Nonautonomous Differential Equations in Banach Space and Nonrectifiable Attractivity in Two-Dimensional Linear Differential Systems

Abstract

We study the asymptotic behaviour on a finite interval of a class of linear nonautonomous singular differential equations in Banach space by the nonintegrability of the first derivative of its solutions. According to these results, the nonrectifiable attractivity on a finite interval of the zero solution of the two-dimensional linear integrable differential systems with singular matrix-elements is characterized.

1. Introduction

Let 𝑡₀>0 and 𝕏 a given real or complex Banach space with the zero 𝟎∈𝕏, and let 𝐿(𝕏) denote the space of all linear bounded operators from 𝕏 into 𝕏. We consider the linear nonautonomous differential equation in 𝕏 as 𝑑𝐱(𝑡)𝑑𝑡=𝐴𝐱,𝑡∈0,𝑡₀,𝐱𝑡₀=𝐱₀∈𝕏, (1) where 𝐱=𝐱(𝑡), 𝐱∈𝐶¹((0,𝑡₀];𝕏), 𝐴=𝐴(𝑡) is an operator-valued function defined on interval (0,𝑡₀] with the value in 𝐿(𝕏) and 𝐴∈𝐶((0,𝑡₀];𝐿(𝕏)). If 𝕏=ℝ^𝑁, then (1) becomes a linear nonautonomous differential system.

As a basic result, in Theorem 1, we state the existence and uniqueness of a solution of (1) which is proved by using a fixed-point theorem in Fréchet spaces for an 𝛼-contraction mapping. Next, in Theorems 2 and 3, we give a necessary and sufficient condition for the nonintegrability of ‖𝐱^′(𝑡)‖_𝕏 on (0,𝑡₀] provided ‖𝐴(𝑡)‖ blows up near 𝑡=0 and ‖𝐴(𝑡)𝑥‖ allows a precise asymptotic behaviour near 𝑡=0. Based on these results, in the case when 𝕏=ℝ², we study the so-called nonrectifiable attractivity of the zero solution (see Definition 4) of the so-called linear integrable two-dimensional system (see Definition 5) as follows:𝑑𝐱=ℎ(𝑡)(𝑡)(𝑡)(𝑡)ℎ(𝑡)𝑑𝑡+𝜇𝑔−𝜈𝑔𝜌𝑔𝐱,𝑡∈0,𝑡₀,𝐱𝑡₀=𝐱₀∈𝕏 (2) with 𝜇, 𝜌, and 𝜈 some real constants.

Precisely, as a kind of singular behaviour near 𝑡=0 of all solutions 𝑥(𝑡) of linear integrable system (2), in Theorem 6, we involve on the matrix elements ℎ(𝑡) and 𝑔(𝑡) a necessary and sufficient condition for the infiniteness of the length of every solution curve of 𝑥(𝑡) (in our best knowledge, it is the first paper dealing with this kind of problems). Theorem 6 is a consequence of the precise asymptotic formula for all solutions 𝑥(𝑡) near 𝑡=0 of integrable differential system (2) presented in Lemmas 11 and 12. Of course, instead of interval (0,𝑡₀] and 𝑡→0, we can also state our main results on [𝑡₀,∞) and 𝑡→∞.

In applications, the last years have seen an increasing interest in the analysis of differential equations and systems on finite time intervals. This is because the Finite-Time Stability (FTS) and the Finite-Time Lyapunov Exponent (FTLE) were introduced, respectively, in the control of systems within a finite time as well as in the Lagrangian Coherent Structure (LCS) on finite-time intervals in fluid, ocean, and atmosphere dynamics [1–3] and in the biological application [4]. It includes the time-varying vector fields known only on a finite-time interval, but not on the whole half line 𝑡≥𝑡₀.

In the theory of differential equations, the importance of studying the different kind of asymptotic behaviours of the nonautonomous linear differential systems comes from their application in the study of asymptotic and oscillatory behaviour of the second and higher order ordinary differential equations. For instance, in [5], authors study the asymptotic behaviour near 𝑡=∞ of oscillatory solutions of the nonlinear second-order differential equation (𝑝(𝑡)𝑢^′)^′=𝑝(𝑡)𝑓(𝑢) via the asymptotic formula for solutions of an auxiliary linear differential system having elements which are absolutely continuous functions on [𝑡₀,∞). In [6], authors derive a precise asymptotic behaviour near 𝑡=∞ of solutions 𝑢(𝑡) of the third-order nonlinear differential equation 𝑢^′′′+((𝑚+1)/2)𝑢𝑢^′′+𝑚(1−𝑢^′2)+𝑀(1−𝑢^′)=0 such that 𝑢(0)=𝑎, 𝑢^′(0)=𝑏, 𝑢^′(∞)=1, and 0<𝑢^′<1 on [0,∞), by using the asymptotic solution formula for the corresponding linear differential system based on Hartman and Wintner’s asymptotic integration (see [7, 8] and for its generalization [9, 10]). Similarly, in [11], authors prove an asymptotic solution formula for the second-order nonlinear differential equations (𝑝𝑢^′)^′+𝑞𝑢=𝑔(𝑡,𝑢,𝑢^′) depending on the asymptotic behaviour of fundamental solutions of the corresponding homogeneous equation (𝑝𝑧^′)^′+𝑞𝑧=0. On certain types of asymptotic behaviour of linear and nonlinear differential and integro-differential systems, we refer reader to some recently published papers [12–14] and references therein. The attractivity of solutions of scalar delay differential equations is widely studied and it is important in mathematical biology; see, for instance, [15] and references therein.

This paper is mainly based on a part of the P.h.D degree thesis [16] of the first author.

2. Statement of the Main Results

Preliminarily, we state the following auxiliary result.

Theorem 1. For each 𝐱₀∈𝕏, there exists a unique solution 𝐱 of (1).

Next, we derive a necessary condition for a kind of singularities of (1) at 𝑡=0 in terms of singular behaviour of the operator norm of 𝐴(𝑡) near 𝑡=0.

Theorem 2. Let there exist a solution 𝐱 of (1) such that ‖𝐱(𝑡)‖_𝕏→0 as 𝑡→0. If ‖𝑑𝐱/𝑑𝑡‖_𝕏∉𝐿¹(0,𝑡₀), then one has: limsup_𝑡→0‖𝐴(𝑡)‖=∞. (3)

Thus, if (3) is not satisfied, then the integrability of ‖𝑑𝐱/𝑑𝑡‖_𝕏 occurs for all solutions 𝐱 of (1), which is out of our interest. All results from this section will be proved in the next sections.

At the second, we impose on the operator-valued function 𝐴(𝑡) the following structural hypotheses:

 (𝐻₁) there are positive constants 𝑐₀, 𝑐₁ and a positive real function 𝜔=𝜔(𝑡) such that 𝑐₀𝜔(𝑡)‖𝑥‖_𝕏≤‖𝐴(𝑡)𝑥‖_𝕏≤𝑐₁𝜔(𝑡)‖𝑥‖_𝕏,∀𝑡∈0,𝑡₀,∀𝑥∈𝕏; (4)  (𝐻₂) the solution of (1) is given by 𝐱(𝑡)=𝑈(𝑡)𝐱₀,𝑡∈0,𝑡₀, (5) where 𝑈=𝑈(𝑡), 𝑈∶(0,𝑡₀]→𝐿(𝕏), is an operator-valued function called the fundamental solution, such that ‖𝑈(𝑡)‖⟶0as𝑡⟶0, (6) and there are positive constants 𝑐₂, 𝑐₃ and a positive real function 𝑎=𝑎(𝑡) such that 𝑐₂𝑎(𝑡)‖𝑥‖_𝕏≤‖𝑈(𝑡)𝑥‖_𝕏≤𝑐₃𝑎(𝑡)‖𝑥‖_𝕏,∀𝑡∈0,𝑡₀,∀𝑥∈𝕏. (7) Such hypotheses on (1) imply the following properties of every solution 𝐱 and its 𝑑𝐱/𝑑𝑡.

Theorem 3. Let the hypotheses (𝐻₁)-(𝐻₂) be fulfilled. Then, for every 𝐱₀∈𝕏, the corresponding solution 𝐱 of (1) satisfies(i)‖𝐱(𝑡)‖_𝕏→0 as 𝑡→0;(ii)𝐱 is an injective function on (0,𝑡₀] if and only if 𝐱₀ is not an eigenvector of 𝑈⁻¹(𝑠)𝑈(𝑡) with an eigenvalue 1 for all 𝑠,𝑡∈(0,𝑡₀], 𝑠≠𝑡;(iii)‖𝑑𝐱/𝑑𝑡‖_𝕏∉𝐿¹(0,𝑡₀) if and only if 𝜔𝑎∉𝐿¹(0,𝑡₀).

The meaning and importance of hypotheses (𝐻₁)-(𝐻₂) as well as the conclusions (i)–(iii) of Theorem 3 will be verified by Theorem 6, where 𝕏=ℝ² and (1) is a large class of two-dimensional linear differential systems.

The second aim of the paper is to use previous theorem in the study of the rectifiable and nonrectifiable attractivity of zero solution of two-dimensional linear differential system as 𝐱^′=𝐴(𝑡)𝐱,𝑡∈0,𝑡₀,𝐱𝑡₀=𝐱₀∈ℝ²,𝐱∈𝐶¹0,𝑡₀;ℝ², (8) where 𝑡₀>0 and the prime denotes 𝑑/𝑑𝑡, 𝐱=𝐱(𝑡)=(𝑥(𝑡),𝑦(𝑡)) and the matrix-valued function 𝐴=𝐴(𝑡), 𝐴∶(0,𝑡₀]→ℳ₂(ℝ) (where ℳ₂(ℝ) denotes the space of real 2×2 matrices), is continuous on (0,𝑡₀]. Therefore, we may apply previous theorems to system (8). As in the general case of 𝕏, the zero solution 𝐱(𝑡)≡𝟎∈ℝ² of system (8) is said to be (global) attractive if for every solution 𝐱(𝑡) of (8), we have ‖𝐱(𝑡)‖_ℝ²→0 as 𝑡→0. On the asymptotic stability and attractivity for several kinds of linear systems in the case when 𝑡→∞, we refer reader to [7–10, 12–14] and references therein.

We know that a plain curve Γ_𝐱 is a Jordan curve in ℝ² if there exists a continuous injection 𝐱=𝐱(𝑡), 𝐱∶[0,𝑡₀]→ℝ², 𝑡₀>0, such that 𝐱([0,𝑡₀])=Γ_𝐱. It is often said that Γ_𝐱 is parametrized by 𝐱(𝑡) or that Γ_𝐱 is associated to 𝐱(𝑡). The length of Γ_𝐱, denoted by length(Γ_𝐱), is defined by lengthΓ_𝐱=sup^𝑚_𝑘=1𝐱𝑡_𝑘−𝐱𝑡_𝑘−1ℝ2, (9) where the supremum is taken over all finite partitions 0=𝑡_𝑚+1<⋯<𝑡₁<𝑡₀ of the interval [0,𝑡₀] (see [17, 18]).

Definition 4. The zero solution of system (8) is rectifiable attractive (resp., nonrectifiable attractive) if it is attractive and the curve Γ_𝐱 of every solution 𝐱 of (8) is a rectifiable (resp., nonrectifiable) Jordan curve in ℝ².

Following [17], we are interested in Jordan curves so that the parametrization of our solutions faithfully represents the length and rectifiability properties (omitting injectivity, we might find solutions that self-intersect on large sets or are just self-winding and, hence, artificially nonrectifiable).

The following well-known fact (see [19]) gives us a main support to relate the singularity in (1) with the nonrectifiability in system (8): 𝑑𝐱𝑑𝑡_ℝ²∈𝐿¹0,𝑡₀ifandonlyiflengthΓ_𝐱<∞. (10)

On the rectifiability of graph of solutions of scalar second-order linear differential equations, we refer reader to [20].

Here the particular attention is paid to the case of the so-called linear integrable systems defined as follows.

Definition 5. Let ℳ₂ denote the linear space of all 2×2 matrix with the elements in ℂ. We say that (8) is a linear integrable system if there exists an invertible matrix 𝑇∈ℳ₂ such that for every 𝑡∈(0,𝑡₀] the matrix Λ(𝑡)=𝑇⁻¹𝐴(𝑡)𝑇∈ℳ₂ is a diagonal matrix for every 𝑡∈(0,𝑡₀].

Note that the matrices 𝐴(𝑡) are themselves always real. As commented in Section 6, we proceed with presentation of our results and arguments in terms of the coefficients of such matrices.

Denoting the matrix elements of 𝐴(𝑡) by 𝐴(𝑡)=𝑒(𝑡)𝑓(𝑡)𝑔(𝑡)ℎ(𝑡),𝑡∈0,𝑡₀, (11) when we have 𝑒(𝑡)≡ℎ(𝑡) and 𝑓(𝑡)≡−𝑔(𝑡), we speak of integrable systems (see for instance [21]). In such a case, both eigenvalues 𝜆(𝑡)=ℎ(𝑡)±𝑔(𝑡)𝑖 of the matrix 𝐴(𝑡) admit eigenvectors (±𝑖,1) that do not depend on variable 𝑡. This motivates us to introduce Definition 5 which gives a more general notion of the integrable system than previous one. It is because (see Lemma 11 in Section 4) the integrable system (8) has the following form: 𝐴(𝑡)=ℎ(𝑡)(𝑡)(𝑡)(𝑡)ℎ(𝑡)+𝜇𝑔−𝜈𝑔𝜌𝑔,𝑡∈0,𝑡₀, (12) where 𝜇,𝜈,𝜌∈ℝ. For 𝜇=0 and 𝜈=𝜌=1 in (8), we have 𝑒(𝑡)≡ℎ(𝑡) and 𝑓(𝑡)≡−𝑔(𝑡); so, the classic integrable system is a special case.

In Section 5, we show that the set of all matrix-valued functions 𝐴=𝐴(𝑡) that satisfy Definition 5 make an algebra.

Furthermore, we show in Section 4 that if (8) is a linear integrable system, then the matrix-valued function 𝐴(𝑡) satisfies the required hypotheses (𝐻₁)-(𝐻₂) in particular for 𝜔(𝑡)=𝑔(𝑡+|ℎ(𝑡,𝑎(𝑡)))|=𝑒^{−∫𝑡0𝑡(ℎ(𝜏)+(𝜇/2)𝑔(𝜏))𝑑𝜏},𝑡∈0,𝑡₀,𝑈(𝑡)=𝑇diag𝑒^{−∫𝑡0𝑡𝜆1(𝜏)𝑑𝜏},𝑒^{−∫𝑡0𝑡𝜆2(𝜏)𝑑𝜏}𝑇⁻¹,𝑡∈0,𝑡₀, (13)(14) where 𝜆₁(𝑡) and 𝜆₂(𝑡) are two eigenvalues of 𝐴(𝑡) for each 𝑡∈(0,𝑡₀]. We also show that the solution curve of every solution of the integrable system (8) is a Jordan curve if for every pair 𝑠,𝑡∈(0,𝑡₀], 𝑠<𝑡, at least one of the statements ^𝑡_𝑠ℎ(𝜏)+𝜇2𝑔(𝜏)𝑑𝜏≠0or4𝜈𝜌−𝜇²4𝜋^𝑡_𝑠𝑔(𝜏)𝑑𝜏∉ℤ (15) holds true, where 4𝜈𝜌−𝜇²>0.

This implies the third main result of the paper.

Theorem 6. Let 𝜇²−4𝜈𝜌<0. One supposes (15) and lim_𝑡→0^𝑡0_𝑡ℎ(𝜏)+12𝜇𝑔(𝜏)𝑑𝜏=∞. (16) The zero solution of an integrable system (8) with 𝐴(𝑡) of form (12) is nonrectifiable (resp., rectifiable) attractive if and only if |ℎ(𝑡)|+𝑔(𝑡)𝑒^{−∫𝑡0𝑡(ℎ(𝜏)+(1/2)𝜇𝑔(𝜏))𝑑𝜏}∉𝐿¹0,𝑡₀resp.,∈𝐿¹0,𝑡₀. (17)

Remark 7. Under the assumptions of Theorem 6, we conclude that if 𝑔(𝑡)≡0, then the zero solution of integrable system (8) is rectifiable attractive.

The proof of the previous theorem is given in Section 4 based on Theorem 3. As an important consequence of preceding Theorem 6, we are able to study the following two-parametric model-system, which is singular at 𝑡=0: 𝑥^′(𝑡)=ℎ₀𝑡^𝑎𝑥(𝑡)−1𝑡^𝑏𝑦(𝑡),𝑦^′(𝑡)=1𝑡^𝑏𝑥(𝑡)+ℎ₀𝑡^𝑎𝑦(𝑡), (18) where 𝑡∈(0,𝑡₀].

Corollary 8. Let ℎ₀>0 and 𝑏∈ℝ. One has that:(i)if 𝑎=1, the zero solution of (18) is rectifiable attractive provided 𝑏<ℎ₀+1 and nonrectifiable attractive provided 𝑏≥ℎ₀+1;(ii)if 𝑎>1, then the zero solution of (18) is rectifiable attractive for all ℎ₀>0 and 𝑏∈ℝ.

As we see, in the case of 𝑎=1, the rectifiable attractivity of (18) depends on the order of growth of the singular term 𝑡^−𝑏 appearing in the antidiagonal coefficients of 𝐴(𝑡). The existence of solutions for the integrable systems (8) and (18) is guaranteed by their explicit forms given in Lemma 11.

3. Proofs of Theorems 1, 2, and 3

In this section, we study the existence and uniqueness of solutions to (1) as well as its attractivity of the zero solution.

In the case when the operator-valued function 𝐴=𝐴(𝑡), 𝐴∶[0,𝑡₀]→𝐿(𝕏) is defined and continuous on the whole interval [0,𝑡₀], then the existence of a classic solution 𝐱=𝐱(𝑡) is well known; see, for instance, [22, Chapter 3], [23, Chapter 2], and [24, Chapter 5.1]; when 𝐴(𝑡) are unbounded linear operators, we refer reader to [24, Chapter 5.4] and some selected papers from [25].

However, our operator-valued function 𝐴=𝐴(𝑡), 𝐴∶(0,𝑡₀]→𝐿(𝕏), is mainly singular and not defined at 𝑡=0. By this reason, we cannot use the existence results and methods obtained and used on [0,𝑡₀]. One way to understand the main difficulties because of the pressumed singular behaviour of 𝐴(𝑡) near 𝑡=0, we can make the reflexion from 𝑡=0 into 𝜏=∞ by the transformation 𝜏=1/𝑡. Then, the existence of a solution of equation 𝐱^′=𝐴(𝑡)𝐱, 𝑡∈(0,𝑡₀], 𝐱(𝑡₀)=𝐱₀, can be related with the existence of a global solution of equation 𝐱^′=𝐴(𝜏)𝐱, 𝜏∈[𝜏₀,∞), where 𝐴(𝜏)=−𝜏⁻²𝐴(𝜏⁻¹), 𝐱(𝜏)=𝐱(1/𝑡), 𝜏₀=1/𝑡₀, and 𝐱(𝑡₀)=𝐱₀. Since [𝜏₀,∞) is an infinite interval, our approach to the existence and uniqueness of solution of (1) is motivated by the works [26–29].

At first, it is clear that (1) can be rewritten in the form of corresponding integral operators equation: 𝑇𝐱=𝐱∈𝐶0,𝑡₀;𝕏, (19) where 𝑇∶𝐶((0,𝑡₀];𝕏)→𝐶((0,𝑡₀];𝕏) is defined by (𝑇𝐱)(𝑡)=𝐱𝑡₀−^𝑡0_𝑡𝐴(𝑠)𝐱(𝑠)𝑑𝑠,𝑡∈0,𝑡₀. (20) The existence and uniqueness of solution of problem (19)-(20) will be explored by the following 𝛼-contraction fixed point result in Fréchet spaces.

Definition 9. Let (𝐹,|⋅|_𝑛) be a Fréchet space. A mapping 𝑇∶𝐹→𝐹 is said to be an 𝛼-contraction on 𝐹 if for each 𝑛∈ℕ there exists 𝛼_𝑛∈[0,1) such that |𝑇𝐱−𝑇𝐲|_𝑛≤𝛼_𝑛|𝐱−𝐲|_𝑛,∀𝐱,𝐲∈𝐹. (21)

In applications, the inequality (21) need not be satisfied for the family of seminorms |⋅|_𝑛, rather only for another family of seminorms ‖⋅‖_𝑛 which is equivalent to |⋅|_𝑛. Hence, the key point of the following 𝛼-contraction principle is that an operator 𝑇 has a fixed point provided (21) is satisfied in ‖⋅‖_𝑛.

Lemma 10. Let (𝐹,|⋅|_𝑛) be a Fréchet space and 𝑇∶𝐹→𝐹 be an 𝛼-contraction on 𝐹 with respect to a family of seminorms ‖⋅‖_𝑛 equivalent to |⋅|_𝑛. Then, 𝑇 has a fixed point on 𝐹.

The previous lemma is a particular case of [29, Theorem 1.2].

Proof of Theorem 1. Let 𝐹=𝐶((0,𝑡₀];𝕏), 𝐱∈𝐹, and |𝐱|_𝑛=sup‖𝐱(𝑡)‖_𝕏1∶𝑡∈𝑛,𝑡₀. (22) It is known that |⋅|_𝑛 is a family of seminorms on 𝐹 and (𝐹,|⋅|_𝑛) is a Fréchet space.
Let 𝑇∶𝐹→𝐹 be an operator defined in (20). Since 𝐴(𝑡) may be singular at 𝑡=0, it is easy to check that in general 𝑇 is not an 𝛼-contraction with respect to the family of seminorms |𝑥|_𝑛 given in (22).
Next, let 𝜔_𝑛 and ℎ_𝑛 be two sequences of real numbers determined by 𝜔_𝑛𝐴(𝑡1=sup‖)‖∶𝑡∈𝑛,𝑡₀ℎ,𝑛∈ℕ,_𝑛>0suchthatℎ_𝑛≥𝜔_𝑛,𝑛∈ℕ. (23)(24) Let us remark that if ‖𝐴(𝑡)‖ satisfies (3), then obviously 𝜔_𝑛→∞ as 𝑛→∞. In respect to these two sequences 𝜔_𝑛 and ℎ_𝑛, we define ‖𝐱‖_𝑛=max𝑒^{−ℎ𝑛(𝑡0−𝑡)}‖𝐱(𝑡)‖_𝕏1∶𝑡∈𝑛,𝑡₀. (25) Such sequence ‖⋅‖_𝑛 is a family of seminorms ‖⋅‖_𝑛 which are equivalent with respect to |⋅|_𝑛 because 𝑒^{−ℎ𝑛(𝑡0−1/𝑛)}|𝐱|_𝑛≤‖𝐱‖_𝑛≤|𝐱|_𝑛,∀𝑛∈ℕ,∀𝐱∈𝐹. (26)
Next, according to (20) and (23), for 𝑡∈[1/𝑛,𝑡₀], we calculate ‖𝑇𝐱(𝑡)−𝑇𝐲(𝑡)‖_𝕏=^𝑡0_𝑡𝐴(𝑠)(𝐱(𝑠)−𝐲(𝑠))𝑑𝑠_𝕏≤^𝑡0_𝑡‖𝐴(𝑠)‖‖𝐱(𝑠)−𝐲(𝑠)‖_𝕏𝑑𝑠≤𝜔_𝑛^𝑡0_𝑡‖𝐱(𝑠)−𝐲(𝑠)‖_𝕏𝑑𝑠. (27) Multiplying previous inequality by 𝑒^{−ℎ𝑛(𝑡0−𝑡)}, where ℎ_𝑛 is defined in (24), we obtain 𝑒^{−ℎ𝑛(𝑡0−𝑡)}‖𝑇𝐱(𝑡)−𝑇𝐲(𝑡)‖_𝕏≤𝜔_𝑛𝑒^{−ℎ𝑛(𝑡0−𝑡)𝑡0}_𝑡‖𝐱(𝑠)−𝐲(𝑠)‖_𝕏𝑑𝑠=𝜔_𝑛^𝑡0_𝑡𝑒^{−ℎ𝑛(𝑠−𝑡)}𝑒^{−ℎ𝑛(𝑡0−𝑠)}‖𝐱(𝑠)−𝐲(𝑠)‖_𝕏𝑑𝑠≤𝜔_𝑛‖𝐱−𝐲‖_𝑛^𝑡0_𝑡𝑒^{−ℎ𝑛(𝑠−𝑡)}𝜔𝑑𝑠≤^𝑛ℎ_𝑛‖𝐱−𝐲‖_𝑛. (28) Taking the maximum in the previous inequality over the interval [1/𝑛,𝑡₀] and using (24) and (25), we conclude that ‖𝑇𝐱−𝑇𝐲‖_𝑛≤𝛼_𝑛‖𝐱−𝐲‖_𝑛,𝛼_𝑛=𝜔_𝑛ℎ_𝑛∈[0,1),∀𝑛∈ℕ,∀𝐱,𝐲∈𝐹. (29) Now, applying Lemma 10, we obtain the existence of 𝐱∈𝐹 satisfying problem (19)-(20). Since 𝐱∈𝐹=𝐶((0,𝑡₀];𝕏) and 𝐴∈𝐶((0,𝑡₀];𝐿(𝕏)), from (19)-(20) follows 𝐱∈𝐶¹((0,𝑡₀];𝕏). Hence, by differentiating (19), we conclude that 𝐱=𝐱(𝑡) is also a solution of equation (1). Thus, the existence of a solution of (1) is proved.
Now, we will show the uniqueness of solution of equation (1). Let 𝐮(𝑡) and 𝐯(𝑡) be two solutions of (1) such that 𝐮(𝑡₀)=𝐯(𝑡₀)=𝐱₀. Integrating (1) with respect to 𝐮(𝑡) and 𝐯(𝑡), we obtain 𝐮=𝑇𝐮 and 𝐯=𝑇𝐯, which yields ‖𝑇𝐮−𝑇𝐯‖_𝑛=‖𝐮−𝐯‖_𝑛. Puting this equality into (29), we conclude that ‖𝐮−𝐯‖_𝑛≤𝛼_𝑛‖𝐮−𝐮‖_𝑛,𝛼_𝑛=𝜔_𝑛ℎ_𝑛∈[0,1),∀𝑛∈ℕ. (30) On the other hand, if the solutions 𝐮(𝑡) and 𝐯(𝑡) are different, then there is 𝑡^∗∈(0,𝑡₀] such that 𝐮(𝑡^∗)≠𝐯(𝑡^∗), which implies that ‖𝐮(𝑡^∗)−𝐯(𝑡^∗)‖_𝕏>0. In particular, there is a big enough 𝑚∈ℕ such that 𝑡^∗∈[1/𝑚,𝑡₀], and, hence, ‖𝐮−𝐯‖_𝑚=max𝑒^{−ℎ𝑚(𝑡0−𝑡)}‖𝐮−𝐯‖_𝕏1∶𝑡∈𝑚,𝑡₀≥𝑒^{−ℎ𝑚(𝑡0−𝑡∗)}𝐮𝑡^∗−𝐯𝑡^∗_𝕏>0; (31) that is, ‖𝐮−𝐯‖_𝑚>0. (32) Putting (32) into (30), we obtain that 𝛼_𝑚≥1. But it is not possible since 𝛼_𝑚∈[0,1). Hence, the assumption about the existence of 𝑡^∗∈(0,𝑡₀] such that 𝐮(𝑡^∗)≠𝐯(𝑡^∗) is not possible. Therefore, we conclude that 𝐮(𝑡)=𝐯(𝑡) for all 𝑡∈(0,𝑡₀], and, thus, the uniqueness of solution of (1) is shown.

Proof of Theorem 2. It is enough to show that if the statement (3) does not hold, then ‖𝑑𝐱/𝑑𝑡‖_𝕏∈𝐿¹(0,𝑡₀) for every 𝐱₀∈𝕏 and the corresponding solution 𝐱 of (1) such that 𝐱(𝑡₀)=𝐱₀. In fact, if we suppose contrary to (3), then there exists 𝑐₀>0 such that ‖𝐴(𝑡)‖≤𝑐₀, 𝑡∈(0,𝑡₀]. Also, because of ‖𝐱(𝑡)‖_𝕏→0 as 𝑡→0, there exists 𝑐₁>0 such that ‖𝐱(𝑡)‖_𝕏≤𝑐₁, 𝑡∈(0,𝑡₀]. Hence, from (1), we obtain 𝑑𝐱𝑑𝑡_𝕏=‖𝐴(𝑡)𝐱(𝑡)‖_𝕏≤‖𝐴(𝑡)‖‖𝐱(𝑡)‖_𝕏≤𝑐₀𝑐₁,𝑡∈0,𝑡₀, (33) which shows that ‖𝑑𝐱/𝑑𝑡‖_𝕏∈𝐿¹(0,𝑡₀).

Proof of Theorem 3. The conclusion (i) immediately follows from the hypothesis (𝐻₂) because ‖𝐱(𝑡)‖_𝕏=‖𝑈(𝑡)𝐱₀‖_𝕏≤‖𝑈(𝑡)‖‖𝐱₀‖_𝕏→0 as 𝑡→0.
Next, we give the proof of the conclusion (ii). Let 𝑠,𝑡∈(0,𝑡₀], 𝑠<𝑡, be such that 𝐱₀ is an eigenvector of 𝑈⁻¹(𝑠)𝑈(𝑡) with an eigenvalue 1. Then, 𝑈⁻¹(𝑠)𝑈(𝑡)𝐱₀=𝐱₀. (34) Applying 𝑈(𝑠) to the pervious formula, we obtain the expression 𝑈(𝑡)𝐱₀=𝑈(𝑠)𝐱₀, which by definition of 𝐱(𝑡) equals to 𝐱(𝑡)=𝐱(𝑠). Hence, 𝐱=𝐱(𝑡) is not an injective function. Note that, since 𝑈(𝑡) is postulated invertible for all 𝑡∈(0,𝑡₀], all the steps of this proof are reversible. Thus, the required equivalence holds.
Finally, the conclusion (iii) follows from the hypotheses (𝐻₁)-(𝐻₂) because 𝑐₀𝑐₂𝑎(𝑡)𝜔(𝑡)𝐱_0𝕏≤𝑐₀𝜔(𝑡)𝑈(𝑡)𝐱_0𝕏=𝑐₀𝜔(𝑡)‖𝐱(𝑡)‖_𝕏≤‖𝐴(𝑡)𝐱(𝑡)‖_𝕏,‖𝐴(𝑡)𝐱(𝑡)‖_𝕏≤𝑐₁𝜔(𝑡)‖𝐱(𝑡)‖_𝕏=𝑐₁𝜔(𝑡)𝑈(𝑡)𝐱_0𝕏≤𝑐₁𝑐₃𝑎(𝑡)𝜔(𝑡)𝐱_0𝕏. (35) Hence, the required equivalence in (iii) holds.

Note that conclusion (ii) does not depend on any of the hypotheses (𝐻₁)-(𝐻₂).

4. Proofs of Theorem 6 and Corollary 8

In this section, we study the rectifiable and nonrectifiable attractivity of the linear integrable systems (8).

At the first, we state the following lemma in which a specific form of the matrix 𝐴(𝑡) of linear integrable systems is proposed.

Lemma 11. Suppose that system (8) is integrable. Then, there exist functions 𝑔,ℎ∶(0,𝑡₀]→ℝ and constants 𝜇,𝜈,𝜌∈ℝ such that 𝐴(𝑡)=ℎ(𝑡)(𝑡)(𝑡)(𝑡)ℎ(𝑡)+𝜇𝑔−𝜈𝑔𝜌𝑔,𝑡∈0,𝑡₀. (36) The eigenvalues 𝜆_1,2∶(0,𝑡₀]→ℂ of 𝐴(𝑡) are given by the formula 𝜆_1,2(𝑡)=ℎ(𝑡)+𝜇2𝑔(𝑡)±𝑔(𝑡)𝜇²4−𝜌𝜈. (37)

Proof. Let 𝜆_1,2∶(0,𝑡₀]→ℂ be two eigenvalues of the matrix-valued function 𝐴(𝑡). Since system (8) is supposed to be an integrable system, from Definition 5, there is a regular matrix 𝑇∈ℳ₂ such that 𝐴(𝑡)=𝑇diag𝜆₁(𝑡),𝜆₂(𝑡)𝑇⁻¹,𝑡∈0,𝑡₀. (38) If det(𝑇)∈ℝ and det(𝑇)<0, then we can change the columns in 𝑇 which causes det(𝑇)>0 and diag[𝜆₂(𝑡),𝜆₁(𝑡)] instead of diag[𝜆₁(𝑡),𝜆₂(𝑡)]. Also, if det(𝑇)∈ℝ and det(𝑇)>0 or det(𝑇)∈ℂ⧵ℝ, then 𝐴(𝑡)=𝑇₀diag𝜆₂(𝑡),𝜆₁(𝑡)𝑇₀⁻¹,det𝑇₀=1,where𝑇₀=1det(𝑇)𝑇. (39) Hence, we may suppose that det(𝑇)=1; that is, for some complex numbers 𝛼, 𝛽, 𝛾, 𝛿 with 𝛼𝛿−𝛽𝛾=1, we may set 𝑇=𝛼𝛽𝛾𝛿,𝑇⁻¹=𝛿−𝛽−𝛾𝛼. (40) Now from (38) and (40), we obtain 𝐴(𝑡)=𝛼𝛿𝜆₁(𝑡)−𝛽𝛾𝜆₂(𝑡)𝛼𝛽𝜆₂(𝑡)−𝜆₁(𝑡)𝛾𝛿𝜆₁(𝑡)−𝜆₂(𝑡)𝛼𝛿𝜆₂(𝑡)−𝛽𝛾𝜆₁(𝑡), (41) where the matrix elements of 𝐴(𝑡) are real numbers for all 𝑡∈(0,𝑡₀].
We denote that ℎ(𝑡)∶=𝛼𝛿𝜆₂(𝑡)−𝛽𝛾𝜆₁(𝑡),𝑡∈0,𝑡₀, (42) and if 𝜆_1,2(𝑡)∈ℝ for all 𝑡∈(0,𝑡₀], then let 𝑔(𝑡)∶=𝜆₁(𝑡)−𝜆₂(𝑡),,𝜌∶=𝛾𝛿,𝜈∶=𝛼𝛽,𝜇∶=−𝛼𝛿+𝛾𝛽 (43) and if 𝜆_1,2(𝑡) form a complex-conjugate pair for all 𝑡∈(0,𝑡₀], then let 𝑔(𝑡)∶=−𝑖𝜆₁(𝑡)−𝜆₂(𝑡).,𝜌∶=𝑖𝛾𝛿,𝜈∶=𝑖𝛼𝛽,𝜇∶=−𝑖𝛼𝛿+𝛾𝛽 (44)
Putting previous notations in (41), we get the desired conclusion (36). Note that all of these functions and constants are real, since ℎ is defined as a matrix-element of a real matrix, 𝑔 is defined real in both cases, and the rest of the constants are necessarily real by virtue of satisfying (36). Now, the conclusion (37) immediately follows from (36).

By Theorem 1, we know that there exists a unique solution 𝐱(𝑡) of system (8). Moreover, in the next lemma, we show that if (8) is a linear integrable system, then the solution 𝐱(𝑡) is explicitly expressed by the matrix elements of 𝐴(𝑡).

Lemma 12. Let 𝐱 be a solution of linear integrable system (8) and let 𝑇∈ℳ₂ be a matrix determined in Definition 5. Then, there exists a matrix-valued function 𝑈=𝑈(𝑡), 𝑈∶(0,𝑡₀]→ℳ₂, such that 𝐱(𝑡)=𝑈(𝑡)𝐱₀ and satisfies the following properties: (1)𝑈(𝑡₀) is the identity matrix;(2)𝑈(𝑡) is invertible for every 𝑡∈(0,𝑡₀];(3)𝑇⁻¹𝑈(𝑡)𝑇 is a diagonal matrix for every 𝑡∈(0,𝑡₀];(4)if 𝜆(𝑡) is an eigenvalue of 𝐴(𝑡), then 𝑒^{−∫𝑡0𝑡𝜆(𝜏)𝑑𝜏} is an eigenvalue of 𝑈(𝑡);(5)𝑈^′(𝑡)=𝐴(𝑡)𝑈(𝑡) for every 𝑡∈(0,𝑡₀].

Proof. Let 𝑇 be the matrix that diagonalizes 𝐴(𝑡). Let Λ_𝐴(𝑡)=𝑇⁻¹𝐴(𝑡)𝑇 be of form Λ_𝐴(𝑡)=𝜆₁(𝑡)00𝜆₂(𝑡). (45) We define Λ_𝑈(𝑡)=𝑒^{−∫𝑡0𝑡𝜆1(𝜏)𝑑𝜏}00𝑒^{−∫𝑡0𝑡𝜆2(𝜏)𝑑𝜏},𝑈(𝑡)=𝑇Λ_𝑈(𝑡)𝑇⁻¹. (46) Now, Λ_𝑈(𝑡₀) is obviously the identity matrix, and thus 𝑈(𝑡₀)=𝑇𝐼𝑇⁻¹=𝐼. Since Λ_𝑈(𝑡) is invertible for all 𝑡∈(0,𝑡₀], so is 𝑈(𝑡). Next, 𝑇⁻¹𝑈(𝑡)𝑇=Λ_𝑈(𝑡) is diagonal by construction. For an eigenvalue 𝜆(𝑡) of 𝐴(𝑡), by construction, exp{−∫^𝑡0_𝑡𝜆(𝜏)𝑑𝜏} is a diagonal element of Λ_𝑈(𝑡); so, it is an eigenvalue of 𝑈(𝑡). Also, 𝑈^′(𝑡)=𝑇Λ^′_𝑈(𝑡)𝑇⁻¹=𝑇Λ_𝐴(𝑡)Λ_𝑈(𝑡)𝑇⁻¹=𝑇Λ_𝐴(𝑡)𝑇⁻¹𝑇Λ_𝑈(𝑡)𝑇⁻¹=𝐴(𝑡)𝑈(𝑡). (47) And finally, the function 𝐱(𝑡)=𝑈(𝑡)𝐱₀ solves the integrable system (8) because 𝐱^′(𝑡)=𝑈^′(𝑡)𝐱₀=𝐴(𝑡)𝑈(𝑡)𝐱₀=𝐴(𝑡)𝐱(𝑡) (48) and has the property 𝐱(𝑡₀)=𝑈(𝑡₀)𝐱₀=𝐱₀.

According to [23, p. 188], the matrix-valued function 𝑈(𝑡) is called the evolution operator.

In the next lemma, we give some necessary and sufficient conditions such that the solution's curve Γ_𝐱 of every solution 𝐱(𝑡) of integrable system (8) is a Jordan curve. Results of that kind are important because Definition 4 requires the Jordan property.

Lemma 13. Let (8) be a linear integrable system such that 𝜇²−4𝜈𝜌<0 and 𝑔(𝑡)≠0, 𝑡∈(0,𝑡₀], where the real numbers 𝜇, 𝜈, 𝜌, and 𝑔(𝑡) are determined by Lemma 11. Then, the solution's curve Γ_𝐱 of every solution 𝐱(𝑡) is a Jordan curve in ℝ² if and only if for every pair 𝑠,𝑡∈(0,𝑡₀], 𝑠<𝑡, at least one of the statements in (15) holds true.

Proof. In order to prove this lemma, we use the equivalence stated in the conclusion (ii) of Theorem 3 (we may use Theorem 3 because of Lemma 12; see also the remark at the end of Theorem 3). According to that, we need to find pairs 𝑠 and 𝑡 such that 1 is an eigenvalue of 𝑈⁻¹(𝑠)𝑈(𝑡). Since the evolution operator 𝑈(𝑡) can be diagonalized into the form Λ_𝑈(𝑡), we compute 𝑈⁻¹(𝑠)𝑈(𝑡)=𝑇Λ_𝑈(𝑠)𝑇⁻¹⁻¹𝑇Λ_𝑈(𝑡)𝑇⁻¹=𝑇Λ_𝑈⁻¹(𝑠)Λ_𝑈(𝑡)𝑇⁻¹. (49) The diagonalized form of 𝑈⁻¹(𝑠)𝑈(𝑡) is then given by the formula Λ_𝑈⁻¹(𝑠)Λ_𝑈(𝑡)=𝑒^{∫𝑡0𝑠𝜆1(𝜏)𝑑𝜏}00𝑒^{∫𝑡0𝑠𝜆2(𝜏)𝑑𝜏}𝑒^{−∫𝑡0𝑡𝜆1(𝜏)𝑑𝜏}00𝑒^{−∫𝑡0𝑡𝜆2(𝜏)𝑑𝜏}=𝑒^{−∫𝑠𝑡𝜆1(𝜏)𝑑𝜏}00𝑒^{−∫𝑠𝑡𝜆2(𝜏)𝑑𝜏}. (50) Now, the eigenvalues of 𝑈⁻¹(𝑠)𝑈(𝑡) are 𝑒^{−∫𝑠𝑡𝜆1(𝜏)𝑑𝜏} and 𝑒^{−∫𝑠𝑡𝜆2(𝜏)𝑑𝜏}. Since 𝜆₂=𝜆₁, we only need to solve the equation 𝑒^{−∫𝑠𝑡𝜆1(𝜏)𝑑𝜏}=𝑒^{−∫𝑠𝑡(ℎ(𝜏)+(𝜇/2)𝑔(𝜏)𝑑𝜏)}1cos24𝜈𝜌−𝜇^2𝑠_𝑡𝑔(𝜏)1𝑑𝜏+𝑖sin24𝜈𝜌−𝜇²×^𝑠_𝑡𝑔(𝜏)𝑑𝜏=1. (51) This equation is solved whenever ^𝑡_𝑠ℎ(𝜏)+𝜇2𝑔(𝜏)1𝑑𝜏=0,24𝜈𝜌−𝜇^2𝑡_𝑠𝑔(𝜏)𝑑𝜏=2𝑘𝜋,forsome𝑘∈ℤ, (52) where the order of integration has been reversed by multiplying by −1.
Since by this argument, 𝑈⁻¹(𝑠)𝑈(𝑡)=𝐼 if and only if 1 is an eigenvalue of 𝑈⁻¹(𝑠)𝑈(𝑡), we get the full equivalence.

Proof of Theorem 6. The key point of this proof is to show that the integrable system (8) satisfies the required hypotheses (𝐻₁)-(𝐻₂), because all conclusions of Theorem 6 follow immediately from the conclusions (i), (ii), and (iii) of Theorem 3. Before we show that, we state the following two propositions which will be proved in Section 5.

Proposition 14. Let 𝑓∶ℝ×ℝ→ℝ be defined by 𝑓(𝑥,𝑦)=𝑥²+𝜇𝑥𝑦+𝜌𝜈𝑦², where 𝜇, 𝜈, 𝜌 are real numbers such that 𝜇²<4𝜈𝜌. Then, there exist constants 𝑐₁,𝑐₂>0 such that 𝑐₁|𝑥|+𝑦≤𝑓𝑥,𝑦≤𝑐₂|𝑥|+𝑦,∀𝑥,𝑦∈ℝ. (53)

Proposition 15. Let Λ∶(0,𝑡₀]→ℳ₂ be a diagonal matrix of the form Λ(𝑡)=𝜆(𝑡)00𝜆(𝑡), (54) for all 𝑡∈(0,𝑡₀]. Then, detΛ(𝑡)≥0, 𝑡∈(0,𝑡₀] and Λ(𝑡)^→𝑧=|𝜆(𝑡)|^→𝑧=detΛ(𝑡)^→𝑧∀𝑡∈0,𝑡₀,^→𝑧∈ℂ². (55)

Now we proceed with the proof of Theorem 6. By definition, 𝐴(𝑡)=𝑇⁻¹Λ_𝐴(𝑡)𝑇, where 𝑇∈ℳ₂ is an invertible matrix and Λ_𝐴∶(0,𝑡₀]→ℳ₂ is a matrix-valued function given in (45). Since 𝑇 is invertible, there exist constants 𝑐₁,𝑐₂>0 such that 𝑐₁^→𝑧≤𝑇^→𝑧≤𝑐₂^→𝑧,∀^→𝑧∈ℂ²,1𝑐₂^→𝑤≤𝑇^→−1𝑤≤1𝑐₁^→𝑤,∀𝑤^→∈ℂ². (56)

Next, from (36), we get detΛ_𝐴(𝑡)=(𝑡)=det𝐴ℎ²(𝑡)+𝜇𝑔(𝑡)ℎ(𝑡)+𝜌𝜈𝑔²(𝑡)=𝑓ℎ(𝑡),𝑔(𝑡),𝑡∈0,𝑡₀, (57) where 𝑓(𝑥,𝑦) is defined in Proposition 14. Hence, by (57) and Propositions 14 and 15 applied to Λ(𝑡)=Λ_𝐴(𝑡), we get two constants 𝑐₃,𝑐₄>0 such that 𝑐₃𝜔(𝑡)^→𝑧≤Λ_𝐴(𝑡)^→𝑧≤𝑐₄𝜔(𝑡)^→𝑧,𝑡∈0,𝑡₀,^→𝑧∈ℂ², (58) where 𝜔(𝑡)=|ℎ(𝑡)|+|𝑔(𝑡)|. Now from (56) and (58), for all 𝑡∈(0,𝑡₀] and ^→𝑥∈ℝ², we have 𝑐₁𝑐₃𝑐₂𝜔(𝑡)^→𝑥≤𝑐₃𝑐₂𝜔(𝑡)𝑇^→𝑥≤1𝑐₂Λ_𝐴(𝑡)𝑇^→𝑥≤𝑇⁻¹Λ_𝐴(𝑡)𝑇^→𝑥=𝐴(𝑡)^→𝑥,𝐴(𝑡)^→𝑥=𝑇⁻¹Λ_𝐴(𝑡)𝑇^→𝑥≤1𝑐₁Λ_𝐴(𝑡)𝑇^→𝑥≤𝑐₄𝑐₁𝜔(𝑡)𝑇^→𝑥≤𝑐₄𝑐₂𝑐₁𝜔(𝑡)^→𝑥, (59) which shows that the hypothesis (𝐻₁) is fulfilled with respect to 𝜔(𝑡) given in (13).

Next, let the evolution operator 𝑈(𝑡) and matrix-valued function Λ_𝑈∶(0,𝑡₀]→ℳ₂ be defined by (46). The required condition (𝐻₂) is satisfied because of the following arguments. First, from Lemma 12 and (37), we have detΛ_𝑈(𝑡)=det𝑈(𝑡)=𝑒^{−∫𝑡0𝑡(ℎ(𝜏)+(𝜇/2)𝑔(𝜏))𝑑𝜏}=∶𝑎(𝑡),𝑡∈0,𝑡₀. (60) Hence, by Proposition 15 applied to Λ(𝑡)=Λ_𝑈(𝑡), we obtain Λ_𝑈(𝑡)^→𝑧=𝑎(𝑡)^→𝑧,𝑡∈0,𝑡₀,^→𝑧∈ℂ². (61) Now from (56) and (61), for all 𝑡∈(0,𝑡₀] and 𝑥∈ℝ², we have 𝑐₁𝑐₂𝑎(𝑡)^→𝑥≤1𝑐₂𝑎(𝑡)𝑇^→𝑥=1𝑐₂Λ_𝑈(𝑡)𝑇^→𝑥≤𝑇⁻¹Λ_𝑈(𝑡)𝑇^→𝑥=𝑈(𝑡)^→𝑥,𝑈(𝑡)^→𝑥=𝑇⁻¹Λ_𝑈(𝑡)𝑇^→𝑥≤1𝑐₁Λ_𝑈(𝑡)𝑇^→𝑥=1𝑐₁𝑎(𝑡)𝑇^→𝑥≤𝑐₂𝑐₁𝑎(𝑡)^→𝑥, (62) which shows that 𝑈(𝑡) satisfied the required inequalities in (𝐻₂) with respect to 𝑎(𝑡) given in (13).

Finally, by Lemma 13, we obtain that the solution's curve Γ_𝐱 of every solution 𝐱 is a Jordan curve. Thus, all assumptions of Theorem 3 are fulfilled, and, therefore, we may apply Theorem 3 here. Consequently, the proof of this theorem is complete.

Proof of Corollary 8. It is enough to show that all assumptions of Theorem 6: 𝜇²−4𝜈𝜌<0, (15), (16), and (17) are fulfilled in particular for ℎ(𝑡)=ℎ₀𝑡^−𝑎,𝑔(𝑡)=𝑡^−𝑏,𝜇=0,𝑣=𝜌=1, (63) where ℎ₀>0, 𝑎≥1, and 𝑏∈ℝ. In fact, the following elementary calculation is shown:𝜇²−4𝜈𝜌=−4<0,_𝑡𝑠ℎ(𝜏)+𝜇2𝑔(𝜏)𝑑𝜏=^𝑡_𝑠ℎ(𝜏)𝑑𝜏=ℎ₀^𝑡_𝑠𝜏^−𝑎𝑑𝜏≠0,𝑠≠𝑡,lim_𝑡→0^𝑡0_𝑡ℎ(𝜏)+12𝜇𝑔(𝜏)𝑑𝜏=ℎ₀lim_𝑡→0^𝑡0_𝑡𝜏^−𝑎𝑑𝜏=∞. (64) Moreover, for 𝑎=1, |ℎ(𝑡)|+𝑔(𝑡)𝑒^{−∫𝑡0𝑡(ℎ(𝜏)+(1/2)𝜇𝑔(𝜏))𝑑𝜏}=𝑡^−ℎ0₀ℎ₀𝑡^−1+ℎ0+𝑡^−𝑏+ℎ0∉𝐿¹0,𝑡₀; (65) so, condition (17) is fulfilled if and only if 𝑏≥ℎ₀+1. Also, for 𝑎>1, |ℎ(𝑡)|+𝑔(𝑡)𝑒^{−∫𝑡0𝑡(ℎ(𝜏)+(1/2)𝜇𝑔(𝜏))𝑑𝜏}=ℎ₀𝑡^−𝑎+𝑡^−𝑏𝑒^{−(ℎ0/(𝑎−1))(𝑡1−𝑎−𝑡01−𝑎)}∈𝐿¹0,𝑡₀. (66) Therefore, we may apply Theorem 6 to system (18) which proves this corollary.

5. Appendix

In this section, we first state a result saying that the set of all matrix-valued functions 𝐴∶(0,𝑡₀]→ℳ₂ which satisfy Definition 5 forms an algebra. In this direction, let 𝒜((0,𝑡₀];ℳ₂) be a set defined by 𝒜0,𝑡₀;ℳ₂=𝐴∶0,𝑡₀⟶ℳ₂∣∃𝑇∈ℳ₂,det𝑇≠0,suchthat𝑇⁻¹𝐴(𝑡)𝑇isdiagonal,∀𝑡∈0,𝑡₀. (67)

Theorem 16. The set 𝒜((0,𝑡₀];ℳ₂) is an algebra with respect to the classic matrix operations + and ∘; that is to say, if 𝐴,𝐵∈𝒜((0,𝑡₀];ℳ₂), then the following properties hold: (1)𝛼𝐴+𝛽𝐵∈𝒜((0,𝑡₀];ℳ₂) for all 𝛼,𝛽∈ℝ; (2)𝐴∘𝐵∈𝒜((0,𝑡₀];ℳ₂); (3)if det𝐴(𝑡)≠0 for all 𝑡∈(0,𝑡₀], then 𝐴⁻¹∈𝒜((0,𝑡₀];ℳ₂);(4)if 𝑓∶ℝ→ℝ is a real analytic function such that 𝑓(𝐴(𝑡)) converges absolutely for all 𝑡∈𝐽, then 𝑓(𝐴)∈𝒜((0,𝑡₀];ℳ₂).

Proof. We denote the diagonal matrices Λ_𝐴=𝑇𝐴𝑇¹ and Λ_𝐵=𝑇𝐵𝑇¹. (1)𝑇(𝛼𝐴+𝛽𝐵)𝑇⁻¹=𝛼𝑇𝐴𝑇⁻¹+𝛽𝑇𝐵𝑇⁻¹=𝛼Λ_𝐴+𝛽Λ_𝐵, which is diagonal. (2)𝑇𝐴𝐵𝑇⁻¹=𝑇𝐴𝑇⁻¹𝑇𝐵𝑇⁻¹=Λ_𝐴Λ_𝐵, which is diagonal. (3)(𝑇𝐴⁻¹𝑇⁻¹)=(𝑇𝐴𝑇⁻¹)⁻¹=Λ_𝐴⁻¹, which is also diagonal. (4)𝑇𝑓(𝐴)𝑇⁻¹=𝑇(∑_𝑛𝑎_𝑛𝐴^𝑛)𝑇⁻¹=∑_𝑛𝑎_𝑛𝑇𝐴^𝑛𝑇⁻¹=∑_𝑛𝑎_𝑛Λ^𝑛_𝐴, which is also diagonal.

Next, we give the proofs of some technical results used in previous sections for proving the main results.

Proof of Proposition 14. Denote by 𝑢∈ℝ² the vector 𝑢=(𝑥,𝑦). It is enough to show that 𝑓=𝑓(𝑢) is a norm as a function 𝑓∶ℝ²→ℝ, since all the norms in ℝ² are equivalent [30, p. 38], which, by definition, is the claim of this lemma. Let 1𝜇𝑀=2𝜇2𝜌𝜈. (68) By direct computation, we see that 𝑓(𝑢)=(𝑀𝑢,𝑢). Since 𝑀 is symmetric and positive definite, it has a unique symmetric positive definite square root 𝑁²=𝑀, so that 𝑓(𝑢)=(𝑁𝑢,𝑁𝑢) [31, p. 231]. Hence, 𝑓(𝑢)=‖𝑣‖_ℝ², where 𝑣=𝑁𝑢; so, the function 𝑓=𝑓(𝑢) is a norm.

Proof of Proposition 15. Let 𝜙(𝑡)∶(0,𝑡₀]→ℝ be such that 𝜆(𝑡)=|𝜆(𝑡)|𝑒^{𝑖𝜙(𝑡)}. Then, 𝜆(𝑡)=|𝜆(𝑡)|𝑒^{−𝑖𝜙(𝑡)}; so, for any 𝑧=(𝑧₁,𝑧₂)∈ℂ², we have ‖Λ(𝑡)𝑧‖=|𝜆(𝑡)|𝑒^{𝑖𝜙(𝑡)}𝑧₁,𝑒^{−𝑖𝜙(𝑡)}𝑧₂=|𝜆(𝑡)|𝑒^{𝑖𝜙(𝑡)}𝑧₁²+𝑒^{−𝑖𝜙(𝑡)}𝑧₂²=|𝜆(𝑡)|‖𝑧‖, (69) since |𝑒^±𝑖𝜙|=1. The remaining equality is true because |𝜆(𝑡)|=𝜆(𝑡)𝜆(𝑡)=detΛ(𝑡). (70)

6. Final Remarks

In our approach to the model over ℝ², we have chosen to emphasize the a priori structure of the real matrices describing the integrable systems. Thus, we have stated and proved this theory in terms of such real functions, complexifying only when absolutely necessary.

We could have taken another approach, complexifying from the start and then easily diagonalizing all the relevant matrices, but this would not essentially simplify any argument except in notation, while we would run the risk of confusing the reader even further about which term or function is real and which one is complex.