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T. S. Doan, S. Siegmund, "Finite-Time Attractivity for Diagonally Dominant Systems with Off-Diagonal Delays", Abstract and Applied Analysis, vol. 2012, Article ID 210156, 10 pages, 2012. https://doi.org/10.1155/2012/210156
Finite-Time Attractivity for Diagonally Dominant Systems with Off-Diagonal Delays
We introduce a notion of attractivity for delay equations which are defined on bounded time intervals. Our main result shows that linear delay equations are finite-time attractive, provided that the delay is only in the coupling terms between different components, and the system is diagonally dominant. We apply this result to a nonlinear Lotka-Volterra system and show that the delay is harmless and does not destroy finite-time attractivity.
Finite-time dynamical systems generated by nonautonomous differential equations which are defined only on a compact interval of time have recently become an active field of research, see, for example, [1–3] and the references therein.
A key ingredient of a qualitative theory is the notion of hyperbolicity of solutions. Finite-time versions of hyperbolicity are introduced and discussed, for example, in [4–8]. Finite-time attractivity is a special case of finite-time hyperbolicity, in case the unstable direction is missing. So far finite-time attractivity has been discussed only for ordinary differential equations, see [9, 10]. For a closely related notion, namely, finite-time stability, we refer to [11, 12] and the references therein for an overview. In this paper we go one step further to extend and investigate finite-time attractivity for delay equations.
For a nonnegative number , let denote the space of all continuous functions . For , the norm on is defined as follows: where for all . Consider a finite-time delay differential equation where is assumed to be continuous and Lipschitz in the second argument. For each , let denote the solution of (2.1) satisfying the initial condition for all . The evolution operator generated by (1.2) is defined as Motivated by recent results on finite-time hyperbolicity (see, e.g., [4, 6, 7]), we introduce in the following an analog notion of finite-time attractivity for delay equations.
Definition 1.1 (finite-time attractivity). The solution is called finite-time attractive on with respect to the norm if there exist positive constants and such that for all with the following estimate holds: for all in the neighborhood of .
Remark 1.2. In the case that is a linear function in the second argument, it is easy to see that the generated semigroup is also linear in the second argument. In particular, for linear systems the following statements are equivalent: (i)there exists a finite-time attractive solution for a ,(ii)for all , the solution is finite-time attractive, and(iii)there exists such that for all we have
In this paper, we prove the finite-time attractivity for linear off-diagonal delay systems in Section 2. Section 3 is devoted to show the finite-time attractivity of the equilibrium for a Lotka-Voltera system.
2. Finite-Time Attractivity for Linear Off-Diagonal Delay Systems
In this section, we consider the following finite-time nonautonomous linear differential equation with off-diagonal delays (see, e.g.,  and the reference therein): where is a given positive constant, , , are continuous functions and for with . Define Note that (2.1) is a special case of (1.2). More precisely, the right hand side of (2.1) equals , where is defined as follows: Let denote the evolution operator of (2.1). From (2.3), we see that the function is linear in the second argument. Therefore, the evolution operator is also linear in the second argument. Our aim in this section is to provide a sufficient condition for the finite-time attractivity for the zero solution of (2.1) and thus for all solutions of (2.1), see Remark 1.2.
Before presenting the main result, we recall the notion of row diagonal dominance. We refer the reader to [14, Definition 7.10] for a discussion of this notion. System (2.1) is called row diagonally dominant if there exists a positive constant such that
Theorem 2.1 (finite-time attractivity for delay equations). Consider system (2.1) on a finite-time interval . Suppose that system (2.1) is row diagonally dominant with a positive constant and for all and . Define
and let be a positive number satisfying that
Then for every , the zero solution of (2.1) is finite-time attractive with respect to the norm with exponent , that is,
Proof. We divide the proof into two steps.
Step 1. We show that for the inequality holds. Suppose the opposite, that is, assume that there exists such that the set is not empty. Define . By continuity of the map , we get that Now, we will show that Indeed, we consider the following two cases: (i) and (ii) .
Case (i). If , then, according to (2.10) and (2.11), we obtain that which proves (2.12) in this case.
Case (ii). If , then, according to (2.10) and the definition of the norm , we obtain that Hence, (2.12) is proved. To conclude the proof of this step, we estimate the norm for all in a neighborhood of in order to show a contradiction to the assumption that the set is not empty. To this end, we define the following set: The continuity of the functions for implies that there exists a neighborhood for some such that By virtue of (2.1), the derivative of the function is estimated as follows: which together with (2.12) and the definition of implies that for all Thus, from the row diagonal dominance (2.4) and bound (2.5), we derive that Using (2.6), we obtain that which yields that there exists a neighborhood for an such that for Thus, for all with , we have which together with (2.10) implies that Consequently, , and this is a contradiction to the definition of and . Thus, (2.8) is proved.
Step 2. Using (2.8) from Step 1, we show (2.7) by considering two cases: (i) and (ii) .
Case (i). If , then by virtue of (2.8) we have which proves (2.7) in this case.
Case (ii). If , then we have
Hence, using (2.8), we obtain that
Thus, (2.7) is proved, and the proof is complete.
3. Finite-Time Attractivity of Lotka-Voltera Systems
Consider a Lotka-Voltera system of the following form: where and for all . Suppose that is row diagonal dominant with for all ; that is, there exists such that We assume in the following that there exists a positive vector such that To shorten the notation, the function for all is also denoted by . The function is a fixed point of the evolution operator generated by (3.1), that is, for all . For system (3.1), the result in [13, Theorem 1] showed that the equilibrium is exponentially attractive on the positive real line ; that is, there exist positive constants and such that However, the constant is usually greater than . Using the result developed in the preceding section, we show in the next theorem that the constant in (3.4) can be chosen to be equal to on the state space with norm for some . As a consequence, the equilibrium solution of system (3.1) is finite-time attractive with respect to these norms.
Theorem 3.1 (finite-time attractive equilibrium of Lotka-Voltera equations). Consider (3.1) on an arbitrary finite-time interval satisfying (3.2) and (3.3). Then, there exists a positive weight factor such that for all the positive equilibrium is finite-time attractive on with respect to the norm .
Proof . The proof is divided into three steps.
Step 1. Construction of the weight factor . Due to compactness of and continuity of solutions of (3.1), there exists such that Then, we have which implies that for all and we have Define Now let be the solution of the following equation:
Step 2. In this step, we show that is the solution of the delay equation with the initial condition for , where Indeed, we have which together with the fact that implies that is a solution of (3.10). Furthermore, we have where denotes the evolution operator generated by (3.10).
Step 3. In this step, we show that for all and we have Choose and fix , and we show that (3.10) fulfills all assumptions of Theorem 2.1. Indeed, using (3.7) and the definition of , we obtain the following upper bound: Combining (3.2) and (3.7), we also get that Therefore, system (3.10) fulfills all assumptions of Theorem 2.1. Then, the zero solution of system (3.10) is finite-time attractive with respect to the norm for all , that is, In particular, substituting we get that which together with (3.13) implies (3.14) and the proof is complete.
In the rest of the paper, we discuss a planar Lotka-Voltera system, for which we can explicitly compute its equilibrium. Consequently, applying Theorem 3.1 yields a sufficient condition for finite-time attractivity of this equilibrium.
Example 3.2. Consider a planar Lotka-Voltera of the following form: where and the coefficients for satisfy the following inequalities for some : Additionally, we assume that the equilibrium point is positive, where According to Theorem 3.1, for any finite-time interval , there exists a positive weight factor such that for all the positive equilibrium is finite-time attractive on with respect to the norm .
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