A Nonlinear Integral Equation Related to Infectious Diseases

The integral equation (2) models the spread of certain infectious diseases with periodic contact rate that varies seasonally (see [1]). Several results related to certain mathematical aspects of (2) have been obtained by many authors (see, e.g., [1–9] and the references therein). In particular, in [3], using the Picard operator technique, the integral equation (2) was investigated regarding the existence and uniqueness of solutions and periodic solutions, lower and upper subsolutions, the data dependence, and the differentiability of solutions with respect to a parameter. In this paper, we are concerned with the integral equation (1). We first investigate the existence and uniqueness of solutions and provide numerical algorithms that converge to the unique solution. Next, we study the lower and upper subsolutions, as well as the data dependence of the solution. The next section is devoted to the main results of this paper. Namely, in Subsection 2.1, we fix some notations that will be used throughout this paper. In Subsection 2.2, we provide some lemmas that will be used in the proofs of our main results. In Subsection 2.3, the existence and uniqueness of solutions and periodic solutions are derived using the Banach contraction principle. Moreover, an iterative algorithm based on Picard iteration for approximating the unique solution is provided. In Subsection 2.4, a Prešic′-type iterative algorithm that converges to the unique solution is provided. Lower and upper subsolutions type results are obtained in Subsection 2.5. Finally, in Subsection 2.6, the data dependence of solutions is studied.


Introduction
We consider the nonlinear integral equation where n ≥ 2 is an integer and τ i > 0, i = 1, 2, ⋯, n. In the case n = 1 and f 1 ≡ 0, (1) reduces to The integral equation (2) models the spread of certain infectious diseases with periodic contact rate that varies seasonally (see [1]). Several results related to certain mathematical aspects of (2) have been obtained by many authors (see, e.g., [1][2][3][4][5][6][7][8][9] and the references therein). In particular, in [3], using the Picard operator technique, the integral equation (2) was investigated regarding the existence and uniqueness of solutions and periodic solutions, lower and upper subsolutions, the data dependence, and the differentiability of solutions with respect to a parameter.
In this paper, we are concerned with the integral equation (1). We first investigate the existence and uniqueness of solutions and provide numerical algorithms that converge to the unique solution. Next, we study the lower and upper subsolutions, as well as the data dependence of the solution.
The next section is devoted to the main results of this paper. Namely, in Subsection 2.1, we fix some notations that will be used throughout this paper. In Subsection 2.2, we provide some lemmas that will be used in the proofs of our main results. In Subsection 2.3, the existence and uniqueness of solutions and periodic solutions are derived using the Banach contraction principle. Moreover, an iterative algorithm based on Picard iteration for approximating the unique solution is provided. In Subsection 2.4, a Prešic ′ -type iterative algorithm that converges to the unique solution is provided. Lower and upper subsolutions type results are obtained in Subsection 2.5. Finally, in Subsection 2.6, the data dependence of solutions is studied.

Results
We first fix some notations.
The functional space X is equipped with the norm ∥·∥ X , where Notice that ðX, k·k X Þ is a Banach space.

Preliminaries.
The following lemma will be useful later. It can be easily proved by induction.
Lemma 1. Let fa n g and fb n g be two real sequences. Then, for all n ≥ 2, We recall the following result due to Prešic ′ [10].

Lemma 2.
Let ðX, dÞ be a complete metric space, k a positive integer and φ : X k ⟶ X a mapping satisfying the following condition: for all x 1 , ⋯, x k+1 ∈ X, where q 1 , q 2 , ⋯, q k are nonnegative constants such that q 1 + q 2 + ⋯+q k < 1. Then, (i) There exists a unique x * ∈ X such that (ii) For all x 1 , x 2 , ⋯, x k ∈ X, the sequence fx p g ⊂ X defined by For more details about the above result, we refer to [11][12][13][14][15].

Existence and Uniqueness
Result. Problem (1) is investigated under the following conditions: (C3) For all i = 1, 2, ⋯, n, there exists a constant L g i > 0 such that for all t ∈ ℝ, We have the following existence and uniqueness result.

Theorem 3.
Under conditions (C 1 )-(C 5 ), problem (1) admits one and only one solution x * ∈ X. Moreover, for all x 0 ∈ X, the sequence fx p g ⊂ X defined by converges uniformly to x * .
Proof. Let us define the operator T : where By (C 1 ), for all i = 1, 2, ⋯, n and t ∈ ℝ, one has which yields Then, using (C 5 ), one deduces that Similarly, by (C 1 ), one has which yields Hence, using (C 5 ), one obtains Journal of Function Spaces Therefore, it follows from (16) and (19) that Moreover, the set of solutions to the integral equation (1) coincides with the set of fixed points of the operator T. Next, by Lemma 1, for all x, y ∈ X and t ∈ ℝ, one has On the other hand, by (C 2 ) and (C 3 ), for all i = 1, 2, ⋯, n, one has Therefore, using (14), (21), and (22), one obtains which yields Finally, using (C 4 ), (20) and (24), the conclusion of the theorem follows from the Banach contraction principle. Now, we consider problem (1) under the additional condition: (C6) There exists ω > 0 such that for all i = 1, 2, ⋯, n, Theorem 4. Under conditions (C 1 )-(C 6 ), problem (1) admits one and only one ω-periodic solution x * ∈ X. Moreover, for any ω-periodic function x 0 ∈ X, the sequence fx p g defined by (11) converges uniformly to x * .
Proof. Let T : X ⟶ X be the operator defined by (12). Notice that from the proof of Theorem 3, we know that under conditions (C 1 )-(C 5 ), one has TX ⊂ X. Let V be the closed subset of X (with respect to the norm k·k X ) defined by For all x ∈ V and t ∈ ℝ, using (C 6 ), one obtains Hence, one has TV ⊂ V. On the other hand, since V ⊂ X, it follows from (24) that Then, the conclusion of the theorem follows from the Banach contraction principle.

Prešic′-Type Approximation of the Unique Solution.
Let us consider the integral equation (1) under conditions (C 1 )-(C 5 ). Notice that by Theorem 3, (1) admits one and only one solution x * ∈ X.

Theorem 5.
Under conditions (C 1 )-(C 5 ), for any x 1 , x 2 , ⋯, x n ∈ X, the sequence fx p g defined by converges uniformly to x * .
Proof. Consider the function φ : X n ⟶ X defined by 3 Journal of Function Spaces that is, where for all i = 1, 2, ⋯, n, the operator T i is defined by (13). Notice that from the considered assumptions, one has φðX n Þ ⊂ X, so φ is well-defined. On the other hand, using Lemma 1, for all x 1 , x 2 , ⋯, x n , x n+1 ∈ X and t ∈ ℝ, on has Next, using (14), it holds that On the other hand, under the considered assumptions, for all k = 1, 2, ⋯, n, one has Hence, one deduces that Finally, using (C 4 ) and Lemma 2, the desired result follows.

Lower and Upper Subsolutions.
We consider problem (1) under conditions (C 1 )-(C 5 ). We recall that by Theorem 3, problem (1) admits one and only one solution x * ∈ X. We suppose also that For all i = 1, 2, ⋯, n and t ∈ ℝ, the functions f i t, · ð Þ: I ⟶ J and g i t, · ð Þ: are nondecreasing.
Theorem 6. Suppose that conditions (C 1 )-(C 5 ) and (C 6 ′ ) are satisfied. If x ∈ Cðℝ, IÞ satisfies Proof. Let T : X ⟶ X be the operator defined by (12). Then, (39) is equivalent to We shall prove that T is a nondecreasing operator, that is, Let u, v ∈ X be such that By (C 6 ′ ), for all i = 1, 2, ⋯, n and t ∈ ℝ, one obtains which yields that is, This proves (40). Next, by (39), it holds that for all nonnegative integer p and t ∈ ℝ, where Hence, it holds that where fx p g is the sequence defined by (11) with x 0 = x.

Journal of Function Spaces
On the other hand, by Theorem 3, one has lim p→∞ x p t ð Þ = x * t ð Þ, t ∈ ℝ: ð48Þ Therefore, passing to the limit as p ⟶ ∞ in (47), (38) follows.