > 0,v k ≤ u 1 , u 2 ≤ w k on N + 0 .Now we take p i = v k+1,i −u 1,i and using the mixed quasi monotone property of f we have Also p i (0) ≤ 0 implies that p i ≤ 0. Thus v k ≤ u 1 .Similarly we can prove u 1 ≤ w k+1 also.By the principle of mathematical induction, it can be proved that, for every m ∈ N + 0 , v m (n) ≤ u 1 (n), u 2 (n) ≤ w m (n).As m → ∞, we have v(n) ≤ u 1 (n), u 2 (n) ≤ w(n).Hence the functions v(n) and w(n) defined on n ∈ N + 0 are minimal and maximal quasi solutions of (3.4) respectively.Since any solution u ∈ [v 0 , w 0 ] can be considered as (u, u) coupled quasi solutions of (3.4), we also have v ≤ u ≤ w on n ∈ N + 0 .This completes the proof.

Definition 2 . 1 .
The backward difference operator ∆ −n is defined as ∆ −n = ε −1 (1 − B) where Bf n = f n−1 is standard backward shift operator and ε is interval length.Henry L Gray and Nien fan Zhang gave a definition of the fractional difference as follows: