Analytical Contribution to a Cubic Functional Integral Equation with Feedback Control on the Real Half Axis

Abstract

Synthetic biology involves trying to create new approaches using design-based approaches. A controller is a biological system intended to regulate the performance of other biological processes. The design of such controllers can be based on the results of control theory, including strategies. Integrated feedback control is central to regulation, sensory adaptation, and long-term effects. In this work, we present a study of a cubic functional integral equation with a general and new constraint that may help in investigating some real problems. We discuss the existence of solutions for an equation that involves a control variable in the class of bounded continuous functions $\mathcal{BC}\left({\mathbb{R}}_{+}\right),$ by applying the technique of measure of noncompactness on ${\mathbb{R}}_{+}$. Furthermore, we establish sufficient conditions for the continuous dependence of the state function on the control variable. Finally, some remarks and discussion are presented to demonstrate our results.

1. Introduction and Background

The measure of noncompactness is essential in fixed point theory (see [1,2]). It can be utilized to demonstrate the existence results related to integral equations of numerous types; for example, we refer to [3,4]. The notion of a measure of noncompactness (MNC) was introduced by Kuratowski [5] in 1930. Darbo’s fixed point theorem [6], which guarantees the existence of a fixed point, is an important application of this measure because it generalizes both the Schauder fixed point and the Banach contraction principle connected with a measure of noncompactness in the Banach space $\mathcal{BC}\left({\mathbb{R}}_{+}\right)$ (of all bounded and continuous functions on ${\mathbb{R}}_{+}$), has been successfully used by J. Banaś (see [7,8,9,10,11,12,13]) to obtain the existence of asymptotically stable solutions of some integral and quadratic integral equations (see [9,10]).

Nonetheless, in the more real problems, biological systems or ecosystems are always perturbed via unpredictable forces. These perturbations usually change the system parameters. In control theory, these perturbation functions can be viewed as control variables. Integrated feedback control is central to regulation, sensory adaptation, and long-term effects Robustness [14,15,16,17].

In [14], Chen obtained some averaged conditions for the permanence and global attractivity of a nonautonomous Lotka–Volterra system with feedback controls by creating an appropriate Lyapunov function (Lyapunov functional).

In [15], Nasertayoob proved the existence, asymptotic stability, and global attractivity of a class of nonlinear functional-integral equations with feedback control, using Darbo’s fixed point theorem [6] associated with a measure of noncompactness. Furthermore, the existence of a positive periodic solution for a nonlinear neutral delay population system with feedback control is examined under certain conditions [16]. The proof depends on the fixed-point theorem of strict-set-contraction operators [16]. El-Sayed et al. [18] were concerned with a functional integral equation involving a control parameter function that satisfies a constraint functional equation. Further existence results were obtained in [19], where they studied a nonlinear functional integral equation that was constrained by a functional equation with a parameter.

Recently, cubic integral equations have gained much attention. Many authors who are concerned with studying quadratic integral equations have extended their results to some special cubic integral equations on a bounded interval, for example, [20,21,22,23].

The aim of this article is to establish existence results for solutions to the following cubic functional integral equation, which is more general than those studied in [20,21,22,23].

$$x\left(\mathfrak{r}\right)=f_1(\mathfrak{r},x\left({\varphi }_1\left(\mathfrak{r}\right)\right)).g_1\left(\mathfrak{r},u\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x\left(s\right))ds\right),\mathfrak{r}\in {\mathbb{R}}_{+}$$

(1)

with a feedback control given by

$$u\left(\mathfrak{r}\right)=f_2(\mathfrak{r},u\left({\varphi }_2\left(\mathfrak{r}\right)\right)).g_2\left(\mathfrak{r},x\left({\varphi }_2\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_2\left(\mathfrak{r}\right)}{h}_2(\mathfrak{r},s,u\left(s\right))ds\right),\mathfrak{r}\in {\mathbb{R}}_{+}.$$

(2)

for $\mathfrak{r},s\in {\mathbb{R}}_{+},$ we prove that problems (1) and (2) have solutions that belong to $\mathcal{BC}\left({\mathbb{R}}_{+}\right)\times \mathcal{BC}\left({\mathbb{R}}_{+}\right).$ Our proof depends on a suitable combination of the technique of measures of noncompactness and Darbo’s fixed point principle on the real half-axis. It is the first attempt to investigate a cubic functional integral equation restricted to a control variable u on the real half-axis. Moreover, we establish some sufficient conditions for the continuous dependence of the state variable x on the control variable $u.$ Finally, two illustrated examples and some particular cases are presented.

Definition 1.

By a solution of the cubic functional integral Equation (1), we mean a state function $\mu \in \mathcal{BC}\left({\mathbb{R}}_{+}\right)$, which satisfies (1) restricted to a control variable $\nu \in \mathcal{BC}\left({\mathbb{R}}_{+}\right)$ given by (2). The solvability of the cubic functional integral Equation (1) with the feedback control (2) is equivalent to finding a pair $(\mu ,\nu )\in \mathcal{BC}\left({\mathbb{R}}_{+}\right)\times \mathcal{BC}\left({\mathbb{R}}_{+}\right)$ that satisfies the system (1) and (2).

The notations mentioned below will be necessary for our work.

For $\mu \in X\subseteq \mathcal{BC}\left({\mathbb{R}}_{+}\right)$ and $ϵ\ge 0$ denote by $w^{\mathcal{T}}(\mu ,ϵ)$, $\mathcal{T}>0$, the modulus of continuity of the function $\mu$ on the interval $[0,\mathcal{T}]$, i.e.,

$${\omega }^{\mathcal{T}}(\mu ,ϵ)=sup[\mid \mu \left(\mathfrak{r}\right)-\mu \left(s\right)\mid :\mathfrak{r},s\in [0,\mathcal{T}],\mid \mathfrak{r}-s\mid \le ϵ],$$

$$w^{\mathcal{T}}(X,ϵ)=sup[w^{\mathcal{T}}(\mu ,ϵ):\mu \in X]$$

and

$$w_0^{\mathcal{T}}\left(X\right)=\underset{ϵ\to 0}{lim}{w}^{\mathcal{T}}(X,ϵ),w_0\left(X\right)=\underset{\mathcal{T}\to \infty }{lim}{w}_0^{\mathcal{T}}\left(X\right).$$

Additionally,

$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}X\left(\mathfrak{r}\right)=sup\{\mid \mu \left(\mathfrak{r}\right)-\nu \left(\mathfrak{r}\right)\mid ,\mu ,\nu \in X\},\alpha \left(X\right)=\underset{\mathfrak{r}\to \infty }{lim}sup\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}X\left(\mathfrak{r}\right)$$

and the measure of noncompactness on $\mathcal{BC}\left({\mathbb{R}}_{+}\right)$ is given by [1]

$$\mu \left(X\right)=w_0\left(X\right)+\alpha \left(X\right)=w_0\left(X\right)+\underset{\mathfrak{r}\to \infty }{lim}sup\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}X\left(\mathfrak{r}\right).$$

Theorem 1

([6]). Let Q be a nonempty bounded closed convex subset of the space E and let $F:Q\to Q$ be a continuous operator such that $\mu \left(FX\right)\le k\mu \left(X\right)$ for any nonempty subset X of Q, where $k\in [0,1)$ is a constant. Then, F has a fixed point in the set Q.

Now, let $E=BC\left({\mathbb{R}}_{+}\right)\times BC\left({\mathbb{R}}_{+}\right),X,Y\subset BC\left({\mathbb{R}}_{+}\right)$ and

$$U=\{u\in U:u=(x,y),x\in X,y\in Y\}=X\times Y.$$

Then, we can introduce the following:

$${\omega }^{\mathcal{T}}(x,ϵ)=sup\{\mid x\left(t\right)-x\left(s\right)\mid :\mathfrak{r},s\in J=[0,\mathcal{T}],\mid \mathfrak{r}-s\mid \le ϵ\},$$

$${\omega }^T(y,ϵ)=sup\{\mid y\left(\mathfrak{r}\right)-y\left(s\right)\mid :\mathfrak{r},s\in J=[0,\mathcal{T}],\mid t-s\mid \le ϵ\}$$

and

$${\omega }^{\mathcal{T}}(u,ϵ)=max\{{\omega }^{\mathcal{T}}(x,ϵ),{\omega }^{\mathcal{T}}(y,ϵ)\}.$$

Then,

$${\omega }^{\mathcal{T}}(U,ϵ)=sup\{{\omega }^{\mathcal{T}}(u,ϵ):u\in U\},$$

$${\omega }_0^{\mathcal{T}}\left(U\right)=\underset{ϵ\to 0}{lim}{\omega }^{\mathcal{T}}(U,ϵ),w_0\left(U\right)=\underset{\mathcal{T}\to \infty }{lim}{w}_0^{\mathcal{T}}\left(U\right).$$

Additionally,

$$\omega \left(F\right(U\left)\right)=\omega \left(F\right(X\times Y\left)\right)\le max\left\{\omega \right(X),\omega (Y\left)\right\},$$

$$diam\left(F\right(U\left)\right)=diam\left(F\right(X\times Y\left)\right)\le max\left\{diam\right(X),diam(Y\left)\right\},$$

$$\underset{\mathfrak{r}\to \infty }{lim}supdiam\left(F\left(U\right)\right)\le max\{\underset{\mathfrak{r}\to \infty }{lim}supdiam\left(X\right),\underset{\mathfrak{r}\to \infty }{lim}supdiam\left(Y\right)\}.$$

and

$$\mu \left(U\right)={\omega }_0\left(U\right)+\alpha \left(U\right)={\omega }_0\left(U\right)+\underset{\mathfrak{r}\to \infty }{lim}supdiamU\left(\mathfrak{r}\right).$$

2. Main Results

2.1. Existence of Solutions

Let $B_r$ be the ball defined by

$$B_r=\{v=(x,u)\in \mathcal{BC}\left({\mathbb{R}}_{+}\right)\times \mathcal{BC}\left({\mathbb{R}}_{+}\right):\parallel x\parallel \le r_2,\parallel u\parallel \le r_1,\parallel (x,u)\parallel \le max\{r_1,r_2\}=r\},$$

where r is a positive solution of the equation

$$\eta lb{r}^3+(a\eta +bN)l{r}^2+(alN-1)r+NM=0.$$

(3)

Due to consideration of the cubic functional integral Equation (1) with the feedback control (2). Based on the following hypothesis:

(i)

The functions ${\varphi }_i:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, (for $i=1,2$) are continuous, and ${\varphi }_i\to \infty$ as $\mathfrak{r}\to \infty$.

(ii)

The functions $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ are continuous non-decreasing and $\psi \left(\mathfrak{r}\right)\le t.$

(iii)

$g_i:{\mathbb{R}}_{+}\times \mathbb{R},i=1,2$ are continuous functions and there exists a positive constant $l_i$ such that

$$\mid g_i(\mathfrak{r},\mu )-g_i(\mathfrak{r},\nu )\mid \le l_i\mid \mu -\nu \mid$$

for each $\mathfrak{r}\in {\mathbb{R}}_{+}$ and for all $\mu ,\nu \in \mathbb{R}$. Moreover, the function $\mathfrak{r}\to g_i(\mathfrak{r},0)\in \mathcal{BC}\left({\mathbb{R}}_{+}\right)$ and we have

$$\mid g_i(\mathfrak{r},u)\mid \le l_i\mid u-v\mid +M_i,\mathrm{where}{M}_i=\underset{\mathfrak{r}\in I}{sup}\mid \{g_i(\mathfrak{r},0)\mid :\mathfrak{r}\in {\mathbb{R}}_{+}\}<\infty .$$

(iv)

$f_i:{\mathbb{R}}_{+}\times \mathbb{R}\to \mathbb{R},i=1,2$ are continuous functions and there exists a positive constant ${\eta }_i$ such that

$$\mid f_i(\mathfrak{r},u)-f_i(\mathfrak{r},v)\mid \le {\eta }_i\mid u-v\mid$$

for each $\mathfrak{r}\in {\mathbb{R}}_{+}$ and for all $u,v\in \mathbb{R}$. Moreover, the function $\mathfrak{r}\to f_i(\mathfrak{r},0)$$\in \mathcal{BC}\left({\mathbb{R}}_{+}\right)$ and we have

$$\mid f_i(\mathfrak{r},u)\mid \le {\eta }_i\mid u\mid +N_i,\mathrm{where}{N}_i=\underset{\mathfrak{r}\in I}{sup}\mid \{f_i(\mathfrak{r},0)\mid :\mathfrak{r}\in {\mathbb{R}}_{+}\}<\infty ,$$

(v)

The function $h_i:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times \mathbb{R},i=1,2$ are a Carathéodory function, which are measurable in $\mathfrak{r},s\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+},$$\forall u,\in \mathbb{R}$ and continuous in $u\in \mathbb{R},\forall \mathfrak{r},s\in {\mathbb{R}}_{+},$ and there exists measurable and bounded functions $a_i,b_i:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to \mathbb{R}$ such that

$$\mid h_i(\mathfrak{r},s,u)\mid \le a_i(\mathfrak{r},s)+b_i(\mathfrak{r},s)\mid u\mid ,\mathfrak{r}\in J$$

and

$$\underset{\mathfrak{r}\in R_{+}}{sup}{\int }_0^{\mathfrak{r}}{a}_i(\mathfrak{r},s)ds=a_i,\underset{\mathfrak{r}\to \infty }{lim}{\int }_0^{\mathfrak{r}}{a}_i(\mathfrak{r},s)ds=0,$$

$$\underset{\mathfrak{r}\in R_{+}}{sup}{\int }_0^{\mathfrak{r}}{b}_i(\mathfrak{r},s)ds=b_i.\underset{\mathfrak{r}\to \infty }{lim}{\int }_0^{\mathfrak{r}}{b}_i(\mathfrak{r},s)ds=0.$$

Without losing generality, taking $l=max\{l_1,l_2\},M=max\{M_1,M_2\},\eta =max\{{\eta }_1,{\eta }_2\},a=max\{a_1,a_2\}$ and $b=max\{b_1,b_2\mid \}.$

(vi)

Now a positive constant c exists such that

$$c=\eta \left[M+lr[a+br]\right]+l[N+\eta ][a+b]<1.$$

Theorem 2.

Assume that assumptions (i)–(vi) hold. Then, the cubic functional integral Equation (1) with a feedback control (2) has a solution in the space $\mathcal{BC}\left({\mathbb{R}}_{+}\right).$

Proof. 

Define an operator $\mathcal{F}$ by

$$\mathcal{F}(x,u)\left(\mathfrak{r}\right)=({\mathcal{F}}_{1}x\left(\mathfrak{r}\right),{\mathcal{F}}_{2}u\left(\mathfrak{r}\right)),$$

where ${\mathcal{F}}_1,{\mathcal{F}}_2$ are given by

$${\mathcal{F}}_{1}x\left(\mathfrak{r}\right)=f_1(t,x\left({\varphi }_1\left(\mathfrak{r}\right)\right)).g_1\left(\mathfrak{r},u\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x\left(s\right))ds\right),\mathfrak{r}\in R_{+},$$

and

$${\mathcal{F}}_{2}u\left(\mathfrak{r}\right)=f_2(\mathfrak{r},u\left({\varphi }_2\left(\mathfrak{r}\right)\right)).g_2\left(\mathfrak{r},x\left({\varphi }_2\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_2\left(\mathfrak{r}\right)}{h}_2(\mathfrak{r},s,u\left(s\right))ds\right),\mathfrak{r}\in R_{+},$$

We shall prove that for some $r>0,\mathcal{F}{B}_r\subset B_r,$ we have

$$\begin{array}{ccc}\hfill \mid {\mathcal{F}}_{1}x\left(\mathfrak{r}\right))\mid & \le & \mid f_1(\mathfrak{r},x\left({\varphi }_1\left(\mathfrak{r}\right)\right))g_1\left(t,u\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\varphi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x\left(s\right))ds\right)\mid \hfill \\ & \le & \mid f_1(\mathfrak{r},x\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid \mid g_1\left(\mathfrak{r},u\left({\varphi }_1\left(\mathfrak{r}\right)\right)){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x\left(s\right))ds\right)\mid \hfill \\ & \le & {\displaystyle [\mid }{f}_1(\mathfrak{r},0)\mid +{\eta }_1\mid x\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[\mid g_1(\mathfrak{r},0)\mid +l_1\mid u\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(\mathfrak{r},s,x\left(s\right))\mid ds\right]\hfill \\ & \le & [\mid f_1(\mathfrak{r},0)\mid +{\eta }_1\mid x\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[\mid g_1(\mathfrak{r},0)\mid +l_1\mid u\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\mid x\left(s\right)\mid ]ds\right]\hfill \\ & \le & [\mid f_1(\mathfrak{r},0)\mid +{\eta }_1\mid x\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[\mid g_1(\mathfrak{r},0)\mid +l_1\mid u\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x\parallel ]ds\right],\hfill \end{array}$$

then for $\mathfrak{r}\in {\mathbb{R}}_{+}$, we have

$$\begin{array}{ccc}\hfill \mid \mid {\mathcal{F}}_{1}x\mid \mid & \le & [N_1+{\eta }_1\parallel x\parallel ]\left[M_1+l_1\parallel u\parallel {\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x\parallel ]ds\right]\hfill \\ \hfill & \le & [N_1+{\eta }_1{r}_2][M_1+l_1{r}_1(a_1+b_1{r}_2)]=r_2.\hfill \end{array}$$

(4)

Similarly, we obtain

$$\mid \mid {\mathcal{F}}_{2}u\mid \mid \le [N_2+{\eta }_2{r}_1][M_2+l_2{r}_2(a_2+b_2{r}_1)]=r_1,$$

We shall prove that for some $r>0,\mathcal{F}{B}_r\subset B_r,$ we have Therefore,

$$\mid \mid \mathcal{F}u\mid \mid =max\{\mid \mid {\mathcal{F}}_{1}x\mid \mid ,\mid \mid {\mathcal{F}}_{2}u\mid \mid \}=max\{r_1,r_2\}=r.$$

This demonstrates that the operator $\mathcal{F}:B_r\to B_r.$

Next, we prove that $\mathcal{F}$ is continuous on the ball $B_r$. Let $u=(x_1,u_1)$ and $v=(x_2,u_2)$ in $B_r.$ For any $\mathfrak{r}\in {\mathbb{R}}_{+},$ such that $\parallel x_1-x_2\parallel \le ϵ$ and $\parallel u_1-u_2\parallel \le ϵ$, then we have

$$\begin{array}{cc}\hfill & \mid {\mathcal{F}}_1{x}_1\left(\mathfrak{r}\right)-{\mathcal{F}}_1{x}_2\left(\mathfrak{r}\right)\mid \hfill \\ \hfill & =\mid f_1(\mathfrak{r},x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right))g_1(\mathfrak{r},u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_1\left(s\right))ds)\hfill \\ \hfill & -f_1(\mathfrak{r},x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right))g_1(\mathfrak{r},u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_2\left(s\right))ds)\mid \hfill \\ \hfill & \le \mid f_1(\mathfrak{r},x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right))-f_1(\mathfrak{r},x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right))\mid \mid g_1(\mathfrak{r},u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_1\left(s\right))ds)\mid \hfill \\ \hfill & +\mid f_1(\mathfrak{r},x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right))\mid \mid g_1(\mathfrak{r},u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_2\left(s\right))ds)-g_1(\mathfrak{r},u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_1\left(s\right))ds)\mid \hfill \\ \hfill & \le {\eta }_1\mid x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(\mid g_1(\mathfrak{r},0)\mid +l_1\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\mid x_1\left({\varphi }_1\left(s\right)\right)\mid ]ds\right]\hfill \\ \hfill & +l_1[\mid f_1(\mathfrak{r},0)\mid +{\eta }_1\mid x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_1\left(s\right))ds-u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_2\left(s\right))ds\mid \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\eta }_1\mid x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u_1\parallel ){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x_2\parallel \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(\mathfrak{r},s,x_1\left(s\right))\mid ds\hfill \\ \hfill & +\mid u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(s,x_1\left(s\right))-h_1(s,x_2\left(s\right))\mid ds]\hfill \\ \hfill & \le {\eta }_1\mid x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u_1\parallel ){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x_2\parallel \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds\hfill \\ \hfill & +\parallel u_2\parallel {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(\mathfrak{r},s,x_1\left(s\right))-h_1(\mathfrak{r},s,x_2\left(s\right))\mid ds]\hfill \\ \hfill & \le {\eta }_1\mid x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u_1\parallel ){\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x_2\parallel \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds\hfill \\ \hfill & +\parallel u_2\parallel {\int }_0^{\mathfrak{r}}\mid h_1(\mathfrak{r},s,x_1\left({\varphi }_1\left(s\right)\right))-h_1(\mathfrak{r},s,x_2\left({\varphi }_1\left(s\right)\right))\mid ds].\hfill \end{array}$$

Let us consider the following two cases:

(1^*)

Choose $\mathcal{T}>0$ such that for $\mathfrak{r}\ge \mathcal{T}$ the following inequalities holds:

$$r{\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds\le {ϵ}_1\mathrm{and}r{\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_2\parallel ]ds\le {ϵ}_2.$$

Then, we have

$$\begin{array}{cc}\hfill & \mid {\mathcal{F}}_1{x}_1(\mathfrak{r})-{\mathcal{F}}_1{x}_2(\mathfrak{r})\mid \le {\eta }_1\parallel x_1-x_2\parallel ]\left[(M_1+l_1\parallel u_1\parallel ){\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x_2\parallel \left]\right[\parallel u_1-u_2\parallel \mid {\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds\hfill \\ \hfill & +\parallel u_2\parallel \left[{\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds+{\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_2\parallel ]ds\right]\hfill \\ \hfill & \le \left({\eta }_1\left[M_1+l_1{r}_1[a_1+b_1{r}_2]\right]+l_1[N_1+{\eta }_1{r}_2][a_1+b_1{r}_2]\right)ϵ+{ϵ}_1+{ϵ}_2\le ϵ+{ϵ}_1+{ϵ}_2\hfill \end{array}$$

(2^*)

For $t\le \mathcal{T}$. Define the function ${\omega }^{\mathcal{T}}(h_1,ϵ)$ where, for $ϵ>0$, we denote

$${\omega }^{\mathcal{T}}(h_1,ϵ)=sup\{\mid h_1(\mathfrak{r},x_1\left(\mathfrak{r}\right))-h_1(\mathfrak{r},x_2\left(\mathfrak{r}\right))\mid :\mathfrak{r}\in [0,\mathcal{T}],\parallel x_1-x_2\parallel \le ϵ\}.$$

Taking into account the uniform continuity of the function $h_1$, we deduce that ${\omega }^{\mathcal{T}}(h_1,ϵ)$$\to 0$ as $ϵ\to 0$. Thus, in this case, by the above estimation, we obtain

$$\begin{array}{cc}\hfill \parallel {\mathcal{F}}_1{x}_1-{\mathcal{F}}_1{x}_2\parallel & \le {\eta }_1\parallel x_1-x_2\parallel \left[M_1+l_1\parallel u_1\parallel [a_1+b_1\parallel x_1\parallel ]\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x_2\parallel ]\left[\parallel u_1-u_2\parallel [a_1+b_1\parallel x_1\parallel ]+\parallel u_2\parallel {\omega }^T(h_1,ϵ)\right]\hfill \\ \hfill & \le \left[{\eta }_1\left[M_1+l{r}_1[a_1+b_1{r}_2]\right]+l_1[N_1+{\eta }_1{r}_2][a_1+b_1{r}_2]\right]ϵ\le ϵ.\hfill \end{array}$$

(5)

Finally, from the two cases $\left(1^{*}\right)and\left(2^{*}\right)$, we can deduce that the operator ${\mathcal{F}}_1$ is a continuous operator by the same way we can prove that

$$\begin{array}{ccc}\hfill \parallel {\mathcal{F}}_2{u}_1-{\mathcal{F}}_2{u}_2\parallel & \le & {\eta }_2\parallel u_1-u_2\parallel \left[M_2+l_2\parallel x_1\parallel [a_2+b_2\parallel x_1\parallel ]\right]\hfill \\ & +& l_2[N_2+{\eta }_2\parallel u_2\parallel ]\left[\parallel x_1-x_2\parallel [a_2+b_2\parallel u_1\parallel ]+\parallel x_2\parallel {\omega }^T(h_2,ϵ)\right]\hfill \\ & \le & ϵ\hfill \end{array}$$

(6)

so operator ${\mathcal{F}}_2$ is also a continuous operator.

Hence, for all $v_1=(x_1,u_1)$, $v_2=(x_2,u_2)\in B_r$, by definition of the operator $\mathcal{F}$, we obtain

$$\begin{array}{cc}\hfill \mathcal{F}{v}_1-\mathcal{F}{v}_2& =\mathcal{F}(x_1,u_1)-\mathcal{F}(x_2,u_2)\hfill \\ \hfill & =({\mathcal{F}}_1{x}_1,{\mathcal{F}}_2{u}_1)-({\mathcal{F}}_1{x}_2,{\mathcal{F}}_2{u}_2)\hfill \\ \hfill & =({\mathcal{F}}_1{x}_1-{\mathcal{F}}_1{x}_2,{\mathcal{F}}_2{u}_1-{\mathcal{F}}_2{u}_2),\hfill \end{array}$$

then

$$\begin{array}{ccc}\hfill \parallel \mathcal{F}{v}_1-\mathcal{F}{v}_1\parallel & =& \parallel ({\mathcal{F}}_1{x}_1-{\mathcal{F}}_1{x}_2,{\mathcal{F}}_2{u}_1-{\mathcal{F}}_2{u}_2)\parallel ,\hfill \\ & \le & max\{\parallel ({\mathcal{F}}_1{x}_1-{\mathcal{F}}_1{x}_2\parallel ,\parallel {\mathcal{F}}_2{u}_1-{\mathcal{F}}_2{u}_2)\parallel \}\hfill \\ & \le & ϵ\hfill \end{array}$$

Hence, $\mathcal{F}:B_r\to \mathcal{BC}\left({\mathbb{R}}_{+}\right)$ is a continuous operator

Now, let us take a nonempty subset $U=X\times Y$ of $B_r.$ Fix $ϵ>0$ and choose $x\in X$ and ${\mathfrak{r}}_1,{\mathfrak{r}}_2\in {\mathbb{R}}_{+}$ such that $\mid {\mathfrak{r}}_2-{\mathfrak{r}}_1\mid \le ϵ,$ without loss of generality we may assume that ${\mathfrak{r}}_1\le {\mathfrak{r}}_2.$ Then,

$$\begin{array}{ccc}& & \mid {\mathcal{F}}_{1}x\left({\mathfrak{r}}_2\right)-{\mathcal{F}}_{1}x\left({\mathfrak{r}}_1\right)\mid \hfill \\ & =& \mid f_1({\mathfrak{r}}_2,x\left({\varphi }_1\left(t_2\left(\mathfrak{r}\right)\right)\right)g_1\left({\mathfrak{r}}_2,u({\varphi }_1\left(\mathfrak{r}{t}_2\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left({\mathfrak{r}}_2\right)}{h}_1({\mathfrak{r}}_2,s,x\left(s\right))ds\right)\hfill \\ & -& f_1({\mathfrak{r}}_1,x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right))g_1\left(\mathfrak{r},u\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right){\int }_0^{{\psi }_1\left({\mathfrak{r}}_1\right)}{h}_1({\mathfrak{r}}_1,s,x\left(s\right))ds\right)\mid \hfill \\ & \le & \mid f_1({\mathfrak{r}}_2,x\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right))-f_1({\mathfrak{r}}_1,x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right))\mid \mid g_1\left({\mathfrak{r}}_2,u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right){\int }_0^{{\psi }_1\left({\mathfrak{r}}_2\right)}{h}_1({\mathfrak{r}}_2,s,x\left(s\right))ds\right)\mid \hfill \\ & +& \mid f_1({\mathfrak{r}}_1,x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right))\mid \hfill \\ & \times & \mid g_1\left({\mathfrak{r}}_2,u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right){\int }_0^{{\psi }_1\left({\mathfrak{r}}_2\right)}{h}_1({\mathfrak{r}}_2,s,x\left(s\right))ds\right)-g_1({\mathfrak{r}}_1,u\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right){\int }_0^{{\psi }_1\left({\mathfrak{r}}_1\right)}{h}_1({\mathfrak{r}}_1,s,x\left(s\right))ds)\mid \hfill \end{array}$$

$$\begin{array}{ccc}& \le & \left[\mid f_1({\mathfrak{r}}_2,x\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right))-f_1({\mathfrak{r}}_1,x\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right))\mid +\mid f_1({\mathfrak{r}}_1,x\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right))-f_1({\mathfrak{r}}_1,x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right))\mid \right]\hfill \\ & \times & \left[\mid g_1({\mathfrak{r}}_2,0)\mid +l_1\mid x\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)\mid {\int }_0^{{\psi }_1\left({\mathfrak{r}}_2\right)}\mid h_1({\mathfrak{r}}_2,s,x\left(s\right))\mid ds\right]\hfill \\ & & +[\mid f_1({\mathfrak{r}}_1,0)\mid +{\eta }_1\mid x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid ]\hfill \\ & \times & [\mid g_1\left({\mathfrak{r}}_2,u({\varphi }_1\left(\left({\mathfrak{r}}_2\right)\right){\int }_0^{{\psi }_1\left({\mathfrak{r}}_2\right)}h({\mathfrak{r}}_2,s,x\left(s\right))ds\right)-g_1\left({\mathfrak{r}}_1,u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)){\int }_0^{{\psi }_1\left({\mathfrak{r}}_2\right)}{h}_1({\mathfrak{r}}_2,s,x\left(s\right))ds\right)\mid \hfill \\ & +& \mid g_1\left({\mathfrak{r}}_1,u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right){\int }_0^{{\psi }_1\left({\mathfrak{r}}_2\right)}{h}_1({\mathfrak{r}}_2,s,x\left(s\right))ds\right)-g_1\left({\mathfrak{r}}_1,u\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right){\int }_0^{{\psi }_1\left({\mathfrak{r}}_1\right)}{h}_1({\mathfrak{r}}_1,s,x\left(s\right))ds\right)\mid ]\hfill \\ & \le & [{\theta }_{f_1}\left(\delta \right)+{\eta }_1\mid x\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)-x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid ]\left[M_1+l_1\mid u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)\mid {\int }_0^{{\psi }_1\left({\mathfrak{r}}_2\right)}[a_1({\mathfrak{r}}_2,s)+b_1({\mathfrak{r}}_2,s)\mid y\left(s\right))\mid ]ds\right]\hfill \\ & +& [N_1+{\eta }_1\mid x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid ]\hfill \\ & \times & \mid {\theta }_{g_1}\left(\delta \right)+l_1[\mid u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right){\int }_0^{{\psi }_1\left({\mathfrak{r}}_2\right)}{h}_1(t_2,s,x\left(s\right))ds-u\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right){\int }_0^{{\psi }_1\left({\mathfrak{r}}_1\right)}\mid h_1(t_1,s,x\left(s\right))ds\mid \hfill \\ & \le & [{\theta }_{f_1}\left(\delta \right)+{\eta }_1\mid x\left({\varphi }_1\left(t_2\right)\right)-x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid ]\left[M_1+l_1\mid u({\varphi }_1\left({\mathfrak{r}}_2\right)\mid {\int }_0^{{\psi }_1\left(t_2\right)}[a_1({\mathfrak{r}}_2,s)+b_1({\mathfrak{r}}_2,s)\mid x\left(s\right))\mid ]ds\right]\hfill \\ & +& [N_1+{\eta }_1\mid x\left({\varphi }_1\left(t_1\right)\right)\mid \left]\right[{\theta }_{g_1}\left(\delta \right)+l_1\mid u\left({\varphi }_1\left(t_2\right)\right)\mid {\int }_0^{{\psi }_1\left(t_2\right)}[h({\mathfrak{r}}_2,s,x\left(s\right))-h({\mathfrak{r}}_1,s,x\left(s\right))]ds\hfill \\ & +& l_1\mid u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)-u\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid {\int }_0^{{\psi }_1\left({\mathfrak{r}}_1\right)}\mid h_1({\mathfrak{r}}_1,s,x\left(s\right))\mid ds+l_1\mid u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)\mid {\int }_{{\psi }_1\left({\mathfrak{r}}_1\right)}^{{\psi }_1\left({\mathfrak{r}}_2\right)}\mid h_1({\mathfrak{r}}_1,s,x\left(s\right))\mid ds]\hfill \\ \\ & \le & [{\theta }_{f_1}\left(\delta \right)+{\eta }_1\mid x\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)-x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid ]\left[M_1+l_1\mid u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)\mid {\int }_0^{{\mathfrak{r}}_2}[a_1({\mathfrak{r}}_2,s)+b_1({\mathfrak{r}}_2,s)\mid x\left(s\right))\mid ]ds\right]\hfill \\ & +& [N_1+{\eta }_1\mid x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid \left]\right[{\theta }_{g_1}\left(\delta \right)+l_1\mid u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)-u\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid {\int }_0^{{\mathfrak{r}}_1}[a_1({\mathfrak{r}}_2,s)+b_1({\mathfrak{r}}_2,s)\mid x\left(s\right))\mid ]ds\hfill \\ & +& l_1\mid u\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid {\int }_{{\mathfrak{r}}_1}^{{\mathfrak{r}}_2}[a_1({\mathfrak{r}}_2,s)+b_1({\mathfrak{r}}_2,s)\mid x\left(s\right))\mid ]ds\hfill \\ & \le & [{\theta }_{f_1}\left(\delta \right)+{\eta }_1\mid x\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)-x\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid ]\left[M_1+l_1\parallel u\parallel {\int }_0^{t_2}[a_1({\mathfrak{r}}_2,s)+b_1({\mathfrak{r}}_2,s)\parallel x\parallel ]ds\right]\hfill \\ & +& [N_1+{\eta }_1\parallel x\parallel \left]\right[{\theta }_{g_1}\left(\delta \right)+l_1\mid u\left({\varphi }_1\left({\mathfrak{r}}_2\right)\right)-u\left({\varphi }_1\left({\mathfrak{r}}_1\right)\right)\mid {\int }_0^{{\mathfrak{r}}_1}[a_1({\mathfrak{r}}_2,s)+b_1({\mathfrak{r}}_2,s)\parallel x\parallel ]ds\hfill \\ & +& l_1\parallel u\parallel {\int }_{t_1}^{t_2}[a_1(t_2,s)+b_2(t_2,s)\parallel x\parallel ]ds\hfill \\ & \le & [{\theta }_{f_1}\left(\delta \right)+{\eta }_1{\omega }^{\mathcal{T}}(x,{\omega }^{\mathcal{T}}({\varphi }_1,ϵ))]\left[M_1+l_1{r}_1[a_1+b_1{r}_2]\right]\hfill \\ & +& [N_1+{\eta }_1{r}_2]\left[{\theta }_{g_1}\left(\delta \right)+l_1{\omega }^{\mathcal{T}}(u,{\omega }^{\mathcal{T}}({\varphi }_1,ϵ))[a_1+b_1{r}_2]\right],\hfill \end{array}$$

where we denoted

$$\begin{array}{cc}\hfill {\theta }_{f_1}\left(\delta \right)& =sup\{\mid f_1({\mathfrak{r}}_2,x)-f_1({\mathfrak{r}}_1,x)\mid :{\mathfrak{r}}_1,{\mathfrak{r}}_2\in [0,\mathcal{T}],{\mathfrak{r}}_1<t_2,\mid {\mathfrak{r}}_2-{\mathfrak{r}}_1\mid <\delta ,\mid x\mid \le r_2\},\hfill \\ \hfill {\theta }_{g_1}\left(\delta \right)& =sup\{\mid g_1({\mathfrak{r}}_2,u)-g_1({\mathfrak{r}}_1,u)\mid :t_1,t_2\in [0,\mathcal{T}],{\mathfrak{r}}_1<{\mathfrak{r}}_2,\mid {\mathfrak{r}}_2-{\mathfrak{r}}_1\mid <\delta ,\mid u\mid \le r_1\}.\hfill \end{array}$$

We therefore arrive at the following estimate:

$$\begin{array}{cc}\hfill \omega {({\mathcal{F}}_{1}x,ϵ)}^{\mathcal{T}}& \le [{\theta }_{f_1}\left(\delta \right)+{\eta }_1{\omega }^{\mathcal{T}}(x,{\omega }^{\mathcal{T}}({\varphi }_1,ϵ))]\left[M_1+l_1{r}_1[a_1+b_1{r}_2]\right]\\ & +[N_1+{\eta }_1{r}_2]\left[{\theta }_{g_1}\left(\delta \right)+l_1{\omega }^{\mathcal{T}}(u,{\omega }^{\mathcal{T}}({\varphi }_1,ϵ))[a_1+b_1{r}_2]\right],\end{array}$$

(7)

then based on the functions $f_1,g_1:[0,\infty )\times B_r\to \mathbb{R},$ are uniform continuity, assumptions (iii) and (iv), we have concluded ${\theta }_{f_1}\left(\delta \right),{\theta }_{g_1}\left(\delta \right)\to 0,$ as $\delta \to 0.$ Moreover, it is obvious that ${\omega }^{\mathcal{T}}({\varphi }_1,ϵ)\to 0,$ as $ϵ\to 0.$ Thus, linking the established facts with the estimate (7), we obtain

$$\begin{array}{ccc}\hfill w_0^{\mathcal{T}}\left({\mathcal{F}}_{1}X\right)& \le & \left[{\eta }_1(M_1+l_1{r}_1(a_1+b_1{r}_2))+l_1[N_1+{\eta }_1{r}_2](a_1+b_1{r}_2)\right]w_0^{\mathcal{T}}\left(X\right).\hfill \end{array}$$

Consequently, and as $\mathcal{T}\to \infty$, we obtain

$$\begin{array}{ccc}\hfill w_0\left({\mathcal{F}}_{1}X\right)& \le & \left[\eta (M+lr(a+br\left)\right)+l[N+\eta r](a+br)\right]w_0\left(X\right)\hfill \end{array}$$

(8)

Then, we derive the inequality

$${\omega }_0\left({\mathcal{F}}_{1}X\right)\le c{\omega }_0\left(X\right).$$

By a similar way, we can deduce that

$${\omega }_0\left({\mathcal{F}}_{2}Y\right)\le c{\omega }_0\left(Y\right).$$

Therefore,

$$\begin{array}{cc}\hfill w_0\left(\mathcal{F}U\right)& =max\{{\omega }_0\left({\mathcal{F}}_{1}X\right),{\omega }_0\left({\mathcal{F}}_{2}Y\right)\}\\ & \le cmax\{{\omega }_0\left(X\right),{\omega }_0\left(Y\right).\}\end{array}$$

Hence,

$$\begin{array}{cc}\hfill w_0\left(\mathcal{F}U\right)& \le c{\omega }_0\left(U\right)\hfill \end{array}$$

(9)

Moreover, for any $v_1=(x_1,u_1),v_2=(x_2,u_2)\in U\subset B_r$ and fixed $\mathfrak{r}\ge 0$, we put

$$\begin{array}{cc}\hfill \mid {\mathcal{F}}_1{x}_1(\mathfrak{r})-{\mathcal{F}}_1{x}_2(\mathfrak{r})\mid & \le {\eta }_1\mid x_1\left(\mathfrak{r}\right)-x_2\left(\mathfrak{r}\right)\mid ]\left[(M_1+l_1\parallel u_1\parallel ){\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x_2\parallel \left]\right[\mid u_1\left(\mathfrak{r}\right)-u_2\left(\mathfrak{r}\right)\mid {\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_2\parallel ]ds\hfill \\ \hfill & +\parallel u_2\parallel \left[{\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_1\parallel ]ds+{\int }_0^{\mathfrak{r}}[a_1(\mathfrak{r},s)+b_1(\mathfrak{r},s)\parallel x_2\parallel ]ds\right]\hfill \end{array}$$

Hence, we can easily derive the following inequality:

$$\begin{array}{cc}\hfill \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left({\mathcal{F}}_{1}X\right)\left(\mathfrak{r}\right)& \le \eta \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}X\left(t\right)\left[M+lr\mid [a+br]\right]\hfill \\ \hfill & +l[N+\eta r]\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}X\left(t\right)[a+br]\hfill \end{array}$$

Now, considering our assumptions, we obtain the following estimate:

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}sup\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}{\mathcal{F}}_{1}X\left(\mathfrak{r}\right)& \le \eta \left[M+lr\mid [a+br]\right]+l[N+\eta r][a+br])\underset{t\to \infty }{lim}sup\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}X\left(\mathfrak{r}\right)\hfill \end{array}$$

$$\underset{t\to \infty }{lim}sup\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left({\mathcal{F}}_{1}X\right)\left(\mathfrak{r}\right)\le c\underset{t\to \infty }{lim}sup\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}X\left(\mathfrak{r}\right),$$

(10)

where we denoted $c=\eta \left[M+lr\mid [a+br]\right]+lk[N+\eta r][a+br])$. Clearly, given assumption (v), we know that $c<1$.

By a similar way as performed above, we can deduce that

$$\begin{array}{cc}\hfill \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left({\mathcal{F}}_{2}Y\right)\left(\mathfrak{r}\right)& \le \eta \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}Y\left(\mathfrak{r}\right)\left[M+lr\mid [a+br]\right]+l[N+\eta r]\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}Y\left(\mathfrak{r}\right)[a+br]\hfill \end{array}$$

$$\underset{t\to \infty }{lim}sup\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left({\mathcal{F}}_{2}Y\right)\left(\mathfrak{r}\right)\le c\underset{t\to \infty }{lim}sup\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}Y\left(\mathfrak{r}\right),$$

(11)

Therefore,

$$\begin{array}{ccc}\hfill \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\mathcal{F}(X,Y)\left(\mathfrak{r}\right)& =& max\{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left({\mathcal{F}}_{1}X\right)\left(\mathfrak{r}\right),\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left({\mathcal{F}}_{2}Y\right)\left(\mathfrak{r}\right)\}\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}sup\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}FU\left(\mathfrak{r}\right)& \le c\underset{t\to \infty }{lim}sup\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}U\left(\mathfrak{r}\right).\hfill \end{array}$$

(12)

Now, from the estimations (9) and (12), and the definition of $\mu$ on U, we obtain

$$\mu \left(FU\right)\le c\mu \left(U\right).$$

Since all conditions of Theorem 1 are satisfied, then F has a fixed point $(x,y)\in U$. Consequently, the cubic functional integral Equation (1) has at least one solution $x\in \mathcal{BC}\left({\mathbb{R}}_{+}\right)$ restricted to the control variable $u\in \mathcal{BC}\left({\mathbb{R}}_{+}\right)$. □

2.2. Asymptotic Stability

The following corollary can now be deduced from the proof of Theorem 2.

Corollary 1.

The solution $x\in \mathcal{BC}\left({\mathbb{R}}_{+}\right)$ of the cubic functional integral Equation (1) with the feedback control (2) is asymptotically stable; that is to say, ∀$ϵ>0$, there exists $\mathcal{T}\left(ϵ\right)>0$ and $r>0$, such that, if any two solutions of the cubic functional integral Equation (1) $x,x_1$, which are restricted to the two control variables $u,u_1$, respectively, then $(x,u),(x_1,u_1)\in U$ satisfy

$$\mid \mid (x,u)-(x_1,u_1)\mid \mid \le ϵ,t\ge \mathcal{T}\left(ϵ\right).$$

This indicates that

$$\mid x\left(\mathfrak{r}\right)-x_1\left(\mathfrak{r}\right)\mid \le ϵand\mid u\left(\mathfrak{r}\right)-u_1\left(\mathfrak{r}\right)\mid \le ϵ,\mathfrak{r}\ge \mathcal{T}\left(ϵ\right).$$

Proof. 

Under the assumptions of Theorem 2 and similarly to how relations (5) and (6) are evaluated, we have

$$\begin{array}{ccc}\hfill \parallel x-x_1\parallel & =& \parallel {\mathcal{F}}_{1}x-{\mathcal{F}}_1{x}_1\parallel \hfill \\ & \le & {\eta }_1ϵ\left[M_1+l_1{r}_1[a_1+b_1{r}_2]\right]\hfill \\ & +& l_1[N_1+{\eta }_1{r}_2]\left[ϵ[a_1+b_1{r}_2]+\parallel u_1\parallel {\omega }^T(h_1,ϵ)\right]\hfill \\ \hfill \parallel u-u_1\parallel & =& \parallel {\mathcal{F}}_2{u}_1-{\mathcal{F}}_2{u}_1\parallel \hfill \\ & \le & {\eta }_2ϵ\left[M_2+l_2{r}_2[a_2+b_2{r}_1]\right]\hfill \\ & +& l_2[N_2+{\eta }_2{r}_1]\left[ϵ[a_2+b_2{r}_2]+\parallel x_1\parallel {\omega }^T(h_2,ϵ)\right]\hfill \\ \hfill \parallel (x,u)-(x_1,u_1){\parallel }_U& =& \parallel ({\mathcal{F}}_{1}x,{\mathcal{F}}_{2}u)-({\mathcal{F}}_1{x}_1,{\mathcal{F}}_2{u}_1)\parallel \hfill \\ & =& \parallel ({\mathcal{F}}_{1}x-{\mathcal{F}}_1{x}_1,{\mathcal{F}}_{2}u-{\mathcal{F}}_2{u}_1)\parallel \hfill \\ & =& max\{\parallel {\mathcal{F}}_{1}x-{\mathcal{F}}_1{x}_1\parallel ,\parallel {\mathcal{F}}_{2}u-{\mathcal{F}}_2{u}_1\parallel \}\hfill \\ & \le & \eta ϵ\left[M+lr[a+br]\right]+l[N+\eta r]\left[ϵ[a+br]+r{\omega }^{\mathcal{T}}\left(ϵ\right)\right],\hfill \end{array}$$

for $\mathfrak{r}>\mathcal{T}\left(ϵ\right)$. □

2.3. Particular Cases and Example

(I)

In case $g_i(\mathfrak{r},x)=1,i=1,2,$ then our problem (1) and (2) has been reduced to the system of functional equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}x\left(\mathfrak{r}\right)=f_1(\mathfrak{r},x\left({\varphi }_1\left(\mathfrak{r}\right)\right),\mathfrak{r}\in {\mathbb{R}}_{+}\\ \\ u\left(\mathfrak{r}\right)=f_2(\mathfrak{r},u\left({\varphi }_2\left(\mathfrak{r}\right)\right),\mathfrak{r}\in {\mathbb{R}}_{+}.\end{array}\right.\end{array}$$

(II)

Let $f_i(\mathfrak{r},x)=1,i=1,2$, then the functional integral equation

$$x\left(\mathfrak{r}\right)=g_1\left(\mathfrak{r},u\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x\left(s\right)))ds\right),\mathfrak{r}\in {\mathbb{R}}_{+}$$

with the feedback control

$$u\left(\mathfrak{r}\right)=g_2\left(\mathfrak{r},x\left({\varphi }_2\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_2\left(\mathfrak{r}\right)}{h}_2(\mathfrak{r},s,u\left(s\right)))ds\right),\mathfrak{r}\in {\mathbb{R}}_{+}.$$

For ${\psi }_i\left(\mathfrak{r}\right)=1,i=1,2,g_1(\mathfrak{r},x)=1+x$ and $g_2(\mathfrak{r},u)=u,$ then we have the quadratic functional integral equation of Urysohn-type

$$x\left(\mathfrak{r}\right)=1+u\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^1{h}_1(\mathfrak{r},s,x\left(s\right))ds,\mathfrak{r}\in {\mathbb{R}}_{+}$$

with the feedback control

$$u\left(\mathfrak{r}\right)=x\left({\varphi }_2\left(\mathfrak{r}\right)\right){\int }_0^1{h}_2(\mathfrak{r},s,u\left(s\right)))ds,\mathfrak{r}\in {\mathbb{R}}_{+}$$

(III)

${\psi }_i\left(\mathfrak{r}\right)=1,i=1,2,g_1(\mathfrak{r},x)=1+x,f_2(\mathfrak{r},x)=1$ and $g_2(\mathfrak{r},u)=u,$ then we obtain the cubic functional integral equation of Urysohn-type

$$x\left(t\right)=f_1(\mathfrak{r},x\left({\varphi }_1\left(\mathfrak{r}\right)\right))\left(1+u\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^1{h}_1(\mathfrak{r},s,x\left(s\right))ds\right),\mathfrak{r}\in {\mathbb{R}}_{+}$$

with the feedback control

$$u\left(\mathfrak{r}\right)=x\left({\varphi }_2\left(\mathfrak{r}\right)\right){\int }_0^1{h}_2(\mathfrak{r},s,u\left(s\right)))ds,\mathfrak{r}\in {\mathbb{R}}_{+}$$

Example 1.

Consider the cubic functional integral equation

$$x\left(\mathfrak{r}\right)=\frac{\mathfrak{r}}{1+{\mathfrak{r}}^2}arctan(\mathfrak{r}+x\left(\mathfrak{r}\right))+\frac{1}{1+\mathfrak{r}}sin(\mathfrak{r}+u\left(\mathfrak{r}\right){\int }_0^{\mathfrak{r}}[\frac{\mathfrak{r}(2\mathfrak{r}-s)}{1+{\mathfrak{r}}^4}+\frac{\mathfrak{r}\mid x\left(s\right)\mid }{2\pi ({\mathfrak{r}}^2+1)(s+1)}]ds),\mathfrak{r}\ge 0$$

(13)

with a feedback control given by

$$u\left(t\right)=\frac{1}{1+\mathfrak{r}}arccot(\mathfrak{r}+u\left(\mathfrak{r}\right))+\frac{1}{1+\mathfrak{r}}cos(\mathfrak{r}+x\left(\mathfrak{r}\right){\int }_0^{\mathfrak{r}}[\frac{\mathfrak{r}(2\mathfrak{r}-s)}{1+{\mathfrak{r}}^4}+\frac{\mid u\left(s\right)\mid }{4\pi ({\mathfrak{r}}^{\frac{2}{3}}+1)(\sqrt{\mathfrak{r}-s}+1)}]ds),\mathfrak{r}\ge 0$$

(14)

So, we investigate the solvability of the cubic functional integral Equation (13) with a feedback control (14) on the space $\mathcal{BC}\left({\mathbb{R}}_{+}\right)$. Considering these functional equations is a particular case of the cubic functional integral Equation (1) with a feedback control (2), where

$$\begin{array}{ccc}\hfill f_1(\mathfrak{r},x\left(\mathfrak{r}\right))& =& \frac{\mathfrak{r}}{1+{\mathfrak{r}}^2}arctan(\mathfrak{r}+x\left(\mathfrak{r}\right)),f_2(\mathfrak{r},x\left(\mathfrak{r}\right))=\frac{1}{1+\mathfrak{r}}arccot(\mathfrak{r}+x\left(\mathfrak{r}\right)),\hfill \\ \hfill g_1(\mathfrak{r},x\left(\mathfrak{r}\right))& =& \frac{1}{1+\mathfrak{r}}sin(\mathfrak{r}+x\left(\mathfrak{r}\right)),g_2(\mathfrak{r},x\left(\mathfrak{r}\right))=\frac{1}{1+\mathfrak{r}}cos(\mathfrak{r}+x\left(\mathfrak{r}\right)),\hfill \\ \hfill h_1(\mathfrak{r},s,x\left(s\right))& =& \frac{\mathfrak{r}(2\mathfrak{r}-s)}{1+{\mathfrak{r}}^4}+\frac{\mathfrak{r}\mid x\left(s\right)\mid }{2\pi ({\mathfrak{r}}^2+1)(s+1)},\hfill \\ \hfill h_2(\mathfrak{r},s,x\left(s\right))& =& \frac{\mathfrak{r}(2\mathfrak{r}-s)}{1+{\mathfrak{r}}^4}+\frac{\mid x\left(s\right)\mid }{4\pi ({\mathfrak{r}}^{\frac{2}{3}}+1)(\sqrt{\mathfrak{r}-s}+1)}.\hfill \end{array}$$

Clearly, functions $f_i,(i=1,2)$ are continuous. Consequently, for any $x,u\in \mathcal{BC}\left({\mathbb{R}}_{+}\right)and\mathfrak{r}\in {\mathbb{R}}_{+}$

$$\begin{array}{ccc}\hfill \mid f_i(\mathfrak{r},x\left(\mathfrak{r}\right))-f_i(\mathfrak{r},u\left(\mathfrak{r}\right))\mid & \le & \frac{1}{2}\mid x\left(\mathfrak{r}\right)-u\left(\mathfrak{r}\right)\mid .\hfill \end{array}$$

This shows that condition $\left(v\right)$ has been satisfied. with $N=\frac{\pi }{2}$ and $\eta =\frac{1}{2},$ with $f_1(\mathfrak{r},0)=\frac{\mathfrak{r}}{1+{\mathfrak{r}}^2}arctan\left(\mathfrak{r}\right),f_2(\mathfrak{r},0)=\frac{1}{1+\mathfrak{r}}arccot\left(\mathfrak{r}\right).$ However, we also have

$$\begin{array}{c}\hfill \mid g_i(\mathfrak{r},x\left(\mathfrak{r}\right))-g_i(\mathfrak{r},u\left(\mathfrak{r}\right))\mid \le \frac{\mid x\left(\mathfrak{r}\right)-u\left(\mathfrak{r}\right)\mid }{2},\end{array}$$

with $l=\frac{1}{2}$, $g_1(\mathfrak{r},0)=\frac{1}{1+\mathfrak{r}}sin\left(t\right),$$g_2(\mathfrak{r},0)=\frac{1}{1+\mathfrak{r}}cos\left(\mathfrak{r}\right)$ and $M=\frac{1}{2}$. Additionally, notice that the function $h_i(\mathfrak{r},s,x)$ met condition (iv), with

$$\mid h_1(\mathfrak{r},s,x\left(s\right))\mid \le \frac{\mathfrak{r}(2\mathfrak{r}-s)}{1+{\mathfrak{r}}^4}+\frac{\mathfrak{r}\mid x\left(s\right)\mid }{2\pi ({\mathfrak{r}}^2+1)(s+1)},$$

and

$$\mid h_2(\mathfrak{r},s,x\left(s\right))\mid \le \frac{\mathfrak{r}(3\mathfrak{r}-s)}{1+{\mathfrak{r}}^4}+\frac{\mid x\left(s\right)\mid }{4\pi ({\mathfrak{r}}^{\frac{2}{3}}+1)(\sqrt{\mathfrak{r}-s}+1)}.$$

Consequently, we can put $a_i(\mathfrak{r},s)=\frac{\mathfrak{r}(2\mathfrak{r}-s)}{2(1+{\mathfrak{r}}^4)},(i=1,2),$$b_1(\mathfrak{r},s)=\frac{\mathfrak{r}}{2\pi ({\mathfrak{r}}^2+1)(s+1)},$ and $b_2(\mathfrak{r},s)=\frac{1}{4\pi ({\mathfrak{r}}^{\frac{3}{4}}+1)(\sqrt{\mathfrak{r}-s}+1)}.$ To confirm the assumption $\left(iv\right)$, notice that

$$\underset{\mathfrak{r}\to \infty }{lim}{\int }_0^{\mathfrak{r}}{a}_i(\mathfrak{r},s)=\underset{\mathfrak{r}\to \infty }{lim}{\int }_0^{\mathfrak{r}}\frac{\mathfrak{r}(2\mathfrak{r}-s)}{2(1+{\mathfrak{r}}^4)}ds=\underset{\mathfrak{r}\to \infty }{lim}{\displaystyle \frac{3{\mathfrak{r}}^3}{4{\mathfrak{r}}^4+4}}=0,$$

$$\underset{\mathfrak{r}\to \infty }{lim}{\int }_0^{\mathfrak{r}}{b}_1(\mathfrak{r},s)=\underset{\mathfrak{r}\to \infty }{lim}{\int }_0^{\mathfrak{r}}\frac{\mathfrak{r}}{2\pi ({\mathfrak{r}}^2+1)(s+1)}ds=\underset{\mathfrak{r}\to \infty }{lim}{\displaystyle \frac{\mathfrak{r}ln\left(\mathfrak{r}+1\right)}{2\pi ·\left({\mathfrak{r}}^2+1\right)}}=0,$$

and

$$\underset{\mathfrak{r}\to \infty }{lim}{\int }_0^{\mathfrak{r}}{b}_2(\mathfrak{r},s)=\underset{\mathfrak{r}\to \infty }{lim}{\int }_0^{\mathfrak{r}}\frac{1}{4\pi ({\mathfrak{r}}^{\frac{2}{3}}+1)(\sqrt{\mathfrak{r}-s}+1)}ds=\underset{\mathfrak{r}\to \infty }{lim}{\displaystyle \frac{\sqrt{\mathfrak{r}}-ln\left(\sqrt{\mathfrak{r}}+1\right)}{2\pi ·({\mathfrak{r}}^{\frac{2}{3}}+1)}}=0.$$

Moreover, we have Easily, we determine $r=0.272031$

$$\eta \left[M+l{r}_0\mid [a+b{r}_0]\right]+l[N+\eta r_0][a+b{r}_0]\simeq 0.492808<1.$$

Consequently, all the conditions of Theorem 2 are satisfied. Hence, we conclude that the cubic functional integral Equation (13) with a feedback control (14) has at least one asymptotically stable solution in the space $BC\left({\mathbb{R}}_{+}\right)$.

3. Uniqueness of the Solution

Let us state the following assumption:

($v^{*}$)

$h_i:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times \mathbb{R},i=1,2$ are continuous functions and there exists a positive constant ${\gamma }_i,$ with $\gamma =max\{{\gamma }_1,{\gamma }_2\}$, such that

$$\mid h_i(\mathfrak{r},s,x)-h_i(\mathfrak{r},s,u)\mid \le {\gamma }_i\mid x-u\mid$$

for each $\mathfrak{r}\in {\mathbb{R}}_{+}$ and for all $u,v\in \mathbb{R}$. Moreover, the function $\mathfrak{r}\to h_i(\mathfrak{r},s,0)$$\in \mathcal{BC}\left({\mathbb{R}}_{+}\right)$ and we have

$$\mid h_i(\mathfrak{r},s,u)\mid \le {\gamma }_i\mid u\mid +H_i,\mathrm{where}{H}_i=\underset{\mathfrak{r}\in I}{sup}\mid \{h_i(\mathfrak{r},s,0)\mid :\mathfrak{r}\in {\mathbb{R}}_{+}\}<\infty ,$$

Theorem 3.

Assume that $\left(i\right)$–$\left(iv\right)$ and $\left(v^{*}\right)$ hold. If the following is true:

$$\eta \left[M+lr[H+\gamma r]\right]+l[N+\eta r][H+\gamma r]+rl\gamma [N+\eta r]<1$$

(15)

then there exists a unique solution for the cubic functional integral Equation (1) with a feedback control (2) in the space $BC\left({\mathbb{R}}_{+}\right).$

Proof. 

For $v_1=(x_1,u_1)$, $v_2=(x_2,u_2)$, two solutions for the problem (1) and (2), we have

$$\begin{array}{cc}\hfill & \mid x_1\left(\mathfrak{r}\right)-x_2\left(\mathfrak{r}\right)\mid \hfill \\ \hfill & =\mid f_1(\mathfrak{r},x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right))g_1(\mathfrak{r},u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_1\left(s\right))ds)\hfill \\ \hfill & -f_1(\mathfrak{r},x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right))g_1(\mathfrak{r},u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_2\left(s\right))ds)\mid \hfill \\ \hfill & \le \mid f_1(\mathfrak{r},x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right))-f_1(\mathfrak{r},x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right))\mid \mid g_1(\mathfrak{r},u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_1\left(s\right))ds)\mid \hfill \\ \hfill & +\mid f_1(\mathfrak{r},x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right))\mid \mid g_1(\mathfrak{r},u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_2\left(s\right))ds)-g_1(\mathfrak{r},u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_1\left(s\right))ds)\mid \hfill \\ \hfill & \le {\eta }_1\mid x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(\mid g_1(\mathfrak{r},0)\mid +l_1\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[H_1+{\gamma }_1\mid x_1\left(s\right)\mid ]ds\right]\hfill \\ \hfill & +l_1[\mid f_1(\mathfrak{r},0)\mid +{\eta }_1\mid x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(t\right)}{h}_1(\mathfrak{r},s,x_1\left(s\right))ds-u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x_2\left(s\right))ds\mid \hfill \\ \hfill & \le {\eta }_1\mid x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u_1\parallel ){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[H_1+{\gamma }_1\parallel x_1\parallel ]ds\right]ds]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x_2\parallel \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(\mathfrak{r},s,x_1\left(s\right))\mid ds\hfill \\ \hfill & +\mid u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(s,x_1\left(s\right))-h_1(s,x_2\left(s\right))\mid ds]\hfill \\ \hfill & \le {\eta }_1\mid x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u_1\parallel ){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[H_1+{\gamma }_1\parallel x_1\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x_2\parallel \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[H_1+{\gamma }_1\parallel x_1\parallel ]ds\hfill \\ \hfill & +\parallel u_2\parallel {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(\mathfrak{r},s,x_1\left(s\right))-h_1(\mathfrak{r},s,x_2\left(s\right))\mid ds]\hfill \\ \hfill & \le {\eta }_1\mid x_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u_1\parallel ){\int }_0^{\mathfrak{r}}[H_1+{\gamma }_1\parallel x_1\parallel ]ds\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +l_1[N_1+{\eta }_1\parallel x_2\parallel \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^t[H_1+{\gamma }_1\parallel x_1\parallel ]ds\hfill \\ \hfill & +\parallel u_2\parallel {\int }_0^{\mathfrak{r}}\mid {\gamma }_1\mid x_1\left(s\right))-x_2\left(s\right)\mid \mid ds]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\eta }_1\parallel x_1-x_2\parallel \left[M_1+l_1\parallel u_1\parallel [H_1+{\gamma }_1\parallel x_1\parallel ]\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x_2\parallel ]\left[\parallel u_1-u_2\parallel [H_1+{\gamma }_1\parallel x_1\parallel ]+\parallel u_2\parallel {\gamma }_1\parallel x_1-x_2\parallel \right]\hfill \end{array}$$

Taking the supremum $\mathfrak{r}\in {\mathbb{R}}_{+}$, we obtain

$$\begin{array}{cc}\hfill \parallel x_1-x_2\parallel & \le \eta \parallel x_1-x_2\parallel \left[M+lr[H+\gamma r]\right]\hfill \\ \hfill & +l[N+\eta r]\left[\parallel u_1-u_2\parallel [H+\gamma r]+r\gamma \parallel x_1-x_2\parallel \right].\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill \mid u_1\left(\mathfrak{r}\right)-u_2\left(\mathfrak{r}\right)\mid & \le {\eta }_2\parallel u_1-u_2\parallel \left[M_2+l_{2}r[H_2+{\gamma }_{2}r]\right]\hfill \\ \hfill & +l_2[N_2+{\eta }_{2}r]\left[\parallel x_1-x_2\parallel [H_2+{\gamma }_{2}r]+r{\gamma }_1\parallel u_1-u_2\parallel \right].\hfill \end{array}$$

Taking the supremum $\mathfrak{r}\in {\mathbb{R}}_{+}$, we obtain

$$\begin{array}{cc}\hfill \parallel u_1-u_2\parallel & \le \eta \parallel u_1-u_2\parallel \left[M+lr[H+\gamma r]\right]\hfill \\ \hfill & +l[N+\eta r]\left[\parallel x_1-x_2\parallel [H+\gamma r]+r\gamma \parallel u_1-u_2\parallel \right].\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \parallel v_1-v_2\parallel & =\parallel (x_1,u_1)-(x_2,u_2)\parallel \hfill \\ \hfill & =\parallel (x_1-x_2,u_1-u_2)\parallel \hfill \\ \hfill & =max\{\parallel x_1-x_2\parallel ,\parallel u_1-u_2\parallel \}\hfill \\ \hfill & \le \eta \parallel v_1-v_2\parallel \left[M+lr[H+\gamma r]\right]\hfill \\ \hfill & +l[N+\eta r]\left[\parallel v_1-v_2\parallel [H+\gamma r]+r{\gamma }_1\parallel v_1-v_2\parallel \right].\hfill \end{array}$$

Then,

$$\begin{array}{c}\hfill \left[1-\left(\eta \left[M+lr[H+\gamma r]\right]+l[N+\eta r][H+\gamma r]+rl\gamma [N+\eta r]\right)\right]\parallel v_1-v_2\parallel \le 0.\end{array}$$

which implies

$$\parallel v_1-v_2\parallel =0\Rightarrow v_1=v_2.$$

This proves the existence of a unique solution for the cubic functional integral Equation (1) with a feedback control (2) in the space $BC\left({\mathbb{R}}_{+}\right).$ □

Example 2.

Consider the cubic functional integral equation

$$x\left(\mathfrak{r}\right)=\left(\frac{2+\mid x\left(\mathfrak{r}\right)\mid }{2e^{\mathfrak{r}+1}\left(1+\mid x\left(\mathfrak{r}\right)\mid \right)}\right)\frac{{\mathfrak{r}}^2}{1+{\mathfrak{r}}^2}arctan\left(\mathfrak{r}+\mu \left(\mathfrak{r}\right){\int }_0^{\mathfrak{r}}\frac{\mathfrak{r}\mid 2+cos\left(x\right(s\left)\right)\mid }{2\sqrt{400+{\mathfrak{r}}^2}}ds\right),\mathfrak{r}\ge 0.$$

(16)

with a feedback control given by

$$\mu \left(\mathfrak{r}\right)=\left(\frac{4+\mid \mu \left(\mathfrak{r}\right)\mid }{4e^{\mathfrak{r}+1}\left(1+\mid x\left(\mathfrak{r}\right)\mid \right)}\right)\frac{{\mathfrak{r}}^2}{1+{\mathfrak{r}}^2}arccot\left(\mathfrak{r}+x\left(\mathfrak{r}\right){\int }_0^{\mathfrak{r}}\frac{{\mathfrak{r}}^2\mid cos\left(\mu \left(s\right)\right)\mid }{\sqrt{1600+{\mathfrak{r}}^2}}ds\right),\mathfrak{r}\ge 0.$$

(17)

Set

$$\begin{array}{ccc}\hfill f_1(\mathfrak{r},x\left(\mathfrak{r}\right))& =& \frac{2+\mid x\left(\mathfrak{r}\right)\mid }{2e^{\mathfrak{r}+1}\left(1+\mid x\left(\mathfrak{r}\right)\mid \right)},f_2(\mathfrak{r},x\left(\mathfrak{r}\right))=\frac{4+\mid x\left(\mathfrak{r}\right)\mid }{4e^{\mathfrak{r}+1}\left(1+\mid x\left(\mathfrak{r}\right)\mid \right)},\hfill \\ \hfill g_1(\mathfrak{r},x\left(\mathfrak{r}\right))& =& \frac{\mathfrak{r}}{1+{\mathfrak{r}}^2}arctan(\mathfrak{r}+x\left(\mathfrak{r}\right)),g_2(\mathfrak{r},x\left(\mathfrak{r}\right))=\frac{\mathfrak{r}}{1+{\mathfrak{r}}^2}arccot(\mathfrak{r}+x\left(\mathfrak{r}\right)),\hfill \\ \hfill h_1(\mathfrak{r},s,x\left(\mathfrak{r}\right))& =& \frac{\mathfrak{r}\mid 2+cos\left(x\right(s\left)\right)\mid }{2\sqrt{400+{\mathfrak{r}}^2}},h_2(\mathfrak{r},s,x\left(\mathfrak{r}\right))=\frac{{\mathfrak{r}}^2\mid cos\left(x\left(s\right)\right)\mid }{\sqrt{1600+{\mathfrak{r}}^2}},\hfill \end{array}$$

Clearly, functions $f_i(i=1,2)$ are continuous. Consequently, for any $x,\mu \in R\mathrm{and}\mathfrak{r}\in [0,1]$

$$\begin{array}{ccc}\hfill \mid f_1(\mathfrak{r},x\left(\mathfrak{r}\right))-f_1(\mathfrak{r},\mu \left(\mathfrak{r}\right))\mid & \le & \frac{1}{2e^2}\mid x\left(\mathfrak{r}\right)-\mu \left(\mathfrak{r}\right)\mid ,\hfill \\ \hfill \mid f_2(\mathfrak{r},x\left(\mathfrak{r}\right))-f_2(\mathfrak{r},\mu \left(\mathfrak{r}\right))\mid & \le & \frac{1}{4e^2}\mid x\left(\mathfrak{r}\right)-\mu \left(\mathfrak{r}\right)\mid \hfill \end{array}$$

This shows that condition $\left(v\right)$ has been satisfied. with ${\eta }_1\left(t\right)=\frac{1}{2e^{t+1}},{\eta }_2\left(t\right)=\frac{1}{4e^{t+1}}$. Additionally, we have $f_i(t,0)=\frac{1}{e^{t+1}},N=\frac{1}{e^2}$, and $\eta =\frac{1}{2e^2}$.

On the other hand, we have

$$\begin{array}{c}\hfill \mid g_i(\mathfrak{r},x\left(\mathfrak{r}\right))-g_i(\mathfrak{r},\mu \left(\mathfrak{r}\right))\mid \le \frac{{\mathfrak{r}}^2}{1+{\mathfrak{r}}^2}\mid x\left(\mathfrak{r}\right)-\mu \left(t\right)\mid \le \frac{\mid x\left(\mathfrak{r}\right)-\mu \left(\mathfrak{r}\right)\mid }{2},(i=1,2),\end{array}$$

where $l=\frac{1}{2}$ and $M=\frac{\pi }{2},$ Further, notice that the function $h(\mathfrak{r},s,x)$ satisfies assumption ($v^{*}$), where

$$\mid h_i(\mathfrak{r},s,x\left(\mathfrak{r}\right))-h_i(\mathfrak{r},s,\mu \left(\mathfrak{r}\right))\mid \le \frac{\mid x\left(\mathfrak{r}\right)-\mu \left(\mathfrak{r}\right)\mid }{40}.$$

Putting $\gamma =0.025,H=0.05$.

We easily determine r = 0.213469, where r is a positive solution of the Equation (3) and

$$\eta \left[M+lr[H+\gamma r]\right]+l[N+\eta r]r\gamma =0.1070914187<1.$$

Thus, in view of Theorem 3, we conclude that the problem (16) and (17) has a unique solution.

3.1. Continuous Dependency of the State Variable on the Control Variable

We now demonstrate that the solution of the problem (1) and (2) is continuously dependent on the control variable $u.$

Definition 2.

The solution of the cubic functional integral Equation (1) depends continuously on the control variable $u,$ if $\forall ϵ>0,\exists \delta >0$, such that

$$\parallel u-u^{*}\parallel \le \delta \Rightarrow \parallel x-x^{*}\parallel \le ϵ$$

where $x^{*}$ and $u^{*}$ satisfy the following

$$x^{*}\left(\mathfrak{r}\right)=f_1(\mathfrak{r},x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)).g_1\left(\mathfrak{r},u^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x^{*}\left(s\right))ds\right),\mathfrak{r}\in {\mathbb{R}}_{+}$$

and

$$u^{*}\left(\mathfrak{r}\right)=f_2(\mathfrak{r},u^{*}\left({\varphi }_2\left(\mathfrak{r}\right)\right)).g_2\left(\mathfrak{r},x^{*}\left({\varphi }_2\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_2\left(\mathfrak{r}\right)}{h}_2(\mathfrak{r},s,u^{*}\left(s\right))ds\right),\mathfrak{r}\in {\mathbb{R}}_{+}.$$

Theorem 4.

Let the assumptions of Theorem 3 be satisfied, if condition (3) is satisfied. Then, the state variable x is continuously dependent on the control variable $u$.

Proof. 

Let $v=(x,u)$, $v^{*}=(x^{*},u^{*})$ be two solutions of the problem (1) and (2). Then, Let $\delta >0$ be given such that $\parallel u-u^{*}\parallel \le \delta ,$ then

$$\begin{array}{cc}\hfill & \mid x\left(\mathfrak{r}\right)-x^{*}\left(\mathfrak{r}\right)\mid \hfill \\ \hfill & \le \mid f_1(\mathfrak{r},x\left({\varphi }_1\left(\mathfrak{r}\right)\right))g_1(\mathfrak{r},u\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x\left(s\right))ds)\hfill \\ \hfill & -f_1(\mathfrak{r},x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right))g_1(\mathfrak{r},u^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x^{*}\left(s\right))ds)\mid \hfill \\ \hfill & \le \mid f_1(\mathfrak{r},x\left({\varphi }_1\left(\mathfrak{r}\right)\right))-f_1(\mathfrak{r},x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right))\mid \mid g_1(\mathfrak{r},u\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x\left(s\right))ds)\mid \hfill \\ \hfill & +\mid f_1(\mathfrak{r},x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right))\mid \mid g_1(\mathfrak{r},u^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x^{*}\left(s\right))ds)-g_1(\mathfrak{r},u\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x\left(s\right))ds)\mid \hfill \\ \hfill & \le {\eta }_1\mid x\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(\mid g_1(\mathfrak{r},0)\mid +l_1\mid u\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[H_1+{\gamma }_1\mid x\left({\varphi }_1\left(s\right)\right)\mid ]ds\right]\hfill \\ \hfill & +l_1[\mid f_1(\mathfrak{r},0)\mid +{\eta }_1\mid x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid \left]\right[\mid u\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}{h}_1(\mathfrak{r},s,x\left(s\right))ds-u^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}h(\mathfrak{r},s,x^{*}\left(s\right))ds\mid \hfill \\ \hfill & \le {\eta }_1\mid x\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u_1\parallel ){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[H_1+{\gamma }_1\parallel x\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x^{*}\parallel \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u_2\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(\mathfrak{r},s,x\left(s\right))\mid ds\hfill \\ \hfill & +\mid u^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(\mathfrak{r},s,x\left(s\right))-h_1(\mathfrak{r},s,x^{*}\left(s\right))\mid ds]\hfill \\ \hfill & \le {\eta }_1\mid x\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u\parallel ){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[H_1+{\gamma }_1\parallel x\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x^{*}\parallel \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[H_1+{\gamma }_1\parallel x\parallel ]ds\hfill \\ \hfill & +\parallel u^{*}\parallel {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(\mathfrak{r},s,x\left(s\right))-h_1(\mathfrak{r},s,x^{*}\left(s\right))\mid ds]\hfill \\ \hfill & \le {\eta }_1\mid x\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u\parallel ){\int }_0^{\mathfrak{r}}[H_1+{\gamma }_1\parallel x\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x^{*}\parallel \left]\right[\mid u\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^t[H_1+{\gamma }_1\parallel x\parallel ]ds\hfill \\ \hfill & +\parallel u^{*}\parallel {\int }_0^{\mathfrak{r}}\mid h_1(\mathfrak{r},s,x\left(s\right))-h_1(\mathfrak{r},s,x^{*}\left(s\right))\mid ds]\hfill \\ \hfill & \le {\eta }_1\mid x\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u\parallel ){\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[H_1+{\gamma }_1\parallel x^{*}\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x^{*}\parallel \left]\right[\mid u\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}[H_1+{\gamma }_1\parallel x\parallel ]ds\hfill \\ \hfill & +\parallel u^{*}\parallel {\int }_0^{{\psi }_1\left(\mathfrak{r}\right)}\mid h_1(\mathfrak{r},s,x\left(s\right))-h_1(\mathfrak{r},s,x^{*}\left(s\right))\mid ds]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\eta }_1\mid x\left({\varphi }_1\left(\mathfrak{r}\right)\right)-x^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid ]\left[(M_1+l_1\parallel u\parallel ){\int }_0^{\mathfrak{r}}[H_1+{\gamma }_1\parallel x\parallel ]ds\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x^{*}\parallel \left]\right[\mid u_1\left({\varphi }_1\left(\mathfrak{r}\right)\right)-u^{*}\left({\varphi }_1\left(\mathfrak{r}\right)\right)\mid {\int }_0^{\mathfrak{r}}[H_1+{\gamma }_1\parallel x\parallel ]ds\hfill \\ \hfill & +\parallel u^{*}\parallel {\int }_0^{\mathfrak{r}}\mid {\gamma }_1\mid x\left(s\right))-x^{*}\left(s\right)\mid \mid ds]\hfill \\ \hfill & \le {\eta }_1\parallel x-x^{*}\parallel \left[M_1+l_1\parallel u\parallel [H_1+{\gamma }_1\parallel x\parallel ]\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x^{*}\parallel ]\left[\parallel u-u^{*}\parallel [H_1+{\gamma }_1\parallel x\parallel ]+\parallel u^{*}\parallel {\gamma }_1\parallel x_1-x^{*}\parallel \right]\hfill \\ \hfill & \le {\eta }_1\parallel x-x^{*}\parallel \left[M_1+l_1\parallel u\parallel [H_1+{\gamma }_1\parallel x\parallel ]\right]\hfill \\ \hfill & +l_1[N_1+{\eta }_1\parallel x^{*}\parallel ]\left[\parallel u-u^{*}\parallel [H_1+{\gamma }_1\parallel x\parallel ]+\parallel u^{*}\parallel {\gamma }_1\parallel x-x^{*}\parallel \right],\hfill \end{array}$$

then we obtain

$$\begin{array}{cc}\hfill \parallel x-x^{*}\parallel & \le \eta \parallel x-x^{*}\parallel \left[M+lr[H+\gamma r]\right]\hfill \\ \hfill & +l[N+\eta r]\left[\parallel u-u^{*}\parallel [H+\gamma r]+r\gamma \parallel x-x^{*}\parallel \right]\hfill \\ \hfill & \le \eta \parallel x-x^{*}\parallel \left[M+lr[H+\gamma r]\right]\hfill \\ \hfill & +l[N+\eta r]\left[\delta [H+\gamma r]+r\gamma \parallel x-x^{*}\parallel \right].\hfill \end{array}$$

Since $\eta \left[M+lr[H+\gamma r]\right]+l[N+\eta r][H+\gamma r]+rl\gamma [N+\eta r]<1,$ then

$\eta \left[M+lr[H+\gamma r]\right]+l[N+\eta r]r\gamma <1.$ Therefore,

$$\parallel x-x^{*}\parallel \le \frac{l[N+\eta r][H+\gamma r]}{1-\left(\eta \left[M+lr[H+\gamma r]\right]+l[N+\eta r]r\gamma \right)}\delta =ϵ.$$

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3.2. Discussions and Conclusions

The present paper aims to propose a feedback control $u\left(t\right)$ to a cubic functional integral Equation (1), and it seems natural to choose the control variable u in the form (2). We have discussed the solvability of a cubic functional integral Equation (1) with feedback control (2) on the real half-axis by applying the procedure linked to measures of noncompactness in $BC\left({\mathbb{R}}_{+}\right)$ by a specific modulus of continuity. Our study has lied in the space of bounded continuous functions $BC\left({\mathbb{R}}_{+}\right).$ This investigation for the problem (1) and (2) is equivalent to that of a coupled system of two functional integral Equations (1) and (2). The authors have initiated applying the technique associated with measures of noncompactness on a coupled system of integral equations in $BC\left({\mathbb{R}}_{+}\right)$ in [24]. This article is a continuation of applying the technique of MNC on a cubic functional integral equation, but in our case, Equation (1) is restricted to a feedback control (2). Furthermore, some particular cases and examples are illustrated. Finally, the continuous dependence of the state variable x on the control variable u is studied.