Solvability of Boundary Value Problems for Differential Equations Combining Ordinary and Fractional Derivatives of Non-Autonomous Variable Order

Abstract

This study introduces a novel approach for investigating the solvability of boundary value problems for differential equations that incorporate both ordinary and fractional derivatives, specifically within the context of non-autonomous variable order. Unlike traditional methods in the literature, which often rely on generalized intervals and piecewise constant functions, we propose a new fractional operator better suited for this problem. We analyze the existence and uniqueness of solutions, establishing the conditions necessary for these properties to hold using the Krasnoselskii fixed-point theorem and Banach’s contraction principle. Our study also addresses the Ulam–Hyers stability of the proposed problems, examining how variations in boundary conditions influence the solution dynamics. To support our theoretical framework, we provide numerical examples that not only validate our findings but also demonstrate the practical applicability of these mixed derivative equations across various scientific domains. Additionally, concepts such as symmetry may offer further insights into the behavior of solutions. This research contributes to a deeper understanding of complex differential equations and their implications in real-world scenarios.

1. Introduction

Fractional calculus, which uses natural numbers to represent rational numbers in the context of derivation operators, has emerged as a powerful tool in mathematical modeling. Its ability to capture the intricacies of various physical phenomena has led to significant advancements in fields such as physics, engineering, and control theory. Among these fields, variable-order fractional calculus has gained particular attention due to its flexibility and applicability in diverse scenarios, providing both numerical and analytical techniques to handle complex problems. Despite the remarkable results achieved with constant-order fractional derivatives (see [1,2,3,4,5,6,7]), research on nonlinear differential equations with variable fractional orders remains limited (see [8,9,10,11,12,13]). The existing literature highlights the need for further exploration of the challenges posed by these equations. Furthermore, more sophisticated approaches to variable-order fractional calculus have been discussed in recent studies (refer to [14,15]), indicating a growing interest in this area.

Boundary value problems for differential equations involving mixed derivatives with non-autonomous variable orders are among the most significant topics in modern mathematical analysis. The term “non-autonomous variable order” refers to a class of fractional differential equations where the fractional order is not constant but varies dynamically depending on both the time and the solution of the problem. This dynamic dependence allows for a more accurate modeling of complex systems, where the behavior evolves with time and is influenced by the system’s state, making it suitable for applications in fields like physics and biology, where such interactions are common. The significance of these equations lies in their ability to accurately model complex dynamical systems, where behavior evolves over time and is influenced by the system’s state. This makes them particularly suitable for applications in fields such as physics and biology, where such interactions are prevalent. Symmetry analysis is a fundamental approach in studying these equations, as symmetry techniques can simplify equations, determine possible solutions, and help reveal the changing behaviors within studied systems.

According to recently published studies (see [16,17,18]), the influence of symmetry on boundary values and solutions of these equations has been extensively explored, aiding in the development of new applications in engineering and physical sciences that require precise mathematical models. It is worth noting that this study extensively relies on the concept of a piecewise constant function, which plays a pivotal role in the analysis. To facilitate this, the interval of existence $[0,\rho ]$ is partitioned as follows: $M:=\{I_1=[0,{\rho }_1],I_2=({\rho }_1,{\rho }_2],I_3=({\rho }_2,{\rho }_3],\dots ,I_{\sigma }=({\rho }_{\sigma -1},\rho ]\},$ where $\sigma$ is a specific natural number. Furthermore, the piecewise constant function $q\left(t\right):[0,\rho ]\to (0,1]$ with respect to P is expressed as

$$q\left(t\right)=\sum _{k=1}^{\sigma }{q}_k{I}_k\left(t\right),t\in [0,\rho ],$$

with constants $0<q_k\le 1$ and $k=1,2,\dots ,\sigma$. Here, ${\rho }_0=0$ and ${\rho }_{\sigma }=\rho$, meaning that $I_k=1$ for $t\in [{\rho }_{k-1},{\rho }_k]$ and $I_k=0$ elsewhere. Most of the referenced results are derived using this method, which first partitions the existence interval into subintervals and then defines differential and integral operators relative to these subintervals. This approach allows researchers to transform fractional problems with variable order into their corresponding conventional fractional problems of constant order.

In [15], A. Razminia et al. examined the existence of solutions for variable-order fractional differential equations of the form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathbb{D}}_{{\varsigma }^{+}}^{\aleph (𝚤,𝚥(𝚤\left)\right)}𝚥(𝚤)=\eta (𝚤,𝚥(𝚤)),\hfill \\ 𝚥\left(h\right)={𝚥}_0,\hfill \end{array}\right.\end{array}$$

where ${\mathbb{D}}_{h^{+}}^{\aleph (𝚤,𝚥(𝚤\left)\right)}$ denotes the Riemann–Liouville fractional derivative of order $\aleph (𝚤,𝚥(𝚤\left)\right)$, and $\eta$ is a given continuous function.

Building on these findings, we address the following non-autonomous variable-order boundary value problem (NAVOITVP):

$$\left\{\begin{array}{c}{\mathbb{D}}_{0^{+}}^{\aleph (𝚤,𝚥(𝚤\left)\right)}𝚥(𝚤)+\varsigma {𝚥}^{″}(𝚤)=\eta (𝚤,𝚥(𝚤)),𝚤\in \widetilde{{\psi }_{\tau }}:=[0,\tilde{\iota }],\hfill \\ 𝚥\left(0\right)=0,𝚥\left(\tilde{\iota }\right)=0,\hfill \end{array}\right.$$

(1)

where $0<\tilde{\iota }<+\infty$, $\varsigma >0$, and functions $\aleph :\widetilde{{\psi }_{\tau }}\times \mathbb{R}\to (1,2)$ and $\eta :\widetilde{{\psi }_{\tau }}\times \mathbb{R}\to \mathbb{R}$ are continuous. Here, ${\mathbb{D}}_{0^{+}}^{\aleph (𝚤,𝚥(𝚤\left)\right)}$ represents the Riemann–Liouville fractional derivative of variable order $\aleph (𝚤,𝚥(𝚤\left)\right)$.

The primary goal of this paper is to propose an innovative method that eliminates the need for the piecewise constant function and partitioning of the existence interval. The cornerstone of our approach is the development of a more flexible operator that requires no extra steps. We propose a new criterion for the existence and uniqueness of solutions to the NAVOITVP defined in (1). We will illustrate our theoretical results with two examples provided at the end of this paper. The structure of our paper is as follows. In Section 2, we first present some preliminary information and characteristics of the non-autonomous variable-order $\aleph (𝚤,𝚥(𝚤\left)\right)$ fractional derivative of Riemann–Liouville. The solutions’ existence and uniqueness requirements for the suggested NAVOITVP (1) are given in Section 3. In Section 4, we further examine the stability criterion of the given solution in the context of Ulam–Hyers stability. The given applications in Section 5 illustrate the theoretical conclusions. Finally, the presentation of the conclusion part in Section 6 completes this paper.

2. Preliminary

This section defines some of the notations, definitions, and basic terms used in this work.

Note that $\mathbb{E}=C(\widetilde{{\psi }_{\tau }},\mathbb{R})$ is a Banach space of continuous functions 𝚥 from $\widetilde{{\psi }_{\tau }}$ into $\mathbb{R}$, with a norm defined as

$${\displaystyle \parallel }𝚥{\displaystyle \parallel }=sup\left\{\right|𝚥(𝚤)|/𝚤\in \widetilde{{\psi }_{\tau }}\}.$$

Definition 1. 

Let $F:\mathbb{E}\to \mathbb{E}$ be an operator.

• F is  continuous  if for every sequence ${\left({𝚥}_n\right)}_{n\in \mathbb{N}}$ in $\mathbb{E}$ such that ${\left({𝚥}_n\right)}_{n\in \mathbb{N}}$ converge to 𝚥 in $\mathbb{E}$, the sequence ${\left(F{𝚥}_n\right)}_{n\in \mathbb{N}}$ converge to $F𝚥$.
• F is  completely continuous  if F is continuous and if the image for every bounded B in $\mathbb{E}$ $F\left(B\right)$ is relatively compact.

Definition 2. 

Let M be a subset of $C(I,\mathbb{R}).$

• M is called  uniformly bounded  if and only if
  there exists $c>0:\mid \mid f\left(t\right)\mid \mid \le c$ for all $t\in I$ and for all $f\in M$.

Definition 3 

([19,20,21]). Let $\aleph :\widetilde{{\psi }_{\tau }}\times \mathbb{R}\to (1,2]$ be a continuous function. The left Riemann–Liouville fractional integral of variable order $\aleph (𝚤,𝚥(𝚤\left)\right)$ for function $𝚥(𝚤)$ is defined by

$${\mathbb{I}}_0^{\aleph (𝚤,𝚥(𝚤\left)\right)}𝚥(𝚤)={\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{\aleph (\propto ,𝚥(\propto \left)\right)-1}}{\Gamma (\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto ,𝚤>0,$$

(2)

where $\Gamma (.)$ is the gamma function.

Definition 4 

([19,20,21]). Let $\aleph :\widetilde{{\psi }_{\tau }}\times \mathbb{R}\to (1,2]$ be a continuous function. The left Riemann–Liouville fractional derivative of variable order $\aleph (𝚤,𝚥(𝚤\left)\right)$ for function $𝚥(𝚤)$ is defined by

$${\mathbb{D}}_0^{\aleph (𝚤,𝚥(𝚤\left)\right)}𝚥(𝚤)={\left({\displaystyle \frac{d}{dt}}\right)}^2{\mathbb{I}}_0^{2-\aleph (𝚤,𝚥(𝚤\left)\right)}𝚥(𝚤)={\left({\displaystyle \frac{d}{dt}}\right)}^2{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto ,𝚤>0.$$

(3)

Remark 1 

([22,23,24]). For general functions $\aleph (𝚤,𝚥(𝚤\left)\right)$ and $v(𝚤,𝚥(𝚤\left)\right)$, we notice that the semigroup property does not hold, i.e.,

$${\mathbb{I}}_{a^{+}}^{\aleph (𝚤,𝚥(𝚤\left)\right)}{\mathbb{I}}_{a^{+}}^{v(𝚤,𝚥(𝚤\left)\right)}𝚥(𝚤)\ne {\mathbb{I}}_{a^{+}}^{\aleph (𝚤,𝚥(𝚤\left)\right)+v(𝚤,𝚥(𝚤\left)\right)}𝚥(𝚤).$$

Lemma 1 

([25]). Let $\aleph :\widetilde{{\psi }_{\tau }}\times \mathbb{R}\to (1,2)$ be a continuous function. Then, for

$y\in C_{\delta }(\widetilde{{\psi }_{\tau }},\mathbb{R})=\{y(𝚤)\in C(\widetilde{{\psi }_{\tau }},\mathbb{R}),{𝚤}^{\delta }y(𝚤)\in C(\widetilde{{\psi }_{\tau }},\mathbb{R}),(0\le \delta <1)\}$, the variable-order fractional integral ${\mathbb{I}}_{0^{+}}^{\aleph (𝚤,𝚥(𝚤\left)\right)}y(𝚤)$ exists $\forall 𝚤\in \widetilde{{\psi }_{\tau }}$.

Lemma 2 

([25]). Let $\aleph \in C(\widetilde{{\psi }_{\tau }}\times \mathbb{R},(1,2])$ be a continuous function. Then, ${\mathbb{I}}_{0^{+}}^{\aleph (𝚤,𝚥(𝚤\left)\right)}y(𝚤)\in C(\widetilde{{\psi }_{\tau }},\mathbb{R})$ for $y\in C(\widetilde{{\psi }_{\tau }},\mathbb{R})$.

Theorem 1 

([26]). Let K be a closed, bounded, and convex subset of a real Banach space ℑ and let $C_1$ and $C_2$ be operators on K satisfying the following conditions:

• $C_1\left(K\right)+C_2\left(K\right)\subset K$;
• $C_1$ is continuous on K, and $C_1\left(K\right)$ is a relatively compact subset of ℑ;
• $C_2$ is a strict contraction on K, i.e, there exists $\tilde{p}\in [0,1)$ such that

  $${\displaystyle \mid \mid }{C}_2\left(𝚥\right)-C_2\left(\kappa \right)\mid \mid \le \tilde{p}\mid \mid 𝚥-\kappa {\displaystyle \mid \mid }$$
  for every $𝚥,\kappa \in K.$
  Then, there exists $𝚥\in K$ such that $C_1\left(𝚥\right)+C_2\left(𝚥\right)=𝚥.$

3. Existence and Uniqueness Criteria

3.1. Results of Existence

Let us introduce the following assumptions:

Hypothesis 1. 

There exist constants $0<\sigma <1$ and $\tilde{p}>0$ such that the function ${𝚤}^{\sigma }\eta$ is a continuous function on $\widetilde{{\psi }_{\tau }}\times \mathbb{R}$ and

${𝚤}^{\sigma }\mid$η (ı, $𝚥(𝚤)$) − η (ı, y$(𝚤)$) $|$ ≤ $\tilde{p}$$\left|𝚥\right(𝚤)$ − y(ı),     $\forall$$𝚥$, y $\in$ $\mathbb{R}$, ı $\in$ $\widetilde{{\psi }_{\tau }}$.

Hypothesis 2. 

ℵ: $\widetilde{{\psi }_{\tau }}\times \mathbb{R}$→(1, ${\aleph }^{*}$] is a continuous function such that $1\le \aleph (𝚤,𝚥(𝚤))\le {\aleph }^{*}<2$.

Remark 2 

([27]).

• The function $\Gamma$(2-ℵ($𝚤,𝚥(𝚤)$)) is continuous as a composition of two continuous functions. We can let $\mathcal{M}=max\mid {\displaystyle \frac{1}{\Gamma (2-\aleph (𝚤,𝚥(𝚤)\left)\right)}}\mid .$
• By the continuity of the function ℵ($𝚤,𝚥(𝚤)$), we let
  ${\tilde{\iota }}^{1-\aleph (𝚤,𝚥(𝚤\left)\right)}$ ≤ 1 if $1\le \tilde{\iota }<\infty ,$ and ${\tilde{\iota }}^{1-\aleph (𝚤,𝚥(𝚤\left)\right)}$ ≤ ${\tilde{\iota }}^{1-{\aleph }^{*}}$ if $0\le \tilde{\iota }\le 1.$
  We conclude that ${\tilde{\iota }}^{1-\aleph (𝚤,𝚥(𝚤\left)\right)}$≤ $\max$ (1,${\tilde{\iota }}^{1-{\aleph }^{*}}$)= ${\tilde{\iota }}^{*}.$

Remark 3 

([28]). If X and Y are two real numbers, then

$$\mid \alpha X-\beta Y\mid \le 2max(\alpha ,\beta )\mid X-Y\mid ,$$

where α and β are positive real numbers.

Lemma 3 

([27]). Let (SY 2) hold and let ${𝚥}_n,𝚥\in C[0,\tilde{\iota }]$. Assume that ${𝚥}_n(𝚤)\to 𝚥(𝚤)$, $𝚤\in [0,\tilde{\iota }]$ as $n\to \infty$; then,

$${\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,{𝚥}_n(\propto ))}}{\Gamma (2-\aleph (\propto ,{𝚥}_n(\propto )))}}{𝚥}_n(\propto )d\propto \to {\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-)}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto ,𝚤\in [0,\tilde{\iota }],$$

as $n\to \infty$.

We will need the following lemma to solve NAVOITVP (1).

Lemma 4. 

The function $𝚥\in \mathbb{E}$ forms a solution of NAVOITVP (1) if and only if $𝚥$ fulfills the integral equation

$$\begin{gathered} 𝚥(𝚤)={\displaystyle \frac{1}{\varsigma }}[-{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto +{\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,𝚥(\propto ))d\propto \\ +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto -{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,𝚥(\propto ))d\propto ]. \end{gathered}$$

(4)

Proof. 

By the definition of the fractional derivative of the variable order provided by (3), it is possible to express NAVOITVP (1) as follows:

$${\displaystyle \frac{d^2}{d{𝚤}^2}}{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto +\varsigma {𝚥}^{″}(𝚤)=\eta (𝚤,𝚥(𝚤)).$$

Then,

$${\displaystyle \frac{d}{d𝚤}}{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto +\varsigma {𝚥}^{\prime }(𝚤)={\int }_0^{𝚤}\eta (\propto ,𝚥(\propto ))d\propto +c_1.$$

Thus,

$${\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto +\varsigma 𝚥(𝚤)={\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,𝚥(\propto ))d\propto +c_1𝚤+c_2.$$

(5)

Evaluating Equation (5) at $𝚤=0$ gives us $c_2=0$, and evaluating Equation (5) at $𝚤=\tilde{\iota }$, we find

$$c_1={\displaystyle \frac{1}{\tilde{\iota }}}\left[{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto -{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,𝚥(\propto ))d\propto \right].$$

Then,

$${\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto +\varsigma 𝚥(𝚤)={\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,𝚥(\propto ))d\propto +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto$$

$$-{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,𝚥(\propto ))d\propto ,$$

So,

$$\begin{gathered} 𝚥(𝚤)={\displaystyle \frac{1}{\varsigma }}[-{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto +{\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,𝚥(\propto ))d\propto \\ +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto -{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,𝚥(\propto ))d\propto ]. \end{gathered}$$

Conversely, by deriving both sides of Equation (4), we have

$$\begin{gathered} {\displaystyle \frac{d}{d𝚤}}\left({\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto \right)+\varsigma {𝚥}^{\prime }(𝚤)={\int }_0^{𝚤}\eta (\propto ,𝚥(\propto ))d\propto \\ +{\displaystyle \frac{1}{\tilde{\iota }}}\left({\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto -{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,𝚥(\propto ))d\propto \right). \end{gathered}$$

By derivation again, we obtain

$${\displaystyle \frac{d^2}{d{𝚤}^2}}\left({\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto \right)+\varsigma {𝚥}^{″}(𝚤)=\eta (𝚤,𝚥(𝚤)),$$

which means NAVOITVP (1).

Theorem 1 forms the basis of the first finding. □

Theorem 2. 

Assume that Hypothesis 1 and Hypothesis 2 hold. If

$${\displaystyle \frac{2\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }}{2-{\aleph }^{*}}}+2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}<\varsigma ,$$

(6)

then NAVOITVP (1) has at least one solution on $\mathbb{E}$.

Proof. 

We construct operators

$$C_1,C_2:\mathbb{E}\to \mathbb{E},$$

as follows:

$$\begin{gathered} C_1𝚥(𝚤)={\displaystyle \frac{1}{\varsigma }}\left[-{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto \right], \\ {C}_2𝚥(𝚤)={\displaystyle \frac{1}{\varsigma }}\left[{\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,𝚥(\propto ))d\propto -{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,𝚥(\propto ))d\propto \right]. \end{gathered}$$

Let

$$R\ge {\displaystyle \frac{\frac{{\eta }^{*}{\tilde{\iota }}^2}{\varsigma }}{1-\frac{1}{\varsigma }\left[\frac{2\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }}{2-{\aleph }^{*}}+2\tilde{p}\frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}\right]}},$$

where

$${\eta }^{*}=\underset{𝚤\in \widetilde{{\psi }_{\tau }}}{sup}\mid \eta (𝚤,0)\mid .$$

We consider the set

$$B_R=\{𝚥\in \mathbb{E},\mid \mid 𝚥\mid \mid \le R\}.$$

Clearly, $B_R$ is a bounded, convex, and closed set.

We now prove that $C_1$ and $C_2$ satisfy the conditions given by Theorem 1. The argument will be put into practice in four steps.

Step 1: $C_1\left(B_R\right)+C_2\left(B_R\right)\subseteq B_R$.

For $𝚥\in B_R$, we obtain

$$\mid C_1𝚥(𝚤)+C_2𝚥(𝚤)\mid$$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{1}{\varsigma }}[{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}\mid 𝚥(\propto )\mid d\propto +{\int }_0^{𝚤}(𝚤-\propto )\mid \eta (\propto ,𝚥(\propto ))\mid d\propto \hfill \\ & +& {\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}\mid 𝚥(\propto )\mid d\propto +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\mid \eta (\propto ,𝚥(\propto ))\mid d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}\mid 𝚥(\propto )\mid d\propto +{\int }_0^{𝚤}(𝚤-\propto )\mid \eta (\propto ,𝚥(\propto ))-\eta (\propto ,0)\mid d\propto \hfill \\ & +& {\int }_0^{𝚤}(𝚤-\propto )\mid \eta (\propto ,0)\mid d\propto +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto ))}}\mid 𝚥(\propto )\mid d\propto \hfill \\ & +& {\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\mid \eta (\propto ,𝚥(\propto ))-\eta (\propto ,0)\mid d\propto +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\mid \eta (\propto ,0)\mid d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[\mathcal{M}{\tilde{\iota }}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}{\int }_0^{𝚤}{\left({\displaystyle \frac{𝚤-\propto }{\tilde{\iota }}}\right)}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}\mid 𝚥(\propto )\mid d\propto \hfill \\ & +& {\int }_0^{𝚤}{(𝚤-\propto )\propto }^{-\sigma }{\propto }^{\sigma }\mid \eta (\propto ,𝚥(\propto ))-\eta (\propto ,0)\mid d\propto \hfill \\ & +& {\int }_0^{𝚤}(𝚤-\propto )\mid \eta (\propto ,0)\mid d\propto +\mathcal{M}{\tilde{\iota }}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}{\int }_0^{\tilde{\iota }}{\left({\displaystyle \frac{\tilde{\iota }-\propto }{\tilde{\iota }}}\right)}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}\mid 𝚥(\propto )\mid d\propto \hfill \\ & +& {\int }_0^{\tilde{\iota }}{\propto }^{-\sigma }{\propto }^{\sigma }(\tilde{\iota }-\propto )\mid \eta (\propto ,𝚥(\propto ))-\eta (\propto ,0)\mid d\propto +{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\mid \eta (\propto ,0)\mid d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[\mathcal{M}{\tilde{\iota }}^{*}{\int }_0^{𝚤}{\left({\displaystyle \frac{𝚤-\propto }{\tilde{\iota }}}\right)}^{1-{\aleph }^{*}}\mid 𝚥(\propto )\mid d\propto \hfill \\ & +& {\int }_0^{𝚤}{(𝚤-\propto )\propto }^{-\sigma }{\propto }^{\sigma }\mid \eta (\propto ,𝚥(\propto ))-\eta (\propto ,0)\mid d\propto \hfill \\ & +& {\int }_0^{𝚤}(𝚤-\propto )\mid \eta (\propto ,0)\mid d\propto +\mathcal{M}{\tilde{\iota }}^{*}{\int }_0^{\tilde{\iota }}{\left({\displaystyle \frac{\tilde{\iota }-\propto }{\tilde{\iota }}}\right)}^{1-{\aleph }^{*}}\mid 𝚥(\propto )\mid d\propto \hfill \\ & +& {\int }_0^{𝚤}{\propto }^{-\sigma }{\propto }^{\sigma }(\tilde{\iota }-\propto )\mid \eta (\propto ,𝚥(\propto ))-\eta (\propto ,0)\mid d\propto +{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\mid \eta (\propto ,0)\mid d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[{\displaystyle \frac{\mathcal{M}{\tilde{\iota }}^{*}}{{\tilde{\iota }}^{1-{\aleph }^{*}}}}{\displaystyle \frac{{\tilde{\iota }}^{2-{\aleph }^{*}}}{2-{\aleph }^{*}}}\mid \mid 𝚥\mid \mid +{\int }_0^{𝚤}{(𝚤-\propto )}^{{\propto }^{-\sigma }}\tilde{p}\mid 𝚥(\propto )\mid d\propto +{\int }_0^{𝚤}(𝚤-\propto ){\eta }^{*}d\propto \hfill \\ & +& {\displaystyle \frac{\mathcal{M}{\tilde{\iota }}^{*}}{{\tilde{\iota }}^{1-{\aleph }^{*}}}}{\displaystyle \frac{{\tilde{\iota }}^{2-{\aleph }^{*}}}{2-{\aleph }^{*}}}\mid \mid 𝚥\mid \mid +{\int }_0^{\tilde{\iota }}{(\tilde{\iota }-\propto )}^{{\propto }^{-\sigma }}\tilde{p}\mid 𝚥(\propto )\mid d\propto +{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto ){\eta }^{*}d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[{\displaystyle \frac{2\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }}{2-{\aleph }^{*}}}\mid \mid 𝚥\mid \mid +\tilde{p}\mid \mid 𝚥\mid \mid {\int }_0^{𝚤}(𝚤-\propto ){\propto }^{-\sigma }d-+{\eta }^{*}{\displaystyle \frac{{𝚤}^2}{2}}\hfill \\ & +& \tilde{p}\mid \mid 𝚥\mid \mid {\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto ){\propto }^{-\sigma }d\propto +{\eta }^{*}{\displaystyle \frac{{\tilde{\iota }}^2}{2}}]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left[{\displaystyle \frac{2\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }}{2-{\aleph }^{*}}}\mid \mid 𝚥\mid \mid +\tilde{p}\mid \mid 𝚥\mid \mid {\displaystyle \frac{{𝚤}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}+{\eta }^{*}{\tilde{\iota }}^2+\tilde{p}\mid \mid 𝚥\mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left[{\displaystyle \frac{2\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }}{2-{\aleph }^{*}}}\mid \mid 𝚥\mid \mid +2\tilde{p}\mid \mid 𝚥\mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}+{\eta }^{*}{\tilde{\iota }}^2\right]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left[{\displaystyle \frac{2\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }}{2-{\aleph }^{*}}}+2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right]\mid \mid 𝚥\mid \mid +{\displaystyle \frac{{\eta }^{*}{\tilde{\iota }}^2.}{\varsigma }}\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left[{\displaystyle \frac{2\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }}{2-{\aleph }^{*}}}+2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right]R+{\displaystyle \frac{{\eta }^{*}{\tilde{\iota }}^2.}{\varsigma }}\hfill \\ & \le & R.\hfill \end{array}$$

which means that $C_1\left(B_R\right)+C_2\left(B_R\right)\subseteq B_R$.

Step 2: $C_1$ is continuous.

Let ${𝚥}_n$ be a sequence satisfying ${𝚥}_n\to 𝚥$ in $\mathbb{E}$. For $𝚤\in \widetilde{{\psi }_{\tau }}$, we estimate

$$\mid C_1{𝚥}_n(𝚤)-C_1𝚥(𝚤)\mid$$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{1}{\varsigma }}\left[\right|{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,{𝚥}_n(\propto ))}}{\Gamma (2-\aleph (\propto ,{𝚥}_n(\propto )))}}{𝚥}_n(\propto )-{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto |\hfill \\ & +& {\displaystyle \frac{𝚤}{\tilde{\iota }}}|{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,{𝚥}_n(\propto ))}}{\Gamma (2-\aleph \left({𝚥}_n(\propto )\right))}}{𝚥}_n\left(\right)-{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto |],\hfill \end{array}$$

Using Lemma 3, we have

$$\mid \mid C_1{𝚥}_n(𝚤)-C_1𝚥(𝚤)\mid \mid \to 0,asn\to \infty .$$

The above relation shows that operator $C_1$ is continuous on $\mathbb{E}$.

Step 3: $C_1$ is compact.

Here, we intend to prove the relative compactness of $C_1\left(B_R\right)$, which means that $C_1$ is compact. Evidently, $C_1\left(B_R\right)$ is uniformly bounded; due to step 1, we saw that

$$C_1\left(B_R\right)=\{C_1\left(𝚥\right):𝚥\in B_R\}\subset C_1\left(B_R\right)+C_2\left(B_R\right)\subset B_R.$$

Hence, for every $𝚥\in B_R$, we obtain $\mid \mid C_1\left(𝚥\right)\mid \mid \le R$, which indicates the uniform boundedness of $C_1\left(B_R\right)$. Firstly, we can know that the function $w(𝚤)=a^{𝚤}-b^{𝚤},𝚤\in (-1,0),0<a<b<1$, is decreasing. Indeed, since $lna<lnb<0$ and $a^{𝚤}>b^{𝚤}>0$, we have that

$$w^{\prime }(𝚤)=a^{𝚤}lna-b^{𝚤}lnb<b^{𝚤}lna-b^{𝚤}lnb=b^{𝚤}(lna-lnb)<0,$$

which implies that $w(𝚤)$ is decreasing. Thus, for $\varkappa (\propto )={\left({\displaystyle \frac{{𝚤}_1-\propto }{\tilde{\iota }}}\right)}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}-{\left({\displaystyle \frac{{𝚤}_2-\propto }{\tilde{\iota }}}\right)}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}$ (where $0<{\displaystyle \frac{{𝚤}_1-\propto }{\tilde{\iota }}}<{\displaystyle \frac{{𝚤}_2-\propto }{\tilde{\iota }}}<1$), we may look at $\varkappa (\propto )$ as the same type as $w(\propto )$; then, $\varkappa (\propto )$ is decreasing with respect to its exponent $1-\aleph (\propto ,𝚥(\propto \left)\right)$. Then, for ${𝚤}_1,{𝚤}_2\in \widetilde{{\psi }_{\tau }}$, ${𝚤}_1<{𝚤}_2$ and $𝚥\in B_R$, we have

$$\mid C_1𝚥\left({𝚤}_2\right)-C_1𝚥\left({𝚤}_1\right)\mid$$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{1}{\varsigma }}\left[\right|{\int }_0^{{𝚤}_2}{\displaystyle \frac{{({𝚤}_2-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto -{\int }_0^{{𝚤}_1}{\displaystyle \frac{{({𝚤}_1-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto |\hfill \\ & +& |{\displaystyle \frac{{𝚤}_2}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto -{\displaystyle \frac{{𝚤}_1}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto |]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left[\right|{\int }_0^{{𝚤}_1}{\displaystyle \frac{{({𝚤}_2-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )-{\displaystyle \frac{{({𝚤}_1-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto |\hfill \\ & +& |{\int }_{{𝚤}_1}^{{𝚤}_2}{\displaystyle \frac{{({𝚤}_2-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto |+|{\displaystyle \frac{1}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto \left[{𝚤}_2-{𝚤}_1\right]|]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[{\int }_0^{{𝚤}_1}\mid {\displaystyle \frac{1}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}\mid \mid {({𝚤}_2-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}-{({𝚤}_1-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}\mid \mid 𝚥(\propto )\mid d\propto \hfill \\ & +& {\int }_{{𝚤}_1}^{{𝚤}_2}{\displaystyle \frac{{({𝚤}_2-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}\mid 𝚥(\propto )\mid d\propto +{\displaystyle \frac{{𝚤}_2-{𝚤}_1}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}\mid 𝚥(\propto )\mid d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[\mathcal{M}\mid \mid 𝚥\mid \mid {\int }_0^{{𝚤}_1}\left[{\left({𝚤}_1-\propto \right)}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}-{\left({𝚤}_2-\propto \right)}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}\right]d\propto \hfill \\ & +& \mathcal{M}\mid \mid 𝚥\mid \mid {\tilde{\iota }}^{*}{\int }_{{𝚤}_1}^{{𝚤}_2}{\left({\displaystyle \frac{{𝚤}_2-\propto }{\tilde{\iota }}}\right)}^{1-{\aleph }^{*}}d\propto +{\displaystyle \frac{\mathcal{M}}{\tilde{\iota }}}({𝚤}_2-{𝚤}_1)\mid \mid 𝚥\mid \mid {\tilde{\iota }}^{*}{\int }_0^{\tilde{\iota }}{\left({\displaystyle \frac{\tilde{\iota }-\propto }{\tilde{\iota }}}\right)}^{1-{\aleph }^{*}}d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[\mathcal{M}\mid \mid 𝚥\mid \mid {\int }_0^{{𝚤}_1}{\tilde{\iota }}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}\left[{\left({\displaystyle \frac{{𝚤}_1-\propto }{\tilde{\iota }}}\right)}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}-{\left({\displaystyle \frac{{𝚤}_2-\propto }{\tilde{\iota }}}\right)}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}\right]d\propto \hfill \\ & +& \mathcal{M}\mid \mid 𝚥\mid \mid {\tilde{\iota }}^{*}{\int }_{{𝚤}_1}^{{𝚤}_2}{\left({\displaystyle \frac{{𝚤}_2-\propto }{\tilde{\iota }}}\right)}^{1-{\aleph }^{*}}d\propto +{\displaystyle \frac{\mathcal{M}{\tilde{\iota }}^{*}}{2-{\aleph }^{*}}}\mid \mid 𝚥\mid \mid ({𝚤}_2-{𝚤}_1)]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[\mathcal{M}\mid \mid 𝚥\mid \mid {\tilde{\iota }}^{*}{\int }_0^{{𝚤}_1}\left[{\left({\displaystyle \frac{{𝚤}_1-}{\tilde{\iota }}}\right)}^{1-{\aleph }^{*}}-{\left({\displaystyle \frac{{𝚤}_2-}{\tilde{\iota }}}\right)}^{1-{\aleph }^{*}}\right]d\propto \hfill \\ & +& \mathcal{M}\mid \mid 𝚥\mid \mid {\tilde{\iota }}^{*}{\int }_{{𝚤}_1}^{{𝚤}_2}{\left({\displaystyle \frac{{𝚤}_2-}{\tilde{\iota }}}\right)}^{1-{\aleph }^{*}}d\propto +{\displaystyle \frac{\mathcal{M}{\tilde{\iota }}^{*}}{2-{\aleph }^{*}}}\mid \mid 𝚥\mid \mid \left({𝚤}_2-{𝚤}_1\right)]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[\left[{\displaystyle \frac{\mathcal{M}\mid \mid 𝚥\mid \mid {\tilde{\iota }}^{*}}{{\tilde{\iota }}^{1-{\aleph }^{*}}(2-{\aleph }^{*})}}\right]\left[{\left({𝚤}_1\right)}^{2-{\aleph }^{*}}-{\left({𝚤}_2\right)}^{2-{\aleph }^{*}}+2{({𝚤}_2-{𝚤}_1)}^{2-{\aleph }^{*}}\right]\hfill \\ & +& \left[{\displaystyle \frac{\mathcal{M}{\tilde{\iota }}^{*}}{2-{\aleph }^{*}}}\mid \mid 𝚥\mid \mid \left({𝚤}_2-{𝚤}_1\right)\right]]\hfill \end{array}$$

Hence, $\mid C_1𝚥\left({𝚤}_2\right)-C_1𝚥\left({𝚤}_1\right)\mid \to 0$ as ${𝚤}_2\to {𝚤}_1$. This implies that $C_1\left(B_R\right)$ is equicontinuous.

Step 4: $C_2$ is a strict contraction.

For 𝚥, $\kappa$$\in \mathbb{E}$ and $𝚤\in \widetilde{{\psi }_{\ell }}$, we obtain

$$\mid C_2𝚥(𝚤)-C_2\kappa (𝚤)\mid$$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{1}{\varsigma }}[{\int }_0^{𝚤}{(𝚤-\propto )\propto }^{-\sigma }{\propto }^{\sigma }\mid \eta (\propto ,𝚥(\propto ))-\eta (\propto ,\kappa (\propto ))\mid d\propto \hfill \\ & +& {\int }_0^{\tilde{\iota }}{(𝚤-\propto )\propto }^{-\sigma }{\propto }^{\sigma }\mid \eta (\propto ,\kappa (\propto ))-\eta (\propto ,𝚥(\propto ))\mid d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left[\tilde{p}\mid \mid 𝚥-\kappa \mid \mid {\int }_0^{𝚤}{(𝚤-\propto )}^{{\propto }^{-\sigma }}d\propto +\tilde{p}\mid \mid 𝚥-\kappa \mid \mid {\int }_0^{\tilde{\iota }}{(\tilde{\iota }-\propto )}^{{\propto }^{-\sigma }}d\propto \right]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left[\tilde{p}\mid \mid 𝚥-\kappa \mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}+\tilde{p}\mid \mid 𝚥-\kappa \mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left[2\tilde{p}\mid \mid 𝚥-\kappa \mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left(2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right)\mid \mid 𝚥-\kappa \mid \mid .\hfill \end{array}$$

Consequently, by (6), operator $C_2$ is a strict contraction.

Therefore, all conditions of Theorem 1 are fulfilled. We infer that NAVOIVP (1) has at least one solution in $\mathbb{E}$. □

3.2. Results of Uniqueness

The next result is based on the Banach contraction theorem.

Theorem 3. 

Let Hypothesis 1 and Hypothesis 2 be satisfied if

$${\displaystyle \frac{1}{\varsigma }}\left(8\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }+2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right)<1,$$

(7)

Then, NAVOITVP (1) has a unique solution on $\mathbb{E}$.

Proof. 

Consider the operator

$$C:\mathbb{E}\to \mathbb{E},$$

defined by

$$\left(C𝚥\right)(𝚤)=\left(C_1𝚥\right)(𝚤)+\left(C_2𝚥\right)(𝚤),for𝚥(𝚤)\in \mathbb{E}.$$

For $𝚥$, ${𝚥}^{*}$ ∈ $\mathbb{E}$, we may write

$$\mid C𝚥(𝚤)-C{𝚥}^{*}(𝚤)\mid$$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{1}{\varsigma }}\left[\right|{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )-{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}{𝚥}^{*}(\propto )d\propto |\hfill \\ & +& {\int }_0^{𝚤}(𝚤-\propto ){\propto }^{-\sigma }{\propto }^{\sigma }\mid \eta (\propto ,𝚥(\propto ))-\eta (\propto ,{𝚥}^{*}(\propto ))\mid d\propto \hfill \\ & +& |{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )-{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}{𝚥}^{*}(\propto )d\propto |\hfill \\ & +& {\int }_0^{\tilde{\iota }}{(𝚤-\propto )\propto }^{-\sigma }{\propto }^{\sigma }\mid \eta (\propto ,{𝚥}^{*}(\propto ))-\eta (\propto ,𝚥(\propto ))\mid d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[2{\int }_0^{𝚤}\left(\underset{𝚤\in \widetilde{{\psi }_{\tau }}}{sup}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}+\underset{𝚤\in \widetilde{{\psi }_{\tau }}}{sup}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,{𝚥}^{*}(\propto ))}}{\Gamma (2-\aleph (\propto ,{𝚥}^{*}(\propto )))}}\right)\mid 𝚥-{𝚥}^{*}\mid d\propto \hfill \\ & +& \tilde{p}\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\int }_0^{𝚤}(𝚤-\propto ){\propto }^{-\sigma }d\propto +\tilde{p}\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto ){\propto }^{-\sigma }d\propto \hfill \\ & +& 2{\int }_0^{\tilde{\iota }}\left(sup{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}+sup{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,{𝚥}^{*}(\propto ))}}{\Gamma (2-\aleph (\propto ,{𝚥}^{*}(\propto )))}}\right)\mid 𝚥-{𝚥}^{*}\mid d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[2\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\int }_0^{𝚤}(\mathcal{M}{\tilde{\iota }}^{*}+\mathcal{M}{\tilde{\iota }}^{*})d\propto +\tilde{p}\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\displaystyle \frac{{𝚤}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\hfill \\ & +& 2\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\int }_0^{\tilde{\iota }}(\mathcal{M}{\tilde{\iota }}^{*}+\mathcal{M}{\tilde{\iota }}^{*})d\propto +\tilde{p}\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[4\mathcal{M}{\tilde{\iota }}^{*}\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\int }_0^{𝚤}d\propto +2\tilde{p}\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\hfill \\ & +& 4\mathcal{M}{\tilde{\iota }}^{*}\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\int }_0^{\tilde{\iota }}d\propto ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[4\mathcal{M}{\tilde{\iota }}^{*}𝚤\mid \mid 𝚥-{𝚥}^{*}\mid \mid +2\tilde{p}\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\hfill \\ & +& 4\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }\mid \mid 𝚥-{𝚥}^{*}\mid \mid ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}[4\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }\mid \mid 𝚥-{𝚥}^{*}\mid \mid +2\tilde{p}\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\hfill \\ & +& 4\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }\mid \mid 𝚥-{𝚥}^{*}\mid \mid ]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left[8\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }\mid \mid 𝚥-{𝚥}^{*}\mid \mid +2\tilde{p}\mid \mid 𝚥-{𝚥}^{*}\mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right]\hfill \\ & \le & {\displaystyle \frac{1}{\varsigma }}\left(8\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }+2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right)\mid \mid 𝚥-{𝚥}^{*}\mid \mid .\hfill \end{array}$$

Consequently by Equation (7), operator C will be a contraction. According to the Banach contraction theorem, operator C has a unique fixed point that is the unique solution of NAVOITVP (1). □

4. Ulam–Hyers Stability

One of the important qualitative specifications of solutions to given NAVOIVPs is their stability, and in the following, we aim to investigate the Ulam–Hyers stability for solutions of the supposed non-autonomous variable-order initial and terminal value problem (NAVOITVP (1)).

Definition 5 

([29]). On account of NAVOITVP (1), consider the inequality

$$|D_{0^{+}}^{\aleph (𝚤,\vartheta (𝚤\left)\right)}\vartheta (𝚤)+\alpha {\vartheta }^{″}(𝚤)-\eta (𝚤,\vartheta (𝚤))|\le ϵ,𝚤\in \widetilde{{\psi }_{\tau }}.$$

(8)

We say that NAVOITVP (1) is Ulam–Hyers stable if there is a $c_{\eta }>0$ such that for any $ϵ>0$ and for any solution $\vartheta \in C(\widetilde{{\psi }_{\tau }},\mathbb{R})$ of (8), there is a solution $𝚥\in C(\widetilde{{\psi }_{\tau }},\mathbb{R})$ of NAVOITVP (1) such that

$$|\vartheta (𝚤)-𝚥(𝚤)|\le c_{\eta }ϵ,𝚤\in \widetilde{{\psi }_{\tau }}.$$

Theorem 4. 

Let Hypothesis 1, Hypothesis 2, and inequality (7) be satisfied. Then, NAVOITVP (1) is Ulam–Hyers stable.

Proof. 

Let $ϵ>0$ be an arbitrary number and the function $\vartheta (𝚤)$ from $C(\widetilde{{\psi }_{\tau }},\mathbb{R})$ satisfy the following inequality:

$$|D_{0^{+}}^{\aleph (𝚤,\vartheta (𝚤\left)\right)}\vartheta (𝚤)+\alpha {\vartheta }^{″}(𝚤)-\eta (𝚤,\vartheta (𝚤))|\le ϵ,𝚤\in \widetilde{{\psi }_{\tau }}.$$

(9)

We integrate both sides of inequality (9) two times to obtain

$$\begin{array}{ccc}\hfill \left|\vartheta \right(𝚤)& +& {\displaystyle \frac{1}{\varsigma }}[{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\vartheta (\propto )d\propto -{\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,\vartheta (\propto ))d\propto \hfill \\ & -& {\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\vartheta (\propto )d\propto +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,\vartheta (\propto ))d\propto \left]\right|\hfill \\ & \le & {\displaystyle \frac{ϵ{\tilde{\iota }}^2}{2}}\hfill \end{array}$$

Let $𝚤\in \widetilde{{\psi }_{\tau }}$. Then,

$$\begin{array}{ccc}\hfill \mid \vartheta (𝚤)-𝚥(𝚤)\mid & =& |\vartheta (𝚤)+{\displaystyle \frac{1}{\varsigma }}[{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto -{\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,𝚥(\propto ))d\propto \hfill \\ & -& {\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,𝚥(\propto ))d\propto \left]\right|\hfill \\ & =& |\vartheta (𝚤)+{\displaystyle \frac{1}{\varsigma }}[{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto -{\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,𝚥(\propto ))d\propto \hfill \\ & -& {\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,𝚥(\propto ))d\propto ]\hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}\left[{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\vartheta (\propto )-{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\vartheta (\propto )d\propto \right]\hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}\left[{\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,\vartheta (\propto ))d\propto -{\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,\vartheta (\propto ))d\propto \right]\hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}\left[{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\vartheta (\propto )d\propto -{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\vartheta (\propto )d\propto \right]\hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}\left[{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,\vartheta (\propto ))d\propto -{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,\vartheta (\propto ))d\propto \right]|\hfill \\ & \le & |\vartheta (𝚤)+{\displaystyle \frac{1}{\varsigma }}[{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\vartheta (\propto )d\propto -{\int }_0^{𝚤}(𝚤-\propto )\eta (\propto ,\vartheta (\propto ))d\propto \hfill \\ & -& {\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\vartheta (\propto )d\propto +{\displaystyle \frac{𝚤}{\tilde{\iota }}}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\eta (\propto ,\vartheta (\propto ))d\propto \left]\right|\hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}\left[|{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto -{\int }_0^{𝚤}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\vartheta (\propto )d\propto |\right]\hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}\left[{\int }_0^{𝚤}(𝚤-\propto )\mid \eta (\propto ,\vartheta (\propto ))-\eta (\propto ,𝚥(\propto ))\mid d\propto \right]\hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}{\displaystyle \frac{𝚤}{\tilde{\iota }}}\left[|{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\vartheta (\propto )d\propto -{\int }_0^{\tilde{\iota }}{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}𝚥(\propto )d\propto |\right]\hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}{\displaystyle \frac{𝚤}{\tilde{\iota }}}\left[{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto )\mid \eta (\propto ,𝚥(\propto ))-\eta (\propto ,\vartheta (\propto ))\mid d\propto \right]\hfill \\ & \le & {\displaystyle \frac{ϵ{\tilde{\iota }}^2}{2}}+{\displaystyle \frac{1}{\varsigma }}2{\int }_0^{𝚤}\left(\underset{𝚤\in \widetilde{{\psi }_{\tau }}}{sup}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}+\underset{𝚤\in \widetilde{{\psi }_{\tau }}}{sup}{\displaystyle \frac{{(𝚤-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}\right)\mid 𝚥-\vartheta \mid d\propto \hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}{\int }_0^{𝚤}(𝚤-\propto ){\propto }^{-\sigma }{\propto }^{\sigma }\sigma \mid \eta (\propto ,\vartheta (\propto ))-\eta (\propto ,𝚥(\propto ))\mid d\propto \hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}2{\int }_0^{\tilde{\iota }}\left(sup{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,\vartheta (\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,\vartheta (\propto )\left)\right)}}+sup{\displaystyle \frac{{(\tilde{\iota }-\propto )}^{1-\aleph (\propto ,𝚥(\propto \left)\right)}}{\Gamma (2-\aleph (\propto ,𝚥(\propto )\left)\right)}}\right)\mid \vartheta -𝚥\mid d\propto \hfill \\ & +& {\displaystyle \frac{1}{\varsigma }}{\int }_0^{\tilde{\iota }}(\tilde{\iota }-\propto ){\propto }^{-\sigma }{\propto }^{\sigma }\sigma \mid \eta (\propto ,𝚥(\propto ))-\eta (\propto ,\vartheta (\propto ))\mid d\propto \hfill \\ & \le & {\displaystyle \frac{ϵ{\tilde{\iota }}^2}{2}}+{\displaystyle \frac{1}{\varsigma }}[2\mid \mid \vartheta -𝚥\mid \mid {\int }_0^{𝚤}(\mathcal{M}{\tilde{\iota }}^{*}+\mathcal{M}{\tilde{\iota }}^{*})d\propto +\tilde{p}\mid \mid \vartheta -𝚥\mid \mid {\displaystyle \frac{{𝚤}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\hfill \\ & +& 2\mid \mid \vartheta -𝚥\mid \mid {\int }_0^{\tilde{\iota }}(\mathcal{M}{\tilde{\iota }}^{*}+\mathcal{M}{\tilde{\iota }}^{*})d\propto +\tilde{p}\mid \mid \vartheta -𝚥\mid \mid {\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}]\hfill \\ & \le & {\displaystyle \frac{ϵ{\tilde{\iota }}^2}{2}}+{\displaystyle \frac{1}{\varsigma }}[4\mathcal{M}{\tilde{\iota }}^{*}\mid \mid \vartheta -𝚥\mid \mid {\int }_0^{𝚤}d\propto +2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\mid \mid \vartheta -𝚥\mid \mid \hfill \\ & +& 4\mathcal{M}{\tilde{\iota }}^{*}\mid \mid \vartheta -𝚥\mid \mid {\int }_0^{\tilde{\iota }}d\propto ]\hfill \\ & \le & {\displaystyle \frac{ϵ{\tilde{\iota }}^2}{2}}+{\displaystyle \frac{1}{\varsigma }}\left[8\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }+2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right]\mid \mid \vartheta -𝚥\mid \mid .\hfill \end{array}$$

Then,

$$\mid \mid \vartheta -𝚥\mid \mid \left(1-{\displaystyle \frac{1}{\varsigma }}\left[8\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }+2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right]\right)\le {\displaystyle \frac{ϵ{\tilde{\iota }}^2}{2}}.$$

We obtain, for each $𝚤\in \widetilde{{\psi }_{\tau }}$,

$$\mid \vartheta (𝚤)-𝚥(𝚤)\mid \le \mid \mid \vartheta -𝚥\mid \mid \le {\displaystyle \frac{\frac{{\tilde{\iota }}^2}{2}}{1-\frac{1}{\varsigma }\left[8\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }+2\tilde{p}\frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}\right]}}ϵ=c_{\eta }ϵ.$$

Then, NAVOITVP (1) is Ulam–Hyers stable. □

5. Numerical Examples

Example 1. 

Consider the following NAVOITVP:

$$\left\{\begin{array}{c}{D}^{{\displaystyle \frac{𝚤}{2}}+1}𝚥(𝚤)+3{𝚥}^{″}(𝚤)={\displaystyle \frac{exp(-𝚤+4)}{𝚤+8}}\left({𝚤}^2\right)+{\displaystyle \frac{1}{3+𝚤}}𝚥,𝚤\in [0,\tilde{\iota }]=[0,{\displaystyle \frac{1}{4}}],\hfill \\ 𝚥\left(0\right)=0,𝚥\left(\tilde{\iota }\right)=0,\hfill \end{array}\right.$$

(10)

We have that $\aleph (𝚤,𝚥(𝚤))={\displaystyle \frac{𝚤}{2}}+1$ is a continuous function with

$1<\aleph (𝚤,𝚥(𝚤))<{\displaystyle \frac{\frac{1}{4}}{2}}+1={\displaystyle \frac{1}{8}}+1={\displaystyle \frac{9}{8}}={\aleph }^{*}<2$, $\varsigma =3$, and for $0\le \sigma <1$, we obtain $\sigma =0$.

$$\begin{array}{ccc}\hfill {𝚤}^{\sigma }\mid \eta (𝚤,𝚥)-\eta (𝚤,y)\mid & =& {𝚤}^{\sigma }\mid {\displaystyle \frac{exp(-𝚤+4)}{𝚤+8}}\left({𝚤}^2\right)+{\displaystyle \frac{1}{3+𝚤}}𝚥-{\displaystyle \frac{exp(-𝚤+4)}{𝚤+8}}\left({𝚤}^2\right)-{\displaystyle \frac{1}{3+𝚤}}y\mid \hfill \\ & =& {𝚤}^{\sigma }\mid {\displaystyle \frac{1}{3+𝚤}}𝚥-{\displaystyle \frac{1}{3+𝚤}}y\mid \hfill \\ & \le & {𝚤}^{\sigma }{\displaystyle \frac{1}{3+𝚤}}\mid 𝚥-y\mid \hfill \\ & \le & {\displaystyle \frac{1}{3}}\mid 𝚥-y\mid .\hfill \end{array}$$

So, Hypothesis 1 and Hypothesis 2 are satisfied with $\tilde{p}={\displaystyle \frac{1}{3}}.$ In addition,

$${\displaystyle \frac{1}{\varsigma }}\left(8\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }+2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right)={\displaystyle \frac{1}{3}}\left[8{\displaystyle \frac{1}{4}}+2{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{{\left(\frac{1}{4}\right)}^2}{2}}\right)\right]={\displaystyle \frac{1}{3}}\left[2+{\displaystyle \frac{2}{3}}{\displaystyle \frac{1}{32}}\right]={\displaystyle \frac{1}{3}}\left[2+{\displaystyle \frac{2}{96}}\right]={\displaystyle \frac{194}{288}}\eqsim 0.67<1.$$

According to Theorem 3, NAVOITVP (10) has a unique solution; by Theorem 4, NAVOITVP (10) is Ulam–Hyers stable.

Example 2. 

Consider the following NAVOITVP:

$$\left\{\begin{array}{c}{D}^{{\displaystyle \frac{𝚤}{5}}+1}𝚥(𝚤)+4{𝚥}^{″}(𝚤)={\displaystyle \frac{8(𝚤+1)}{3\sqrt{2}\pi }}+(exp\left(\sqrt{𝚤+1}\right))+{\displaystyle \frac{\pi }{3}}𝚥,𝚤\in [0,\tilde{\iota }]=[0,{\displaystyle \frac{1}{4}}],\hfill \\ 𝚥\left(0\right)=0,𝚥\left(\tilde{\iota }\right)=0,\hfill \end{array}\right.$$

(11)

We have that $\aleph (𝚤,𝚥(𝚤))={\displaystyle \frac{𝚤}{5}}+1$ is a continuous function with

$1<\aleph (𝚤,𝚥(𝚤))<{\displaystyle \frac{\frac{1}{4}}{5}}+1={\displaystyle \frac{1}{20}}+1={\displaystyle \frac{21}{20}}={\aleph }^{*}<2$, $\varsigma =4$, and for $0\le \sigma <1$, we obtain $\sigma =0$.

$$\begin{array}{ccc}\hfill {𝚤}^{\sigma }\mid \eta (𝚤,𝚥)-\eta (𝚤,y)\mid & =& {𝚤}^{\sigma }\mid {\displaystyle \frac{8(𝚤+1)}{3\sqrt{2}\pi }}+(exp\left(\sqrt{𝚤+1}\right))+{\displaystyle \frac{\pi }{3}}𝚥-{\displaystyle \frac{8(𝚤+1)}{3\sqrt{2}\pi }}-(exp\left(\sqrt{𝚤+1}\right))-{\displaystyle \frac{\pi }{3}}y\mid \hfill \\ & =& {𝚤}^{\sigma }\mid {\displaystyle \frac{\pi }{3}}𝚥-{\displaystyle \frac{\pi }{3}}y\mid \hfill \\ & \le & {𝚤}^{\sigma }{\displaystyle \frac{\pi }{3}}\mid 𝚥-y\mid \hfill \\ & \le & {\displaystyle \frac{\pi }{3}}\mid 𝚥-y\mid .\hfill \end{array}$$

So, Hypothesis 1 and Hypothesis 2 are satisfied with $\tilde{p}={\displaystyle \frac{\pi }{3}}.$ In addition,

$${\displaystyle \frac{1}{\varsigma }}\left(8\mathcal{M}{\tilde{\iota }}^{*}\tilde{\iota }+2\tilde{p}{\displaystyle \frac{{\tilde{\iota }}^{-\sigma +2}}{(-\sigma +1)(-\sigma +2)}}\right)={\displaystyle \frac{1}{4}}\left[8{\displaystyle \frac{1}{4}}+2{\displaystyle \frac{\pi }{3}}\left({\displaystyle \frac{{\left(\frac{1}{4}\right)}^2}{2}}\right)\right]={\displaystyle \frac{1}{4}}\left[2+{\displaystyle \frac{2\pi }{3}}{\displaystyle \frac{1}{32}}\right]={\displaystyle \frac{1}{4}}\left[2+{\displaystyle \frac{2\pi }{96}}\right]={\displaystyle \frac{192+2\pi }{384}}\eqsim 0.52<1.$$

According to Theorem 3, NAVOITVP (11) has a unique solution; so, by Theorem 4, NAVOITVP (11) is Ulam–Hyers stable.

6. Conclusions

In this study, we explored the existence and uniqueness of solutions for non-autonomous variable-order differential equations that incorporate both ordinary and fractional mixed derivatives, specifically addressing the boundary value problem known as NAVOITVP (1). Our analysis was based on the framework established with the Ulam–Hyers stability theorem (Theorem 4) and two pivotal fixed-point theorems (Theorems 2 and 3) that provided a solid theoretical foundation for our results.

Our findings go beyond demonstrating the existence of solutions, emphasizing the complex interaction between variable-order derivatives and stability in dynamic systems. By elucidating these connections, we enhance the understanding of how such equations can be effectively applied to model real-world phenomena.

To strengthen our theoretical contributions, we provide several numerical examples that illustrate the practical applications of our results. These examples not only validate our theoretical assertions but also highlight the potential of variable-order fractional calculus across various scientific disciplines.

In conclusion, this work marks a significant advancement in the developing field of variable-order fractional calculus, paving the way for future research and applications. We expect that our findings will encourage further exploration into the complexities of mixed derivative equations and their role in modeling intricate systems across multiple domains.