Non-Decreasing Solutions for (k,Υ)-Fractional Quadratic Integral Equations of Urysohn–Volterra Type

Abstract

In this paper, we investigate a $(k,{\rm Y})$ fractional quadratic integral equation in the Banach space of real-valued continuous functions on $[0,1]$. By using a measure of noncompactness associated with monotonicity and Darbo’s fixed point theorem, we provide sufficient conditions for the existence of at least one monotonic solution and analyze its stability. Finally, an illustrative example is presented to demonstrate the theoretical results, including several particular cases.

1. Introduction and Preliminaries

Fractional calculus is an essential and useful branch of mathematical analysis that investigates derivatives and integrals of fractional order. In [1,2], there were many definitions for fractional integral operators, such as Riemann–Liouville, Hadamard, Katugampola, and Erdelyi-Kober fractional integral operators. Recently, in 2017, Almeida [3] proposed a new definition of the fractional integral and called this operator the ${\rm Y}$-Caputo integral. This new definition is more generalized than the Riemann–Liouville, Hadamard, Erdelyi Kober, and Caputo operator kinds. Moreover, fractional differential and integral equations have become a central topic in applied mathematics due to their ability to model complex phenomena in physics, engineering, and biology. In recent years, various stability and solution properties of fractional systems have been extensively studied. For instance, stability analysis of multi-point boundary conditions with non-instantaneous integral impulses has been investigated [4], while fractional exponential stability for nonlinear conformable fractional-order delayed systems with delayed impulses was analyzed in [5]. Similarly, stability and bifurcation analysis of fractional-order single-gene regulatory models under novel PD $\alpha$ control laws [6], input-to-state stability of fractional-order systems [7], and Turing–Hopf bifurcation in diffusive Leslie–Gower models with weak Allee effects and fear effects on predators [8] have been considered. Exponentially stable multi-stage epidemic systems with discontinuous incidence were also explored [9], and the physical meaning and constraint conditions of the function ${\rm Y}\left(\zeta \right)$ were given by the continuous time random walk theory [10] and the boundedness theorem [11].

Integral equations introduce significant advancements in the domain of functional analysis. This is because integral equations are very important and are employed in many fields, especially those that come from real-life situations like nuclear energy [12], heat conduction [13], and electromagnetics [14].

Quadratic integral equations are frequently utilized in the domains of radiative transfer theory, kinetic gas theory, neutron transport theory, and traffic theory. The quadratic integral equation of the Chandrasekhar type is frequently encountered in numerous applications. Conversely, the concept of quadratic integral equations is extensively researched and has several applications in addressing real-world situations. This theory was initiated by examining a quadratic integral equation of the Chandrasekhar type (see [15,16,17]). Banas et al. [18,19], Darwish et al. [20,21,22], W. G. El-Sayed [23], and El-Borai, et al. [24,25] investigated several forms of quadratic integral equations, employing the measure of noncompactness as the fundamental instrument in their proofs.

The notion of measure of noncompactness is among the most valuable concepts in general topology. This concept is defined in various ways ([26,27,28]). Certain authors have attempted to present an axiomatic definition of the measure of noncompactness [29]. These measures are defined in Banach spaces ([26,27,28]), within a metric space [26], or in a locally convex space [29].

In recent years, numerous authors have investigated the utilization of strategies related to the measure of noncompactness to enhance outcomes in resolving nonlinear integral equations. This approach specifically depends on employing Darbo-type fixed point theorems via the Hausdorff or Kuratowski measures of noncompactness.

A generalization of the Euler gamma function, the k-gamma function ${\mathrm{\Gamma }}_k(.)$ was introduced by Daz and Pariguan in [30]. The k-Riemann–Liouville fractional integral operator was previously proposed by Habibullah and Mubeen in [31]. In this study, we focus on a $(k,{\rm Y})$-fractional quadratic integral equation of the Urysohn–Volterra type

$$z\left(\zeta \right)=\mathbb{G}(\zeta ,z\left(\zeta \right))+\frac{\mathbb{F}(\zeta ,z(\zeta \left)\right)}{k{\mathrm{\Gamma }}_k\left(\omega \right)}{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu (\zeta ,\eta ,z(\eta \left)\right)}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta ,$$

(1)

where $\zeta \in \mathcal{J}=[0,1]$ and $\omega \in (0,1)$. The functions ${\rm Y}:\mathcal{J}\to \mathbb{R}$, $\mathbb{G}:\mathcal{J}\times \mathbb{R}\to \mathbb{R}$, $\mathbb{F}:\mathcal{J}\times \mathbb{R}\to \mathbb{R}$ and $\nu :\mathcal{J}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are appropriate functions specified later. The aim of this paper is to establish the existence of at least one monotonic solution in the Banach space of real-valued continuous functions on the interval $[0,1]$ for (1). Throughout this article, $C\left(\mathcal{J}\right)$ represents the Banach space of continuous functions defined on the interval $\mathcal{J}$ endowed with the standard norm $\parallel z\parallel =\underset{\zeta \in \mathcal{J}}{max}\left|z\left(\zeta \right)\right|$. Also, we will let $L_p\left(\mathcal{J}\right)$, $(1\le p<\infty )$ indicate the Banach space of p-integrable functions on $\mathcal{J}$ endowed with the standard norm ${\parallel ·\parallel }_p$.

2. (k, Υ)-Riemann–Liouville Fractional Integrals and Derivatives

Definition 1 

([30]).  For $z\in \mathbb{C}$ with $Re\left(z\right)>0$ and $k>0,(k\in \mathbb{R})$, the k -gamma function ${\mathrm{\Gamma }}_k(.)$ is defined by

$${\mathrm{\Gamma }}_k\left(\tau \right)={\int }_0^{\infty }{\varsigma }^{\tau -1}{e}^{\frac{-{\varsigma }^k}{k}}d\varsigma .$$

(2)

It is well known that

$$\mathrm{\Gamma }(\tau )=\underset{k\to 1}{lim}{\mathrm{\Gamma }}_k(\tau ),{\mathrm{\Gamma }}_k(\tau )=k^{{\displaystyle \frac{\tau }{k}}-1}\mathrm{\Gamma }\left({\displaystyle \frac{\tau }{k}}\right)\mathrm{and}{\mathrm{\Gamma }}_k(\tau +k)=\tau {\mathrm{\Gamma }}_k(\tau ).$$

Definition 2 

([2]). Let $z\in L^1(\mathcal{J},\mathbb{R})$ and ${\rm Y}\left(·\right)$ be an increasing and positive continuous function on $\mathcal{J},$ having a continuous derivatives ${{\rm Y}}^{\prime }\left(·\right)\ne 0$ on $\left(0,1\right)$ such that ${\rm Y}\left(0\right)=0$. Then the Υ- Riemann–Liouville fractional integrals of a function z with respect to another function Υ on $\mathcal{J}$,

$${\mathcal{I}}_0^{\omega ;{\rm Y}}z(\zeta ):=\frac{1}{\mathrm{\Gamma }(\omega )}{\int }_0^{\zeta }\frac{z(\eta ))}{{({\rm Y}(\zeta )-{\rm Y}(\eta )}^{1-\omega }}{{\rm Y}}^{\prime }(\eta )d\eta ,\hspace{1em}\zeta \in \mathcal{J},\hspace{1em}\omega \in (0,1).$$

(3)

For completeness, we define ${\mathcal{I}}_a^{\omega ,{\rm Y}}z\left(0\right):=$ 0.

Definition 3 

([31]). Let $z\in L^1(\mathcal{J},\mathbb{R})$, $k>0$ and ${\rm Y}\left(·\right)$ be an increasing and positive continuous function on $\mathcal{J},$ having a continuous derivatives ${{\rm Y}}^{\prime }\left(·\right)\ne 0$ on $\left(0,1\right)$ such that ${\rm Y}\left(0\right)=0$. Then the $(k,{\rm Y})$- Riemann–Liouvilion Υ on $\mathcal{J}$,

$$^k{\mathcal{I}}_0^{\omega ;{\rm Y}}z\left(\zeta \right):=\frac{1}{k{\mathrm{\Gamma }}_k\left(\omega \right)}{\int }_0^{\zeta }\frac{\omega (\eta )}{{({\rm Y}(\zeta )-{\rm Y}(\eta )}^{1-\frac{\omega }{k}}}{{\rm Y}}^{\prime }(\eta )d\eta ,\hspace{1em}\zeta \in \mathcal{J},\hspace{1em}\omega \in (0,1).$$

(4)

Definition 4  

([32]). Let $\stackrel{-}{\alpha },k\in {\mathbb{R}}^{+}=(0,\infty ),\stackrel{-}{\beta }\in [0,1],{\rm Y}\in C^n([c,d],\mathbb{R}),{{\rm Y}}^{\prime }\left(\zeta \right)\ne 0,\zeta \in [c,d]$ and $z\in C^n([c,d],\mathbb{R})$. Then the $(k,{\rm Y})$-Hilfer fractional derivative of the function z of order $\stackrel{-}{\alpha }$ and type $\stackrel{-}{\beta }$, is defined by

$$^{k,H}{\mathcal{D}}_{c+}^{\stackrel{-}{\alpha },\stackrel{-}{\beta };{\rm Y}}z\left(\zeta \right)=^k{\mathcal{I}}_{c+}^{\stackrel{-}{\beta }(nk-\stackrel{-}{\alpha });{\rm Y}}{\left({\displaystyle \frac{k}{{{\rm Y}}^{\prime }(\zeta )}}{\displaystyle \frac{d}{d\zeta }}\right)}^n^k{\mathcal{I}}_{c+}^{(1-\stackrel{-}{\beta })(nk-\stackrel{-}{\alpha });{\rm Y}}z\left(\zeta \right),n=⌈{\displaystyle \frac{\stackrel{-}{\alpha }}{k}}⌉.$$

(5)

For $\stackrel{-}{\beta }=0$ and $\stackrel{-}{\beta }=1$, (14) reduces to

$$^{k,RL}{\mathcal{D}}_{c+}^{\stackrel{-}{\alpha };{\rm Y}}z(\zeta )={\left({\displaystyle \frac{k}{{{\rm Y}}^{\prime }(\zeta )}}{\displaystyle \frac{d}{d\zeta }}\right)}^n^k{\mathcal{I}}_{c+}^{(nk-\stackrel{-}{\alpha });{\rm Y}}z\left(\zeta \right),$$

(6)

and

$$^{k,C}{\mathcal{D}}_{c+}^{\stackrel{-}{\alpha };{\rm Y}}z\left(\zeta \right)=^k{\mathcal{I}}_{c+}^{nk-\stackrel{-}{\alpha };{\rm Y}}{\left({\displaystyle \frac{k}{{{\rm Y}}^{\prime }\left(\zeta \right)}}{\displaystyle \frac{d}{d\zeta }}\right)}^{n}z\left(\zeta \right),$$

(7)

which are, respectively, the $(k,{\rm Y})$-Riemann–Liouville and $(k,{\rm Y})$-Caputo fractional derivatives.

Now, let E be an infinite-dimensional Banach space equipped with the norm $∥.∥$. Denote by $\mathcal{B}(z,\rho )$ the closed ball centered at z with radius ρ, and write ${\mathcal{B}}_{\rho }$ for $\mathcal{B}(\theta ,\rho )$, where θ is the zero element of E. For a nonempty subset Z of E, we denote by $\stackrel{-}{Z}$ its closure and by $ConvZ$ its convex closure of Z ($ConvZ$ is defined as the smallest convex closed set containing Z). Furthermore, let ${\mathbb{M}}_E$ denote the family of all nonempty and bounded subsets of E, and let ${\mathbb{N}}_E$ be the subfamily consisting of all nonempty and relatively compact subsets of E.

Definition 5 

([33]). A function ${\mu }_1:{\mathbb{M}}_E\to [0,\infty )$ is considered a measure of noncompactness in $E$ if it fulfills the following conditions:

1. 

The family $ker{\mu }_1=\{Z\in {\mathbb{M}}_E:{\mu }_1\left(Z\right)=0\}$ is nonempty and $ker{\mu }_1\subset {\mathbb{N}}_E$;

2. 

$Z_1\subset Z_2\Rightarrow {\mu }_1\left(Z_1\right)\le {\mu }_1\left(Z_2\right)$;

3. 

${\mu }_1\left(\overline{Z}\right)={\mu }_1\left(ConvZ\right)={\mu }_1\left(Z\right)$;

4. 

${\mu }_1(\beta Z_1+(1-\beta )Z_2)\le \beta {\mu }_1\left(Z_1\right)+(1-\beta ){\mu }_1\left(Z_2\right)$ for $\beta \in [0,1]$;

5. 

If ${\left(Z_m\right)}_{m\in \mathbb{N}}$ is a sequence of closed sets from ${\mathbb{M}}_E$

such that $Z_{m+1}\subset Z_m,$ for $m=1,2,3,\dots ,$ and if

$$\underset{m\to \infty }{\lim }{\mu }_1(Z_m)=0,then{Z}_{\infty }={\cap }_{m=1}^{\infty }{Z}_{m}is not empty.$$

The family mentioned in 1. is termed the kernel of the measure of noncompactness ${\mu }_1.$

Next, we define the measure of noncompactness used in the following section (see [33,34]). Let Z be a nonempty, bounded subset of $\mathcal{J}$. For $z\in Z$ and $ϵ\ge 0$, let $\varpi (z,ϵ)$ denote the modulus of continuity of z, i.e.,

$$\varpi (z,ϵ)=\sup \left\{\right|z\left(\zeta \right)-z\left(\eta \right)|:\zeta ,\eta \in \mathcal{J},|\zeta -\eta |\le ϵ\}.$$

In addition, we set

$$\varpi (Z,ϵ)=\sup \left\{\varpi \right(z,ϵ):z\in Z\},$$

and

$${\varpi }_0(Z)=\underset{ϵ\to 0}{lim}\varpi (Z,ϵ).$$

Define

$${\mu }_2(z)=\sup \left\{\right|z\left(\eta \right)-z\left(\zeta \right)|-\left[z\right(\eta )-z(\zeta \left)\right]:\zeta ,\eta \in \mathcal{J},\zeta \le \eta \},$$

and

$${\mu }_2\left(Z\right)=\sup \left\{{\mu }_2\right(z):z\in Z\}.$$

All functions belonging to Z are nondecreasing on $\mathcal{J}$ if and only if ${\mu }_2\left(Z\right)=0$. Define the function ${\mu }_1$ on the family ${\mathbb{M}}_{C\left(\mathcal{J}\right)}$ as follows:

$${\mu }_1\left(Z\right)={\varpi }_0\left(Z\right)+{\mu }_2\left(Z\right).$$

The function ${\mu }_1$ quantifies noncompactness in the space $C\left(\mathcal{J}\right)$. We will utilize the subsequent fixed-point theorem established by Darbo [35]. To articulate this theorem, we require the subsequent formulation.

Definition 6 

([35]).Let U be a nonempty subset of a Banach space E and let $\mathcal{L}:U\to E$ be a continuous operator that maps bounded sets onto bounded ones. We state that $\mathcal{L}$ satisfies the Darbo condition (with a constant $l\ge 0$) relative to a measure of noncompactness ${\mu }_1$ if for any bounded subset Z of U, the following holds:

$${\mu }_1\left(\mathcal{L}Z\right)\le l{\mu }_1\left(Z\right).$$

If $\mathcal{L}$ satisfies the Darbo condition with $l<1$, then it is called a contraction operator with respect to ${\mu }_1$.

Theorem 1 

([35]). Let $\mathfrak{Q}$ be a nonempty, bounded, closed, and convex subset of the space E, and let $\mathcal{L}:\mathfrak{Q}\to \mathfrak{Q}$ be a contraction concerning the measure of noncompactness ${\mu }_1$. Consequently, $\mathcal{L}$ has a fixed point in the set $\mathfrak{Q}$.

Remark 1. 

Under the assumptions outlined in the above theorem, it can be demonstrated that the set $\mathrm{Fix}\mathcal{L}$ of fixed points of $\mathcal{L}$ belonging to $\mathfrak{Q}$ is an element of $\ker {\mu }_1$.

Remark 2. 

Under the assumptions outlined in the above theorem, it can be demonstrated that the set $\mathrm{Fix}\mathcal{L}$ of fixed points of $\mathcal{L}$ belonging to $\mathfrak{Q}$ is an element of $\ker {\mu }_1$.

3. Results

We need to divide the proof into a few steps. In fact, we will prove the following claims:

(c1)

$\mathbb{G}:\mathcal{J}\times \mathbb{R}\to \mathbb{R}$ is continuous, $\mathbb{G}:\mathcal{J}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ and ∃ a constant $a\ge 0$ such that $|\mathbb{G}(\zeta ,z_1)-\mathbb{G}(\zeta ,z_2)|\le a|z_1-z_2|$$\forall \zeta \in \mathcal{J}$ and $z_1,z_2\in \mathbb{R}$.

(c2)

The superposition operator $\mathfrak{G}$ defined by $\left(\mathfrak{G}z\right)\left(\zeta \right)=\mathbb{G}(\zeta ,z(\zeta \left)\right)$ satisfies that for any nonnegative function z, ${\mu }_2\left(\mathfrak{G}z\right)\le a{\mu }_2\left(z\right)$ where a is the same constant appearing in (c1).

(c3)

$\mathbb{F}:\mathcal{J}\times \mathbb{R}\to \mathbb{R}$ is continuous, $\mathbb{F}:\mathcal{J}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ and ∃ a constant $b\ge 0$ such that $|\mathbb{F}(\zeta ,z_1)-\mathbb{F}(\zeta ,z_2)|\le b|z_1-z_2|$ $\forall \zeta \in \mathcal{J}$ and $z_1,z_2\in \mathbb{R}$.

(c4)

The superposition operator $\mathfrak{F}$ defined by $\left(\mathfrak{F}z\right)\left(\zeta \right)=\mathbb{F}(\zeta ,z(\zeta \left)\right)$ satisfies that for any nonnegative function z, ${\mu }_2(\mathfrak{F}z)\le b{\mu }_2(z)$ where b is the same constant appearing in (c3).

(c5)

$\nu :\mathcal{J}\times \mathcal{J}\times \mathbb{R}\to \mathbb{R}$ is a continuous function, $\nu :\mathcal{J}\times \mathcal{J}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ and $\nu (\zeta ,\eta ,z)$ is nondecreasing relative to each variable ζ, η, and z, separately, ∃ a nondecreasing function $\chi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that $\left|\nu \right(\zeta ,\eta ,z\left)\right|\le \chi \left(\right|z\left|\right)$ $\forall \zeta ,\eta \in \mathcal{J}$ and $z\in \mathbb{R}$.

(c6)

The function ${\rm Y}:\mathcal{J}\to \mathbb{R}$ belongs to $C^1\left(\mathcal{J}\right)$ is nondecreasing.

(c7)

∃ a positive number ${\rho }_0$ that satisfies

$$(a{\rho }_0+g^{*}){\mathrm{\Gamma }}_k(\omega +k)+(b{\rho }_0+f^{*}){({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}\chi ({\rho }_0)\le {\rho }_0{\mathrm{\Gamma }}_k(\omega +k),$$

(8)

where $a{\mathrm{\Gamma }}_k(\omega +k){({\rm Y}(1)-{\rm Y}(0))}^{{\displaystyle \frac{\omega }{k}}}\chi ({\rho }_0)<{\mathrm{\Gamma }}_k(\omega +k)$,

$$g^{*}=\underset{\zeta \in \mathcal{J}}{\max }\mathbb{G}(\zeta ,0),$$

and

$$f^{*}=\underset{\zeta \in \mathcal{J}}{\max }\mathbb{F}(\zeta ,0).$$

Having established the necessary preliminaries, we are now prepared to state and demonstrate our principal result in this paper.

Theorem 2. 

If requirements (c1) through (c7) are satisfied, Equation (1) possesses at least one solution that is continuous and nondecreasing on the interval $\mathcal{J}$.

Proof. 

Let $\mathcal{T}$ be the operator corresponding to the right-hand side of Equation (1), namely, $\mathcal{T}z=z$ where

$$\left(\mathcal{T}z\right)\left(\zeta \right)=\left(\mathfrak{G}z\right)\left(\zeta \right)+\left(\mathfrak{F}z\right)\left(\zeta \right)\left(\mathcal{U}z\right)\left(\zeta \right),\zeta \in \mathcal{J},$$

(9)

and

$$\left(\mathcal{U}z\right)\left(\zeta \right)={\displaystyle \frac{1}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z(\eta \left)\right)}{{\left({\rm Y}\right(\zeta )-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta ,\zeta \in \mathcal{J}=[0,1],\omega \in (0,1).$$

(10)

For easy of presentation, we partition the proof into a series of steps.

To prove the assertion in Step (I), we show that $\mathcal{T}$ maps the space $C\left(\mathcal{J}\right)$ into itself.

Considering (c1) and (c3), it is sufficient to demonstrate that $\mathcal{U}$ maps $C\left(\mathcal{J}\right)$ into itself. Fix $ϵ>0$, take ${\zeta }_1,{\zeta }_2\in \mathcal{J}$ with $|{\zeta }_2-{\zeta }_1|\le ϵ$, and assume without loss of generality that ${\zeta }_2\ge {\zeta }_1$. Then, we have

$$\begin{array}{cc}\hfill |\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathcal{U}z\right)\left({\zeta }_1\right)|& =\left|{\displaystyle \frac{1}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_2,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{{\zeta }_1}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\left|\nu ({\zeta }_2,\eta ,z\left(\eta \right))-\nu ({\zeta }_1,\eta ,z\left(\eta \right))\right|}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_{{\zeta }_1}^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\left|\nu ({\zeta }_1,\eta ,z\left(\eta \right))\right|}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{{\zeta }_1}{{\rm Y}}^{\prime }\left(\eta \right)\left|\nu ({\zeta }_1,\eta ,z\left(\eta \right))\right|\hfill \\ \hfill & \times \left[{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{{\displaystyle \frac{\omega }{k}}-1}-{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{{\displaystyle \frac{\omega }{k}}-1}\right]d\eta .\hfill \end{array}$$

Now, let

$$\begin{array}{cc}\hfill {\varpi }_d(\nu ,ϵ)=\sup \{& |\nu ({\zeta }_2,\eta ,h)-\nu ({\zeta }_1,\eta ,h)|:\hfill \\ \hfill & \eta ,{\zeta }_1,{\zeta }_2\in \mathcal{J},\eta \le {\zeta }_1,\eta \le {\zeta }_2,|{\zeta }_2-{\zeta }_1|\le ϵ,h\in [-d,d]\}.\hfill \end{array}$$

Utilizing the fact that $({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(0\right))\ge ({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(0\right))$, we obtain

$$\begin{array}{cc}\hfill |\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathcal{U}z\right)\left({\zeta }_1\right)|& \le {\displaystyle \frac{1}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right){\varpi }_{\chi \left(∥z∥\right)}(\nu ,ϵ)}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_{{\zeta }_1}^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\chi \left(∥z∥\right)}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\displaystyle \frac{1}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{{\zeta }_1}{{\rm Y}}^{\prime }\left(\eta \right)\chi \left(∥z∥\right)[{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{{\displaystyle \frac{\omega }{k}}-1}\hfill \\ \hfill & -{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{{\displaystyle \frac{\omega }{k}}-1}]d\eta \hfill \\ \hfill & \le {\displaystyle \frac{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}}{{\mathrm{\Gamma }}_k(\omega +k)}}{\varpi }_{\chi \left(∥z∥\right)}(\nu ,ϵ)+{\displaystyle \frac{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left({\zeta }_1\right))}^{{\displaystyle \frac{\omega }{k}}}}{{\mathrm{\Gamma }}_k(\omega +k)}}\chi \left(∥z∥\right)\hfill \\ \hfill & +{\displaystyle \frac{\chi \left(∥z∥\right)}{{\mathrm{\Gamma }}_k(\omega +k)}}[{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}-{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}\hfill \\ \hfill & +{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left({\zeta }_1\right))}^{{\displaystyle \frac{\omega }{k}}}]\hfill \\ \hfill & \le {\displaystyle \frac{{({\rm Y}\left(t_2\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}}{{\mathrm{\Gamma }}_k(\omega +k)}}{\varpi }_{\chi \left(∥z∥\right)}(\nu ,ϵ)\hfill \\ \hfill & +{\displaystyle \frac{\chi \left(∥z∥\right)}{{\mathrm{\Gamma }}_k(\omega +k)}}[{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}-{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}\hfill \\ \hfill & +2{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left({\zeta }_1\right))}^{{\displaystyle \frac{\omega }{k}}}]\hfill \\ \hfill & \le {\displaystyle \frac{{({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}}{{\mathrm{\Gamma }}_k(\omega +k)}}{\varpi }_{\chi \left(∥z∥\right)}(\nu ,ϵ)+{\displaystyle \frac{2{\left[\varpi ({\rm Y},ϵ)\right]}^{\frac{\omega }{k}}}{{\mathrm{\Gamma }}_k(\omega +k)}}\chi \left(∥z∥\right).\hfill \end{array}$$

Thus,

$$\varpi (\mathcal{U}z,ϵ)\le {\displaystyle \frac{1}{{\mathrm{\Gamma }}_k(\omega +k)}}[{({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}{\varpi }_{\chi \left(∥z∥\right)}(\nu ,ϵ)+2{[\varpi ({\rm Y},ϵ)]}^{\frac{\omega }{k}}\chi \left(∥z∥\right)].$$

(11)

If $ϵ\to 0$ we have $\varpi ({\rm Y},ϵ)\to 0$ and ${\varpi }_{\chi \left(∥z∥\right)}(\nu ,ϵ)\to 0$ due to the uniform continuity of the function ν on $\mathcal{J}\times \mathcal{J}\times [-\chi \left(∥z∥\right),\chi \left(∥z∥\right)]$. Consequently, the function $\mathcal{U}z$ is continuous on $\mathcal{J}$.

To prove the assertion in Step (II), $\mathcal{T}$ maps the ball ${\mathcal{B}}_{{\rho }_0}$ into itself.

For $\zeta \in \mathcal{J}$, we have

$$\begin{array}{cc}\hfill \left|\right(\mathcal{T}z\left)\right(\zeta \left)\right|& \le \left|\mathbb{G}(\zeta ,z\left(\zeta \right))\right|+{\displaystyle \frac{\left|\mathbb{F}\right(\zeta ,z\left(\zeta \right)\left)\right|}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}\left|{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu (\zeta ,\eta ,z\left(\eta \right))}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \right|\hfill \\ \hfill & \le \left|\mathbb{G}\right(\zeta ,z\left(\zeta \right))-\mathbb{G}(\zeta ,0\left)\right|+\left|\mathbb{G}\right(\zeta ,0\left)\right|\hfill \\ \hfill & +{\displaystyle \frac{\left|\mathbb{F}\right(\zeta ,z\left(\zeta \right))-\mathbb{F}(\zeta ,0\left)\right|+\left|\mathbb{F}\right(\zeta ,0\left)\right|}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\left|v(\zeta ,\eta ,z\left(\eta \right))\right|}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & \le (a∥z∥+g^{*})+{\displaystyle \frac{b∥z∥+f^{*}}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}\chi \left(∥z∥\right){\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & =(a∥z∥+g^{*})+{\displaystyle \frac{(b∥z∥+f^{*}){({\rm Y}\left(\zeta \right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}}{{\mathrm{\Gamma }}_k(\omega +k)}}\chi \left(∥z∥\right).\hfill \end{array}$$

And so

$$∥\mathcal{T}z∥\le (a∥z∥+g^{*})+{\displaystyle \frac{(b∥z∥+f^{*}){({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}}{{\mathrm{\Gamma }}_k(\omega +k)}}\chi \left(∥z∥\right).$$

(12)

If $∥z∥\le {\rho }_0$ then by (c7), inequality (12) yields

$$∥\mathcal{T}z∥\le (a{\rho }_0+g^{*})+{\displaystyle \frac{(b{\rho }_0+f^{*}){({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}}{{\mathrm{\Gamma }}_k(\omega +k)}}\chi \left({\rho }_0\right)\le {\rho }_0.$$

Consequently, the operator $\mathcal{T}$ maps ${\mathcal{B}}_{{\rho }_0}$ into itself.

To prove the assertion in Step (III), $\mathcal{T}$ maps the set

$${\mathcal{B}}_{{\rho }_0}^{+}=\{z\in {\mathcal{B}}_{{\rho }_0}:z\left(\zeta \right)\ge 0,\zeta \in \mathcal{J}\}$$

into itself.

Notice that the set ${\mathcal{B}}_{{\rho }_0}^{+}$ is nonempty, bounded, closed, and convex. Consequently, based on our assumptions, we observe that $\mathcal{T}$ maps ${\mathcal{B}}_{{\rho }_0}^{+}$ into itself.

To prove the assertion in Step (IV), we show that $\mathcal{T}$ is continuous on ${\mathcal{B}}_{{\rho }_0}^{+}$.

Fix $ϵ>0$ and take $z_1,z_2\in {\mathcal{B}}_{{\rho }_0}^{+}$ with $∥z_1-z_2∥\le ϵ$. Then, for $\zeta \in \mathcal{J}$, it can easily seen that

$$\begin{array}{cc}\hfill |\left(\mathcal{T}{z}_1\right)\left(\zeta \right)-\left(\mathcal{T}{z}_2\right)\left(\zeta \right)|& \le |\mathbb{G}(\zeta ,z_1\left(\zeta \right))-\mathbb{G}(\zeta ,z_2\left(\zeta \right))|\hfill \\ \hfill & +{\displaystyle \frac{1}{K{\mathrm{\Gamma }}_k\left(\omega \right)}}\left|\mathbb{F}(\zeta ,z_1\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z_1\left(\eta \right))}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right.\hfill \\ \hfill & -\left.\mathbb{F}(\zeta ,z_2\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z_2\left(\eta \right))}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right|\hfill \\ \hfill & \le a|z_1\left(\zeta \right)-z_2\left(\zeta \right)|\hfill \\ \hfill & +{\displaystyle \frac{1}{K{\mathrm{\Gamma }}_k\left(\omega \right)}}\left|\mathbb{F}(\zeta ,z_1\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z_1\left(\eta \right))}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right.\hfill \\ \hfill & -\left.\mathbb{F}(\zeta ,z_2\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z_1\left(\eta \right))}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right|\hfill \\ \hfill & +{\displaystyle \frac{1}{K{\mathrm{\Gamma }}_k\left(\omega \right)}}\left|\mathbb{F}(\zeta ,z_2\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z_1\left(\eta \right))}{{\left({\rm Y}\left(\zeta \right)\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right.\hfill \\ \hfill & -\left.\mathbb{F}(\zeta ,z_2\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z_2\left(\eta \right))}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right|\hfill \\ \hfill & \le a|z_1\left(\zeta \right)-z_2\left(\zeta \right)|\hfill \\ \hfill & +{\displaystyle \frac{|\mathbb{F}(\zeta ,z_1\left(\zeta \right))-\mathbb{F}(\zeta ,z_2\left(\zeta \right))|}{K{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\left|\nu (\zeta ,\eta ,z_1\left(\eta \right))\right|}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\displaystyle \frac{\left|\mathbb{F}\right(\zeta ,z_2\left(\zeta \right)\left)\right|}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)|\nu (\zeta ,\eta ,z_1\left(\eta \right))-\nu (\zeta ,\eta ,z_2\left(\eta \right))|}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & \le a|z_1\left(\zeta \right)-z_2\left(\zeta \right)|+{\displaystyle \frac{b|z_1\left(\zeta \right)-z_2\left(\zeta \right)|}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\chi \left(∥z_1∥\right)}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\displaystyle \frac{\left|\mathbb{F}\right(\zeta ,z_2\left(\zeta \right))-\mathbb{F}(\zeta ,0)|+\left|\mathbb{F}(\zeta ,0)\right|}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right){\delta }_{\nu }\left(ϵ\right)}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & \le a|z_1\left(\zeta \right)-z_2\left(\zeta \right)|+{\displaystyle \frac{b|z_1\left(\zeta \right)-z_2\left(\zeta \right)|}{{\mathrm{\Gamma }}_k(\omega +k)}}{({\rm Y}\left(\zeta \right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}\chi \left(∥z_1∥\right)\hfill \\ \hfill & +{\displaystyle \frac{b|z_2\left(\zeta \right)|+f^{*}}{{\mathrm{\Gamma }}_k(\omega +k)}}{({\rm Y}\left(\zeta \right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}{\delta }_{\nu }\left(ϵ\right).\hfill \end{array}$$

Where we denoted

$${\delta }_{\nu }\left(ϵ\right)=sup\left\{|\nu (\zeta ,\eta ,u_1)-\nu (\zeta ,\eta ,u_2)|:\zeta ,\eta \in \mathcal{J},u_1,u_2\in [0,\chi \left({\rho }_0\right)],∥u_1-u_2∥\le ϵ\right\}.$$

Therefore,

$$\begin{array}{ccc}\hfill ∥\mathcal{T}{z}_1-\mathcal{T}{z}_2∥& \le & \left(a+{\displaystyle \frac{b}{{\mathrm{\Gamma }}_k(\omega +k)}}{({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}\chi \left({\rho }_0\right)\right)∥z_1-z_2∥\hfill \\ & +& {\displaystyle \frac{b{\rho }_0+f^{*}}{{\mathrm{\Gamma }}_k(\omega +k)}}{({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}{\delta }_{\nu }\left(ϵ\right).\hfill \end{array}$$

(13)

As $ϵ\to 0$, we have ${\delta }_{\nu }\left(ϵ\right)\to 0$ since ν is uniformly continuous on the set $\mathcal{J}\times \mathcal{J}\times [0,\chi \left({\rho }_0\right)]$. Consequently, it follows from (13) that $\mathcal{T}$ is continuous on ${\mathcal{B}}_{{\rho }_0}^{+}$.

To prove the assertion in Step (V), we estimate $\mathcal{T}$ with respect to the monotonic term ${\mu }_2$.

We take $\varnothing \ne Z\subset {\mathcal{B}}_{{\rho }_0}^{+}$ and fix an arbitrary $z\in Z$ and ${\zeta }_1,{\zeta }_2\in \mathcal{J}$ with ${\zeta }_1\le {\zeta }_2$. Consequently, considering our assumptions, we obtain

$$\begin{array}{cc}\hfill {\mu }_2\left(\mathcal{T}z\right)& \le |\left(\mathcal{T}z\right)\left({\zeta }_2\right)-\left(\mathcal{T}z\right)\left({\zeta }_1\right)|-[\left(\mathcal{T}z\right)\left({\zeta }_2\right)-\left(\mathcal{T}z\right)\left({\zeta }_1\right)]\hfill \\ \hfill & \le |\left(\mathfrak{G}z\right)\left({\zeta }_2\right))-\left(\mathfrak{G}z\right)\left({\zeta }_1\right)|-[\left(\mathfrak{G}z\right)\left({\zeta }_2\right)-\left(\mathfrak{G}z\right)\left({\zeta }_1\right)]\hfill \\ \hfill & +|\left(\mathfrak{F}z\right)\left({\zeta }_2\right)\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left(\mathcal{U}z\right)\left({\zeta }_1\right)|\hfill \\ \hfill & -[\left(\mathfrak{F}z\right)\left({\zeta }_2\right)\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left(\mathcal{U}z\right)\left({\zeta }_1\right)]\hfill \\ \hfill & \le {\mu }_2\left(\mathfrak{G}z\right)+|\left(\mathfrak{F}z\right)\left({\zeta }_2\right)\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left(\mathcal{U}z\right)\left({\zeta }_2\right)|\hfill \\ \hfill & +|\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left(\mathcal{U}z\right)\left({\zeta }_1\right)|\hfill \\ \hfill & -[\left(\mathfrak{F}z\right)\left({\zeta }_2\right)\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left(\mathcal{U}z\right)\left({\zeta }_2\right)]\hfill \\ \hfill & -[\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left(\mathcal{U}z\right)\left({\zeta }_1\right)]\hfill \\ \hfill & \le {\mu }_2\left(\mathfrak{G}z\right)+{\displaystyle \frac{{\mu }_2\left(\mathfrak{F}z\right)}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_2,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\displaystyle \frac{\left(\mathfrak{F}z\right)\left({\zeta }_1\right)}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}\left\{\left|{\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_2,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta -{\int }_0^{{\zeta }_1}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \right|\right.\hfill \\ \hfill & \left.-\left[{\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_2,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta -{\int }_0^{{\zeta }_1}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \right]\right\}.\hfill \end{array}$$

Next, we will show that

$${\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_2,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \ge {\int }_0^{{\zeta }_1}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta .$$

Indeed, using the monotonicity of ν and Υ, for ${\zeta }_2\ge {\zeta }_1$, we have

$$\begin{array}{cc}\hfill & {\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_2,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta -{\int }_0^{{\zeta }_1}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & ={\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_2,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta -{\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta -{\int }_0^{{\zeta }_1}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\int }_0^{{\zeta }_1}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta -{\int }_0^{{\zeta }_1}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & \ge {\int }_{{\zeta }_1}^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\int }_0^{{\zeta }_1}{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,\eta ,z\left(\eta \right))[{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{{\displaystyle \frac{\omega }{k}}-1}-{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{{\displaystyle \frac{\omega }{k}}-1}]d\eta \hfill \\ \hfill & \ge {\int }_{{\zeta }_1}^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,{\zeta }_1,z\left({\zeta }_1\right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\int }_0^{{\zeta }_1}{{\rm Y}}^{\prime }\left(\eta \right)\nu ({\zeta }_1,{\zeta }_1,z\left({\zeta }_1\right))[{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{{\displaystyle \frac{\omega }{k}}-1}-{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{{\displaystyle \frac{\omega }{k}}-1}]d\eta \hfill \\ \hfill & =\nu ({\zeta }_1,{\zeta }_1,z\left({\zeta }_1\right))\left[{\int }_0^{{\zeta }_2}{{\rm Y}}^{\prime }\left(\eta \right){({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{\frac{\omega }{k}-1}d\eta -{\int }_0^{{\zeta }_1}{{\rm Y}}^{\prime }\left(\eta \right){({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(\eta \right))}^{{\displaystyle \frac{\omega }{k}}-1}d\eta \right]\hfill \\ \hfill & =\nu ({\zeta }_1,{\zeta }_1,z\left({\zeta }_1\right)){\displaystyle \frac{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}-{({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}}{\frac{\omega }{k}}}\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

where

$${({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}\ge {({\rm Y}\left({\zeta }_1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}.$$

Therefore,

$$\begin{array}{cc}\hfill {\mu }_2\left(\mathcal{T}z\right)& \le {\mu }_2\left(\mathfrak{G}z\right)+{\displaystyle \frac{{\mu }_2\left(\mathfrak{F}z\right)}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{{\zeta }_2}{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)v({\zeta }_2,\eta ,z\left(\eta \right))}{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & \le {\mu }_2\left(\mathfrak{G}z\right)+{\displaystyle \frac{{({\rm Y}\left({\zeta }_2\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}}{{\mathrm{\Gamma }}_k(\omega +k)}}\chi \left({\rho }_0\right){\mu }_2\left(\mathfrak{F}z\right)\hfill \\ \hfill & \le \left(a+{\displaystyle \frac{b{({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}}{{\mathrm{\Gamma }}_k(\omega +k)}}\chi \left({\rho }_0\right)\right){\mu }_2\left(z\right).\hfill \end{array}$$

And consequently

$${\mu }_2\left(\mathcal{T}Z\right)\le \left(a+{\displaystyle \frac{b{({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}}{{\mathrm{\Gamma }}_k(\omega +k)}}\chi \left({\rho }_0\right)\right){\mu }_2\left(Z\right).$$

(14)

To prove the assertion in Step (VI), we derive an estimate of $\mathcal{T}$ with respect to ${\varpi }_0$.

Fix $ϵ>0$, take $z\in Z$ and ${\zeta }_1,{\zeta }_2\in \mathcal{J}$ with $|{\zeta }_2-{\zeta }_1|\le ϵ$, and assume, without loss of generality, that ${\zeta }_1\le {\zeta }_2$. Furthermore, based on our assumptions, we obtain

$$\begin{array}{cc}\hfill \varpi (\mathcal{T}z,ϵ)& =|\left(\mathcal{T}z\right)\left({\zeta }_2\right)-\left(\mathcal{T}z\right)\left({\zeta }_1\right)|\hfill \\ \hfill & \le |\left(\mathfrak{G}z\right)\left({\zeta }_2\right)-\left(\mathfrak{G}z\right)\left({\zeta }_1\right)|+|\left(\mathfrak{F}z\right)\left({\zeta }_2\right)\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left(\mathcal{U}z\right)\left({\zeta }_1\right)|\hfill \\ \hfill & \le |\mathbb{G}({\zeta }_2,z\left({\zeta }_2\right))-\mathbb{G}({\zeta }_1,z\left({\zeta }_2\right))|+|\mathbb{G}({\zeta }_1,z\left({\zeta }_2\right))-\mathbb{G}({\zeta }_1,z\left({\zeta }_1\right))|\hfill \\ \hfill & +|\left(\mathfrak{F}z\right)\left({\zeta }_2\right)\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathfrak{F}z\right)\left({\zeta }_2\right)\left(\mathcal{U}z\right)\left({\zeta }_1\right)|\hfill \\ \hfill & +|\left(\mathfrak{F}z\right)\left({\zeta }_2\right)\left(\mathcal{U}z\right)\left({\zeta }_1\right)-\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left(\mathcal{U}z\right)\left({\zeta }_1\right)|\hfill \\ \hfill & \le |\mathbb{G}({\zeta }_2,z\left({\zeta }_2\right))-\mathbb{G}({\zeta }_1,z\left({\zeta }_2\right))|+|\mathbb{G}({\zeta }_1,z\left({\zeta }_2\right))-\mathbb{G}({\zeta }_1,z\left({\zeta }_1\right))|\hfill \\ \hfill & +|\left(\mathfrak{F}z\right)\left({\zeta }_2\right)\left|\right|\left(\mathcal{U}z\right)\left({\zeta }_2\right)-\left(\mathcal{U}z\right)\left({\zeta }_1\right)|+|\left(\mathfrak{F}z\right)\left({\zeta }_2\right)-\left(\mathfrak{F}z\right)\left({\zeta }_1\right)\left|\right|\left(\mathcal{U}z\right)\left({\zeta }_1\right)|\hfill \\ \hfill & \le {\delta }_{\mathbb{G}}\left(ϵ\right)+a\varpi (z,ϵ)+{\displaystyle \frac{b{\rho }_0+f^{*}}{{\mathrm{\Gamma }}_k(\omega +k))}}{[({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}{\varpi }_{\chi \left({\rho }_0\right)}(\nu ,ϵ)\hfill \\ \hfill & +2{\left[\varpi ({\rm Y},ϵ)\right]}^{{\displaystyle \frac{\omega }{k}}}\chi \left({\rho }_0\right)]+{\displaystyle \frac{b\varpi (z,ϵ)+{\delta }_{\mathbb{F}}\left(ϵ\right)}{{\mathrm{\Gamma }}_k(\omega +k)}}\chi \left({\rho }_0\right){({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}},\hfill \end{array}$$

where ${\delta }_{\mathbb{H}}\left(ϵ\right)=sup\left\{\right|\mathbb{H}({\zeta }_2,y)-\mathbb{H}({\zeta }_1,y)|:{\zeta }_1,{\zeta }_2\in \mathcal{J},\mathbb{H}\in [0,{\rho }_0],|{\zeta }_2-{\zeta }_1|\le ϵ\}$.

Taking the supremum over $z\in Z$ and the limit as $ϵ\to 0$, we get

$${\varpi }_0\left(\mathcal{T}Z\right)\le \left(a+{\displaystyle \frac{b{({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}\chi \left({\rho }_0\right)}{{\mathrm{\Gamma }}_k(\omega +k)}}\right){\varpi }_0\left(Z\right).$$

(15)

To prove the assertion in Step (VII), we show that $\mathcal{T}$ is a contraction with respect to ${\mu }_1$. Noncompactness measure definition and inequality proofs (14) and (15) provide

$${\mu }_1\left(\mathcal{T}Z\right)\le \left(a+{\displaystyle \frac{b{({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}\chi \left({\rho }_0\right)}{{\mathrm{\Gamma }}_k(\omega +k)}}\right){\mu }_1\left(Z\right).$$

Since

$$a{\mathrm{\Gamma }}_k(\omega +k)+b\chi \left({\rho }_0\right){({\rm Y}\left(1\right)-{\rm Y}\left(0\right))}^{\frac{\omega }{k}}<{\mathrm{\Gamma }}_k(\omega +k).$$

Therefore, $\mathcal{T}$ is a contraction operator relative to ${\mu }_1$.

To prove the assertion in Step (VIII), we apply the Darbo fixed point theorem.

Considering the above steps, we may employ Theorem 1 to conclude that $\mathcal{T}$ possesses at least one fixed point, or, equivalently, that Equation (1) has at least one nondecreasing solution in ${\mathcal{B}}_{{\rho }_0}$. This completes the proof. □

Theorem 3. 

If requirements (c1) through (c7) are satisfied, Equation (1) is asymptotically stable.

Proof. 

Let z be a solution of (1) and let $z^{*}$ be another solution of (1). Then we have

$$\begin{array}{cc}\hfill |z\left(\zeta \right)-z^{*}\left(\zeta \right)|& \le |\mathbb{G}(\zeta ,z\left(\zeta \right))-\mathbb{G}(\zeta ,z^{*}\left(\zeta \right))|\hfill \\ \hfill & +{\displaystyle \frac{1}{K{\mathrm{\Gamma }}_k\left(\omega \right)}}\left|\mathbb{F}(\zeta ,z\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z(\eta \left)\right)}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right.\hfill \\ \hfill & -\left.\mathbb{F}(\zeta ,z^{*}\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z^{*}\left(\eta \right))}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right|.\hfill \end{array}$$

From the conditions (c1)–(c5), we get

$$\begin{array}{cc}\hfill |z\left(\zeta \right)-z^{*}\left(\zeta \right)|& \le a|z\left(\zeta \right)-z^{*}\left(\zeta \right)|+{\displaystyle \frac{1}{K{\mathrm{\Gamma }}_k\left(\omega \right)}}\left|\mathbb{F}(\zeta ,z\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z(\eta \left)\right)}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right.\hfill \\ \hfill & -\left.\mathbb{F}(\zeta ,z^{*}\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z(\eta \left)\right)}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right|\hfill \\ \hfill & +{\displaystyle \frac{1}{K{\mathrm{\Gamma }}_k\left(\omega \right)}}\left|\mathbb{F}(\zeta ,z^{*}\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z(\eta \left)\right)}{{\left({\rm Y}\left(\zeta \right)\right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right.\hfill \\ \hfill & -\left.\mathbb{F}(\zeta ,z^{*}\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\nu (\zeta ,\eta ,z^{*}\left(\eta \right))}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}{{\rm Y}}^{\prime }\left(\eta \right)d\eta \right|\hfill \\ \hfill & \le a|z\left(\zeta \right)-z^{*}\left(\zeta \right)|+{\displaystyle \frac{|\mathbb{F}(\zeta ,z\left(\zeta \right))-\mathbb{F}(\zeta ,z^{*}\left(\zeta \right))|}{K{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\left|\nu (\zeta ,\eta ,z\left(\eta \right))\right|}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\displaystyle \frac{\left|\mathbb{F}\right(\zeta ,z^{*}\left(\zeta \right)\left)\right|}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)|\nu (\zeta ,\eta ,z\left(\eta \right))-\nu (\zeta ,\eta ,z^{*}\left(\eta \right))|}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & \le a|z\left(\zeta \right)-z^{*}\left(\zeta \right)|+{\displaystyle \frac{b|z\left(\zeta \right)-z^{*}\left(\zeta \right)|}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right)\chi \left(∥z∥\right)}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & +{\displaystyle \frac{\left|\mathbb{F}\right(\zeta ,z^{*}\left(\zeta \right))-\mathbb{F}(\zeta ,0)|+\left|\mathbb{F}(\zeta ,0)\right|}{k{\mathrm{\Gamma }}_k\left(\omega \right)}}{\int }_0^{\zeta }{\displaystyle \frac{{{\rm Y}}^{\prime }\left(\eta \right){\delta }_{\nu }\left(ϵ\right)}{{({\rm Y}\left(\zeta \right)-{\rm Y}\left(\eta \right))}^{1-\frac{\omega }{k}}}}d\eta \hfill \\ \hfill & \le a|z\left(\zeta \right)-z^{*}\left(\zeta \right)|+{\displaystyle \frac{b|z\left(\zeta \right)-z^{*}\left(\zeta \right)|}{{\mathrm{\Gamma }}_k(\omega +k)}}{({\rm Y}\left(\zeta \right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}\chi \left(∥z∥\right)\hfill \\ \hfill & +{\displaystyle \frac{b|z^{*}\left(\zeta \right)|+f^{*}}{{\mathrm{\Gamma }}_k(\omega +k)}}{({\rm Y}\left(\zeta \right)-{\rm Y}\left(0\right))}^{{\displaystyle \frac{\omega }{k}}}{\delta }_{\nu }\left(ϵ\right).\hfill \end{array}$$

Thus, we obtain that

$$\left|z\left(\zeta \right)-z^{*}\left(\zeta \right)\right|\le \left(a+bM\chi \left({\rho }_0\right)\right)\left|z\left(\zeta \right)-z^{*}\left(\zeta \right)\right|+(b{\rho }_0+f^{*})M{\delta }_{\nu }\left(ϵ\right),$$

where

$$M={\displaystyle \frac{{\left({\rm Y}\left(1\right)-{\rm Y}\left(0\right)\right)}^{\frac{\omega }{k}}}{{\mathrm{\Gamma }}_k(\omega +k)}}.$$

It follows that

$$\begin{array}{cc}\hfill |z\left(\zeta \right)-z^{*}\left(\zeta \right)|& \le {\displaystyle \frac{(b{\rho }_0+f^{*})M{\delta }_{\nu }\left(ϵ\right)}{1-(a+bM\chi \left({\rho }_0\right))}}=ϵ.\hfill \end{array}$$

Hence, we have $|z\left(\zeta \right)-z^{*}\left(\zeta \right)|\le ϵ$, and (1) is asymptotically stable. □

4. Applications and Special Cases

In order to encompass the full scope of this paper, we provide examples showing how the obtained results can be applied to particular problems.

Example 1. 

Consider (1) having the form

$$z\left(\zeta \right)={\displaystyle \frac{\zeta z\left(\zeta \right)}{1+{\zeta }^2}}+{\displaystyle \frac{4z\left(\zeta \right)e^{-2\zeta }}{3{\mathrm{\Gamma }}_{\frac{1}{4}}\left(\frac{1}{2}\right)}}{\int }_0^{\zeta }{\displaystyle \frac{(1+\zeta +\eta )}{(1+\eta )}}ln\left({\displaystyle \frac{1+\zeta }{1+\eta }}\right)z\left(\eta \right)d\eta ,\zeta \in \mathcal{J},$$

(16)

Equation (16) represents a particular case of Equation (1) with $\omega ={\displaystyle \frac{1}{2}}$, $k={\displaystyle \frac{1}{4}}$, ${\rm Y}\left(\zeta \right)=ln(1+\zeta )$ and $\nu (\zeta ,\eta ,z)=(1+\zeta +\eta )z$, we have $\mathbb{G}(\zeta ,z\left(\zeta \right))={\displaystyle \frac{\zeta z\left(\zeta \right)}{1+{\zeta }^2}}$ and this function fulfills assumption (c1) and

$$|\mathbb{G}(\zeta ,z_1)-\mathbb{G}(\zeta ,z_2)|\le {\displaystyle \frac{1}{2}}|z_1-z_2|.$$

The function $\mathbb{F}(\zeta ,z\left(\zeta \right))={\displaystyle \frac{z\left(\zeta \right)e^{-2\zeta }}{3}}$ fulfills assumptions (c3) and (c4) with $b={\displaystyle \frac{1}{3}}$.

$$|\mathbb{F}(\zeta ,z_1)-\mathbb{F}(\zeta ,z_2)|\le {\displaystyle \frac{1}{3}}|z_1-z_2|.$$

Additionally, $\nu (\zeta ,\eta ,z)=(1+\zeta +\eta )z$ and this function fulfills the assumption (c5). Indeed,

$$\left|\nu \right(\zeta ,\eta ,z\left)\right|\le 3\left|z\right|.$$

Therefore, for any $\zeta ,\eta \in \mathcal{J}$ and $z\in \mathbb{R}$, (c5) is satisfied with $\chi \left(\rho \right)=3\rho$.

Ultimately, the inequality (c7) takes the form

$$\left({\displaystyle \frac{1}{2}}\right)\left({\displaystyle \frac{\rho }{8}}\right)+{\displaystyle \frac{{\rho }^2}{3}}\left(3{ln}^2\left(2\right)\right)\le {\displaystyle \frac{\rho }{8}},$$

and this admits ${\rho }_0=0.5$ as a positive solution. Moreover,

$$a{\mathrm{\Gamma }}_{{\displaystyle \frac{1}{4}}}\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}\right)+b{\left({\rm Y}\left(1\right)-{\rm Y}\left(0\right)\right)}^2\chi \left({\rho }_0\right)<{\mathrm{\Gamma }}_{{\displaystyle \frac{1}{4}}}\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}\right)$$

$${\displaystyle \frac{a}{8}}+{\left(3bln\left(2\right)\right)}^2\chi \left({\rho }_0\right)<{\displaystyle \frac{1}{8}}.$$

This proves that (c7) is fulfilled. Therefore, by Theorem 2, Equation (16) has at least one continuous and nondecreasing solution $z\left(\zeta \right)$ with $∥z∥\le {\rho }_0$.

Example 2. 

Consider (1) having the form

$$z\left(\zeta \right)={\displaystyle \frac{\zeta arctan\left(z\right(\zeta \left)\right)}{3+{\zeta }^2}}+{\displaystyle \frac{2\zeta sin\left(z\right(\zeta \left)\right)}{3(2+{\zeta }^2){\mathrm{\Gamma }}_{\frac{3}{4}}\left(\frac{3}{2}\right)}}{\int }_0^{\zeta }{\displaystyle \frac{(\sqrt{\zeta }-\sqrt{\eta })}{\sqrt{\eta }}}arctan\left({\displaystyle \frac{\zeta z\left(\eta \right)}{{\eta }^2+1}}\right)d\eta ,\zeta \in \mathcal{J},$$

(17)

Equation (17) represents a particular case of Equation (1) with $\omega ={\displaystyle \frac{3}{2}}$, $k={\displaystyle \frac{3}{4}}$, ${\rm Y}\left(\zeta \right)=\sqrt{\zeta }$ and

$\nu (\zeta ,\eta ,z)=arctan\left({\displaystyle \frac{\zeta z}{{\eta }^2+1}}\right)$,

we have $\mathbb{G}(\zeta ,z\left(\zeta \right))={\displaystyle \frac{\zeta arctan\left(z\right(\zeta \left)\right)}{3+{\zeta }^2}}$ and this function fulfills assumption (c1) and

$$|\mathbb{G}(\zeta ,z_1)-\mathbb{G}(\zeta ,z_2)|\le {\displaystyle \frac{1}{4}}|z_1-z_2|.$$

The function $\mathbb{F}(\zeta ,z\left(\zeta \right))={\displaystyle \frac{\zeta sin\left(z\right(\zeta \left)\right)}{(2+{\zeta }^2)}}$ fulfills assumptions (c3) and (c4) with $b={\displaystyle \frac{1}{3}}$.

$$|\mathbb{F}(\zeta ,z_1)-\mathbb{F}(\zeta ,z_2)|\le {\displaystyle \frac{1}{3}}|z_1-z_2|.$$

Additionally, $\nu (\zeta ,\eta ,z)=arctan\left({\displaystyle \frac{\zeta z}{{\eta }^2+1}}\right)$ and this function fulfills the assumption (c5). Indeed,

$$\left|\nu \right(\zeta ,\eta ,z\left)\right|\le \left|z\right|.$$

Therefore, for any $\zeta ,\eta \in \mathcal{J}$ and $z\in \mathbb{R}$, (c5) is satisfied with $\chi \left(\rho \right)=\rho$.

Ultimately, the inequality (c7) takes the form

$$\left({\displaystyle \frac{1}{4}}\right)\left({\displaystyle \frac{9\rho }{8}}\right)+{\displaystyle \frac{{\rho }^2}{3}}\le {\displaystyle \frac{9\rho }{8}},$$

and this admits ${\rho }_0=0.25$ as a positive solution. Moreover,

$$a{\mathrm{\Gamma }}_{{\displaystyle \frac{3}{4}}}\left({\displaystyle \frac{3}{2}}+{\displaystyle \frac{3}{4}}\right)+b{\left({\rm Y}\left(1\right)-{\rm Y}\left(0\right)\right)}^2\chi \left({\rho }_0\right)<{\mathrm{\Gamma }}_{{\displaystyle \frac{3}{4}}}\left({\displaystyle \frac{3}{2}}+{\displaystyle \frac{3}{4}}\right),$$

$${\displaystyle \frac{9a}{8}}+b\chi \left({\rho }_0\right)<{\displaystyle \frac{9}{8}}.$$

This proves that (c7) is fulfilled. Therefore, by Theorem 2, Equation (17) has at least one continuous and nondecreasing solution $z\left(\zeta \right)$ with $\parallel z\parallel \le {\rho }_0$.

Some special cases

Equation (1) is a very general fractional integral equation. Consequently, it includes the following special cases:

(1)

Putting $\omega =k=1$, then we have the following equation:

$$z\left(\zeta \right)=\mathbb{G}(\zeta ,z\left(\zeta \right))+\mathbb{F}(\zeta ,z\left(\zeta \right)){\int }_0^{\zeta }\nu (\zeta ,\eta ,z\left(\eta \right))d\eta ,\zeta \in \mathcal{J}=[0,1].$$

(18)

(2)

Letting ${\rm Y}\left(\zeta \right)=\zeta$, $\omega \in (0,1)$ and $k=1$, then we get the following integral equation of Riemann Liouville kernel kind

$$z\left(\zeta \right)=\mathbb{G}(\zeta ,z\left(\zeta \right))+\mathbb{F}(\zeta ,z\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{{(\zeta -\eta )}^{\omega -1}}{\mathrm{\Gamma }\left(\omega \right)}}\nu (\zeta ,\eta ,z\left(\eta \right))d\eta ,\zeta \in \mathcal{J}=[0,1].$$

(19)

(3)

Letting ${\rm Y}\left(\zeta \right)=ln\left(\zeta \right)$, $\omega \in (0,1)$, $\mathcal{J}=[1,e]$ and $k=1$, then we get the following weakly singular integral equations of Hadamard kernel kind

$$z\left(\zeta \right)=\mathbb{G}(\zeta ,z\left(\zeta \right))+\mathbb{F}(\zeta ,z\left(\zeta \right)){\int }_0^{\zeta }{ln\left({\displaystyle \frac{\zeta }{\eta }}\right)}^{\omega -1}{\displaystyle \frac{1}{\eta \mathrm{\Gamma }\left(\omega \right)}}\nu (\zeta ,\eta ,z\left(\eta \right))d\eta ,\zeta \in \mathcal{J}=[1,e].$$

(20)

(4)

Letting ${\rm Y}\left(\zeta \right)={\zeta }^{\sigma }$, $\sigma \in {\mathbb{R}}_{+}$, $\omega \in (0,1)$ and $k=1$, then we get the following integral equation of Erdely–Kober kernel kind

$$z\left(\zeta \right)=\mathbb{G}(\zeta ,z\left(\zeta \right))+\mathbb{F}(\zeta ,z\left(\zeta \right)){\int }_0^{\zeta }{\displaystyle \frac{\sigma {\zeta }^{\sigma -1}{({\zeta }^{\sigma }-{\eta }^{\sigma })}^{\omega -1}}{\mathrm{\Gamma }\left(\omega \right)}}\nu (\zeta ,\eta ,z\left(\eta \right))d\eta ,\zeta \in \mathcal{J}=[0,1].$$

(21)

Finally, we can state the following results for the above special cases.

Theorem 4. 

Under conditions (c1)–(c7), each of the integral Equations (19)–(21) has at least one asymptotically stable solution in $C\left(\mathcal{J}\right)$.

5. Conclusions

In this article, we investigated a quadratic integral equation involving a highly general $(k,{\rm Y})$-fractional orders and established the existence of at least one monotonic solution in the Banach space of real-valued continuous functions defined on the interval $[0,1]$. To study the solvability of the proposed equation, we employed the measure of noncompactness associated with monotonicity together with Darbo’s fixed-point theorem. Moreover, the stability of the proposed fractional quadratic integral equation was analyzed under appropriate assumptions. Ultimately, the results presented in this work extend several known results available in the literature and emphasize the generality of the proposed fractional quadratic integral equation. Some special cases were also discussed to illustrate the wide applicability of the obtained results, and an illustrative example was provided to support the theoretical findings. We hope these techniques and results developed in this paper will be useful for further investigations on fractional quadratic integral equations and related nonlinear problems.