Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag

1.   Introduction

Fractional difference calculus is a tool used to explain many phenomena in physics, control problems, modeling, chaotic dynamical systems, and various fields of engineering and applied mathematics. In this direction, different kinds of methods and techniques, including numerical and analytical methods, have been utilized by researchers to discuss given fractional discrete and continuous mathematical models and boundary value problems (BVPs) ^[1,2,3,4]. For some recent developments on the existence, uniqueness, and stability of solutions for fractional differential equations, see, for example, ^{[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]} and the references therein.

Discrete fractional calculus and difference equations open a new study context for mathematicians. For this reason, they have received increasing attention in recent years. Some real-world processes and phenomena are analyzed with the aid of discrete fractional operators, since such operators provide an accurate tool to describe memory. A large number of research articles dealing with difference equations and discrete fractional boundary value problems (FBVPs) can be found in ^{[24,25,26,27,28,29,30,31,32]}.

In 2020, Selvam et al. ^[33] proved the existence of a solution to a discrete fractional difference equation formulated as

for $\xi\in[0, T]_{\mathbb{N}_{0}} = [0, 1, 2, \ldots, T]$, $T\in\mathbb{N}$, $\eta\in[\varrho-1, T+\varrho-1]_{\mathbb{N}_{\varrho-1}}$, $M_1$ and $M_2$ constants, $\Phi:[\varrho-2, \varrho+T]_{\mathbb{N}_{\varrho-2}}\times\mathbb{R}\longrightarrow\mathbb{R}$ continuous, and where $^{c}\Delta_{\xi}^{\varrho}$ denotes the $\varrho$th-Caputo difference. Here, motivated by the discrete model (1.1), we shall consider two generalized discrete problems.

Our first goal consists to study existence and uniqueness of solutions to the following discrete fractional equation that involves Caputo discrete derivatives:

for $0 < \beta\leq1$, $\xi\in[0, T]_{\mathbb{N}_{0}} = [0, 1, 2, \ldots, T]$, $T\in\mathbb{N}$, $\eta\in[\varrho-1, T+\varrho-1]_{\mathbb{N}_{\varrho-1}}$, $\lambda$, $A_{1}$ and $A_{2}$ constants, and where $\Phi:[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}\times\mathbb{R}\longrightarrow\mathbb{R}$ is continuous.

The second goal is to study the stability of solutions to the discrete Riemann-Liouville fractional problem

for $0 < \beta\leq1$, $\xi\in[0, T]_{\mathbb{N}_{0}} = [0, 1, 2, \ldots, T]$ and $\eta\in[\varrho-1, T+\varrho-1]_{\mathbb{N}_{\varrho-1}}$, where $^{RL}\Delta_{\xi}^{\varrho}$ is the Riemann-Liouville difference operator.

The organization of the paper is as follows. In Section 2, we collect some fundamental definitions available from the literature. In Section 3, we prove the existence and uniqueness results for the discrete FBVP (1.2). Hyers-Ulam and Hyers-Ulam-Rassias stability of the solution for the FBVP (1.3) is established in Section 4. In Section 5, two examples are given to illustrate the obtained results. We end with Section 6 of conclusions.

2.   Preliminaries

We begin by recalling some necessary definitions and essential lemmas that will be used throughout the paper.

Definition 2.1. (See ^[27]) Let $\varrho > 0$. The $\varrho$-order fractional sum of $\Phi$ is defined by

where $\xi\in\mathbb{N}_{a+\varrho}: = \{a+\varrho, a+ \varrho+1, \ldots\}$ and $\xi^{(\varrho)}: = \frac{\Gamma(\xi+1)}{\Gamma(\xi+1-\varrho)}$.

Definition 2.2. (See ^[27]) Let $\varrho > 0$ and $\Phi$ be defined on $\mathbb{N}_{a}$. The $\varrho$-order Caputo fractional difference of $\Phi$ is defined by

while the Riemann-Liouville fractional difference of $\Phi$ is defined by

where $\xi\in\mathbb{N}_{a+n-\varrho}$ and $n-1 < \varrho\leq n$.

Lemma 2.1. (See ^[24,27]) For $\varrho > 0$,

where $C_i \in \mathbb{R}$, $i = 1, 2, \ldots, N-1$, $\Phi$ is defined on $\mathbb{N}_a$, and $0\leq N-1 < \varrho\leq N$.

Lemma 2.2. (See (^[32]) Let $0\leq N-1 < \varrho\leq N$ and $\Phi$ be defined on $\mathbb{N}_a$. Then,

for $B_1, \dots, B_N \in \mathbb{R}$.

Lemma 2.3. (See ^[31]) Let $\varrho$ and $\xi$ be any arbitrary real numbers. Then,

Lemma 2.4. (See ^[31]) For $\zeta\in \mathbb{R}\setminus (\mathbb{Z}^- \setminus \{ 0\})$, we have

3.   Existence and uniqueness of solutions

In this section, we prove the existence and uniqueness of solution for the Caputo three-point discrete fractional problem (1.2). To accomplish this, we denote by $C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$ the collection of all continuous functions $\chi$ with the norm

Lemma 3.1. Let $2 < \varrho\leq 3$ and $\Phi:[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}\longrightarrow \mathbb{R}$. A function $\chi(\xi)$ $(\xi\in[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}})$ that satisfies the discrete FBVP

is given by

with

and

Proof. Let $\chi(\xi)$ be a solution to (3.1). Applying Lemma 2.1 and Definition 2.1, we find that

for $\xi\in[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}$, where $C_{0}, C_{1}, C_{2}\in\mathbb{R}$. By using the difference of order 1 for (3.5), we have

and

Now, from conditions $\Delta\chi(\varrho-3) = A_{1}$ and $\Delta^{2}\chi(\varrho-3) = A_{2}$, we obtain that

Therefore,

for $\xi\in[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}$. By using formula (3.5), one has

The other condition of (3.1) gives

We have

where $K_{1}$ and $K_{2}$ are defined by (3.3) and (3.4), and one obtains (3.2) by substituting the value of $C_{0}$ into (3.6).

Now, let us consider the operator $H:C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R}) \rightarrow C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$ defined by

Theorem 3.1. Assume that:

$(\mathcal{H}1)$ Function $\Phi$ satisfies $\vert\Phi(\xi, \chi_{1})-\Phi(\xi, \chi_{2})\vert\leq K\vert\chi_{1}-\chi_{2}\vert$, where $K > 0$, $\forall \xi\in\mathbb{N}_{\varrho-3, \varrho+T}$ and $\chi_{1}, \chi_{2}\in C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$. The discrete FBVP (3.1) has a unique solution on $C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$ provided

with

Proof. Let $\chi_{1}, \chi_{2}\in C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$. Then, for each $\xi\in\mathbb{N}_{\varrho-3, \varrho+T}$, we have

It follows that

From (3.8), we conclude that $H$ is a contraction. Then, by the Banach contraction principle, the discrete problem (3.1) has a unique solution on $C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$.

Theorem 3.2. Suppose that $\Phi:[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}} \times\mathbb{R}\longrightarrow\mathbb{R}$ is a continuous function and

The discrete problem (3.1) has a solution provided that

Proof. Let $G = \lbrace\chi\in C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R}); \Vert\chi\Vert\leq2|K_{2}|\rbrace$. For $\chi(\xi)\in G$, we get

From (3.10), we have $\Vert H\chi\Vert\leq2|K_{2}|$, which implies that $H:G\to G$. By Brouwer's fixed point theorem, we know that the discrete problem (3.1) has a solution.

4.   Stability analysis

In this section, we study the Hyers-Ulam and Hyers-Ulam-Rassias stability for the solutions of the discrete Riemann-Liouville (RL) FBVP

for $\xi\in[0, 1, 2, \ldots, T] = [0, T]_{\mathbb{N}_{0}}$ and $\eta\in[\varrho-1, T+\varrho-1]_{\mathbb{N}_{\varrho-1}}$, where $^{RL}\Delta_{\xi}^{\varrho}$ is the RL fractional difference operator. We begin by proving the following lemma.

Lemma 4.1. Suppose that $2 < \varrho\leq 3$ and $\Phi:[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}\longrightarrow\mathbb{R}$. A function $\chi$ satisfies the discrete problem

if, and only if, $\chi(\xi)$, $\xi\in[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}$, has the form

where

Proof. Let $\chi(\xi)$ be a solution to (4.2). Applying Lemma 2.2 and Definition 2.1, we obtain that the general solution of (4.2) is given by

$\xi\in[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}$, where $B_{1}, B_{2}, B_{3}\in\mathbb{R}$. The first order difference of (4.7) is

while

From the conditions $\Delta\chi(\varrho-3) = A_{1}$ and $\Delta^{2}\chi(\varrho-3) = A_{2}$, we obtain that

and

Now, by using the difference of order $\beta$ for (4.7), it follows that

Based on condition (4.1), we have

Then,

where $h_{\varrho}$, $d_{\varrho}$ and $f_{\Phi}$ are defined by (4.4)-(4.6), respectively. Substituting the values of the constants $B_{1}$, $B_{2}$ and $B_{3}$ into (4.7), we obtain (4.3) and our proof is complete.

From Lemma 4.1, the solution of the discrete RL problem (4.1) is given by the formula

where $d_\rho$ and $h_\rho$ are defined by (4.4) and (4.5), respectively, and

Lemma 4.2. If $\chi$ is a solution of (4.3), then

where

and

Proof. Take $\chi$ as a solution of (4.2). For $\xi\in[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}$, then

The proof is complete.

Definition 4.1. We say that the discrete RL problem (4.1) is Hyers-Ulam stable if for each function $\nu\in C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$ of

and $\epsilon > 0$, there exists $\chi\in C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$ solution of (4.1) and $\delta > 0$ such that

Definition 4.2. We say that the discrete RL problem (4.1) is Hyers-Ulam–Rassias stable if for each function $\nu\in C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$ of

and $\epsilon > 0$, there exists $\chi\in C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$ solution of (4.1) and $\delta_{2} > 0$ such that

Remark 4.1. A function $\chi(\xi)\in C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$ is a solution of (4.12) if, and only if, there exists $\mu:[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}\longrightarrow\mathbb{R}$ satisfying:

$(\mathcal{H}2)$ $\big|\mu(\xi+\varrho-1)\big|\leq\epsilon, \quad \xi\in [0, T]_{\mathbb{N}_{0}}$;

$(\mathcal{H}3)$ $^{RL}\Delta_{\xi}^{\varrho}\nu(\xi) = \Phi(\xi+\varrho-1, \nu(\xi+\varrho-1))+\mu(\xi+\varrho-1), \quad \xi\in [0, T]_{\mathbb{N}_{0}}$.

Remark 4.2. A function $\chi(\xi)\in C(\mathbb{N}_{\varrho-3, \varrho+T}, \mathbb{R})$ is a solution of (4.14) if, and only if, there exists $\mu:[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}\longrightarrow\mathbb{R}$ satisfying

$(\mathcal{H}4)$ $\big|\mu(\xi+\varrho-1)\big|\leq\epsilon\theta(\xi+\varrho-1), \quad \xi\in [0, T]_{\mathbb{N}_{0}}$,

$(\mathcal{H}5)$ $^{RL}\Delta_{\xi}^{\varrho}\nu(\xi) = \Phi(\xi+\varrho-1, \nu(\xi+\varrho-1))+\mu(\xi+\varrho-1), \quad \xi\in [0, T]_{\mathbb{N}_{0}}$.

Lemma 4.3. If $\nu$ satisfies (4.12), then

Proof. Using our hypothesis, and based on Remark 4.1, the solution to $(\mathcal{H}3)$ satisfies

Hence,

and the desired inequality is derived.

Theorem 4.1. If condition $(\mathcal{H}1)$ holds and (4.13) is satisfied, then the discrete RL problem (4.1) is Hyers-Ulam stable under the condition

where $M_{1} = \max(1, |K(\xi)|)$.

Proof. Let $\xi\in[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}}$. From Lemma 8, we have

It follows that

Therefore,

Moreover, $\Vert\nu-\chi\Vert\leq\epsilon\delta$, where

Thus, the discrete RL problem (4.1) is Hyers-Ulam stable.

Lemma 4.4. If $v$ solves (4.14) under the condition:

$(\mathcal{H}6)$ The function $\theta:[\varrho-3, \varrho+T]_{\mathbb{N}_{\varrho-3}} \longrightarrow\mathbb{R}$ is increasing and there exists a constant $\gamma > 0$ such that

then

Proof. Let $\nu$ satisfy (4.14). From Remark 4.2, the solution to $(\mathcal{H}5)$ satisfies

Hence,

and the desired inequality is derived.

Remark 4.3. About the restrictiveness of hypotheses $(\mathcal{H}1)$–$(\mathcal{H}6)$, and the bounds imposed on the family of discrete systems that satisfy them, one should note that such hypotheses are the usual conditions for proving existence, uniqueness or stability of solutions. In fact, the conditions $(\mathcal{H}2)$–$(\mathcal{H}6)$ are considered as the fundamental conditions in the definition of Hyers-Ulam-Rassias stability. Condition $(\mathcal{H}1)$ is a standard Lipschitz condition while other constants are computed based on the given fractional system. Therefore, these conditions are natural and, in real systems, with specified numerical data, their expressions are reduced to numerical bounds.

Theorem 4.2. If the inequality (4.16) and the hypotheses $(\mathcal{H}1)$ and $(\mathcal{H}6)$ are satisfied, then the discrete RL problem (4.1) is Hyers-Ulam-Rassias stable.

Proof. From Lemmas 4.4 and 2.3, we obtain that

where

Thus, the discrete RL problem (4.1) is Hyers-Ulam-Rassias stable.

5.   Illustrative examples

In this section, we consider two examples to illustrate the obtained results.

Example 5.1. Let

where $^{*}\Delta_{\xi}^{\varrho}$ denotes the operator $^{c}\Delta_{\xi}^{\varrho}$ or $^{RL}\Delta_{\xi}^{\varrho}$. Set $\beta = 0.5$, $T = 4$, $\lambda = 0.7$, $A_1 = 1$, $A_2 = 0$, and

$\bullet$ If $^{*}\Delta_{\xi}^{\varrho}\chi(\xi) = {^{c}\Delta}_{\xi}^{\frac{5}{2}}\chi(\xi)$ and $\eta = \frac{5}{2}$, then we obtain that

Hence, the inequality (3.8) takes the form

such that

From Theorem 3.1, the discrete problem (5.1) has a unique solution.

$\bullet$ In the case $^{*}\Delta_{\xi}^{\varrho}\chi(\xi) = {^{RL}\Delta}_{\xi}^{3}\chi(\xi)$ and $\eta = 3$, we obtain

Also,

If $K = 0.0014 < 0.0075$ and

holds, then, by Theorem 4.1, the discrete RL problem (5.1) is Hyers-Ulam stable.

Example 5.2. Let

After some calculations, we find that

We define the following Banach space:

Note that

It is clear that $\vert\Phi(t, \chi)\vert \leq 0.0586\leq 0.3262$ whenever $\chi\in [-32.5927, 32.5927]$. Therefore, by Theorem 3.2, we find out that the discrete FBVP (5.5) has a solution.

6.   Conclusions

We proved existence and uniqueness of solution to discrete fractional boundary value problems (FBVPs) involving fractional difference operators via the Brouwer fixed point theorem and the Banach contraction principle. Different versions of stability criteria were obtained for a discrete FBVP involving Riemann-Liouville difference operators. The results were illustrated by suitable examples. The approach of this paper is new and can be a beginning method for discussing different real-world models in the context of discrete behavior structures. In particular, our results can contribute for the development of discrete fractional boundary value problems describing discrete dynamics of some physical applications. In future works, we plan to extend our approach to other types of discrete differential inclusions or fully-hybrid discrete fractional differential equations.

Acknowledgments

The third and fourth authors would like to thank Azarbaijan Shahid Madani University. The fifth author was supported by the Portuguese Foundation for Science and Technology (FCT) and CIDMA through project UIDB/04106/2020. This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation (Grant number B05F650018).

Conflict of interest

The authors declare no conflict of interest.