Existence of (ω, c)-Periodic Solutions for a Class of Nonlinear Functional Integral Equations and Applications

Abstract

We provide sufficient conditions for the existence of $(\omega ,c)$-periodic solutions of a general class of nonlinear functional integral equations. This study extends and generalizes previous contributions in the literature. As an application of the developed theory, we establish the existence of $(\omega ,c)$-periodic solutions for recurrent neural networks with time-varying coefficients and mixed delays, as well as for a class of nonlinear Volterra–Stieltjes integral equations with infinite delay.

1. Introduction

A wide variety of problems in mathematical physics, biology, mechanics, engineering, vehicular traffic theory, and queuing theory, together with numerous real-world applications, can be formulated in terms of integral equations; see, for example [1,2,3,4,5,6,7]. The theory of functional integral equations is highly developed and constitutes a significant branch of analysis. To date, numerous research papers have been published, containing, among other results, many existence theorems for functional integral equations of various types.

In this paper we are going to consider the nonlinear functional integral equation of type

$$y\left(t\right)=\mathcal{Q}(t,y_t)+{\int }_{t_0}^t\mathcal{G}(s,y_s)\mathrm{d}g\left(s\right),$$

(1)

where the integral on the right-hand side is taken in the Perron–Stieltjes sense with respect to the function g, where $\mathcal{G}:\mathbb{R}\times C(\mathbb{R},{\mathbb{C}}^n)\to {\mathbb{C}}^n$, $\mathcal{Q}:\mathbb{R}\times C(\mathbb{R},{\mathbb{C}}^n)\to {\mathbb{C}}^n$ and $g:\mathbb{R}\to \mathbb{R}$ are functions, and $C(\mathbb{R},{\mathbb{C}}^n)$ represents the space of all continuous functions defined on $\mathbb{R}$ and with values in ${\mathbb{C}}^n.$ Given a function $y\in C(\mathbb{R},{\mathbb{C}}^n),$ the usual Krasovsky notation [8] (Chapter VI) $y_t\in C(\mathbb{R},{\mathbb{C}}^n),$ with $t\in \mathbb{R},$ is employed to denote the function

$$y_t\left(\theta \right):=y(t+\theta ),\theta \in \mathbb{R}.$$

Note that Equation (1) includes several types of integral or functional integral equations. For example:

• If $\mathcal{Q}(t,y_t)=a\left(t\right)+\alpha \left(t\right){\int }_0^{\infty }v(s,y\left(s\right))\mathrm{d}s$, $\mathcal{G}(s,y_s)=k\left(s\right)u(s,y\left(s\right))$ and $g\left(s\right)=s,$ we obtain the mixed Volterra–Fredholm integral equation:

  $$y\left(t\right)=a\left(t\right)+\alpha \left(t\right){\int }_0^{\infty }v(s,y\left(s\right))\mathrm{d}s+{\int }_0^{t}k\left(s\right)u(s,y\left(s\right))\mathrm{d}s.$$
• If $\mathcal{Q}(t,y_t)=y\left(t_0\right)+{\int }_{-r}^0{\mathrm{d}}_{\theta }\left[\mu (t,\theta )\right]y(t+\theta )-{\int }_{-r}^0{\mathrm{d}}_{\theta }\left[\mu (t_0,\theta )\right]y(t_0+\theta )$ and $\mathcal{G}(s,y_s)=f(s,y_s),$ we obtain a neural-type measure functional differential equation:

  $$y\left(t\right)=y\left(t_0\right)+{\int }_{-r}^0{\mathrm{d}}_{\theta }\left[\mu (t,\theta )\right]y(t+\theta )-{\int }_{-r}^0{\mathrm{d}}_{\theta }\left[\mu (t_0,\theta )\right]y(t_0+\theta )+{\int }_{t_0}^{t}f(s,y_s)\mathrm{d}g\left(s\right).$$
• If $\mathcal{Q}(t,y_t)=y\left(0\right)$ and $\mathcal{G}(s,y_s)=f(s,y\left(s\right)),$ we obtain the classical measure differential equation:

  $$y\left(t\right)=y\left(0\right)+{\int }_0^{t}f(s,y\left(s\right))\mathrm{d}g\left(s\right).$$
• If $\mathcal{Q}(t,y_t)=\ell \left(t\right),$ $\mathcal{G}(s,y_s)=\lambda \left(s\right)u(s,y\left(s\right))$ and $g\left(s\right)=s,$ we recover a nonlinear Volterra and Fredholm integral:

  $$y\left(t\right)=\ell \left(t\right)+{\int }_0^t\lambda \left(s\right)u(s,y\left(s\right))\mathrm{d}s.$$

In addition, the inclusion of a non-decreasing function g in the integral on the right-hand side of Equation (1) allows for the consideration of a broader class of equations. It is a known fact (see [9] (Subsection 3.2)) that depending on the definition of the function g, it is possible to include an impulsive functional integral equation of type

$$y\left(t\right)=\tilde{\mathcal{Q}}(t,y_t)+{\int }_{t_0}^t\tilde{\mathcal{G}}(s,y_s)\mathrm{d}\tilde{g}\left(s\right)+\sum _{\begin{array}{c}k\in \{1,\dots ,m\}\\ t_k<t\end{array}}{I}_k\left(y\left(t_k\right)\right),$$

as a special case of Equation (1). Meanwhile, it is also possible (following the approach in [9] (Subsection 3.3)) to include a functional integral on time scales as a special case of Equation (1) by setting $g\left(t\right)=t^{*}=inf\left\{s\in \mathbb{T}:s⩾t\right\}$.

A wide variety of processes observed in nature can be described by mathematical models incorporating periodic, anti-periodic, and quasi-periodic functions. In particular, it is well known that several applications of classical Hamiltonian perturbation theory rely on the study of periodic and quasi-periodic motions [10,11]. Indeed, the study of periodic phenomena has been of fundamental importance since the pioneering work of H. Poincaré on planetary orbits in celestial mechanics (see [12]).

The purpose of the present paper is to investigate a general theory of “periodic” solutions, known as $(\omega ,c)$-periodic solutions, in the context of nonlinear functional integral equations of type (1). The theory of $(\omega ,c)$-periodic functions was introduced in the literature by Álvarez, Gómez and Pinto [13] and, through the years, this theory has attracted the attention of many mathematicians (see [14,15,16] and the references therein). Let $c\in \mathbb{C}\setminus \left\{0\right\}$ and $\omega >0.$ A function $y:\mathbb{R}\to {\mathbb{C}}^n$ is said to be $(\omega ,c)$-periodic if $y(t+\omega )=cy\left(t\right)$ for all $t\in \mathbb{R}$. Note that the space of all $(\omega ,c)$-periodic functions include those that are periodic (when $c=1$), anti-periodic (when $c=-1$), and unbounded $(\omega ,c)$-periodic functions (when $\left|c\right|\ne 1$). In addition, Bloch functions (or Bloch waves) are likewise $(\omega ,c)$-periodic.

It is well known that fixed-point theorems are frequently employed to obtain results on the existence of periodic solutions for various types of integral equations. As shown in [17,18], the topological transversality theorem (which is a variant of the nonlinear alternative of the Leray–Schauder theorem) and Krasnoselskii’s fixed-point theorem have been applied to prove the existence of periodic solutions for various types of integral equations. It is also worth mentioning that D. O’Regan, in [19] (Theorem 2.1), obtained a variation of a fixed-point theorem as an application of the nonlinear alternative of Leray–Schauder for condensing mappings. This theorem extends Krasnoselskii’s fixed-point theorem. Hence, inspired by this result and by the works [17,18], we obtain more general conditions, through the theory of regulated functions and the theory of Perron–Stieltjes integration, to guarantee the existence of a $(\omega ,c)$-periodic solution for the nonlinear functional integral Equation (1).

It is worth emphasizing that the Perron–Stieltjes integral framework plays a crucial role in Equation (1), as it allows one to impose significantly weaker regularity assumptions on the mapping $s\mapsto \mathcal{G}(s,y_s)$. In particular, this last function may exhibit highly oscillatory behavior, possess multiple discontinuities, and need not be of bounded variation.

Beyond the abstract theoretical framework, we illustrate the strength and relevance of our approach by means of concrete and significant applications. As a first application, we consider a recurrent neural network with time-varying coefficients and mixed delays, described by the following equation:

$${\displaystyle \frac{d}{dt}}\left(y\left(t\right)+d\left(t\right)u\left(y(t-{\tau }_1)\right)\right)=a\left(t\right)h\left(y(t-{\tau }_2)\right)+{\int }_{-\infty }^{t}k(t-s)q\left(y\left(s\right)\right)\mathrm{d}s+I\left(t\right),$$

where $y\left(t\right)={(y_1\left(t\right),y_2\left(t\right))}^T$ is the state vector of the neural network at time t, $d\left(t\right)={\left(d_{ij}\left(t\right)\right)}_{2\times 2}$ and $a\left(t\right)={\left(a_{ij}\left(t\right)\right)}_{2\times 2}$ are the interconnection matrices representing the weight coefficients of neurons, $u(·),$ $h(·)$, and $q(·)$ are activation functions, $k\left(t\right)=\mathrm{diag}(k_1\left(t\right),k_2\left(t\right))$ is the delay kernel, ${\tau }_1,{\tau }_2>0$ are fixed delays, and $I\left(t\right)$ is an external input.

As a second illustration, we study the following nonlinear Volterra–Stieltjes integral equation with infinite delay of the form

$${\displaystyle y\left(t\right)={\int }_{-\infty }^{t}k(s,y\left(t\right))\mathrm{d}g\left(s\right)+{\int }_0^t\left({\int }_{-\infty }^{s}a(s,\xi )\mu (s,y\left(\xi \right))\mathrm{d}\xi \right)\mathrm{d}g\left(s\right),}$$

where $a:\mathbb{R}\times \mathbb{R}\to \mathbb{R},$ $k:\mathbb{R}\times {\mathbb{C}}^n\to {\mathbb{C}}^n$, $g:\mathbb{R}\to \mathbb{R}$, and $\mu :\mathbb{R}\times {\mathbb{C}}^n\to {\mathbb{C}}^n$ are functions that satisfy certain technical conditions, which allow us to develop a qualitative theory.

2. Preliminaries

2.1. Regulated Functions and Auxiliary Results

Let ${\mathbb{C}}^n$ be the n-dimensional complex space with norm $\parallel ·\parallel ,$ $a,b\in \mathbb{R}$ be real numbers with $a<b$ and $\mathbb{C}$ be equipped with the usual norm $|·|$. A function $x:[a,b]\to {\mathbb{C}}^n$ is called regulated if the lateral limits

$${\displaystyle x\left(t^{-}\right):=\underset{s\to t^{-}}{lim}x\left(s\right)\mathrm{for}t\in (a,b]\mathrm{and}x\left(t^{+}\right):=\underset{s\to t^{+}}{lim}x\left(s\right)\mathrm{for}t\in [a,b)}$$

exist. The space of all regulated functions $x:[a,b]\to {\mathbb{C}}^n$ will be denoted by $G([a,b],{\mathbb{C}}^n).$ It is a known fact that $G([a,b],{\mathbb{C}}^n)$ endowed with the usual supremum norm

$${\parallel x\parallel }_{\infty }:=\underset{s\in [a,b]}{sup}\parallel x\left(s\right)\parallel$$

is a Banach space (see [20] (Theorem 3.6)).

On the other hand, $G(\mathbb{R},{\mathbb{C}}^n)$ denotes the space of all functions $x:\mathbb{R}\to {\mathbb{C}}^n$ such that the restriction ${x|}_{[a,b]}$ belongs to the space $G([a,b],{\mathbb{C}}^n)$ for all real numbers $a,b$ with $a<b$. The next result is a special case of [9] (Corollary 1.19).

Theorem 1.

The following conditions are equivalent:

(i) 

$\mathcal{A}\subset (G([a,b],{\mathbb{C}}^n),\parallel ·{\parallel }_{\infty })$ is relatively compact.

(ii) 

$\mathcal{A}$ is bounded and there exists a non-decreasing function $v:[a,b]\to \mathbb{R}$ such that

$$\parallel x\left(t_2\right)-x\left(t_1\right)\parallel ⩽v\left(t_2\right)-v\left(t_1\right)$$

for all $x\in \mathcal{A}$.

The following fixed-point theorem was established by O’Regan in [19] (Theorem 2.1).

Theorem 2.

Let U be a open set in a closed convex set C of a Banach space $(E,\parallel ·{\parallel }_E).$ Suppose that $0\in U,$ $T\left(U\right)$ is bounded and $T:\overline{U}\to E$ is given by $T:=T_1+T_2,$ where $T_1:\overline{U}\to E$ is a nonlinear contraction (i.e., there exists a continuous non-decreasing function $\gamma :[0,\infty )\to [0,\infty )$, with $\gamma \left(t\right)<t$ for $t>0$, such that $\parallel T_1\left(x\right)-T_1{\left(y\right)\parallel }_E⩽{\gamma (\parallel x-y\parallel }_E)$ for $x,y\in \overline{U}$), and $T_2:\overline{U}\to E$ is completely continuous. Then either

(i) 

T has a fixed point on $\overline{U}$;

(ii) 

There are points $y\in \partial U$ and $\lambda \in (0,1)$ with $\lambda y=T\left(y\right),$ where $\partial U$ denotes the boundary of U in $E.$

2.2. The Perron–Stieltjes Integral

In this section, we recall the definition of the Perron–Stieltjes integral. For further details, the reader is referred to [21,22].

A tagged division of interval $\left[a,b\right]\subset \mathbb{R}$ is a finite collection $D=\left({\tau }_i,[s_{i-1},s_i]\right)$, where $a=s_0⩽s_1⩽\dots ⩽s_{\left|D\right|}=b$ is a division of $\left[a,b\right]$ and ${\tau }_i\in \left[s_{i-1},s_i\right]$ for $i=1,2,\dots ,\left|D\right|$. Here, the symbol $\left|D\right|$ denotes the number of subintervals in which $[a,b]$ is divided.

A gauge on $\left[a,b\right]$ is any function $\delta :[a,b]\to (0,\infty ).$ Given a gauge $\delta$ on $\left[a,b\right]$, we say that a tagged division $D=\left({\tau }_i,[s_{i-1},s_i]\right)$ is $\delta$-fine if for all $i\in \left\{1,2,\dots ,\left|D\right|\right\}$, we get

$$\left[s_{i-1},s_i\right]\subset ({\tau }_i-\delta \left({\tau }_i\right),{\tau }_i+\delta \left({\tau }_i\right)).$$

Definition 1.

We say that a function $f:[a,b]\to {\mathbb{C}}^n$ is Perron–Stieltjes integrable on $[a,b]$ with respect to a function $g:[a,b]\to \mathbb{R}$ if there is $I\in {\mathbb{C}}^n$ such that for each $\epsilon >0,$ there is a gauge $\delta :[a,b]\to (0,\infty )$ such that

$$∥\sum _{i=1}^{\left|D\right|}f\left({\tau }_i\right)\left(g\left(s_i\right)-g\left(s_{i-1}\right)\right)-I∥<\epsilon$$

for all δ-fine tagged divisions of $[a,b].$ In this case, I is called the Perron–Stieltjes integral of f with respect to g over $[a,b]$ and it will be denoted by ${\int }_a^{b}f\left(s\right)\mathrm{d}g\left(s\right).$ In particular, if $g\left(s\right)=s$, one obtains the Perron integral.

Recall that the classical properties of linearity, additivity over adjacent intervals, and integrability on subintervals hold for the Perron–Stieltjes integral.

Remark 1.

The Perron–Stieltjes integral is equivalent to the integral of Kurzweil–Henstock–Stieltjes for functions taking values in a finite-dimensional space; see [9] (Theorem 3.1.3). As presented in [9] (Chapter 2), the Perron–Stieltjes integral can be extended to unbounded intervals.

3. Nonlinear Functional Integral Equation: Existence of $(\mathit{\omega },\mathit{c})$-Periodic Solutions

The purpose of this section is to establish sufficient conditions for the existence of $(\omega ,c)$-periodic solutions to the following nonlinear functional integral equation:

$$y\left(t\right)=\mathcal{Q}(t,y_t)+{\int }_0^t\mathcal{G}(s,y_s)\mathrm{d}g\left(s\right)$$

(2)

for all $t\in \mathbb{R},$ where $g:\mathbb{R}\to \mathbb{R}$, $\mathcal{G}:\mathbb{R}\times C(\mathbb{R},{\mathbb{C}}^n)\to {\mathbb{C}}^n$ and $\mathcal{Q}:\mathbb{R}\times C(\mathbb{R},{\mathbb{C}}^n)\to {\mathbb{C}}^n$ are functions with $\mathcal{Q}(0,·)=0.$ We recall that $C(\mathbb{R},{\mathbb{C}}^n)$ denotes the space of continuous functions $z:\mathbb{R}\to {\mathbb{C}}^n$.

The following definition can be found in [13] (Definition 2.1).

Definition 2.

Let $c\in \mathbb{C}\setminus \left\{0\right\}$ and $\omega >0.$ A function $h:\mathbb{R}\to {\mathbb{C}}^n$ is said to be $(\omega ,c)$-periodic if $h(t+\omega )=ch\left(t\right)$ for all $t\in \mathbb{R}$. In this case, ω is referred to as the c-period of h.

We recall that $G_{\omega c}:=(\mathbb{R},{\mathbb{C}}^n)\{h\in G(\mathbb{R},{\mathbb{C}}^n):h\mathrm{is}(\omega ,c)-\mathrm{periodic}\},$ which is a Banach space endowed with the norm

$${\parallel x\parallel }_{\omega c}:=\underset{s\in [0,\omega ]}{sup}\parallel x\left(s\right)\parallel .$$

Likewise, $C_{\omega c}(\mathbb{R},{\mathbb{C}}^n)$ denotes the space of continuous $(\omega ,c)$-periodic functions.

To establish the existence of $(\omega ,c)$-periodic solutions to Equation (2), we define, for $\rho >0$,

$${\mathcal{C}}_{\rho }^0:=\{x\in C_{\omega c}(\mathbb{R},{\mathbb{C}}^n){:\parallel x\parallel }_{\omega c}⩽\rho \mathrm{and}x\left(0\right)=0\}.$$

(3)

Notice that ${\mathcal{C}}_{\rho }^0$ is a nonempty closed convex subset of $G_{\omega c}(\mathbb{R},{\mathbb{C}}^n).$ We assume the following general conditions:

(H1)

g is a non-decreasing function. Moreover, there exist $\beta \in \mathbb{R},$ $c\in \mathbb{C}\setminus \left\{0\right\}$ and $\omega >0$ such that

$$g(t+\omega )=\beta +g\left(t\right),\mathcal{G}(t+\omega ,c\phi )=c\mathcal{G}(t,\phi ),\mathcal{Q}(t+\omega ,c\phi )=c\mathcal{Q}(t,\phi ),$$

for all $t\in \mathbb{R}$ and $\phi \in C(\mathbb{R},{\mathbb{C}}^n).$

(H2)

The Perron–Stieltjes integral ${\int }_{{\tau }_1}^{{\tau }_2}\mathcal{G}(s,y_s)\mathrm{d}g\left(s\right)$ exists for all $y\in C_{\omega c}(\mathbb{R},{\mathbb{C}}^n)$ with ${\parallel y\parallel }_{\omega c}⩽\rho ,$ ${\tau }_1,{\tau }_2\in \mathbb{R}.$

(H3)

There exists a locally Perron–Stieltjes integrable function $M:\mathbb{R}\to \mathbb{R}$ with respect to g such that

$$∥\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}\mathcal{G}(s,x_s)\mathrm{d}g\left(s\right)∥⩽\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}M\left(s\right)\mathrm{d}g\left(s\right)$$

and

$$∥\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}[\mathcal{G}(s,x_s)-\mathcal{G}(s,z_s)]\mathrm{d}g\left(s\right)∥⩽\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}M\left(s\right){∥x-z∥}_{\omega c}\mathrm{d}g\left(s\right)$$

for all $x,z\in C_{\omega c}(\mathbb{R},{\mathbb{C}}^n)$ with ${\parallel x\parallel ,\parallel z\parallel }_{\omega c}⩽\rho$ and $0⩽{\tau }_1⩽{\tau }_2⩽\omega .$

(H4)

$t\mapsto \mathcal{Q}(t,y_t)$ is regulated for $y\in C_{\omega c}(\mathbb{R},{\mathbb{C}}^n),$ and there is $0<\eta <1$ and a non-decreasing function $\ell :[0,\infty )\to [0,\infty ),$ $\ell \left(t\right)⩽\eta t$ for $t⩾0$ such that

$$\parallel \mathcal{Q}(t,x_t)-\mathcal{Q}(t,z_t){\parallel ⩽\ell (\parallel x-z\parallel }_{\omega c})$$

for all $t\in [0,\omega ]$ and $x,z\in C_{\omega c}(\mathbb{R},{\mathbb{C}}^n)$.

The following theorem constitutes the main result of this work.

Theorem 3.

Suppose that conditions (H1)–(H4) hold. If there exists $\rho >0$ such that

$$\ell \left(\rho \right)+\underset{t\in [0,\omega ]}{sup}\parallel \mathcal{Q}(t,0)\parallel +{\int }_0^{\omega }M\left(s\right)\mathrm{d}g\left(s\right)<\rho ,$$

then the nonlinear functional integral Equation (2) admits at least one $(\omega ,c)$-periodic solution $y\in {\mathcal{C}}_{\rho }^0$. Here, ℓ is given by (H4) and M by (H3).

Proof. 

Let $\rho >0$ and assume that

$$\ell \left(\rho \right)+\underset{t\in [0,\omega ]}{sup}\parallel \mathcal{Q}(t,0)\parallel +{\int }_0^{\omega }M\left(s\right)\mathrm{d}g\left(s\right)<\rho .$$

(4)

Define the mapping $T:{\mathcal{C}}_{\rho }^0\to G_{\omega c}(\mathbb{R},{\mathbb{C}}^n)$ by $T\left(y\right):=T_1\left(y\right)+T_2\left(y\right),$ where $T_1,T_2:{\mathcal{C}}_{\rho }^0\to G_{\omega c}(\mathbb{R},{\mathbb{C}}^n)$ are defined by

$$T_1\left(y\right)\left(t\right):=\mathcal{Q}(t,y_t),t\in \mathbb{R},$$

and

$$T_2\left(y\right)\left(t\right):={\int }_{k\omega }^t\mathcal{G}(s,x_s)\mathrm{d}g\left(s\right),t\in (k\omega ,(k+1)\omega ],k\in \mathbb{Z},$$

where ${\mathcal{C}}_{\rho }^0$ is given by (3).

T is well-defined. In fact, let $y\in {\mathcal{C}}_{\rho }^0.$ According to condition (H4) and [9] (Corollary 2.14), we have $T\left(y\right)\in G(\mathbb{R},{\mathbb{C}}^n).$ On the other hand, let $t\in \mathbb{R}$ and $k\in \mathbb{Z}$ such that $k\omega <t⩽(k+1)\omega .$ Then, by condition (H1) and properties of the integral, we have

$$\begin{array}{cc}\hfill T_2\left(y\right)(t+\omega )& ={\int }_{(k+1)\omega }^{t+\omega }\mathcal{G}(s,y_s)\mathrm{d}g\left(s\right)\hfill \\ \hfill & ={\int }_{k\omega }^t\mathcal{G}(s+\omega ,y_{s+\omega })\mathrm{d}g(s+\omega )\hfill \\ \hfill & ={\int }_{k\omega }^t\mathcal{G}(s+\omega ,c{y}_s)\mathrm{d}g\left(s\right)\hfill \\ \hfill & =c{T}_2\left(y\right)\left(t\right)\hfill \end{array}$$

and

$$T_1\left(y\right)(t+\omega )=\mathcal{Q}(t+\omega ,y_{t+\omega })=\mathcal{Q}(t+\omega ,c{y}_t)=c{T}_1\left(y\right)\left(t\right).$$

Thus, T is well-defined.

Step 1. $T\left({\mathcal{C}}_{\rho }^0\right)$ is bounded. Indeed, let $y\in {\mathcal{C}}_{\rho }^0.$ Then,

$$\begin{array}{cc}\hfill \parallel T_1{\left(y\right)\parallel }_{\omega c}& =\underset{t\in (0,\omega ]}{sup}\parallel T_1\left(y\right)\left(t\right)\parallel \hfill \\ \hfill & =\underset{t\in (0,\omega ]}{sup}\parallel \mathcal{Q}(t,y_t)-\mathcal{Q}(t,0)+\mathcal{Q}(t,0)\parallel \hfill \\ \hfill & ⩽\ell \left(\rho \right)+\mu ,\hfill \end{array}$$

(5)

where $\mu :={sup}_{t\in (0,\omega ]}\parallel \mathcal{Q}(t,0)\parallel$. Notice that, as $t\mapsto \mathcal{Q}(t,0)$ is regulated, ${sup}_{t\in (0,\omega ]}\parallel \mathcal{Q}(t,0)\parallel$ $<\infty$. On the other hand, by (H3), the property of the supremum and the fact that $T_2\left(y\right)\left(\omega \right)=c{T}_2\left(y\right)\left(0\right),$ we obtain

$$\begin{array}{cc}\hfill \parallel T_2{\left(y\right)\parallel }_{\omega c}& =\underset{t\in [0,\omega ]}{sup}\parallel T_2\left(y\right)\left(t\right)\parallel \hfill \\ \hfill & =max\left\{\underset{t\in (0,\omega ]}{sup}\parallel T_2\left(y\right)\left(t\right)\parallel ,\parallel c^{-1}{T}_2\left(y\right)\left(\omega \right)\parallel \right\}\hfill \\ \hfill & =max\left\{\underset{t\in (0,\omega ]}{sup}∥{\int }_0^t\mathcal{G}(s,x_s)\mathrm{d}g\left(s\right)∥,∥c^{-1}{\int }_0^{\omega }\mathcal{G}(s,x_s)\mathrm{d}g\left(s\right)∥\right\}\hfill \\ \hfill & ⩽max\left\{{\int }_0^{\omega }M\left(s\right)\mathrm{d}g\left(s\right),\left|c^{-1}\right|{\int }_0^{\omega }M\left(s\right)\mathrm{d}g\left(s\right)\right\}.\hfill \end{array}$$

(6)

Hence, $T\left({\mathcal{C}}_{\rho }^0\right)$ is bounded on $(G_{\omega c}(\mathbb{R},{\mathbb{C}}^n),\parallel ·{\parallel }_{\omega c})$.

Step 2. $T_1$ is a nonlinear contraction. Let $x,y\in {\mathcal{C}}_{\rho }^0$ and $t\in [0,\omega ].$ Then, by (H4) we obtain

$$\parallel T_1\left(y\right)\left(t\right)-T_1\left(z\right)\left(t\right)\parallel =\parallel \mathcal{Q}(t,y_t)-\mathcal{Q}(t,z_t){\parallel ⩽\ell (\parallel y-z\parallel }_{\omega c}{)⩽\eta \parallel y-z\parallel }_{\omega c}=\gamma (\parallel y-z{\parallel }_{\omega c}),$$

where $\gamma \left(\xi \right):=\eta \xi$ for $\xi \in [0,\infty ).$ Note that $\gamma$ is a continuous non-decreasing function with $\gamma \left(t\right)<t$ for $t>0,$ since by hypotheses $0<\eta <1,$ which gives the desired result.

Step 3. $T_2$ is completely continuous. In fact, note that by (5) and (6), the set $\{T_2\left(y\right):y\in {\mathcal{C}}_{\rho }^0\}$ is uniformly bounded. On the other hand, for every $y\in {\mathcal{C}}_{\rho }^0$, we have

$$\parallel T_2\left(y\right)\left(\tau \right)-T_2\left(y\right)\left(t\right)\parallel =∥{\int }_t^{\tau }\mathcal{G}(s,x_s)\mathrm{d}g\left(s\right)∥⩽{\int }_t^{\tau }M\left(s\right)\mathrm{d}g\left(s\right)=h\left(\tau \right)-h\left(t\right),$$

where $h\left(\xi \right):={\int }_0^{\xi }M\left(s\right)\mathrm{d}g\left(s\right)$ for all $\xi \in [0,\omega ].$ Moreover, by (H3), h is non-decreasing. Hence, according to Theorem 1, the set $T_2\left({\mathcal{C}}_{\rho }^0\right)$ is relatively compact in $(G_{\omega c}(\mathbb{R},{\mathbb{C}}^n),{\parallel ·\parallel }_{\omega c}).$ Notice that the compactness is reduced to $[0,\omega ]$ via $(\omega ,c)$-periodicity. Meanwhile, for $x,y\in {\mathcal{C}}_{\rho }^0$ we have that (H3) implies

$$\begin{array}{cc}\hfill & \parallel T_2\left(x\right)-T_2{\left(y\right)\parallel }_{\omega c}\hfill \\ \hfill & =max\left\{\underset{t\in (0,\omega ]}{sup}\parallel T_2\left(x\right)\left(t\right)-T_2\left(y\right)\left(t\right)\parallel ,\parallel c^{-1}(T_2\left(x\right)\left(\omega \right)-T_2\left(y\right)\left(\omega \right))\parallel \right\}\hfill \\ \hfill & =max\left\{\underset{t\in (0,\omega ]}{sup}∥{\int }_0^t(\mathcal{G}(s,x_s)-\mathcal{G}(s,y_s))\mathrm{d}g\left(s\right)∥,∥c^{-1}{\int }_0^{\omega }(\mathcal{G}(s,x_s)-\mathcal{G}(s,y_s))\mathrm{d}g\left(s\right)∥\right\}\hfill \\ \hfill & ⩽max\left\{{\int }_0^{\omega }M\left(s\right)\mathrm{d}g\left(s\right),\left|c^{-1}\right|{\int }_0^{\omega }M\left(s\right)\mathrm{d}g\left(s\right)\right\}{\parallel x-y\parallel }_{\omega c},\hfill \end{array}$$

which implies that $T_2$ is continuous on ${\mathcal{C}}_{\rho }^0.$ Thus, $T_2$ is completely continuous.

From the above arguments and Theorem 2, it follows that either T has a fixed point in ${\mathcal{C}}_{\rho }^0$ or there exist $y\in \partial {\mathcal{C}}_{\rho }^0$ and $\lambda \in (0,1)$ such that $\lambda y=T\left(y\right).$

Assume that there exist $y\in \partial {\mathcal{C}}_{\rho }^0$ and $\lambda \in (0,1)$ such that $\lambda y=T\left(y\right).$ Then, ${\parallel y\parallel }_{\omega c}=\rho$ and $y\left(0\right)=0.$ Let $\tau \in [0,\omega ]$ such that $y\left(\tau \right)=\rho .$ We assume without loss of generality that $\tau \in (0,\omega ]$. Hence, by (H4) and (4), we have that

$$\begin{array}{cc}\hfill \rho =\parallel y\left(\tau \right)\parallel & =\lambda ∥\mathcal{Q}(\tau ,y_{\tau })+{\int }_0^{\tau }\mathcal{G}(s,y_s)\mathrm{d}g\left(s\right)∥\hfill \\ \hfill & ⩽\parallel \mathcal{Q}(\tau ,y_{\tau })-\mathcal{Q}(\tau ,0)\parallel +\parallel \mathcal{Q}(\tau ,0)\parallel +∥{\int }_0^{\tau }\mathcal{G}(s,y_s)\mathrm{d}g\left(s\right)∥\hfill \\ \hfill & ⩽\ell \left(\rho \right)+\underset{\tau \in [0,\omega ]}{sup}\parallel \mathcal{Q}(\tau ,0)\parallel +{\int }_0^{\omega }M\left(s\right)\mathrm{d}g\left(s\right)\hfill \\ \hfill & <\rho ,\hfill \end{array}$$

which is a contradiction. It follows that T has a fixed point $\mathfrak{z}\in {\mathcal{C}}_{\rho }^0$. Thus,

$$\mathfrak{z}\left(t\right)=\mathcal{Q}(t,{\mathfrak{z}}_t)+{\int }_{k\omega }^t\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right),$$

(7)

for all $t\in (k\omega ,(k+1\left)\omega \right]$ and $k\in \mathbb{Z}.$

We claim that ${\int }_{k\omega }^t\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right)={\int }_0^t\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right).$ In fact, since $\mathfrak{z}\in {\mathcal{C}}_{\rho }^0,$ by (3), $\mathfrak{z}\left(0\right)=0$ and $\mathfrak{z}(t+\omega )=c\mathfrak{z}\left(t\right)$ for all $t\in \mathbb{R}.$ In particular, for $t=0,$ we have $\mathfrak{z}\left(\omega \right)=c\mathfrak{z}\left(0\right)$. Thus, by (H1) and the fact that $\mathcal{Q}(0,·)=0,$

$$\begin{array}{cc}\hfill 0=\mathfrak{z}\left(0\right)& =c^{-1}\mathfrak{z}\left(\omega \right)\hfill \\ \hfill & =c^{-1}\left(\mathcal{Q}(\omega ,{\mathfrak{z}}_{\omega })+{\int }_0^{\omega }\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right)\right)\hfill \\ \hfill & =c^{-1}\left(c\mathcal{Q}(0,{\mathfrak{z}}_0)+{\int }_0^{\omega }\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right)\right)\hfill \\ \hfill & =c^{-1}{\int }_0^{\omega }\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right),\hfill \end{array}$$

which implies that

$${\int }_0^{\omega }\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right)=0.$$

Using this fact, (H1) and properties of the integral, for $k\in \mathbb{Z},$ we have

$${\int }_{k\omega }^{(k+1)\omega }\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right)={\int }_0^{\omega }\mathcal{G}(s+k\omega ,{\mathfrak{z}}_{s+k\omega })\mathrm{d}g(s+k\omega )=c^k{\int }_0^{\omega }\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right)=0.$$

(8)

Now, let $t\in (k\omega ,(k+1\left)\omega \right]$ and $k\in \mathbb{Z}.$ Without loss of generality, let us assume that $k>0$. Then, by (8), we have

$${\int }_0^t\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right)=\sum _{j=0}^{k-1}{\int }_{j\omega }^{(j+1)\omega }\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right)+{\int }_{k\omega }^t\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right)={\int }_{k\omega }^t\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right),$$

obtaining the claim. Therefore, by (7), we obtain

$$\mathfrak{z}\left(t\right)=\mathcal{Q}(t,{\mathfrak{z}}_t)+{\int }_0^t\mathcal{G}(s,{\mathfrak{z}}_s)\mathrm{d}g\left(s\right),$$

for all $t\in \mathbb{R},$ which implies that $\mathfrak{z}$ is a $(\omega ,c)$-periodic solution of Equation (2). □

Remark 2.

Condition (4) is satisfied, for instance, when

$${\int }_0^{\omega }M\left(s\right)\mathrm{d}g\left(s\right)⩽K<\rho -\ell \left(\rho \right)-\underset{t\in [0,\omega ]}{sup}\parallel \mathcal{Q}(t,0)\parallel .$$

4. Applications

In this section, we present two applications of our main result.

4.1. Application to Recurrent Neural Networks with Time-Varying Coefficient and Mixed Delays

We consider the following recurrent neural networks with time-varying coefficients and mixed delays of type

$${\displaystyle \frac{d}{dt}}(y\left(t\right)+d\left(t\right)u\left(y(t-{\tau }_1)\right))=a\left(t\right)h\left(y(t-{\tau }_2)\right)+{\int }_{-\infty }^{t}k(t-s)q\left(y\left(s\right)\right)\mathrm{d}s+I\left(t\right),$$

(9)

with initial condition $y\left(0\right)=0,$ where $y\left(t\right)={(y_1\left(t\right),y_2\left(t\right))}^T$ is the neuron state of neural networks at time $t,$ $d\left(t\right)={\left(d_{ij}\left(t\right)\right)}_{2\times 2}$ and $a\left(t\right)={\left(a_{ij}\left(t\right)\right)}_{2\times 2}$ are the interconnection matrices representing the weight coefficients of neurons, with $d\left(0\right)=0,$ $u(·),h(·)$ and $q(·)$ are the neuron activation functions, $k\left(t\right)=\mathrm{diag}(k_1\left(t\right),k_2\left(t\right))$ is the delay kernel, ${\tau }_1,{\tau }_2>0$ represent positive delays and $I\left(t\right)$ is an external input. For further details on this model, see [23,24]. The integral form of (9) is given by

$$y\left(t\right)=-d\left(s\right)u\left(y(s-{\tau }_1)\right)+\underset{0}{\overset{t}{\int }}\left[a\left(s\right)h\left(y(s-{\tau }_2)\right)+\underset{-\infty }{\overset{s}{\int }}k(s-\tau )q\left(y\left(\tau \right)\right)d\tau +I\left(s\right)\right]\mathrm{d}s.$$

(10)

Let $\rho >0$ be fixed. We will assume the following conditions:

(N0)

$d,a,k\in G(\mathbb{R},M_{2\times 2}\left(\mathbb{R}\right))$ and ${sup}_{t\in [0,\omega ]}{\parallel d\left(t\right)\parallel }_0<\eta$ with $\eta \in (0,1),$ where ${\parallel ·\parallel }_0$ is the norm in $M_{2\times 2}\left(\mathbb{R}\right).$ Here, $M_{2\times 2}\left(\mathbb{R}\right)$ denotes the set of $2\times 2$ matrices over $\mathbb{R}$.

(N1)

$u,h,q\in C({\mathbb{R}}^2,{\mathbb{R}}^2)$. Also, there exist $c\in \mathbb{R}\setminus \left\{0\right\}$ and $\omega >0$ such that $d(t+\omega )=d\left(t\right),$ $a(t+\omega )=a\left(t\right),$ $k(t+\omega )=k\left(t\right),$ $I(t+\omega )=cI\left(t\right),$ $u\left(cv\right)=cu\left(v\right),$ $q\left(cv\right)=cq\left(v\right)$ and $h\left(cv\right)=ch\left(v\right)$ for all $t\in \mathbb{R}$ and $v\in {\mathbb{R}}^2.$

(N2)

For all $t\in \mathbb{R}$, $y\in C_{\omega c}(\mathbb{R},{\mathbb{R}}^2)$ with ${\parallel y\parallel }_{\omega c}⩽\rho$, the function $\tau \mapsto k(s-\tau )q\left(y\right(\tau \left)\right)$ is Perron integrable on $(-\infty ,s]$, and the function $s\mapsto {\int }_{-\infty }^{s}k(s-\tau )q\left(y\left(\tau \right)\right)d\tau$ is locally Perron integrable on $\mathbb{R}.$

(N3)

For each $y,z\in C_{\omega c}(\mathbb{R},{\mathbb{R}}^2)$ with ${\parallel y\parallel }_{\omega c},{\parallel z\parallel }_{\omega c}⩽\rho ,$

$$∥{\int }_{{\tau }_1}^{{\tau }_2}\left({\int }_{-\infty }^{s}k(s-\tau )q\left(y\left(\tau \right)\right)\mathrm{d}\tau \right)\mathrm{d}s∥⩽{\int }_{{\tau }_1}^{{\tau }_2}{\theta }_1\left(s\right){\parallel y\parallel }_{\omega c}\mathrm{d}s$$

and

$$∥{\int }_{{\tau }_1}^{{\tau }_2}\left({\int }_{-\infty }^{s}k(s-\tau )[q\left(y\left(\tau \right)\right)-q\left(z\left(\tau \right)\right)]\mathrm{d}\tau \right)\mathrm{d}s∥⩽{\int }_{{\tau }_1}^{{\tau }_2}{\theta }_2\left(s\right){\parallel y-z\parallel }_{\omega c}\mathrm{d}s,$$

where $0⩽{\tau }_1⩽{\tau }_2⩽\omega$ and ${\theta }_i:\mathbb{R}\to (0,+\infty ),$ $i=1,2,$ are locally Perron integrable on $\mathbb{R}.$

(N4)

For all $y,z\in C(\mathbb{R},{\mathbb{R}}^2)$ with ${\parallel y\parallel }_{\omega c}⩽\rho ,$ we have

$$∥{\int }_0^{t}a\left(s\right)h\left(y(s-{\tau }_2)\right)\mathrm{d}s∥⩽{\int }_0^t{r}_1\left(s\right){\parallel y\parallel }_{\omega c}\mathrm{d}s$$

and

$$∥{\int }_0^{t}a\left(s\right)(h\left(y(s-{\tau }_2)\right)-h\left(z(s-{\tau }_2)\right))\mathrm{d}s∥⩽{\int }_0^t{r}_2\left(s\right){\parallel y-z\parallel }_{\omega c}\mathrm{d}s,$$

where $0⩽t⩽\omega$ and $r_i:\mathbb{R}\to (0,+\infty ),$ $i=1,2,$ are locally Perron integrable on $\mathbb{R}.$

(N5)

For all $t\in [0,\omega ],$ $y,z\in C_{\omega c}(\mathbb{R},{\mathbb{R}}^2)$ with ${\parallel y\parallel }_{\omega c}⩽\rho ,$ we have

$$\parallel u\left(y(t-{\tau }_1)\right)-u\left(z(t-{\tau }_1)\right){\parallel ⩽\alpha \parallel y-z\parallel }_{\omega c},$$

where $0<\alpha <{\displaystyle \frac{1}{\eta }}$, with $\eta$ given in (N0).

(N6)

$\alpha \eta +{\int }_0^{\omega }({\theta }_1+r_1)\left(s\right)\mathrm{d}s<1$, where $\alpha$ is given in (N5), $\eta$ in (N0), ${\theta }_1$ in (N3) and $r_1$ in (N4).

Now, define $\mathcal{G}:\mathbb{R}\times C(\mathbb{R},{\mathbb{R}}^2)\to {\mathbb{R}}^2$ and $\mathcal{Q}:\mathbb{R}\times C(\mathbb{R},{\mathbb{R}}^2)\to {\mathbb{R}}^2$ by

$$\mathcal{Q}(t,\phi ):=-d\left(t\right)u\left(\phi (-{\tau }_1)\right),$$

$$\mathcal{G}(s,\phi ):=a\left(s\right)h\left(\phi (-{\tau }_2)\right)+{\int }_{-\infty }^{s}k(s-\tau )q\left(\phi (\tau -s)\right)d\tau +I\left(s\right)$$

and $g\left(s\right):=s$ for all $s\in \mathbb{R}$.

Note that $\mathcal{Q}(t,y_t)=-d\left(t\right)u\left(y(t-{\tau }_1)\right)$ and

$$\mathcal{G}(s,y_s)=a\left(s\right)h\left(y(s-{\tau }_2)\right)+{\int }_{-\infty }^{s}k(s-\tau )q\left(y\left(\tau \right)\right)d\tau +I\left(s\right),$$

and therefore, Equation (10) is in the form of (2).

Let us show that conditions (H1)–(H4) hold. In fact, according to condition (N1), we have

$$\mathcal{Q}(t+\omega ,c\phi )=-d(t+\omega )u\left(c\phi (-{\tau }_1)\right)=-cd\left(t\right)u\left(\phi (-{\tau }_1)\right)=c\mathcal{Q}(t,\phi )$$

and

$$\begin{array}{cc}\hfill & \mathcal{G}(t+\omega ,c\phi )\hfill \\ \hfill & =a(t+\omega )h\left(c\phi (-{\tau }_2)\right)+{\int }_{-\infty }^{t+\omega }k(t+\omega -\tau )q\left(c\phi (\tau -(t+\omega ))\right)d\tau +I(t+\omega )\hfill \\ \hfill & =ca\left(t\right)h\left(\phi (-{\tau }_2)\right)+{\int }_{-\infty }^{t}ck(t-\tau )q\left(\phi (\tau -t)\right)d\tau +cI\left(t\right)\hfill \\ \hfill & =c\mathcal{G}(t,\phi ),\hfill \end{array}$$

and noting that $g(t+\omega )=\omega +g\left(t\right)$ we obtain (H1). On the other hand, under assumptions (N2)–(N4), conditions (H2) and (H3) are satisfied with $M\left(s\right)=\rho ({\theta }_1\left(s\right)+r_1\left(s\right))+{\theta }_2\left(s\right)+r_2\left(s\right)$.

Let us check that condition (H4) is satisfied. Let $y,z\in C_{\omega c}(\mathbb{R},{\mathbb{R}}^2)$ and $t\in [0,\omega ].$ Notice that $t\mapsto \mathcal{Q}(t,y_t)$ is regulated, and by conditions (N0) and (N5),

$$\begin{array}{cc}\hfill \parallel \mathcal{Q}(t,y_t)-\mathcal{Q}(t,z_t)\parallel & =\parallel d\left(t\right)u\left(y(t-{\tau }_1)\right)-d\left(t\right)u\left(z(t-{\tau }_1)\right)\parallel \hfill \\ \hfill & ⩽{\alpha \parallel d\left(t\right)\parallel }_0{\parallel y-z\parallel }_{\omega c}\hfill \\ \hfill & ⩽\ell (\parallel y-z{\parallel }_{\omega c}),\hfill \end{array}$$

where $\ell \left(\xi \right):=\alpha \eta \xi .$ If, in addition, $\rho >{\displaystyle \frac{{sup}_{t\in [0,\omega ]}\parallel d\left(t\right)u\left(0\right)\parallel +{\int }_0^{\omega }({\theta }_2+r_2)\left(s\right)\mathrm{d}s}{1-\alpha \eta -{\int }_0^{\omega }({\theta }_1+r_1)\left(s\right)\mathrm{d}s}},$ then by (N6), condition (4) is satisfied, and therefore, according to Theorem 3, Equation (10) admits a $(\omega ,c)$-periodic solution in ${\mathcal{C}}_{\rho }^0.$

4.2. Nonlinear Volterra–Stieltjes Integral Equation with Infinite Delay

In this section, we consider the nonlinear Volterra–Stieltjes integral equation with infinite delay:

$${\displaystyle y\left(t\right)={\int }_{-\infty }^{t}k(s,y\left(t\right))\mathrm{d}g\left(s\right)+{\int }_0^t\left({\int }_{-\infty }^{s}a(s,\xi )\mu (s,y\left(\xi \right))\mathrm{d}\xi \right)\mathrm{d}g\left(s\right),}$$

(11)

where $a:\mathbb{R}\times \mathbb{R}\to \mathbb{R},$ $k:\mathbb{R}\times {\mathbb{C}}^n\to {\mathbb{C}}^n$, $g:\mathbb{R}\to \mathbb{R}$, and $\mu :\mathbb{R}\times {\mathbb{C}}^n\to {\mathbb{C}}^n$ are functions.

Let $\rho >0$. We will assume the following conditions:

(P1)

Assume that $g:\mathbb{R}\to \mathbb{R}$ is non-decreasing and that there exist $\beta \in \mathbb{R},$ $c\in \mathbb{C}\setminus \left\{0\right\}$ and $\omega >0$ such that $k(t+\omega ,cu)=ck(t,u),$ $\mu (t+\omega ,cu)=c\mu (t,u),$ $a(t+\omega ,v+\omega )=a(t,v),$ and $g(t+\omega )=\beta +g\left(t\right)$ for all $t,v\in \mathbb{R}$ and $u\in {\mathbb{C}}^n.$

(P2)

For all $t\in \mathbb{R}$, $\phi \in C_{\omega c}(\mathbb{R},{\mathbb{C}}^n)$ with ${\parallel \phi \parallel }_{\omega c}⩽\rho$, the function $s\mapsto k(s,\phi \left(0\right))$ is Perron–Stieltjes integrable with respect to g on $(-\infty ,t]$ and ${sup}_{t\in [0,\omega ]}\parallel {\int }_{-\infty }^{t}k(s,0)\mathrm{d}g\left(s\right)\parallel <\infty$. Moreover, $∥k(s,u)-k(s,w)∥⩽L_k\left(s\right)∥u-w∥$ for $s\in \mathbb{R}$ and $u,w\in {\mathbb{C}}^n,$ where $L_k:\mathbb{R}\to (0,+\infty )$ is Perron–Stieltjes integrable with respect to g on $(-\infty ,t]$.

(P3)

For each $y,z\in C_{\omega c}(\mathbb{R},{\mathbb{C}}^n)$ with ${\parallel y\parallel }_{\omega c},{\parallel z\parallel }_{\omega c}⩽\rho ,$ the function $\xi \mapsto a(s,\xi )\mu (s,y(\xi \left)\right)$ is Perron integrable on $(-\infty ,s]$, the function $s\mapsto {\int }_{-\infty }^{s}a(s,\xi )\mu (s,y\left(\xi \right))\mathrm{d}\xi$ is locally Perron–Stieltjes integrable with respect to g on $\mathbb{R},$

$$∥{\int }_{{\tau }_1}^{{\tau }_2}\left({\int }_{-\infty }^{s}a(s,\xi )\mu (s,y\left(\xi \right))\mathrm{d}\xi \right)\mathrm{d}g\left(s\right)∥⩽{\int }_{{\tau }_1}^{{\tau }_2}{\lambda }_1\left(s\right){\parallel y\parallel }_{\omega c}\mathrm{d}g\left(s\right)$$

and

$$∥{\int }_{{\tau }_1}^{{\tau }_2}\left({\int }_{-\infty }^{s}a(s,\xi )[\mu (s,y\left(\xi \right))-\mu (s,z\left(\xi \right))]\mathrm{d}\xi \right)\mathrm{d}g\left(s\right)∥⩽{\int }_{{\tau }_1}^{{\tau }_2}{\lambda }_2\left(s\right){\parallel y-z\parallel }_{\omega c}\mathrm{d}g\left(s\right)$$

for all $0⩽{\tau }_1⩽{\tau }_2⩽\omega ,$ where ${\lambda }_i:\mathbb{R}\to (0,+\infty ),$ $i=1,2,$ are locally Perron–Stieltjes integrable with respect to g on $\mathbb{R}.$

(P4)

${\int }_{-\infty }^{\omega }{L}_k\left(s\right)\mathrm{d}g\left(s\right)+{\int }_0^{\omega }{\lambda }_1\left(s\right)\mathrm{d}g\left(s\right)<1$, where $L_k$ is given in (P2) and ${\lambda }_1$ in (P3).

First, we need to rewrite the nonlinear Volterra–Stieltjes integral Equation (11) in the form of Equation (2). For this purpose, define $\mathcal{G}:\mathbb{R}\times C(\mathbb{R},{\mathbb{C}}^n)\to {\mathbb{C}}^n$ and $\mathcal{Q}:\mathbb{R}\times C(\mathbb{R},{\mathbb{C}}^n)\to {\mathbb{C}}^n$ by

$$\mathcal{Q}(\tau ,\phi ):={\int }_{-\infty }^{\tau }k(s,\phi \left(0\right))\mathrm{d}g\left(s\right)\mathrm{and}\mathcal{G}(\tau ,\phi ):={\int }_{-\infty }^{\tau }a(\tau ,\xi )\mu (\tau ,\phi (\xi -\tau ))\mathrm{d}\xi ,$$

for each $\tau \in \mathbb{R}$ and $\phi \in C(\mathbb{R},{\mathbb{C}}^n).$

Note that, by (P1) and properties of the integral, we have

$$\mathcal{Q}(t+\omega ,c\phi )={\int }_{-\infty }^{t+\omega }k(s,c\phi \left(0\right))\mathrm{d}g\left(s\right)={\int }_{-\infty }^{t}k(s+\omega ,c\phi )\mathrm{d}g(s+\omega )=c\mathcal{Q}(t,\phi )$$

and

$$\begin{array}{cc}\hfill \mathcal{G}(t+\omega ,c\phi )& ={\int }_{-\infty }^{t+\omega }a(t+\omega ,\xi )\mu (t+\omega ,c\phi (\xi -(t+\omega )))\mathrm{d}\xi \hfill \\ \hfill & ={\int }_{-\infty }^{t}a(t+\omega ,\xi +\omega )\mu (t+\omega ,c\phi (\xi +\omega -(t+\omega )))\mathrm{d}\xi \hfill \\ \hfill & =c\mathcal{G}(t,\phi )\hfill \end{array}$$

for all $t\in \mathbb{R}$ and $\phi \in C(\mathbb{R},{\mathbb{C}}^n)$. Moreover, by the hypotheses, g is non-decreasing and satisfies $g(t+\omega )=\beta +g\left(t\right)$. Therefore, condition (H1) holds.

On the other hand, condition (P2) implies that (H2) is satisfied.

Now, for $y,z\in C_{\omega c}(\mathbb{R},{\mathbb{C}}^n)$ with ${\parallel y\parallel }_{\omega c},{\parallel z\parallel }_{\omega c}⩽\rho ,$ and $0⩽{\tau }_1⩽{\tau }_2⩽\omega ,$ by (P3), we have

$$∥\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}\mathcal{G}(s,y_s)\mathrm{d}g\left(s\right)∥⩽\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}{\lambda }_1\left(s\right){\parallel y\parallel }_{\omega c}\mathrm{d}g\left(s\right)⩽\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}\rho {\lambda }_1\left(s\right)\mathrm{d}g\left(s\right),$$

and

$$∥\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}[\mathcal{G}(s,y_s)-\mathcal{G}(s,z_s)]\mathrm{d}g\left(s\right)∥⩽\underset{{\tau }_1}{\overset{{\tau }_2}{\int }}{\lambda }_2\left(s\right){\parallel y-z\parallel }_{\omega c}\mathrm{d}g\left(s\right).$$

Therefore, condition (H3) is also true with $M\left(s\right):=\rho {\lambda }_1\left(s\right)+{\lambda }_2\left(s\right)$.

Finally, for all $t\in [0,\omega ],$ by condition (P2), we have

$$\begin{array}{cc}\hfill \parallel \mathcal{Q}(t,y_t)-\mathcal{Q}(t,z_t)\parallel & =∥{\int }_{-\infty }^t(k(s,y\left(t\right))-k(s,z\left(t\right)))\mathrm{d}g\left(s\right)∥\hfill \\ \hfill & ⩽{\int }_{-\infty }^t{L}_k\left(s\right){\parallel y-z\parallel }_{\omega c}\mathrm{d}g\left(s\right)\hfill \\ \hfill & ⩽\ell (\parallel y-z{\parallel }_{\omega c}),\hfill \end{array}$$

where $\ell \left(\xi \right):=\eta \xi$ and $\eta :={\int }_{-\infty }^{\omega }{L}_k\left(s\right)\mathrm{d}g\left(s\right),$ which gives (H4).

Thus, by Theorem 3 it follows that Equation (11) admits a $(\omega ,c)$-periodic solution in ${\mathcal{C}}_{\rho }^0$ for $\rho >{\displaystyle \frac{{sup}_{t\in [0,\omega ]}\parallel {\int }_{-\infty }^{t}k(s,0)\mathrm{d}g\left(s\right)\parallel +{\int }_0^{\omega }{\lambda }_2\left(s\right)\mathrm{d}g\left(s\right)}{1-{\int }_{-\infty }^{\omega }{L}_k\left(s\right)\mathrm{d}g\left(s\right)-{\int }_0^{\omega }{\lambda }_1\left(s\right)\mathrm{d}g\left(s\right)}}$. Notice that by (P4), the condition (4) is satisfied.