**1. Introduction**

In the year 1940, Ulam [1] put forward an abstract problem: under what conditions is the exact solution of an equation closed to the approximate solution? In the year 1941, in order to solve the problem raised by Ulam, Hyers [2] studied the functional equation in Banach space and gave the definition of Hyers–Ulam stability. In the year 1978, based on the work of Hyers, Rassias [3] gave the definition of Hyers–Ulam–Rassias stability. These two kinds of stability are called Ulam stability. After that, scholars began to study the Ulam stability of some solvable equations. See [4–7] and the references therein. Recently, the Ulam stability of delay differential equations and delay integro-differential equations has been discussed. See [8–13]. There are many results about the Ulam stability of delay differential equations. However, there are a few results about the Ulam stability of delay integro-differential equations.

In fact, delay integro-differential equations are usually used to describe many natural phenomena in the fields of thermodynamics, mechanics, mechanical engineering and control. See [14–17]. Mechanical processes, such as rigid heat conduction process [18] and the motion of charged particles with a delayed interaction [19], can be modeled by delay integro-differential equations. Furthermore, the delay integro-differential equation is an appropriate model for studying the effect of tire dynamics on a vehicle shimmy [20] and the optimal control of a size-structured population [21], which is one of its important applications. The mathematical model related to the delay integro-differential equation is an interesting memory effect model. However, new difficulties may arise when delay and integro-differential equations are introduced simultaneously. Some topics about delay integro-differential equations, such as the existence and uniqueness of solutions and Ulam stability, have attracted the attention of many scholars. See [12,14,15,22].

In [23], Otrocol studied Ulam stability of a first-order delay differential equation:

$$u'(t) = \mathcal{g}(t, u(t), u(\pi(t))), \ t \in [t\_0, t\_1]\_\prime$$

where *<sup>g</sup>* <sup>∈</sup> *<sup>C</sup>*([*t*0, *<sup>t</sup>*1] <sup>×</sup> <sup>R</sup>2, <sup>R</sup>); delay function *<sup>τ</sup>*(*t*) <sup>≤</sup> *<sup>t</sup>*, *<sup>τ</sup>* <sup>∈</sup> *<sup>C</sup>*([*t*0, *<sup>t</sup>*1], [*t*<sup>0</sup> <sup>−</sup> *<sup>l</sup>*, *<sup>t</sup>*1]), *<sup>l</sup>* <sup>&</sup>gt; 0.

**Citation:** Wang, S.; Meng, F. Ulam Stability of an *n*-th Order Delay Integro-Differential Equations. *Mathematics* **2021**, *9*, 3029. https:// doi.org/10.3390/math9233029

Academic Editor: Alberto Cabada

Received: 20 October 2021 Accepted: 19 November 2021 Published: 26 November 2021

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In 2015, Kendre [24] discussed the existence of a solution for an integro-differential equation:

$$\begin{cases} u'(t) = \mathbf{g}\left(t, u(u(t)), \int\_{t\_0}^{t} k(t, s)u(u(s))ds\right), & t \in [t\_0, t\_1], \\ u(t\_0) = u\_{0\prime} \end{cases} \tag{1}$$

where *<sup>G</sup>* = [*t*0, *<sup>t</sup>*1], *<sup>g</sup>* ∈ *<sup>C</sup>*(*G*3, *<sup>G</sup>*); *<sup>k</sup>* ∈ *<sup>C</sup>*(*G*2, *<sup>G</sup>*); *<sup>t</sup>*0, *<sup>u</sup>*<sup>0</sup> ∈ *<sup>G</sup>*.

In 2016, Sevgin [25] investigated the Ulam stability of the Volterra integro-differential equation:

$$u'(t) = f(t, u(t)) + \int\_0^t g(t, s, u(s))ds, \ t \in [0, t\_0]\_\tau$$

where *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>t</sup>*0] <sup>×</sup> <sup>R</sup>, <sup>R</sup>); *<sup>g</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>t</sup>*0] <sup>×</sup> [0, *<sup>t</sup>*0] <sup>×</sup> <sup>R</sup>, <sup>R</sup>).

In 2018, Kishor [26] established the Ulam stability of the semilinear Volterra integrodifferential equation:

$$u'(t) = Tu(t) + \left(t, u(t), \int\_0^t z(t, s, u(s)) ds\right), \ t \in [0, t\_0]\_{\mathcal{I}}$$

where *<sup>T</sup>* : *<sup>U</sup>* <sup>→</sup> *<sup>U</sup>* is the infinitesimal generator; *<sup>U</sup>* is Banach space; *<sup>g</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>t</sup>*0] <sup>×</sup> <sup>R</sup>2, <sup>R</sup>); *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>t</sup>*0] <sup>×</sup> <sup>R</sup>2, <sup>R</sup>).

In 2019, Zada [27] obtained the Ulam stability for the following *n*-th order delay differential equation:

$$\begin{cases} u^{(n)}(s) = z(s, \{u^{(0)}\}, \{u^{(1)}\}, \dots, \{u^{(n-1)}\}), \ s \in [s\_0, s\_0 + \eta],\\ u(s) = \chi(s), \ s \in [s0 - \zeta, s0], \end{cases} \tag{2}$$

where {*u*(*j*)} <sup>=</sup> {*u*(*j*)(*s*), *<sup>u</sup>*(*j*)(*λ*1(*s*)), *<sup>u</sup>*(*j*)(*λ*2(*s*)), ... , *<sup>u</sup>*(*j*)(*λk*(*s*))}, *<sup>j</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>* <sup>−</sup> 1; delay functions *<sup>λ</sup><sup>k</sup>* <sup>∈</sup> *<sup>C</sup>*([*s*0,*s*<sup>0</sup> <sup>+</sup> *<sup>η</sup>*], [*s*<sup>0</sup> <sup>−</sup> *<sup>ζ</sup>*,*s*<sup>0</sup> <sup>+</sup> *<sup>η</sup>*]), *<sup>λ</sup>k*(*s*) <sup>≤</sup> *<sup>s</sup>*, *<sup>k</sup>* <sup>∈</sup> <sup>Z</sup>+; *<sup>ζ</sup>* <sup>&</sup>gt; 0, *<sup>η</sup>* <sup>&</sup>gt; 0, *<sup>s</sup>*<sup>0</sup> are constants; *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*(*B*˜, <sup>R</sup>), *<sup>B</sup>*˜ <sup>⊂</sup> [*s*0,*s*<sup>0</sup> <sup>+</sup> *<sup>η</sup>*] <sup>×</sup> <sup>R</sup>*n*(*k*+1) is closed set; *<sup>χ</sup>*: [*s*<sup>0</sup> <sup>−</sup> *<sup>ζ</sup>*,*s*0] <sup>→</sup> <sup>R</sup>.

However, the existence and uniqueness of solutions and Ulam stability for *n*-th delay integro-differential equations have not been studied hitherto. Inspired by [24,25,27], we study Ulam stability for the following *n*-th order delay integro-differential equation:

$$\begin{cases} \boldsymbol{\mu}^{(n)}(\mathbf{s}) = \boldsymbol{z} \Big( \mathbf{s}, \boldsymbol{\mu}^{(0)}, \dots, \boldsymbol{\mu}^{(n-1)}, \boldsymbol{\mu}^{(0)}(\boldsymbol{\lambda}), \dots, \boldsymbol{\mu}^{(n-1)}(\boldsymbol{\lambda}), \int\_{\boldsymbol{s}\_{0}}^{\boldsymbol{s}} \boldsymbol{g}(\tau, \boldsymbol{\mu}^{(0)}, \dots, \boldsymbol{\mu}^{(n-1)}) d\tau \Big), \\\boldsymbol{\mu}(\mathbf{s}) = \boldsymbol{\chi}(\mathbf{s}), \ s \in [\mathbf{s}\_{0} - \boldsymbol{\zeta}, \mathbf{s}], \\\ \boldsymbol{\mu}^{(j)}(\mathbf{s}\_{0}) = \boldsymbol{\chi}^{(j)}(\mathbf{s}\_{0}), \ j = 1, \dots, n - 1, \end{cases} \tag{3}$$

where the definition domain of the first formula of Equation (3) is [*s*0,*s*<sup>0</sup> + *η*], where *u*(*j*) = *<sup>u</sup>*(*j*)(*s*), *<sup>u</sup>*(*j*)(*λ*) = *<sup>u</sup>*(*j*)(*λ*(*s*)), *<sup>j</sup>* <sup>=</sup> 0, ... , *<sup>n</sup>* <sup>−</sup> 1; delay function *<sup>λ</sup>*(*s*) <sup>≤</sup> *<sup>s</sup>*, *<sup>λ</sup>* <sup>∈</sup> *<sup>C</sup>*([*s*0,*s*<sup>0</sup> <sup>+</sup> *<sup>η</sup>*], [*s*<sup>0</sup> <sup>−</sup> *<sup>ζ</sup>*,*s*<sup>0</sup> <sup>+</sup> *<sup>η</sup>*]); *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*(*B*, <sup>R</sup>), *<sup>B</sup>* <sup>⊂</sup> [*s*0,*s*<sup>0</sup> <sup>+</sup> *<sup>η</sup>*] <sup>×</sup> <sup>R</sup>2*n*+<sup>1</sup> is closed set; *<sup>g</sup>* <sup>∈</sup> *<sup>C</sup>*(*H*, <sup>R</sup>), *<sup>H</sup>* <sup>⊂</sup> [*s*0,*s*<sup>0</sup> <sup>+</sup> *<sup>η</sup>*] <sup>×</sup> <sup>R</sup>*<sup>n</sup>* is closed set; *<sup>χ</sup>* <sup>∈</sup> *<sup>C</sup>n*([*s*<sup>0</sup> <sup>−</sup> *<sup>ζ</sup>*,*s*0], <sup>R</sup>).

The aim of our paper is to study the Ulam stability and the existence and uniqueness of solutions for Equation (3). The main tools used in this paper are Lipschitz conditions and Gronwall–Bellman inequality.

The remainder of the paper is organized as follows: In Section 2, we give definitions and lemmas, which are essential for Section 3. In Section 3, we state some Lipschitz conditions, which will be helpful to prove the existence and uniqueness results for a delay integro-differential equation; then the Ulam stability for the delay integro-differential equation is given. In Section 4, we give two examples to illustrate main results.
