Abstract
We investigated the convergence of iterative sequences of approximate solutions to a class of periodic boundary value problem of hybrid system with causal operators and established two sequences of approximate solutions that converge to the solution of the problem with rate of order .
1. Introduction
Recently, the problem of qualitative theory of dynamic systems with causal operators has attracted much attention since such systems include several types of dynamic systems, such as ordinary differential equations, integrodifferential equations, differential equations with finite or infinite delay, Volterra integral equations, and neutral equations. Therefore, the study of the theory of causal systems becomes very important. This is because a single result involving causal operators covers interesting corresponding results from many categories of dynamic systems, thus avoiding duplication and monotony of repetition. For more details, we can refer to the monographs [1–9] and the references cited therein. Since it is difficult to find the solutions of differential equations with causal operators, we need to look for the approximate solutions. Quasilinearization combined with the technique of upper and lower solutions is an effective and fruitful technique for obtaining approximate solutions to a wide variety of nonlinear problems. The main advantages of the method are the practicality of finding successive approximations of the unknown solution as well as the quadratic convergence rate. Some recent results in the development of the method and its real-world applications can be found in [10–19].
Hybrid systems have also attracted much attention in recent years. Hybrid systems are dynamical systems that evolute continuously in time but have formatting changes, called modes, at a sequence of discrete times. Some recent works on hybrid systems are included in [20–26]. However, to our best knowledge, very few results have been achieved on hybrid system with causal operators; particularly methods for finding approximate solutions with rapid convergence are yet to be developed. Hence, the purpose of this paper is to develop the method of quasilinearization for the periodic boundary value problem of hybrid system with causal operators. We will prove that the problem has solutions which can be approximated via monotone sequences with rate of convergence of order .
2. Preliminaries
In this section, we present the following definition and lemma which will help to prove our main result.
Let , where , is an appropriate positive constant, and .
Definition 1 (see [2]). The operator is said to be a causal or nonanticipatory operator if the following property is satisfied: for each couple of elements , of such that for , one also has for with , being arbitrary.
Let the points be fixed such that , and , .
We consider the following periodic boundary value problem (PBVP) of hybrid system with causal operators: where is a continuous causal operator, the functions are increasing, and there exist constants such that, for any points and , the following equalities or inequalities are satisfied: and if , then ; that is, , in which , .
The function is called a lower solution of the PBVP (1) if the following inequalities are satisfied:
Analogously, we can define an upper solution of the PBVP (1) by introducing the inequalities in (3) in opposite directions.
Let the functions be such that . Consider the sets
Similar to the proof of Theorem 3.2.1 in [2], we have the following lemma.
Lemma 2. Let be lower and upper solutions of the PBVP (1) satisfying , . Suppose that the operator is bounded on . Then, there exists a solution of (1) in the closed set , such that ,
3. Main Result
Consider the Banach space with the usual norm . For a given sequence , we say that converges to with order of convergence , if converges to in and there exist and such that
Theorem 3. Let the following conditions hold:The functions , are lower and upper solutions to the PBVP (1) and for There exist continuous functions , , and constants and such that Then there exist two monotone sequences and with and , which converge uniformly to the unique solution of the PBVP (1), and the convergence is of order
Proof. Firstly, we note that the condition implies that the PBVP (1) has a unique solution between and . To construct the sequence , for given where , , define the following function: in which the function . Using , (7), and (8), we getNow, consider the following boundary value problem:It follows from (9) that That is, and are lower and upper solutions of (10), respectively.
Thus, using Lemma 2, we conclude that problem (10) has the unique solution and
Now, suppose that , where is the unique solution of In this case, we have We conclude, using again Lemma 2, that there exists a unique solution forThus, we know that is a nondecreasing sequence and is bounded in . According to the standard arguments (see [12]), the Ascoli-Arzela Theorem guarantees the existence of a subsequence which converges uniformly to a continuous function .
Since we have and is the unique solution of the PBVP (1) in .
Now, we prove that the convergence is of order . For this purpose, using (7), we haveOn the other hand, by (8) and (15), it is verified that, for , Let and ; then , for all and . Thus, we have The continuity of and in implies that there exist and such that Finally, as for all , , we get that where , , and Since converges uniformly to in , (21) implies that there exists and , such that , , for , and . Then, there exists a continuous function on such that or equivalently where is the Green function associated with the following linear boundary value problem: From [4], we have that is positive on , since the solution of problem (26) is given by where , . We can thus conclude that, for any and , where . Hence, for all , and
Similarly, to construct the sequence , define the following function:where the function , and , and are nonnegative constants given by (6) and (21), respectively. Similar to the discussion of above, we have Now, let ; for , we define by induction, as the unique solution of the following boundary value problem: We can obtain . Similar to the discussion of , is a nonincreasing sequence and is bounded in . Then converges uniformly in to the continuous function . Since we have Therefore, is the unique solution of the PBVP (1) in . Furthermore, we prove that the convergence is of order . For this purpose, using (7), we have On the other hand, by (30) and (32), it is verified that, for , if is odd, then while if is even, then Let , and . Then, we have that if is odd, then while if is even, then Furthermore for all and . We can write that if is odd, then while if is even, then where , , , and Since converges uniformly to in , (21) implies that there exist and , such that , , for and . Thus, there exists a continuous function on such that if is odd, if is even, Or equivalently, if is odd, then while if is even, then where is the same with the above.
We conclude that, for every and , if is odd, then for all and , while if is even, then and hence for all and
The proof is complete.
Conflict of Interests
The authors declare that they have no competing interests.
Authors’ Contribution
All authors completed the paper together. All authors read and approved the final paper.
Acknowledgments
The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).