**3. Existence and Stability Results for the Delay Integro-Differential Equation**

Before stating the main theorems, we give the following Lipschitz conditions: (*S*1):

$$\begin{aligned} &|z(s,\boldsymbol{u}^{(0)},\ldots,\boldsymbol{u}^{(n-1)},\overline{\boldsymbol{u}}^{(0)},\ldots,\overline{\boldsymbol{u}}^{(n-1)},\boldsymbol{u})-z(s,\boldsymbol{v}^{(0)},\ldots,\boldsymbol{v}^{(n-1)},\overline{\boldsymbol{v}}^{(0)},\ldots,\overline{\boldsymbol{v}}^{(n-1)},\boldsymbol{v})| \\ &\leq K\_{z}\sum\_{j=0}^{n-1}\left[|\boldsymbol{u}^{(j)}-\boldsymbol{v}^{(j)}|+|\overline{\boldsymbol{u}}^{(j)}-\overline{\boldsymbol{v}}^{(j)}|\right]+M|\overline{\boldsymbol{u}}-\overline{\boldsymbol{v}}|,\end{aligned}$$

where *Kz* > 0, *M* > 0, *u*(*j*) = *u*(*j*)(*λ*), *v*(*j*) = *v*(*j*)(*λ*). (*S*2):

$$|\lg(s, u^{(0)}, \dots, u^{(n-1)}) - \lg(s, v^{(0)}, \dots, v^{(n-1)})| \le N \sum\_{j=0}^{n-1} |u^{(j)} - v^{(j)}|\_{\nu}$$

where *N* > 0. (*S*3):

$$\begin{aligned} &|z(s,\boldsymbol{u}^{(0)},\ldots,\boldsymbol{u}^{(n-1)},\overline{\boldsymbol{u}}^{(0)},\ldots,\overline{\boldsymbol{u}}^{(n-1)},\boldsymbol{u})-z(s,\boldsymbol{v}^{(0)},\ldots,\boldsymbol{v}^{(n-1)},\overline{\boldsymbol{v}}^{(0)},\ldots,\overline{\boldsymbol{v}}^{(n-1)},\boldsymbol{v})| \\ &\leq c(s)|\boldsymbol{u}^{(0)}-\boldsymbol{v}^{(0)}|^{l}+k(s)|\overline{\boldsymbol{u}}^{(0)}-\overline{\boldsymbol{v}}^{(0)}|^{m}+M(s)|\overline{\boldsymbol{u}}-\overline{\boldsymbol{v}}|\_{\prime} \end{aligned}$$

$$\begin{aligned} & \text{where } e(s), k(s), M(s) > 0, s \in J\_1; \\ & \text{I}\_{(1)}(s) \end{aligned} = \begin{aligned} & \text{I}\_1; \text{I}\_1, m \in (0, 1]; \overline{\text{u}}^{(0)} = u^{(0)}(\lambda), \overline{v}^{(0)} = v^{(0)}(\lambda). \\ & \text{I}\_4; \end{aligned}$$

$$|\lg(\mathbf{s}, u^{(0)}, \dots, u^{(n-1)}) - \lg(\mathbf{s}, v^{(0)}, \dots, v^{(n-1)})| \le N(\mathbf{s}) |u^{(0)} - v^{(0)}|^{n}\_{\text{-}} $$

where *N*(*s*) > 0 , *s* ∈ *J*1; *n* ∈ (0, 1].

(*S*5):

Assume *σ*(*s*) is a function from *J*<sup>2</sup> to R<sup>+</sup> and there exists *L<sup>σ</sup>* > 0 such that

$$\int\_{\tau}^{s} \sigma(r) dr \le L\_{\sigma} \cdot \sigma(s), \ s \in J\_2.$$

Firstly, we give the existence and uniqueness of a solution for (3).

**Theorem 2.** *Assume that* (*S*1) *and* (*S*2) *hold. If <sup>η</sup><sup>n</sup>* (*n*−1)! (2*Kz* + *MNη*) < 1*, then Equation (3) has a unique solution.*

**Proof.** (i) We define the operator *γ* as follows:

$$(\gamma u)(s) = \begin{cases} \sum\_{j=0}^{n-1} \frac{(s-s\_0)\_j^j \chi^{(j)}(s\_0)}{j!} + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} \\\cdot z \left(\tau, u^{(0)}, \dots, u^{(n-1)}(\lambda), \int\_{s\_0}^\tau \lg(r, u^{(0)}, \dots, u^{(n-1)}) dr\right) d\tau, \; s \in J\_{2, \epsilon} \\\chi(s), \; s \in J\_3. \end{cases} \tag{8}$$

Since *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*(*B*, <sup>R</sup>), *<sup>γ</sup>* is well defined. Let *<sup>u</sup>*1(*s*), *<sup>u</sup>*2(*s*) <sup>∈</sup> *<sup>C</sup>*(*J*1, <sup>R</sup>) *Cn*(*J*2, R), for any *s* ∈ *J*3. Then we have

$$|(\gamma \mu\_1)(s) - (\gamma \mu\_2)(s)| = 0.$$

For all *s* ∈ *J*2, by condition (*S*1) and (*S*2), we have


where *<sup>u</sup>*(*j*) *<sup>i</sup>* <sup>=</sup> *<sup>u</sup>*(*j*) *<sup>i</sup>* (*λ*), *i* = 1, 2, *j* = 0, 1, . . . , *n*.

Since *<sup>η</sup><sup>n</sup>* (*n*−1)! (2*Kz* <sup>+</sup> *MNη*) <sup>&</sup>lt; 1, for *<sup>u</sup>*1(*s*), *<sup>u</sup>*2(*s*) <sup>∈</sup> *<sup>C</sup>*(*J*1, <sup>R</sup>) *Cn*(*J*2, R), the operator *γ* is a Banach contraction. By Banach contraction principle, the operator *γ* has a unique fixed point *<sup>u</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>C</sup>*(*J*1, <sup>R</sup>) *Cn*(*J*2, R); thus, Equation (3) has a unique solution.

Next, we obtain the following Ulam stability results.

**Theorem 3.** *If the assumptions of the Theorem 2 are satisfied, Equation (3) is Hyers–Ulam stable on J*1*.*

**Proof.** Let *<sup>u</sup>*(*s*) <sup>∈</sup> *<sup>C</sup>*(*J*1, <sup>R</sup>) *Cn*(*J*2, R) be a unique solution of delay integro-differential equation

$$\begin{cases} \boldsymbol{u}^{(n)}(\boldsymbol{s}) = \boldsymbol{z}\left(\boldsymbol{s}, \boldsymbol{u}^{(0)}, \dots, \boldsymbol{u}^{(0)}(\boldsymbol{\lambda}), \dots, \boldsymbol{u}^{(n-1)}(\boldsymbol{\lambda}), \int\_{\boldsymbol{s}\_{0}}^{\boldsymbol{s}} \boldsymbol{g}(\tau, \boldsymbol{u}^{(0)}, \dots, \boldsymbol{u}^{(n-1)}) d\tau\right), \; \mathbf{s} \in \boldsymbol{f}\_{2,\boldsymbol{s}}\\\boldsymbol{u}(\boldsymbol{s}) = \boldsymbol{\chi}(\boldsymbol{s}), \; \mathbf{s} \in \boldsymbol{f}\_{3,\boldsymbol{s}}\\\boldsymbol{u}^{(j)}(\boldsymbol{s}\_{0}) = \boldsymbol{\chi}^{(j)}(\boldsymbol{s}\_{0}), \; j = 1, \ldots, n - 1. \end{cases} \tag{9}$$

Since *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*(*J*<sup>2</sup> <sup>×</sup> <sup>R</sup>2*n*<sup>+</sup>1, <sup>R</sup>), *<sup>λ</sup>* <sup>∈</sup> *<sup>C</sup>*(*J*2, *<sup>J</sup>*1), from Theorem 1, we have

$$u(s) = \begin{cases} \sum\_{j=0}^{n-1} \frac{(s-s\_0)^j \chi^{(j)}(s\_0)}{j!} + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} \\\\ \cdot z \Big(\tau, u^{(0)}, \dots, u^{(0)}(\lambda), \dots, u^{(n-1)}(\lambda), \int\_{s\_0}^\tau g(r, u^{(0)}, \dots, u^{(n-1)}) dr\Big) dr, \ s \in \mathbb{I}\_{2^n} \\\\ \chi(s), \ s \in \mathbb{I}\_3. \end{cases} \tag{10}$$

Let *<sup>v</sup>*(*s*) <sup>∈</sup> *<sup>C</sup>*(*J*1, <sup>R</sup>) *Cn*(*J*2, R) satisfy the following inequality:

$$\left| \left\{ \left| v^{(n)}(s) - z(s, v^{(0)}, \dots, v^{(n-1)}(\lambda), \int\_{s\_0}^{s} g(\tau, v^{(0)}, \dots, v^{(n-1)}) d\tau \right) \right| \le \theta, \ s \in I\_{2'} \right. \tag{11} $$

Let

$$v^{(n)}(s) = z(s, v^{(0)}, \dots, v^{(n-1)}(\lambda), \int\_{s\_0}^s g(\tau, v^{(0)}, \dots, v^{(n-1)}) d\tau) + F(s),$$

from (11), this implies that

$$|F(s)| \le \theta\_\prime \text{ s} \in J\_2.$$

By Theorem 1, we have

$$v(s) = \sum\_{j=0}^{n-1} \frac{(s-s\_0)^j \chi^{(j)}(s\_0)}{j!} + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} z(\tau, v^{(0)}, \dots, v^{(n-1)}(\lambda),$$

$$\int\_{s\_0}^\tau g(r, v^{(0)}, \dots, v^{(n-1)}) dr \Big) d\tau + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} F(\tau) d\tau,$$

then

$$\begin{aligned} & \left| v(s) - \sum\_{j=0}^{n-1} \frac{(s - s\_0)^j \chi^{(j)}(s\_0)}{j!} - \frac{1}{(n-1)!} \int\_{s\_0}^s (s - \tau)^{n-1} z(\tau, v^{(0)}, \dots, v^{(n-1)}(\lambda), \tau) \right| \\ & \left| \int\_{s\_0}^\tau g(r, v^{(0)}, \dots, v^{(n-1)}) dr \right| \le \left| \frac{1}{(n-1)!} \int\_{s\_0}^s (s - \tau)^{n-1} F(\tau) d\tau \right| \end{aligned}$$

$$\begin{aligned} &\leq \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} |F(\tau)| d\tau \\ &\leq \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} d\tau \cdot \theta \\ &\leq \eta^n \theta. \end{aligned}$$

For all *s* ∈ *J*3,

$$|v(s) - u(s)| = 0.$$

For any *s* ∈ *J*2,


where *u*(*j*) = *u*(*j*)(*λ*), *v*(*j*) = *v*(*j*)(*λ*).

From the above inequality, we define the operator *A* as follows: for all *s* ∈ *J*2,

$$\left(Ay^{(j)}\right)(s) = \eta^n \theta + \frac{1}{(n-1)!} \int\_{s\_0}^{s} (s-\tau)^{n-1} \left[K\_{\overline{z}} \sum\_{j=0}^{n-1} (y^{(j)} + \overline{y}^{(j)}) + MN \int\_{s\_0}^{\tau} \sum\_{j=0}^{n-1} y^{(j)} d\tau\right] d\tau,$$

for all *s* ∈ *J*3,

$$\left(Ay^{(j)}\right)(\mathbf{s}) = \mathbf{0}\_{\prime}$$

where *y*¯(*j*) = *y*(*j*)(*λ*), *j* = 0, 1, . . . , *n*. For all *s* ∈ *J*2,

$$\begin{aligned} & \left| \left( Ay\_1^{\cdot (j)} \right) (s) - \left( Ay\_2^{\cdot (j)} \right) (s) \right| \\ & \le \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} \left[ K\_z \sum\_{j=0}^{n-1} \left( |y\_1^{\cdot (j)} - y\_2^{\cdot (j)}| + |\overline{y\_1}^{(j)} - \overline{y\_2}^{(j)}| \right) \right] \\ & \qquad + MN \int\_{s\_0}^\tau \sum\_{j=0}^{n-1} |y\_1^{\cdot (j)} - y\_2^{\cdot (j)}| d\tau \\ & \le \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} ds (2K\_z u + MN\eta n) ||y\_1^{\cdot (j)} - y\_2^{\cdot (j)}|| \\ & \le \frac{\eta^n}{(n-1)!} (2K\_z + MN\eta) ||y\_1^{\cdot (j)} - y\_2^{\cdot (j)}|| \end{aligned}$$

where *y*¯ (*j*) *<sup>i</sup>* = *y* (*j*) *<sup>i</sup>* (*λ*), *i* = 1, 2, *j* = 0, 1, . . . , *n*. Since

$$\frac{\eta^n}{(n-1)!}(2K\_z + MN\eta) < 1,$$

*A* is a strict contraction operator.

By the contraction mapping theorem, *<sup>A</sup>* has a unique fixed point {*ω*(*j*)}, so

$$\omega^{(j)}(s) = \eta^n \theta + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} \left[ \mathcal{K}\_z \sum\_{j=0}^{n-1} \left( \omega^{(j)} + \overline{\omega}^{(j)} \right) + \text{MN} \int\_{s\_0}^\tau \sum\_{j=0}^{n-1} \omega^{(j)} dr \right] d\tau, s \in I\_2.$$

Since *ω*(*j*)(*s*) <sup>≥</sup> 0, *<sup>ω</sup>*(*j*)(*s*) is a nondecreasing function, we have

$$
\omega^{(j)}(\lambda(s)) = \overline{\omega}^{(j)}(s) \le \omega^{(j)}(s),
$$

$$\begin{split} \omega^{(j)}(s) &\leq \eta^{n}\theta + \frac{1}{(n-1)!} \int\_{s\_{0}}^{s} (s-\tau)^{n-1} \left(2K\_{z} \sum\_{j=0}^{n-1} \omega^{(j)} + MN \int\_{s\_{0}}^{\tau} \sum\_{j=0}^{n-1} \omega^{(j)} dr\right) d\tau \\ &\leq \eta^{n}\theta + \frac{1}{(n-1)!} \sum\_{j=0}^{n-1} (2K\_{z} + MN\eta) \int\_{s\_{0}}^{s} (s-\tau)^{n-1} \omega^{(j)} d\tau \\ &\leq \eta^{n}\theta + \frac{n}{(n-1)!} (2K\_{z} + MN\eta) \int\_{s\_{0}}^{s} (s-\tau)^{n-1} \omega^{(j)} d\tau. \end{split}$$

From Lemma 3, we obtain

$$
\omega^{(j)}(s) \le \mathbb{C} \cdot \theta \text{ } \mathbb{C} = \eta^n \exp\left(\frac{(2K\_z + MN\eta)\eta^n}{(n-1)!}\right).
$$

Since

$$|v(s) - \mu(s)| \le \left(Ay^{(j)}\right)(s),$$

then

$$|v(s) - u(s)| \le \left( Aw^{(j)} \right)(s) = w^{(j)}(s) \le \mathcal{C} \cdot \theta.$$

From Definition 1, Equation (3) is Hyers–Ulam stable.

**Theorem 4.** *Assume that* (*S*3) *and* (*S*4) *hold; then Equation (3) is Hyers–Ulam stable on J*1*.*

**Proof.** Let *<sup>u</sup>*(*s*) <sup>∈</sup> *<sup>C</sup>*(*J*1, <sup>R</sup>) *Cn*(*J*2, R) be a solution of delay integro-differential equation

$$\begin{cases} u^{(n)}(s) = z \left( s, u^{(0)}, \dots, u^{(0)}(\lambda), \dots, u^{(n-1)}(\lambda), \int\_{s\_0}^{s} \mathbf{g}(\tau, \mu^{(0)}, \dots, \mu^{(n-1)}) d\tau \right), \text{ s} \in \mathfrak{I}\_2, \\ u(s) = \chi(s), \; s \in \mathfrak{I}\_2, \\ u^{(j)}(s\_0) = \chi^{(j)}(s\_0), \; j = 1, \dots, n - 1. \end{cases} \tag{12}$$

Since *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*(*J*<sup>2</sup> <sup>×</sup> <sup>R</sup>2*n*<sup>+</sup>1, <sup>R</sup>), *<sup>λ</sup>* <sup>∈</sup> *<sup>C</sup>*(*J*2, *<sup>J</sup>*1), from Theorem 1, we have

$$u(s) = \begin{cases} \sum\_{j=0}^{n-1} \frac{(s-s\_0)^j \chi^{(j)}(s\_0)}{j!} + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} \\\\ \therefore \left(\mathbb{1}, u^{(0)}, \dots, u^{(0)}(\lambda), \dots, u^{(n-1)}(\lambda), \int\_{s\_0}^\tau \mathbb{g}(r, u^{(0)}, \dots, u^{(n-1)}) dr\right) dr, \text{ s} \in \mathfrak{I}\_2, \\\ \chi(s), \text{ } s \in \mathfrak{I}\text{s}. \end{cases} \tag{13}$$

Let *<sup>v</sup>*(*s*) <sup>∈</sup> *<sup>C</sup>*(*J*1, <sup>R</sup>) *Cn*(*J*2, R) satisfying inequality

$$\left| \left\{ \left| v^{(n)}(s) - z(s, v^{(0)}, \dots, v^{(n-1)}(\lambda), \int\_{s\_0}^{s} g(\tau, v^{(0)}, \dots, v^{(n-1)}) d\tau \right| \right| \le \theta, \ s \in I\_{2\prime} \right. \tag{14}$$

Let

$$v^{(n)}(s) = z(s, v^{(0)}, \dots, v^{(n-1)}(\lambda), \int\_{s\_0}^s g(\tau, v^{(0)}, \dots, v^{(n-1)}) d\tau) + F(s),$$

from (14), this implies that

$$|F(s)| \le \theta\_\prime \text{ s} \in J\_2.$$

By Theorem 1, we have

$$\begin{aligned} v(s) &= \sum\_{j=0}^{n-1} \frac{(s-s\_0)^j \chi^{(j)}(s\_0)}{j!} + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} z\left(\tau, v^{(0)}, \dots, v^{(n-1)}(\lambda), \tau\right) \\ &\int\_{s\_0}^\tau g(r, v^{(0)}, \dots, v^{(n-1)}) dr \Big) d\tau + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} F(\tau) d\tau, \end{aligned}$$

then

$$\begin{aligned} & \left| v(s) - \sum\_{j=0}^{n-1} \frac{(s-s\_0)^j \chi^{(j)}(s\_0)}{j!} - \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} z(\tau, v^{(0)}, \dots, v^{(n-1)}(\Lambda), \tau) d\tau \right| \\ & \quad \left| \int\_{s\_0}^\tau g(r, v^{(0)}, \dots, v^{(n-1)}) dr \right| \cdot d\tau \Bigg| \\ & = \left| \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} F(\tau) d\tau \right| \\ & \leq \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} |F(\tau)| d\tau \\ & \leq \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} d\tau \cdot \theta \\ & \leq \eta^n \theta . \end{aligned}$$

For all *s* ∈ *J*3,


For any *s* ∈ *J*2,

$$\begin{split} &|v(s)-u(s)| \\ &=\left|v(s)-\sum\_{j=0}^{n-1}\frac{(s-s\_0)^j\chi^{(j)}(s\_0)}{j!}-\frac{1}{(n-1)!}\int\_{s\_0}^{s}(s-\tau)^{n-1}z\left(\tau,\mu^{(0)},\ldots,\mu^{(n-1)},\mu^{(0)}(\lambda),\ldots,\mu^{(n-1)},\mu^{(0)}(\lambda)\right)d\tau\right| \\ &\quad\ldots,\mu^{(n-1)}(\lambda),\int\_{s\_0}^{\tau}\mathcal{G}(r,\mu^{(0)},\ldots,\mu^{(n-1)})dr\right|d\tau\Big|\\ &=\left|v(s)-\sum\_{j=0}^{n-1}\frac{(s-s\_0)^j\chi^{(j)}(s\_0)}{j!}-\frac{1}{(n-1)!}\int\_{s\_0}^{s}(s-\tau)^{n-1}z\left(\tau,\nu^{(0)},\ldots,\nu^{(n-1)},\nu^{(0)}(\lambda),\ldots,\lambda^{(n-1)},\mu^{(0)}(\lambda)\right)d\tau\right|d\tau\Big|\\ &\quad\ldots,\mathcal{O}^{(n-1)}(\lambda),\int\_{s\_0}^{\tau}\mathcal{G}(r,\nu^{(0)},\ldots,\nu^{(n-1)})dr\Big|+\left|\frac{1}{(n-1)!}\int\_{s\_0}^{s}(s-\tau)^{n-1}z\left(\tau,\nu^{(0)},\ldots,\nu^{(n-1)},\nu^{(0)}(\lambda),\ldots,\lambda^{(n-1)},\ldots,\nu^{(n-1)}\right)d\tau\right|\end{split}$$

..., *v*(*n*−1) , *v*(0) (*λ*),..., *v*(*n*−1) (*λ*), *τ s*0 *g*(*r*, *v*(0) ,..., *v*(*n*−1) )*dr dτ* <sup>−</sup> <sup>1</sup> (*n* − 1)! *s s*0 (*<sup>s</sup>* <sup>−</sup> *<sup>τ</sup>*)*n*−1*<sup>z</sup> τ*, *u*(0) ,..., *u*(*n*−1) , *u*(0) (*λ*), ..., *u*(*n*−1) (*λ*), *τ s*0 *g*(*r*, *u*(0) ,..., *u*(*n*−1) )*dr dτ* ≤ *<sup>v</sup>*(*s*) <sup>−</sup> *n*−1 ∑ *j*=0 (*<sup>s</sup>* − *<sup>s</sup>*0)*<sup>j</sup> χ*(*j*)(*s*0) *<sup>j</sup>*! <sup>−</sup> <sup>1</sup> (*n* − 1)! *s s*0 (*<sup>s</sup>* <sup>−</sup> *<sup>τ</sup>*)*n*−1*<sup>z</sup> τ*, *v*(0) ,..., *v*(*n*−1) , *v*(0) (*λ*),..., *v*(*n*−1) (*λ*), *τ s*0 *g*(*r*, *v*(0) ,..., *v*(*n*−1) )*dr dτ* + 1 (*n* − 1)! *s s*0 (*<sup>s</sup>* <sup>−</sup> *<sup>τ</sup>*)*n*−<sup>1</sup> *<sup>e</sup>*(*τ*)|*v*(0) <sup>−</sup> *<sup>u</sup>*(0) | *<sup>l</sup>* <sup>+</sup> *<sup>k</sup>*(*τ*)|*v*(0) (*λ*) <sup>−</sup> *<sup>u</sup>*(0) (*λ*)| *m* + *τ s*0 *<sup>M</sup>*(*τ*)*N*(*r*)|*v*(0) <sup>−</sup> *<sup>u</sup>*(0) | *ndr dτ* ,

then

$$\begin{aligned} |v(s) - u(s)| \leq & \eta^n \theta + \frac{1}{(n-1)!} \int\_{s\_0}^{s} (s - \tau)^{n-1} \Big[ e(\tau) |v - u|^l + k(\tau) |v(\lambda) - u(\lambda)|^m \\ & + \int\_{s\_0}^{\tau} M(\tau) N(r) |v - u|^n dr \Big] d\tau. \end{aligned}$$

From Lemma 4, set *<sup>h</sup>* = 1, *<sup>a</sup>* = *<sup>m</sup>*, *<sup>b</sup>* = *<sup>l</sup>*, *<sup>c</sup>* = *<sup>n</sup>*, *<sup>w</sup>*(*s*) = |*v*(*s*) − *<sup>u</sup>*(*s*)|, *<sup>d</sup>*(*s*) = *<sup>η</sup>nθ*, *l*(*s*) = <sup>1</sup> (*n*−1)! , *<sup>m</sup>*(*τ*)=(*<sup>s</sup>* − *<sup>τ</sup>*)*n*−1*k*(*τ*), *<sup>n</sup>*(*τ*)=(*<sup>s</sup>* − *<sup>τ</sup>*)*n*−1*e*(*τ*),*z*(*r*)=(*<sup>s</sup>* − *<sup>τ</sup>*)*n*−1*M*(*τ*)*N*(*r*); we have

$$\begin{aligned} |v(s) - u(s)| &\le \eta^n \theta + \frac{1}{(n-1)!} \mathcal{U}(s) \exp\left(\int\_{s\_0}^s V(\tau)d\tau\right), \\\\ |v(s) - u(s)| &\le \left[\eta^n + \frac{1}{(n-1)!} \frac{\mathcal{U}(s)}{\theta} \exp\left(\int\_{s\_0}^s V(\tau)d\tau\right)\right] \cdot \theta \le K\_{z,\theta} \cdot \theta, \end{aligned}$$

where for any *H* > 0,

$$K\_{z, \theta} = \max\_{s \in f\_1} \left\{ \eta^{\mathrm{II}} + \frac{1}{(n-1)!} \frac{\mathcal{U}(s)}{\theta} \exp\left(\int\_{s\_0}^s V(\tau)d\tau\right) \right\},$$

$$\begin{split} \mathcal{U}(s) &= \int\_{s\_0}^{s} \left[ (s-\tau)^{n-1} k(\tau) \left( m H^{m-1} \eta^n \theta + (1-m) H^m \right) + \mathfrak{g}(\tau, s) \left( l H^{l-1} \eta^n \theta + (1-l) H^l \right) \right] \\ &+ \int\_{s\_0}^{\tau} (s-\tau)^{n-1} M(\tau) N(r) \left( n H^{n-1} \eta^n \theta + (1-n) H^n \right) dr, \\ \mathcal{V}(s) &= \frac{1}{(n-1)!} \int\_{s\_0}^{s} (s-\tau)^{n-1} M(\tau) N(\tau) n H^{n-1} d\tau. \end{split}$$

From Definition 1, Equation (3) is Hyers–Ulam stable.

**Theorem 5.** *Assume that* (*S*3)*,* (*S*4) *and* (*S*5) *hold; then Equation (3) is Hyers–Ulam–Rassias stable with respect to σ*(*s*) *on J*1*.*

**Proof.** Let *<sup>u</sup>*(*s*) <sup>∈</sup> *<sup>C</sup>*(*J*1, <sup>R</sup>) *Cn*(*J*2, R) be a unique solution of delay integro-differential equation

$$\begin{cases} u^{(n)}(s) = z \left( s, u^{(0)}, \dots, u^{(0)}(\lambda), \dots, u^{(n-1)}(\lambda), \int\_{s\_0}^{s} \mathbf{g}(\tau, u^{(0)}, \dots, u^{(n-1)}) d\tau \right), \text{ s} \in [\mathfrak{z}, \\ u(s) = \chi(s), \ s \in J\_{\mathfrak{z}} \\ u^{(j)}(s\_0) = \chi^{(j)}(s\_0), \ j = 1, \dots, n - 1. \end{cases} \tag{15}$$

Since *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*(*J*<sup>2</sup> <sup>×</sup> <sup>R</sup>2*n*<sup>+</sup>1, <sup>R</sup>), *<sup>λ</sup>* <sup>∈</sup> *<sup>C</sup>*(*J*2, *<sup>J</sup>*1), from Theorem 1, we have

$$u(s) = \begin{cases} \sum\_{j=0}^{n-1} \frac{(s-s\_0)^j \chi^{(j)}(s\_0)}{j!} + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} \\\\ \therefore \left(\tau, u^{(0)}, \dots, u^{(0)}(\lambda), \dots, u^{(n-1)}(\lambda), \int\_{s\_0}^\tau g(r, u^{(0)}, \dots, u^{(n-1)}) dr\right) dr, \ s \in \mathfrak{I}\_2, \\\ \chi(s), \ s \in \mathfrak{I}\_3. \end{cases} \tag{16}$$

Let *<sup>v</sup>*(*s*) <sup>∈</sup> *<sup>C</sup>*(*J*1, <sup>R</sup>) *Cn*(*J*2, R) satisfy the following inequality

$$\left| \left\{ \left| v^{(n)}(s) - z(s, v^{(0)}, \dots, v^{(n-1)}(\lambda), \int\_{s\_0}^{s} g(\tau, v^{(0)}, \dots, v^{(n-1)}) d\tau \right| \right| \le \sigma(s), \ s \in J\_{2'} \right. \tag{17} $$

Let

$$v^{(n)}(s) = z(s, v^{(0)}, \dots, v^{(n-1)}(\lambda), \int\_{s\_0}^s g(\tau, v^{(0)}, \dots, v^{(n-1)}) d\tau) + F(s),$$

from (17), this implies that

$$|F(s)| \le \sigma(s), \ s \in J\_2.$$

By Theorem 1, we have

$$\begin{aligned} v(s) &= \sum\_{j=0}^{n-1} \frac{(s-s\_0)^j \chi^{(j)}(s\_0)}{j!} + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} z\left(\tau, v^{(0)}, \dots, v^{(n-1)}(\lambda), \tau\right) \\ &\int\_{s\_0}^\tau g(r, v^{(0)}, \dots, v^{(n-1)}) dr \Big) d\tau + \frac{1}{(n-1)!} \int\_{s\_0}^s (s-\tau)^{n-1} F(\tau) d\tau; \end{aligned}$$

then

$$\begin{aligned} & \left| v(s) - \sum\_{j=0}^{n-1} \frac{(s - s\_0)^j \chi^{(j)}(s\_0)}{j!} - \frac{1}{(n-1)!} \int\_{s\_0}^s (s - \tau)^{n-1} z(\tau, v^{(0)}, \dots, v^{(n-1)}(\lambda), \tau) d\tau \right| \\ & \left| \int\_{s\_0}^\tau g(r, v^{(0)}, \dots, v^{(n-1)}) dr \right| \le \\ & \left| \frac{1}{(n-1)!} \int\_{s\_0}^s (s - \tau)^{n-1} F(\tau) d\tau \right| \\ & \le \frac{1}{(n-1)!} \int\_{s\_0}^s (s - \tau)^{n-1} |F(\tau)| d\tau \\ & \le \frac{1}{(n-1)!} \int\_{s\_0}^s (s - \tau)^{n-1} \sigma(\tau) d\tau. \end{aligned}$$

By Lemma 1 and condition (*S*5), we obtain

$$\begin{aligned} & \left| v(s) - \sum\_{j=0}^{n-1} \frac{(s - s\_0)^j \chi^{(j)}(s\_0)}{j!} - \frac{1}{(n-1)!} \int\_{s\_0}^s (s - \tau)^{n-1} z(\tau, v^{(0)}, \dots, v^{(n-1)}(\lambda), \tau) d\tau \right| \\ & \left| \int\_{s\_0}^\tau g(r, v^{(0)}, \dots, v^{(n-1)}) dr \right| \end{aligned}$$

$$\begin{aligned} & \leq \int\_{s\_0}^{s} \int\_{s\_0}^{s\_1} \int\_{s\_0}^{s\_2} \int\_{s\_0}^{s\_3} \dots \int\_{s\_0}^{s\_{n-1}} \sigma(s\_n) ds\_n ds\_{n-1} \dots \, ds\_2 ds\_1 \\ & \leq L\_{\sigma} \,^n \cdot \sigma(s). \end{aligned}$$
For all  $s \in J\_3$ .
$$|v(s) - u(s)| = 0.$$

For any *s* ∈ *J*2,


then

$$\begin{aligned} &|v(s) - u(s)| \\ \leq &L\_{\sigma}^{n} \sigma(s) + \frac{1}{(n-1)!} \int\_{s\_{0}}^{s} (s - \tau)^{n-1} \left[ e(\tau)|v - u|^{l} + k(\tau)|v(\lambda) - u(\lambda)|^{m} \right] \\ &+ \int\_{s\_{0}}^{\tau} M(\tau)N(r)|v - u|^{n} dr \Big] d\tau. \end{aligned}$$

From Lemma 4, set *h* = 1, *a* = *m*, *b* = *l*, *c* = *n*, *w*(*s*) = |*v*(*s*) − *u*(*s*)|, *d*(*s*) = *L<sup>σ</sup> <sup>n</sup>σ*(*s*), *l*(*s*) = <sup>1</sup> (*n*−1)! , *<sup>m</sup>*(*τ*)=(*<sup>s</sup>* − *<sup>τ</sup>*)*n*−1*k*(*τ*), *<sup>n</sup>*(*τ*)=(*<sup>s</sup>* − *<sup>τ</sup>*)*n*−1*e*(*τ*), *<sup>z</sup>*(*r*)=(*<sup>s</sup>* − *<sup>τ</sup>*)*n*−1*M*(*τ*)*N*(*r*); thus, we have

$$|v(s) - u(s)| \le L\_{\sigma}"\sigma(s) + \frac{1}{(n-1)!} \mathcal{U}(s) \exp\left(\int\_{s\_0}^{s} V(\tau)d\tau\right),$$

$$|v(s) - u(s)| \le \left[ L\_{\sigma}^{\;\;\;\;\;u} + \frac{1}{(n-1)!} \frac{\mathcal{U}(s)}{\sigma(s)} \exp\left( \int\_{s\_0}^{s} V(\tau) d\tau \right) \right] \cdot \sigma(s) \le K\_{\sigma} \cdot \sigma(s),$$

where for any *H* > 0,

$$\begin{split} \mathcal{K}\_{\tau} &= \max\_{s \in I\_{1}} \left\{ L\_{\tau}^{-n} + \frac{1}{(n-1)!} \frac{\mathcal{U}(s)}{\sigma(s)} \exp\left(\int\_{s\_{0}}^{s} V(\tau) d\tau\right) \right\}, \\ \mathcal{U}(s) &= \int\_{s\_{0}}^{s} \left[ (s-\tau)^{n-1} k(\tau) \left( m H^{m-1} L\_{\tau}{}^{n} \sigma(\tau) + (1-m) H^{m} \right) + \mathcal{g}(\tau, s) \left( l H^{l-1} L\_{\tau}{}^{n} \sigma(\tau) \right) \right. \\ &\left. + (1-l) H^{l} \right) + \int\_{s\_{0}}^{\tau} (s-\tau)^{n-1} M(\tau) N(\tau) \left( n H^{m-1} L\_{\tau}{}^{n} \sigma(\tau) + (1-n) H^{n} \right) d\tau, \\ V(s) &= \frac{1}{(n-1)!} \int\_{s\_{0}}^{s} (s-\tau)^{n-1} M(\tau) N(\tau) n H^{n-1} d\tau. \end{split}$$

From Definition 2, Equation (3) is Hyers–Ulam–Rassias stable.
