Finite-Time Attractivity for Diagonally Dominant Systems with Off-Diagonal Delays

Abstract

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.

1. Introduction

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 𝑟≥0, let 𝒞∶=𝐶([−𝑟,0],ℝ^𝑑) denote the space of all continuous functions 𝜑∶[−𝑟,0]→ℝ^𝑑. For 𝛾∈ℝ, the norm ‖⋅‖_𝛾,∞ on 𝒞 is defined as follows: 𝜑_𝛾,∞∶=max𝑒^𝛾𝑠𝜑(𝑠)_∞∶𝑠∈[−𝑟,0], (1.1) where ‖𝑥‖_∞=max{|𝑥_𝑖|∶𝑖=1,…,𝑑} for all 𝑥=(𝑥₁,…,𝑥_𝑑)^𝑇∈ℝ^𝑑. Consider a finite-time delay differential equation 𝑥̇=𝑓𝑡,𝑥_𝑡for𝑡∈[0,𝑇], (1.2) where 𝑓∶[0,𝑇]×𝒞→ℝ^𝑑 is assumed to be continuous and Lipschitz in the second argument. For each 𝜑∈𝒞, let 𝑥(⋅,𝜑) denote the solution of (2.1) satisfying the initial condition 𝑥(𝑠)=𝜑(𝑠) for all 𝑠∈[−𝑟,0]. The evolution operator 𝑆∶[0,𝑇]×𝒞→𝒞 generated by (1.2) is defined as 𝑆(𝑡)𝜑(𝑠)=𝑥𝑡+𝑠,𝜑∀𝑠∈[−𝑟,0],𝑡∈[0,𝑇]. (1.3) 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 𝑆(⋅,𝜑) is called finite-time attractive on [0,𝑇] with respect to the norm ‖⋅‖_𝛾,∞ if there exist positive constants 𝛼 and 𝜂 such that for all 𝑡,𝑠∈[0,𝑇] with 𝑠≤𝑡 the following estimate holds: 𝑆𝑡,𝜑−𝑆𝑡,𝜓_𝛾,∞≤𝑒^{−𝛼(𝑡−𝑠)}𝑆𝑠,𝜑−𝑆𝑠,𝜓_𝛾,∞ (1.4) for all 𝜓 in the neighborhood 𝐵_𝜂(𝜑) of 𝜑.

Remark 1.2. In the case that 𝑓∶[0,𝑇]×𝒞→ℝ^𝑑 is a linear function in the second argument, it is easy to see that the generated semigroup 𝑆∶[0,𝑇]×𝒞→𝒞 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 𝑆(⋅,𝜑) for a 𝜑∈𝒞,(ii)for all 𝜑∈𝒞, the solution 𝑆(⋅,𝜑) is finite-time attractive, and(iii)there exists 𝛼>0 such that for all 0≤𝑠≤𝑡≤𝑇 we have 𝑆𝑡,𝜑_𝛾,∞≤𝑒^{−𝛼(𝑡−𝑠)}𝑆𝑠,𝜑_𝛾,∞∀𝜑∈𝒞. (1.5)
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., [13] and the reference therein): 𝑥̇_𝑖(𝑡)=𝑎_𝑖𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗for𝑖=1,…,𝑑,𝑡∈[0,𝑇], (2.1) where 𝑇 is a given positive constant, 𝑎_𝑖𝑗∶ℝ→ℝ, 𝑖,𝑗=1,…,𝑑, are continuous functions and 𝜏_𝑖𝑗>0 for 𝑖,𝑗=1,…,𝑑 with 𝑖≠𝑗. Define 𝑟∶=max𝜏_𝑖𝑗∶𝑖,𝑗=1,…,𝑑,𝑖≠𝑗. (2.2) Note that (2.1) is a special case of (1.2). More precisely, the right hand side of (2.1) equals 𝑓(𝑡,𝑥_𝑡), where 𝑓=(𝑓₁,…,𝑓_𝑑)∶[0,𝑇]×𝒞→ℝ^𝑑 is defined as follows: 𝑓_𝑖𝑡,𝜑∶=𝑎_𝑖𝑖(𝑡)𝜑(0)+^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)𝜑_𝑗−𝜏_𝑖𝑗. (2.3) Let 𝑆∶[0,𝑇]×𝒞→𝒞 denote the evolution operator of (2.1). From (2.3), we see that the function 𝑓 is linear in the second argument. Therefore, the evolution operator 𝑆 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 𝛿 such that 𝑎_𝑖𝑖(𝑡)≥^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)+𝛿for𝑡∈[0,𝑇]. (2.4)

Theorem 2.1 (finite-time attractivity for delay equations). Consider system (2.1) on a finite-time interval [0,𝑇]. Suppose that system (2.1) is row diagonally dominant with a positive constant 𝛿 and 𝑎_𝑖𝑖(𝑡)<0 for all 𝑖=1,…,𝑑 and 𝑡∈[0,𝑇]. Define 𝑀∶=max^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)∶𝑖=1,…,𝑑,𝑡∈[0,𝑇], (2.5) and let 𝛾^∗ be a positive number satisfying that 𝛾^∗2+𝑒^𝛾∗𝑟−1𝑀=𝛿. (2.6)
Then for every 𝛾∈[0,𝛾^∗], the zero solution of (2.1) is finite-time attractive with respect to the norm ‖⋅‖_𝛾,∞ with exponent −𝛾/2, that is, 𝑆𝑡,𝜑_𝛾,∞≤𝑒^{−(𝛾/2)(𝑡−𝑠)}𝑆𝑠,𝜑_𝛾,∞for0≤𝑠≤𝑡≤𝑇. (2.7)

Proof. We divide the proof into two steps.
Step 1. We show that for 𝜑∈𝒞 the inequality 𝑥𝑡,𝜑_∞≤𝑒^{−(𝛾/2)(𝑡−𝑠)}𝑆𝑠,𝜑_𝛾,∞∀0≤𝑠≤𝑡≤𝑇 (2.8) holds. Suppose the opposite, that is, assume that there exists 𝑠∈[0,𝑇] such that the set 𝒩∶=𝑡∈[𝑠,𝑇)∶𝑥𝑡,𝜑_∞>𝑒^{−(𝛾/2)(𝑡−𝑠)}𝑆𝑠,𝜑_𝛾,∞ (2.9) is not empty. Define 𝑡_inf=inf{𝑡∶𝑡∈𝒩}. By continuity of the map 𝑡↦‖𝑥(𝑡,𝜑)‖_∞, we get that 𝑥𝑡_inf,𝜑_∞=𝑒^{−(𝛾/2)(𝑡inf−𝑠)}𝑆𝑠,𝜑_𝛾,∞,𝑥𝑡,𝜑_∞≤𝑒^{−(𝛾/2)(𝑡−𝑠)}𝑆𝑠,𝜑_𝛾,∞∀𝑡∈𝑠,𝑡_inf. (2.10)(2.11) Now, we will show that 𝑥𝑡_inf,𝜑_∞≥𝑒^−𝛾𝑟𝑥𝑡,𝜑_∞∀𝑡∈𝑡_inf−𝑟,𝑡_inf. (2.12) Indeed, we consider the following two cases: (i) 𝑡∈[𝑠,𝑡_inf]∩[𝑡_inf−𝑟,𝑡_inf] and (ii) 𝑡∈(−∞,𝑠]∩[𝑡_inf−𝑟,𝑡_inf].
Case (i). If 𝑡∈[𝑠,𝑡_inf]∩[𝑡_inf−𝑟,𝑡_inf], then, according to (2.10) and (2.11), we obtain that 𝑥𝑡_inf,𝜑_∞=𝑒^{−(𝛾/2)(𝑡inf−𝑠)}𝑆𝑠,𝜑_𝛾,∞≥𝑒^{−(𝛾/2)(𝑡inf−𝑠)}𝑒^{(𝛾/2)(𝑡−𝑠)}𝑥𝑡,𝜑_∞≥𝑒^−𝛾𝑟𝑥𝑡,𝜑_∞, (2.13) which proves (2.12) in this case.
Case (ii). If 𝑡∈(−∞,𝑠]∩[𝑡_inf−𝑟,𝑡_inf], then, according to (2.10) and the definition of the norm ‖⋅‖_𝛾,∞, we obtain that 𝑥𝑡_inf,𝜑_∞=𝑒^{−(𝛾/2)(𝑡inf−𝑠)}max_{𝜔∈[−𝑟,0]}𝑒^𝛾𝜔𝑆𝑠,𝜑(𝜔)_∞=max_{[]𝜔∈−𝑟,0}𝑒^{−(𝛾/2)(𝑡inf−𝑠−2𝜔)}𝑥𝑠+𝜔,𝜑_∞≥𝑒^{−(𝛾/2)(𝑡inf+𝑠−2𝑡)}𝑥𝑡,𝜑_∞≥𝑒^−𝛾𝑟𝑥𝑡,𝜑_∞. (2.14) Hence, (2.12) is proved. To conclude the proof of this step, we estimate the norm ‖𝑥(𝑡,𝜑)‖_∞ for all 𝑡 in a neighborhood of 𝑡_inf in order to show a contradiction to the assumption that the set 𝒩 is not empty. To this end, we define the following set: 𝐼∶=𝑖∈{1,…,𝑑}∶𝑥_𝑖𝑡_inf,𝜑=𝑥𝑡_inf,𝜑_∞. (2.15) The continuity of the functions 𝑡↦𝑥_𝑖(𝑡,𝜑) for 𝑖=1,…,𝑑 implies that there exists a neighborhood (𝑡_inf−𝜀,𝑡_inf+𝜀) for some 𝜀>0 such that 𝑥𝑡,𝜑_∞=max_𝑖∈𝐼𝑥_𝑖𝑡,𝜑∀𝑡∈𝑡_inf−𝜀,𝑡_inf+𝜀. (2.16) By virtue of (2.1), the derivative of the function 𝑡↦𝑥²_𝑖(𝑡,𝜑) is estimated as follows: 12𝑑𝑥𝑑𝑡^2𝑖𝑡,𝜑_𝑡=𝑡inf=𝑥_𝑖𝑡_inf,𝜑𝑎_𝑖𝑖𝑡_inf𝑥_𝑖𝑡_inf,𝜑+^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑡_inf𝑥_𝑗𝑡_inf−𝜏_𝑖𝑗,𝜑≤𝑎_𝑖𝑖𝑡_inf𝑥_𝑖𝑡_inf,𝜑²+𝑥_𝑖𝑡_inf,𝜑^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑡_inf𝑥𝑡_inf−𝜏_𝑖𝑗,𝜑_∞, (2.17) which together with (2.12) and the definition of 𝐼 implies that for all 𝑖∈𝐼12𝑑𝑥𝑑𝑡^𝑖𝑡,𝜑²_𝑡=𝑡inf≤𝑥_𝑖𝑡_inf,𝜑²𝑎_𝑖𝑖𝑡_inf+𝑒^𝛾𝑟𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑡_inf. (2.18) Thus, from the row diagonal dominance (2.4) and bound (2.5), we derive that 12𝑑𝑥𝑑𝑡^𝑖𝑡,𝜑²_𝑡=𝑡inf≤−𝛿+𝑒^𝛾𝑟−1𝑀𝑥²_𝑖𝑡_inf,𝜑. (2.19) Using (2.6), we obtain that 12𝑑𝑥𝑑𝑡^2𝑖𝑡,𝜑_𝑡=𝑡inf𝛾<−2𝑥²_𝑖𝑡_inf,𝜑, (2.20) which yields that there exists a neighborhood (𝑡_inf,𝑡_inf+𝜀₂) for an 𝜀₂>0 such that for 𝑖∈𝐼𝑥_𝑖𝑡,𝜑≤𝑒^{−(𝛾/2)(𝑡−𝑡inf)}𝑥_𝑖𝑡_inf,𝜑∀𝑡∈𝑡_inf,𝑡_inf+𝜀₂. (2.21) Thus, for all 𝑡∈(𝑡_inf,𝑡_inf+𝜀) with 𝜀∶=min{𝜀₁,𝜀₂}, we have 𝑥𝑡,𝜑_∞≤𝑒^{−(𝛾/2)(𝑡−𝑡inf)}𝑥𝑡_inf,𝜑_∞, (2.22) which together with (2.10) implies that 𝑥𝑡,𝜑_∞≤𝑒^{−(𝛾/2)(𝑡−𝑠)}𝑆𝑠,𝜑_𝛾,∞. (2.23) Consequently, 𝒩∩(𝑡_inf,𝑡_inf+𝜀)=∅, and this is a contradiction to the definition of 𝒩 and 𝑡_inf. Thus, (2.8) is proved.
Step 2. Using (2.8) from Step 1, we show (2.7) by considering two cases: (i) 𝑡∈[𝑠+𝑟,𝑇]∩[𝑠,𝑇] and (ii) 𝑡∈[𝑠,𝑠+𝑟]∩[𝑠,𝑇].
Case (i). If 𝑡∈[𝑠+𝑟,𝑇]∩[𝑠,𝑇], then by virtue of (2.8) we have 𝑆𝑡,𝜑_𝛾,∞=sup_{[]𝜔∈−𝑟,0}𝑒^𝛾𝜔𝑥𝑡+𝜔,𝜑_∞≤sup_{[]𝜔∈−𝑟,0}𝑒^𝛾𝜔𝑒^{−(𝛾/2)(𝑡+𝜔−𝑠)}𝑆𝑠,𝜑_𝛾,∞≤𝑒^{−(𝛾/2)(𝑡−𝑠)}𝑆𝑠,𝜑_𝛾,∞, (2.24) which proves (2.7) in this case.
Case (ii). If 𝑡∈[𝑠,𝑠+𝑟]∩[𝑠,𝑇], then we have 𝑆𝑡,𝜑_𝛾,∞=𝑚𝑎𝑥sup_{[−()]𝜔∈𝑡−𝑠,0}𝑒^𝛾𝜔𝑥𝑡+𝜔,𝜑_∞,sup_{[(𝜔∈−𝑟,−𝑡−𝑠)]}𝑒^𝛾𝜔𝑥𝑡+𝜔,𝜑_∞. (2.25)
Hence, using (2.8), we obtain that 𝑆𝑡,𝜑_𝛾,∞≤𝑚𝑎𝑥sup_{[−()]𝜔∈𝑡−𝑠,0}𝑒^(𝛾/2)𝜔𝑒^{−(𝛾/2)(𝑡−𝑠)}𝑆𝑠,𝜑_𝛾,∞,𝑒^{−𝛾(𝑡−𝑠)}𝑆𝑠,𝜑_𝛾,∞. (2.26)
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: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖+^𝑑_𝑗=1𝑎_𝑖𝑗𝑥_𝑗𝑡−𝑟_𝑖𝑗,𝑖=1,…,𝑑, (3.1) where 𝑟_𝑖𝑗≥0 and 𝑟_𝑖𝑖=0 for all 𝑖=1,…,𝑑. Suppose that 𝐴=(𝑎_𝑖𝑗)_{𝑖,𝑗=1,…,𝑑} is row diagonal dominant with 𝑎_𝑖𝑖<0 for all 𝑖=1,…,𝑑; that is, there exists 𝛿>0 such that 𝑎_𝑖𝑖≥^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗+𝛿. (3.2) We assume in the following that there exists a positive vector 𝑥^∗∈ℝ^𝑑₊ such that 𝐫+𝐴𝑥^∗=0where𝐫=𝑟₁,…,𝑟_𝑑^𝑇. (3.3) To shorten the notation, the function 𝑡↦𝑥^∗ for all 𝑡∈[−𝑟,0] is also denoted by 𝑥^∗. The function 𝑥^∗ is a fixed point of the evolution operator 𝑆(𝑡,⋅) generated by (3.1), that is, 𝑆(𝑡,𝑥^∗)=𝑥^∗ for all 𝑡. For system (3.1), the result in [13, Theorem 1] showed that the equilibrium 𝑥^∗ is exponentially attractive on the positive real line ℝ₊; that is, there exist positive constants 𝐾,𝛼 and 𝜂 such that 𝑆𝑡,𝜑−𝑥^∗_0,∞≤𝐾𝑒^−𝛼𝑡𝜑−𝑥^∗∀𝜑∈𝐵_𝜂𝑥^∗. (3.4) However, the constant 𝐾 is usually greater than 1. Using the result developed in the preceding section, we show in the next theorem that the constant 𝐾 in (3.4) can be chosen to be equal to 1 on the state space 𝒞 with norm ‖⋅‖_𝛾,∞ for some 𝛾≥0. 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 [0,𝑇] satisfying (3.2) and (3.3). Then, there exists a positive weight factor 𝛾^∗ such that for all 𝛾∈[0,𝛾^∗] the positive equilibrium 𝑥^∗ is finite-time attractive on [0,𝑇] with respect to the norm ‖⋅‖_𝛾,∞.

Proof . The proof is divided into three steps.
Step 1. Construction of the weight factor 𝛾^∗. Due to compactness of [0,𝑇] and continuity of solutions of (3.1), there exists 𝜂^∗>0 such that 𝑆𝑡,𝜑−𝑥^∗_0,∞≤min_{𝑖=1,…,𝑑}𝑥^∗_𝑖2for𝑡∈[0,𝑇],𝜑∈𝐵_𝜂∗𝑥^∗. (3.5) Then, we have 𝑥_𝑖𝑡,𝜑−𝑥^∗_𝑖≤𝑆𝑡,𝜑−𝑥^∗_0,∞≤min_{𝑖=1,…,𝑑}𝑥^∗_𝑖2, (3.6) which implies that for all 𝑡∈[0,𝑇] and 𝜑∈𝐵_𝜂∗(𝑥^∗) we have min_{𝑖=1,…,𝑑}𝑥^∗_𝑖2≤𝑥_𝑖𝑡,𝜑≤min_{𝑖=1,…,𝑑}𝑥^∗_𝑖2+max_{𝑖=1,…,𝑑}𝑥^∗_𝑖. (3.7) Define 𝛿^∗∶=min_{𝑖=1,…,𝑑}𝑥^∗_𝑖2𝛿,𝑀∶=min_{𝑖=1,…,𝑑}𝑥^∗_𝑖2+max_{𝑖=1,…,𝑑}𝑥^∗_𝑖max_{𝑖=1,…,𝑑}^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗. (3.8) Now let 𝛾^∗>0 be the solution of the following equation: 𝛾^∗2+𝑒^𝛾∗𝑟−1𝑀=𝛿^∗. (3.9)
Step 2. In this step, we show that 𝑦(𝑡)∶=𝑥(𝑡,𝜑)−𝑥^∗ is the solution of the delay equation 𝑦_𝑖̇(𝑡)=𝑎_𝑖𝑖(𝑡)𝑦_𝑖(𝑡)+^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝑟_𝑖𝑗for𝑖=1,…,𝑑 (3.10) with the initial condition 𝑦(𝑠)=𝜑(𝑠)−𝑥^∗ for 𝑠∈[−𝑟,0], where 𝑎_𝑖𝑗(𝑡)∶=𝑎_𝑖𝑗𝑥_𝑖𝑡,𝜑for𝑖,𝑗=1,…,𝑑. (3.11) Indeed, we have 𝑦_𝑖̇(𝑡)=𝑥_𝑖𝑡,𝜑𝑟_𝑖+^𝑑_𝑗=1𝑎_𝑖𝑗𝑥_𝑗𝑡−𝑟_𝑖𝑗,𝜑=𝑥_𝑖𝑡,𝜑𝑟_𝑖+^𝑑_𝑗=1𝑎_𝑖𝑗𝑥^∗_𝑗+^𝑑_𝑗=1𝑎_𝑖𝑗𝑦_𝑗𝑡−𝑟_𝑖𝑗, (3.12) which together with the fact that 𝐫+𝐴𝑥^∗=0 implies that 𝑦(𝑡) is a solution of (3.10). Furthermore, we have 𝑆^∗𝑡,𝜑−𝑥^∗=𝑆𝑡,𝜑−𝑥^∗∀𝑡∈[0,𝑇], (3.13) where 𝑆^∗ denotes the evolution operator generated by (3.10).
Step 3. In this step, we show that for all 𝛾∈[0,𝛾^∗] and 𝜑∈𝐵_𝜂∗(𝑥^∗) we have 𝑆𝑡,𝜑−𝑥^∗_𝛾,∞≤𝑒^{−𝛾(𝑡−𝑠)/2}𝑆𝑠,𝜑−𝑥^∗_𝛾,∞∀0≤𝑠≤𝑡≤𝑇. (3.14) Choose and fix 𝜑∈𝐵_𝜂∗(𝑥^∗), and we show that (3.10) fulfills all assumptions of Theorem 2.1. Indeed, using (3.7) and the definition of 𝑀, we obtain the following upper bound: max_{[]𝑡∈0,𝑇,𝑖=1,…,𝑑}^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)≤min_{𝑖=1,…,𝑑}𝑥^∗_𝑖2+max_{𝑖=1,…,𝑑}𝑥^∗_𝑖max_{𝑖=1,…,𝑑}^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗≤𝑀. (3.15) Combining (3.2) and (3.7), we also get that 𝑎_𝑖𝑖(𝑡)−^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)=𝑥_𝑖𝑡,𝜑𝑎_𝑖𝑖−^𝑑_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗≥min_{𝑖=1,…,𝑑}𝑥^∗_𝑖2𝛿=𝛿^∗. (3.16) 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 ‖⋅‖_𝛾,∞ for all 𝛾∈[0,𝛾^∗], that is, 𝑆^∗𝑡,𝜓_𝛾,∞≤𝑒^{−𝛾(𝑡−𝑠)/2}𝑆^∗𝑠,𝜓_𝛾,∞∀0≤𝑠≤𝑡≤𝑇,𝜓∈C. (3.17) In particular, substituting 𝜓=𝜑−𝑥^∗ we get that 𝑆^∗𝑡,𝜑−𝑥^∗_𝛾,∞≤𝑒^{−(𝛾(𝑡−𝑠))/2}𝑆^∗𝑠,𝜑−𝑥^∗_𝛾,∞∀0≤𝑠≤𝑡≤𝑇, (3.18) 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: 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑟₁+𝑎₁₁𝑥₁(𝑡)+𝑎₁₂𝑥₂𝑡−𝜏₁₂,𝑥̇₂(𝑡)=𝑥₂(𝑡)𝑟₂+𝑎₂₁𝑥₁𝑡−𝜏₂₁+𝑎₂₂𝑥₂(𝑡), (3.19) where 𝑟₁,𝑟₂,𝜏₁₂,𝜏₂₁>0 and the coefficients 𝑎_𝑖𝑗 for 𝑖,𝑗=1,2 satisfy the following inequalities for some 𝜀>0: 𝑎₁₁,𝑎₂₂<0,𝑎₁₁≥𝑎₁₂+𝜀,𝑎₂₂≥𝑎₂₁+𝜀. (3.20) Additionally, we assume that the equilibrium point 𝑥^∗=(𝑥^∗₁,𝑥^∗₂) is positive, where 𝑥^∗₁=𝑟₂𝑎₁₂−𝑟₁𝑎₂₂𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁,𝑥^∗₂=𝑟₁𝑎₂₁−𝑟₂𝑎₁₁𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁. (3.21) According to Theorem 3.1, for any finite-time interval [0,𝑇], there exists a positive weight factor 𝛾^∗ such that for all 𝛾∈[0,𝛾^∗] the positive equilibrium 𝑥^∗ is finite-time attractive on [0,𝑇] with respect to the norm ‖⋅‖_𝛾,∞.