In this section, we are begin to study the existence condition of solution for linear homogeneous equation with constant coefficients

${\mathfrak{D}}_{q,\omega }^{n}Y\left(s\right)+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{P}_i{\mathfrak{D}}_{q,\omega }^{n-i}Y\left(s\right)=0,s\in I_1.$ (2)

For Equation (2), we get corresponding characteristic equation

${\lambda }^n+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{P}_i{\lambda }^{n-i}=\underset{j=1}{\overset{k}{{\displaystyle \prod }}}{\left(\lambda -{\lambda }_j\right)}^{r_j}=0$

where ${\lambda }_j,j=1,\cdots ,k$ are eigenvalues and ${\displaystyle {\sum }_{j=1}^k{r}_j}=n$.

Theorem 3.1. Equation (2) has a solution on $I_1$ if and only if the eigenvalues ${\lambda }_j,j=1,\cdots ,k$ satisfy

${\lambda }_j\left(s-\sigma \left(s\right)\right)\ne 1,\forall s\in I_1,j=1,2,\cdots ,k.$ (3)

Proof. Let $Z\left(s\right)={\left(Y\left(s\right),{\mathfrak{D}}_{q,\omega }Y\left(s\right),\cdots ,{\mathfrak{D}}_{q,\omega }^{n-1}Y\left(s\right)\right)}^{\text{T}}$. Then, (2) can be turned into

${\mathfrak{D}}_{q,\omega }Z\left(s\right)=AZ\left(s\right),A=\left(\begin{array}{cccc}0& 1& \cdots & 0\\ 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 1\\ -P_n& -P_{n-1}& \cdots & -P_1\end{array}\right).$ (4)

Clearly, (4) has a solution on $I_1$ if and only if $E-A\left(s-\sigma \left(s\right)\right)$ is invertible on $I_1$, where $E$ is identity matrix. Thus, the conclusion holds.

In the following lemma, we supplement the Theorem 6.1 from paper [12] by adding the existential condition for the solution of Equation (2).

Lemma 3.2. With the Condition (3), Equation (2) has a fundamental set $\left\{Y_{j,i},j=1,2,\cdots ,k;i=0,1,\cdots ,r_j-1\right\}$ of solutions as follows

$Y_{j,i}=\{\begin{array}{l}{e}_{{\lambda }_j}\left(s\right),i=0,\hfill \\ \frac{1}{{\lambda }_j^i}{\displaystyle {\sum }_{n=i}^{\infty }\frac{n\left(n-1\right)\cdots \left(n-i+1\right){\left({\lambda }_j\left(s-\sigma \left(s\right)\right)\right)}^n}{i!{\left(q:q\right)}_n}},i=1,2,\cdots ,r_j-1,{\lambda }_j\ne 0,\hfill \end{array}$

and $Y_{j,i}=\frac{{\left(s-\sigma \left(s\right)\right)}^i}{{\left(q:q\right)}_i},{\lambda }_j=0,i=0,1,\cdots ,r_j-1$.

In the following theorem, we get the general solutions for Equation (1) by using the method of constant variation.

Theorem 3.3. With the Condition (3), Equation (1) has a general solution

$X\left(s\right)=\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}{a}_{j,i}{Y}_{j,i}\left(s\right)+{\displaystyle {\int }_{{\omega }_0}^s\frac{W\left(\sigma \left(t\right),s\right)F\left(t\right)}{W\left(\sigma \left(t\right)\right)}{d}_{q,\omega }t},$ (5)

where $W\left(\sigma \left(t\right)\right)=W\left(Y_{1,0}\left(\sigma \left(t\right)\right),\cdots ,Y_{1,r_1-1}\left(\sigma \left(t\right)\right),\cdots ,Y_{k,0}\left(\sigma \left(t\right)\right),\cdots ,Y_{k,r_k-1}\left(\sigma \left(t\right)\right)\right)$ and $W\left(\sigma \left(t\right),s\right)$ can be obtained by replacing nth row of $W\left(\sigma \left(t\right)\right)$ with $\left(Y_{1,0}\left(s\right),\cdots ,Y_{1,r_1-1}\left(s\right),\cdots ,Y_{k,0}\left(s\right),\cdots ,Y_{k,r_k-1}\left(s\right)\right)$.

Proof. By Lemma 3.2, we get the general solution of Equation (2): $Y\left(s\right)={\displaystyle {\sum }_{i=0}^{r_j-1}{\displaystyle {\sum }_{j=1}^k{a}_{j,i}{Y}_{j,i}\left(s\right)}}$. Then we use the method of variation of parameters to get a particular solution $Y_0\left(s\right)$ of Equation (1)

$Y_0\left(s\right)=\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}{a}_{j,i}\left(s\right)Y_{j,i}\left(s\right),$ (6)

where $a_{j,i}\left(s\right)\left(j=1,2,\cdots ,k;i=0,1,\cdots ,r_j-1\right):I_1\to ℝ$. By finding the Hahn derivative on both sides of the Equation (6), we get

${\mathfrak{D}}_{q,\omega }{Y}_0\left(s\right)=\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}\left({\mathfrak{D}}_{q,\omega }\left(a_{j,i}\left(s\right)\right)Y_{j,i}\left(\sigma \left(s\right)\right)+a_{j,i}\left(s\right){\mathfrak{D}}_{q,\omega }\left(Y_{j,i}\left(s\right)\right)\right).$

Let ${\displaystyle {\sum }_{i=0}^{r_j-1}{\displaystyle {\sum }_{j=1}^k{\mathfrak{D}}_{q,\omega }\left(a_{j,i}\left(s\right)\right)Y_{j,i}\left(\sigma \left(s\right)\right)}}=0$. We get

${\mathfrak{D}}_{q,\omega }^2{Y}_0\left(s\right)=\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}\left({\mathfrak{D}}_{q,\omega }\left(a_{j,i}\left(s\right)\right){\mathfrak{D}}_{q,\omega }\left(Y_{j,i}\left(\sigma \left(s\right)\right)\right)+a_{j,i}\left(s\right){\mathfrak{D}}_{q,\omega }^2\left(Y_{j,i}\left(s\right)\right)\right).$

Let ${\displaystyle {\sum }_{i=0}^{r_j-1}{\displaystyle {\sum }_{j=1}^k{\mathfrak{D}}_{q,\omega }\left(a_{j,i}\left(s\right)\right){\mathfrak{D}}_{q,\omega }\left(Y_{j,i}\left(\sigma \left(s\right)\right)\right)}}=0$. We have

${\mathfrak{D}}_{q,\omega }^3{Y}_0\left(s\right)=\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}\left({\mathfrak{D}}_{q,\omega }\left(a_{j,i}\left(s\right)\right){\mathfrak{D}}_{q,\omega }^2\left(Y_{j,i}\left(\sigma \left(s\right)\right)\right)+a_{j,i}\left(s\right){\mathfrak{D}}_{q,\omega }^3\left(Y_{j,i}\left(s\right)\right)\right).$

Similarly, by sequentially letting

$\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}{\mathfrak{D}}_{q,\omega }\left(a_{j,i}\left(s\right)\right){\mathfrak{D}}_{q,\omega }^{m-1}\left(Y_{j,i}\left(\sigma \left(s\right)\right)\right)=0,1\le m\le n-1,$ (7)

and finding the Hahn derivative, we get

${\mathfrak{D}}_{q,\omega }^m{Y}_0\left(s\right)=\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}{a}_{j,i}\left(s\right){\mathfrak{D}}_{q,\omega }^m\left(Y_{j,i}\left(s\right)\right),1\le m\le n-1,$

and

${\mathfrak{D}}_{q,\omega }^n{Y}_0\left(s\right)=\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}\left({\mathfrak{D}}_{q,\omega }\left(a_{j,i}\left(s\right)\right){\mathfrak{D}}_{q,\omega }^{n-1}\left(Y_{j,i}\left(\sigma \left(s\right)\right)\right)+a_{j,i}\left(s\right){\mathfrak{D}}_{q,\omega }^n\left(Y_{j,i}\left(s\right)\right)\right).$

Thus, we have

${\mathfrak{D}}_{q,\omega }^n{Y}_0\left(s\right)+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{P}_i{\mathfrak{D}}_{q,\omega }^{n-i}{Y}_0\left(s\right)=\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}{\mathfrak{D}}_{q,\omega }\left(a_{j,i}\left(s\right)\right){\mathfrak{D}}_{q,\omega }^{n-1}\left(Y_{j,i}\left(\sigma \left(s\right)\right)\right)=F\left(s\right).$ (8)

For (7) and (8), we can obtain

${\mathfrak{D}}_{q,\omega }{a}_{j,i}\left(s\right)=\frac{W_z\left(\sigma \left(s\right)\right)F\left(s\right)}{W\left(\sigma \left(s\right)\right)},z=\underset{t=1}{\overset{j-1}{{\displaystyle \sum }}}{r}_t+i+1,1\le z\le n,$

where $W_z\left(\sigma \left(s\right)\right)$ can be obtained by replacing $z$ th column of $W\left(\sigma \left(s\right)\right)$ with $\left(0,\cdots ,0,1\right)$. Then, we obtain

$a_{j,i}\left(s\right)={\displaystyle {\int }_{{\omega }_0}^s\frac{W_z\left(\sigma \left(t\right)\right)F\left(t\right)}{W\left(\sigma \left(t\right)\right)}{d}_{q,\omega }t},1\le z\le n,$

and

$Y_0\left(s\right)=\underset{z=1}{\overset{n}{{\displaystyle \sum }}}{\displaystyle {\int }_{{\omega }_0}^s\frac{W_z\left(\sigma \left(t\right)\right)F\left(t\right)Y_{j,i}\left(s\right)}{W\left(\sigma \left(t\right)\right)}{d}_{q,\omega }t}={\displaystyle {\int }_{{\omega }_0}^s\frac{W\left(\sigma \left(t\right),s\right)F\left(t\right)}{W\left(\sigma \left(t\right)\right)}{d}_{q,\omega }t}.$

Thus, Equation (1) has a general Solution (5).

In the subsection, we study UHS and UHRS of Equation (1) by using the general solutions of Equation (1).

Definition 3.4. Equation (1) is called UHS on $I_1$, if there is a number $C>0$, for $\forall$ $\epsilon >0$ and $\forall$ $X$ satisfies

$\left|{\mathfrak{D}}_{q,\omega }^{n}X\left(s\right)+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{P}_i{\mathfrak{D}}_{q,\omega }^{n-i}X\left(s\right)-F\left(s\right)\right|\le \epsilon ,s\in I_1,$ (9)

there exists a solution $Y$ of Equation (1) such that $\left|X\left(s\right)-Y\left(s\right)\right|\le C\epsilon ,s\in I_1$.

Definition 3.5. Equation (1) is called generalized UHS if there is a function $\theta :{ℝ}_{+}\to {ℝ}_{+}$, for $\forall$ $\epsilon >0$ and $\forall$ $X$ satisfies Inequality (9), there is a solution $Y$ of Equation (1) such that

$\left|X\left(s\right)-Y\left(s\right)\right|\le \theta \left(\epsilon \right),s\in I_1.$

Definition 3.6. Equation (1) is called UHRS on $I_1$ with regard to a continuous function $\eta :I_1\to {ℝ}_{+}$, if there is a number $C>0$, for $\forall$ $\epsilon >0$ and $\forall$ $X$ satisfies

$\left|{\mathfrak{D}}_{q,\omega }^{n}X\left(s\right)+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{P}_i{\mathfrak{D}}_{q,\omega }^{n-i}X\left(s\right)-F\left(s\right)\right|\le \epsilon \eta \left(s\right),s\in I_1,$ (10)

there exists a solution $Y$ of Equation (1) such that $\left|X\left(s\right)-Y\left(s\right)\right|\le C\epsilon \eta \left(s\right),s\in I_1$.

Definition 3.7. Equation (1) is called generalized UHRS with regard to $\eta$ if there is a number $C>0$, for $\forall$ $X$ satisfies

$\left|{\mathfrak{D}}_{q,\omega }^{n}X\left(s\right)+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{P}_i{\mathfrak{D}}_{q,\omega }^{n-i}X\left(s\right)-F\left(s\right)\right|\le \eta \left(s\right),s\in I_1,$

there is a solution $Y$ of Equation (1) such that

$\left|X\left(s\right)-Y\left(s\right)\right|\le C\eta \left(s\right),s\in I_1.$

Remark 3.8. A function $X$ satisfies Inequality (9) if and only if there is a function $H:ℝ\to ℝ$ such that

(i) $\left|H\left(s\right)\right|\le \epsilon ,\forall s\in I_1$;

(ii) ${\mathfrak{D}}_{q,\omega }^{n}X\left(s\right)+{\displaystyle {\sum }_{i=1}^n{P}_i{\mathfrak{D}}_{q,\omega }^{n-i}X\left(s\right)}=F\left(s\right)+H\left(s\right),\forall s\in I_1$.

Theorem 3.9. With the Condition (3), Equation (1) maintains UHS on $I_1$.

Proof. Assume $X:I_1\to ℝ$ satisfy Equation (9). According to Remark 3.8, we have

${\mathfrak{D}}_{q,\omega }^{n}X\left(s\right)+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{P}_i\left(s\right){\mathfrak{D}}_{q,\omega }^{n-i}X\left(s\right)=F\left(s\right)+H\left(s\right).$ (11)

Then, by Theorem 3.3, Equation (11) has a solution

$X\left(s\right)=\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}{a}_{j,i}{Y}_{j,i}\left(s\right)+{\displaystyle {\int }_{{\omega }_0}^s\frac{W\left(\sigma \left(t\right),s\right)\left(F\left(t\right)+H\left(t\right)\right)}{W\left(\sigma \left(t\right)\right)}{d}_{q,\omega }t}.$ (12)

We define a function $Y:I_1\to ℝ$ by

$Y\left(s\right)=\underset{i=0}{\overset{r_j-1}{{\displaystyle \sum }}}\underset{j=1}{\overset{k}{{\displaystyle \sum }}}{a}_{j,i}{Y}_{j,i}\left(s\right)+{\displaystyle {\int }_{{\omega }_0}^s\frac{W\left(\sigma \left(t\right),s\right)F\left(t\right)}{W\left(\sigma \left(t\right)\right)}{d}_{q,\omega }t}.$

By Theorem 3.3, we know $Y$ is a solution of Equation (1). Thus, we get

$\begin{array}{l}\begin{array}{c}\left|X\left(s\right)-Y\left(s\right)\right|\le \epsilon {\displaystyle {\int }_{{\omega }_0}^s\left|\frac{W\left(\sigma \left(t\right),s\right)}{W\left(\sigma \left(t\right)\right)}\right|d_{q,\omega }t}\\ =\epsilon \left(s\left(1-q\right)-\omega \right)\underset{k=0}{\overset{\infty }{{\displaystyle \sum }}}{q}^k\left|\frac{W\left({\sigma }^{k+1}\left(s\right),s\right)}{W\left({\sigma }^{k+1}\left(s\right)\right)}\right|.\end{array}\hfill \end{array}$

Since $\left|\frac{W\left({\sigma }^{k+1}\left(s\right),s\right)}{W\left({\sigma }^{k+1}\left(s\right)\right)}\right|$ is a continuous function, then it is bounded on $I_1$. Therefore, ${\displaystyle {\sum }_{k=0}^{\infty }{q}^k\left|\frac{W\left({\sigma }^{k+1}\left(s\right),s\right)}{W\left({\sigma }^{k+1}\left(s\right)\right)}\right|}$ is convergent and continuous on $I_1$. Then

there is a constant $C$ such that $\left|X\left(s\right)-Y\left(s\right)\right|\le C\epsilon$.

Corollary 3.10. With the Condition (3), Equation (1) maintains generalized UHS on $I_1$.

Now we make the assumption:

(A₁) Let $\eta :\to {ℝ}_{+}$ be increasing function.

Theorem 3.11. With the Condition (3) and (A₁), Equation (1) maintains UHRS with regard to $\eta$ on $I_1$.

Proof. Assume $X:I_1\to ℝ$ satisfy Equation (10). Similar to the proof of Theorem 3.9, according to Condition (3) and (A₁), we have

$\left|X\left(s\right)-Y\left(s\right)\right|\le {\displaystyle {\int }_{{\omega }_0}^s\left|\frac{W\left(\sigma \left(t\right),s\right)}{W\left(\sigma \left(t\right)\right)}\right|\epsilon \eta \left(t\right)d_{q,\omega }t}\le \epsilon \eta \left(s\right){\displaystyle {\int }_{{\omega }_0}^s\left|\frac{W\left(\sigma \left(t\right),s\right)}{W\left(\sigma \left(t\right)\right)}\right|d_{q,\omega }t}.$

Thus, Equation (1) maintains UHRS with regard to $\eta$ on $I_1$.

Corollary 3.12. With the Condition (3) and (A₁), Equation (1) maintains generalized UHRS with regard to $\eta$ on $I_1$.

In the subsection, we give two examples to demonstrate the theoretical results.

Example 3.13. Considering the following equation

${\mathfrak{D}}_{\frac{1}{2},10}^{2}Y\left(s\right)-\frac{5{\mathfrak{D}}_{\frac{1}{2},10}Y\left(s\right)-Y\left(s\right)}{6}=e_{\frac{1}{3}}\left(s\right),s\in \left[20,21\right],$ (13)

and the following inequalities

$\left|{\mathfrak{D}}_{\frac{1}{2},10}^{2}Y\left(s\right)-\frac{5{\mathfrak{D}}_{\frac{1}{2},10}Y\left(s\right)-Y\left(s\right)}{6}-e_{\frac{1}{3}}\left(s\right)\right|\le \epsilon ,$

$\left|{\mathfrak{D}}_{\frac{1}{2},10}^{2}Y\left(s\right)-\frac{5{\mathfrak{D}}_{\frac{1}{2},10}Y\left(s\right)-Y\left(s\right)}{6}-e_{\frac{1}{3}}\left(s\right)\right|\le \epsilon e_{\frac{1}{4}}\left(s\right).$

We get that characteristic equation ${\lambda }^2-\frac{5\lambda }{6}+\frac{1}{6}=0$ has characteristic roots

${\lambda }_1=\frac{1}{2}$ and ${\lambda }_2=\frac{1}{3}$. Then homogeneous equation

${\mathfrak{D}}_{\frac{1}{2},10}^{2}Y\left(s\right)-\frac{5{\mathfrak{D}}_{\frac{1}{2},10}Y\left(s\right)-Y\left(s\right)}{6}=0$ has the general solution

$Y\left(s\right)=a_1{e}_{\frac{1}{2}}\left(s\right)+a_2{e}_{\frac{1}{3}}\left(s\right),a_1,a_2\in ℝ.$

We also get

$W\left(\sigma \left(t\right)\right)=\frac{e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)e_{\frac{1}{3}}\left(\sigma \left(t\right)\right)}{6},$

and

$W\left(\sigma \left(t\right),s\right)=e_{\frac{1}{2}}\left(s\right)e_{\frac{1}{3}}\left(\sigma \left(t\right)\right)-e_{\frac{1}{3}}\left(s\right)e_{\frac{1}{2}}\left(\sigma \left(t\right)\right).$

Then, we obtain

$\begin{array}{l}{\displaystyle {\int }_{20}^s\frac{W\left(\sigma \left(t\right),s\right)F\left(t\right)}{W\left(\sigma \left(t\right)\right)}{d}_{q,\omega }t}\\ ={\displaystyle {\int }_{20}^s\frac{6\left(e_{\frac{1}{2}}\left(s\right)e_{\frac{1}{3}}\left(\sigma \left(t\right)\right)-e_{\frac{1}{3}}\left(s\right)e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)\right)e_{\frac{1}{3}}\left(t\right)}{e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)e_{\frac{1}{3}}\left(\sigma \left(t\right)\right)}{d}_{q,\omega }t}\end{array}$

$\begin{array}{l}=6e_{\frac{1}{2}}\left(s\right){\displaystyle {\int }_{20}^s\frac{e_{\frac{1}{3}}\left(t\right)}{e_{\frac{1}{2}}\left(t\right)\left(1-\frac{1}{2}\left(t-\sigma \left(t\right)\right)\right)}{d}_{q,\omega }t}-6e_{\frac{1}{3}}\left(s\right){\displaystyle {\int }_{20}^{s}1-\frac{1}{3}\left(t-\sigma \left(t\right)\right)d_{q,\omega }t}\\ =6e_{\frac{1}{2}}\left(s\right){\displaystyle {\int }_{20}^s\frac{1}{1-\frac{1}{2}\left(t-\sigma \left(t\right)\right)}{e}_{-\frac{1}{6\left(1-\frac{1}{2}\left(t-\sigma \left(t\right)\right)\right)}}\left(t\right)d_{q,\omega }t}-6e_{\frac{1}{3}}\left(s\right)\left(s-20\right)+\frac{2e_{\frac{1}{3}}\left(s\right){\left(s-20\right)}^2}{3}\\ =-36e_{\frac{1}{2}}\left(s\right)e_{-\frac{1}{6\left(1-\frac{1}{2}\left(s-\sigma \left(s\right)\right)\right)}}\left(s\right)+36e_{\frac{1}{2}}\left(s\right)-6e_{\frac{1}{3}}\left(s\right)\left(s-20\right)+\frac{2e_{\frac{1}{3}}\left(s\right){\left(s-20\right)}^2}{3}.\end{array}$

Thus, Equation (13) has a general solution

$\begin{array}{c}Y\left(s\right)=a_1{e}_{\frac{1}{2}}\left(s\right)+a_2{e}_{\frac{1}{3}}\left(s\right)-36e_{\frac{1}{2}}\left(s\right)e_{-\frac{1}{6\left(1-\frac{1}{2}\left(s-\sigma \left(s\right)\right)\right)}}\left(s\right)+36e_{\frac{1}{2}}\left(s\right)\\ -6e_{\frac{1}{3}}\left(s\right)\left(s-20\right)+\frac{2e_{\frac{1}{3}}\left(s\right){\left(s-20\right)}^2}{3},\end{array}$

where $a_1,a_2\in ℝ$. Additionally, we get

$\begin{array}{l}{\displaystyle {\int }_{20}^s\left|\frac{W\left(\sigma \left(t\right),s\right)}{W\left(\sigma \left(t\right)\right)}\right|d_{q,\omega }t}\\ =6{\displaystyle {\int }_{20}^s\left|\frac{e_{\frac{1}{2}}\left(s\right)e_{\frac{1}{3}}\left(\sigma \left(t\right)\right)-e_{\frac{1}{3}}\left(s\right)e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)}{e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)e_{\frac{1}{3}}\left(\sigma \left(t\right)\right)}\right|d_{q,\omega }t}\\ =6{\displaystyle {\int }_{20}^s\left|\frac{e_{\frac{1}{2}}\left(s\right)}{e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)}-\frac{e_{\frac{1}{3}}\left(s\right)}{e_{\frac{1}{3}}\left(\sigma \left(t\right)\right)}\right|d_{q,\omega }t}\\ =3\left(s-20\right)\underset{k=0}{\overset{\infty }{{\displaystyle \sum }}}{\left(\frac{1}{2}\right)}^k\left|\frac{e_{\frac{1}{2}}\left(s\right)}{e_{\frac{1}{2}}\left({\sigma }^{k+1}\left(s\right)\right)}-\frac{e_{\frac{1}{3}}\left(s\right)}{e_{\frac{1}{3}}\left({\sigma }^{k+1}\left(s\right)\right)}\right|\\ =3\left(s-20\right)\underset{k=0}{\overset{\infty }{{\displaystyle \sum }}}{\left(\frac{1}{2}\right)}^k\left|\frac{1}{{{\displaystyle \prod }}_{i=0}^{k}1-\frac{1}{2}{\left(\frac{1}{2}\right)}^i\left(s-\sigma \left(s\right)\right)}-\frac{1}{{{\displaystyle \prod }}_{i=0}^{k}1-\frac{1}{3}{\left(\frac{1}{2}\right)}^i\left(s-\sigma \left(s\right)\right)}\right|\\ =3\left(s-20\right)\underset{k=0}{\overset{\infty }{{\displaystyle \sum }}}{\left(\frac{1}{2}\right)}^k\left(\frac{1}{{{\displaystyle \prod }}_{i=0}^{k}1-\frac{1}{2}{\left(\frac{1}{2}\right)}^i\left(s-\sigma \left(s\right)\right)}-\frac{1}{{{\displaystyle \prod }}_{i=0}^{k}1-\frac{1}{3}{\left(\frac{1}{2}\right)}^i\left(s-\sigma \left(s\right)\right)}\right)\\ =6{\displaystyle {\int }_{20}^s\left(\frac{e_{\frac{1}{2}}\left(s\right)}{e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)}-\frac{e_{\frac{1}{3}}\left(s\right)}{e_{\frac{1}{3}}\left(\sigma \left(t\right)\right)}\right)d_{q,\omega }t}\end{array}$

$\begin{array}{l}=12e_{\frac{1}{2}}\left(s\right)-18e_{\frac{1}{3}}\left(s\right)+6\\ \le 12\left(e_{\frac{1}{2}}\left(b\right)-1\right)\le \mathrm{8.9.}\end{array}$

Hence, Equation (13) has Ulam-Hyers stability and Ulam-Hyers-Rassias stability with regard to $e_{\frac{1}{4}}\left(s\right)$ on [20, 21].

Example 3.14. Considering the following equation

${\mathfrak{D}}_{\frac{1}{3},12}^{3}Y\left(s\right)-\frac{11}{12}{\mathfrak{D}}_{\frac{1}{3},12}^{2}Y\left(s\right)+\frac{1}{4}{\mathfrak{D}}_{\frac{1}{3},12}Y\left(s\right)-\frac{1}{48}Y\left(s\right)=e_{\frac{1}{8}}\left(s\right),s\in \left[18,19\right],$ (14)

and the following inequalities

$\left|{\mathfrak{D}}_{\frac{1}{3},12}^{3}Y\left(s\right)-\frac{11}{12}{\mathfrak{D}}_{\frac{1}{3},12}^{2}Y\left(s\right)+\frac{1}{4}{\mathfrak{D}}_{\frac{1}{3},12}Y\left(s\right)-\frac{1}{48}Y\left(s\right)-e_{\frac{1}{8}}\left(s\right)\right|\le \epsilon ,$

$\left|{\mathfrak{D}}_{\frac{1}{3},12}^{3}Y\left(s\right)-\frac{11}{12}{\mathfrak{D}}_{\frac{1}{3},12}^{2}Y\left(s\right)+\frac{1}{4}{\mathfrak{D}}_{\frac{1}{3},12}Y\left(s\right)-\frac{1}{48}Y\left(s\right)-e_{\frac{1}{8}}\left(s\right)\right|\le \epsilon s^3.$

Characteristic equation ${\lambda }^3-\frac{11}{12}{\lambda }^2+\frac{1}{4}\lambda -\frac{1}{48}=0$ has characteristic roots

${\lambda }_1=\frac{1}{2}$, ${\lambda }_2=\frac{1}{4}$ and ${\lambda }_3=\frac{1}{6}$. Then homogeneous equation of (14) has a general

solution

$Y\left(s\right)=a_1{e}_{\frac{1}{2}}\left(s\right)+a_2{e}_{\frac{1}{4}}\left(s\right)+a_3{e}_{\frac{1}{6}}\left(s\right),a_1,a_2,a_3\in ℝ.$

Additionally, we get

$W\left(\sigma \left(t\right)\right)=\frac{1}{144}{e}_{\frac{1}{6}}\left(\sigma \left(t\right)\right)e_{\frac{1}{4}}\left(\sigma \left(t\right)\right)e_{\frac{1}{2}}\left(\sigma \left(t\right)\right),$

and

$\begin{array}{c}W\left(\sigma \left(t\right),s\right)=\frac{e_{\frac{1}{6}}\left(\sigma \left(t\right)\right)e_{\frac{1}{4}}\left(\sigma \left(t\right)\right)e_{\frac{1}{2}}\left(s\right)}{12}-\frac{e_{\frac{1}{6}}\left(\sigma \left(t\right)\right)e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)e_{\frac{1}{4}}\left(s\right)}{3}\\ +\frac{e_{\frac{1}{6}}\left(s\right)e_{\frac{1}{4}}\left(\sigma \left(t\right)\right)e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)}{4}.\end{array}$

Then, we have

$\begin{array}{l}{\displaystyle {\int }_{18}^s\frac{W\left(\sigma \left(t\right),s\right)F\left(t\right)}{W\left(\sigma \left(t\right)\right)}{d}_{q,\omega }t}\\ ={\displaystyle {\int }_{18}^s\left(\frac{12e_{\frac{1}{2}}\left(s\right)}{e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)}-\frac{48e_{\frac{1}{4}}\left(s\right)}{e_{\frac{1}{4}}\left(\sigma \left(t\right)\right)}+\frac{36e_{\frac{1}{6}}\left(s\right)}{e_{\frac{1}{6}}\left(\sigma \left(t\right)\right)}\right)e_{\frac{1}{8}}\left(t\right)d_{q,\omega }t}\\ =-32e_{\frac{1}{2}}\left(s\right)e_{-\frac{3}{8\left(1-\frac{1}{2}\left(s-\sigma \left(s\right)\right)\right)}}\left(s\right)+32e_{\frac{1}{2}}\left(s\right)-384e_{\frac{1}{4}}\left(s\right)\\ +384e_{\frac{1}{4}}\left(s\right)e_{-\frac{1}{8\left(1-\frac{1}{4}\left(s-\sigma \left(s\right)\right)\right)}}\left(s\right)+864e_{\frac{1}{6}}\left(s\right)-864e_{\frac{1}{6}}\left(s\right)e_{-\frac{1}{24\left(1-\frac{1}{6}\left(s-\sigma \left(s\right)\right)\right)}}\left(s\right).\end{array}$

Thus, Equation (14) has a general solution

$\begin{array}{c}Y\left(s\right)=a_1{e}_{\frac{1}{2}}\left(s\right)+a_2{e}_{\frac{1}{4}}\left(s\right)+a_3{e}_{\frac{1}{6}}\left(s\right)-32e_{\frac{1}{2}}\left(s\right)e_{-\frac{3}{8\left(1-\frac{1}{2}\left(s-\sigma \left(s\right)\right)\right)}}\left(s\right)\\ +32e_{\frac{1}{2}}\left(s\right)-384e_{\frac{1}{4}}\left(s\right)+384e_{\frac{1}{4}}\left(s\right)e_{-\frac{1}{8\left(1-\frac{1}{4}\left(s-\sigma \left(s\right)\right)\right)}}\left(s\right)\\ +864e_{\frac{1}{6}}\left(s\right)-864e_{\frac{1}{6}}\left(s\right)e_{-\frac{1}{24\left(1-\frac{1}{6}\left(s-\sigma \left(s\right)\right)\right)}}\left(s\right).\end{array}$

Next, we get

$\begin{array}{l}{\displaystyle {\int }_{18}^s\left|\frac{W\left(\sigma \left(t\right),s\right)}{W\left(\sigma \left(t\right)\right)}\right|d_{q,\omega }t}\\ ={\displaystyle {\int }_{18}^s\left|\frac{12e_{\frac{1}{2}}\left(s\right)}{e_{\frac{1}{2}}\left(\sigma \left(t\right)\right)}-\frac{48e_{\frac{1}{4}}\left(s\right)}{e_{\frac{1}{4}}\left(\sigma \left(t\right)\right)}+\frac{36e_{\frac{1}{6}}\left(s\right)}{e_{\frac{1}{6}}\left(\sigma \left(t\right)\right)}\right|d_{q,\omega }t}\\ \le 24e_{\frac{1}{2}}\left(b\right)+192e_{\frac{1}{4}}\left(b\right)+216e_{\frac{1}{6}}\left(b\right)-432\le \mathrm{118.9.}\end{array}$