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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

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 [

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 [

In this section, we present the following definition and lemma which will help to prove our main result.

Let

The operator

Let the points

We consider the following periodic boundary value problem (PBVP) of hybrid system with causal operators:

The function

Analogously, we can define an upper solution of the PBVP (

Let the functions

Similar to the proof of Theorem 3.2.1 in [

Let

Consider the Banach space

Let the following conditions hold:

The functions

There exist continuous functions

Firstly, we note that the condition

Now, consider the following boundary value problem:

It follows from (

Thus, using Lemma

Now, suppose that

In this case, we have

We conclude, using again Lemma

Thus, we know that

Since

Now, we prove that the convergence is of order

On the other hand, by (

Let

Since

Similarly, to construct the sequence

Now, let

Therefore,

On the other hand, by (

We conclude that, for every

The proof is complete.

The authors declare that they have no competing interests.

All authors completed the paper together. All authors read and approved the final paper.

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).