Skip to main content

Theory and Modern Applications

Monotonicity results for h-discrete fractional operators and application

Abstract

In this article, we formulate nabla fractional sums and differences of order \(0 < \alpha \leq 1\) on the time scale \(h\mathbb{Z}\), where \(0 < h \leq 1\). Then, we prove that if the nabla h-Riemann–Liouville (RL) fractional difference operator \(({}_{a}\nabla_{h}^{\alpha }y)(t) > 0 \), then \(y(t)\) is α-increasing. Conversely, if \(y(t)\) is α-increasing and \(y(a)>0\), then \(({}_{a}\nabla_{h}^{\alpha }y)(t)>0\). The monotonicity results for the nabla h-Caputo fractional difference operator are also concluded by using the relation between h-nabla RL and Caputo fractional difference operators. It is observed that the reported monotonicity coefficient is not affected by the step h. We formulate a nabla h-fractional difference initial value problem as well. Finally, we furniture our results by proving a fractional difference version of the Mean Value Theorem (MVT) on \(h\mathbb{Z}\).

1 Introduction

Due to their successful applications in many branches of science and engineering, techniques of fractional calculus have been under focus by many researchers in the past and in the present decades [110]. The theory of fractional sums with delta operator and the fractional differences with nabla operators were firstly introduced in [11]. Extensive development of the theory can be found in [1231]. The monotonicity properties of delta and nabla fractional operators were studied in [3236]. It is worth mentioning that Atıcı et al. in [34] studied the monotonicity properties of delta fractional differences and obtained a delta-fractional difference version of the mean-value theorem. In [37], interesting monotonicity results were provided using the dual identities related to delta and nabla fractional operators. In [38], the authors studied the relationship between the discrete sequential fractional operators and monotonicity. Recently, in [3945], fractional operators with Mittag–Leffler and exponential “non-singular” kernels have been studied together with their discrete versions, and the monotonicity has been investigated. Motivated by all the above-mentioned works, we prove interesting monotonicity results with mean value theorem as an application in this paper for nabla fractional differences in the time scale \(h \mathbb{Z} \), \(0< h \leq 1 \) (called h-nabla fractional differences), which is considered as a generalized form of the monotonicity results when \(h=1\). Also, working on \(h\mathbb{Z} \), \(0< h<1\) guarantees more accurate approximations for the solutions of fractional dynamical systems.

The article is organized as follows. In Sect. 2, we present the main definitions and important information needed in this study, and in the third section we present the monotonicity analysis of the fractional difference operator. In Sect. 4 we formulate an initial value h-fractional difference problem in the sense of Riemann. Finally, we have the application on the mean value theorem and then the conclusions are presented.

2 Definitions and preliminary results

Definition 2.1

The backward difference operator on \(h\mathbb{Z}\) is defined by

$$ \nabla_{h}f(t)=\frac{f(t)- f(t-h)}{h}, $$

and the forward difference operator on \(h\mathbb{Z}\) is defined by

$$ \Delta_{h}f(t)=\frac{f(t+h)-f(t)}{h}. $$

Definition 2.2

The backward jump operator on the time scale \(h\mathbb{Z}\) is defined by \(\rho_{h}(t)=t-h\) and the forward jump operator is defined by \(\sigma_{h}(t)= t+h\).

For \(a,b \in \mathbb{R}\) with \(a< b\), \(\frac{b-a}{h}\in \mathbb{N}\) and \(0 < h\leq 1\), we use the notations \(\mathbb{N}_{a,h} = \lbrace a,a+h, a+2h,\ldots \rbrace \) and \({}_{b,h}\mathbb{N} = \lbrace b,b-h,b-2h,\ldots \rbrace \).

Definition 2.3

Let \(\alpha \in \mathbb{R}\) and \(0 < h\leq 1\), the nabla h-factorial of t is defined by

$$ t^{\overline{\alpha }}_{h}= h^{\alpha }\frac{\Gamma ( \frac{t}{h} + \alpha ) }{\Gamma ( \frac{t}{h} ) } $$

such that \(t \in \mathbf{R}- \lbrace \ldots,-2h,-h,0\rbrace \), \(0^{\overline{\alpha }}_{h} = 0\) and dividing by poles leads to zero.

Lemma 2.1

For \(\alpha>0\) and \(h>0\), \(t^{\overline{\alpha }}_{h}\) is increasing on \(\mathbb{N}_{0,h}\).

Proof

$$\begin{aligned} \nabla_{h} t^{\overline{\alpha }}_{h} &=\frac{t^{\overline{ \alpha }}_{h} - (t-h)^{\overline{\alpha }}_{h} }{h} = \frac{1}{h} \bigl( t^{\overline{\alpha }}_{h} - (t-h)^{\overline{\alpha }}_{h} \bigr) =\frac{1}{h} \biggl( h^{\alpha } \frac{\Gamma ( \frac{t}{h} + \alpha ) }{\Gamma ( \frac{t}{h} ) } - h^{\alpha } \frac{\Gamma ( \frac{(t-h)}{h} +\alpha ) }{ \Gamma ( \frac{t-h}{h} ) } \biggr) \\ & = \frac{h^{\alpha }}{h} \biggl( \frac{\Gamma ( \frac{t}{h} + \alpha ) }{\Gamma ( \frac{t}{h} ) } - \frac{\Gamma ( \frac{t}{h} +\alpha - 1 ) }{\Gamma ( \frac{t}{h} - 1 ) } \biggr) = h^{\alpha - 1} \biggl( \frac{\Gamma ( \frac{t}{h} + \alpha - 1 ) }{\Gamma ( \frac{t}{h} - 1 ) } \biggr) \biggl( \frac{\frac{t}{h} + \alpha - 1}{\frac{t}{h} - 1} - 1 \biggr) \\ & = h^{\alpha - 1} \biggl( \frac{\Gamma ( \frac{t}{h} + \alpha - 1 ) }{\Gamma ( \frac{t}{h} - 1 ) } \biggr) \biggl( \frac{ \frac{t}{h} + \alpha - 1 - \frac{t}{h} + 1}{ \frac{t}{h} -1} \biggr) = h^{\alpha - 1} \biggl( \frac{\Gamma ( \frac{t}{h} + \alpha - 1 ) }{\Gamma ( \frac{t}{h} - 1 ) } \biggr) \biggl( \frac{ \alpha }{ \frac{t}{h} -1} \biggr) \\ & =\alpha h^{\alpha - 1} \frac{\Gamma ( \frac{t}{h} + ( \alpha - 1 ) ) }{\Gamma ( \frac{t}{h} ) } = \alpha t^{\overline{ \alpha - 1} }_{h}. \end{aligned}$$

Notice that since \(\alpha ,h>0\), then \(\nabla_{h} t^{\overline{ \alpha }}_{h} =\frac{t^{\overline{\alpha }}_{h} - (t-h)^{\overline{ \alpha }}_{h} }{h} = \alpha t^{\overline{\alpha - 1} }_{h} \geq 0 \), and hence the proof is completed. □

Definition 2.4

(Nabla h-fractional sums)

For a function \(f : \mathbb{N}_{a,h} ={\lbrace a, a+h, a+2h,\ldots \rbrace } \rightarrow \mathbb{R} \), the nabla left h-fractional sum of order \(\alpha >0\) is defined by

$$\begin{aligned} \bigl( {}_{a}\nabla^{- \alpha }_{h} f \bigr) (t) &= \frac{1}{\Gamma (\alpha )} \int^{t}_{a} \bigl(t - \rho_{h} (s) \bigr)_{h} ^{\overline{\alpha -1}} f(s) \nabla_{h} s \\ & = \frac{1}{\Gamma (\alpha )} \sum^{t/h}_{k=a/h + 1 } \bigl(t - \rho_{h} (kh)\bigr)_{h} ^{ \overline{\alpha -1}} f(kh)h, \quad t\in \mathbb{N}_{a,h}. \end{aligned}$$

For a function \(f : {}_{b,h }\mathbb{N} = {\lbrace b, b-h, b-2h,\ldots\rbrace } \rightarrow \mathbb{R} \), the nabla right h-fractional sum of order \(\alpha >0\) is defined by

$$\begin{aligned} \bigl( {}_{h} \nabla^{- \alpha }_{b} f \bigr) (t)& = \frac{1}{\Gamma (\alpha )} \int^{b}_{t} \bigl( s - \rho_{h} (t) \bigr)_{h} ^{\overline{\alpha -1}} f(s) \Delta_{h} s \\ &= \frac{1}{\Gamma (\alpha )} \sum^{b/h - 1}_{k=t/h} \bigl(kh - \rho_{h} (t)\bigr)_{h} ^{ \overline{\alpha -1}} f(kh)h, \quad t \in{}_{b,h }\mathbb{N}. \end{aligned}$$

Definition 2.5

(Nabla h-RL fractional differences)

  • The nabla left h-fractional difference of order \(0<\alpha \leq 1\) (starting from a) is defined by

    $$\begin{aligned}& \bigl( {}_{a}\nabla_{h} ^{ \alpha } f \bigr) (t)= \bigl( \nabla_{h}\, {}_{a}\nabla^{- ( 1 - \alpha )}_{h} f \bigr) (t), \quad \text{which is} \\& \bigl( {}_{a}\nabla_{h} ^{ \alpha } f \bigr) (t) = \frac{1}{ \Gamma (1 - \alpha ) } \nabla_{h} \sum_{k = a/h + 1} ^{t/h} \bigl(t - \rho_{h} (kh)\bigr)^{\overline{- \alpha }}_{h} f(kh) h, \quad t\in \mathbb{N}_{a+h , h}. \end{aligned}$$
  • The nabla right h-fractional difference of order \(0<\alpha \leq 1\) (ending at b) is defined by

    $$\begin{aligned}& \bigl( {}_{h} \nabla_{b} ^{ \alpha } f \bigr) (t)= \bigl( - \Delta_{h} \,{}_{h} \nabla^{- ( 1 - \alpha )}_{b} f \bigr) (t), \quad \text{which is} \\& \bigl( {}_{h} \nabla_{b} ^{ \alpha } f \bigr) (t) = \frac{-1}{ \Gamma (1 - \alpha ) } \Delta_{h} \sum_{k = t/h} ^{b/h - 1} \bigl(kh - \rho_{h} (t)\bigr)^{\overline{- \alpha }}_{h} f(kh) h,\quad t\in{}_{b-h ,h} \mathbb{N}. \end{aligned}$$

Definition 2.6

(Nabla h-Caputo fractional differences)

Assume that \(0<\alpha \leq 1\), \(0< h\leq 1 \), \(a,b \in \mathbb{R} \), and \(a< b\), f is defined on \(\mathbb{N}_{a,h} = \lbrace a, a+h, a+2h,\ldots \rbrace \) and on \({}_{b,h} \mathbb{N} =\lbrace b, b-h, b-2h,\ldots \rbrace \). Then:

  • the left h-Caputo fractional difference of order α starting at a is defined by

    $$ \bigl( {}_{a} ^{C} \nabla_{h} ^{\alpha } f \bigr) (t)= \bigl( {}_{a}\nabla _{h} ^{ -( 1 - \alpha )} \nabla_{h} f \bigr) (t), \quad t \in \mathbb{N}_{a+h , h}; $$
  • the right h-Caputo fractional difference of order α ending at b is defined by

    $$ \bigl({}_{h} ^{C} \nabla_{b} ^{\alpha } f \bigr) (t)= \bigl({}_{h} \nabla _{b} ^{ -( 1 - \alpha )} ( - \Delta_{h} f )\bigr) (t) ,\quad t \in {}_{b-h , h} \mathbb{N}. $$

Proposition 2.2

(The relation between nabla h-RL fractional difference and h-Caputo fractional difference)

$$\begin{aligned}& (\mathrm{i})\quad \bigl( {}_{a} ^{C} \nabla_{h} ^{\alpha } f \bigr) (t) = \bigl( {}_{a}\nabla_{h} ^{\alpha } f \bigr) (t) - \frac{1}{ \Gamma ( 1 - \alpha )} ( t - a )_{h} ^{ \overline{ - \alpha }} f(a); \\& (\mathrm{ii}) \quad \bigl( {}_{h} ^{C} \nabla_{b} ^{\alpha } f \bigr) (t) = \bigl( {}_{h} \nabla_{b} ^{\alpha } f \bigr) (t) - \frac{1}{ \Gamma ( 1 - \alpha )} ( b - t )_{h} ^{ \overline{ - \alpha }} f(b). \end{aligned}$$

Proof

$$\begin{aligned}& (\mathrm{i}) \quad \bigl( {}_{a} ^{C} \nabla_{h} ^{\alpha } f \bigr) (t) \\& \hphantom{(\mathrm{i})}\qquad= \bigl({}_{a} \nabla_{h} ^{ -( 1 - \alpha )} \nabla_{h} f \bigr) (t) \\& \hphantom{(\mathrm{i})}\qquad = {}_{a}\nabla_{h} ^{ -( 1 - \alpha )} \biggl( \frac{f(t)-f(t-h)}{h} \biggr) \\& \hphantom{(\mathrm{i})}\qquad = {}_{a}\nabla_{h} ^{ -( 1 - \alpha )} \biggl( \frac{f(t)}{h} \biggr) - {}_{a}\nabla_{h} ^{ -( 1 - \alpha )} \biggl( \frac{f(t-h)}{h} \biggr) \\& \hphantom{(\mathrm{i})}\qquad = \frac{1}{\Gamma ( 1 - \alpha )} \sum^{t/h}_{k=a/h + 1 } \bigl(t - \rho _{h} (kh)\bigr)_{h} ^{\overline{ - \alpha }} \frac{f(kh)}{h} h \\& \hphantom{(\mathrm{i})}\qquad\quad{} - \frac{1}{\Gamma ( 1 - \alpha )} \sum^{(t-h)/h}_{k=(a-h)/h + 1 } \bigl(t-h - \rho_{h} (kh)\bigr)_{h} ^{\overline{ - \alpha }} \frac{f(kh)}{h} h \\& \hphantom{(\mathrm{i})}\qquad = \frac{1}{h \Gamma ( 1 - \alpha )} \sum^{t/h}_{k=a/h + 1 } \bigl(t - \rho_{h} (kh)\bigr)_{h} ^{\overline{ - \alpha }} f(kh) h \\& \hphantom{(\mathrm{i})}\qquad\quad{} - \frac{1}{ h \Gamma ( 1 - \alpha )} \sum^{t/h - 1}_{k=a/h} \bigl((t-h)- (kh-h)\bigr)_{h} ^{\overline{ - \alpha }} f(kh) h \\& \hphantom{(\mathrm{i})}\qquad = \frac{1}{h \Gamma ( 1 - \alpha )} \sum^{t/h}_{k=a/h + 1 } \bigl(t - \rho_{h} (kh)\bigr)_{h} ^{\overline{ - \alpha }} f(kh) h \\& \hphantom{(\mathrm{i})}\qquad\quad{} - \frac{1}{ h \Gamma ( 1 - \alpha )} \sum^{t/h - 1}_{k=a/h + 1} \bigl(t-h- \rho_{h} (kh)\bigr)_{h} ^{\overline{ - \alpha }} f(kh) h \\& \hphantom{(\mathrm{i})}\qquad\quad{} -\frac{1}{h \Gamma ( 1 - \alpha )} \biggl( t - h - \frac{a}{h} (h) + h \biggr)_{h} ^{ \overline{ - \alpha } } f\biggl( \frac{a}{h} h \biggr) h \\& \hphantom{(\mathrm{i})}\qquad =\frac{1}{\Gamma ( 1 - \alpha )} \\& \hphantom{(\mathrm{i})}\qquad\quad{}\times \biggl( \frac{ \sum^{t/h}_{k=a/h + 1} (t- \rho_{h} (kh))_{h} ^{ \overline{ - \alpha }} f(kh) h - \sum^{t/h-1}_{k=a/h+1} ( (t-h) - \rho_{h} (kh) )_{h} ^{ \overline{ - \alpha } } f(kh) h }{h} \biggr) \\& \hphantom{(\mathrm{i})}\qquad\quad{} - \frac{1}{\Gamma (1 - \alpha )} (t-a)_{h} ^{\overline{- \alpha }} f(a) \\& \hphantom{(\mathrm{i})}\qquad = \frac{1}{\Gamma ( 1 - \alpha )} \nabla_{h} \sum^{t/h}_{k=a/h+1} \bigl( t-\rho_{h} (kh) \bigr)_{h} ^{ \overline{ - \alpha } } f(kh) h - \frac{1}{\Gamma (1 - \alpha )} (t-a)_{h} ^{\overline{- \alpha } } f(a) \\& \hphantom{(\mathrm{i})}\qquad = \bigl( {}_{a}\nabla_{h} ^{\alpha } f \bigr) (t) - \frac{1}{\Gamma ( 1 - \alpha )} (t-a)_{h} ^{\overline{ - \alpha }} f(a). \end{aligned}$$

(ii) The proof is similar to that in (i), and hence we omit it. □

The following lemma is a generalization of Lemma 3.3 in [20].

Lemma 2.3

Let \(\alpha >0\), \(\mu >-1\), \(h>0\), and \(t\in \mathbb{N}_{a,h} \). Then

$$ {}_{a}\nabla_{h}^{-\alpha } (t-a)_{h}^{\overline{\mu }} = \frac{ \Gamma (\mu +1)}{\Gamma (\mu +1+\alpha )}(t-a)_{h}^{\overline{\alpha +\mu }}. $$
(1)

Proof

Using Lemma 3.3 in [20], we have

$$\begin{aligned}& \nabla_{a/h}^{-\alpha } \biggl( \frac{t}{h} - \frac{a}{h} \biggr) ^{\overline{\mu }} = \frac{\Gamma (\mu + 1)}{\Gamma (\mu + \alpha + 1 )} \biggl( \frac{t}{h} - \frac{a}{h} \biggr) ^{\overline{ \alpha + \mu }} , \\& \frac{1}{\Gamma (\alpha )} \sum_{a/h + 1} ^{t/h} \biggl( \frac{t}{h} - \rho (k) \biggr) ^{\overline{\alpha -1}} \biggl( k - \frac{a}{h} \biggr) ^{\overline{\mu }} = \frac{\Gamma (\mu + 1)}{\Gamma (\mu + \alpha + 1)} \biggl( \frac{t-a}{h} \biggr) ^{\overline{\alpha + \mu }}, \\& \frac{1}{\Gamma (\alpha )} \sum_{a/h + 1} ^{t/h} \biggl( \frac{t}{h} - k + 1 \biggr) ^{\overline{\alpha -1}} \biggl( k - \frac{a}{h} \biggr) ^{\overline{\mu }} = \frac{\Gamma (\mu + 1)}{\Gamma (\mu + \alpha + 1)} \frac{\Gamma ( \frac{t-a}{h} + \alpha + \mu ) }{\Gamma ( \frac{t-a}{h} ) }, \\& \frac{1}{\Gamma (\alpha )} \sum_{a/h + 1} ^{t/h} \biggl( \frac{t - kh + h}{h} \biggr) ^{\overline{\alpha -1}} \biggl( \frac{kh-a}{h} \biggr) ^{\overline{\mu }} = \frac{\Gamma (\mu + 1)}{\Gamma (\mu + \alpha + 1)} \frac{ ( t - a )_{h} ^{\overline{\alpha +\mu }}}{h^{\alpha + \mu } } , \\& \frac{1}{\Gamma (\alpha )} \sum_{a/h + 1} ^{t/h} \frac{\Gamma ( \frac{t - kh + h}{h} + \alpha - 1 ) }{\Gamma ( \frac{t - kh + h}{h} ) } \frac{\Gamma ( \frac{kh - a}{h} + \mu ) }{\Gamma ( \frac{kh-a}{h} ) } = \frac{\Gamma (\mu + 1)}{ \Gamma (\mu + \alpha + 1)} \frac{ ( t - a )_{h} ^{\overline{ \alpha +\mu }}}{h^{\alpha + \mu } } , \\& \frac{1}{\Gamma (\alpha )} \sum_{a/h + 1} ^{t/h} \frac{ ( t - kh + h )_{h} ^{\overline{\alpha - 1 }} }{ h ^{\alpha - 1 }} \frac{ ( kh - a )_{h} ^{\overline{\mu }} }{ h^{\mu } } = \frac{ \Gamma (\mu + 1)}{\Gamma (\mu + \alpha + 1)} \frac{ ( t - a )_{h} ^{\overline{\alpha +\mu }}}{h^{\alpha } h^{\mu } }, \\& \frac{1}{\Gamma (\alpha )} \sum_{a/h + 1} ^{t/h} \bigl( t - \rho (kh) \bigr) ^{\overline{\alpha -1}}_{h} ( kh - a ) ^{\overline{ \mu }}_{h} h = \frac{\Gamma (\mu + 1)}{\Gamma (\mu + \alpha + 1)} ( t-a ) ^{\overline{\alpha + \mu }}_{h} , \\& {}_{a}\nabla_{h}^{-\alpha } (t-a)_{h}^{\overline{\mu }} = \frac{\Gamma (\mu +1)}{\Gamma (\mu +1+\alpha )}(t-a)_{h}^{\overline{\alpha +\mu }}. \end{aligned}$$

 □

3 The monotonicity results

The following two definitions are the \(h\mathbb{Z}\) versions of monotonicity definitions given in [34].

Definition 3.1

Let \(y: \mathbb{N}_{a,h} \rightarrow \mathbb{R}\) be a function satisfying \(y(a)\geq 0 \), and let \(0 \leq h < 1 \). Then \(y(t) \) is called an α-increasing function on \(\mathbb{N}_{a,h} \) if \(y(t+h)\geq \alpha y(t)\) \(\forall t\in \mathbb{N}_{a,h} \).

Note that if \(y(t)\) is increasing on \(\mathbb{N}_{a,h}\) (\(y(t+h) \geq y(t)\) \(\forall t\in \mathbb{N}_{a,h} \)), then \(y(t) \) is an α-increasing function on \(\mathbb{N}_{a,h} \), and if \(\alpha = 1 \), then the increasing and α-increasing concepts coincide.

Definition 3.2

Let \(y: \mathbb{N}_{a,h} \rightarrow \mathbb{R}\) be a function satisfying \(y(a) \leq 0 \), and \(0 \leq h < 1 \). Then \(y(t) \) is called an α-decreasing function on \(\mathbb{N}_{a,h} \) if \(y(t+h) \leq \alpha y(t)\) \(\forall t\in \mathbb{N}_{a,h} \).

Note that if \(y(t)\) is decreasing on \(\mathbb{N}_{a,h}\) (\(y(t+h) \leq y(t)\) \(\forall t\in \mathbb{N}_{a,h}\)), then \(y(t) \) is an α-decreasing function on \(\mathbb{N}_{a,h} \), and if \(\alpha = 1 \), then the decreasing and α-decreasing concepts coincide.

Theorem 3.1

Let \(y: \mathbb{N}_{a-h,h} \rightarrow \mathbb{R}\), and suppose that (\({}_{a-h} \nabla^{\alpha }_{h} y \)) \((t) \geq 0\) for \(0 < \alpha \leq 1 \), and \(0 < h \leq 1 \), \(t \in \mathbb{N}_{a-h,h} \). Then \(y(t) \) is α-increasing.

Proof

First we recall that

$$ \begin{aligned} \bigl( {}_{a-h} \nabla_{h} ^{ \alpha } y \bigr) (t)&= \frac{1}{ \Gamma (1 - \alpha ) } \nabla_{h} \sum_{k = (a-h)/h + 1} ^{t/h} \bigl(t - \rho _{h} (kh)\bigr)^{\overline{- \alpha }}_{h} y(kh) h \\ &= \frac{1}{ \Gamma (1 - \alpha ) } \nabla_{h} \sum _{k = a/h } ^{t/h} \bigl(t - \rho _{h} (kh) \bigr)^{\overline{- \alpha }}_{h} y(kh) h. \end{aligned} $$

Let

$$ S(t) = \sum_{k = a/h} ^{t/h} \bigl(t - \rho_{h} (kh)\bigr)^{\overline{- \alpha }}_{h} y(kh) h. $$

Then, from the assumption, we have \(\nabla_{h} S(t)\geq 0\). That is,

$$\begin{aligned} \nabla_{h} S(t)& = \frac{S(t)-S(t-h)}{h} \\ & = \frac{\sum_{k = a/h} ^{t/h} (t - \rho_{h} (kh))^{\overline{- \alpha }}_{h} y(kh) h - \sum_{k = a/h} ^{t/h - 1} (t - h - \rho_{h} (kh))^{\overline{- \alpha }}_{h} y(kh) h }{h} \\ &= \frac{\sum_{k = a/h} ^{t/h - 1} (t - \rho_{h} (kh))^{\overline{- \alpha }}_{h} y(kh) h - \sum_{k = a/h} ^{t/h - 1} (t - h - \rho_{h} (kh))^{\overline{- \alpha }}_{h} y(kh) h }{h} \\ &\quad{} + \frac{( t - \rho_{h} (t) )^{\overline{- \alpha }}_{h} y(t) h }{h} \\ & = ( t - t + h )_{h} ^{\overline{- \alpha }} y(t) + \sum _{k=a/h} ^{t/h -1} \biggl( \frac{ (t - \rho_{h} (kh))^{\overline{- \alpha } }_{h} - (t - h - \rho_{h} (kh))^{\overline{- \alpha }}_{h} }{h} \biggr) y(kh) h \\ & = h _{h} ^{\overline{-\alpha }} y(t) + \sum_{k=a/h} ^{t/h -1} y(kh) h \nabla_{h} \bigl(t - \rho_{h} (kh) \bigr)^{\overline{- \alpha } }_{h} \\ & = h _{h} ^{\overline{-\alpha }} y(t) + \sum_{k=a/h} ^{t/h -1} y(kh) h \frac{ ( t-\rho_{h} (kh) )_{h} ^{\overline{- \alpha }} - ( t-h - \rho_{h} (kh) )_{h} ^{\overline{- \alpha }} }{h} \\ & = h _{h} ^{\overline{-\alpha }} y(t) + \sum_{k=a/h} ^{t/h -1} y(kh) h \frac{ ( t-kh + h )_{h} ^{\overline{-\alpha }} - ( t-h - kh + h )_{h} ^{\overline{-\alpha }} }{h} \\ & = h _{h} ^{\overline{-\alpha }} y(t) + \sum_{k=a/h} ^{t/h -1} y(kh) h \biggl( \frac{\Gamma ( \frac{t-kh+h}{h}-\alpha ) }{\Gamma ( \frac{t-kh+h}{h} ) } h ^{-\alpha } - \frac{\Gamma ( \frac{t-kh}{h} - \alpha ) }{\Gamma ( \frac{t-kh}{h} ) } h^{-\alpha } \biggr) \frac{1}{h} \\ & = h _{h} ^{\overline{-\alpha }} y(t) + \sum_{k=a/h} ^{t/h -1} y(kh) h \biggl( \frac{\Gamma ( \frac{t}{h}-k+1-\alpha ) }{\Gamma ( \frac{t}{h} -k + 1 ) } h ^{-\alpha } - \frac{\Gamma ( \frac{t}{h} -k - \alpha ) }{\Gamma ( \frac{t}{h} -k ) } h^{-\alpha } \biggr) \frac{1}{h} \\ & = h _{h} ^{\overline{-\alpha }} y(t) + \sum_{k=a/h} ^{t/h -1} y(kh) h \biggl( \frac{ ( \frac{t}{h}-k-\alpha ) }{ ( \frac{t}{h} -k ) } - 1 \biggr) \frac{\Gamma ( \frac{t}{h} -k - \alpha ) }{\Gamma ( \frac{t}{h} -k ) } \frac{h^{-\alpha } }{h} \\ & = h _{h} ^{\overline{-\alpha }} y(t) + \sum_{k=a/h} ^{t/h -1} y(kh) h \biggl( \frac{ \frac{t}{h}-k-\alpha - \frac{t}{h} +k }{ \frac{t}{h} -k } \biggr) \frac{\Gamma ( \frac{t}{h} -k - \alpha ) }{\Gamma ( \frac{t}{h} -k ) } h^{-\alpha - 1} \\ & = h _{h} ^{\overline{-\alpha }} y(t) + \sum_{k=a/h} ^{t/h -1} y(kh) h \biggl( \frac{ -\alpha }{ \frac{t}{h} -k } \biggr) \frac{ \Gamma ( \frac{t}{h} -k - \alpha ) }{\Gamma ( \frac{t}{h} -k ) } h^{-\alpha - 1} \\ & = h _{h} ^{\overline{-\alpha }} y(t) + \sum_{k=a/h} ^{t/h -1} y(kh) h ( -\alpha ) \frac{\Gamma ( \frac{t}{h} -k +1+ ( - \alpha -1) ) }{\Gamma ( \frac{t}{h} -k + 1 ) } h^{-\alpha - 1} \\ & = h _{h} ^{\overline{-\alpha }} y(t) + \sum_{k=a/h} ^{t/h -1} y(kh) h ( -\alpha ) \bigl(t - \rho_{h} (kh) \bigr)^{ \overline{- \alpha -1} }_{h} \\ & = h _{h} ^{\overline{-\alpha }} y(t) - \alpha \sum _{k=a/h} ^{t/h -1} \bigl(t - \rho_{h} (kh) \bigr)^{\overline{- \alpha -1} }_{h} y(kh) h \geq 0. \end{aligned}$$

Therefore

$$ \nabla_{h} S(t) = h _{h} ^{\overline{-\alpha }} y(t) - \alpha \sum_{k=a/h} ^{t/h -1} \bigl(t - \rho_{h} (kh)\bigr)^{\overline{- \alpha -1} }_{h} y(kh) h \geq 0. $$
(2)

When \(t=a\), we have

$$ \nabla_{h} S(a) = h _{h} ^{\overline{-\alpha }} y(a) = h ^{ - \alpha } \Gamma ( 1-\alpha ) y(a) \geq 0, \quad \text{and hence } y(a)\geq 0. $$

When \(t=a+h\), we get

$$\begin{aligned} \nabla_{h} S(a+h) =& h _{h} ^{\overline{-\alpha }} y(a+h) - \alpha \bigl( a + h - \rho_{h} (a)\bigr)^{\overline{-\alpha -1} }_{h} y(a) h \\ =& h _{h} ^{\overline{-\alpha }} y(a+h) - \alpha ( a + h - a + h)^{\overline{- \alpha -1} }_{h} y(a) h \\ =& h _{h} ^{\overline{-\alpha }} y(a+h) - \alpha ( 2h)^{\overline{- \alpha -1} }_{h} y(a) h \\ =& h ^{- \alpha } \frac{ \Gamma ( \frac{h}{h} - \alpha ) }{ \Gamma ( \frac{h}{h} ) } y(a+h) - \alpha h ^{- \alpha - 1 } \frac{ \Gamma ( \frac{2h}{h} - \alpha - 1 ) }{ \Gamma ( \frac{2h}{h} ) } y(a) h \\ =& h ^{- \alpha } \frac{ \Gamma ( 1 - \alpha )}{ \Gamma ( 1 ) } y(a+h) - \alpha h ^{- \alpha } \frac{ \Gamma ( 2 - \alpha - 1 )}{ \Gamma ( 2 ) } y(a) \\ =& h ^{- \alpha } \Gamma ( 1 - \alpha ) y(a+h) - \alpha h ^{- \alpha } \Gamma (1- \alpha ) y(a) \geq 0, \end{aligned}$$

hence, \(y(a+h) \geq \alpha y(a) \).

Now we follow inductively to show that

$$ y(t+h) \geq \alpha y(t) , \quad \forall t \in \mathbb{N}_{a,h}. $$

Assume \(y(k+h) \geq \alpha y(k) \geq 0 \), \(\forall k < t \) such that \(k,t \in \mathbb{N}_{a,h}\). We need to show that \(y(t+h) \geq \alpha y(t)\). We know that

$$ \nabla_{h} S(t) = h _{h} ^{\overline{-\alpha }} y(t) - \alpha \sum_{k=a/h} ^{t/h -1} \bigl(t - \rho_{h} (kh)\bigr)^{\overline{- \alpha -1} }_{h} y(kh) h \geq 0. $$
(3)

In (3) replace t by \(t+h \). Then we have

$$\begin{aligned}& h _{h} ^{\overline{-\alpha }} y(t+h) - \alpha \sum _{k=a/h} ^{t/h } \bigl(t + h - \rho_{h} (kh) \bigr)^{\overline{- \alpha -1} }_{h} y(kh) h \geq 0 , \quad \text{which implies that} \\& \begin{aligned} & h _{h} ^{\overline{-\alpha }} y(t+h) - \alpha \bigl( \bigl(t + h - \rho_{h} (a)\bigr)^{\overline{- \alpha -1} }_{h} y(a) h + \bigl(t + h - \rho_{h} (a+h)\bigr)^{\overline{- \alpha -1} }_{h} y(a+h) h \\ &\quad{} + \cdots + \bigl(t + h - \rho_{h} (t)\bigr)^{\overline{- \alpha -1} }_{h} y(t) h \bigr) \geq 0, \end{aligned} \end{aligned}$$

or

$$\begin{aligned} h _{h} ^{\overline{-\alpha }} y(t+h) \geq & \alpha \bigl( \bigl(t + h - \rho _{h} (a)\bigr)^{\overline{- \alpha -1} }_{h} y(a) h + \bigl(t + h - \rho_{h} (a+h)\bigr)^{\overline{- \alpha -1} }_{h} y(a+h) h \\ &{}+\cdots + \bigl(t + h - \rho_{h} (t) \bigr)^{\overline{- \alpha -1} }_{h} y(t) h \bigr) \\ =& \alpha \bigl(t + h - \rho_{h} (a)\bigr)^{\overline{- \alpha -1} }_{h} y(a) h + \cdots + \alpha \bigl(t + h - \rho_{h} (t) \bigr)^{\overline{- \alpha -1} }_{h} y(t) h \\ \geq & \alpha \bigl(t + h - \rho_{h} (t) \bigr)^{\overline{- \alpha -1} }_{h} y(t) h \\ =&\alpha (2h)^{\overline{- \alpha -1} }_{h} y(t) h. \end{aligned}$$
(4)

Then from Definition 2.3 it follows that

$$ h^{-\alpha } \Gamma ( 1 -\alpha ) y(t+h) \geq \alpha h ^{-\alpha } \Gamma ( 1 -\alpha ) y(t) . $$

Hence, \(y(t+h) \geq \alpha y(t) \), and the proof is completed. □

Using Proposition 2.2 and Theorem 3.1, we can state the following h-Caputo fractional difference monotonicity result.

Corollary 3.2

Let \(y: \mathbb{N}_{a-h,h} \rightarrow \mathbb{R}\), and suppose that

$$ \bigl( {}_{a-h} ^{C} \nabla^{\alpha }_{h} y \bigr) (t) \geq \frac{-1}{ \Gamma ( 1 - \alpha )} ( t - a + h )_{h} ^{ \overline{ - \alpha }} y(a - h) , \quad t \in \mathbb{N}_{a-h,h} $$

for \(0 < \alpha \leq 1 \), and \(0 < h \leq 1 \) then \(y(t) \) is α-increasing.

Theorem 3.3

Assume that the function \(y: \mathbb{N}_{a-h,h} \rightarrow \mathbb{R}\) satisfies \(y(a)\geq 0 \) and assume \(0 < \alpha \leq 1 \) and \(0 < h \leq 1 \). If y is increasing on \(\mathbb{N}_{a,h} \), then we have

$$ \bigl( {}_{a-h} \nabla^{\alpha }_{h} y \bigr) (t) \geq 0 ,\quad \forall t \in \mathbb{N}_{a-h,h}. $$

Proof

Since we have

$$ \bigl( {}_{a-h} \nabla_{h} ^{ \alpha } y \bigr) (t)= \frac{1}{ \Gamma (1 - \alpha ) } \nabla_{h} S(t), \quad t\in \mathbb{N}_{a-h , h}, $$

it is enough to show that

$$ S(t) = \sum_{k = (a-h)/h+1} ^{t/h} \bigl(t - \rho_{h} (kh)\bigr)^{\overline{- \alpha }}_{h} y(kh)h= \sum _{k = a/h} ^{t/h} \bigl(t - \rho_{h} (kh)\bigr)^{\overline{- \alpha }}_{h} y(kh)h $$

is increasing on \(\mathbb{N}_{a,h}\). In reference to the proof of Theorem 3.1, when \(t=a\) we have

$$ \nabla_{h} S(a)= h _{h} ^{\overline{-\alpha }} y(a)= h^{- \alpha } \Gamma ( 1 - \alpha ) y(a), $$

and since \(y(a)\geq 0 \), \(h ^{- \alpha } >0 \), and \(\Gamma (1- \alpha )>0 \), then

$$ \nabla_{h} S(a)= h^{-\alpha } \Gamma ( 1 - \alpha ) y(a) \geq 0. $$

Assume that \(\nabla_{h} S(i)\geq 0 \), \(\forall i < t \). We shall show that \(\nabla_{h} S(t) \geq 0 \).

From the assumption that \(y(t) \) is increasing, it follows that \(y(t) \geq y(t-h) \geq y(a)\geq 0 \), \(\forall t \in \mathbb{N}_{a,h} \).

From (2), we recall that

$$ \nabla_{h} S(t) = h _{h} ^{\overline{-\alpha }} y(t) - \alpha \sum_{k=a/h} ^{t/h -1} \bigl(t - \rho_{h} (kh)\bigr)^{\overline{- \alpha -1} }_{h} y(kh) h \geq 0. $$

Then we have

$$\begin{aligned} \nabla_{h} S(t) =& h _{h} ^{\overline{-\alpha }} y(t) - \alpha \bigl(t - \rho_{h} (t-h)\bigr)^{\overline{- \alpha -1} }_{h} y(t-h) h - \alpha \sum_{k=a/h} ^{t/h -2} \bigl(t - \rho_{h} (kh)\bigr)^{\overline{- \alpha -1} }_{h} y(kh) h \\ =& h _{h} ^{\overline{-\alpha }} y(t) - \alpha (t - t + h + h)^{\overline{- \alpha -1} }_{h} y(t-h) h - \alpha \sum _{k=a/h} ^{t/h -2} \bigl(t - \rho_{h} (kh) \bigr)^{\overline{- \alpha -1} }_{h} y(kh) h \\ =& h _{h} ^{\overline{-\alpha }} y(t) - \alpha ( 2h )^{\overline{- \alpha -1} }_{h} y(t-h) h - \alpha \sum_{k=a/h} ^{t/h -2} \bigl(t - \rho _{h} (kh)\bigr)^{\overline{- \alpha -1} }_{h} y(kh) h \\ &{}- \alpha \sum_{k=a/h} ^{t/h -2} \bigl(t - \rho_{h} (kh)\bigr)^{\overline{- \alpha -1} }_{h} y(t-h) h + \alpha \sum_{k=a/h} ^{t/h -2} \bigl(t - \rho _{h} (kh)\bigr)^{\overline{- \alpha -1} }_{h} y(t-h) h \\ =& h _{h} ^{\overline{-\alpha }} y(t) - \alpha ( 2h )^{\overline{- \alpha -1} }_{h} y(t-h) h + \alpha \sum_{k=a/h} ^{t/h -2} \bigl(t - \rho _{h} (kh)\bigr)^{\overline{- \alpha -1} }_{h} \bigl( y(t-h) - y(kh) \bigr) h \\ &{}- \alpha \sum_{k=a/h} ^{t/h -2} \bigl(t - \rho_{h} (kh)\bigr)^{\overline{- \alpha -1} }_{h} y(t-h) h . \end{aligned}$$

Note that since \(y(t) \) is increasing, then \(y(t-h) - y(kh) \geq 0\), \(\forall k=\frac{a}{h},\frac{a}{h} +1,\ldots,\frac{t}{h}-2 \), from which it follows that

$$\begin{aligned} \nabla_{h} S(t) \geq & h _{h} ^{\overline{-\alpha }} y(t) - \alpha ( 2h )^{\overline{- \alpha -1} }_{h} y(t-h) h - \alpha \sum _{k=a/h} ^{t/h -2} \bigl(t - \rho_{h} (kh) \bigr)^{\overline{- \alpha -1} }_{h} y(t-h) h \\ =& h _{h} ^{\overline{-\alpha }} y(t) - \alpha \sum _{k=a/h} ^{t/h -1} \bigl(t - \rho_{h} (kh) \bigr)^{\overline{- \alpha -1} }_{h} y(t-h) h \\ =& h _{h} ^{\overline{-\alpha }} y(t) - h _{h} ^{\overline{-\alpha }} y(t-h) + h _{h} ^{\overline{-\alpha }} y(t-h) - \alpha y(t-h) h \sum _{k=a/h} ^{t/h -1} \bigl(t - \rho_{h} (kh) \bigr)^{ \overline{- \alpha -1} }_{h} \\ =& h _{h} ^{\overline{-\alpha }} \bigl( y(t) - y(t-h) \bigr) + y(t-h) \Biggl( h _{h} ^{\overline{-\alpha }} - \alpha h \sum _{k=a/h} ^{t/h -1} \bigl( t - \rho_{h} (kh) \bigr)^{\overline{- \alpha -1} }_{h} \Biggr) \\ \geq& y(t-h) \Biggl( h _{h} ^{\overline{-\alpha }} - \alpha h \sum _{k=a/h} ^{t/h -1} \bigl( t - \rho_{h} (kh) \bigr)^{\overline{- \alpha -1} }_{h} \Biggr) \\ =& y(t-h) \bigl( h _{h} ^{\overline{-\alpha }} - \alpha h \bigl[ \bigl( t - \rho_{h} (a) \bigr)^{\overline{- \alpha -1} }_{h} + \bigl( t - \rho_{h} (a+h) \bigr)^{\overline{- \alpha -1} }_{h} \\ &{}+ \cdots + \bigl( t - \rho_{h} (t-h) \bigr)^{\overline{- \alpha -1} }_{h} \bigr] \bigr) \\ =& y(t-h) \bigl( h _{h} ^{\overline{-\alpha }} - \alpha h \bigl[ ( t - a + h )^{\overline{- \alpha -1} }_{h} + ( t - a )^{\overline{- \alpha -1} }_{h} + \cdots + (2h)^{\overline{- \alpha -1} }_{h} \bigr] \bigr) \\ =& y(t-h) h ^{-\alpha } \biggl[ \frac{\Gamma ( 1-\alpha ) }{\Gamma (1)} - \alpha \frac{\Gamma ( \frac{t-a}{h} - \alpha ) }{\Gamma ( \frac{t-a}{h} + 1 ) } ) - \alpha \frac{\Gamma ( \frac{t-a}{h} - \alpha - 1 ) }{\Gamma ( \frac{t-a}{h} ) } ) - \cdots - \alpha \frac{\Gamma (1 - \alpha )}{\Gamma (2)} \biggr] \\ =& y(t-h) h ^{-\alpha } \\ &\quad{}\times \biggl[ \frac{\Gamma (1-\alpha )}{\Gamma (1)} - \alpha \frac{\Gamma (1 - \alpha )}{\Gamma (2)} - \alpha \frac{\Gamma (2 - \alpha )}{\Gamma (3)} - \alpha \frac{\Gamma (3 - \alpha )}{\Gamma (4)} - \cdots - \alpha \frac{\Gamma ( \frac{t-a}{h} - \alpha ) }{\Gamma ( \frac{t-a}{h} + 1 ) } \biggr] \\ =& y(t-h) h ^{-\alpha } \biggl[ \frac{\Gamma (1-\alpha )}{ 1 \Gamma (1)} ( 1 - \alpha ) - \alpha \frac{\Gamma (2 - \alpha )}{\Gamma (3)} - \alpha \frac{\Gamma (3 - \alpha )}{\Gamma (4)} - \cdots - \alpha \frac{ \Gamma ( \frac{t-a}{h} - \alpha ) }{\Gamma ( \frac{t-a}{h} + 1 ) } \biggr] \\ =& y(t-h) h ^{-\alpha } \biggl[ \frac{ 2 \Gamma (2-\alpha )}{ 2 \Gamma (2)} - \alpha \frac{\Gamma (2 - \alpha )}{\Gamma (3)} - \alpha \frac{\Gamma (3 - \alpha )}{\Gamma (4)} - \cdots \alpha \frac{ \Gamma ( \frac{t-a}{h} - \alpha ) }{\Gamma ( \frac{t-a}{h} + 1 ) } \biggr] \\ =& y(t-h) h ^{-\alpha } \biggl[ \frac{\Gamma (2-\alpha )}{ \Gamma (3)} ( 2 - \alpha ) - \alpha \frac{ \Gamma (3 - \alpha )}{\Gamma (4)} - \cdots - \alpha \frac{\Gamma ( \frac{t-a}{h} - \alpha ) }{\Gamma ( \frac{t-a}{h} + 1 ) } \biggr] \\ =& y(t-h) h ^{-\alpha } \biggl[ \frac{ 3 \Gamma (3-\alpha )}{ 3 \Gamma (3)} - \alpha \frac{\Gamma (3 - \alpha )}{\Gamma (4)} - \cdots - \alpha \frac{\Gamma ( \frac{t-a}{h} - \alpha ) }{ \Gamma ( \frac{t-a}{h} + 1 ) } \biggr] . \end{aligned}$$

If we continue in the same manner, we conclude that

$$\begin{aligned} \nabla_{h} S(t) \geq & y(t-h) h ^{-\alpha } \biggl( \frac{\Gamma ( \frac{t-a}{h} - \alpha ) }{\Gamma ( \frac{t-a}{h} ) } - \alpha \frac{\Gamma ( \frac{t-a}{h} - \alpha ) }{\Gamma ( \frac{t-a}{h} + 1 ) } \biggr) \\ =& y(t-h) h ^{-\alpha } \frac{\Gamma ( \frac{t-a}{h} - \alpha ) }{\Gamma ( \frac{t-a}{h} ) } \biggl( 1 - \alpha \frac{ 1 }{ ( \frac{t-a}{h} ) } \biggr) \\ =& y(t-h) h ^{-\alpha } \frac{ ( \frac{t-a}{h} - \alpha ) \Gamma ( \frac{t-a}{h} - \alpha ) }{ ( \frac{t-a}{h} ) \Gamma ( \frac{t-a}{h} ) } = y(t-h) \biggl( h ^{-\alpha } \frac{ \Gamma ( \frac{t-a+h}{h} - \alpha ) }{ \Gamma ( \frac{t-a+h}{h} ) } \biggr) \\ =&y(t-h) (t-a+h)_{h} ^{\overline{-\alpha }} \geq 0,\quad \text{which completes the proof}. \end{aligned}$$

 □

The proof of the following theorem is similar to that in Theorem 3.3.

Theorem 3.4

Let a function \(y: \mathbb{N}_{a-h,h} \rightarrow \mathbb{R}\) satisfy \(y(a)> 0 \) and be strictly increasing on \(\mathbb{N}_{a,h} \), where \(0 < \alpha \leq 1 \) and \(0 < h \leq 1 \). Then

$$ \bigl({}_{a-h} \nabla^{\alpha }_{h} y \bigr) (t) > 0. $$

Theorem 3.5

Let \(y: \mathbb{N}_{a-h,h} \rightarrow \mathbb{R}\), and suppose that (\({}_{a-h} \nabla^{\alpha }_{h} y \)) \((t) \leq 0\) for \(0 < \alpha \leq 1 \), and \(0 < h \leq 1 \), \(t \in \mathbb{N}_{a-h,h} \). Then \(y(t) \) is α-decreasing.

Proof

Let \(g: \mathbb{N}_{a-h,h} \rightarrow \mathbb{R}\) be a function such that \(g(t)= - y(t)\), hence

$$ \bigl( {}_{a-h} \nabla^{\alpha }_{h} g \bigr) (t)= \bigl( {}_{a-h} \nabla^{\alpha }_{h} (-y) \bigr) (t)= -\bigl( {}_{a-h} \nabla^{\alpha }_{h} y \bigr) (t) \geq 0. $$

Then the proof follows by applying Theorem 3.1 to \(g(t)\). □

Theorem 3.6

Let a function \(y: \mathbb{N}_{a-h,h} \rightarrow \mathbb{R}\) satisfy \(y(a) \leq 0 \) and be decreasing on \(\mathbb{N}_{a,h} \). Then, for \(0 < \alpha \leq 1 \) and \(0 < h \leq 1 \), we have

$$ \bigl( {}_{a-h} \nabla^{\alpha }_{h} y \bigr) (t) \leq 0,\quad \forall t \in \mathbb{N}_{a-h,h}. $$

Proof

The proof follows by applying Theorem 3.3 to \(g(t)=-y(t)\). □

4 Riemann-type fractional difference initial value problem

The following results are essential to proceed for the mean value theorem.

Lemma 4.1

For any \(0 < \alpha \leq 1\), \(0 < h \leq 1 \), and \(f: \mathbb{N}_{a+h,h} \rightarrow \mathbb{R} \), the following equality holds:

$$ {}_{a}\nabla_{h} ^{ - \alpha } \nabla_{h} f(t) = \nabla_{h}\, {}_{a}\nabla^{ - \alpha }_{h} f(t) - \frac{ ( t-a )_{h} ^{\overline{ \alpha - 1 }} }{ \Gamma (\alpha ) } f(a). $$

Proof

Recalling that

$$ {}_{a}\nabla_{h} ^{ - \alpha } f(t) = \frac{1}{\Gamma ( \alpha )} \sum_{k=a/h + 1} ^{t/h} \bigl( t - \rho_{h} (kh)\bigr)_{h} ^{\overline{\alpha - 1 }} f(kh) h, $$

we have

$$\begin{aligned}& \nabla_{h}\, {}_{a}\nabla^{ - \alpha }_{h} f(t) \\& \quad = \frac{1}{h} \Biggl( \frac{1}{\Gamma ( \alpha )} \sum _{k=a/h + 1} ^{ t/h } \bigl(t - \rho_{h} (kh) \bigr)_{h} ^{\overline{\alpha - 1 }} f(kh) h - \frac{1}{ \Gamma ( \alpha )} \sum _{k=a/h+1} ^{(t-h)/h } \bigl( t - h - \rho_{h} (kh) \bigr)_{h} ^{\overline{\alpha - 1}} f(kh) h \Biggr) \\& \quad = \frac{h}{\Gamma ( \alpha )} \sum_{k=a/h + 1} ^{ t/h } \frac{(t - \rho_{h} (kh) )_{h} ^{\overline{\alpha - 1 }}}{h} f(kh) - \frac{h}{ \Gamma ( \alpha )} \sum _{k=a/h+1} ^{t/h } \frac{( t - h - \rho_{h} (kh) )_{h} ^{\overline{\alpha - 1}}}{h} f(kh) \\& \quad\quad{} + \frac{h}{\Gamma (\alpha )} \bigl(t - h - \rho_{h} (t) \bigr)_{h} ^{\overline{ \alpha - 1}} f(t) \\& \quad = \frac{h}{\Gamma ( \alpha )} \sum_{k=a/h + 1} ^{ t/h } f(kh) \nabla _{h} \bigl(t - \rho_{h} (kh) \bigr)_{h} ^{\overline{\alpha - 1 }}. \end{aligned}$$

On the other hand, we have

$$\begin{aligned}& {}_{a}\nabla_{h} ^{ - \alpha } \nabla_{h} f(t) \\& \quad = \frac{1}{\Gamma ( \alpha )} \sum_{k=a/h+1} ^{t/h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{ \overline{\alpha - 1} } \nabla_{h} f(kh) h \\& \quad = \frac{1}{\Gamma ( \alpha )} \sum_{k=a/h+1} ^{t/h} \bigl( t - \rho _{h} (kh) \bigr)_{h} ^{ \overline{\alpha - 1} } \frac{f(kh)-f(kh-h)}{h} h \\& \quad = \frac{1}{\Gamma ( \alpha )} \sum_{k=a/h+1} ^{t/h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{ \overline{\alpha - 1} } f(kh) - \frac{1}{\Gamma ( \alpha )} \sum_{k=a/h+1} ^{t/h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{ \overline{ \alpha - 1} } f(kh-h) \\& \quad = \frac{1}{\Gamma ( \alpha )} \sum_{k=a/h+1} ^{t/h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{ \overline{\alpha - 1} } f(kh) - \frac{1}{\Gamma ( \alpha )} \sum_{k=a/h} ^{t/h - 1} \bigl( t - \rho_{h} (kh+h) \bigr)_{h} ^{ \overline{ \alpha - 1} } f(kh-h+h) \\& \quad = \frac{1}{\Gamma ( \alpha )} \sum_{k=a/h+1} ^{t/h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{ \overline{\alpha - 1} } f(kh) - \frac{1}{\Gamma ( \alpha )} \sum_{k=a/h + 1} ^{t/h} \bigl( t - h - \rho_{h} (kh) \bigr)_{h} ^{ \overline{ \alpha - 1} } f(kh) \\& \quad\quad {}+ \frac{1}{\Gamma (\alpha )} \bigl(t - h - \rho_{h} (t) \bigr)_{h} ^{\overline{ \alpha - 1}} f(t) - \frac{1}{\Gamma (\alpha ) } \bigl( t - h - \rho _{h} (a) \bigr)_{h} ^{ \overline{\alpha - 1} } f(a) \\& \quad = \frac{h}{\Gamma ( \alpha )} \sum_{k=a/h+1} ^{t/h} \frac{( t - \rho_{h} (kh) )_{h} ^{ \overline{\alpha - 1} } - ( t - h - \rho _{h} (kh) )_{h} ^{ \overline{\alpha - 1} }}{h} f(kh) -\frac{1}{ \Gamma (\alpha ) } ( t - a )_{h} ^{ \overline{\alpha - 1} } f(a) \\& \quad = \frac{h}{\Gamma ( \alpha )} \sum_{k=a/h+1} ^{t/h} \nabla_{h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{ \overline{\alpha - 1} } f(kh) - \frac{( t - a )_{h} ^{ \overline{\alpha - 1} }}{\Gamma (\alpha ) } f(a) \\& \quad = \nabla_{h} \,{}_{a}\nabla_{h} ^{ - \alpha } f(t) - \frac{( t - a )_{h} ^{ \overline{ \alpha - 1} }}{\Gamma (\alpha ) } f(a). \end{aligned}$$

 □

Lemma 4.2

For any \(0 < \alpha \leq 1\), \(0 < h \leq 1 \), and \(y: \mathbb{N}_{a+h,h} \rightarrow \mathbb{R} \), the following equality holds:

$$ {}_{a-h} \nabla_{h} ^{\alpha } y(t) = {}_{a} \nabla_{h} ^{\alpha } y(t) + \frac{(t-a+h)_{h} ^{\overline{-\alpha - 1 }}}{\Gamma ( - \alpha )} y(a)h. $$

Proof

From the definition and the proof of Lemma 2.1, we have

$$\begin{aligned} {}_{a-h} \nabla_{h} ^{\alpha } y(t) =& \nabla_{h}\, {}_{a-h} \nabla_{h} ^{ -(1- \alpha )} y(t) \\ =& \nabla_{h} \Biggl( \frac{1}{\Gamma ( 1 - \alpha ) } \sum _{k=a/h} ^{t/h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{\overline{-\alpha }} y(kh)h \Biggr) \\ =& \frac{1}{h\Gamma ( 1 - \alpha )} \Biggl( \sum_{k=a/h} ^{t/h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{\overline{-\alpha }} y(kh)h - \sum_{k=a/h} ^{t/h - 1 } \bigl( t - h - \rho_{h} (kh) \bigr)_{h} ^{\overline{-\alpha }} y(kh)h \Biggr) \\ =&\frac{1}{\Gamma ( 1 - \alpha ) } \sum_{k=a/h} ^{t/h} \nabla_{h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{\overline{-\alpha }} y(kh)h + \frac{1}{ \Gamma ( 1 - \alpha ) } ( t - h - t + h) )_{h} ^{\overline{-\alpha }} y(t) \\ =&\frac{1}{\Gamma ( 1 - \alpha ) } \sum_{k=a/h} ^{t/h} ( - \alpha ) \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{\overline{-\alpha - 1 }} y(kh)h \\ =&\frac{ - \alpha }{ - \alpha \Gamma ( - \alpha ) } \sum_{k=a/h} ^{t/h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{\overline{-\alpha - 1 }} y(kh)h \\ =&\frac{ 1 }{ \Gamma ( - \alpha ) } \sum_{k=a/h + 1 } ^{t/h} \bigl( t - \rho_{h} (kh) \bigr)_{h} ^{\overline{-\alpha - 1 }} y(kh)h + \frac{ 1 }{ \Gamma ( - \alpha ) } \bigl( t - \rho_{h} (a) \bigr)_{h} ^{\overline{-\alpha - 1 }} y(a)h \\ =&{}_{a}\nabla_{h} ^{\alpha } y(t) + \frac{ ( t - a + h )_{h} ^{\overline{-\alpha - 1 }} }{ \Gamma ( - \alpha ) } y(a) h. \end{aligned}$$
(5)

 □

Theorem 4.3

For any \(0 < \alpha \leq 1\), \(0 < h \leq 1 \), and \(y: \mathbb{N}_{a+h,h} \rightarrow \mathbb{R} \), the following equality holds:

$$ {}_{a}\nabla^{-\alpha }_{h} \,{}_{a-h} \nabla^{\alpha }_{h} y (t) = y(t) - \frac{ h^{1- \alpha } }{\Gamma (\alpha )} (t-a+h)_{h} ^{\overline{\alpha - 1}} y(a). $$

Proof

By the help of Lemma 4.2, we have

$$\begin{aligned} {}_{a}\nabla^{-\alpha }_{h} \bigl( {}_{a-h} \nabla^{\alpha }_{h} y (t) \bigr) =& {}_{a}\nabla_{h} ^{ - \alpha } \biggl( {}_{a} \nabla_{h} ^{\alpha } y(t) + \frac{(t-a+h)_{h} ^{\overline{- \alpha - 1}}}{\Gamma (- \alpha )} y(a)h \biggr) \\ =&{}_{a}\nabla_{h} ^{ - \alpha } {}_{a}\nabla_{h} ^{\alpha } y(t) + {}_{a} \nabla_{h} ^{ - \alpha } \biggl( \frac{(t-a+h)_{h} ^{\overline{- \alpha - 1}}}{\Gamma (- \alpha )} y(a)h \biggr) \\ =& y(t) + {}_{a}\nabla_{h} ^{ - \alpha } \biggl( \frac{(t-a+h)_{h} ^{\overline{- \alpha - 1}}}{\Gamma (- \alpha )} y(a)h \biggr) \\ =& y(t) + {}_{a}\nabla_{h} ^{ - \alpha } \nabla_{h} \biggl( \frac{(t-a+h)_{h} ^{\overline{- \alpha }}}{\Gamma ( 1 - \alpha )} \biggr) y(a)h. \end{aligned}$$
(6)

Moreover, by the help of Lemma 4.1 and that \(h_{h}^{\overline{- \alpha }}=h^{-\alpha }\Gamma (1-\alpha )\), we have

$$\begin{aligned}& \begin{aligned}[b] &{}_{a}\nabla_{h} ^{ - \alpha } \nabla_{h} \biggl( \frac{(t-a+h)_{h} ^{\overline{- \alpha }}}{\Gamma ( 1 - \alpha )} y(a)h \biggr) \\ &\quad = \nabla_{h} \,{}_{a} \nabla_{h} ^{ - \alpha } \biggl( \frac{(t-a+h)_{h} ^{\overline{- \alpha }}}{\Gamma ( 1 - \alpha )} y(a)h \biggr) - \frac{ ( t-a )_{h} ^{\overline{\alpha - 1 }} }{ \Gamma (\alpha ) } y(a)h ^{1-\alpha }. \end{aligned} \end{aligned}$$
(7)

Applying the identity

$$ {}_{a}\nabla_{h}^{-\alpha }g(t)={}_{a-h} \nabla_{h}^{-\alpha }g(t)-\frac{(t-a+h)_{h} ^{\overline{\alpha -1}} g(a)h}{\Gamma (\alpha )} $$

to the function \(g(t)=\frac{(t-a+h)_{h} ^{\overline{- \alpha }}}{ \Gamma (1 - \alpha )} y(a) h\), we obtain

$$ \begin{aligned}[b] & {}_{a}\nabla_{h} ^{- \alpha } \biggl( \frac{(t-a+h)_{h} ^{\overline{- \alpha }}}{\Gamma (1 - \alpha )} y(a) h \biggr) \\&\quad = {}_{a-h} \nabla_{h} ^{- \alpha } \biggl( \frac{(t-a+h)_{h} ^{\overline{- \alpha }}}{\Gamma (1 - \alpha )} y(a) h \biggr) - \frac{(t-a+h)_{h} ^{\overline{\alpha - 1}}}{\Gamma ( \alpha )} y(a)h^{2-\alpha }.\end{aligned} $$
(8)

Hence, by substituting (8) in (7) and by making use of Lemma 2.3 with \(\mu =-\alpha \), we obtain

$$\begin{aligned}& {}_{a}\nabla_{h} ^{ - \alpha } \nabla_{h} \biggl( \frac{(t-a+h)_{h} ^{\overline{- \alpha }}}{\Gamma ( 1 - \alpha )} y(a)h \biggr) \\& \quad = \nabla _{h} \biggl\lbrace {}_{a} \nabla_{h} ^{ - \alpha } \biggl( \frac{(t-(a-h))_{h} ^{\overline{- \alpha }}}{\Gamma ( 1 - \alpha )} y(a)h \biggr) \biggr\rbrace - \frac{ ( t-a )_{h} ^{\overline{\alpha - 1 }} }{ \Gamma (\alpha ) } y(a)h ^{1-\alpha } \\& \quad = \nabla_{h} \biggl\lbrace {}_{a-h} \nabla_{h} ^{- \alpha } \biggl( \frac{(t-(a-h))_{h} ^{\overline{- \alpha }}}{\Gamma (1 - \alpha )} y(a) h \biggr) - \frac{(t-(a-h))_{h} ^{\overline{\alpha - 1}}}{ \Gamma ( \alpha )} y(a)h^{2-\alpha } \biggr\rbrace \\& \quad\quad{} - \frac{ ( t-a )_{h} ^{\overline{\alpha - 1 }} }{ \Gamma (\alpha ) } y(a)h^{1-\alpha } \\& \quad = \nabla_{h} \bigl(y(a)h\bigr) - \frac{ h^{1- \alpha } }{\Gamma (\alpha )} (t-a+h)_{h} ^{\overline{\alpha - 1}} y(a) = -\frac{ h^{1- \alpha } }{\Gamma (\alpha )} (t-a+h)_{h} ^{\overline{\alpha - 1}} y(a). \end{aligned}$$

We have used that

$$ \nabla_{h} \frac{(t-(a-h))_{h} ^{\overline{\alpha - 1}}}{\Gamma ( \alpha )} y(a)h^{2-\alpha }= \frac{ h^{1- \alpha } }{\Gamma (\alpha )} (t-a+h)_{h} ^{\overline{ \alpha - 1}} y(a)- \frac{ ( t-a )_{h} ^{\overline{\alpha - 1 }} }{ \Gamma (\alpha ) } y(a)h^{1-\alpha }. $$

Hence,

$$ {}_{a}\nabla^{-\alpha }_{h} \,{}_{a-h} \nabla^{\alpha }_{h} y (t) = y(t) - \frac{ h^{1- \alpha } }{\Gamma (\alpha )} (t-a+h)_{h} ^{\overline{\alpha - 1}} y(a). $$

 □

Consider the following initial fractional difference equation:

$$\begin{aligned}& {}_{a-h}\nabla_{h}^{\alpha }y(t) = f \bigl(t, y(t)\bigr) \quad \text{for }t = a+h, a+2h,\ldots, \end{aligned}$$
(9)
$$\begin{aligned}& {}_{a-h}\nabla_{h}^{-(1-\alpha )}y(t)\big|_{t=a} = h^{1-\alpha }y(a) = c, \end{aligned}$$
(10)

where \(0 < \alpha ,h < 1\) and a is any real number.

By means of Theorem 4.3, we can state the following theorem.

Theorem 4.4

y is a solution of the initial value problem, (9), (10) if and only if it has the representation

$$ y(t) = \frac{(t-a+h)_{h}^{\overline{\alpha -1}}}{\Gamma (\alpha )}c+ {}_{a} \nabla_{h}^{-\alpha } f\bigl(t, y(t)\bigr). $$
(11)

5 Application: Mean Value Theorem (MVT)

First, for the sake of simplification, depending on Theorem 4.3, we shall write

$$ {}_{a}\nabla^{-\alpha }_{h}\, {}_{a-h} \nabla^{\alpha }_{h} y (t) = y(t) - R_{h} (\alpha , t , a) y(a), $$
(12)

where \(R_{h} (\alpha , t , a) = \frac{ h^{1- \alpha } }{ \Gamma (\alpha )} (t-a+h)_{h} ^{\overline{\alpha - 1}} \).

Theorem 5.1

(The h-fractional difference MVT)

Let f and g be functions defined on \(\mathbb{N}_{a,h} \cap {}_{b,h} \mathbb{N} = \lbrace a , a+h, a+2h,\ldots, b-2h, b-h, b \rbrace \), where \(b=a+kh \) for some \(k \in \mathbb{N} \). Assume that g is strictly increasing, \(g(a)>0 \), and \(0 < \alpha < 1 \), \(0 < h \leq 1\). Then there exist \(s_{1}, s_{2} \in \mathbb{N}_{a,h} \cap {}_{b,h} \mathbb{N} \) such that

$$ \frac{( {}_{a-h} \nabla^{\alpha }_{h} f )(s_{1})}{ ( {}_{a-h} \nabla^{\alpha }_{h} g )(s_{1}) } \leq \frac{ f(b) - R_{h} (\alpha , b , a) f(a) }{ g(b) - R _{h} (\alpha , b , a) g(a) } \leq \frac{( {}_{a-h} ^{R} \nabla^{\alpha }_{h} f )(s_{2})}{ ( _{a-h} ^{R} \nabla^{\alpha }_{h} g )(s_{2}) }. $$
(13)

Proof

First we need to show that \(g(b)-R_{h}(\alpha ,b,a)g(a)>0 \). Since g is strictly increasing, then by Theorem 3.4 we have

$$ \bigl( {}_{a-h} \nabla^{\alpha }_{h} g \bigr) (t) > 0 \quad \forall t \in \mathbb{N}_{a,h} \cap {}_{b,h} \mathbb{N}. $$

Applying the fractional sum operator on both sides of the inequality, by means of (12), we get

$$ {}_{a}\nabla_{h} ^{-\alpha } \bigl( {}_{a-h} \nabla^{\alpha }_{h} g \bigr) (t) > {}_{a} \nabla_{h} ^{-\alpha } ( 0 ) \quad \forall t \in \mathbb{N}_{a,h} \cap {}_{b,h} \mathbb{N}, $$

or by means of (12) we have

$$ g(t) - R_{h} (\alpha , t,a) g(a) > 0 \quad \forall t \in \mathbb{N}_{a,h} \cap {}_{b,h} \mathbb{N}. $$

For \(t=b \), we get

$$ g(b) - R_{h} (\alpha , b,a) g(a) > 0. $$

To prove the theorem, we use contradiction. Assume that (13) is not true, then either

$$ \frac{ f(b) - R_{h} (\alpha , b , a) f(a) }{ g(b) - R _{h} (\alpha , b , a) g(a) } < \frac{( {}_{a-h} \nabla^{\alpha }_{h} f )(t)}{ ({}_{a-h} \nabla^{\alpha }_{h} g )(t) } , \quad \forall t \in \mathbb{N}_{a,h} \cap {}_{b,h} \mathbb{N}, $$
(14)

or

$$ \frac{ f(b) - R_{h} (\alpha , b , a) f(a) }{ g(b) - R _{h} (\alpha , b , a) g(a) } > \frac{( {}_{a-h} \nabla^{\alpha }_{h} f )(t)}{ ( {}_{a-h} ^{R} \nabla^{\alpha }_{h} g )(t) } , \quad \forall t \in \mathbb{N}_{a,h} \cap {}_{b,h} \mathbb{N}. $$
(15)

Again, since g is strictly increasing, then by Theorem 3.4 we conclude that

$$ \bigl( {}_{a-h} \nabla^{\alpha }_{h} g \bigr) (t) > 0 \quad \forall t \in \mathbb{N}_{a,h} \cap {}_{b,h} \mathbb{N}. $$

Hence (14) becomes

$$ \frac{ f(b) - R_{h} (\alpha , b , a) f(a) }{ g(b) - R _{h} (\alpha , b , a) g(a) } \bigl({}_{a-h} \nabla^{\alpha }_{h} g \bigr) (t) < \bigl( {}_{a-h} \nabla^{\alpha }_{h} f \bigr) (t) ,\quad \forall t \in \mathbb{N}_{a,h} \cap {}_{b,h} \mathbb{N}. $$

Applying the fractional sum operator on both sides of the inequality at \(t=b\) and by making use of (12), we see that

$$ \frac{ f(b) - R_{h} (\alpha , b , a) f(a) }{ g(b) - R _{h} (\alpha , b , a) g(a) } \bigl( g(b) - R _{h} (\alpha , b , a) g(a) \bigr) < \bigl( f(b) - R_{h} (\alpha , b,a) f(a) \bigr), $$

and hence \(f(b)< f(b)\), which is a contradiction. In a similar way, (15) will lead to contradiction. □

Remark 5.1

If we let \(h=1\) in Theorem 5.1, then we reobtain the results in [34] via using the dual identities presented in [30, 31], or else we refer to [37].

6 Conclusions

The contributions of this paper can be concluded as follows:

  1. 1.

    Nabla fractional sums and differences of order \(0 < \alpha \leq 1\) on the time scale \(h\mathbb{Z}\) have been formulated.

  2. 2.

    Riemann–Liouville and Caputo discrete fractional operators on the time scale \(h \mathbb{Z} \) have been defined.

  3. 3.

    The relation between nabla h-RL and h-Caputo fractional differences has been detected.

  4. 4.

    If \(( {}_{a}\nabla_{h}^{\alpha }y)(t) > 0 \), then \(y(t)\) is α-increasing.

  5. 5.

    If \(y(t)\) is α-increasing and \(y(a)>0\), then \(({}_{a}\nabla _{h}^{\alpha }y)(t)>0\).

  6. 6.

    The monotonicity factor, which is α, has not been affected by the discretization step h.

  7. 7.

    A Riemann-type fractional difference initial value problem has been formulated and solved, and hence we generalize the representation obtained in [20].

  8. 8.

    A monotonicity result for the nabla h-Caputo fractional difference operator has been proved as well.

  9. 9.

    As an application, a fractional difference version of the Mean Value Theorem on \(h\mathbb{Z}\) has been proved.

  10. 10.

    Working on \(h\mathbb{Z}\), \(h\in (0,1)\) rather than on \(\mathbb{Z}\) makes it possible to guarantee the convergence of solutions for a larger class of fractional difference initial value problems.

References

  1. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  2. Samko, S., Kilbas, A., Marichev, A.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)

    MATH  Google Scholar 

  3. Kilbas, A., Srivastava, M., Trujillo, J.: Theory and Application of Fractional Differential Equations. North Holland Mathematics Studies, vol. 204 (2006)

    MATH  Google Scholar 

  4. Abdeljawad, T., Baleanu, D., Jarad, F.: Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives. J. Math. Phys. 49, 083507 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Silva, M., Machado, J., Lopes, A.: Modeling and simulation of artificial locomotion systems. Robotica 23, 595–606 (2005)

    Article  Google Scholar 

  6. Abdeljawad, T., Jarad, F., Baleanu, D.: On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives. Sci. China Ser. A, Math. 51(10), 1775–1786 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bozkurt, F., Abdeljawad, T., Hajji, M.: Stability analysis of a fractional order differential equation model of a brain tumor growth depending on the density. Appl. Comput. Math. 14(1), 50–62 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Jarad, F., Abdeljawad, T., Baleanu, D.: Higher order variational optimal control problems with delayed arguments. Appl. Math. Comput. 218(18), 9234–9240 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Jarad, F., Abdeljawad, T., Baleanu, D.: Fractional variational optimal control problems with delayed arguments. Nonlinear Dyn. 62(3), 609–614 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sadati, S., Baleanu, D., Ranjbar, A., Ghaderi, R., Abdeljawad, T.: Mittag-Leffler stability theorem for fractional-nonlinear systems with delay. Abstr. Appl. Anal. 2010, Article ID 108651 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Miller, K., Ross, B.: Fractional difference calculus. In: Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, pp. 139–152 (1989)

    Google Scholar 

  12. Atıcı, F., Eloe, P.: Discrete fractional calculus with the nabla operator. Electron. J. Qual. Theory Differ. Equ. 2009, 3 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62(3), 1602–1611 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Abdeljawad, T., Baleanu, D.: Fractional differences and integration by parts. J. Comput. Anal. Appl. 13(3), 574–582 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Jarad, F., Abdeljawad, T., Baleanu, D., Biçen, K.: On the stability of some discrete fractional nonautonomous systems. Abstr. Appl. Anal. 2012, Article ID 476581 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Jarad, F., Abdeljawad, T., Gündog̃du, E., Baleanu, D.: On the Mittag-Leffler stability of q-fractional nonlinear dynamical systems. Proc. Rom. Acad. 12(4), 309–314 (2011)

    MathSciNet  Google Scholar 

  17. Abdeljawad, T., Baleanu, D.: Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 16(12), 4682–4688 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Atıcı, F., Eloe, P.: A transform method in discrete fractional calculus. Int. J. Difference Equ. 2(2), 165–176 (2007)

    MathSciNet  Google Scholar 

  19. Atıcı, F., Eloe, P.: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137(3), 981–989 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Abdeljawad, T., Atıcı, F.: On the definitions of nabla fractional differences. Abstr. Appl. Anal. 2012, Article ID 406757 (2012)

    MATH  Google Scholar 

  21. Atıcı, F., Eloe, P.: Linear systems of fractional nabla difference equations. Rocky Mt. J. Math. 41(2), 353–370 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Atıcı, F., Eloe, P.: Gronwall’s inequality on discrete fractional calculus. Comput. Math. Appl. 64(10), 3193–3200 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ferreira, R., Torres, D.: Fractional h-difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 5(1), 110–121 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Goodrich, C.: Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23(9), 1050–1055 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bastos, N., Ferreira, R., Torres, D.: Discrete-time fractional variational problems. Signal Process. 91(3), 513–524 (2011)

    Article  MATH  Google Scholar 

  26. Anastassiou, G.: Nabla discrete fractional calculus and nabla inequalities. Math. Comput. Model. 51(5–6), 562–571 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wu, G., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Goodrich, C., Peterson, A.: Discrete Fractional Calculus. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  29. Hein, J., McCarthy, Z., Gaswick, N., McKain, B., Speer, K.: Laplace transforms for the nabla-difference operator. Panam. Math. J. 21(3), 79–97 (2011)

    MathSciNet  MATH  Google Scholar 

  30. Abdeljawad, T.: Dual identities in fractional difference calculus within Riemann. Adv. Differ. Equ. 2013, 36 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Abdeljawad, T.: On delta and nabla Caputo fractional differences and dual identities. Discrete Dyn. Nat. Soc. 2013, Article ID 406910 (2013)

    MathSciNet  Google Scholar 

  32. Dahal, R., Goodrich, C.: A monotonicity result for discrete fractional difference operators. Arch. Math. (Basel) 102, 293–299 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Jia, B., Erbe, L., Peterson, A.: Two monotonicity results for nabla and delta fractional differences. Arch. Math. (Basel) 104(6), 589–597 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Atıcı, F., Uyanık, M.: Analysis of discrete fractional operators. Appl. Anal. Discrete Math. 9, 139–149 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Erbe, L., Goodrich, C., Jia, B., Peterson, A.: Monotonicity results for delta and nabla fractional differences revisited. Math. Slovaca 67(4), 895–906 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Goodrich, C.: A convexity result for fractional differences. Appl. Math. 35, 58–62 (2014)

    MathSciNet  MATH  Google Scholar 

  37. Abdeljawad, T., Abdalla, B.: Monotonicity results for delta and nabla Caputo and Riemann fractional differences via dual identities. Filomat 31(12), 3671–3683 (2017)

    Article  MathSciNet  Google Scholar 

  38. Dahal, R., Goodrich, C.: An almost sharp monotonicity result for discrete sequential fractional delta differences. J. Differ. Equ. Appl. 23(7), 1190–1203 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Abdeljawad, T., Baleanu, D.: On fractional derivatives with exponential kernel and their discrete versions. Rep. Math. Phys. 80(1), 11–27 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)

    Article  Google Scholar 

  41. Abdeljawad, T., Baleanu, D.: Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. J. Nonlinear Sci. Appl. 10(3), 1098–1107 (2017)

    Article  MathSciNet  Google Scholar 

  42. Abdeljawad, T., Baleanu, D.: Discrete fractional differences with non-singular discrete Mittag-Leffler kernels. Adv. Differ. Equ. 2016, 232 (2016)

    Article  Google Scholar 

  43. Abdeljawad, T., Al-Mdallal, Q.M.: Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall’s inequality. J. Comput. Appl. Math. 339, 218–230 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Abdeljawad, T., Baleanu, D.: Monotonicity results for fractional difference operators with discrete exponential kernels. Adv. Differ. Equ. 2017, 78 (2017)

    Article  MathSciNet  Google Scholar 

  45. Abdeljawad, T., Baleanu, D.: Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel. Chaos Solitons Fractals 102, 106–110 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The third author would like to thank Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

Author information

Authors and Affiliations

Authors

Contributions

All the authors participated in obtaining the main results of this manuscript and drafted the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Thabet Abdeljawad.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Suwan, I., Owies, S. & Abdeljawad, T. Monotonicity results for h-discrete fractional operators and application. Adv Differ Equ 2018, 207 (2018). https://doi.org/10.1186/s13662-018-1660-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-018-1660-5

Keywords