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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 952601, 20 pages
Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability
1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech Republic
2Department of Mathematics and Statistics, Faculty of Science, Masaryk University, 611 37 Brno, Czech Republic
Received 20 February 2012; Accepted 4 April 2012
Academic Editor: Miroslava Růžičková
Copyright © 2012 Zdeněk Šmarda and Josef Rebenda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The asymptotic behaviour of a real two-dimensional differential system with unbounded nonconstant delays satisfying is studied under the assumption of instability. Here, , and are supposed to be matrix functions and a vector function. The conditions for the instable properties of solutions and the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a Lyapunov-Krasovskii functional and the suitable Ważewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.
Consider the real two-dimensional system where are real functions, , are real square matrices, and is a real vector function, , . It is supposed that the functions , are locally absolutely continuous on , are locally Lebesgue integrable on , and the function satisfies Carathéodory conditions on .
There are a lot of papers dealing with the stability and asymptotic behaviour of -dimensional real vector equations with delay. Among others we should mention the recent results [1–13]. Since the plane has special topological properties different from those of -dimensional space, where or , it is interesting to study the asymptotic behaviour of two-dimensional systems by using tools that are typical and effective for two-dimensional systems. The convenient tool is the combination of the method of complexification and the method of Lyapunov-Krasovskii functional. For the case of instability, it is useful to add to this combination the version of Ważewski topological principle formulated by Rybakowski in the papers [14, 15]. Using these techniques, we obtain new and easy applicable results on stability, asymptotic stability, instability, or boundedness of solutions of the system (1.1).
The main idea of the investigation, the combination of the method of complexification and the method of Lyapunov-Krasovskii functional, was introduced for ordinary differential equations in the paper by Ráb and Kalas  in 1990. The principle was transferred to differential equations with delay by Kalas and Baráková  in 2002. The results in the case of instability were obtained for ODEs by Kalas and Osička  in 1994 and for delayed differential equations by Kalas  in 2005.
We extend such type of results to differential equations with a finite number of nonconstant delays. We introduce the transformation of the considered real system to one equation with complex-valued coefficients. We present sufficient conditions for the instability of a solution and for the existence of a bounded solution. The applicability of the results is demonstrated with several examples.
At the end of this introduction we append a brief overview of notation used in the paper and the transformation of the real system to one equation with complex-valued coefficients. is the set of all real numbers, the set of all positive real numbers, the set of all nonnegative real numbers, the set of all negative real numbers, the set of all nonpositive real numbers, the set of all complex numbers, the class of all continuous functions , the class of all locally absolutely continuous functions , the class of all locally Lebesgue integrable functions , the class of all functions satisfying Carathéodory conditions on , the real part of , the imaginary part of , and the complex conjugate of .
Introducing complex variables , ,..., , we can rewrite the system (1.1) into an equivalent equation with complex-valued coefficients where for , , , , .
We consider (1.2) in the case when and study the behavior of solutions of (1.2) under this assumption. This situation corresponds to the case when the equilibrium 0 of the autonomous homogeneous system where is supposed to be regular constant matrix, is a centre or a focus. See  for more details.
Regarding (2.1) and since the delay functions satisfy , there are numbers , , and such that
Notice that the above-defined function need not be positive.
Since and , the inequality is valid for . It can be easily verified that .
For the rest of this section we will denote
The instability and boundedness of solutions are studied subject to suitable subsets of the following assumptions. (i)The numbers , , and are such that (2.3) holds.(ii)There exist functions such that for , , where is continuous on .(iin)There exist numbersand functionssuch that for.()is a function satisfying where is defined for by (iiin) is a function satisfying where is defined for by (ivn) is a real locally Lebesgue integrable function satisfying the inequalities , for almost all , where is defined by
Obviously, if , , , and are locally absolutely continuous on and , , the choice is admissible in (iii). Similarly, if , , , and are locally absolutely continuous on and , , the choice is admissible in (iiin).
Denote From assumption (i) it follows that therefore the function is locally Lebesgue integrable on , assuming that (i) holds true. If the relations , , and for almost all together with conditions (i) and (iin) are fulfilled, then we can choose for in (ivn).
Theorem 3.1. Let assumptions (i), (ii0), (iii0), and (iv0) be fulfilled for some . Suppose there exist and such that If is any solution of (1.2) satisfying where then for all , for which is defined.
In the proof we use the following Lemma.
Lemma 3.2. Let , , , , . Then, for , .
Proof of Theorem 3.1. Let be any solution of (1.2) satisfying (3.2). Consider the Lyapunov functional
For brevity we shall denote and we shall write the function of variable simply without indicating the variable , for example, instead of .
In view of (3.6), we have for almost all for which is defined and exists. Put exists,. Clearly for . The derivative exists for almost all .
Since is a solution of (1.2), we obtain for almost all . Taking into account we get
By the use of Lemma 3.2, we get
The last inequality together with (2.12), taken for , assumption (ii0), and the relation yields for almost all .
Consequently, for almost all . Inequality (3.15) together with relation (3.8) gives
Using (2.11) and (2.13) for , we obtain
Hence, in view of (iv0), for almost all .
Multiplying (3.18) by and integrating over , we get on any interval , where the solution exists and satisfies the inequality . Now, with respect to (3.6), (3.7), and , we have If (3.2) is fulfilled, there is such that . By virtue of (3.1), and (3.2), we can easily see that for all , for which is defined.
To obtain results on the existence of bounded solutions, we shall suppose that (1.2) satisfies the uniqueness property of solutions. Moreover, we suppose that the delays are bounded, that is, that the functions satisfy the condition where is a constant. Our assumptions imply the existence of numbers , , and such that
In view of this, we replace (2.3) in assumption (i) with (). All other assumptions we keep in validity.
In the proof of the following theorem we shall utilize Ważewski topological principle for retarded functional differential equations of Carathéodory type. Details of this theory can be found in the paper of Rybakowski .
Theorem 3.3. Let conditions (i), (ii), and (iii) be fulfilled, and let , be continuous functions such that the inequality holds a. e. on , where is defined by (2.14). Suppose that is a continuous function such that for and some constant . Then, there exist and a solution of (1.2) satisfying for .
Proof. Write (1.2) in the form
where is defined by
and is the element of defined by the relation , . Let . Put
It can be easily verified that is a polyfacial set generated by the functions , (see Rybakowski [15, page 134]). It holds that . As , we have for . It holds that Let and be such that and for all . If , then Therefore, provided that the derivatives , exist and that . Thus, Using (3.10), (3.13), and (ii), similarly to the proof of Theorem 3.1, we obtain and consequently, with respect to (iii), for almost all and for sufficiently close to . Replacing and by and , respectively, in the last expression, we get Therefore, in view of the continuity, holds for sufficiently close to and almost all sufficiently close to . Hence, is a regular polyfacial set with respect to ().
Choose , where is fixed. It can be easily verified that is a retract of , but is not a retract of . Let be such that and for . Define the mapping for by the relation The mapping is continuous, and it holds that Since we have for and . Clearly, inequality (3.39) holds also for and .
Using a topological principle for retarded functional differential equations (see Rybakowski [15, Theorem 2.1]), we see that there is a solution of (1.2) such that for all for which the solution exists. Obviously, exists for all and Hence
Theorem 3.4. Suppose that hypotheses (i), (ii), (iin), (iii), (iiin), and (ivn) are fulfilled for and , where , . Let , be continuous functions satisfying the inequality a. e. on , where is defined by (2.14). Assume that is a continuous function such that for and some constant . Suppose that for , where and . Then, there exists a solution of (1.2) such that
Proof. By the use of Theorem 3.3 we observe that there is a and a solution of (1.2) with property for . Suppose that (3.46) is not satisfied. Then, there is such that Choose such that It holds that for some . In view of (3.44), we can suppose that Therefore, taking into account (2.5), (3.47), (3.50), and (3.51) and the nonpositiveness of , we have Moreover, (3.45) implies that By Theorem 3.1, we obtain an estimation for all , being defined by Relation (3.47) together with (3.54) yields that is for . However, the last inequality contradicts (3.43) and Theorem 3.4 is proved.
From Theorem 3.1 we easily obtain several corollaries.
Corollary 3.5. Let the assumptions of Theorem 3.1 be fulfilled with . If then for any , , there is such that for all , for which is defined.
Proof. Without loss of generality we can assume that . Choose , such that . In view of (3.58), there is such that for . Hence, for . Estimation (3.4) together with (3.2) now yields for all , for which is defined.
Example 3.7. Consider (1.2) where , , , for , , .
Obviously, and . Suppose that and . Then, , . Further,