Monotonicity and positivity analyses for two discrete fractional-order operator types with exponential and Mittag–Leffler kernels

Definition 2.1

Definition 2.1 see Abdeljawad, 2018; Abdeljawad and Baleanu, 2017a

Let $\mathtt{Q}\in {\mathcal{F}}_{\mathtt{p}{}_0,0}$ and $\mathcal{C}\left(\phi \right)>0$ is a multiplier. Then the following operators:

$$\left({}_{\mathtt{p}{}_0}{}^{\mathit{ABC}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)≔\frac{\mathcal{C}\left(\phi \right)}{-\phi +1}{\displaystyle \sum _{r=\mathtt{p}{}_0+1}^{\tau }}\left({\nabla }_r\mathtt{Q}\right)\left(r\right){\mathsf{E}}_{\bar{\phi }}(\xi ,1+\tau -r)\left\{\tau \in {\mathbb{N}}_{\mathtt{p}{}_0+1}\right\},$$ and

(2.2)

$$\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFC}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)≔\mathcal{C}\left(\phi \right){\displaystyle \sum _{r=\mathtt{p}{}_0+1}^{\tau }}\left({\nabla }_r\mathtt{Q}\right)\left(r\right){(-\phi +1)}^{\tau -r}\left\{\tau \in {\mathbb{N}}_{\mathtt{p}{}_0+1}\right\}$$ are called the discrete Atangana-Baleanu and Caputo-Fabrizo fractional operators in the Liouville-Caputo sense, respectively. Also, the following operators:

(2.3)

$$\left({}_{\mathtt{p}{}_0}{}^{\mathit{ABR}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)≔\frac{\mathcal{C}\left(\phi \right)}{-\phi +1}{\nabla }_{\tau }{\displaystyle \sum _{r=\mathtt{p}{}_0+1}^{\tau }}\mathtt{Q}\left(r\right){\mathsf{E}}_{\bar{\phi }}(\xi ,\tau -r+1)\left\{\tau \in {\mathbb{N}}_{\mathtt{p}{}_0+1}\right\},$$ and

(2.4)

$$\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)≔\mathcal{C}\left(\phi \right){\nabla }_{\tau }{\displaystyle \sum _{r=\mathtt{p}{}_0+1}^{\tau }}\mathtt{Q}\left(r\right){(-\phi +1)}^{\tau -r}\left\{\tau \in {\mathbb{N}}_{\mathtt{p}{}_0+1}\right\}$$ are called the discrete Atangana-Baleanu and Caputo-Fabrizo fractional operators in the Riemann–Liouville sense, respectively.

Definition 2.2

Definition 2.2 see Abdeljawad et al., 2017; Abdeljawad and Madjidi, 2017

For $\mathtt{Q}\in {\mathcal{F}}_{\mathtt{p}{}_0,\kappa }$ with $\kappa <\phi \leqq \kappa +1$ , the discrete Atangana-Baleanu and Caputo-Fabrizo fractional difference operators can be expressed as follows:

$$\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFC}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)=\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFC}}\nabla ^{\phi -\kappa }{\nabla }^{\kappa }\mathtt{Q}\right)\left(\tau \right)\mathrm{and}\left({}_{\mathtt{p}{}_0}{}^{\mathit{ABC}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)=\left({}_{\mathtt{p}{}_0}{}^{\mathit{ABC}}\nabla ^{\phi -\kappa }{\nabla }^{\kappa }\mathtt{Q}\right)\left(\tau \right),$$ in the Liouville-Caputo sense, and

(6)

$$\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)=\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi -\kappa }{\nabla }^{\kappa }\mathtt{Q}\right)\left(\tau \right),\mathrm{and}\left({}_{\mathtt{p}{}_0}{}^{\mathit{ABR}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)=\left({}_{\mathtt{p}{}_0}{}^{\mathit{ABR}}\nabla ^{\phi -\kappa }{\nabla }^{\kappa }\mathtt{Q}\right)\left(\tau \right),$$ in the Riemann–Liouville sense, for each $\tau \in {\mathbb{N}}_{\mathtt{p}{}_0+1}$ .

Remark 2.1

The above definitions contain the one-parameter Mittag–Leffler function $E_{\alpha }\left(z\right)$ , which is defined here as follows (see Mohammed and Abdeljawad, 2020; see also Srivastava, 2021a; Srivastava, 2021b for much more general families of Mittag–Leffler type functions):

$${\mathsf{E}}_{\bar{\phi }}(\xi ,\tau )≔{\displaystyle \sum _{k=0}^{\infty }}{\xi }^k\frac{{\tau }^{\overline{k\phi }}}{\mathrm{\Gamma }(k\phi +1)}=:E_{\bar{\phi }}\left(\xi {\tau }^{\bar{\phi }}\right)$$ for any $\xi \in \mathbb{R}$ such that $\left|\xi \right|<1$ , $\phi ,\tau \in \mathbb{C}$ with $\mathrm{Re}\left(\phi \right)>0$ . and ${\tau }^{\bar{\phi }}$ is the rising function defined by

(8)

$${\tau }^{\bar{\phi }}=\frac{\mathrm{\Gamma }\left(\tau +\phi \right)}{\mathrm{\Gamma }\left(\tau \right)},$$ for $\phi$ in $\mathbb{R}$ and $\tau$ in $\mathbb{R}⧹\{\cdots ,-2,-1,0\}$ .

Proposition 2.1

Assume that $Q\in {\mathcal{F}}_{p_0,0}$ . Then, for $\kappa <\phi \leqq \kappa +1$ , it is asserted that $$\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)=\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFC}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)+\frac{\mathcal{C}(\phi -\kappa )}{\kappa +-\phi +1}{(\kappa +-\phi +1)}^{\tau -\mathtt{p}{}_0}\left({\nabla }^{\kappa }\mathtt{Q}\right)\left({\mathtt{p}}_0\right)\text{.}$$ Furthermore, for $\kappa <\phi \leqq \kappa +\frac{1}{2}$ , it is asserted that $$\left({}_{\mathtt{p}{}_0}{}^{\mathit{ABR}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)=\left({}_{\mathtt{p}{}_0}{}^{\mathit{ABC}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)+\frac{\mathcal{C}(\phi -\kappa )}{\kappa +-\phi +1}{\mathsf{E}}_{\overline{\phi -\kappa }}({\xi }_{\kappa },\tau -\mathtt{p}{}_0)\left({\nabla }^{\kappa }\mathtt{Q}\right)\left({\mathtt{p}}_0\right),$$ for each $\tau \in {\mathbb{N}}_{p_0+1}$ , where $${\xi }_{\kappa }=-\frac{\phi -\kappa }{\kappa +-\phi +1}\text{.}$$

Proof

The first part of the proof can be deduced by referencing Definition 2.2 and Abdeljawad (2018, Proposition 8). Similarly, the second part of the proof can be established by relying on Definition 2.2 and Abdeljawad (2018, Theorem 10).  □

Remark 2.2

We note that, for $\xi =-\frac{\phi }{-\phi +1}$ with $0<\phi <\frac{1}{2}$ , it is known that $\left|\xi \right|<1$ . Moreover, since $$0<\phi -\kappa \leqq \frac{1}{2},$$ we see that $\left|{\xi }_{\kappa }\right|<1$ for $${\xi }_{\kappa }=-\frac{\phi -\kappa }{\kappa +-\phi +1}$$ with $$\kappa <\phi \leqq \kappa +\frac{1}{2}\text{.}$$

Lemma 3.1

Assume that $Q\in {\mathcal{F}}_{p_0,0},1<\phi <2$ and $\left({}_{p_0}{}^{\mathit{CFR}}\nabla ^{\phi }Q\right)\left(\tau \right)⩾q0$ $\forall$ $\tau \in {\mathbb{N}}_{p_0+1}$ . Then $$\left(\nabla \mathtt{Q}\right)\left(\tau \right)⩾q(\phi -1)\left\{{(-\phi +2)}^{\tau -\mathtt{p}{}_0-2}\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)+{\displaystyle \sum _{r=\mathtt{p}{}_0+2}^{\tau -1}}\left({\nabla }_r\mathtt{Q}\right)\left(r\right){(-\phi +2)}^{\tau -r-1}\right\},$$ for each $\tau \in {\mathbb{N}}_{p_0+2}$ .

Proof

From Definition 2.1 with $1<\phi <2$ , one can see, for each $\tau \in {\mathbb{N}}_{\mathtt{p}{}_0+2}$ , that

$$\begin{array}{cc}\hfill \left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)& =\mathcal{C}(\phi -1)\left\{{\displaystyle \sum _{r=\mathtt{p}{}_0+1}^{\tau }}\left({\nabla }_r\mathtt{Q}\right)\left(r\right){(-\phi +2)}^{\tau -r}-{\displaystyle \sum _{r=\mathtt{p}{}_0+1}^{\tau -1}}\left({\nabla }_r\mathtt{Q}\right)\left(r\right){(-\phi +2)}^{\tau -r-1}\right\}\hfill \\ \hfill & =\mathcal{C}(\phi -1)\left\{\left(\nabla \mathtt{Q}\right)\left(\tau \right)+{\displaystyle \sum _{r=\mathtt{p}{}_0+1}^{\tau -1}}\left({\nabla }_r\mathtt{Q}\right)\left(r\right)\left[{(-\phi +2)}^{\tau -r}-{(-\phi +2)}^{\tau -r-1}\right]\right\}\hfill \\ \hfill & =\mathcal{C}(\phi -1)\left\{\left(\nabla \mathtt{Q}\right)\left(\tau \right)+(-\phi +2){(-\phi +1)}^{\tau -\mathtt{p}{}_0-2}\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)\right.\hfill \\ \hfill & \left.+(-\phi +1){\displaystyle \sum _{r=\mathtt{p}{}_0+2}^{\tau -1}}\left({\nabla }_r\mathtt{Q}\right)\left(r\right){(-\phi +2)}^{\tau -r-1}\right\},\hfill \end{array}$$ which, together with the fact that $\mathcal{C}(\phi -1)>0$ and the assumption that $\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)⩾q0$ , can be rearranged to the derive the desired results. □

Corollary 3.1

Assume that $Q\in {\mathcal{F}}_{p_0,0},1<\phi <2$ and $$\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFC}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)⩾q-\frac{\mathcal{C}(\phi -1)}{-\phi +2}{(-\phi +2)}^{\tau -\mathtt{p}{}_0}\left(\nabla \mathtt{Q}\right)\left({\mathtt{p}}_0\right)$$ for each $\tau \in {\mathbb{N}}_{p_0+1}$ . Then $$\left(\nabla \mathtt{Q}\right)\left(\tau \right)⩾q(-1+\phi )\left\{{(2-\phi )}^{\tau -\mathtt{p}{}_0-2}\left(\nabla \mathtt{Q}\right)(1+\mathtt{p}{}_0)+{\displaystyle \sum _{r=\mathtt{p}{}_0+2}^{\tau -1}}\left({\nabla }_r\mathtt{Q}\right)\left(r\right){(-\phi +2)}^{\tau -r-1}\right\},$$ for each $\tau \in {\mathbb{N}}_{p_0+2}$ .

Proof

The deduction of the outcome can be made directly by referring to Proposition 2.1 and Lemma 3.1. □

Lemma 3.2

Suppose that $Q\in {\mathcal{F}}_{p_0,0},1<\phi <\frac{3}{2},{\xi }_1=-\frac{\phi -1}{-\phi +2}$ and $\left({}_{p_0}{}^{\mathit{ABR}}\nabla ^{\phi }Q\right)\left(\tau \right)⩾q0$ for each $\tau \in {\mathbb{N}}_{p_0+1}$ . Then $$\left(\nabla \mathtt{Q}\right)\left(\tau \right)⩾q\frac{1}{-\phi +2}\left\{\left[{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -\mathtt{p}{}_0-1)-{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -\mathtt{p}{}_0)\right]\left(\nabla \mathtt{Q}\right)(1+\mathtt{p}{}_0)+{\displaystyle \sum _{r=\mathtt{p}{}_0+2}^{\tau -1}}\left[{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -r)-{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -r+1)\right]\left({\nabla }_r\mathtt{Q}\right)\left(r\right)\right\},$$ for each $\tau \in {\mathbb{N}}_{p_0+2}$ .

Proof

According to Definition 2.1 when $1<\phi <\frac{3}{2}$ , we find for each $\tau \in {\mathbb{N}}_{\mathtt{p}{}_0+2}$ that

$$\begin{array}{cc}\hfill \left({}_{\mathtt{p}{}_0}{}^{\mathit{ABR}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)& =\frac{\mathcal{C}(\phi -1)}{-\phi +2}\left\{{\displaystyle \sum _{r=\mathtt{p}{}_0+1}^{\tau }}{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -r+1)\left({\nabla }_r\mathtt{Q}\right)\left(r\right)-{\displaystyle \sum _{r=\mathtt{p}{}_0+1}^{\tau -1}}{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -r)\left({\nabla }_r\mathtt{Q}\right)\left(r\right)\right\}\hfill \\ \hfill & =\frac{\mathcal{C}(\phi -1)}{2-\phi }\left\{(-\phi +2)\left(\nabla \mathtt{Q}\right)\left(\tau \right)+{\displaystyle \sum _{r=\mathtt{p}{}_0+1}^{\tau -1}}\left[{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -r+1)-{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -r)\right]\left({\nabla }_r\mathtt{Q}\right)\left(r\right)\right\}\hfill \\ \hfill & =\frac{\mathcal{C}(\phi -1)}{2-\phi }\left\{(-\phi +2)\left(\nabla \mathtt{Q}\right)\left(\tau \right)+\left[{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -\mathtt{p}{}_0)-{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -\mathtt{p}{}_0-1)\right]\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)\right.\hfill \\ \hfill & \left.+{\displaystyle \sum _{r=\mathtt{p}{}_0+2}^{\tau -1}}\left[{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -r+1)-{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -r)\right]\left({\nabla }_r\mathtt{Q}\right)\left(r\right)\right\}\text{.}\hfill \end{array}$$ Considering this together with the fact that $\mathcal{C}(\phi -1)>0$ and the assumption $\left({}_{\mathtt{p}{}_0}{}^{\mathit{ABR}}\nabla ^{\phi }\mathtt{Q}\right)\left(\tau \right)⩾q0$ , we arrive at the intended result. □

Corollary 3.2

Assume that $Q\in {\mathcal{F}}_{p_0,0},1<\phi <\frac{3}{2},{\xi }_1=-\frac{\phi -1}{-\phi +2}$ and $\left({}_{p_0}{}^{\mathit{ABC}}\nabla ^{\phi }Q\right)\left(\tau \right)⩾q-\frac{\mathcal{C}(\phi -1)}{-\phi +2}{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -p_0)\left(\nabla Q\right)\left(p_0\right)$ for each $\tau \in {\mathbb{N}}_{p_0+1}$ . Then, we have $$\left(\nabla \mathtt{Q}\right)\left(\tau \right)⩾q\frac{1}{2-\phi }\left\{\left[{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -\mathtt{p}{}_0)-{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -\mathtt{p}{}_0-1)\right]\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)+{\displaystyle \sum _{r=\mathtt{p}{}_0+2}^{\tau -1}}\left[{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -r+1)-{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -r)\right]\left({\nabla }_r\mathtt{Q}\right)\left(r\right)\right\},$$ for each $\tau \in {\mathbb{N}}_{p_0+2}$ .

Proof

The proof can be straightforwardly derived from the application of Proposition 2.1 and Lemma 3.2.  □

Theorem 3.1

Suppose that $Q\in {\mathcal{F}}_{p_0,0}$ , $1<\phi <2$ and $\left({}_{p_0}{}^{\mathit{CFR}}\nabla ^{\phi }Q\right)\left(\tau \right)⩾q0$ $\forall$ $\tau \in {\mathbb{N}}_{p_0+1}$ . Then $\left(\nabla Q\right)\left(\tau \right)⩾q0$ $\forall$ $\tau \in {\mathbb{N}}_{p_0+1}$ .

Proof

We proceed with the proof by mathematical induction. By using Definition (6) for $1<\phi <2$ at $\tau =\mathtt{p}{}_0+1$ , one can have $$\begin{array}{cc}\hfill \left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi }\mathtt{Q}\right)({\mathtt{p}}_0+1)& =\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi -1}\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)\hfill \\ \hfill & =\mathcal{C}(\phi -1)\left\{{\displaystyle \sum _{r=1+\mathtt{p}{}_0}^{1+\mathtt{p}{}_0}}\left(\nabla \mathtt{Q}\right)\left(r\right){(-\phi +2)}^{\mathtt{p}{}_0+1-r}-{\displaystyle \sum _{r=1+\mathtt{p}{}_0}^{\mathtt{p}{}_0}}\left(\nabla \mathtt{Q}\right)\left(r\right){(2-\phi )}^{\mathtt{p}{}_0-r}\right\}\hfill \\ \hfill & =\mathcal{C}(\phi -1)\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1),\hfill \end{array}$$ which implies that $\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)⩾q0$ by the hypothesis and the fact that $\mathcal{C}(\phi -1)>0$ . At $\tau =\mathtt{p}{}_0+2$ , we have $$\begin{array}{cc}\hfill \left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi }\mathtt{Q}\right)({\mathtt{p}}_0+2)& =\mathcal{C}(\phi -1)\left\{{\displaystyle \sum _{r=1+\mathtt{p}{}_0}^{2+\mathtt{p}{}_0}}\left(\nabla \mathtt{Q}\right)\left(r\right){(-\phi +2)}^{\mathtt{p}{}_0+2-r}-{\displaystyle \sum _{r=1+\mathtt{p}{}_0}^{1+\mathtt{p}{}_0}}\left(\nabla \mathtt{Q}\right)\left(r\right){(-\phi +2)}^{\mathtt{p}{}_0+1-r}\right\}\hfill \\ \hfill & =\mathcal{C}(\phi -1)\left\{(-\phi +1)\left(\nabla \mathtt{Q}\right)(1+\mathtt{p}{}_0)+\left(\nabla \mathtt{Q}\right)(2+\mathtt{p}{}_0)\right\}⩾q0,\hfill \end{array}$$ which leads to $$\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+2)⩾q\underset{>0}{\underbrace{(\phi -1)}}\underset{⩾q0}{\underbrace{\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)}}⩾q0\text{.}$$ Suppose now that $\kappa ⩾q2$ and that $\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+ı)⩾q0$ for $ı=2,3,\cdots ,\kappa$ . Then, by using Lemma 3.1 at $\tau =\mathtt{p}{}_0+\kappa +1$ , we have $$\begin{array}{cc}\hfill \left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+\kappa +1)& ⩾q{\displaystyle \underset{>0}{\underbrace{(\phi -1)}}}\left\{{\displaystyle \underset{>0}{\underbrace{{(2-\phi )}^{\kappa -1}}}}{\displaystyle \underset{⩾q0}{\underbrace{\left(\nabla \mathtt{Q}\right)(1+\mathtt{p}{}_0)}}}+{\displaystyle \sum _{r=\mathtt{p}{}_0+2}^{\mathtt{p}{}_0+\kappa }}{\displaystyle \underset{⩾q0\text{by inductive assumption}}{\underbrace{\left({\nabla }_r\mathtt{Q}\right)\left(r\right)}}}{\displaystyle \underset{>0}{\underbrace{{(-\phi +2)}^{\kappa +\mathtt{p}{}_0-r}}}}\right\}\hfill \\ \hfill & ⩾q0,\hfill \end{array}$$ which completes the proof of the induction process. Hence, the proof is done.  □

Corollary 3.3

Suppose $Q\in {\mathcal{F}}_{p_0,0},1<\phi <2$ and $\left({}_{p_0}{}^{\mathit{CFC}}\nabla ^{\phi }Q\right)\left(\tau \right)⩾q-\frac{\mathcal{C}(\phi -1)}{-\phi +2}{(-\phi +2)}^{\tau -p_0}\left(\nabla Q\right)\left(p_0\right)$ $\forall$ $\tau \in {\mathbb{N}}_{p_0+1}$ . Then, $\left(\nabla Q\right)\left(\tau \right)⩾q0$ $\forall$ $\tau \in {\mathbb{N}}_{p_0+1}$ .

Theorem 3.2

Suppose that $Q\in {\mathcal{F}}_{p_0,0}$ , $1<\phi <\frac{3}{2},{\xi }_1=-\frac{\phi -1}{-\phi +2}$ and $\left({}_{p_0}{}^{\mathit{ABR}}\nabla ^{\phi }Q\right)\left(\tau \right)⩾q0$ $\forall$ $\tau \in {\mathbb{N}}_{p_0+1}$ . Then $\left(\nabla Q\right)\left(\tau \right)⩾q0$ $\forall$ $\tau \in {\mathbb{N}}_{p_0+1}$ .

Proof

Again, we use mathematical induction to proceed with the proof of Theorem 3.2. Indeed, by using Definition (6) for $1<\phi <\frac{3}{2}$ at $\tau =\mathtt{p}{}_0+1$ , and Remark 2.1, we see that $$\begin{array}{cc}\hfill \left({}_{\mathtt{p}{}_0}{}^{\mathit{ABR}}\nabla ^{\phi }\mathtt{Q}\right)({\mathtt{p}}_0+1)& =\left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi -1}\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)\hfill \\ \hfill & =\frac{\mathcal{C}(\phi -1)}{-\phi +2}\left\{{\displaystyle \sum _{r=1+\mathtt{p}{}_0}^{1+\mathtt{p}{}_0}}\left(\nabla \mathtt{Q}\right)\left(r\right){\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\mathtt{p}{}_0+2-r)-{\displaystyle \sum _{r=1+\mathtt{p}{}_0}^{\mathtt{p}{}_0}}\left(\nabla \mathtt{Q}\right)\left(r\right){\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\mathtt{p}{}_0-r+1)\right\}\hfill \\ \hfill & =\frac{\mathcal{C}(\phi -1)}{-\phi +2}\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1){\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,1)=\mathcal{C}(\phi -1)\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)⩾q0,\hfill \end{array}$$ which implies that $\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)⩾q0$ , where we have used the fact that $\mathcal{C}(\phi -1)>0$ . Moreover, at $\tau =\mathtt{p}{}_0+2$ , we have $$\begin{array}{cc}\hfill \left({}_{\mathtt{p}{}_0}{}^{\mathit{ABR}}\nabla ^{\phi }\mathtt{Q}\right)({\mathtt{p}}_0+2)& =\frac{\mathcal{C}(\phi -1)}{-\phi +2}\left\{{\displaystyle \sum _{r=1+\mathtt{p}{}_0}^{2+\mathtt{p}{}_0}}\left(\nabla \mathtt{Q}\right)\left(r\right){\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\mathtt{p}{}_0-r+3)-{\displaystyle \sum _{r=1+\mathtt{p}{}_0}^{\mathtt{p}{}_0+1}}\left(\nabla \mathtt{Q}\right)\left(r\right){\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\mathtt{p}{}_0-r+2)\right\}\hfill \\ \hfill & =\frac{\mathcal{C}(\phi -1)}{-\phi +2}\left\{\left(\nabla \mathtt{Q}\right)(1+\mathtt{p}{}_0){\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,2)+\left(\nabla \mathtt{Q}\right)(2+\mathtt{p}{}_0){\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,1)-\left(\nabla \mathtt{Q}\right)(1+\mathtt{p}{}_0){\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,1)\right\}\hfill \\ \hfill & =\mathcal{C}(\phi -1)\left\{-{(\phi -1)}^2\left(\nabla \mathtt{Q}\right)(1+\mathtt{p}{}_0)+\left(\nabla \mathtt{Q}\right)(2+\mathtt{p}{}_0)\right\}⩾q0,\hfill \end{array}$$ which implies that $$\left(\nabla \mathtt{Q}\right)(2+\mathtt{p}{}_0)⩾q\underset{>0}{\underbrace{{(\phi -1)}^2}}\underset{⩾q0}{\underbrace{\left(\nabla \mathtt{Q}\right)(1+\mathtt{p}{}_0)}}⩾q0\text{.}$$ Assume that $\kappa ⩾q2$ and that $\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+ı)⩾q0$ for $ı=2,3,\cdots ,\kappa$ . Then, by making use of Lemma 3.2 at $\tau =\mathtt{p}{}_0+\kappa +1$ , we get $$\begin{array}{cc}\hfill \left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+\kappa +1)& ⩾q{\displaystyle \underset{>0}{\underbrace{\frac{1}{-\phi +2}}}}\left\{{\displaystyle \underset{>0}{\underbrace{\left[{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\kappa )-{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\kappa +1)\right]}}}{\displaystyle \underset{⩾q0}{\underbrace{\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)}}}\right.\hfill \\ \hfill & \left.+{\displaystyle \sum _{r=\mathtt{p}{}_0+2}^{\mathtt{p}{}_0+\kappa }}\left.{\displaystyle \underset{>0}{\underbrace{\left[{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\mathtt{p}{}_0+\kappa -r+1)-{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\mathtt{p}{}_0+\kappa -r+2)\right.}}}\right]{\displaystyle \underset{⩾q0\text{by inductive assumption}}{\underbrace{\left({\nabla }_r\mathtt{Q}\right)\left(r\right)}}}\right\}\hfill \\ \hfill & ⩾q0,\hfill \end{array}$$ which completes the induction process, where we used that ${\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\kappa )$ is monotonically decreasing according to Remark 2.1. Hence, the proof is complete.  □

Corollary 3.4

Suppose $Q\in {\mathcal{F}}_{p_0,0},1<\phi <\frac{3}{2},{\xi }_1=-\frac{\phi -1}{2-\phi }$ and $\left({}_{p_0}{}^{\mathit{ABC}}\nabla ^{\phi }Q\right)\left(\tau \right)⩾q-\frac{\mathcal{C}(\phi -1)}{2-\phi }{\mathsf{E}}_{\overline{\phi -1}}({\xi }_1,\tau -p_0)\left(\nabla Q\right)\left(p_0\right)$ $\forall$ $\tau \in {\mathbb{N}}_{p_0+1}$ . Then, $\left(\nabla Q\right)\left(\tau \right)⩾q0$ $\forall$ $\tau \in {\mathbb{N}}_{p_0+1}$ .

Example 3.1

By considering Eq. (3.1) for $\tau ≔\mathtt{p}{}_0+3$ , we get $$\begin{array}{c}\hfill \left({}_{\mathtt{p}{}_0}{}^{\mathit{CFR}}\nabla ^{\phi }\mathtt{Q}\right)({\mathtt{p}}_0+3)=\mathcal{C}(\phi -1)\left\{\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+3)+(2-\phi )(1-\phi )\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+1)\right.\\ \hfill \left.+(1-\phi ){\displaystyle \sum _{r=\mathtt{p}{}_0+2}^{\mathtt{p}{}_0+2}}\left({\nabla }_r\mathtt{Q}\right)\left(r\right){(2-\phi )}^{\mathtt{p}{}_0+2-r}\right\}\text{.}\end{array}$$ For $a=0$ , it follows that $$\begin{array}{cc}\hfill \left({}_0{}^{\mathit{CFR}}\nabla ^{\phi }\mathtt{Q}\right)\left(3\right)& =\mathcal{C}(\phi -1)\left\{\left(\nabla \mathtt{Q}\right)\left(3\right)+(2-\phi )(-\phi +1)\left(\nabla \mathtt{Q}\right)\left(1\right)+(1-\phi ){\displaystyle \sum _{r=2}^2}\left({\nabla }_r\mathtt{Q}\right)\left(r\right){(2-\phi )}^{2-r}\right\}\hfill \\ \hfill & =\mathcal{C}(\phi -1)\left\{\left(\nabla \mathtt{Q}\right)\left(3\right)+(2-\phi )(1-\phi )\left(\nabla \mathtt{Q}\right)\left(1\right)+(1-\phi )\left(\nabla \mathtt{Q}\right)\left(2\right)\right\}\hfill \\ \hfill & =\mathcal{C}(\phi -1)\left\{\mathtt{Q}\left(3\right)-\mathtt{Q}\left(2\right)+(2-\phi )(1-\phi )\left[\mathtt{Q}\left(1\right)-\mathtt{Q}\left(0\right)\right]+(1-\phi )\left[\mathtt{Q}\left(2\right)-\mathtt{Q}\left(1\right)\right]\right\}\text{.}\hfill \end{array}$$ Let’s consider $\mathtt{Q}\left(0\right)=\frac{1}{2},\mathtt{Q}\left(1\right)=1,\mathtt{Q}\left(2\right)=\frac{3}{2},\mathtt{Q}\left(3\right)=2$ and $\phi =\frac{3}{2}$ , we see that $$\begin{array}{cc}\hfill \left({}_0{}^{\mathit{CFR}}\nabla ^{1.5}\mathtt{Q}\right)\left(3\right)& =\mathcal{C}\left(0.5\right)\left\{0.5+(-0.5)\left(0.5\right)\left(0.5\right)+(-0.5)\left(0.5\right)\right\}\hfill \\ \hfill & =0.1250\mathcal{C}\left(0.5\right)>0\text{.}\hfill \end{array}$$ Thus, clearly, Theorem 3.1 confirms that $\left(\nabla \mathtt{Q}\right)\left(3\right)>0$ .

Example 3.2

By using Eq. (2) for $t≔\mathtt{p}{}_0+3$ , we see that $$\begin{array}{cc}\hfill \left({}_{\mathtt{p}{}_0}{}^{\mathit{ABR}}\nabla ^{\phi }\mathtt{Q}\right)({\mathtt{p}}_0+3)& =\mathcal{C}(\phi -1)\left\{\left(\nabla \mathtt{Q}\right)({\mathtt{p}}_0+3)+\frac{1}{2-\phi }\left[{\mathsf{E}}_{\overline{\phi -1}}(\xi ,3)-{\mathsf{E}}_{\overline{\phi -1}}(\xi ,2)\right]\left(\nabla \mathtt{Q}\right)(1+\mathtt{p}{}_0)\right.\hfill \\ \hfill & \left.+\frac{1}{2-\phi }{\displaystyle \sum _{r=2+\mathtt{p}{}_0}^{2+\mathtt{p}{}_0}}\left[{\mathsf{E}}_{\overline{\phi -1}}(\xi ,\mathtt{p}{}_0+4-r)-{\mathsf{E}}_{\overline{\phi -1}}(\xi ,\mathtt{p}{}_0+3-r)\right]\left({\nabla }_r\mathtt{Q}\right)\left(r\right)\right\}\text{.}\hfill \end{array}$$ For $\mathtt{p}{}_0=0$ , it follows that $$\begin{array}{cc}\hfill \left({}_0{}^{\mathit{ABR}}\nabla ^{\phi }\mathtt{Q}\right)\left(3\right)& =\mathcal{C}(\phi -1)\left\{\left(\nabla \mathtt{Q}\right)\left(3\right)+\frac{1}{2}{(\phi -1)}^2\left(2{\phi }^2-5\phi +2\right)\left(\nabla \mathtt{Q}\right)\left(1\right)\right.\hfill \\ \hfill & \left.+\frac{1}{-\phi +2}{\displaystyle \sum _{r=2}^2}\left[{\mathsf{E}}_{\overline{\phi -1}}(\xi ,4-r)-{\mathsf{E}}_{\overline{\phi -1}}(\xi ,3-r)\right]\left({\nabla }_r\mathtt{Q}\right)\left(r\right)\right\}\hfill \\ \hfill & =\mathcal{C}(\phi -1)\left\{\left(\nabla \mathtt{Q}\right)\left(3\right)+\frac{1}{2}{(\phi -1)}^2\left(2{\phi }^2-5\phi +2\right)\left(\nabla \mathtt{Q}\right)\left(1\right)-{(\phi -1)}^2\left(\nabla \mathtt{Q}\right)\left(2\right)\right\}\hfill \\ \hfill & =\mathcal{C}(\phi -1)\left\{\mathtt{Q}\left(3\right)-\mathtt{Q}\left(2\right)+\frac{1}{2}{(\phi -1)}^2\left(2{\phi }^2-5\phi +2\right)\left[\mathtt{Q}\left(1\right)-\mathtt{Q}\left(0\right)\right]-{(\phi -1)}^2\left[\mathtt{Q}\left(2\right)-\mathtt{Q}\left(1\right)\right]\right\}\text{.}\hfill \end{array}$$ By considering $\mathtt{Q}\left(0\right)=\frac{1}{2},\mathtt{Q}\left(1\right)=1,\mathtt{Q}\left(2\right)=\frac{3}{2},\mathtt{Q}\left(3\right)=2$ and $\phi =1.4$ , we find that $$\left({}_0{}^{\mathit{ABR}}\nabla ^{1.4}\mathtt{Q}\right)\left(3\right)=0.3768\mathcal{C}\left(0.4\right)>0\text{.}$$ Thus, obviously, Theorem 3.2 confirms that $\left(\nabla \mathtt{Q}\right)\left(3\right)>0$ .