Existence and uniqueness of a class of uncertain Liouville-Caputo fractional difference equations

Definition 13

An uncertain fractional difference equation is a fractional difference equation which is driven by an uncertain sequence. Moreover, an uncertain fractional forward difference equation in the Liouville-Caputo sense (UFLCDE) is the uncertain fractional difference equation with the Liouville-Caputo forward difference.

Lemma 4

The initial-value problem (1.3) with the initial conditions (1.4) is equivalent to the following uncertain fractional sum equation:

$$\mathcal{p}\left(z\right)=c_0+\frac{1}{\mathrm{\Gamma }\left(\varphi \right)}{\displaystyle \sum _{r=0}^{z-\varphi }}{\left[z-\sigma \left(r\right)\right]}^{(\varphi -1)}[{\mathcal{H}}_1\left(r+\varphi -1,\mathcal{p}(r+\varphi -1)\right)+{\mathcal{H}}_2\left(r+\varphi -1,\mathcal{p}(r+\varphi -1)\right){\epsilon }_{r+\varphi -1}](z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}\left]\right)\text{.}$$

Proof 1

By applying ${}_0{\mathrm{\Delta }}^{-\varphi }$ on IVP (1.3) and, by using (1.4), we can directly obtain the desired result. Moreover, the readers can see Abdeljawad, 2011, Example 17 and Baleanu et al., 2020, Lemma 2.10 for more details.

Theorem 1

For any $z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]$ and $\left|\lambda \right|<1$ , the linear UFLCDE (3.2) with the initial condition (3.3) has a solution given by

$$\mathcal{p}\left(z\right)={\mathrm{E}}_{\left(\varphi \right)}(\lambda ,z)c_0+{\epsilon }_z,$$

Proof 2

Applying ${}_0{\mathrm{\Delta }}^{-\varphi }$ on the Eq. (3.2), we get

$${}_0{\mathrm{\Delta }}_{\varphi }^{-\varphi }\left[{}^C{}_{\varphi -1}{\mathrm{\Delta }}^{\varphi }\mathcal{p}\left(z\right)\right]=\lambda {}_0{\mathrm{\Delta }}^{-\varphi }\mathcal{p}(z+\varphi -1)+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }{\epsilon }_{z+\varphi -1}(z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}\left]\right)\text{.}$$ Thus, by making use of Lemma 3, it follows that

(3.6)

$$\mathcal{p}\left(z\right)=c_0+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }\mathcal{p}(z+\varphi -1)+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }{\epsilon }_{z+\varphi -1}(z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}\left]\right),$$ which is the solution of the given UFLCDE (3.4).

To obtain an explicit solution, we use the method of the Picard approximation with a starting point ${\mathcal{p}}_0\left(z\right)=c_0$ . In addition, we can obtain the other components by using the following recurrence relation:

$${\mathcal{p}}_q\left(z\right)=c_0+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }{\mathcal{p}}_q(z+\varphi -1)+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }{\epsilon }_{z+\varphi -1}(z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]\text{;}q=0,1,\cdots )\text{.}$$ We can now write ${\epsilon }_{z+\varphi -1}=\epsilon$ in the distribution, since ${\epsilon }_{\varphi -1},{\epsilon }_{\varphi },\cdots ,{\epsilon }_{\mathcal{T}+\varphi -1}$ are the IID uncertain variables. By making use of Lemma 1 and the fact that the linear combination of finite independent uncertain variables is also an uncertain variable with positive linear combination coefficient (see, for details, Liu, 2010, Theorems 1.21 to 1.24), we can obtain $${\mathcal{p}}_1\left(z\right)=c_0+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }{\mathcal{p}}_0\left(z\right)+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }\epsilon =\left[1+\lambda \frac{z^{\left(\varphi \right)}}{\mathrm{\Gamma }(\varphi +1)}\right]c_0+\lambda \frac{z^{\left(\varphi \right)}}{\mathrm{\Gamma }(\varphi +1)}\epsilon$$ $$\begin{array}{cc}{\mathcal{p}}_2\left(z\right)\hfill & =\frac{{(z-\varphi +1)}^{\overline{\varphi -1}}}{\mathrm{\Gamma }\left(\varphi \right)}{c}_0+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }{\mathcal{p}}_1\left(z\right)+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }\epsilon ,\hfill \\ \hfill & =\left[1+\lambda \frac{z^{\left(\varphi \right)}}{\mathrm{\Gamma }(\varphi +1)}+{\lambda }^2\frac{z^{\left(2\varphi \right)}}{\mathrm{\Gamma }(2\varphi +1)}\right]c_0+\left[\lambda \frac{z^{\left(\varphi \right)}}{\mathrm{\Gamma }(\varphi +1)}+{\lambda }^2\frac{z^{\left(2\varphi \right)}}{\mathrm{\Gamma }(2\varphi +1)}\right]\epsilon \hfill \\ \hfill & ⋮\hfill \end{array}$$ and so on. Continuing the process up to the qth term, we find that $${\mathcal{p}}_q\left(z\right)=c_0{\displaystyle \sum _{k=0}^q}{\lambda }^k\frac{{\left[z+k(\varphi -1)\right]}^{\left(k\varphi \right)}}{\mathrm{\Gamma }(k\varphi +1)}+{\displaystyle \sum _{k=1}^q}{\lambda }^k\frac{{\left[z+(k-1)(\varphi -1)\right]}^{\left(k\varphi \right)}}{\mathrm{\Gamma }(k\varphi +1)}\epsilon ,$$ for $z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]$ . Now, by letting $q\to \infty$ , we get the following solution:

(3.8)

$$\begin{array}{cc}\widehat{\mathcal{p}}\left(z\right):={\displaystyle \underset{q\to \infty }{\lim }}{\mathcal{p}}_q\left(z\right)\hfill & =c_0{\displaystyle \sum _{k=0}^{\infty }}{\lambda }^k\frac{{\left[z+k(\varphi -1)\right]}^{\left(k\varphi \right)}}{\mathrm{\Gamma }(k\varphi +1)}+{\displaystyle \sum _{k=1}^{\infty }}{\lambda }^k\frac{{\left[z+(k-1)(\varphi -1)\right]}^{\left(k\varphi \right)}}{\mathrm{\Gamma }(k\varphi +1)}\epsilon \hfill \\ \hfill & =c_0{\mathrm{E}}_{\left(\varphi \right)}(\lambda ,z)+\lambda {\mathrm{E}}_{(\varphi ,\varphi +1)}(\lambda ,z)\epsilon ,z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]\text{.}\hfill \end{array}$$ The solution $\widehat{\mathcal{p}}\left(z\right)$ exists, since the Mittag-Liffler functions ${\mathrm{E}}_{\left(\varphi \right)}(\lambda ,z)$ and ${\mathrm{E}}_{(\varphi ,\varphi +1)}(\lambda ,t+\varphi -1)$ are absolutely convergent for $\left|\lambda \right|<1$ .

In addition, if we take ${\mathit{lim}}_{q\to \infty }$ on both sides of (3.7), we get $$\widehat{\mathcal{p}}\left(z\right)=c_0+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }\widehat{\mathcal{p}}(z+\varphi -1)+\lambda {}_0{\mathrm{\Delta }}^{-\varphi }{\epsilon }_{z+\varphi -1}(z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}\left]\right)\text{.}$$ This means that $\widehat{\mathcal{p}}\left(z\right)$ satisfies the Eq. (3.6). Hence, clearly, $\widehat{\mathcal{p}}\left(z\right)$ is a solution of the Eqs. 3.2,3.3. Our proof of Theorem 1 is thus completed.

Theorem 2

Theorem 2 Existence and Uniqueness

Let ${\mathcal{H}}_1(z,{\mathcal{p}}_1)$ and ${\mathcal{H}}_2(z,{\mathcal{p}}_1)$ be two real-valued functions in (1.3) and satisfy the following Lipschitz condition:

$$\left|{\mathcal{H}}_1\right(z,{\mathcal{p}}_1)-{\mathcal{H}}_1(z,{\mathcal{p}}_2\left)\right|+\left|{\mathcal{H}}_2\right(z,{\mathcal{p}}_1)-{\mathcal{H}}_2(z,{\mathcal{p}}_2\left)\right|\leqq L|{\mathcal{p}}_1-{\mathcal{p}}_2|,$$ with the Lipschitz constant L that satisfies the inequality given by

(3.10)

$$L<\frac{\mathrm{\Gamma }\left(\varphi +1\right)\mathrm{\Gamma }\left(\mathcal{T}+1-\varphi \right)}{\mathrm{\Gamma }\left(\mathcal{T}+1\right)(\vartheta +1)},$$ where $\vartheta =\left|{\ell }_1\right|\vee \left|{\ell }_2\right|$ . Then the UFLCDE (1.3)–(1.4) have a unique solution $\mathcal{p}\left(z\right)$ for all $z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]$ almost surely.

Proof 3

Let us define $l_{\varphi }^k$ (the set of all finite real sequences ${\left\{\mathcal{p}\right(z\left)\right\}}_{\varphi }^k$ ) with the norm $\Vert \mathcal{p}\Vert$ as follows: $$l_{\varphi }^k:=\left\{\mathcal{p}\text{;}\mathcal{p}={\left\{\mathcal{p}\right(z\left)\right\}}_{\varphi }^k,k\in {\mathcal{N}}_1\right\}$$ and $$\Vert \mathcal{p}\Vert :={\displaystyle \underset{z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]}{\max }}\left|\mathcal{p}\right(z\left)\right|,$$ which has k terms. It is easy to see that $\left(l_{\varphi }^k,\Vert ·\Vert \right)$ is a Banach space (see, for details, Sacks, 2017, Chapter 4).

We now define the operator $\mathcal{D}$ for ${\mathcal{p}}_z\in l_{\varphi }^k$ as follows: $${\mathcal{Dp}}_z=c_0+\frac{1}{\mathrm{\Gamma }\left(\varphi \right)}{\displaystyle \sum _{r=0}^{z-\varphi }}{\left[z-\sigma \left(r\right)\right]}^{(\varphi -1)}\left[{\mathcal{H}}_1(r+\varphi -1,{\mathcal{p}}_{r+\varphi -1})+{\mathcal{H}}_2(r+\varphi -1,{\mathcal{p}}_{r+\varphi -1}){\epsilon }_{r+\varphi -1}\right]\text{.}$$ We also assume that $\chi$ represents the universal set on the uncertainty space. Clearly, $\mathcal{M}\left\{\right({\epsilon }_z<a)\cup ({\epsilon }_z>b\left)\right\}=0$ , since ${\epsilon }_z(z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}\left]\right)$ is an uncertain variable at each time t with the linear uncertainty distribution $\mathbf{L}({\ell }_1,{\ell }_2)$ . In addition, the inequality ${\epsilon }_z\left(\gamma \right)\leqq \vartheta$ , where $\vartheta =\left|{\ell }_1\right|\vee \left|{\ell }_2\right|$ , holds true almost surely for each $\gamma$ given by $$\gamma \in \chi \left\{({\epsilon }_z<{\ell }_1)\cup ({\epsilon }_z>{\ell }_2)\text{;}z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]\right\}\text{.}$$ For any $x_z,{\mathcal{p}}_z\in l_{\varphi }^k$ , we then obtain $$\begin{array}{c}\hfill ‖\mathcal{D}{x}_z\left(\gamma \right)-{\mathcal{Dp}}_z\left(\gamma \right)‖={\displaystyle \underset{z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]}{\max }}\left|\mathcal{D}{x}_z\left(\gamma \right)-{\mathcal{Dp}}_z\left(\gamma \right)\right|\\ \hfill \leqq \frac{1}{\mathrm{\Gamma }\left(\varphi \right)}{\displaystyle \underset{z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]}{\max }}{\displaystyle \sum _{r=0}^{z-\varphi }}{\left[z-\sigma \left(r\right)\right]}^{\overline{\varphi -1}}(\left|{\mathcal{H}}_1\left(r+\varphi -1,x_{r+\varphi -1}\left(\gamma \right)\right)-{\mathcal{H}}_1\left(r+\varphi -1,{\mathcal{p}}_{r+\varphi -1}\left(\gamma \right)\right)\right|+\left|\left[{\mathcal{H}}_2\left(r+\varphi -1,x_{r+\varphi -1}\left(\gamma \right)\right)-{\mathcal{H}}_2\left(r+\varphi -1,{\mathcal{p}}_{r+\varphi -1}\left(\gamma \right)\right)\right]{\epsilon }_r\right|)\\ \hfill \leqq \frac{1}{\mathrm{\Gamma }\left(\varphi \right)}{\displaystyle \underset{z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]}{\max }}{\displaystyle \sum _{r=0}^{z-\varphi }}{\left[z-\sigma \left(r\right)\right]}^{\overline{\varphi -1}}(\left|{\mathcal{H}}_1\left(r+\varphi -1,x_{r+\varphi -1}\left(\gamma \right)\right)-{\mathcal{H}}_1\left(r+\varphi -1,{\mathcal{p}}_{r+\varphi -1}\left(\gamma \right)\right)\right|+\vartheta \left|{\mathcal{H}}_2\left(r+\varphi -1,x_{r+\varphi -1}\left(\gamma \right)\right)-{\mathcal{H}}_2\left(r+\varphi -1,{\mathcal{p}}_{r+\varphi -1}\left(\gamma \right)\right)\right|)\end{array}$$ Thus, by the help of the assumptions and Lemma 1, we get $$\begin{array}{c}\hfill ‖\mathcal{D}{x}_z\left(\gamma \right)-{\mathcal{Dp}}_z\left(\gamma \right)‖\leqq L(1+\vartheta )\frac{1}{\mathrm{\Gamma }\left(\varphi \right)}{\displaystyle \underset{z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]}{\max }}{\displaystyle \sum _{r=0}^{z-\varphi }}{\left(z-\sigma \left(r\right)\right)}^{\overline{\varphi -1}}\left|x_r\left(\gamma \right)-{\mathcal{p}}_r\left(\gamma \right)\right|\\ \hfill \leqq L(1+\vartheta )‖x_z\left(\gamma \right)-{\mathcal{p}}_z\left(\gamma \right)‖{\displaystyle \underset{z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]}{\max }}\left({}_0{\mathrm{\Delta }}^{-\varphi }{z}^{\left(0\right)}\right)\\ \hfill =L(1+\vartheta )‖x_z\left(\gamma \right)-{\mathcal{p}}_z\left(\gamma \right)‖{\displaystyle \underset{z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]}{\max }}\left(\frac{1}{\mathrm{\Gamma }(\varphi +1)}{z}^{\left(\varphi \right)}\right)\\ \hfill \leqq \frac{L(1+\vartheta ){\mathcal{T}}^{\left(\varphi \right)}}{\mathrm{\Gamma }(\varphi +1)}‖x_z\left(\gamma \right)-{\mathcal{p}}_z\left(\gamma \right)‖\\ \hfill =\frac{L(1+\vartheta )\mathrm{\Gamma }(\mathcal{T}+1)}{\mathrm{\Gamma }(\varphi +1)\mathrm{\Gamma }(\mathcal{T}-\varphi +1)}‖x_z\left(\gamma \right)-{\mathcal{p}}_z\left(\gamma \right)‖\text{.}\end{array}$$ The mapping $\mathcal{D}$ is a contraction in $l_{\varphi }^k$ almost surely such that (see Sacks, 2017, Chapter 4) $$0<L<\frac{\mathrm{\Gamma }(\varphi +1)\mathrm{\Gamma }(\mathcal{T}-\varphi +1)}{(1+\vartheta )\mathrm{\Gamma }(\mathcal{T}+1)}\text{.}$$ Therefore, by using the Banach contraction mapping theorem (see Sacks, 2017, Chapter 4), we obtain a unique fixed point ${\mathcal{p}}_z\left(\gamma \right)$ of $\mathcal{D}$ in $l_{\varphi }^k$ almost surely. Furthermore, we have $${\mathcal{p}}_z\left(\gamma \right)={\displaystyle \underset{j\to \infty }{\lim }}{\mathcal{p}}_z^j\left(\gamma \right),$$ where $${\mathcal{p}}_z^j\left(\gamma \right)=\mathcal{D}\left({\mathcal{p}}_z^{j-1}\left(\gamma \right)\right)$$ with $${\mathcal{p}}_z^0\left(\gamma \right)=\frac{{(z-\varphi )}^{\overline{\varphi -1}}}{\mathrm{\Gamma }\left(\varphi \right)}{c}_0\text{.}$$ On the other hand, the operator $\mathcal{D}$ is measurable for any $z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]$ , since ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are Lipschitz continuous functions. Since $${\mathcal{p}}_z^1\left(\gamma \right),{\mathcal{p}}_z^2\left(\gamma \right),\cdots ,{\mathcal{p}}_z^j\left(\gamma \right),\cdots$$ are uncertain variables and since ${\mathcal{p}}_z^0\left(\gamma \right)$ is a real-valued measurable function of uncertain variables, ${\mathcal{p}}_z^0\left(\gamma \right)$ is seen to be an uncertain variable by using Liu, 2010, Theorem 1.10. Hence, clearly, ${\mathcal{p}}_z={\lim }_{j\to \infty }{\mathcal{p}}_z^j$ is an uncertain variable by using Zhu, 2015, Theorem 3. Therefore, the UFLCDE (3.2) subject to the initial condition (3.3) has a unique solution ${\mathcal{p}}_z$ for $z\in {\mathcal{N}}_{\varphi }\cap [0,\mathcal{T}]$ almost surely. We thus have completed the existence and uniqueness asserted by Theorem 2.

Example 3

Consider the following UFLCDE:

$${}^C{}_{-0.5}{\mathrm{\Delta }}^{0.5}\mathcal{p}\left(z\right)=\left\{\begin{array}{c}\frac{\ln \left(\left|\mathcal{p}\right(z-0.5\left)\right|+1\right)}{64{(z-0.5)}^3}+0.5{\epsilon }_{z-0.5}(z\in {\mathcal{N}}_0\cap [0,3\left]\right),\hfill \\ \mathcal{p}\left(\frac{1}{2}\right)=1,\hfill \end{array}\right.$$ where ${\epsilon }_{-0.5},{\epsilon }_{0.5},{\epsilon }_{1.5},{\epsilon }_{2.5}$ are four IID uncertain variables with the uncertainty distribution $\mathbf{L}(-1,2)$ .

According to Lemma 4 with $\varphi =\frac{1}{2}$ , the inverse uncertainty distribution of the solution for the UFLCDE (4.1) is the solution of the following sum equation: $$\mathcal{p}\left(z\right)=c_0+\frac{1}{\mathrm{\Gamma }\left(0.5\right)}{\displaystyle \sum _{r=0}^{z-0.5}}{\left[z-\sigma \left(r\right)\right]}^{(-0.5)}\left(\frac{\ln \left(\left|\mathcal{p}\right(r-0.5\left)\right|+1\right)}{4{(r-0.5)}^3}+0.25{\epsilon }_{r-0.5}\right)(z\in {\mathcal{N}}_0\cap [0,3\left]\right)\text{.}$$ Thus, for $z\in {\mathcal{N}}_0\cap [0,3]$ , we have $$\begin{array}{c}\hfill \left|{\mathcal{H}}_1\right(z,{\mathcal{p}}_1)-{\mathcal{H}}_1(z,{\mathcal{p}}_2\left)\right|+\left|{\mathcal{H}}_2\right(z,{\mathcal{p}}_1)-{\mathcal{H}}_2(z,{\mathcal{p}}_2\left)\right|=\left|\frac{\ln \left(\right|{\mathcal{p}}_1|+1)}{64{(z-0.5)}^3}-\frac{\ln \left(\right|{\mathcal{p}}_2|+1)}{64{(z-0.5)}^3}\right|\\ \hfill =\frac{1}{64{(z-0.5)}^3}\left|\ln \left(\right|{\mathcal{p}}_1|+1)-\ln \left(\right|{\mathcal{p}}_2|+1)\right|\\ \hfill \leqq \frac{1}{64{\left(\frac{1}{2}\right)}^3}\left|\left|{\mathcal{p}}_1\right|-\left|{\mathcal{p}}_2\right|\right|\\ \hfill \leqq \frac{\left|{\mathcal{p}}_1-{\mathcal{p}}_2\right|}{8}\end{array}$$ and $$\frac{\mathrm{\Gamma }(0.5+1)\mathrm{\Gamma }(3+1-0.5)}{3\mathrm{\Gamma }(3+1)}\approx 0.1636>\frac{1}{8}=0.125\text{.}$$ Therefore, in view of Theorem 2, this confirms that the UFLCDE (4.1) has a unique solution almost surely.