Topology degree results on a G-ABC implicit fractional differential equation under three-point boundary conditions

Abstract

This research manuscript aims to study a novel implicit differential equation in the non-singular fractional derivatives sense, namely Atangana-Baleanu-Caputo ($\mathcal{A}\mathcal{B}\mathcal{C}$) of arbitrary orders belonging to the interval (2, 3] with respect to another positive and increasing function. The major results of the existence and uniqueness are investigated by utilizing the Banach and topology degree theorems. The stability of the Ulam-Hyers ($\mathbb{U}\mathbb{H}$) type is analyzed by employing the topics of nonlinear analysis. Finally, two examples are constructed and enhanced with some special cases as well as illustrative graphics for checking the influence of major outcomes.

1 Introduction

The derivatives of non-integer orders, or fractional derivatives, are a mathematical concept that extends differentiation beyond integer orders. These types of derivatives have a large domain of applications in numerous fields, including physics, engineering, and finance, for additional details see these manuscripts [1–8]. In the realm of fractional calculus has seen the advent of a fresh fractional operator by Atangana and Baleanu ($\mathcal{A}\mathcal{B}$) [9], which is free from singularity kernels. This operator is established through the Mittag-Lefler function in the Caputo and Riemann-Liouville contexts. The non-singular fractional operators are well-behaved and allow for more accurate modeling of the underlying system. The $\mathcal{A}\mathcal{B}$ operator can be used to describe phenomena such as diffusion, wave propagation, and viscoelasticity, and has applications in many fields such as image processing, signal analysis, and control theory. Non-singular fractional operators are important tools for researchers and practitioners seeking to understand and manipulate complex systems in a variety of contexts and they have stimulated a great deal of interest among researchers in their applicability to diverse problems, we indicate the readers to these works [10–19] and the references therein. Subsequently, authors of this work [20] popularized the $\mathcal{A}\mathcal{B}$ definition to contain differentiation and integration with respect to non-negative, non-decreasing function, leading to the development of the $\mathcal{G}-\mathcal{A}\mathcal{B}$ operator. In 2023, Abdeljawad et al. [21] expanded this operator to higher-order fractional derivatives and integrals. Furthermore, this type of fractional derivative is a generalization of the traditional derivative, where the derivative is taken with respect to a function rather than a variable. This type of derivative is commonly used in fractional calculus with various operators to describe complex systems with non-integer order dynamics. In fact, by taking this fractional derivative, the researchers can better model the behavior of these systems and gain a deeper understanding of their underlying dynamics, see for example [22–24] and references cited therein.

An implicit differential equation involving a fractional derivative of an unknown function that appears implicitly in the equation has several benefits, including the ability to model complex systems with memory effects, non-local interactions, and anomalous diffusion. These equations also accurately describe physical phenomena such as transport in porous media or viscoelastic materials [25–28]. In particular, Thabet and Kedim [29] studied a Hilfer fractional snap dynamic system on an infinite interval. Authors [30] discussed stability analysis of fractional pantograph implicit differential equations with initial boundary and impulsive conditions. Also, $\mathcal{A}\mathcal{B}$ fractional derivative used to investigate the stability of implicit differential problem by authors [31].

Recently in 2022, Shah et al. [32] used degree theory to establish qualitative results for the following differential equation:

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle {\mathrm{y}}^{‴}\left(\upsilon \right)+\varpi (\upsilon ,\text{y}(\upsilon \left)\right)=0,\upsilon \in \text{J}=[\iota ,\rho ],}\hfill \\ {\displaystyle \text{y}\left(\iota \right)=0,{\text{y}}^{\prime }\left(\iota \right)=0,\text{y}\left(\rho \right)=\xi \text{y}\left(\mathfrak{s}\right),\mathfrak{s}\in (\iota ,\rho ).}\hfill \end{array}\end{array}$$ (1.1)

Very recently, authors [33] extended the above equation (1.1) to the Caputo fractional order derivative for discussing the qualitative results and some types of $\mathbb{U}\mathbb{H}$ stability as in the following form:

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle {{}^{\mathfrak{C}}\mathfrak{D}}_{\iota }^{\mu }\text{y}\left(\upsilon \right)+\varpi (\upsilon ,\text{y}(\upsilon \left)\right)=0,\upsilon \in \text{J}=[\iota ,\rho ],\mu \in (2,3],}\hfill \\ {\displaystyle \text{y}\left(\iota \right)=0,{\text{y}}^{\prime }\left(\iota \right)=0,\text{y}\left(\rho \right)=\xi \text{y}\left(\mathfrak{s}\right),\mathfrak{s}\in (\iota ,\rho ).}\hfill \end{array}\end{array}$$ (1.2)

Inspired by the research mentioned above articles, in this current work, we study the qualitative properties of the solution for $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ implicit fractional differential equation (IFDE) of the following form:

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle {{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)=\varpi (\upsilon ,\text{y}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)),\upsilon \in \text{J}=[\iota ,\rho ],}\hfill \\ {\displaystyle \text{y}\left(\iota \right)=0,{\left[\text{y}\left(\iota \right)\right]}_{\mathcal{G}}^{\prime }=0,\text{y}\left(\rho \right)=\xi \text{y}\left(\mathfrak{s}\right),\mathfrak{s}\in (\iota ,\rho ),}\hfill \end{array}\end{array}$$ (1.3)

where ${{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}$ denotes the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ fractional derivatives of arbitrary order μ ∈ (2, 3] and the function $\varpi :\text{J}\times {\mathbb{R}}^2\to \mathbb{R}$ is continuous. Additionally, $\mathcal{G}:[\iota ,\rho ]\to {\mathbb{R}}^{+}$ be a non-decreasing and non-negative function with $\mathcal{G}\left(\upsilon \right)\in {\mathfrak{C}}^1((\iota ,\rho ),{\mathbb{R}}^{+})$, such that $\left[\text{y}\right(\upsilon ){]}_{\mathcal{G}}^{\prime }=(\frac{1}{{\mathcal{G}}^{\prime }\left(\upsilon \right)}\frac{d}{d\upsilon })\text{y}\left(\upsilon \right)$ and ${\mathcal{G}}^{\prime }\left(\upsilon \right)\ne 0,\forall \upsilon \in \text{J}.$ Furthermore, $\left(\mathcal{G}\right(\rho )-\mathcal{G}(\iota ){)}^2\ne \xi (\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right){)}^2,$ and $\xi \in \mathbb{R}$.

In this situation, we would like to indicate that our contributions are interesting and the Eq (1.3) is new in the framework of $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ fractional order derivatives which include $\mathcal{A}\mathcal{B}\mathcal{C}$ derivative as a special case when $\mathcal{G}\left(\upsilon \right)=\upsilon$. Moreover, an approach analysis in this work is different about methods used in these works [32, 33], and the Eq (1.3) covers many problems available in the literature studies, for instance,

(i) the Eq (1.3) reduces to problem (1.1) if μ → 3, and the implicit term omitted;

(ii) the Eq (1.3) returns to problem (1.2) if we replace the operator ${{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}$ by ${{}^{\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu }$ with omitting the implicit term.

The remaining parts of this paper are arranged as follows: Sec.2 is devoted to recalling the basic background materials related to fractional calculus and nonlinear analysis. Sec.3 discusses the existence and uniqueness theorems by using $\mathbb{F}\mathbb{P}\mathbb{T}s$. Sec.4 is investigated $\mathbb{U}\mathbb{H}$ stability. Finally, Sec.5 is dedicated to testing the effectiveness of main outcomes.

2 Preliminaries

In this situation, we present essential background material. Consider the space of continuous functions denoted by $\Omega ≔\mathfrak{C}(\text{J}=[\iota ,\rho ],\mathbb{R})$ which is Banach space gifted with the norm ‖y‖ = sup_υ∈[ι,ρ]|y(υ)|.

Definition 2.1 [34] Let $\text{y}:[\iota ,\rho ]\to \mathbb{R}$ and μ > 0, then the equation

$$\begin{array}{c}\hfill {{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)=\frac{1}{\Gamma \left(\mu \right)}{\int }_{\iota }^{\upsilon }{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(z\right))}^{\mu -1}{\mathcal{G}}^{\prime }\left(z\right)\text{y}\left(z\right)\text{d}z,\end{array}$$

is μ^th order of $\mathcal{G}$-Riemann–Liouville fractional integral, where Γ is Gamma function.

Definition 2.2 [21]. The $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ fractional derivatives of ${\text{y}}^{(m+1)}\in {\mathcal{H}}^1(\iota ,\rho )$ with order μ ∈ (m, m + 1], ν = μ − m, m = 0, 1, 2, …, is defined as

$$\begin{array}{cc}\left({{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\right)\left(\upsilon \right)\hfill & =\left({{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\nu ,\mathcal{G}}{\text{y}}_{\mathcal{G}}^{\left(m\right)}\right)\left(\upsilon \right)\hfill \\ \hfill & =\frac{\Phi \left(\mu -m\right)}{m+1-\mu }{{\displaystyle \int }}_{\iota }^{\upsilon }{\mathcal{G}}^{\prime }\left(z\right){\mathbb{E}}_{\mu -m}\left(\frac{-\left(\mu -m\right)}{m+1-\mu }{\left(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(z\right)\right)}^{\mu -m}\right){\text{y}}_{\mathcal{G}}^{\left(m+1\right)}\left(z\right)dz,\hfill \end{array}$$

where ${\text{y}}_{\mathcal{G}}^{\left(m\right)}\left(\upsilon \right)=({\displaystyle \frac{1}{{\mathcal{G}}^{\prime }\left(\upsilon \right)}}{\displaystyle \frac{d}{d\upsilon }}{)}^m\text{y}\left(\upsilon \right)$ and ${\text{y}}_{\mathcal{G}}^{\left(0\right)}\left(\upsilon \right)=\text{y}\left(\upsilon \right)$. If $\mu =k\in \mathbb{N}$, then $\left({{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\right)\left(\upsilon \right)={\text{y}}_{\mathcal{G}}^{\left(k\right)}\left(\upsilon \right)$. Furthermore, ${\mathbb{E}}_{\mu }$ is the Mittag-Leffler function

$$\begin{array}{c}\hfill {\mathbb{E}}_{\mu }\left(\text{x}\right)=\sum _{j=0}^{\infty }\frac{{\text{x}}^j}{\Gamma (\mu j+1)},Re\left(\mu \right)>0,\text{x}\in \mathbb{C},\end{array}$$

and Φ(μ) denotes the normalization function endowed by Φ(0) = Φ(1) = 1.

Definition 2.3 [21]. The following relation:

$$\begin{array}{cc}({{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\text{y})\left(\upsilon \right)\hfill & =\left({{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{m,\mathcal{G}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{v,\mathcal{G}}\text{y}\right)\left(\upsilon \right)=\left({{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{v,\mathcal{G}}{{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{m,\mathcal{G}}\text{y}\right)\left(\upsilon \right)\hfill \\ \hfill & =\frac{m+1-\mu }{\Phi \left(\mu -m\right)}{{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{m,\mathcal{G}}\text{y}\left(\upsilon \right)+\frac{\left(\mu -m\right)}{\Phi \left(\mu -m\right)}{{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right),\hfill \end{array}$$

is $\mathcal{G}-\mathcal{A}\mathcal{B}$ fractional integral of a function y with order μ ∈ (m, m + 1], ν = μ − m, m = 0, 1, 2, …, where ${{}^{\mathcal{R}\mathcal{L}}\Im }_{\iota }^{m,\mathcal{G}}$ is $\mathcal{G}$-Riemann–Liouville fractional integral.

Lemma 2.4 [21]. For μ ∈ (m, m + 1], ν = μ − m, m = 0, 1, 2, …, and $\text{y}\in {\mathcal{H}}^1(\text{J},\mathbb{R}),$ and $\mathcal{G}\in {\mathfrak{C}}^m(\text{J},\mathbb{R})$. Then

$$\begin{array}{c}\hfill \left({{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\right)\left(\upsilon \right)=\text{y}\left(\upsilon \right)-\sum _{r=0}^m{d}_r\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota ){)}^r,d_r\in \mathbb{R}.\end{array}$$

Lemma 2.5 [21]. For μ ∈ (m, m + 1], ν = μ − m, m = 0, 1, 2, …, ϵ > 0, and $\mathcal{G}\in {\mathfrak{C}}^m(\text{J},\mathbb{R}),$ with ${\mathcal{G}}^{\prime }\left(\upsilon \right)\ne 0.$ Then,

• (i) ${{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}{[\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right)]}^{ϵ}=\frac{(m+1-\mu )\Gamma (ϵ+1){[\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right)]}^{ϵ+m}}{\Phi (\mu -m)\Gamma (m+ϵ+1)}+\frac{(\mu -m)\Gamma (ϵ+1){[\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right)]}^{ϵ+\mu }}{\Phi (\mu -m)\Gamma (\mu +ϵ+1)}$ ;

• (ii) $\left({{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}1\right)\left(\upsilon \right)=\frac{(m+1-\mu ){[\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right)]}^m}{\Phi (\mu -m)\Gamma (m+1)}+\frac{(\mu -m){[\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -m)\Gamma (\mu +1)}$ .

Now, we about to introduce a definition of the Kuratowski’s measure of noncompactness χ(⋅) as follows:

$$\begin{array}{c}\hfill \chi \left(\mathcal{F}\right)=\text{inf}\{\epsilon >0:\mathcal{F}={\cup }_{i=1}^n{\mathcal{F}}_i\text{and}diam\left({\mathcal{F}}_i\right)\le \epsilon ,n\in \mathbb{N}\},\end{array}$$

where $diam\left({\mathcal{F}}_i\right)=sup\left\{\right|\text{y}-\widehat{\text{y}}|:\text{y},\widehat{\text{y}}\in {\mathcal{F}}_i\}$, and $\mathcal{F}$ is a bounded subset of the Banach space Ω. It is clear that $0\le \chi \left(\mathcal{F}\right)\le diam\left(\mathcal{F}\right)<+\infty$ [35].

Definition 2.6 [35] Let $\mathcal{D}:\mathcal{N}\to \mathcal{H}$ be bounded and continuous with $\mathcal{N}\subset \mathcal{H}$. Then, $\mathcal{D}$ will be χ-Lipschitz if ∃ ϵ ≥ 0, so that

$$\begin{array}{c}\hfill \chi \left(\mathcal{D}\right(B\left)\right)<ϵ\chi \left(B\right),\forall boundedB\subset \mathcal{N}.\end{array}$$

As well as, $\mathcal{D}$ is named as strict χ-contraction when ϵ < 1 holds.

Definition 2.7 [35] A function $\mathcal{D}$ is χ-condensing if

$$\begin{array}{c}\hfill \chi \left(\mathcal{D}\right(B\left)\right)<\chi \left(B\right),\forall B\subset \mathcal{N}bounded,\text{with}\chi \left(B\right)>0.\end{array}$$

So, $\chi \left(\mathcal{D}\right(B\left)\right)\ge \chi \left(B\right)$ gives χ(B) = 0. Also, $\mathcal{D}:\mathcal{N}\to \mathcal{H}$ is Lipschitz for ϵ > 0 such that

$$\begin{array}{c}\hfill \Vert \mathcal{D}\left(\text{y}\right)-\mathcal{D}\left(\widehat{\text{y}}\right)\Vert \le ϵ\Vert \text{y}-\widehat{\text{y}}\Vert forall\text{y},\widehat{\text{y}}\in \mathcal{N}.\end{array}$$

If ϵ < 1, in this case $\mathcal{D}$ is called a strict contraction.

Lemma 2.8 [35] $\mathcal{D}$ is χ-Lipschitz with constant ϵ = 0 iff $\mathcal{D}:\mathcal{N}\to \mathcal{H}$ is compact.

Lemma 2.9 [35] A function $\mathcal{D}$ is χ-Lipschitz with constant ϵ iff $\mathcal{D}:\mathcal{N}\to \mathcal{H}$ is Lipschitz with Lipschitz constant ϵ.

Theorem 2.10 [36] Let $\mathcal{D}:\Omega \to \Omega$ be an χ-condensing and

$$\begin{array}{c}\hfill \mathbb{W}=\{\text{y}\in \Omega :\zeta \in [0,1]existssothat\text{y}=\zeta \mathcal{D}(\text{y}\left)\right\}.\end{array}$$

If $\mathbb{W}$ is a bounded subset contained in Ω, i.e., a constant k > 0 exists with $\mathbb{W}\subset B_k\left(0\right),$ then $\mathfrak{d}\mathfrak{e}\mathfrak{g}(I-\zeta \mathcal{D},B_k\left(0\right),0)=1$ for all ζ ∈ [0, 1]. Therefore, $\mathcal{D}$ has a fixed point and the set $\mathbb{F}\mathbb{I}\mathbb{X}\left(\mathcal{D}\right)$ belongs to B_k(0).

3 Existence and uniqueness analysis

We introduce an equivalent integral fractional equation of the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3). Regarding this, we first derive the following lemma:

Lemma 3.1 Let $\mu \in (2,3],,{\text{y}}^{\left(3\right)}\in {\mathcal{H}}^1(\iota ,\rho ),\mathfrak{q}\in \Omega$ and $\mathcal{Z}=\left(\mathcal{G}\right(\rho )-\mathcal{G}(\iota ){)}^2-\xi (\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right){)}^2\ne 0$ . Then, the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ fractional differential problem:

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle {{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)=\mathfrak{q}\left(\upsilon \right),\upsilon \in \text{J}=[\iota ,\rho ],\mu \in (2,3],}\hfill \\ {\displaystyle \text{y}\left(\iota \right)=0,{\left[\text{y}\left(\iota \right)\right]}_{\mathcal{G}}^{\prime }=0,\text{y}\left(\rho \right)=\xi \text{y}\left(\mathfrak{s}\right),\mathfrak{s}\in (\iota ,\rho ),}\hfill \end{array}\end{array}$$ (3.1)

is equivalent to

$$\begin{array}{c}\text{y}\left(\upsilon \right)=\frac{\xi {(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\left(\mathfrak{s}\right)-\frac{{\left(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right)\right)}^2}{\mathcal{Z}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\left(\rho \right)+{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\left(\upsilon \right).\end{array}$$

Proof At the beginning, we apply ${{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}$ on both sides of Eq (3.1) and using Lemma 2.4, we get

$$\begin{array}{cc}\hfill \text{y}\left(\upsilon \right)& =e_0+e_1\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota \left)\right)+e_2\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota ){)}^2+{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}(\upsilon )\hfill \\ & =e_0+e_1\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota \left)\right)+e_2\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota ){)}^2\hfill \\ & +\frac{3-\mu }{\Phi (\mu -2)}{{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{2,\mathcal{G}}\mathfrak{q}\left(\upsilon \right)+\frac{(\mu -2)}{\Phi (\mu -2)}{{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\left(\upsilon \right).\hfill \end{array}$$

So, due to the condition y(ι) = 0, we deduce that e₀ = 0. Thus, by substituting the value of e₀ and by taking the first derivative with respect to a function $\mathcal{G}$, we find

$${\left[\text{y}\left(\upsilon \right)\right]}_{\mathcal{G}}^{\prime }=e_1+2e_2\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota \left)\right)+\frac{3-\mu }{\Phi (\mu -2)}{{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{1,\mathcal{G}}\mathfrak{q}\left(\upsilon \right)+\frac{(\mu -2)}{\Phi (\mu -2)}{{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{\mu -1,\mathcal{G}}\mathfrak{q}\left(\upsilon \right),$$

and due to the boundary condition ${\left[\text{y}\left(\iota \right)\right]}_{\mathcal{G}}^{\prime }=0$, we have e₁ = 0, which yields that

$$\begin{array}{c}\hfill \text{y}\left(\upsilon \right)=e_2\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota ){)}^2+\frac{3-\mu }{\Phi (\mu -2)}{{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{2,\mathcal{G}}\mathfrak{q}(\upsilon )+\frac{(\mu -2)}{\Phi (\mu -2)}{{}^{\mathcal{R}\mathcal{L}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}(\upsilon ).\end{array}$$

Next, by applying the condition $\text{y}\left(\rho \right)=\xi \text{y}\left(\mathfrak{s}\right),$ one has

$$\begin{array}{c}\hfill e_2\left(\mathcal{G}\right(\rho )-\mathcal{G}(\iota ){)}^2+{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}(\rho )=e_2\xi (\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right){)}^2+\xi {{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\left(\mathfrak{s}\right),\end{array}$$

which implies that

$$\begin{array}{c}\hfill e_2={\displaystyle \frac{1}{\mathcal{Z}}}\left[\xi {{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\right(\mathfrak{s})-{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}(\rho \left)\right].\end{array}$$

Hence, we deduce that

$$\begin{array}{cc}\hfill \text{y}\left(\upsilon \right)& ={\displaystyle \frac{{\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota \left)\right)}^2}{\mathcal{Z}}}\left[\xi {{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\right(\mathfrak{s})-{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}(\rho \left)\right]+{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\left(\upsilon \right)\hfill \\ & ={\displaystyle \frac{\xi {\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota \left)\right)}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\left(\mathfrak{s}\right)-{\displaystyle \frac{{\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota \left)\right)}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\left(\rho \right)+{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\mathfrak{q}\left(\upsilon \right).\hfill \end{array}$$

Therefore, the proof is finished.

As a consequence of the above lemma, we present the following essential result:

Lemma 3.2 Let $\mu \in (2,3],{\text{y}}^{\left(3\right)}\in {\mathcal{H}}^1(\iota ,\rho )$ and $\mathcal{Z}=\left(\mathcal{G}\right(\rho )-\mathcal{G}(\iota ){)}^2-\xi (\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right){)}^2\ne 0$ . Then, the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3) has a solution equivalent to

$$\begin{array}{cc}\hfill \text{y}\left(\upsilon \right)& ={\displaystyle \frac{\xi {\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota \left)\right)}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\mathfrak{s},\text{y}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\mathfrak{s}\right))\hfill \\ & -{\displaystyle \frac{{\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota \left)\right)}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\rho ,\text{y}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\rho \right))\hfill \\ & +{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\upsilon ,\text{y}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)).\hfill \end{array}$$ (3.2)

Now, to achieve the required existence and uniqueness theorems, according to Lemma 3.2, the solution of the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3) is a fixed point of the operator ℵ: Ω → Ω which is defined as:

$$\begin{array}{cc}\hfill \left(\aleph \text{y}\right)\left(\upsilon \right)& ={\displaystyle \frac{\xi {\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota \left)\right)}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\mathfrak{s},\text{y}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\mathfrak{s}\right))\hfill \\ & -{\displaystyle \frac{{\left(\mathcal{G}\right(\upsilon )-\mathcal{G}(\iota \left)\right)}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\rho ,\text{y}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\rho \right))\hfill \\ & +{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\upsilon ,\text{y}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)).\hfill \end{array}$$ (3.3)

For working analysis, we state the following conditions:

• (AS₁)
  There are the constants δ₁, δ₂, δ₃ > 0, such that

  $$\begin{array}{c}\hfill |\varpi (\upsilon ,\text{y}\left(\upsilon \right),\widehat{\text{y}}\left(\upsilon \right))|\le {\delta }_1+{\delta }_2\left|\text{y}\left(\upsilon \right)\right|+{\delta }_3\left|\widehat{\text{y}}\left(\upsilon \right)\right|,\end{array}$$

  for any $\text{y},\widehat{\text{y}}\in \Omega$ and υ ∈ J.
• (AS₂)
  There are the constants ℓ₁ > 0 and ℓ₂ ∈ (0, 1), for any ${\text{y}}_1,{\text{y}}_2,{\widehat{\text{y}}}_1,{\widehat{\text{y}}}_2\in \Omega$ and υ ∈ J, satisfy

  $$\begin{array}{c}\hfill |\varpi (\upsilon ,{\text{y}}_1,{\text{y}}_2)-\varpi (\upsilon ,{\widehat{\text{y}}}_1,{\widehat{\text{y}}}_2)|\le {\ell }_1|{\text{y}}_1-{\text{y}}_2|+{\ell }_2|{\widehat{\text{y}}}_1-{\widehat{\text{y}}}_2|.\end{array}$$

For simplicity, we set

$$\begin{array}{cc}\hfill {\Pi }_1& ={\displaystyle \frac{\left|\xi \right|{\left(\mathcal{G}\right(\rho )-\mathcal{G}(\iota \left)\right)}^2}{\left|\mathcal{Z}\right|}}{\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & +[{\displaystyle \frac{{\left(\mathcal{G}\right(\rho )-\mathcal{G}(\iota \left)\right)}^2}{\left|\mathcal{Z}\right|}}+1]{\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}];\hfill \\ \hfill {\Pi }_2& ={\displaystyle \frac{\left|\xi \right|{\left(\mathcal{G}\right(\rho )-\mathcal{G}(\iota \left)\right)}^2}{\left|\mathcal{Z}\right|}}{\displaystyle \frac{{\delta }_2}{1-{\delta }_3}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & +[{\displaystyle \frac{{\left(\mathcal{G}\right(\rho )-\mathcal{G}(\iota \left)\right)}^2}{\left|\mathcal{Z}\right|}}+1]{\displaystyle \frac{{\delta }_2}{1-{\delta }_3}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}].\hfill \end{array}$$

Theorem 3.3 Under assumption (AS₁) with δ₃ ≠ 1. The mapping ℵ: Ω → Ω is continuous and satisfy the growth condition ‖ℵy‖ ≤ Π₁ + Π₂‖y‖.

Proof We define a bounded ball ${\mathbb{D}}_r$ as ${\mathbb{D}}_r=\{\text{y}\in \Omega :\Vert \text{y}\Vert \le r\}$. Regarding to show the continuity of ℵ, let us taking the convergence sequence ${\left\{{\text{y}}_n\right\}}_{n\in \mathbb{N}}$ to y in the ball ℧_ς as n → ∞. Thus, by continuity of ϖ and by applying Lebesgue dominated convergence theorem, one has

$$\begin{array}{cc}\hfill \underset{n\to \infty }{\text{lim}}\left(\aleph {\text{y}}_n\right)\left(\upsilon \right)& ={\displaystyle \frac{\xi {(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\underset{n\to \infty }{\text{lim}}\varpi (\mathfrak{s},{\text{y}}_n\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}{\text{y}}_n\left(\mathfrak{s}\right))\hfill \\ & -{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\underset{n\to \infty }{\text{lim}}\varpi (\rho ,{\text{y}}_n\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}{\text{y}}_n\left(\rho \right))\hfill \\ & +{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\underset{n\to \infty }{\text{lim}}\varpi (\upsilon ,{\text{y}}_n\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}{\text{y}}_n\left(\upsilon \right))=\left(\aleph \text{y}\right)\left(\upsilon \right).\hfill \end{array}$$

Hence, ℵ is continuous.

Next, regarding to the growth condition, by applying (AS₁), we find

$$\begin{array}{cc}\hfill \left|\left(\aleph \text{y}\right)\left(\upsilon \right)\right|& \le {\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\left|\varpi (\mathfrak{s},\text{y}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\mathfrak{s}\right))\right|\hfill \\ & +{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\left|\varpi (\rho ,\text{y}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\rho \right))\right|\hfill \\ & +{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\left|\varpi (\upsilon ,\text{y}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right))\right|\hfill \\ & \le {\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}({\delta }_1+{\delta }_2\left|\text{y}\left(\mathfrak{s}\right)\right|+{\delta }_3\left|{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\mathfrak{s}\right)\right|)\hfill \\ & +{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}({\delta }_1+{\delta }_2\left|\text{y}\left(\rho \right)\right|+{\delta }_3\left|{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\rho \right)\right|)\hfill \\ & +{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}({\delta }_1+{\delta }_2\left|\text{y}\left(\upsilon \right)\right|+{\delta }_3\left|{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)\right|).\hfill \end{array}$$ (3.4)

Since, ${{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)=\varpi (\upsilon ,\text{y}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)),$ then

$$\begin{array}{c}\hfill |{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)|\le {\delta }_1+{\delta }_2\left|\text{y}\left(\upsilon \right)\right|+{\delta }_3\left|{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)\right|,\end{array}$$

which implies that

$$\begin{array}{c}\hfill |{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right)|\le {\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}+{\displaystyle \frac{{\delta }_2}{1-{\delta }_3}}\left|\text{y}\left(\upsilon \right)\right|.\end{array}$$ (3.5)

Therefore, in view of the Eqs (3.4) and (3.5), and taking supremum, one has

$$\begin{array}{cc}& \Vert \aleph \text{y}\Vert \hfill \\ & \le {\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}({\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}+{\displaystyle \frac{{\delta }_2}{1-{\delta }_3}}\Vert \text{y}\Vert )[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & +{\displaystyle \frac{{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}({\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}+{\displaystyle \frac{{\delta }_2}{1-{\delta }_3}}\Vert \text{y}\Vert )[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & +({\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}+{\displaystyle \frac{{\delta }_2}{1-{\delta }_3}}\Vert \text{y}\Vert )[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & \le {\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & +[{\displaystyle \frac{{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}+1]{\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & +{\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{\displaystyle \frac{{\delta }_2}{1-{\delta }_3}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\Vert \text{y}\Vert \hfill \\ & +[{\displaystyle \frac{{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}+1]{\displaystyle \frac{{\delta }_2}{1-{\delta }_3}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\Vert \text{y}\Vert .\hfill \end{array}$$

Hence, ‖ℵy‖ ≤ Π₁ + Π₂‖y‖ and this finishes the proof.

Theorem 3.4 Under assumption (AS₁) with δ₃ ≠ 1, the mapping ℵ: Ω → Ω is compact and consequently is χ-Lipschitz with the Lipschitz’s constant zero.

Proof The boundedness of ℵ implied from Theorem 3.3. It remains to prove that ℵ is an equi-continuous mapping. Therefore, by the assumption (AS₁), for any $\text{y}\in {\mathbb{D}}_r$ and υ₁, υ₂ ∈ J with υ₁ < υ₂, we get

$$\begin{array}{cc}& |\left(\aleph \text{y}\right)\left({\upsilon }_2\right)-\left(\aleph \text{y}\right)\left({\upsilon }_1\right)|\hfill \\ & \le {\displaystyle \frac{\left|\xi \right||{(\mathcal{G}\left({\upsilon }_2\right)-\mathcal{G}\left(\iota \right))}^2-{(\mathcal{G}\left({\upsilon }_1\right)-\mathcal{G}\left(\iota \right))}^2|}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\left|\varpi (\mathfrak{s},\text{y}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\mathfrak{s}\right))\right|\hfill \\ & +{\displaystyle \frac{|{(\mathcal{G}\left({\upsilon }_2\right)-\mathcal{G}\left(\iota \right))}^2-{(\mathcal{G}\left({\upsilon }_1\right)-\mathcal{G}\left(\iota \right))}^2|}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\left|\varpi (\rho ,\text{y}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\rho \right))\right|\hfill \\ & +|{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi ({\upsilon }_2,\text{y}\left({\upsilon }_2\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left({\upsilon }_2\right))-{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi ({\upsilon }_1,\text{y}\left({\upsilon }_1\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left({\upsilon }_1\right))|\hfill \\ & \le {\displaystyle \frac{\left|\xi \right||{(\mathcal{G}\left({\upsilon }_2\right)-\mathcal{G}\left(\iota \right))}^2-{(\mathcal{G}\left({\upsilon }_1\right)-\mathcal{G}\left(\iota \right))}^2|}{\left|\mathcal{Z}\right|}}({\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}+{\displaystyle \frac{r{\delta }_2}{1-{\delta }_3}})\hfill \\ & \times [{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & +{\displaystyle \frac{|{(\mathcal{G}\left({\upsilon }_2\right)-\mathcal{G}\left(\iota \right))}^2-{(\mathcal{G}\left({\upsilon }_1\right)-\mathcal{G}\left(\iota \right))}^2|}{\left|\mathcal{Z}\right|}}({\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}+{\displaystyle \frac{r{\delta }_2}{1-{\delta }_3}})\hfill \\ & \times [{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & +({\displaystyle \frac{{\delta }_1}{1-{\delta }_3}}+{\displaystyle \frac{r{\delta }_2}{1-{\delta }_3}})\left|\right[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left({\upsilon }_2\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left({\upsilon }_2\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & -[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left({\upsilon }_1\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left({\upsilon }_1\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]|.\hfill \end{array}$$

Obviously, |(ℵy)(υ₂) − (ℵy)(υ₁)| → 0 whenever υ₂ → υ₁ and thus $\aleph \left({\mathbb{D}}_r\right)$ is equi-continuous. Hence, due to Arzelá-Ascoli theorem, $\aleph \left({\mathbb{D}}_r\right)$ is compact and in view of Lemma 2.8, the mapping ℵ is χ-Lipschitz with the Lipschitz’s constant ϵ = 0.

Theorem 3.5 Under assumption (AS₂), the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3) possesses an one solution on condition of

$$\begin{array}{cc}\hfill {\Pi }_3& ≔\frac{\left|\xi \right|{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}\frac{{\ell }_1}{1-{\ell }_2}[\frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}+\frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}]\hfill \\ & +(\frac{{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}+1)\frac{{\ell }_1}{1-{\ell }_2}[\frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}+\frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}]<1.\hfill \end{array}$$ (3.6)

Proof Let us take the mapping ℵ as given in (3.3). For any $\text{y},\widehat{\text{y}}\in \Omega$ and υ ∈ J, we find

$$\begin{array}{cc}& |\left(\aleph \text{y}\right)\left(\upsilon \right)-\left(\aleph \widehat{\text{y}}\right)\left(\upsilon \right)|\hfill \\ & \le {\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}|\varpi (\mathfrak{s},\text{y}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\mathfrak{s}\right))-\varpi (\mathfrak{s},\widehat{\text{y}}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\mathfrak{s}\right))|\hfill \\ & +{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}|\varpi (\rho ,\text{y}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\rho \right))-\varpi (\rho ,\widehat{\text{y}}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\rho \right))|\hfill \\ & +{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}|\varpi (\upsilon ,\text{y}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right))-\varpi (\upsilon ,\widehat{\text{y}}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right))|\hfill \\ & \le {\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{\displaystyle \frac{{\ell }_1}{1-{\ell }_2}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\Vert \text{y}-\widehat{\text{y}}\Vert \hfill \\ & +{\displaystyle \frac{{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{\displaystyle \frac{{\ell }_1}{1-{\ell }_2}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\Vert \text{y}-\widehat{\text{y}}\Vert \hfill \\ & +{\displaystyle \frac{{\ell }_1}{1-{\ell }_2}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\Vert \text{y}-\widehat{\text{y}}\Vert \hfill \\ & \le {\Pi }_3\Vert \text{y}-\widehat{\text{y}}\Vert .\hfill \end{array}$$

Thus, $\Vert \aleph \text{y}-\aleph \widehat{\text{y}}\Vert \le {\Pi }_3\Vert \text{y}-\widehat{\text{y}}\Vert$. Therefore, by condition (3.6), ℵ is contraction mapping and based on the Banach contraction theorem, ℵ possesses a unique fixed point which is a solution of the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3).

Theorem 3.6 Under the assumptions (AS₁) and (AS₂), the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3) admits a solution such that Π₂ < 1. Furthermore, the set containing solutions of the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3) is bounded.

Proof According Theorem 3.5, ℵ is Lipschitz mapping and by Lemma 2.9, ℵ is χ-Lipschitz which implies that ℵ is χ-condensing.

Now, due to Theorem 2.10, it remains to show that the set $\mathbb{W}$ is bounded, where

$$\begin{array}{c}\hfill \mathbb{W}=\{\text{y}\in \Omega :\text{y}=\zeta \aleph \left(\text{y}\right),\text{for}\text{some}\zeta \in [0,1\left]\right\}.\end{array}$$

For end this, let $\text{y}\in \mathbb{W}$, therefore for each υ ∈ J for some ζ ∈ [0, 1], and by Theorem 3.3, we can derive that

$$\begin{array}{c}\hfill \Vert \text{y}\Vert =\Vert \zeta \aleph \left(\text{y}\right)\Vert \le {\Pi }_1+{\Pi }_2\Vert \text{y}\Vert .\end{array}$$

Thus, $\Vert \text{y}\Vert \le {\displaystyle \frac{{\Pi }_1}{1-{\Pi }_2}},$ which implies that $\mathbb{W}$ is a bounded set contained in Ω. In view of Theorem 2.10, implies that ℵ has at least one fixed point, which are act solutions of the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3), and consequently $\mathbb{W}$ contains solutions of the Eq (1.3) is a bounded subset of Ω.

4 Stability analysis

Here, we will discuss the stability of $\mathbb{U}\mathbb{H}$ type. So, we need to state the definitions of $\mathbb{U}\mathbb{H}$ stability:

Definition 4.1 [37] Let there is a real constant Ξ_ϖ > 0, such that for all ς > 0. Then the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3), is called $\mathbb{U}\mathbb{H}$ stable when $\widehat{\text{y}}\in \Omega$ is satisfying the relation

$$\begin{array}{c}\hfill |{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right)-\varpi (\upsilon ,\widehat{\text{y}}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right))|\le \varsigma ,\upsilon \in [\iota ,\rho ],\end{array}$$ (4.1)

hence there is one function y ∈ Ω satisfying the Eq (1.3), provided

$$\begin{array}{c}\hfill |\widehat{\text{y}}\left(\upsilon \right)-\text{y}\left(\upsilon \right)|\le {\Xi }_{\varpi }\varsigma ,\upsilon \in [\iota ,\rho ].\end{array}$$ (4.2)

Moreover, the solution y ∈ Ω of the Eq (1.3) is called generalized $\mathbb{U}\mathbb{H}$ ($\mathbb{G}\mathbb{U}\mathbb{H}$) stable, if there is a function $\Theta \in C({\mathbb{R}}^{+},{\mathbb{R}}^{+}),\Theta \left(0\right)=0$ satisfied

$$\begin{array}{c}\hfill |\widehat{\text{y}}\left(\upsilon \right)-\text{y}\left(\upsilon \right)|\le {\Xi }_{\varpi }\Theta \left(\varsigma \right),\upsilon \in [\iota ,\rho ].\end{array}$$ (4.3)

Remark 4.2 The function $\widehat{\text{y}}\in \Omega$ satisfying the inequality (4.1), iff there is a function σ ∈ Ω, where

1) |σ(υ)| ≤ ς, υ ∈ [ι, ρ], ς > 0;

2) ${{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right)=\varpi (\upsilon ,\widehat{\text{y}}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right))+\sigma \left(\upsilon \right)$ .

Theorem 4.3 Let the arguments of Theorem 3.5 are satisfied. Then, the solution of $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3) is $\mathbb{U}\mathbb{H}$ and consequently $\mathbb{G}\mathbb{U}\mathbb{H}$ stable.

Proof Suppose that $\widehat{\text{y}}\in \Omega$ satisfying the Ineq. (4.1), then by applying (4.2), we get

$$\begin{array}{c}\hfill {{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right)=\varpi (\upsilon ,\widehat{\text{y}}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right))+\sigma \left(\upsilon \right),\forall \upsilon \in [\iota ,\rho ].\end{array}$$

According to Eq (3.2), one has

$$\begin{array}{cc}\hfill \widehat{\text{y}}\left(\upsilon \right)& ={\displaystyle \frac{\xi {(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\mathfrak{s},\widehat{\text{y}}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\mathfrak{s}\right))+{\displaystyle \frac{\xi {(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\sigma \left(\mathfrak{s}\right)\hfill \\ & -{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\rho ,\widehat{\text{y}}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\rho \right))-{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\sigma \left(\rho \right)\hfill \\ & +{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\upsilon ,\widehat{\text{y}}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right))+{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\sigma \left(\upsilon \right),\hfill \end{array}$$ (4.4)

which gives

$$\begin{array}{cc}& |\widehat{\text{y}}\left(\upsilon \right)-{\displaystyle \frac{\xi {(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\mathfrak{s},\widehat{\text{y}}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\mathfrak{s}\right))\hfill \\ & +{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\rho ,\widehat{\text{y}}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\rho \right))-{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\upsilon ,\widehat{\text{y}}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right))|\hfill \\ & \le {\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\left|\sigma \left(\mathfrak{s}\right)\right|+{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\left|\sigma \left(\rho \right)\right|+{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\left|\sigma \left(\upsilon \right)\right|\hfill \\ & \le {\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\varsigma \hfill \\ & +[{\displaystyle \frac{{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}+1][{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\varsigma .\hfill \end{array}$$ (4.5)

Next, for $\text{y},\widehat{\text{y}}\in \Omega$, by utilizing Eqs (4.4) and (4.5) and (AS₂), we have

$$\begin{array}{cc}& \left|\widehat{\text{y}}\left(\upsilon \right)-\text{y}\left(\upsilon \right)\right|\hfill \\ & =|\widehat{\text{y}}\left(\upsilon \right)-{\displaystyle \frac{\xi {(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\mathfrak{s},\text{y}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\mathfrak{s}\right))\hfill \\ & +{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\rho ,\text{y}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\rho \right))-{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\upsilon ,\text{y}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right))|\hfill \\ & \le |\widehat{\text{y}}\left(\upsilon \right)-{\displaystyle \frac{\xi {(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\mathfrak{s},\widehat{\text{y}}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\mathfrak{s}\right))\hfill \\ & +{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\mathcal{Z}}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\rho ,\widehat{\text{y}}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\rho \right))-{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}\varpi (\upsilon ,\widehat{\text{y}}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right))|\hfill \\ & +{\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}|\varpi (\mathfrak{s},\widehat{\text{y}}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\mathfrak{s}\right))-\varpi (\mathfrak{s},\text{y}\left(\mathfrak{s}\right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\mathfrak{s}\right))|\hfill \\ & +{\displaystyle \frac{{(\mathcal{G}\left(\upsilon \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}|\varpi (\rho ,\widehat{\text{y}}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\rho \right))-\varpi (\rho ,\text{y}\left(\rho \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\rho \right))|\hfill \\ & +{{}^{\mathcal{A}\mathcal{B}}\mathfrak{I}}_{\iota }^{\mu ,\mathcal{G}}|\varpi (\upsilon ,\widehat{\text{y}}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\widehat{\text{y}}\left(\upsilon \right))-\varpi (\upsilon ,\text{y}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right))|\hfill \\ & \le {\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\varsigma \hfill \\ & +[{\displaystyle \frac{{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}+1][{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\varsigma \hfill \\ & +{\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{\displaystyle \frac{{\ell }_1}{1-{\ell }_2}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\Vert \widehat{\text{y}}-\text{y}\Vert \hfill \\ & +{\displaystyle \frac{{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}{\displaystyle \frac{{\ell }_1}{1-{\ell }_2}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\Vert \widehat{\text{y}}-\text{y}\Vert \hfill \\ & +{\displaystyle \frac{{\ell }_1}{1-{\ell }_2}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\Vert \widehat{\text{y}}-\text{y}\Vert \hfill \\ & \le {\Pi }_4\varsigma +{\Pi }_3\Vert \widehat{\text{y}}-\text{y}\Vert ,\hfill \end{array}$$

which further implies

$$\begin{array}{c}\hfill \Vert \widehat{\text{y}}-\text{y}\Vert \le {\displaystyle \frac{{\Pi }_4}{1-{\Pi }_3}}\varsigma ,\end{array}$$ (4.6)

where

$$\begin{array}{cc}\hfill {\Pi }_4:& ={\displaystyle \frac{\left|\xi \right|{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}[{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}]\hfill \\ & +[{\displaystyle \frac{{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}}+1][{\displaystyle \frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}}+{\displaystyle \frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}}].\hfill \end{array}$$

Thus, yields that

$$\begin{array}{c}\hfill \Vert \widehat{\text{y}}-\text{y}\Vert \le {\Xi }_{\varpi }\varsigma ;{\Xi }_{\varpi }≔{\displaystyle \frac{{\Pi }_4}{1-{\Pi }_3}}.\end{array}$$

Hence, the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ fractional implicit differential problem (1.3) is $\mathbb{U}\mathbb{H}$ stable. In addition, there is a non-decreasing function Θ: (0, ∞) → (0, ∞), $\Theta \left(\varsigma \right)=\varsigma$ where Θ(0) = 0, so by (4.6), we find

$$\begin{array}{c}\hfill \Vert \widehat{\text{y}}-\text{y}\Vert \le {\Xi }_{\varpi }\Theta \left(\varsigma \right).\end{array}$$

Therefore, the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (1.3) is $\mathbb{G}\mathbb{U}\mathbb{H}$ stable.

5 Applications

This section concerns the applications of the essential results using two comprehensive examples with illustrative graphics and tables.

Example 5.1 Consider the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE as follows:

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle {{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{2.3,\mathcal{G}}\text{y}\left(\upsilon \right)=2{\upsilon }^3+\frac{0.3\left|cos\left(\upsilon \right)\right|}{{\upsilon }^2+6^2}.cos\left(\text{y}\left(\upsilon \right)\right)+\frac{e^{-\upsilon }}{{\upsilon }^3+3^3}sin\left({{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{2.3,\mathcal{G}}\text{y}\left(\upsilon \right)\right)}\hfill \\ {\displaystyle \text{y}\left(1\right)=0,{\left[\text{y}\left(1\right)\right]}_{\mathcal{G}}^{\prime }=0,\text{y}\left(e\right)=\frac{1}{25}\text{y}(2.3),2.3=\mathfrak{s}\in (1,e),\upsilon \in \text{J}=[1,e].}\hfill \end{array}\end{array}$$ (5.1)

Here, $\mu =2.3,\iota =1,\rho =e,\xi =\frac{1}{25},\mathfrak{s}=2.3,$ and

$$\begin{array}{c}\hfill \varpi (\upsilon ,\text{y}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}\text{y}\left(\upsilon \right))=2{\upsilon }^3+\frac{0.3\left|cos\left(\upsilon \right)\right|}{{\upsilon }^2+6^2}.cos\left(\text{y}\left(\upsilon \right)\right)+\frac{e^{-\upsilon }}{{\upsilon }^3+3^3}sin\left({{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.3,\mathcal{G}}\text{y}\left(\upsilon \right)\right).\end{array}$$

Thus, we get

$$\begin{array}{cc}& |\varpi (\upsilon ,{\text{y}}_1\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.3,\mathcal{G}}{\text{y}}_1\left(\upsilon \right))-\varpi (\upsilon ,{\text{y}}_2\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.3,\mathcal{G}}{\text{y}}_2\left(\upsilon \right))|\\ & \le 0.0065|{\text{y}}_1\left(\upsilon \right)-{\text{y}}_2\left(\upsilon \right)|+0.0131385|{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.3,\mathcal{G}}{\text{y}}_1\left(\upsilon \right)-{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.3,\mathcal{G}}{\text{y}}_2\left(\upsilon \right)|,\end{array}$$

hence, ℓ₁ = 0.0065, ℓ₂ = 0.0131385. So, we have

$$\begin{array}{cc}\hfill {\Pi }_3& ≔\frac{\left|\xi \right|{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}\frac{{\ell }_1}{1-{\ell }_2}[\frac{(3-\mu ){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}+\frac{(\mu -2){[\mathcal{G}\left(\mathfrak{s}\right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}]\hfill \\ & +(\frac{{(\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right))}^2}{\left|\mathcal{Z}\right|}+1)\frac{{\ell }_1}{1-{\ell }_2}[\frac{(3-\mu ){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^2}{\Phi (\mu -2)\Gamma \left(3\right)}+\frac{(\mu -2){[\mathcal{G}\left(\rho \right)-\mathcal{G}\left(\iota \right)]}^{\mu }}{\Phi (\mu -2)\Gamma (\mu +1)}]\hfill \\ & \approx \{\begin{array}{c}0.3029277<1,when\mathcal{G}\left(\upsilon \right)={\upsilon }^2;\hfill \\ 0.1501725<1,when\mathcal{G}\left(\upsilon \right)=2^{\upsilon };\hfill \\ 0.0268289<1,when\mathcal{G}\left(\upsilon \right)=ln\left({\upsilon }^2\right).\hfill \end{array}\hfill \end{array}$$

Then, according to Theorem 3.5, the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (5.1) has one solution. Furthermore, based on Theorem 4.3 the such solution is $\mathbb{U}\mathbb{H}$ stable with

$$\begin{array}{c}\hfill {\Xi }_{\varpi }≔{\displaystyle \frac{{\Pi }_4}{1-{\Pi }_3}}\approx \{\begin{array}{c}64.979638,when\mathcal{G}\left(\upsilon \right)={\upsilon }^2;\hfill \\ 26.422625,when\mathcal{G}\left(\upsilon \right)=2^{\upsilon };\hfill \\ 4.1222040,when\mathcal{G}\left(\upsilon \right)=ln\left({\upsilon }^2\right),\hfill \end{array}\end{array}$$

and consequently is $\mathbb{G}\mathbb{U}\mathbb{H}$ stable.

Additionally, Fig 1, represents the graphics of Π₃, which are less than 1, and Table 1, shows the computation values of Π₃, and Ξ_ϖ, whenever the function $\mathcal{G}\left(\upsilon \right)={\upsilon }^2,2^{\upsilon },ln\left({\upsilon }^2\right)$, on υ ∈ [1, e], for the problem (5.1). Also, Fig 2, represents the graphics of Π₃, which are less than 1, and Table 2, shows the computation values of Π₃ whenever the function $\mathcal{G}\left(\upsilon \right)={\upsilon }^2,2^{\upsilon },ln\left({\upsilon }^2\right)$, and various μ ∈ (2, 3] on υ ∈ [1, e] for problem (5.1).

Fig 1. Shows the values of Π₃ < 1, for various functions $\mathcal{G}$ and υ ∈ [1, e] for problem (5.1).

Table 1. Numerical results of Π₃ and Ξ_ϖ at various functions $\mathcal{G}\left(\upsilon \right)={\upsilon }^2$, $\mathcal{G}\left(\upsilon \right)=2^{\upsilon }$ and $\mathcal{G}\left(\upsilon \right)=ln\left({\upsilon }^2\right)$ on [1, e] for problem 5.1.

υ	$\mathcal{G}\left(\upsilon \right)={\upsilon }^2$		$\mathcal{G}\left(\upsilon \right)=2^{\upsilon }$		$\mathcal{G}\left(\upsilon \right)=ln\left({\upsilon }^2\right)$
	Π3 < 1	Ξϖ	Π3 < 1	Ξϖ	Π3 < 1	Ξϖ
1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
1.12450	0.0019	0.2800	0.0008	0.1237	0.0012	0.1752
1.4900	0.0097	1.4678	0.0041	0.6227	0.0040	0.6047
1.7350	0.0275	4.2231	0.0116	1.7557	0.0079	1.1861
1.9800	0.0599	9.5295	0.0256	3.9348	0.0123	1.8612
2.2250	0.1128	19.0147	0.0499	7.8485	0.0171	2.5959
2.4700	0.1929	35.7502	0.0896	14.7260	0.0220	3.3688
2.7150	0.3081	66.5950	0.1529	26.9840	0.0271	4.1664

Fig 2. Shows the values of Π₃ < 1, for some functions $\mathcal{G}$ and various μ ∈ (2, 3] for problem (5.1).

Table 2. Numerical results of Π₃ at various μ ∈ (2, 3] and some functions $\mathcal{G}\left(\upsilon \right)$ on [1, e] for problem 5.1.

μ	2.1	2.2	2.3	2.4	2.5	2.6	2.7	2.8	2.9	3
Π3 at $\mathcal{G}\left(\upsilon \right)={\upsilon }^2$	0.2870	0.2955	0.3099	0.3308	0.3584	0.3930	0.4349	0.4842	0.5410	0.6055
Π3 at $\mathcal{G}\left(\upsilon \right)=2^{\upsilon }$	0.1470	0.1496	0.1539	0.1598	0.1672	0.1761	0.1862	0.1976	0.2099	0.2232
Π3 at $\mathcal{G}\left(\upsilon \right)=ln\left({\upsilon }^2\right)$	0.0278	0.0276	0.0272	0.0266	0.0258	0.0248	0.0236	0.0221	0.0204	0.0186

Example 5.2 Consider the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE as follows:

$$\begin{array}{c}\hfill \{\begin{array}{c}{\displaystyle {{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.8,\mathcal{G}}\text{y}\left(\upsilon \right)=\frac{e^{-2\upsilon }}{9^2+e^{-2\upsilon }}.\frac{1}{1+\left|\text{y}\left(\upsilon \right)\right|+|{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.8,\mathcal{G}}\text{y}\left(\upsilon \right)|}+\frac{e^{-3\upsilon }}{7^2}{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.8,\mathcal{G}}\text{y}\left(\upsilon \right),}\hfill \\ {\displaystyle \text{y}\left(0\right)=0,{\left[\text{y}\left(0\right)\right]}_{\mathcal{G}}^{\prime }=0,\text{y}\left(2\right)=\frac{1}{15}\text{y}(1.3),1.3=\mathfrak{s}\in (1,2),\upsilon \in \text{J}=[1,2],}\hfill \end{array}\end{array}$$ (5.2)

where, $\mu =2.8,\iota =1,\rho =2,\mathfrak{s}=1.3,\xi =\frac{1}{15},$ and

$$\begin{array}{c}\hfill \varpi (\upsilon ,\text{y}\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.8,\mathcal{G}}\text{y}\left(\upsilon \right))=\frac{e^{-2\upsilon }}{9^2+e^{-2\upsilon }}.\frac{1}{1+\left|\text{y}\left(\upsilon \right)\right|+|{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.8,\mathcal{G}}\text{y}\left(\upsilon \right)|}+\frac{e^{-3\upsilon }}{7^2}{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.8,\mathcal{G}}\text{y}\left(\upsilon \right).\end{array}$$

Thus, we get

$$\begin{array}{cc}& |\varpi (\upsilon ,{\text{y}}_1\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}{\text{y}}_1\left(\upsilon \right))-\varpi (\upsilon ,{\text{y}}_2\left(\upsilon \right),{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}{\text{y}}_2\left(\upsilon \right))|\\ & \le 0.0123456|{\text{y}}_1\left(\upsilon \right)-{\text{y}}_2\left(\upsilon \right)|+0.0327538|{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.8,\mathcal{G}}{\text{y}}_1\left(\upsilon \right)-{{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_1^{2.8,\mathcal{G}}{\text{y}}_2\left(\upsilon \right)|,\end{array}$$

hence, ℓ₁ = 0.0123456, ℓ₂ = 0.0327538. So, we have

$$\begin{array}{cc}\hfill {\Pi }_3& \approx \{\begin{array}{c}0.3821305<1,when\mathcal{G}\left(\upsilon \right)=e^{\upsilon };\hfill \\ 0.8552526<1,when\mathcal{G}\left(\upsilon \right)=0.9{\upsilon }^3;\hfill \\ 0.1590327<1,when\mathcal{G}\left(\upsilon \right)=\text{sin}\left(\frac{\upsilon }{2}\right)+{\upsilon }^2.\hfill \end{array}\end{array}$$

Then, in view of Theorem 3.5, the $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$ IFDE (5.2) has one solution. Furthermore, based on Theorem 4.3 the such solution is $\mathbb{U}\mathbb{H}$ stable with

$$\begin{array}{c}\hfill {\Xi }_{\varpi }≔{\displaystyle \frac{{\Pi }_4}{1-{\Pi }_3}}\approx \{\begin{array}{c}48.454820,when\mathcal{G}\left(\upsilon \right)=e^{\upsilon };\hfill \\ 400.62919,when\mathcal{G}\left(\upsilon \right)=0.9{\upsilon }^3;\hfill \\ 14.815947,when\mathcal{G}\left(\upsilon \right)=\text{sin}\left(\frac{\upsilon }{2}\right)+{\upsilon }^2,\hfill \end{array}\end{array}$$

and consequently is $\mathbb{G}\mathbb{U}\mathbb{H}$ stable.

Moreover, Fig 3, represents the graphics of Π₃, which are less than 1, and Table 3, shows the computation values of Π₃, and Ξ_ϖ, whenever the function $\mathcal{G}\left(\upsilon \right)=e^{\upsilon },0.9{\upsilon }^3,\text{sin}\left(\frac{\upsilon }{2}\right)+{\upsilon }^2$, on υ ∈ [1, 2], for the problem (5.2). In addition, Fig 4, represents the graphics of Π₃, which are less than 1, and Table 4, shows the computation values of Π₃ whenever the function $\mathcal{G}\left(\upsilon \right)=e^{\upsilon },0.9{\upsilon }^3,\text{sin}\left(\frac{\upsilon }{2}\right)+{\upsilon }^2$, and various μ ∈ (2, 3] on υ ∈ [1, 2] for problem (5.2). According to Fig 4 and Table 4, we observe that Π₃ ≥ 1 for some values μ at function $\mathcal{G}\left(\upsilon \right)=0.9{\upsilon }^3$, thus for this reason and only at these values we can’t say that the problem (5.2) has one solution.

Fig 3. Shows the values of Π₃ < 1, for various functions $\mathcal{G}$ and υ ∈ [1, 2] of the problem (5.2).

Table 3. Numerical results of Π₃ and Ξ_ϖ of some functions $\mathcal{G}\left(\upsilon \right)=e^{\upsilon }$, $\mathcal{G}\left(\upsilon \right)=0.9{\upsilon }^3$, and $\mathcal{G}\left(\upsilon \right)=\text{sin}\left(\frac{\upsilon }{2}\right)+{\upsilon }^2$, on [1, 2] for problem 5.2.

υ	$\mathcal{G}\left(\upsilon \right)=e^{\upsilon }$		$\mathcal{G}\left(\upsilon \right)=0.9{\upsilon }^3$		$\mathcal{G}\left(\upsilon \right)=\text{sin}\left(\frac{\upsilon }{2}\right)+{\upsilon }^2$
	Π3 < 1	Ξϖ	Π3 < 1	Ξϖ	Π3 < 1	Ξϖ
1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
1.1250	0.0006	0.0493	0.0007	0.0558	0.0005	0.0369
1.2500	0.0039	0.3059	0.0050	0.3974	0.0027	0.2143
1.3750	0.0126	1.0026	0.0186	1.4848	0.0083	0.6538
1.5000	0.0313	2.5335	0.0516	4.2628	0.0190	1.5188
1.6250	0.0668	5.6122	0.1215	10.8329	0.0373	3.0384
1.7500	0.1297	11.6766	0.2563	26.9974	0.0661	5.5489
1.8750	0.2358	24.1681	0.4993	78.1367	0.1089	9.5792
2.0000	0.4086	54.1394	0.9152	845.0222	0.1699	16.0365

Fig 4. Shows the values of Π₃ < 1, for some functions $\mathcal{G}$ and various μ ∈ (2, 3] of problem (5.2).

Table 4. Numerical results of Π₃ at various μ ∈ (2, 3] and some functions $\mathcal{G}\left(\upsilon \right)$ on [1, 2] for problem 5.2.

μ	2.1	2.2	2.3	2.4	2.5	2.6	2.7	2.8	2.9	3
Π3 at $\mathcal{G}\left(\upsilon \right)=e^{\upsilon }$	0.3002	0.3057	0.3148	0.3273	0.3432	0.3622	0.3841	0.4086	0.4355	0.4645
Π3 at $\mathcal{G}\left(\upsilon \right)=0.9{\upsilon }^3$	0.5479	0.5636	0.5908	0.6298	0.6813	0.7458	0.8237	0.9152	1.0205	1.1398
Π3 at $\mathcal{G}\left(\upsilon \right)=\text{sin}\left(\frac{\upsilon }{2}\right)+{\upsilon }^2$	0.1550	0.1562	0.1579	0.1601	0.1626	0.1651	0.1676	0.1699	0.1718	0.1732

6 Conclusions

This manuscript dealt with a new class of $\mathcal{G}-\mathcal{A}\mathcal{B}\mathcal{C}$-IFDE (1.3) with higher orders belonging to the interval (2, 3]. The fundamental conditions of the existence and uniqueness of the solution for Eq (1.3) were established by Banach and topology degree theories. Moreover, the $\mathbb{U}\mathbb{H}$ stability with its generalized was discussed. Finally, two application examples with illustrative graphics and tables were provided to check the effectiveness of the main results with compare the main parameters.

The results of this study can be employed in new problems as special cases of the main Eq (1.3) by taking various functions of $\mathcal{G}$. Furthermore, the $\mathcal{G}\mathcal{A}\mathcal{B}\mathcal{C}$-IFDE (1.3) covers some problems are existing in the literature; for instance (i) the Eq (1.3) can be reduced to problem (1.1) if μ → 3 and the implicit term omitted; (ii) the Eq (1.3) can be returned to problem (1.2) if we replace the operator ${{}^{\mathcal{A}\mathcal{B}\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu ,\mathcal{G}}$ by ${{}^{\mathcal{C}}\mathfrak{D}}_{\iota }^{\mu }$ with omitting the implicit term.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Funding Statement

Pontificia Universidad Cat´olica del Ecuador, Proyecto T´ıtulo: “Algunos resultados Cualitativos sobre Ecuaciones diferenciales fraccionales y desigualdades integrales” (Cod UIO2022 to M. V-C). This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1445 to I. K.).