A nonlinear delay integral equation related to infectious diseases

1.   Introduction

We are concerned with the study of the integral equation

where $m$ is a positive integer and $\mu_i > 0$, $i = 1, 2, \cdots, m$, are constants. Namely, we study the well-posedness of (1.1), the upper and lower solutions and the data dependence of solutions w.r.t. small perturbations of the functions $\iota_j$, $j = 1, 2, \cdots, m$.

In the particular case $m = 1$, (1.1) reduces to

The above equation has been used as a model of the propagation of some infectious diseases with a rate of contact that depends on seasons, see e.g., ^[1,2]. Due to the importance of (1.2) in Biosciences, several mathematical studies of this equation have been done. Namely, different numerical methods for solving integral equations of the form (1.2) have been developed (see e.g., ^[3,4,5,6] and the references therein). Moreover, various results concerning the qualitative behavior of solutions have been obtained. For instance, in ^[1], Eq (1.2) has been studied, where $\iota_1$ is continuous and periodic in $t$, and satisfies $\iota_1(t, 0) = 0$. Namely, a threshold theorem has been established in the following sense:

(i) If $\mu_1 > 0$ is small enough, then every nonnegative solution to (1.2) tends to $0$ as $s\to +\infty$;

(ii) If $\mu_1$ is sufficiently large, then (1.2) admits a positive periodic solution with the same period as $\iota_1$.

In ^[7], sufficient conditions for the existence of a positive periodic solution to (1.2) have been obtained using the theory of fixed point index. In ^[8], the existence of positive almost periodic solutions to (1.2) has been studied when $\mu_1 = \mu_1(t)$. In ^[9], the same question has been studied by means of Hilbert's projective metric. Using the theory of Picard operators developed by Rus and his collaborators (see ^{[10,11,12,13,14,15]}), Dobriţoiu et al. ^[16] provided a detailed study of (1.2) concerning the well-posedness, lower/upper solutions and the data dependence.

The goal of this work is to extend the study made in ^[16] to the integral equation (1.1). Namely, in Section 2, the existence/uniqueness of solutions is established making use of Prešić fixed point theorem ^[17]. We also provide and iterative process for approximating the solution. Next, some estimates involving lower/upper solutions to (1.1) are obtained in Section 3. In Section 4, we study the data dependence of solutions w.r.t. small perturbations of the functions $\iota_j$, $j = 1, 2, \cdots, m$.

As we mentioned above, the proof of our well-posedness result makes use of Prešić fixed point theorem. We recall below this result.

Lemma 1.1 (see ^[17]). Let $(V, \delta)$ be a metric space. Let $\Gamma: V^m\to V$, where $m$ is a positive integer. Assume that

● $(V, \delta)$ is complete;

● There exists a finite sequence $\{\zeta_i\}_{i = 1}^m\subset [0, \infty)$ with $0 < \sum_{i = 1}^m \zeta_i < 1$ such that

for every $\{v_j\}_{j = 0}^m\subset V$.

Then, the equation

has a unique solution $v^*\in V$. Moreover, for any $\{v_j\}_{j = 0}^{m-1}\subset V$, the sequence

converges to $v^*$.

2.   Well-posedness

Assume that the conditions below hold:

(i) $\iota_j\in C(\mathbb{R}\times \mathbb{I}, \mathbb{J})$, $j = 1, 2, \cdots, m$, $m\geq 2$, where $\mathbb{I}$ and $\mathbb{J}$ are two closed and bounded intervals of $\mathbb{R}$;

(ii) for all $j = 1, 2, \cdots, m$, there exists $L_{\iota_j} > 0$ such that

for all $t\in \mathbb{R}$ and $w, z\in \mathbb{I}$;

(iii) $0 < \sum\limits_{j = 1}^m L_{\iota_j} \prod\limits_{k = 1, k\neq j}^m\|\iota_k\| < \left(\prod\limits_{i = 1}^m\mu_i\right)^{-1}$, where $\|\iota_k\| = \sup_{(t, z)\in \mathbb{R}\times \mathbb{I}}|\iota_k(t, z)|;$

(iv) there exists a closed subset $X$ of $C(\mathbb{R}, \mathbb{I})$ such that $\Gamma(X^m)\subset X$, where

for all $v_1, v_2, \cdots, v_m\in X$ and $C(\mathbb{R}, \mathbb{I})$ is equipped with the norm

We have the following result.

Theorem 2.1. Assume that $\rm (i)–(iv)$ hold. Then,

$(I)$ (1.1) admis a unique solution $\upsilon^*\in X$;

$(II)$ for all $v_0, v_1, \cdots, v_{m-1}\in X$, the sequence $\{v_j\}\subset X$ defined by

for all $j\geq m-1$, converges uniformly to $\upsilon^*$, that is,

Proof. From (iv), the mapping

is well-defined. Furthermore, for all $v_0, v_1, v_2, \cdots, v_m\in X$ and $s\in \mathbb{R}$, we have

which yields

Hence, it holds that

where

On the other hand, by (iii), we have

Hence, by Lemma 1.1, there exists a unique $\upsilon^*\in X$ such that

that is, $\upsilon^*$ is the unique solution to (1.1) in $X$, which proves (I). Finally, (II) follows from the convergence result in Lemma 1.1. □

We now take $\mathbb{I} = [a, b]$ and $\mathbb{J} = [c, d]$, where $a < b$ and $0 < c < d$.

Corollary 2.2. Assume that (i)–(iii) hold, and

Then the statements (I) and (II) of Theorem 2.1 hold true, where $X = C(\mathbb{R}, \mathbb{I})$.

Proof. We have just to show that condition (iv) is satisfied. Then, from Theorem 2.1, (I) and (II) follow. Let $v_1, v_2, \cdots, v_m\in C(\mathbb{R}, \mathbb{I})$. For all $j = 1, 2, \cdots, m$, one has

which implies that

Then, for all $s\in \mathbb{R}$, it holds that

Taking into consideration (2.4), we obtain

which shows that $\Gamma(v_1, v_2, \cdots, v_m)\in C(\mathbb{R}, \mathbb{I})$. Consequently, we have $\Gamma(X^m)\subset X$, where $X = C(\mathbb{R}, \mathbb{I})$. □

We provide below an example to illustrate Theorem 2.1.

Example 2.3. Consider the integral equation

where $\mu_1, \mu_2 > 0$ are constants and

Let $X = C(\mathbb{R}, [0, 1])$. We claim that

(I) (2.5) admis a unique solution $\upsilon^*\in X$;

(II) for all $v_0, v_1\in X$, the sequence $\{v_j\}\subset X$ defined by

for all $j\geq 1$, converges uniformly to $\upsilon^*$, that is,

Indeed, for all $k = 1, 2$, let

be the functions defined by

Then (2.5) is a special case of (1.1) with $m = 2$.

Let $\mathbb{I} = [0, 1]$ and $\mathbb{J} = [0, \ln 2]$. For all $t\in \mathbb{R}$ and $w, z\in \mathbb{I}$, by the mean value theorem, we have

where $L_{\iota_1} = 1$. Similarly,

where $L_{\iota_2} = 4$. Furthermore, by (2.6), we have

On the other hand, for all $v_1, v_2\in X$, we have

which shows that $\Gamma(X\times X)\subset X$. Hence, all the assumptions (i)–(iv) of Theorem 2.1 are satisfied. Then the claims (I) and (II) follow from Theorem 2.1.

3.   Lower/upper solutions

We say that $\upsilon$ is a lower solution to (1.1), if

If $\upsilon$ verifies

then $\upsilon$ is called an upper solution to (1.1).

Let us consider (1.1) under conditions (i)–(iv). Then, by Theorem 2.1, (1.1) admits a unique solution $\upsilon^*\in X$.

We have the following result.

Theorem 3.1. Assume that $\rm (i)–(iv)$ hold. Suppose also that $\mathbb{J}\subset [0, +\infty)$ and for all $t\in \mathbb{R}$ and $j = 1, 2, \cdots, m$, we have

If $\upsilon\in X$ is a lower solution to (1.1), then

Proof. Let $\Gamma: X^m\to X$ be the mapping defined by (2.1). By (3.1) and since $\mathbb{J}\subset [0, +\infty)$, if $w\in X$ and $(v_1, \cdots, v_{j-1}, v_j, v_{j+1}, \cdots v_m)\in X^m$ are such that $v_j(s)\leq w(s)$ for all $s\in \mathbb{R}$, then

Let $\upsilon\in X$ be a lower solution to (1.1). Then, for all $s\in \mathbb{R}$,

Let $v_0 = v_1 = \cdots = v_{m-1} = \upsilon$ and

We claim that

By (3.1) and (3.3), we have

which shows that (3.4) holds true for $j = m-1$. On the other hand, by (3.1) and (3.5), we have

Then, (3.5) and (3.6) yield

which shows that (3.4) holds true for $j = m$. Repeating the same argument, by the induction principle, we obtain (3.4). Now, taking the limit as $j\to +\infty$ in (3.4) and using the convergence result provided by part (II) of Theorem 2.1, we obtain (3.2). □

Proceeding as in the proof of Theorem 3.1, we obtain the following result.

Theorem 3.2. Assume that $\rm (i)–(iv)$ hold. Suppose also that $\mathbb{J}\subset [0, +\infty)$ and for all $t\in \mathbb{R}$ and $j = 1, 2, \cdots, m$, (3.1) holds. If $\upsilon\in X$ is an upper solution to (1.1), then

We now take $\mathbb{I} = [a, b]$ and $\mathbb{J} = [c, d]$, where $a < b$ and $0 < c < d$. For all $j = 1, 2, \cdots, m$, let $\iota_{j1}, \iota_{j2}, \iota_{j3}\in C(\mathbb{R}\times \mathbb{I}, \mathbb{J})$. Assume that conditions below hold:

(c$_1$) for all all $j = 1, 2, \cdots, m$ and $i = 1, 2, 3$, there exists $L_{\iota_{ji}} > 0$ such that

for all $t\in \mathbb{R}$ and $w, z\in I$;

(c$_2$) for all $i = 1, 2, 3$, we have

(c$_3$) $a\leq c^m \prod_{i = 1}^m\mu_i, \, \, \, b\geq d^m \prod_{i = 1}^m\mu_i$;

(c$_4$) for all $j = 1, 2, \cdots, m$, we have

(c$_5$)for all $j = 1, 2, \cdots, m$ and $t\in \mathbb{R}$, we have

Observe that under the above conditions, thanks to Theorem 2.1 and Corollary 2.2, for all $i = 1, 2, 3$, the problem

admits a unique solution $\upsilon_i^*\in C(\mathbb{R}, \mathbb{I})$.

The following result holds.

Corollary 3.3. Under conditions (c$_1$)–(c$_5$), it holds that

Proof. We have

for all $s\in \mathbb{R}$. Due to (c$_4$), we obtain

which shows that $\upsilon_1^*$ is a lower solution to (3.7) with $i = 2$. Hence, by Theorem 3.1, we get $\upsilon_1^*(s)\leq \upsilon_2^*(s), \, \, s\in \mathbb{R}$. Similarly, we have

for all $s\in \mathbb{R}$. Due to (c$_4$), we obtain

which shows that $\upsilon_3^*$ is an upper solution to (3.7) with $i = 2$. Then, by Theorem 3.2, we obtain $\upsilon_3^*(s)\geq \upsilon_2^*(s), \, \, s\in \mathbb{R}$. □

4.   Data dependence of solutions

We now consider the perturbed problem

where $m\geq 2$ is an integer, $\mu_i > 0$, $i = 1, 2, \cdots, m$, ${\widetilde{\iota}}_j\in C(\mathbb{R}\times \mathbb{I}, \mathbb{J})$, $j = 1, 2, \cdots, m$, $\mathbb{I}$ and $\mathbb{J}$ are two closed and bounded intervals of $\mathbb{R}$.

We have the following dependence data result.

Theorem 4.1. Assume that $\rm (i)–(iv)$ hold, and let $\upsilon^*\in X$ be the unique solution to (1.1). Suppose that for all $j = 1, 2, \cdots, m$, there exits $\eta_j > 0$ such that

for all $(t, w)\in \mathbb{R}\times \mathbb{I}$. If ${\widetilde{\upsilon}}^*\in X$ is a solution to (4.1), then

where

Proof. Let

For all $s\in \mathbb{R}$, we have

On the other hand, by (2.3), we have

where

Furthermore, we have

which implies by (4.2) that

that is,

Therefore, from (4.4)–(4.6), it follows that

which yields (4.3). □

5.   Conclusions

The nonlinear integral equation (1.1) is investigated. We first studied the well-posedness of the problem in $C(\mathbb{R}, \mathbb{I})$, where $\mathbb{I}$ is a closed and bounded subset of $\mathbb{R}$ (see Theorem 2.1). Namely, by means of Prešić fixed point theorem (see Lemma 1.1), we proved that under conditions (i)–(iv), (1.1) admits a unique solution that can be approximated by the iterative sequence (2.2). We next established some estimates involving upper and lower solutions to (1.1) (see Theorems 3.1 and 3.2). Finally, we studied the dependence of the solution to (1.1) with respect to perturbations of the functions $\iota_j$, $j = 1, 2, \cdots, m$ (see Theorem 4.1).

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number RI-44-0714.

Conflict of interest

The authors declare there is no conflict of interest.