Existence and stability of nonlinear discrete fractional initial value problems with application to vibrating eardrum

1.   Introduction

Understanding the methodology of various physical systems by direct observation consumes time and does not provide clear insight into their working. That is where mathematical descriptions of real-life events play a major role in analyzing their behaviour and nature. Differential equations prove to be ideal tools for modelling phenomena by considering the laws of physics they follow ^[1,2]. Qualitative and quantitative studies of natural occurrences by engineers and researchers contribute a lot to developing technology and guidance in the treatment of physical activity.

The derivatives of arbitrary order were initially proposed in the $17^{th}$ century. It took over three centuries for the expansion in the mathematical theory of fractional calculus. Due to its accuracy in interpreting the system's performance and complexities, arbitrary order calculus proves to be an imperative tool in defining societal problems in various fields of engineering, biotechnology, chemistry, physics, etc., ^[3,4,5]. It was in the last few decades when the discrete version of the non-integer order calculus had gained popularity among the researchers ^[6]. The majority of the research papers on discrete-time calculus of fractional order were dedicated to the existence theory. However, rare results have dealt with the stabilization theory.

The theory of fractional calculus of discrete-time has originated from the ground-works of Atici and Eloe ^[7,8,9,10], Goodrich ^[11] and Miller and Ross ^[12]. Many researchers since then have considered further relevant results ^{[13,14,15,16,17]}. The stability analysis of nonlinear fractional difference equations is more delicated than the continuous fractional calculus's stability notion. The investigation of asymptotic stability by Chen et al. in ^[18,19] has drawn the attention of mathematicians and scientists of various fields. Hyers answers to Ulam's questions have initiated the stability analysis of functional equations ^[20,21]. There were significant contributions towards the study of Ulam stability of classical integer order equations and fractional order equations by Niazi et al. ^[22], Wang et al. ^[23,24,25], and Ibrahim ^[26]. Ulam stability discussion for integrodifferential boundary value problems and Hilfer–Hadamard type coupled system of fractional equations are made in ^[27,28]. The study of Ulam stability in ^{[29,30,31,32]} enriched the qualitative theory of discrete fractional equations.

Oriented by the above work and motivated by the desire to investigate further applications of discrete fractional equations, this paper considers the following nonlinear discrete fractional initial value problem:

where $\Delta^{\alpha}_{*}$, $\alpha \in \{\rho, \vartheta\}$, denotes the $\alpha$-th order Caputo-like forward difference operator and $\mathfrak{h}:\mathfrak{D} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$. Here $\mathfrak{D} = [\kappa + 2, \kappa + \mathcal{T}] \cap \mathbb{N}_{\kappa + 2}$, where $\mathcal{T} \in \mathbb{N}$ and $\mathbb{N}_{\kappa} = \left\{\kappa, \kappa + 1, \ldots \right\}, \kappa \in \mathbb{R}$. We establish sufficient conditions on existence, uniqueness and Hyers-Ulam stability of solutions of Eq (1.1). We claim that our results in this paper are new and are even different from the ones reported in the continuous cases ^[33,34,35].

The rest of the paper adheres to the following plan: Section 2 presents supporting definitions and lemmas which are essential for subsequent sections. The main results including existence, uniqueness and stability results are conferred in Sections 3–5, respectively. In Section 6, an exciting application on the damped eardrum equation along with numerical simulation is considered and will be analyzed. We end the paper with a conclusion.

2.   Basic requisites

This section provides some fundamental mathematical results used throughout this work.

Definition 2.1. ^[9] ($\vartheta-th$ Fractional Sum) The $\vartheta-th$ fractional sum of $\mathfrak{p}$ is

where $\vartheta > 0$, $\mathfrak{p}$ is defined for $\ell \equiv \kappa$ mod(1) and $\varsigma^{(\vartheta)} = \dfrac{\Gamma(\varsigma+1)}{\Gamma(\varsigma-\vartheta+1)}.$

Definition 2.2. ^[9] Let $q > 0$ and $j-1 < q < j$, where $j \in \mathbb{N}_1$, $j = \left\lceil q \right\rceil$. Set $\vartheta = j- q$. The $q-th$ fractional difference is

where $\Delta^{j}$ is the $j-th$ order forward difference operator.

Lemma 2.3. ^[6] For non-integer $q > 0$, $j = \left\lceil q \right\rceil, \vartheta = j- q$, we have

where $\mathfrak{h}$ is defined on $\mathbb{N}_{\kappa}$ with $\kappa \in \mathbb{Z}^{+}$. If $1 < q < 2$ and $\kappa = 0$, then we have

where $\mathfrak{h}$ is defined on $\mathbb{N}_1$.

Lemma 2.4. $\mathfrak{u} : \mathfrak{D} \rightarrow \mathbb{R}$ is a solution of Eq (1.1) if and only if $\mathfrak{u}$ solves the fractional Taylor's difference formula given by

Proof. Let $\mathfrak{u}$ be a solution of Eq (1.1). Then, from Eq (2.3), we have

Therefore, Eq (2.4) holds.

Conversely, let $\mathfrak{u}$ be a solution of Eq (2.4). The comparison of Eqs (2.3) and (2.4) gives

which takes the form

for $\varsigma \in \mathfrak{D}$. Letting $\varsigma = 1, 2, \cdots$, we have

Evidently, $\mathfrak{u}$ is solution of Eq (1.1).

Lemma 2.5. The following identity holds

Proof. Let $r, w \in \mathbb{R}$. For $r > -1, w > -1$ and $r > w,$ we have

This completes the proof.

3.   Existence of solution

First, we state the Krasnoselskii's fixed point theorem.

Theorem 3.1. ^[36] Let $B$ be a nonempty closed convex subset of a Banach space $(X, \|\cdot\|)$. Suppose that $\Lambda_1$ and $\Lambda_2$ map $B$ into $X$ such that

(I) for any $x$, $y \in B$, $\Lambda_1 x + \Lambda_2 y \in B$;

(II) $\Lambda_1$ is a contraction;

(III) $\Lambda_2$ is continuous and $\Lambda_2(B)$ is contained in a compact set.

Then, there exists $z \in B$ such that $z = \Lambda_1 z + \Lambda_2 z$.

Now, we establish sufficient conditions on the existence of solutions for the initial value problem (1.1) using Krasnoselskii's fixed point theorem.

Define the operators $\mathfrak{T}_{1}$ and $\mathfrak{T}_{2}$ by

Let $\mathbb{S} = \left\{\mathbb{C}\left[\mathfrak{D}, \mathbb{R}\right]\right\}$ be the space of all functions $\mathfrak{u}$ with the norm defined by

Consider the set

Then, $\mathcal{C}$ is a closed, convex subset of the Banach space $(\mathbb{S}, \|\cdot\|)$. Clearly, $\mathfrak{T}_{1}$, $\mathfrak{T}_{2} : \mathcal{C} \rightarrow \mathcal{S}$.

Now we consider the following assumptions:

$(H_1)$ $\mathfrak{h} : \mathfrak{D} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ is continuous.

$(H_2)$ There exists $\mathcal{M} > 0$ such that for all $\mathfrak{u} \in \mathcal{C}$ and $\varsigma \in \mathfrak{D}$ the following relation holds:

$(H_3)$ There exists $\mathcal{L} > 0$ such that for all $\mathfrak{u}$, $\psi$, $z$, $v \in \mathcal{C}$ and $\varsigma \in \mathfrak{D}$, the following relation holds:

Theorem 3.2. Assume $(H_1)$–$(H_3)$ hold and

If we choose

where

and

then the initial value problem (1.1) has a solution in $\mathcal{C}$.

Proof. The proof of the result is divided into three steps:

Step 1: For any $\mathfrak{u}$, $\mathfrak{v} \in \mathcal{C}$, $\mathfrak{T}_{1}\mathfrak{u} + \mathfrak{T}_{2}\mathfrak{v} \in \mathcal{C}$.

For each $\varsigma \in \mathfrak{D}$, we have

This implies that

Further, we have

implying that

Thus, we have

and hence $\mathfrak{T}_{1}\mathfrak{u} + \mathfrak{T}_{2}\mathfrak{v} \in \mathcal{C}$.

Step 2: $\mathfrak{T}_{1}$ is a contraction.

Let $\mathfrak{u}$, $\psi \in \mathcal{C}$ and for each $\varsigma \in \mathfrak{D}$, we have

Moreover,

Since

we have

implying that

It is clear that $\mathfrak{T}_{1}$ is a contraction.

Step 3: $\mathfrak{T}_{2}$ is continuous and $\mathfrak{T}_{2}(\mathcal{C})$ is contained in a compact set.

For each $\mathfrak{u} \in \mathcal{C}$ and $\varsigma \in \mathfrak{D}$, $\mathfrak{T}_{2}\mathfrak{u}(\varsigma) = \mathcal{A} + \varsigma \mathcal{B}$ is a linear function of $\varsigma$ implying that $\mathfrak{T}_{2}$ is continuous and $\mathfrak{T}_{2}(\mathcal{C})$ is contained in a compact set. Hence, by Theorem 3.1, the initial value problem (1.1) has at least one solution in $\mathcal{C}$.

4.   Uniqueness of solution

The following suppositions are needed:

$(H_1):$ $\mathfrak{h}:\mathfrak{D} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ is continuous

$(H_4):$ There exists $\mathcal{N} > 0$ such that for all $\mathfrak{u}$, $\psi$, $z$, $v \in \mathbb{R}$ and $\varsigma \in \mathfrak{D}$, the following relation hold:

Theorem 4.1. Assume that the conditions $(H_1)$ and $(H_4)$ hold. Then, Eq (1.1) has a unique solution if

Proof. Define the operator

We show that the operator $\mathfrak{T}:\mathbb{S} \rightarrow \mathbb{S}$ has a unique solution. Let $\mathfrak{u}, \psi \in \mathbb{S}$. Then, for each $\varsigma \in \mathfrak{D}$, we have

Moreover,

Since

we see that,

Therefore,

It is clear that $\mathfrak{T}$ is a contraction. Therefore, Banach fixed point theorem guarantees that the unique fixed point of $\mathfrak{T}$ is the unique solution of Eq (1.1).

5.   Hyers-Ulam stability

In this section, we study the stability results of Eq (1.1).

Definition 5.1. ^[30] The discrete time fractional order initial value problem (1.1) is Hyers-Ulam Stable, if a constant $\mathbb{U} > 0$ exists such that for every $\epsilon > 0$, $\psi(\varsigma) \in \mathbb{R}$ satisfy

with $\psi(0) = \mathcal{A}, \Delta(\psi(0)) = \mathcal{B}$ then the solution $\mathfrak{u}(\varsigma)$ of Eq (1.1) exists such that

Remark 5.2. $\psi(\varsigma) \in \mathbb{R}$ solves Eq (5.1) if and only if there exists a function $\mathfrak{g}: \mathfrak{D} \times \mathbb{R} \rightarrow \mathbb{R}$ such that

A1: $\left|\mathfrak{g}(\varsigma+\rho, \psi(\varsigma+\rho))\right| \leq \epsilon \qquad \varsigma \in \mathfrak{D}.$

A2: $\Delta^{\rho}_{*} [\psi(\varsigma)]+\mathfrak{h}(\varsigma+\rho, \psi(\varsigma+\rho), \Delta^{\vartheta}_{*}[\psi(\varsigma+\rho -\vartheta)]) = \mathfrak{g}(\varsigma+\rho, \psi(\varsigma+\rho)).$

Lemma 5.3. If $\psi(\varsigma)$ solves Eq (5.1), then

for $\varsigma \in \mathfrak{D}.$

Proof. If $\psi$ solves Eq (5.1), using Remark (5.2) and Eq (2.3), then the solution of $(A2)$ satisfies

for $\varsigma \in \mathfrak{D}.$ Hence,

This completes the proof.

Remark 5.4. As a direct outcome of Lemma Eq (5.3), we get

Theorem 5.5. Assume that $(H_4)$ and Eq (4.2) hold. Let $\psi(\varsigma) \in \mathbb{R}$ solve Eq (5.1) for some $\epsilon > 0$ and let $\mathfrak{u}(\varsigma) \in \mathbb{R}$ be the solution of

for $\varsigma \in [0, \mathcal{T}]\cap\mathbb{N}_{2-\rho}$, $1 < \rho\leq 2$, $0 < \vartheta\leq 1$. Then Eq (1.1) is Hyers Ulam Stable.

Proof. In view of Lemma (2.4), the solution $\mathfrak{u}$ of Eq (5.4) satisfies

Therefore,

Furthermore, we have

Therefore,

where $\nu$ is defined in Eq (4.2) and

Thus, Eq (1.1) is Hyers Ulam stable with $\mathbb{U} = \dfrac{\mathbb{V}}{(1-\mu)}$. This completes the proof.

6.   A numerical example

The eardrum is a vibrating membrane inside the ear that converts the vibrations caused by waves hitting them into nerve signals that are sent to the brain. The frequency response produced by the vibrations produced when hit by sound waves is the hearing range. The hearing ranges gets lowered with age, and the normal range of hearing frequency ranges between 20–20,000 Hz. Most of the biological models like the vibrating eardrum follow the 2$^{nd}$ Newton's law ^[37]. In the eardrum, the tympanic membrane with one-dimensional vibration is considered a mechanical system, and the sound wave that enters with different pressure is its forcing factor. Transmission of waves of different frequencies can be modeled with nonlinearity in order to understand frequency responses. It is necessary to analyze the qualitative behavior of the vibrating eardrum. This section establishes the stability of a nonlinear vibrating eardrum equation with a driving force.

Example 6.1. Consider the discrete fractional damped eardrum equation

where $\varsigma \in [0, 8] \cap N_{0.1}$, and $\lambda$ is the damping coefficient. We shall now prove the stability of Eq (6.1). Let the parameters take the values $\lambda = \dfrac{0.1}{3}, a = 0.02, b = 0.03, c = 0.01$.

By checking, we find that

satisfies $(H_2)$ with $\max\limits_{\varsigma \in \mathfrak{D}} \left|\mathfrak{u}(\varsigma)\right| = 0.1$. Moreover, we get

where $\varsigma \in [2, 8]\cap N_{2}$ and $\mathcal{L} = 0.033$. Thus, $(H_3)$ holds and $\mathfrak{h}$ is Lipschitz continuous. Further, it is clear that $\mu = \max\left\{0.8169, 0.3169\right\} < 1.$ By the conclusion of Theorem (3.2), the initial value problem (6.1) has a solution.

Let $\epsilon = 0.82$ and $\mathfrak{u}(\varsigma) = \dfrac{\varsigma^{(2)}}{20}, \varsigma \in [2, 8]\cap N_{2}.$ We make sure that the inequality (5.1) holds.

Under this, we conclude that Theorem (5.5) confirms the Hyers-Ulam stability of Eq (6.1) with constant $\mathbb{U}$.

The values of $\mu$ given in Theorem (4.1) for different fractional orders are tabulated in Table 1 and are plotted in Figure 1.

7.   Conclusions

In this paper, we prove the existence, uniqueness, and stability of the nonlinear discrete fractional initial value problem using the technique of fixed point hypothesis. The stability results are presented in the sense of Hyers-Ulam for the proposed initial value problem. As an application, the vibrating eardrum model is considered for illustrating the efficiency of the stability results. The conditions for the stability for different order of the equations are numerically calculated and are presented in a table and a graph. We claim that the proposed results of this paper are new and are different from the results given in the literature such as the results reported in ^[33,34,35].

Results reported in this paper have verified that the descretization of some physical models provides an efficient and practical tool to study their qualitative behavior. We believe that descretizing and investigating some different models can be a promising topic in the future.

Acknowledgments

J. Alzabut and K. Abodayeh would like to thank Prince Sultan University for supporting this work.

Conflict of interest

The authors declare that they have no competing interests.