## 5 Ulam-Hyers Stability  
Here, the Ulam-Hyers stability for the recommended fractional system of HBV infection will be examined. The idea of this stability was initially presented by [PERSON] in 1940; [PERSON] developed it [26,27]. Many researchers have embraced the Ulam-Hyers stability hypothesis in a variety of disciplines of study [28,29]. Its basic ideas and definitions include the following. Assume that the operator \(\mathcal{H}:X\to X\) behaves in a way that  
\[\mathcal{HY}=\mathcal{Y}\;\;for\;\;\mathcal{Y}\in X. \tag{23}\]  
**Definition 3**. : _The above (23) is Ulam-Hyers stable (UHS) if for \(\varepsilon>0\), and takes any solution \(\mathcal{Y}\in X\) of_  
\[||\mathcal{Y}-\mathcal{HY}||\leq\varepsilon,\;\;\;for\;\;\;r\in[0,\tau]. \tag{24}\]  
_Then, there exists a unique solution \(\bar{\mathcal{Y}}\) of (23), with \(C_{q}>0\) fulfilling the following_  
\[||\bar{\mathcal{Y}}-\mathcal{Y}||\leq C_{q}\varepsilon,\;\;\;r\in[0,\tau]. \tag{25}\]  
**Definition 4**. : _The aforementioned (23) is a generalized Ulam-Hyers stable (GUHS), if every solution \(\mathcal{Y}\) of (24) and any other \(\bar{\mathcal{Y}}\) of (23) satisfies_  
\[||\bar{\mathcal{Y}}-\mathcal{Y}||\leq\mathcal{U}(\varepsilon), \tag{26}\]  
_where zero is the image of zero, i.e., \(\mathcal{U}(0)=0\) and \(\mathcal{U}\;\in\;C(R,R)\)._  
**Remark 2**. : _The solution \(\bar{\mathcal{Y}}\in X\) fulfills (24) if (1) \(|\mathcal{Y}(r)|\leq\varepsilon,\;\;\forall\;r\in[0,\tau]\;\;and\;\;Y\in C([0, \tau];R)\), (2) \({}^{LC}D_{0+}^{\beta}\bar{\mathcal{Y}}(r)=\mathcal{P}(r,\bar{\mathcal{Y}}(r))+ \mathcal{Y}(r)\)._  
After a minor perturbation, system (10) can be expressed as follows:  
\[\left\{ \begin{array}{cc}{}^{LC}D_{0+}^{\beta}\mathcal{Y}(r)=&\mathcal{P}(r, \mathcal{Y}(r))+\mathcal{Y}(r),\\ \mathcal{Y}(0)=\mathcal{Y}_{0}.\end{array} \right. \tag{27}\]  
**Lemma 2**. : _The above system (27) satisfies the condition_  
\[|\mathcal{Y}(r)-T\mathcal{Y}(r)|\leq c\Omega(t)\epsilon,\;\;\mbox{where}\;\;c= \frac{\tau^{\beta}}{\Gamma(\beta+1)}. \tag{28}\]  
Proof. : From the second part of Remark 2 for \(\mathcal{Y}\in X\), we have  
\[{}^{LC}D_{0^{+}}^{\beta}\mathcal{Y}(r)=\mathcal{P}(r,\mathcal{Y}(r))+\mathrm{Y }(r),\]  
applying fractional theory [21], we have  
\[\mathcal{Y}(r) = \mathcal{Y}_{0}+\frac{1}{\Gamma(\beta)}\int_{0}^{r}(r-\tau)^{ \beta-1}\mathcal{P}(\tau,\mathcal{Y}(\tau))d\tau\] \[+ \frac{1}{\Gamma(\beta)}\int_{0}^{r}(r-\tau)^{\beta-1}\mathrm{Y}( \tau)d\tau,\]  
simplifying, we have  
\[|\mathcal{Y}(r) - \mathcal{Y}_{0}-\frac{1}{\Gamma(\beta)}\int_{0}^{r}(r-\tau)^{ \beta-1}\mathcal{P}(\tau,\mathcal{Y}(\tau))d\tau|\] \[\leq \frac{1}{\Gamma(\beta)}\int_{0}^{r}(r-\tau)^{\beta-1}|\mathrm{Y }(\tau)|d\tau.\]  
Using the definition of \(T\mathcal{Y}(r)\) and the first part of the above Remark 2, we have  
\[|\mathcal{Y}(r)-T\mathcal{Y}(r)|\leq c\Omega(t)\epsilon,\;\;\mbox{where}\;\;c= \frac{\tau^{\beta}}{\Gamma(\beta+1)}.\]  
Thus, the required result is proved. **Theorem 6**. : _The solution of system (10) of HBV infection is UHS and GUHS on Lemma 2 if \(\frac{\tau^{\beta}L_{P}}{\Gamma(\beta+1)}<1\) holds true._  
Proof of Theorem 6.: To prove that the solution of system (10) of HBV infection is UHS and GUHS, we assume that \(\mathcal{Y}\in X\) is any solution and \(\tilde{\mathcal{Y}}\in X\) is a unique solution of system (10), then  
\[|\mathcal{Y}(r)-\tilde{\mathcal{Y}}(r)| = |\mathcal{Y}(r)-T\tilde{\mathcal{Y}}(r)| \tag{29}\] \[\leq |\mathcal{Y}(r)-T\mathcal{Y}(r)|+|T\mathcal{Y}(r)-T\tilde{ \mathcal{Y}}(r)|.\]  
Applying Lemma 2, we have  
\[|\mathcal{Y}(r)-\tilde{\mathcal{Y}}(r)| \leq c\epsilon+\frac{\tau^{\beta}L_{P}}{\Gamma(\beta+1)}|\mathcal{Y}( r)-\tilde{\mathcal{Y}}(r)| \tag{30}\] \[{}^{LC}D_{0^{+}}^{\beta}\mathcal{Y}(r)=\mathcal{P}(r,\mathcal{Y}(r))+\mathrm{Y }(r),\]0  
This implies that the solution of system (10) of HBV infection is both UHS and GUHS. **Definition 5**.