Comparison Theorems and Asymptotic Behavior of Solutions of Discrete Fractional Equations

Consider the following ν-th order nabla and delta fractional difference equations ∇ ν ρ(a) x(t) = c(t)x(t), t ∈ N a+1 , x(a) > 0. (*) and ∆ ν a+ν−1 x(t) = c(t)x(t + ν − 1), t ∈ N a , x(a + ν − 1) > 0. (* *) We establish comparison theorems by which we compare the solutions x(t) of (*) and (* *) with the solutions of the equations ∇ ν ρ(a) x(t) = bx(t) and ∆ ν a+ν−1 x(t) = bx(t + ν − 1), respectively, where b is a constant. We obtain four asymptotic results, one of them extends the recent result [F. These results show that the solutions of two fractional difference equations ∇ ν ρ(a) x(t) = cx(t), 0 < ν < 1, and ∆ ν a+ν−1 x(t) = cx(t + ν − 1), 0 < ν < 1, have similar asymptotic behavior with the solutions of the first order difference equations ∇x(t) = cx(t), |c| < 1 and ∆x(t) = cx(t), |c| < 1, respectively.


Introduction
Discrete fractional calculus has generated much interest in recent years. Some of the work has employed the fractional forward and delta difference operators. We refer the readers to [1,4], for example, and more recently [6,8]. Probably more work has been developed for the backward or nabla difference operator and we refer the readers to [5,7]. There has been some work to develop relations between the forward and backward fractional operators, ∆ ν a and ∇ ν a (see [2]) and fractional calculus on time scales (see [4]). This work is motivated by F. Atici and P. Eloe [3] who obtained asymptotic results for the fractional difference equation ∇ ν ρ(a) x(t) = bx(t), 0.5 ≤ ν ≤ 1, t ∈ N a with 0 < b < 1, x(a) > 0. We shall consider the following ν-th order nabla and delta fractional difference equations x(a) > 0. (1.1) and x(a + ν − 1) > 0. (1.2) We establish comparison theorems by which we compare the solutions x(t) of (1.1) and (1.2) with the solutions of the equations ∇ ν ρ(a) x(t) = bx(t) and ∆ ν a+ν−1 x(t) = bx(t + ν − 1), respectively, where b is a constant. We obtain the following asymptotic results in which Theorem A extends the recent result of Atici and Eloe [3]. This shows that the solutions of two fractional difference equations ∇ ν ρ(a) x(t) = cx(t), 0 < ν < 1, and ∆ ν a+ν−1 x(t) = cx(t + ν − 1), 0 < ν < 1, have similar asymptotic behavior with the solutions of the first order difference equations ∇x(t) = cx(t), |c| < 1 and ∆x(t) = cx(t), |c| < 1, respectively.
2 Asymptotic behavior, nabla case, 0 < b ≤ c(t) < 1 Let Γ(x) denote the gamma function. Then we define the rising function (see [10]) by for those values of t and r such that the right-hand side of this equation is well defined. We also use the standard extensions of their domains to define these functions to be zero when the numerator is well defined, but the denominator is not defined. We will be interested in functions defined on sets of the form where a ∈ R. The delta and the nabla integral of a function f : where b ∈ N a . We will use elementary properties of these integrals throughout this paper (see Goodrich and Peterson [8] for these properties). The nabla fractional Taylor monomial of degree ν based at ρ(a) := a − 1 (see [8]) is defined by The following definition of the discrete Mittag-Leffler function is given in Atici and Eloe [3] (see also [8]). Definition 2.1. For |p| < 1, 0 < α < 1, we define the discrete Mittag-Leffler function by To study the asymptotic behavior of the solutions of (2.3) for the case 0.5 ≤ ν ≤ 1, the authors in [3] used the Laplace transformation, the convolution theorem and the properties of a hypergeometric function. They proved that the solutions of the fractional difference equation A natural question arises: if 0 < ν < 0.5 and |b| < 1, then how about the asymptotic behavior of the solutions of equation (2.3)? In this paper we will answer this question and related questions. First we will establish a useful comparison theorem. We will use the following lemma which appears in [8].
for t ∈ N a .
Proof. From Lemma 2.2, we have (2.1) In the following, we first prove that the infinite series for each fixed t is uniformly convergent for s ∈ [ρ(a), t].
We will first show that Also consider Note that for large k it follows that we get by the Root Test that for each fixed t the infinite series (2.2) is uniformly convergent for s ∈ [ρ(a), t]. So from (2.1), integrating term by term, we get, (using ∇ ν ρ(a) H νk+ν−1 (s, ρ(a))) = H νk−1 (s, ρ(a))), This completes the proof.
Atici and Eloe [3] gave a formal proof of the following result using Laplace transforms. With the aid of Lemma 2.3 we now give a rigorous proof of this result. .
This completes the proof.
The following comparison theorem plays an important role in proving our main results.
respectively, for t ∈ N a+1 satisfying x(a) ≥ y(a) > 0, then Proof. For simplicity, we let a = 0. From Lemma 2.2, we have for t = k Using (2.4) and (2.5), we have that We will prove x(k) ≥ y(k) ≥ 0 for k ∈ N 0 by using the principle of strong induction. When i = 0, from the assumption, the result holds. Suppose that x(i) ≥ y(i) ≥ 0, for i = 0, 1, . . . , k − 1. Since This completes the proof.
Proof. From Lemma 2.4, we have and From the comparison theorem (Theorem 2.5), we get that This completes the proof.
The following lemma is from [11, page 4].
The following lemma gives an asymptotic property concerning the nabla fractional Taylor monomial.
Proof. Taking t = a + 1 + n, n ≥ 0, we have Using Lemma 2.7, we have Using (2.11), we complete the proof.
Since there are only a finite number of k which satisfy k < 1−ν ν , from Lemma 2.8 and the definition of E b,ν,ν−1 (t, ρ(a)), we obtain the following theorem.

1)
for t ∈ N a−N+1 (note by our convention on sums the second term on the right-hand side is zero when N=1).
Proof. Using the power rule ( and integrating by parts, we have By applying integration by parts N − 1 more times, we get Using Leibniz's rule N − 1 more times, we get  Taking N = 1 in Lemma 3.1, we get that the following corollary holds.
Proof. Applying the operator ∇ −ν a to each side of equation (3.5) we obtain which can be written in the form Using Corollary 3.2, we get that H −ν (a, ρ(s))y(s)∇s = H −ν (a, ρ(a))y(a) = y(a), we get that
From Lemma 2.4 and Theorem B, we can obtain the following corollary.

Corollary 3.3. Assume that
is the discrete Mittag-Leffler function.
Remark 3.4. The above corollary is not obvious, since E −b,ν,ν−1 (t, ρ(a)) is an infinite series whose terms change sign.
Note that if we let x(t) be a solution of the ν-th order fractional nabla equation

Asymptotic behavior, delta case, c(t) ≥ b > 0
In this section we will be concerned with the asymptotic behavior of solutions of the ν-th order delta fractional difference equation Let Γ(x) denote the gamma function. Then we define the falling function (see [10]) by respectively, for those values of t and r such that the right-hand sides of these equations are well defined. We also use the standard extensions of their domains to define these functions to be zero when the numerators are well defined, but the denominator is not defined. The delta fractional Taylor monomial of degree ν based at a (see [8]) is defined by First we will establish a useful comparison theorem. The following lemma is from [8].  The following comparison theorem plays an important role in proving our main result.
Note that if we let ν = 1 in Theorem 4.3 we get the known result that y(t) = a 0 e b (t, a) is the unique solution of the IVP ∆y(t) = by(t), t ∈ N a y(a) = a 0 .
In the following corollary (see [6]) we give a simplification of the formula for the solution given in Theorem 4.3.
Corollary 4.5. Assume 0 < ν < 1, b is a constant and a 0 ∈ R. Then the solution of the IVP (4.7), (4.8) is given by Proof. From Theorem 4.3 we have that the solution of the IVP (4.7), (4.8) is given by and since i ≥ t − a − ν + 2 implies that the integer t − a − i − ν + 2 ≤ 0 and the numerator in this last expression is well defined.
Theorem C. Assume 0 < b ≤ c(t), 0 < ν < 1 and x(t) is the solution of the initial value problem Then lim t→∞ x(t) = ∞.
Letting t = k + ν − 1, for fixed i, then when k > i, we have (4.15) From Lemma 2.7, we have for the real part of ν(i + 1) > 0. Using this formula for ν(i + 1) > 0 we have (4.16) Take i sufficiently large, such that ν(i + 1) > 1. From (4.15) and (4.16), we get that when Since x(t) ≥ y(t) we get the desired result lim t→∞ x(t) = ∞ and the proof is complete.
This completes the proof.
From Theorem 4.3 and Theorem D, we can obtain the following corollary.
Corollary 5.2. Assume that −ν < −b < 0, 0 < ν < 1. Then for t ∈ N a+ν−1 , we have Now we consider solutions of the following ν-th order fractional delta equation satisfying y(a + ν − 1) < 0. By making the transformation x(t) = −y(t) and using Theorem C and Theorem D, we can get the following theorems.