Abstract
A system of integral equations related to an epidemic model is investigated. Namely, we derive sufficient conditions for the existence and uniqueness of global solutions to the considered system. The proof is based on Perov’s fixed point theorem and some integral inequalities.
1. Introduction
Many phenomena related to infectious diseases can be modeled as an integral equation (see e.g., [1–4] and the references therein). In [3], Gripenberg investigated the large time behavior of solutions to the integral equation which arises in the study of the spread of an infectious disease that does not induce permanent immunity. Namely, sufficient conditions were provided so that (1) admits nonnegative, continuous, and bounded solution. Using the comparison method and some integral estimates, Pachpatte [5] established the convergence of solutions to (1) to as . In [6], Brestovanská studied the integral equation for all . Namely, sufficient criteria for the global existence and uniqueness of global solutions to (2) were derived. Moreover, under certain conditions, the convergence of solutions to (2) to as was proved. In [7], using weakly Picard technique operators in a gauge space, Olaru investigated the qualitative behavior of solutions to the integral equation
In this paper, we consider the system of integral equations where and . Namely, we are concerned with the global existence of solutions to the considered system. Using Perov’s fixed point theorem, sufficient conditions are derived for which the system (4) admits one and only one continuous global solution.
The rest of the paper is organized as follows. In Section 2, we recall some notions on fixed point theory including Perov’s fixed point theorem. In Section 3, we state and prove our main result.
2. Preliminaries
Let be a positive natural number and define the partial order in by for all . We denote by the zero vector in , i.e.,
Let be a nonempty set and be a given mapping. We say that is a vector-valued metric on (see, e.g., [8]), if for all , (i)(ii)(iii)(iv)
In this case, we say that is a generalized metric space. In such spaces, the notions of convergent sequence, Cauchy sequence, and completeness are similar to those for usual metric spaces.
Let be set of square matrices of size with nonnegative coefficients. Given , we denote by its spectral radius.
Lemma 1 (Perov’s fixed point theorem, see [9]). Let be a complete generalized metric space and be a given mapping. Suppose that there exists with such that for all . Then, the mapping admits a unique fixed point in .
3. Global Existence
The system (4) is investigated under the following assumptions: (i) and , (ii), , are bounded functions(iii), , for some (iv)For all , there exist positive constants and such thatfor all and (v)For all , there exist positive constants , and such thatfor all and (vi)For all , there exist positive constants and such thatfor all and (vii)For all , there exist positive constants , and such thatfor all and (viii)There exist positive constants and satisfyingwhere
Remark 2. Notice that from (i) and (ii), one has . Moreover, by (iii), one has .
Our main result is given by the following theorem.
Theorem 3. Under assumptions (i)–(viii), system (4) admits one and only one solution satisfying and , for all .
Proof. Let be an arbitrary positive number and . For , let
We introduce the mapping defined by
where
for all .
Let . For all and , using (i), (ii), (iii), and (v), and taking in consideration Remark 2, one obtains
Therefore, using (12), it holds that
which yields
Similarly, for all and , using (i), (ii), (iii), and (vii), and taking in consideration Remark 2, one obtains
Hence, using (13), it holds that
which yields
Therefore, it follows from (24) and (27) that the mapping maps the set into itself, i.e.,
Next, let us introduce the metric
defined by
where will be specified later. Moreover, we introduce the vector-valued metric
defined by
It can be easily seen that is a complete generalized metric space. On the other hand, for all , and , using (22), one has
Moreover, using (iii), (iv), (20) and Hölder’s inequality, for all , one obtains
where
Notice that by (iii), one has . Hence, it holds that
Therefore, by (33), one obtains
which yields
where
Similarly, using (iii), (vi), (21), and (25), one obtains
where
Therefore, it follows from (19), (39), and (42) that
where is the square matrix of size defined by
On the other hand, one has
Therefore, taking
one obtains
Then, by Lemma 1, one deduces that the mapping defined by (19) admits a unique fixed point in , which is the unique solution to (4) in . On the other hand, since the real number is arbitrary, it holds that (4) admits a unique continuous global solution satisfying and , for all . The proof is completed.
We end the paper with the following example.
Example 4. Consider the system of integral equations System (50) is a special case of System (4), where Let us check the validity of assumptions (i)–(viii). It can be easily seen that Moreover, one has Therefore, assumptions (i)–(iii) of Theorem 3 are satisfied. For all and , one has which shows that assumption (iv) of Theorem 3 is satisfied with Similarly, one has which shows that assumption (vi) of Theorem 3 is satisfied with For all and , one has which shows that assumption (v) of Theorem 3 is satisfied with Similarly, one has which shows that assumption (VII) is satisfied with From the above estimates, one deduces that the system of inequalities (12) and (13) is equivalent to Taking , (76) reduces to On the other hand, one observes easily that Therefore, any satisfying is a solution to (77). In particular, for , one has
Therefore, is a solution to (77), which shows that assumption (viii) of Theorem 3 is satisfied with .
Finally, by Theorem 3, one deduces that system (50) admits one and only one solution satisfying
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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 IFKSURG-237.