Instability of a Fifth-Order Nonlinear Vector Delay Differential Equation with Multiple Deviating Arguments

Abstract

We study a fifth-order nonlinear vector delay differential equation with multiple deviating arguments. Some criteria for guaranteeing the instability of zero solution of the equation are given by using the Lyapunov-Krasovskii functional approach. Comparing with the previous literature, our result is new and complements some known results.

1. Introduction

Lyapunov functions and functionals have been successfully used and are still being used to obtain stability, instability, boundedness, and the existence of periodic solutions of differential equations, differential equations with functional delays, and functional differential equations [1–18]).

It should be noted that in 2000, using the Lyapunov function approach, Li and Duan [2] discussed the instability of the solutions of the fifth-order nonlinear scalar differential equation without delay:𝑥⁽⁵⁾(𝑡)+𝑓₅𝑥(𝑡),𝑥′(𝑡),…,𝑥⁽⁴⁾(𝑡)𝑥⁽⁴⁾(𝑡)+𝑓₄𝑥(𝑡),𝑥′(𝑡),...,𝑥⁽⁴⁾(𝑡)𝑥⁽³⁾(𝑡)+𝑓₃𝑥′′(𝑡)+𝑓₂𝑥′(𝑡)+𝑎₁𝑥(𝑡)=0. (1)

Later, in 2011, using the same technique, Tunç [12] studied the instability of the solutions for the fifth-order nonlinear scalar delay differential equation:𝑥⁽⁵⁾(𝑡)+𝑓₅𝑥(𝑡−𝑟),𝑥′(𝑡−𝑟),…,𝑥⁽⁴⁾(𝑡−𝑟)𝑥⁽⁴⁾(𝑡)+𝑓₄𝑥(𝑡−𝑟),𝑥′(𝑡−𝑟),…,𝑥⁽⁴⁾(𝑡−𝑟)𝑥⁽³⁾(𝑡)+𝑓₃𝑥′′(𝑡−𝑟)+𝑓₂𝑥′(𝑡)+𝑎₁𝑥(𝑡)=0. (2)

In this paper, we consider the fifth-order nonlinear vector delay differential equation with multiple-deviating arguments: 𝑋⁽⁵⁾(𝑡)+𝐹𝑋(𝑡),𝑋𝑡−𝜏₁,…,𝑋𝑡−𝜏_𝑛𝑋,…,₍₄₎(𝑡),…,𝑋⁽⁴⁾𝑡−𝜏_𝑛𝑋⁽⁴⁾(𝑡)+𝐺𝑋(𝑡),𝑋𝑡−𝜏₁,…,𝑋𝑡−𝜏_𝑛𝑋,…,₍₄₎(𝑡),…,𝑋⁽⁴⁾𝑡−𝜏_𝑛𝑋⁽³⁾(𝑡)+^𝑛_𝑖=1𝐻_𝑖𝑋′′𝑡−𝜏_𝑖+Ψ𝑋′(𝑡)+𝐴𝑋(𝑡)=0, (3) where the primes in (3) denote differentiation with respect to 𝑡, 𝑡∈ℜ₊, ℜ₊=[0,∞); 𝑋∈ℜ^𝑛, 𝜏_𝑖 are certain positive constants, the fixed delays, 𝑡−𝜏_𝑖>0,𝐴 is a constant 𝑛×𝑛-symmetric matrix, 𝐹 and 𝐺 are continuous 𝑛×𝑛-symmetric matrix functions for the respective arguments, 𝐻_𝑖∶ℜ^𝑛→ℜ^𝑛,Ψ∶ℜ^𝑛→ℜ^𝑛 with 𝐻_𝑖(0)=Ψ(0)=0 are continuous for the respective arguments. Let 𝐽_𝐻𝑖(𝑋′′) and 𝐽_Ψ(𝑋′) denote the linear operators from 𝐻_𝑖(𝑋′′) and Ψ(𝑋′) to 𝐽_𝐻1𝑋′′=𝜕ℎ_1𝑖𝜕𝑥_𝑗^′′,…,𝐽_𝐻𝑛𝑋′′=𝜕ℎ_𝑛𝑖𝜕𝑥_𝑗^′′,𝐽_Ψ𝑋^′=𝜕𝜓_𝑖𝜕𝑥^′_𝑗,𝑖,𝑗=1,2,…,𝑛, (4) where (𝑥^′₁,…,𝑥^′_𝑛), (𝑥₁^′′,…,𝑥_𝑛^′′), (ℎ_1𝑖),…,(ℎ_𝑛𝑖), and (𝜓₁,…,𝜓_𝑛) are the components of 𝑋^′, 𝑋^′′, 𝐻_𝑖, and Ψ, respectively. In what follows, it is assumed that 𝐽_𝐻𝑖(𝑋′′) and 𝐽_Ψ(𝑋′) exist and are symmetric and continuous.

Equation (3) is the vector version for systems of real nonlinear differential equations of the fifth order: 𝑥_𝑖⁽⁵⁾+^𝑛_𝑘=1𝑓_𝑖𝑘𝑥,𝑥𝑡−𝜏₁,…,𝑥𝑡−𝜏_𝑛𝑥,…,₍₄₎𝑡−𝜏₁,…,𝑥⁽⁴⁾𝑡−𝜏_𝑛𝑥_𝑘⁽⁴⁾+^𝑛_𝑘=1𝑔_𝑖𝑘𝑥,𝑥𝑡−𝜏₁,…,𝑥𝑡−𝜏_𝑛𝑥,…,₍₄₎𝑡−𝜏₁,…,𝑥⁽⁴⁾𝑡−𝜏_𝑛𝑥_𝑘⁽³⁾+ℎ_1𝑖𝑥₁^′′𝑡−𝜏₁,…,𝑥_𝑛^′′𝑡−𝜏_𝑛+⋯+ℎ_𝑛𝑖𝑥₁^′′𝑡−𝜏₁,…,𝑥_𝑛^′′𝑡−𝜏_𝑛+𝜓_𝑖𝑥^′₁,…,𝑥^′_𝑛+^𝑛_𝑘=1𝑎_𝑖𝑘𝑥_𝑘=0,(𝑖=1,2,…,𝑛). (5)

Instead of (3), we consider the corresponding differential system ̇̇̇̇̇𝑋=𝑌,𝑌=𝑍,𝑍=𝑊,𝑊=𝑈,𝑈=−𝐹𝑋,…,𝑋𝑡−𝜏_𝑛,…,𝑈,…,𝑈𝑡−𝜏_𝑛𝑈−𝐺𝑋,…,𝑋𝑡−𝜏_𝑛,…,𝑈,…,𝑈𝑡−𝜏_𝑛𝑊−^𝑛_𝑖=1𝐻_𝑖(𝑍)+^𝑛_𝑖=1^𝑡_{𝑡−𝜏𝑖}𝐽_𝐻𝑖(𝑍(𝑠𝑊(𝑠)(𝑌)))𝑑𝑠−Ψ−𝐴𝑋, (6) which was obtained by setting 𝑋̇=𝑌,𝑋̈=𝑍,𝑋⁽³⁾=𝑊, and 𝑋⁽⁴⁾=𝑈 from (3).

However, a review to date of the literature indicates that the instability of solutions of vector differential equations of the fifth order with a deviating argument has not been investigated up to now. This paper is the first known work regarding the instability of solutions for the nonlinear vector delay differential equations of the fifth order with multiple-deviating arguments. The motivation of this paper comes from the above papers done on scalar nonlinear differential equations of the fifth order without and with delay and the vector differential equations of the fifth order without delay. By this work, we improve the results in [2, 12] to a vector delay differential equation of the fifth order with multiple-deviating arguments. Defining the Lyapunov-Krasovskii functional and taking into account the Krasovskiĭ’s criteria [19], we prove our main result on the subject. The result to be obtained is new, has a contribution to the topic, and may be useful for the researchers working on the qualitative behaviors of solutions of the differential equations.

Definition 1. The zero solution, 𝑥=0, of 𝑥̇=𝐹(𝑥_𝑡) is stable if for each 𝜀>0 there exists 𝛿=𝛿(𝜀)>0 such that ‖𝜙‖<𝛿 implies that |𝑥(𝜙)(𝑡)|<𝜀 for all 𝑡≥0. The zero solution is said to be unstable if it is not stable.
The symbol ⟨𝑋,𝑌⟩ corresponding to any pair 𝑋,𝑌 in ℜ^𝑛 stands for the usual scalar product ∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖, that is, ⟨𝑋,𝑌⟩=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖; thus ⟨𝑋,𝑋⟩=‖𝑋‖² and 𝜆_𝑖(Ω),(𝑖=1,2,…,𝑛) are the eigenvalues of the real symmetric 𝑛×𝑛-matrix Ω. The matrix Ω is said to be negative-definite, when ⟨Ω𝑋,𝑋⟩≤0 for all nonzero 𝑋 in ℜ^𝑛.

2. Main Result

Before stating the main result, we need the following result.

Lemma 2. Let 𝐴 be a real symmetric 𝑛×𝑛-matrix and 𝑎′≥𝜆_𝑖(𝐴)≥𝑎>0,(𝑖=1,2,...,𝑛), (7) where 𝑎′ and 𝑎 are constants.
Then 𝑎′⟨𝑋,𝑋⟩≥⟨𝐴𝑋,𝑋⟩≥𝑎⟨𝑋,𝑋⟩,𝑎′²⟨𝑋,𝑋⟩≥⟨𝐴𝑋,𝐴𝑋⟩≥𝑎²⟨𝑋,𝑋⟩. (8) See [20].
Let 𝑟≥0 be given and let 𝐶=𝐶([−𝑟,0],ℜ^𝑛) with𝜙=max_{−𝑟≤𝑠≤0}𝜙(𝑠),𝜙∈𝐶. (9)
For 𝐻>0 define 𝐶_𝐻⊂𝐶 by 𝐶_𝐻=𝜙∈𝐶∶𝜙<𝐻. (10)
If 𝑥∶[−𝑟,𝐴)→ℜ^𝑛 is continuous, 0<𝐴≤∞, then, for each 𝑡 in [0,𝐴),𝑥_𝑡 in 𝐶 is defined by 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠),−𝑟≤𝑠≤0,𝑡≥0. (11)
Let 𝐺 be an open subset of 𝐶 and consider the general autonomous delay differential system with finite delay: 𝑥̇=𝐹𝑥_𝑡(0)𝑥,𝐹=0,_𝑡=𝑥(𝑡+𝜃),−𝑟≤𝜃≤0,𝑡≥0, (12) where 𝐹∶𝐺→ℜ^𝑛 is continuous and maps closed and bounded sets into bounded sets. It follows from these conditions on 𝐹 that each initial value problem, 𝑥̇=𝐹𝑥_𝑡,𝑥₀=𝜙∈𝐺, (13) has a unique solution defined on some interval [0,𝐴),0<𝐴≤∞. This solution will be denoted by 𝑥(𝜙)(⋅) so that 𝑥₀(𝜙)=𝜙.

The main result of this paper is given by the following theorem.

Theorem 3. In addition to the basic assumptions imposed on 𝐹,𝐺,𝐻_𝑖,Ψ, and 𝐴 that appear in (3), we assume that there exist positive constants 𝑎₁,𝑎₄,𝛽_𝑖, and 𝛿 such that the following conditions hold:
𝐹(⋅),𝐺(⋅),𝐽_𝐻𝑖(⋅),𝐽_Ψ(⋅), and 𝐴 are symmetric, 𝐻_𝑖(0)=0,𝐻_𝑖(𝑍)(),Ψ(0)(𝑌)(),𝜆≠0,𝑍≠0=0,Ψ≠0,𝑌≠0_𝑖(𝐴)≤−𝑎₁,𝜆_𝑖𝐽_Ψ(𝑌)≥𝑎₄,𝜆_𝑖𝐽_𝐻𝑖(𝑍)≤𝛽_𝑖,14𝜆_𝑖(𝐹(⋅))²+𝜆_𝑖(𝐺(⋅))≤−𝛿. (14) If 𝛿𝜏<√𝑛𝛽₁+⋯+𝛽_𝑛, (15) then the zero solution of (3) is unstable.

Proof. We define the Lyapunov-Krasovskii functional 𝑉(⋅)=𝑉(𝑋_𝑡,𝑌_𝑡,𝑍_𝑡,𝑊_𝑡,𝑈_𝑡)∶𝑉(⋅)=⟨⟩+⟨⟩−1𝑊,𝑈𝐴𝑋,𝑍2⟨⟩+⟨Ψ(𝑌)⟩+𝐴𝑌,𝑌,𝑍₁₀𝐻₁(𝜎𝑍),𝑍𝑑𝜎+⋯+¹₀𝐻_𝑛()−𝜎𝑍,𝑍𝑑𝜎_𝑛𝑖=1𝜇_𝑖⁰_−𝜏𝑖^𝑡_𝑡+𝑠‖𝑊(𝜃)‖²𝑑𝜃𝑑𝑠, (16) where 𝜇_𝑖 are certain positive constants and will be determined later in the proof.
It is clear that 𝑉(0,0,0,0,0)=0.
Let𝜀=𝜀₂₁,...,𝜀_2𝑛. (17)
Using the assumption 𝜆_𝑖(𝐴)≤−𝑎₁, we have 𝑉(0,)1𝜀,0,0,0=−2⟨𝐴𝜀,𝜀⟩≥12𝑎₁‖𝜀‖²>0 (18) for all arbitrary 𝜀≠0, 𝜀∈ℜ^𝑛, which verifies the property (𝑃₁) of Krasovskiĭ [19].
Using a basic calculation, the time derivative of 𝑉 along solutions of (6) results in 𝑑𝑉(⋅)=⟨⟩−⟨𝐹(⋅)⟩−⟨𝐺(⋅)⟩+𝐽𝑑𝑡𝑈,𝑈𝑊,𝑈𝑊,𝑊_Ψ(𝑌)+𝑍,𝑍𝑊,^𝑡_𝑡−𝜏1𝐽_𝐻1(𝑍(𝑠))𝑊(𝑠)𝑑𝑠+⋯+𝑊,^𝑡_{𝑡−𝜏𝑛}𝐽_𝐻𝑛(𝑍(𝑠𝑊(𝑠)−𝜇))𝑑𝑠₁𝜏₁𝑊,𝑊−⋯−𝜇_𝑛𝜏_𝑛𝑊,𝑊+𝜇₁^𝑡_𝑡−𝜏1‖𝑊(𝜃)‖²𝑑𝜃+⋯+𝜇_𝑛^𝑡_{𝑡−𝜏𝑛}‖𝑊(𝜃)‖²𝑑𝜃. (19) Under the assumptions of the theorem and Schwarz inequality, it can be easily seen that 𝑊,^𝑡_𝑡−𝜏1𝐽_𝐻1(𝑍(𝑠𝑊(𝑠)‖𝑊))𝑑𝑠≥−‖^𝑡_𝑡−𝜏1𝐽_𝐻1(𝑍(𝑠𝑊(𝑠)√))𝑑𝑠≥−𝑛𝛽₁‖𝑊‖^𝑡_𝑡−𝜏1𝑊(𝑠√)𝑑𝑠≥−𝑛𝛽₁‖𝑊‖^𝑡_𝑡−𝜏1‖𝑊(𝑠1)‖𝑑𝑠≥−2√𝑛𝛽₁𝜏₁‖𝑊‖²−12√𝑛𝛽₁^𝑡_𝑡−𝜏1‖𝑊(𝑠)‖²𝑑𝑠,𝑊,^𝑡_𝑡−𝜏2𝐽_𝐻2(𝑍(𝑠𝑊(𝑠)‖𝑊))𝑑𝑠≥−‖^𝑡_𝑡−𝜏2𝐽_𝐻2(𝑊(𝑠𝑊(𝑠)√))𝑑𝑠≥−𝑛𝛽₂‖𝑊‖^𝑡_𝑡−𝜏2𝑊(𝑠√)𝑑𝑠≥−𝑛𝛽₂‖𝑊‖^𝑡_𝑡−𝜏2‖𝑊(𝑠1)‖𝑑𝑠≥−2√𝑛𝛽₂𝜏₂‖𝑊‖²−12√𝑛𝛽₂^𝑡_𝑡−𝜏2‖𝑊(𝑠)‖²⋮𝑑𝑠,𝑊,^𝑡_{𝑡−𝜏𝑛}𝐽_𝐻𝑛(𝑍(𝑠𝑊(𝑠)‖𝑊))𝑑𝑠≥−‖^𝑡_{𝑡−𝜏𝑛}𝐽_𝐻𝑛(𝑍(𝑠𝑊(𝑠)√))𝑑𝑠≥−𝑛𝛽_𝑛‖𝑊‖^𝑡_{𝑡−𝜏𝑛}𝑊(𝑠√)𝑑𝑠≥−𝑛𝛽_𝑛‖𝑊‖^𝑡_{𝑡−𝜏𝑛}‖𝑊(𝑠1)‖𝑑𝑠≥−2√𝑛𝛽_𝑛𝜏_𝑛‖𝑊‖²−12√𝑛𝛽_𝑛^𝑡_{𝑡−𝜏𝑛}‖𝑊(𝑠)‖²𝑑𝑠. (20) Hence 𝑉̇(⋅)=𝑈−2⁻¹𝐹(⋅)𝑊²−14⟨𝐹(⋅)(⋅)𝑊⟩−⟨𝐺(⋅)⟩+𝐽𝑊,𝐹𝑊,𝑊_Ψ(𝑌)𝑍,𝑍−𝜇₁𝜏₁‖𝑊‖²−⋯−𝜇_𝑛𝜏_𝑛‖𝑊‖²−12√𝑛𝛽₁𝜏₁‖𝑊‖²1−⋯−2√𝑛𝛽_𝑛𝜏_𝑛‖𝑊‖²+𝜇₁−12√𝑛𝛽₁^𝑡_𝑡−𝜏1‖𝑊(𝑠)‖²𝜇𝑑𝑠+⋯+_𝑛−12√𝑛𝛽_𝑛^𝑡_{𝑡−𝜏𝑛}‖𝑊(𝑠)‖²𝑑𝑠. (21)
Let 𝜇_𝑖=(1/2)√𝑛𝛽_𝑖. Then, using the assumptions of the theorem and the estimate 𝜏=max𝜏_𝑖,we get 𝑉̇(⋅)=𝑈−2⁻¹𝐹(⋅)𝑊²−14⟨𝐹(⋅)(⋅)𝑊⟩−⟨𝐺(⋅)⟩+𝐽𝑊,𝐹𝑊,𝑊_Ψ(𝑌)𝑍,𝑍−√𝑛𝛽₁𝜏₁‖𝑊‖²−⋅⋅⋅−√𝑛𝛽_𝑛𝜏_𝑛‖𝑊‖²1≥−4⟨𝐹(⋅)𝑊,𝐹(⋅)𝑊⟩−⟨𝐺(⋅)𝑊,𝑊⟩+𝐽_Ψ(𝑌)−√𝑍,𝑍𝑛𝛽₁𝜏₁‖𝑊‖²−⋅⋅⋅−√𝑛𝛽_𝑛𝜏_𝑛‖𝑊‖²≥𝑎₄‖𝑍‖²+𝛿−√𝑛𝛽₁+⋅⋅⋅+𝛽_𝑛𝜏‖𝑊‖²>0. (22)
If 𝜏<𝛿/√𝑛(𝛽₁+⋯+𝛽_𝑛), then, for some positive constant 𝑘₁,we have 𝑉̇(⋅)≥𝑎₄‖𝑍‖²+𝑘₁‖𝑊‖²>0, (23) which verifies that 𝑉(⋅) has the property (𝑃₂) of Krasovskiĭ [19].
Besides, 𝑉̇(⋅)=0⟺𝑍=𝑊=𝑈=0. (24) The substitution of this estimate into system (6) results in Ψ(𝑌)+𝐴𝑋=0. (25) That is, Ψ𝑋′+𝐴𝑋=0. (26) Because 𝑍=𝑋′′=0,  𝑋′= constant vector, for all 𝑡>0.Hence, since 𝐴 is not the zero matrix, we have 𝑋=constant vector, for all 𝑡>0. But, in view of the assumptions of the theorem, this implies that 𝑋′=0 and thus also, by  Ψ(𝑋′)+𝐴𝑋=0, that 𝑋=0,  forall𝑡>0. These estimates result in 𝑋=𝑌=𝑍=𝑊=𝑈=0. Hence, the property (𝑃₃)of Krasovskiĭ [19] holds for the Lyapunov-Krasovskii functional 𝑉(⋅).
The proof of the theorem is completed.