**4. Examples**

**Example 1.** *We consider the delay integro-differential equation*

$$\frac{du(s)}{ds} = e^s \left[ \frac{1}{|u(s)| + 8} + \frac{1}{|u(\lambda(s))| + 8} + \frac{1}{64} \int\_0^s e^{-\tau} \frac{1}{|u(\tau)| + 8} d\tau \right]$$

*and the inequality*

$$\left| \frac{du(s)}{ds} - e^{\varepsilon} \left[ \frac{1}{|u(s)| + 8} + \frac{1}{|u(\lambda(s))| + 8} + \frac{1}{64} \int\_{0}^{s} e^{-\tau} \frac{1}{|u(\tau)| + 8} d\tau \right] \right| \le \sigma(s).$$

$$\text{Let } z(s, u(s), u(\lambda(s)), \eta) = e^{\varepsilon} \left[ \frac{1}{1 + \frac{1}{1 - \eta} + \frac{1}{1 - \eta} + \frac{1}{2}\eta} + \frac{1}{2}\eta(s, u(s)) - \varrho^{-s} \frac{1}{1 - \eta} \right]$$

$$\begin{array}{rcl} \text{Set } z(s, u(s), u(\lambda(s)), v) & = & \varepsilon^s \left| \frac{1}{\overline{u(s)} | + 8} + \frac{1}{\overline{u(\lambda(s))} | + 8} + \frac{1}{64} v \right| \,, \text{ } g(s, u(s)) & = & \varepsilon^{-s} \frac{1}{|\overline{u(s)}| + 8}, \\ s \in \left[ 0, 2 \right]. \end{array}$$

*For any s* ∈ [0, 2]*, we obtain*

$$\begin{aligned} &|z(s,\mu\_1(s),\mu\_1(\lambda(s)),v\_1)-z(s,\mu\_2(s),\mu\_2(\lambda(s)),v\_2)| \\ &=\varepsilon^{\varepsilon}\left|\frac{(|\mu\_2(s)|-|\mu\_1(s)|)}{(|\mu\_1(s)|+8)(|\mu\_2(s)|+8)}+\frac{(|\mu\_2(\lambda(s))|-|\mu\_1(\lambda(s))|)}{(|\mu\_1(\lambda(s))|+8)(|\mu\_2(\lambda(s))|+8)}+\frac{1}{64}(v\_1-v\_2)\right|^{\frac{1}{2}} \\ &\leq\frac{\varepsilon^{\varepsilon}}{64}|\mu\_1(s)-\mu\_2(s)|+\frac{\varepsilon^{\varepsilon}}{64}|\mu\_1(\lambda(s))-\mu\_2(\lambda(s))|+\frac{1}{64}|v\_1(s)-v\_2(s)|.\end{aligned}$$

*Here, e*(*s*) = *<sup>e</sup><sup>s</sup> , k*(*s*) = *<sup>e</sup><sup>s</sup> , M*(*s*) = *<sup>e</sup><sup>s</sup> , l* = *m* = 1*.*

$$|\mathcal{g}(\mathbf{s},\boldsymbol{\mu}\_{1}(\mathbf{s})) - \mathcal{g}(\mathbf{s},\boldsymbol{\mu}\_{2}(\mathbf{s}))| = e^{-s} \left| \frac{|\boldsymbol{\mu}\_{2}(\mathbf{s})| - |\boldsymbol{\mu}\_{1}(\mathbf{s})|}{(|\boldsymbol{\mu}\_{1}(\mathbf{s})| + 8)(|\boldsymbol{\mu}\_{2}(\mathbf{s})| + 8)} \right| \le \frac{e^{-s}}{64} |\boldsymbol{\mu}\_{1}(\mathbf{s}) - \boldsymbol{\mu}\_{2}(\mathbf{s})|.$$

*Here, N*(*s*) = *<sup>e</sup><sup>s</sup>* <sup>64</sup> *, n* = 1*.*

*Thus,* (*S*1) *and* (*S*2) *hold, <sup>η</sup><sup>n</sup>* (*n*−1)! (2*Kz* + *MNη*) = <sup>23</sup> <sup>50</sup> < 1*. From Theorem 2, Equation has a unique solution*

$$u(s) = \int\_0^s e^{\tau} \left[ \frac{1}{|u(\tau)| + 8} + \frac{1}{|u(\lambda(\tau))| + 8} + \frac{1}{64} \int\_0^\tau e^{-t} \frac{1}{|u(t)| + 8} dt \right] d\tau.$$

*Let σ*(*s*) = *e<sup>s</sup> , s* <sup>0</sup> *<sup>σ</sup>*(*τ*) = *<sup>s</sup>* <sup>0</sup> *<sup>e</sup><sup>τ</sup>* <sup>=</sup> *<sup>e</sup><sup>s</sup>* <sup>−</sup> <sup>1</sup> <sup>≤</sup> *<sup>e</sup><sup>s</sup> , we have L<sup>σ</sup>* = 1 > 0*. As v*(*s*) *satisfies the inequality*

$$\left| \left| \frac{dv(s)}{ds} - e^s \right| \frac{1}{|v(s)| + 8} + \frac{1}{|v(\lambda(s))| + 8} + \frac{1}{64} \int\_0^s e^{-\tau} \frac{1}{|v(\tau)| + 8} d\tau \right| \le e^s, \quad \forall v$$

*we have*

[0, 2]*.*

$$\left| v(s) - \int\_0^s \varepsilon^{\tau} \left[ \frac{1}{|v(\tau)| + 8} + \frac{1}{|v(\lambda(\tau))| + 8} + \frac{1}{64} \int\_0^{\tau} \varepsilon^{-t} \frac{1}{|v(t)| + 8} dt \right] d\tau \right| \le \varepsilon^s. $$

*Since* (*S*3)*,* (*S*4) *and* (*S*5) *hold, from Theorem 5, we have*

$$\begin{aligned} &|v(s) - u(s)| \\ &\leq \varepsilon^{s} + \int\_{0}^{s} \left[ \frac{\epsilon^{\tau}}{64} |v(\tau) - u(\tau)| + \frac{\epsilon^{\tau}}{64} |v(\lambda(\tau)) - u(\lambda(\tau))| + \int\_{0}^{\tau} \frac{\epsilon^{-r}}{64^{2}} |v(\tau) - u(\tau)| d\tau \right] d\tau \\ &\leq \frac{57}{500} \varepsilon^{s} .\end{aligned}$$

*Hence, the equation is Hyers–Ulam–Rassias stable.*
