Global Existence of Solutions to a System of Integral Equations Related to an Epidemic Model

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 0 as t⟶∞. In [6], Brestovanská studied the integral equation


Introduction
Many phenomena related to infectious diseases can be modeled as an integral equation (see e.g., [1][2][3][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 0 as t ⟶ ∞. In [6], Brestovanská studied the integral equation for all t ≥ 0. Namely, sufficient criteria for the global exis-tence and uniqueness of global solutions to (2) were derived. Moreover, under certain conditions, the convergence of solutions to (2) to 0 as t ⟶ ∞ 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 f i , A i , g i , B i ∈ Cð½0,∞ÞÞ and F i , G i ∈ Cð½0,∞Þ × ℝ × ℝÞ. 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.

Preliminaries
Let n be a positive natural number and define the partial order ≺ n in ℝ n by for all y, z ∈ ℝ n . We denote by 0 ℝ n the zero vector in ℝ n , i.e., Let S be a nonempty set and d : S × S ⟶ S be a given mapping. We say that d is a vector-valued metric on S (see, e.g., [8]), if for all x, y, z ∈ S, (i) 0 ℝ n ≺ n dðx, yÞ (ii) dðx, yÞ = 0 ℝ n ⇔ x = y (iii) dðx, yÞ = dðy, xÞ (iv) dðx, zÞ≺ n dðx, yÞ + dðy, zÞ In this case, we say that ðS, dÞ 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 M n ðℝ + Þ be set of square matrices of size n with nonnegative coefficients. Given K ∈ M n ðℝ + Þ, we denote by ρðKÞ its spectral radius.
Lemma 1 (Perov's fixed point theorem, see [9]). Let ðS, dÞ be a complete generalized metric space and H : S ⟶ S be a given mapping. Suppose that there exists K ∈ M n ðℝ + Þ with ρðKÞ < 1 such that for all x, y ∈ S. Then, the mapping H admits a unique fixed point in S.
Our main result is given by the following theorem.
Proof. Let T be an arbitrary positive number and I T = ½0, T.
We introduce the mapping H : where for all i = 1, 2.
We end the paper with the following example.