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

and Applied Analysis 3 when t → ∞, we refer reader to [7–10, 12–14] and references therein. We know that a plain curve Γx is a Jordan curve in R if there exists a continuous injection x = x(t), x : [0, t 0 ] → R, t 0 > 0, such that x([0, t 0 ]) = Γx. It is often said that Γx is parametrized by x(t) or that Γx is associated to x(t).The length of Γx, denoted by length (Γx), is defined by length (Γx) = sup m


Introduction
Let  0 > 0 and X a given real or complex Banach space with the zero 0 ∈ X, and let (X) denote the space of all linear bounded operators from X into X.We consider the linear nonautonomous differential equation in X as x  =  () x,  ∈ (0, 0 ] , x where x = x(), x ∈  1 ((0,  0 ]; X),  = () is an operatorvalued function defined on interval (0,  0 ] with the value in (X) and  ∈ ((0,  0 ]; (X)).If X = R  , 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 -contraction mapping.Next, in Theorems 2 and 3, we give a necessary and sufficient condition for the nonintegrability of ‖x  ()‖ X on (0,  0 ] provided ‖()‖ blows up near  = 0 and ‖()‖ allows a precise asymptotic behaviour near  = 0. Based on these results, in the case when X = R 2 , we study the so-called nonrectifiable attractivity of the zero solution (see Definition 4) of the so-called linear integrable twodimensional system (see Definition 5) as follows: with , , and ] some real constants.Precisely, as a kind of singular behaviour near  = 0 of all solutions () of linear integrable system (2), in Theorem 6, we involve on the matrix elements ℎ() and () a necessary and sufficient condition for the infiniteness of the length of every solution curve of () (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 () near  = 0 of integrable differential system (2) presented in Lemmas 11 and 12.Of course, instead of interval (0,  0 ] and  → 0, we can also state our main results on [ 0 , ∞) and  → ∞.
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][2][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  ≥  0 .
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  = ∞ of oscillatory solutions of the nonlinear second-order differential equation (()  )  = ()() via the asymptotic formula for solutions of an auxiliary linear differential system having elements which are absolutely continuous functions on [ 0 , ∞).In [6], authors derive a precise asymptotic behaviour near  = ∞ of solutions () of the third-order nonlinear differential equation ), 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 (  )  +  = (, ,   ) depending on the asymptotic behaviour of fundamental solutions of the corresponding homogeneous equation (  )  +  = 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][13][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.

Statement of the Main Results
Preliminarily, we state the following auxiliary result.Theorem 1.For each x 0 ∈ X, there exists a unique solution x of (1).
Next, we derive a necessary condition for a kind of singularities of (1) at  = 0 in terms of singular behaviour of the operator norm of () near  = 0. Theorem 2. Let there exist a solution x of (1) such that ‖x()‖ X → 0 as  → 0. If ‖x/‖ X ∉  1 (0,  0 ), then one has: Thus, if (3) is not satisfied, then the integrability of ‖x/‖ 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.
The meaning and importance of hypotheses ( 1 )-( 2 ) as well as the conclusions (i)-(iii) of Theorem 3 will be verified by Theorem 6, where X = R 2 and (1) is a large class of twodimensional 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 where  0 > 0 and the prime denotes /, x = x() = ((), ()) and the matrix-valued function  = (),  : (0,  0 ] → M 2 (R) (where M 2 (R) denotes the space of real 2 × 2 matrices), is continuous on (0,  0 ].Therefore, we may apply previous theorems to system (8).As in the general case of X, the zero solution x() ≡ 0 ∈ R 2 of system ( 8) is said to be (global) attractive if for every solution x() of (8), we have ‖x()‖ R 2 → 0 as  → 0. On the asymptotic stability and attractivity for several kinds of linear systems in the case when  → ∞, we refer reader to [7][8][9][10][12][13][14] and references therein.
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 R 2 .
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): On the rectifiability of graph of solutions of scalar secondorder linear differential equations, we refer reader to [20].
Here the particular attention is paid to the case of the socalled linear integrable systems defined as follows.
Definition 5. Let M 2 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  ∈ M 2 such that for every  ∈ (0,  0 ] the matrix Λ() =  −1 () ∈ M 2 is a diagonal matrix for every  ∈ (0,  0 ]. Note that the matrices () 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 () by when we have () ≡ ℎ() and () ≡ −(), we speak of integrable systems (see for instance [21]).In such a case, both eigenvalues () = ℎ() ± () of the matrix () admit eigenvectors (±, 1) that do not depend on variable .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: where , ],  ∈ R. For  = 0 and ] =  = 1 in (8), we have () ≡ ℎ() and () ≡ −(); so, the classic integrable system is a special case.
In Section 5, we show that the set of all matrix-valued functions  = () 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 () satisfies the required hypotheses ( 1 )-( 2 ) in particular for where  1 () and  2 () are two eigenvalues of () for each  ∈ (0,  0 ].We also show that the solution curve of every solution of the integrable system ( 8) is a Jordan curve if for every pair ,  ∈ (0,  0 ],  < , at least one of the statements holds true, where 4] −  2 > 0. This implies the third main result of the paper. Theorem 6.Let  2 − 4] < 0. One supposes (15) and The zero solution of an integrable system (8) with () of form (12) is nonrectifiable (resp., rectifiable) attractive if and only if Remark 7.Under the assumptions of Theorem 6, we conclude that if () ≡ 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 twoparametric model-system, which is singular at  = 0: where  ∈ (0,  0 ].
As we see, in the case of  = 1, the rectifiable attractivity of (18) depends on the order of growth of the singular term  − appearing in the antidiagonal coefficients of ().The existence of solutions for the integrable systems ( 8) and ( 18) is guaranteed by their explicit forms given in Lemma 11.

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.
At first, it is clear that (1) can be rewritten in the form of corresponding integral operators equation: where  : ((0,  0 ]; X) → ((0,  0 ]; X) is defined by The existence and uniqueness of solution of problem ( 19)-( 20) will be explored by the following -contraction fixed point result in Fréchet spaces.
In applications, the inequality (21) need not be satisfied for the family of seminorms | ⋅ |  , rather only for another family of seminorms ‖ ⋅ ‖  which is equivalent to | ⋅ |  .Hence, the key point of the following -contraction principle is that an operator  has a fixed point provided (21) is satisfied in ‖ ⋅ ‖  .Proof of Theorem 1.Let  = ((0,  0 ]; X), x ∈ , and It is known that |⋅|  is a family of seminorms on  and (, |⋅|  ) is a Fréchet space.
Let  :  →  be an operator defined in (20).Since () may be singular at  = 0, it is easy to check that in general  is not an -contraction with respect to the family of seminorms ||  given in (22).
Note that, since () is postulated invertible for all  ∈ (0,  0 ], all the steps of this proof are reversible.Thus, the required equivalence holds.Finally, the conclusion (iii) follows from the hypotheses Hence, the required equivalence in (iii) holds.
Note that conclusion (ii) does not depend on any of the hypotheses ( 1 )-( 2 ).

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 () of linear integrable systems is proposed.Lemma 11.Suppose that system (8) is integrable.Then, there exist functions , ℎ : (0,  0 ] → R and constants , ],  ∈ R such that The eigenvalues  1,2 : (0,  0 ] → C of () are given by the formula Proof.Let  1,2 : (0,  0 ] → C be two eigenvalues of the matrix-valued function ().Since system (8) is supposed to be an integrable system, from Definition 5, there is a regular matrix  ∈ M 2 such that If det() ∈ R and det() < 0, then we can change the columns in  which causes det() > 0 and diag[ 2 (),  1 ()] Hence, we may suppose that det() = 1; that is, for some complex numbers , , ,  with  −  = 1, we may set Now from ( 38) and (40), we obtain where the matrix elements of () are real numbers for all  ∈ (0,  0 ].We denote that and if  1,2 () ∈ R for all  ∈ (0,  0 ], then let 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,  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).
According to [23, p. 188], the matrix-valued function () 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() of integrable system (8) is a Jordan curve.Results of that kind are important because Definition 4 requires the Jordan property.
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  and  such that 1 is an eigenvalue of  −1 ()().Since the evolution operator () can be diagonalized into the form Λ  (), we compute Abstract and Applied Analysis 7 The diagonalized form of  −1 ()() is then given by the formula Now, the eigenvalues of  −1 ()() are  − ∫    1 () and  − ∫    2 () .Since  2 =  1 , we only need to solve the equation This equation is solved whenever where the order of integration has been reversed by multiplying by −1.
Since by this argument,  −1 ()() =  if and only if 1 is an eigenvalue of  −1 ()(), 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 ( 1 )-( 2 ), 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.
which shows that the hypothesis ( 1 ) is fulfilled with respect to () given in (13).

Final Remarks
In our approach to the model over R 2 , 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.

Lemma 10 .
Let (, | ⋅ |  ) be a Fréchet space and  :  →  be an -contraction on  with respect to a family of seminorms ‖ ⋅ ‖  equivalent to | ⋅ |  .Then,  has a fixed point on .The previous lemma is a particular case of [29, Theorem 1.2].