Fractional Difference and Differential Operators and their Applications in Nonlinear SystemsView this Special Issue
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.
In recent decades, the integro-differential equations have developed rapidly because the models described by various of integro-differential equations have appeared in a number of fields such as fluid dynamics, biology, economics, and control theory, for details and examples, we can refer to references [1–7] and cited therein. Meanwhile, the qualitative theory of integral differential equations creates an branch of nonlinear analysis. Boundary value problems for various first-order integral differential equations have been studied by several researchers, and there are some results on the existence of solutions and extremal solutions, the controllability problem controllability of integral boundary value conditions, and antiperiodic boundary value conditions, such as ordinary differential equations [8–12], difference equations [13, 14], fractional differential equations [15–20], impulsive differential equations [9, 14, 21, 22], integro-differential equations, and impulsive functional differential equations [18, 23–26].
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 ). 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]. The 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 [30–32]. In terms of applications, it is important to pay attention to the high-order convergence of sequences of approximate solutions. The high-order convergence results of various differential equations can be found in [33–39].
In this paper, we consider the following first-order integro-differential equations with antiperiodic boundary value conditions (APBVP):where .
The 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. Then, 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 . Finally, we give an example to illustrate our main results.
2. Comparison Theorems
In this section, we begin with some comparison principles that will be useful in later discussions.
Lemma 1. Assume that there exist constants , , and , such that
If there exists a function , such thatthen on .
Proof. Suppose the conclusion is not true, and we consider the following two cases, where and , respectively.
Case 2. When , there are two cases: for or there exist , , such that and for .
Case 4. If there exist and , such that and , we have , where , . Then, equation (4) holds. Integrating inequality (4) from to , we havewhich is also a contradiction. The proof of Lemma 1 is completed.
Next, consider the linear APBVP:
Corollary 1. Assume that and , then APBVP (8) has at most one solution.
Proof. Let be any solution of APBVP (8), , and , thenIf , then it follows from (9) that . By Lemma 1, we have , that is a contradiction. On the contrary, if , we have . By the proof of Lemma 1, we have , that is also a contradiction. Therefore, we have . Furthermore, by Lemma 1, we have , that is, for . The proof of Corollary 1 is completed.
Similar to the proof of Lemma 1, we have the following lemma.
Lemma 2. Assume that there exist integrable functions , , such that
If there exist functions , , such thatwhere
Then, and on .
Proof. We just prove that the case of . Suppose that the conclusion is not true, we can consider the following two cases, where and , respectively.
Case 6. When , there are two cases: for or there exist and , such that and for .
Case 7. When , , if , by the proof of Lemma 1, we have , that implies , which is a contradiction.
If , we have . Then, there are two cases: for and there exist and , such that and , respectively.
Case 8. When for all , we have ; hence, is decreasing. By and equation (11), imply ; then, is decreasing and , which is a contradiction.
Case 9. For another case, we have , where , . Equation (11) implies thatIntegrating inequality (13) from to , we haveThus,which is also a contradiction.
The proof of Case 4 is analogous to the proof of Lemma 1, and we omit its details here. This completes the proof of Lemma 2.
3. Linear APBVP
In this section, we consider the linear APBVP:
We can get the result of the existence and unique solution of equation (16).
Theorem 1. Assume that , , and . Then, APBVP (16) possesses a unique solution.
Proof. For any , denoting . LetWe define an operator as follows:where .
It is easy to see that is a closed, bounded, and convex set. Furthermore, for any , we havewhich implies that , that is, and is uniformly bounded. Furthermore, for any , , we haveSince and are bounded, thus is uniformly continuous. According to Ascoli–Arzela’s theorem, there exists the subsequences converging uniformly on to the continuous functions and , then, we can see that is compact. Therefore, there exists a solution of APBVP (16) by Schauder’s fixed point theorem. The uniqueness of solutions of APBVP (16) follows from Corollary 1. The proof is completed.
4. Nonlinear APBVP
In this section, we give the existence and uniqueness of the solutions of APBVP (1).
Definition 1. The functions are said to be a pair of coupled lower and upper solutions for APBVP (1) if the following inequalitieshold, where , and .
Theorem 2. Assume that the following conditions hold.
are a pair of coupled lower and upper solutions of APBVP (1) such that on
There exist constants such that , , andwhile for .
Then, APBVP (1) has a unique solution .
Proof. We construct iterative sequences as follows, on , and for , and are the solutions ofThe existence and uniqueness of the solution can be obtained by standard arguments for IVP (24) and (25).
We next prove that .
Let , by the condition of , and we have , , andwhere and .
By Lemma 2, we have on .
Let , by the condition of , and we haveBy using similar arguments of Lemma 1, we have . Therefore, it is easy to see that these sequences satisfyThen, we have two monotone sequences which are bounded, and there exist and , which satisfy , and . Moreover, the convergence is uniform on .
Set , then we obtainBy Lemma 1, we have for . Hence, for , and we can conclude , in which is the solution of APBVP (1). The proof of Theorem 2 is completed.
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 with the usual maximum norm . For any , we call that a given sequence converges to with order of convergence , if converges to in and there exist and such that for all .
Theorem 3. Assume that the conditions of hold.
and exist and are continuous for , andwhere and are constants withwhere .
Then, there exist monotone sequences of approximate solutions converging to the unique solution of (1) with rate of convergence of order .
Proof. Let the functionwhere , . Consider the following linear equation:Setting , by , we haveSimilarly, setting , we obtainThen, and are lower and upper solutions of equation (33), respectively. Furthermore, for and , we haveUsing Theorem 2, we know that problem (33) has a solution in .
Let be a solution of the mentioned problem, with , and we suppose , where is the solution ofwhere and are lower and upper solutions, respectively, for the following problem:Similarly, we know that satisfies the conditions of Theorem 2, then problem (33) has a solution in . Let be a solution of the mentioned problem, with . Hence, the constructed sequence is nondecreasing and bounded. In the same way, we can construct the sequence which is nonincreasing and bounded. Therefore, we obtain the two monotone sequences converge uniformly.
Let , and we can get by using Theorem 2, in which is the solution of (1).
Now, we show that the convergence of to is of order . LetFirstly, we note thatwhere . In sequence,In view of condition , by the continuity of and in , we havewhere , andSimilarly, we haveThe boundary conditions areLet , and we obtainUsing Gronwall’s inequality for (46), we haveLet , and we havewhich impliesthat is,where .
Since , we get the desired convergence. The proof is completed.
6. An Example
In this section, we will provide an example which demonstrates the application of Theorem 3.
Example 1. Consider the following APBVP:It is easy to check that and are lower and upper solutions of (51), respectively, which satisfies condition of Theorem 3. And we can show thatSetting , , , , , , , , and , satisfy conditions and of Theorem 3. Then, convergence of the iterative sequences of approximate solutions for APBVP (51) are of order .
In this paper, we discussed the problem of rapid convergence for the first-order integro-differential equations with antiperiodic boundary value conditions. By using the particular definitions of the coupled lower and upper related solutions, which are new and some new comparison principles, we obtained the existence and uniqueness of solution of the problems. Meanwhile, by using the method of quasilinearization, we obtained the monotone sequences of approximate solutions, converging to the unique solution of such problems with the rate of convergence of order . Finally, we give an example to illustrate our main results.
No data were used to support the findings of this study.
Conflicts of Interest
The authors declare that they have no competing interest.
All authors read and approved the final manuscript.
This paper was supported by the National Natural Science Foundation of China (11771115) and Scientific Research start-up project of Shaoguan University (99000608).
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