**Example 2.** *Consider the equation*

$$\frac{du(s)}{ds} = -\frac{1}{8} \frac{1}{|u(s)| + 1} - \frac{1}{8} \frac{1}{|u(\lambda(s))| + 1} - \frac{1}{64} \int\_0^s \frac{1}{|u(\tau)| + 1} d\tau.$$
  $\text{Set } z(s, u(s), u(\lambda(s)), v) = -\frac{1}{8} \frac{1}{|u(s)| + 1} - \frac{1}{8} \frac{1}{|u(\lambda(s))| + 1} - \frac{1}{64} v, \text{ } g(s, u(s)) = \frac{1}{|u(\tau)| + 1}, s \in \mathbb{C}.$ 

*For any s* ∈ [0, 2]*, we obtain*

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

$$|\mathcal{g}(s,\mu\_1(s)) - \mathcal{g}(s,\mu\_2(s))| = \left| \frac{|\mu\_2(s)| - |\mu\_1(s)|}{(|\mu\_1(s)| + 1)(|\mu\_2(s)| + 1)} \right| \le |\mu\_1(s) - \mu\_2(s)|.$$

*Thus,* (*S*1) *and* (*S*2) *hold, <sup>η</sup><sup>n</sup>* (*n*−1)! (2*Kz* + *MNη*) = <sup>9</sup> <sup>16</sup> < 1*. From Theorem 2, the equation has a unique solution:*

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

*Let v*(*s*) = *e<sup>s</sup> and choose θ* = <sup>9</sup> <sup>32</sup> *. We have*

$$\begin{aligned} & \left| \frac{dv(s)}{ds} - \left[ -\frac{1}{8} \frac{1}{|v(s)| + 1} - \frac{1}{8} \frac{1}{|v(\lambda(s))| + 1} - \frac{1}{64} \int\_0^s \frac{1}{|v(\tau)| + 1} d\tau \right] \right| \\ &= \left| \frac{1}{8} \frac{1}{|v(s)| + 1} + \frac{1}{8} \frac{1}{|v(\lambda(s))| + 1} + \frac{1}{64} \int\_0^s \frac{1}{|v(\tau)| + 1} d\tau - e^{-s} \right| \le \frac{9}{32} = \theta. \end{aligned}$$

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

$$\begin{aligned} &|v(s) - u(s)| \\ &\le 2\theta + \int\_0^s \left[ \frac{1}{8} |v(\tau) - u(\tau)| + \frac{1}{8} |v(\lambda(\tau)) - u(\lambda(\tau))| + \int\_0^\tau \frac{1}{64} |v(\tau) - u(\tau)| d\tau \right] d\tau \\ &\le \frac{16}{5}\theta. \end{aligned}$$

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