Convergence of Antiperiodic Boundary Value Problems for First-Order Integro-Differential Equations

In this paper, we investigate the convergence of approximate solutions for a class of first-order integro-differential equations with antiperiodic boundary value conditions. By introducing the definitions of the coupled lower and upper solutions which are different from the former ones and establishing some new comparison principles, the results of the existence and uniqueness of solutions of the problem are given. Finally, we obtain the uniform and rapid convergence of the iterative sequences of approximate solutions via the coupled lower and upper solutions and quasilinearization method. In addition, an example is given to illustrate the feasibility of the method.

However, we found that most of these known results concerned with the existence and uniformly convergence results of solutions and extremal solutions via the method of upper and lower solutions coupled with the monotone iterative technique (see [27]). It is well known that the method of quasilinearization (QSL) provides a powerful tool for obtaining convergence of approximate solutions of nonlinear problems [28,29]. e technique of upper and lower solutions coupled with the QSL have been applied successfully to obtain monotone sequences of approximate solutions converging uniformly and quadratically to the unique solution of integro-differential equations with antiperiodic boundary value conditions [30][31][32]. In terms of applications, it is important to pay attention to the high-order convergence of sequences of approximate solutions. e high-order convergence results of various differential equations can be found in [33][34][35][36][37][38][39].
In this paper, we consider the following first-order integro-differential equations with antiperiodic boundary value conditions (APBVP): e aim of this paper is to investigate the convergence of approximate solutions of the problem. We give the particular definitions of the coupled lower and upper related solutions which are new and establish some new comparison principles in order to discuss the existence and uniqueness of the solutions. en, by using the method of quasilinearization, we obtain the two monotone sequences of approximate solutions converging to the unique solution of the problem with rate of convergence of order k. Finally, we give an example to illustrate our main results.

Comparison Theorems
In this section, we begin with some comparison principles that will be useful in later discussions.
Case 2. When p(0) > 0, there are two cases: p(t) > 0 for t ∈ J or there exist t, t, such that p(t) ≤ 0 and p(t) > 0 for t, t ∈ J.

Case 4.
If there exist t and t, such that p(t) ≤ 0 and p(t) > 0, which is also a contradiction. e proof of Lemma 1 is completed.
Similar to the proof of Lemma 1, we have the following lemma.
If there exist functions where en, p 1 (t) ≤ 0 and p 2 (t) ≤ 0 on J.

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Proof. We just prove that the case of p 1 (t) ≤ 0. Suppose that the conclusion is not true, we can consider the following two cases, where p 1 (0) ≤ 0 and p 1 (0) > 0, respectively.
Case 9. For another case, we have Integrating inequality (13) from t * to T, we have us, which is also a contradiction. e proof of Case 4 is analogous to the proof of Lemma 1, and we omit its details here. is completes the proof of Lemma 2.

Linear APBVP
In this section, we consider the linear APBVP: We can get the result of the existence and unique solution of equation (16). Proof. For any x ∈ C(J, R), denoting ‖x‖ � max t∈J |x(t)|. Let ω 0 � max t∈J |σ(t)|, We define an operator S: E ⟶ C(J, R) as follows: where It is easy to see that E is a closed, bounded, and convex set. Furthermore, for any x ∈ E, we have which implies that ‖S(x)‖ ≤ ω 1 , that is, S(E) ⊂ E and S is uniformly bounded. Furthermore, for any t 1 , t 2 ∈ J, we have Discrete Dynamics in Nature and Society Since σ and x are bounded, thus S is uniformly continuous. According to Ascoli-Arzela's theorem, there exists the subsequences Sx n converging uniformly on J to the continuous functions Sx and Sx ∈ E, then, we can see that S is compact. erefore, there exists a solution of APBVP (16) by Schauder's fixed point theorem. e uniqueness of solutions of APBVP (16) follows from Corollary 1. e proof is completed.

Nonlinear APBVP
In this section, we give the existence and uniqueness of the solutions of APBVP (1).

Definition 1.
e functions v, w ∈ C ′ (J, R) are said to be a pair of coupled lower and upper solutions for APBVP (1) .

Theorem 2.
Assume that the following conditions hold.
while v ≤ u ≤ u ≤ w and Tv ≤ Tu ≤ Tu ≤ Tw for t ∈ J. en, APBVP (1) has a unique solution x ∈ [v, w].
Proof. We construct iterative sequences v n , w n ⊂ C ′ (J, R) as follows, v 1 � v and w 1 � w on J, and for n > 1, v n and w n are the solutions of 4 Discrete Dynamics in Nature and Society e existence and uniqueness of the solution can be obtained by standard arguments for IVP (24) and (25).
Let p 1 � v 1 − v 2 and p 2 � w 2 − w 1 , by the condition of (H 2 ), and we have p 1 (0) ≤ p 2 (T), p 2 (0) ≤ p 1 (T), and where and By Lemma 2, we have Let p � v 2 − w 2 , by the condition of (H 2 ), and we have By using similar arguments of Lemma 1, we have erefore, it is easy to see that these sequences satisfy v n ≤ v n+1 ≤ w n+1 ≤ w n , n ≥ 1.
en, we have two monotone sequences which are bounded, and there exist ρ and μ, which satisfy lim n⟶∞ v n � ρ, lim n⟶∞ w n � μ, and ρ ≤ μ. Moreover, the convergence is uniform on J.
Set p � μ − ρ, then we obtain By Lemma 1, we have p(t) ≤ 0 for t ∈ J. Hence, ρ ≡ μ for t ∈ J, and we can conclude ρ ≡ μ ≡ x, in which x is the solution of APBVP (1). e proof of eorem 2 is completed.

Quasilinearization
In this section, we apply the quasilinesrization method in order to obtain the result on convergence of the iterative sequences of approximate solutions for APBVP (1).
Consider the Banach space C(J, R) with the usual maximum norm ‖x‖ 1 � max t∈J |x(t)|. For any x ∈ C(J, R), we call that a given sequence x n converges to x with order of convergence k, if x n converges to x in C(J, R) and there exist n 0 ∈ N and k > 0 such that ‖x m+1 − x‖ 1 ≤ ‖x m − x‖ k 1 for all m ≥ n 0 .
where M i and N i are constants with where (t, u, Tu) ∈ Ω � (t, u, Tu): v ≤ u ≤ w { }. en, there exist monotone sequences v n , w n of approximate solutions converging to the unique solution of (1) with rate of convergence of order k.
Let v 1 be a solution of the mentioned problem, with v 1 ∈ [v, w], and we suppose v � v 0 ≤ v 1 ≤ · · · ≤ v n ≤ w, where v n ∈ [v n− 1 , w] is the solution of u ′ (t) � g t, u, Tu; v n− 1 , Tv n− 1 , t ∈ J, where v n− 1 and w are lower and upper solutions, respectively, for the following problem: