On the Unstable Solutions to Functional Vector Differential Equations of the Seventh Order

Abstract

This paper studies the instability of the zero solution for a certain nonlinear functional vector differential equation of the seventh order with multiple deviating arguments. Under sufficient conditions, we prove a result on the instability of the zero solution. This work contributes and complements to previously known results in the literature.

1. Introduction

More than 100 years ago, the world famous mathematician Lyapunov established the Lyapunov direct method to study stability problems. From then on, the Lyapunov’s direct method was also widely used to study the instability of solutions of ordinary differential equations and functional differential equations, see for example, Bereketoğlu [1], Sadek [2], Tejumola [3], Tunç [4–7], C. Tunç and E. Tunç [8], and the references therein. However, a review to date of the literature indicates that the instability of the solutions to the nonlinear functional vector differential equation of the seventh order has not been investigated. This paper is the first work on the subject. It is also known that the expressions of Lyapunov-Krasovskii functional are very complicated and hard to construct. In this paper, we define a Lyapunov-Krasovskii functional [9] and base on the Krasovskii criteria to prove a new theorem on the topic for the nonlinear functional vector differential equation of the seventh order. In this paper, intend to make a contribution to the subject since the functional differential equations have an important place in various fields of science and engineering.

Meanwhile, some respective contributions on the topic can be summarized as follows.

In 2000, Tejumola [3] discussed the instability of the zero solution of the seventh order scalar nonlinear differential equation without a deviating argument 𝑥⁽⁷⁾+𝑎₁𝑥⁽⁶⁾+𝑎₂𝑥⁽⁵⁾+𝑎₃𝑥⁽⁴⁾+𝜓₄𝑥,.𝑥^′,…,𝑥⁽⁶⁾𝑥^′′′+𝜓₅𝑥^′𝑥^′′+𝜓₆𝑥,𝑥^′,…,𝑥⁽⁶⁾+𝜓₇(𝑥)=0. (1)

Later, in 2003, Sadek [2] proved an instability theorem for the seventh order scalar nonlinear differential equation without a deviating argument 𝑥⁽⁷⁾+𝑎₁𝑥⁽⁶⁾+𝑎₂𝑥⁽⁵⁾+𝑎₃𝑥⁽⁴⁾+𝑎₄𝑥^′′′+𝑓𝑥^′𝑥^′′+𝑔(𝑥)𝑥^′+ℎ(𝑥)=0. (2)

Recently, Tunç [6, 7] discussed the same problem for the seventh order scalar nonlinear delay differential equations of the form: 𝑥⁽⁷⁾+𝑎₁𝑥⁽⁶⁾+𝑎₂𝑥⁽⁵⁾+𝑎₃𝑥⁽⁴⁾+𝑎₄𝑥^′′′+𝑓𝑥^′𝑥^′′+𝑔(𝑥)𝑥^′+ℎ𝑥,𝑥(𝑡−𝑟),𝑥^′,…,𝑥⁽⁶⁾()𝑥𝑡−𝑟=0,₍₇₎+𝑎₁𝑥⁽⁶⁾+𝑎₂𝑥⁽⁵⁾+𝑎₃𝑥⁽⁴⁾+𝜓₄𝑥,𝑥(𝑡−𝑟),𝑥^′,…,𝑥⁽⁶⁾(𝑡−𝑟)𝑥^′′′+𝜓₅𝑥^′𝑥^′′+𝜓₆𝑥,𝑥(𝑡−𝑟),𝑥^′,…,𝑥⁽⁶⁾(𝑡−𝑟)+𝜓₇(𝑥(𝑡−𝑟))=0. (3)(4)

In this paper, instead of (4), we consider the seventh order nonlinear functional vector differential equation with 𝑛-deviating arguments, 𝜏_𝑖, (𝑖=1,2,…,𝑛): 𝑋⁽⁷⁾+𝐴₁𝑋⁽⁶⁾+𝐴₂𝑋⁽⁵⁾+𝐴₃𝑋⁽⁴⁾+Ψ₄𝑋,𝑋𝑡−𝜏₁,…,𝑋𝑡−𝜏_𝑛,…,𝑋⁽⁶⁾𝑡−𝜏_𝑛×𝑋^′′′+Ψ₅𝑋^′𝑋^′′+Ψ₆×𝑋,𝑋𝑡−𝜏₁,…,𝑋𝑡−𝜏_𝑛,…,𝑋⁽⁶⁾𝑡−𝜏_𝑛𝑋^′+^𝑛_𝑖=1𝐻_𝑖𝑋𝑡−𝜏_𝑖=0. (5) Setting 𝑋=𝑋₁, 𝑋^′=𝑋₂, 𝑋^′′=𝑋₃, 𝑋^′′′=𝑋₄, 𝑋⁽⁴⁾=𝑋₅, 𝑋⁽⁵⁾=𝑋₆, 𝑋⁽⁶⁾=𝑋₇, we can write (5) in system form as follows 𝑋^′₁=𝑋₂,𝑋^′₂=𝑋₃,𝑋^′₃=𝑋₄,𝑋^′₄=𝑋₅,𝑋^′₅=𝑋₆,𝑋^′₆=𝑋₇,𝑋^′₇=−𝐴₁𝑋₇−𝐴₂𝑋₆−𝐴₃𝑋₅−Ψ₄𝑋₁,…,𝑋₁𝑡−𝜏_𝑛,…,𝑋₇𝑡−𝜏_𝑛𝑋₄−Ψ₅𝑋₂𝑋₃−Ψ₆𝑋₁,…,𝑋₁𝑡−𝜏_𝑛,…,𝑋₇𝑡−𝜏_𝑛𝑋₂−^𝑛_𝑖=1𝐻_𝑖𝑋₁+^𝑛_𝑖=1^𝑡_{𝑡−𝜏𝑖}𝐽_𝐻𝑖𝑋₁(𝑠)𝑋₂(𝑠)𝑑𝑠, (6) where 𝜏_𝑖 are certain positive constants, the fixed delays, 𝑡−𝜏_𝑖>0, 𝐴₁, 𝐴₂,  and 𝐴₃ are constant 𝑛×𝑛-symmetric matrices, the primes in (5) denote differentiation with respect to 𝑡, 𝑡∈ℜ₊, ℜ₊=[0,∞); Ψ₄, Ψ₅, and Ψ₆ are continuous 𝑛×𝑛-symmetric matrix functions for the arguments displayed explicitly, 𝐻_𝑖∶ℜ^𝑛→ℜ^𝑛, 𝐻_𝑖(0)=0, 𝐻_𝑖(𝑋₁)≠0, (𝑋₁≠0), and 𝐻_𝑖 are continuous for all the respective arguments. The Jacobian matrices of 𝐻_𝑖(𝑋) are given by 𝐽_𝐻1(𝑋)=𝜕ℎ_1𝑖𝜕𝑥_𝑗,…,𝐽_𝐻𝑛(𝑋)=𝜕ℎ_𝑛𝑖𝜕𝑥_𝑗𝑖,𝑗=1,2,…,𝑛, (7) where (𝑥₁,…,𝑥_𝑛) and (ℎ_1𝑖),…,(ℎ_𝑛𝑖) are the components of 𝑋 and 𝐻_𝑖, respectively. It is also assumed that the Jacobian matrices 𝐽_𝐻𝑖(𝑋) exist and are continuous. The existence and uniqueness of the solutions of (5) are assumed (see Èl’sgol’ts [10, pages 14, 15]). Throughout what follows that 𝑋₁(𝑡),…,𝑋₇(𝑡) are abbreviated as 𝑋₁,…,𝑋₇, respectively.

Let 𝑟≥0 be given, and let 𝐶=𝐶([−𝑟,0],ℜ^𝑛) with 𝜙=max_{−𝑟≤𝑠≤0}𝜙(𝑠),𝜙∈𝐶. (8)

For 𝐻>0 define 𝐶_𝐻⊂𝐶 by 𝐶_𝐻=𝜙∈𝐶∶𝜙<𝐻. (9)

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. (10)

Let 𝐺 be an open subset of 𝐶 and consider the general autonomous delay differential system 𝑥̇=𝐹𝑥_𝑡,𝑥_𝑡=𝑥(𝑡+𝜃),−𝑟≤𝜃≤0,𝑡≥0, (11) where 𝐹∶𝐺→ℜ^𝑛 is continuous and maps closed and bounded sets into bounded sets. It follows from the conditions on 𝐹 that each initial value problem 𝑥̇=𝐹𝑥_𝑡,𝑥₀=𝜙∈𝐺 (12) 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.

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.

Consider the linear constant coefficient differential equation of the seventh order: 𝑥⁽⁷⁾+𝑎₁𝑥⁽⁶⁾+𝑎₂𝑥⁽⁵⁾+𝑎₃𝑥⁽⁴⁾+𝑎₄𝑥⃛+𝑎₅𝑥̈+𝑎₆𝑥̇+𝑎₇𝑥=0. (13) It is known from the qualitative behavior of solutions of linear differential equations that the zero solution of (13) is unstable if and only if, the associated auxiliary equation: 𝜓(𝜆)≡𝜆⁷+𝑎₁𝜆⁶+𝑎₂𝜆⁵+𝑎₃𝜆⁴+𝑎₄𝜆³+𝑎₅𝜆²+𝑎₆𝜆+𝑎₇=0 (14) has at least one root with a positive real part. The existence of such a root naturally depends on (though not always all of) the coefficients 𝑎₁,𝑎₂,…,𝑎₇ in (14). For example, if 𝑎₁<0 (15) then it is clear from a consideration of the fact that the sum of the roots of (14) equals 𝑎₁ and that at the least one root of (14) has a positive real part for arbitrary values of 𝑎₂, 𝑎₃, 𝑎₄, 𝑎₅, 𝑎₆, and 𝑎₇. An analogue consideration, combined with the fact that the product of the roots (14) equals (−𝑎₇) will verify that at least one root of (14) have a positive real part if 𝑎₁=0,𝑎₇≠0 (16) for arbitrary 𝑎₂, 𝑎₃, 𝑎₄, 𝑎₅ and 𝑎₆. The condition 𝑎₁=0 here in (16) is, however, superfluous when 𝑎₇<0; (17) then for 𝜓(0)=𝑎₇<0 and 𝜓(𝑅)>0 if 𝑅>0 is sufficiently large, thus showing that there is a positive real root of (14) subject to (17) and for arbitrary 𝑎₁, 𝑎₂, 𝑎₃, 𝑎₄, 𝑎₅ and 𝑎₆. Moreover, a necessary and sufficient condition for (14) to has a purely imaginary root 𝜆=𝑖𝛽 (𝛽 real) is that the two equations −𝑎₁𝛽⁶+𝑎₃𝛽⁴−𝑎₅𝛽²+𝑎₇=0,−𝛽⁶+𝑎₂𝛽⁴−𝑎₄𝛽²+𝑎₆=0 (18) are satisfied at the same time. If 𝑎₁≤0,𝑎₃≥0,𝑎₅≤0,𝑎₇>0 (19) or 𝑎₁≥0,𝑎₃≤0,𝑎₅≥0,𝑎₇<0, (20) then (14) cannot have any purely imaginary root whatever.

It should be noted that there are no restrictions on the constants 𝑎₂, 𝑎₄, and 𝑎₆ in (13).

2. Main Result

First, we give the following lemma.

Lemma 2. Let 𝐴 be a real symmetric 𝑛×𝑛-matrix and 𝑎^′≥𝜆_𝑖(𝐴)≥𝑎>0,(𝑖=1,2,…,𝑛), (21) where 𝑎^′and 𝑎 are constants, and 𝜆_𝑖(𝐴) are the eigenvalues of the matrix 𝐴.
Then 𝑎^′⟨⟩≥⟨⟩⟨⟩,𝑎𝑋,𝑋𝐴𝑋,𝑋≥𝑎𝑋,𝑋_′2⟨𝑋,𝑋⟩≥⟨𝐴𝑋,𝐴𝑋⟩≥𝑎²⟨𝑋,𝑋⟩ (22) (Bellman [11]).

Let 𝜏=max𝜏_𝑖,(𝑖=1,2,…,𝑛).

The following theorem is our main result.

Theorem 3. In addition to all the assumptions imposed to 𝐴₁, 𝐴₂, 𝐴₃, Ψ₄, Ψ₅, Ψ₆, and 𝐻_𝑖 that appear in (5), we assume that there exist constants 𝑎₂<0, 𝛼_𝑖>0, 𝛽_𝑖>0, and 𝛿>0 such that the conditions 𝜆_𝑖𝐴₂≤𝑎₂,𝛽₁≤𝜆_𝑖𝐽_𝐻1𝑋₁≤𝛼₁,…,𝛽_𝑛≤𝜆_𝑖𝐽_𝐻𝑛𝑋₁≤𝛼_𝑛,14𝑎₂𝜆_𝑖Ψ₄(⋅)²−𝜆_𝑖Ψ₆(⋅)≥𝛿>0 (23) hold. If 𝜏<𝛿/√𝑛(𝛼₁+⋯+𝛼_𝑛), then the zero solution of (5) is unstable.

Remark 4. For the proof of the theorem, under the conditions sated in the theorem, it suffices to find that there exists a continuous Lyapunov-Krasovskii functional 𝑉(⋅)=𝑉(𝑋_1𝑡,…,𝑋_7𝑡), which has the Krasovskii properties [9], say (𝐾₁), (𝐾₂), and (𝐾₃). (𝐾₁) In every neighborhood of (0,0,0,0,0,0,0) there exists a point (𝜉₁,…,𝜉₇) such that 𝑉(𝜉₁,…,𝜉₇)>0.(𝐾₂) The time derivative 𝑉̇(⋅)=(𝑑/𝑑𝑡)𝑉(𝑋_1𝑡,…,𝑋_7𝑡) along solution paths of (6) is positive semidefinite. (𝐾₃) The only solution (𝑋₁,…,𝑋₇)=(𝑋₁(𝑡),…,𝑋₇(𝑡)) of (6) which satisfies (𝑑/𝑑𝑡)𝑉(𝑋_1𝑡,…,𝑋_7𝑡)=0,(𝑡≥0), is the trivial solution (0,0,0,0,0,0,0).

Proof. Consider the Lyapunov-Krasovskii functional 𝑉(⋅)=𝑉(𝑋_1𝑡,…,𝑋_7𝑡) defined by 𝑉(⋅)=𝑋₇,𝑋₂+𝐴₁𝑋₂,𝑋₆+𝐴₂𝑋₂,𝑋₅+𝐴₃𝑋₂,𝑋₄−𝑋₃,𝑋₆−𝐴₁𝑋₃,𝑋₅−𝐴₂𝑋₃,𝑋₄+𝑋₄,𝑋₅−12𝐴₃𝑋₃,𝑋₃+12𝐴₁𝑋₄,𝑋₄+¹₀𝐻₁𝜎𝑋₁,𝑋₁𝑑𝜎+⋯+¹₀𝐻_𝑛𝜎𝑋₁,𝑋₁+𝑑𝜎₁₀Ψ₅𝜎𝑋₂𝑋₂,𝑋₂−𝑑𝜎_𝑛𝑖=1𝜆_𝑖⁰_−𝜏𝑖^𝑡_𝑡+𝑠𝑋₂(𝜃)²𝑑𝜃𝑑𝑠, (24) where 𝑠 is a real variable such that the integrals ∫⁰_−𝜏𝑖∫^𝑡_𝑡+𝑠‖𝑋₂(𝜃)‖²𝑑𝜃𝑑𝑠 are nonnegative, and 𝜆_𝑖 are certain positive constants to be determined later in the proof.
We see that 𝑉(0,0,0,0,0,0,0)=0. (25)
Since (𝜕/𝜕𝜎)𝐻_𝑖(𝜎𝑋₁)=𝐽_𝐻𝑖(𝜎𝑋₁)𝑋₁, 𝐻_𝑖(0)=0, then ¹₀𝐻₁𝜎𝑋₁,𝑋₁=𝑑𝜎₁₀𝜎₁𝐽_𝐻1𝜎₁𝜎₂𝑋₁𝑋₁,𝑋₁𝑑𝜎₂𝑑𝜎₁≥𝛽₁𝑋₁²,₁₀𝐻₂𝜎𝑋₁,𝑋₁=𝑑𝜎₁₀𝜎₁𝐽_𝐻2𝜎₁𝜎₂𝑋₁𝑋₁,𝑋₁𝑑𝜎₂𝑑𝜎₁≥𝛽₂𝑋₁²,₁₀⋮𝐻_𝑛𝜎𝑋₁,𝑋₁=𝑑𝜎₁₀𝜎₁𝐽_𝐻𝑛𝜎₁𝜎₂𝑋₁𝑋₁,𝑋₁𝑑𝜎₂𝑑𝜎₁≥𝛽_𝑛𝑋₁². (26) Let 𝜀=𝜀₁₁,…,𝜀_1𝑛. (27) Then, we get 𝑉(𝜀,0,0,0,0,0,0)=^𝑛_𝑖=1𝛽_𝑖‖𝜀‖²>0 (28) for all 𝜀,𝜀∈ℜ, so that every neighborhood of the origin in the (𝑋₁,…,𝑋₇)-space contains points (𝜉₁,…,𝜉₇) such that 𝑉(𝜉₁,…,𝜉₇)>0.
Let 𝑋₁,…,𝑋₇=𝑋₁(𝑡),…,𝑋₇(𝑡) (29) be an arbitrary solution of (6).
Calculating the time derivative of the Lyapunov-Krasovskii functional 𝑉 along this solution, we obtain 𝑉̇(⋅)=𝑋₅,𝑋₅−𝐴₂𝑋₄,𝑋₄−Ψ₄(⋅)𝑋₂,𝑋₄−Ψ₆(⋅)𝑋₂,𝑋₂+𝑋₂,^𝑡_𝑡−𝜏1𝐽_𝐻1𝑋₁(𝑠)𝑋₂(𝑠)𝑋𝑑𝑠+⋯+₂,^𝑡_{𝑡−𝜏𝑛}𝐽_𝐻𝑛𝑋₁(𝑠)𝑋₂(𝑠)−𝜆𝑑𝑠₁𝜏₁𝑋₂,𝑋₂−⋯−𝜆_𝑛𝜏_𝑛𝑋₂,𝑋₂+𝜆₁^𝑡_𝑡−𝜏1𝑋₂(𝜃)²𝑑𝜃+⋯+𝜆_𝑛^𝑡_{𝑡−𝜏𝑛}𝑋₂(𝜃)²𝑑𝜃. (30) Using the assumption 0<𝜆_𝑖(𝐽_𝐻𝑖(𝑋₁))≤𝛼_𝑖 and the Schwarz inequality, we have 𝑋₂,^𝑡_𝑡−𝜏1𝐽_𝐻1𝑋₁(𝑠)𝑋₂(𝑠)𝑋𝑑𝑠≥−₂^𝑡_𝑡−𝜏1𝐽_𝐻1𝑋₁(𝑠)𝑋₂(𝑠)√𝑑𝑠≥−𝑛𝛼₁𝑋₂^𝑡_𝑡−𝜏1𝑋₂(𝑠√)𝑑𝑠≥−𝑛𝛼₁𝑋₂^𝑡_𝑡−𝜏1𝑋₂(𝑠1)𝑑𝑠≥−2√𝑛𝛼₁𝜏₁𝑋₂²−12√𝑛𝛼₁^𝑡_𝑡−𝜏1𝑋₂(𝑠)²𝑋𝑑𝑠,₂,^𝑡_𝑡−𝜏2𝐽_𝐻2𝑋₁(𝑠)𝑋₂(𝑠)𝑋𝑑𝑠≥−₂^𝑡_𝑡−𝜏2𝐽_𝐻2𝑋₁(𝑠)𝑋₂(𝑠)√𝑑𝑠≥−𝑛𝛼₂𝑋₂^𝑡_𝑡−𝜏2𝑋₂(𝑠√)𝑑𝑠≥−𝑛𝛼₂𝑋₂^𝑡_𝑡−𝜏2𝑋₂(𝑠1)𝑑𝑠≥−2√𝑛𝛼₂𝜏₂𝑋₂²−12√𝑛𝛼₂^𝑡_𝑡−𝜏2𝑋₂(𝑠)²⋮𝑋𝑑𝑠,₂,^𝑡_{𝑡−𝜏𝑛}𝐽_𝐻𝑛𝑋₁(𝑠)𝑋₂(𝑠)𝑋𝑑𝑠≥−₂^𝑡_{𝑡−𝜏𝑛}𝐽_𝐻2𝑋₁(𝑠)𝑋₂(𝑠)√𝑑𝑠≥−𝑛𝛼_𝑛𝑋₂^𝑡_{𝑡−𝜏𝑛}𝑋₂(𝑠√)𝑑𝑠≥−𝑛𝛼_𝑛𝑋₂^𝑡_{𝑡−𝜏𝑛}𝑋₂(𝑠1)𝑑𝑠≥−2√𝑛𝛼_𝑛𝜏_𝑛𝑋₂²−12√𝑛𝛼_𝑛^𝑡_{𝑡−𝜏𝑛}𝑋₂(𝑠)²𝑑𝑠. (31) Hence 𝑉̇(⋅)≥𝑋₅²−𝑎₂𝑋₄+2⁻¹𝑎₂⁻¹Ψ₄(⋅)𝑋₂²+14𝑎₂Ψ₄(⋅)𝑋₂,Ψ₄(⋅)𝑋₂−Ψ₆(⋅)𝑋₂,𝑋₂−𝜆₁+12√𝑛𝛼₁𝜏₁𝑋₂²−⋯−𝜆_𝑛+12√𝑛𝛼_𝑛𝜏_𝑛𝑋₂²+𝜆₁−12√𝑛𝛼₁^𝑡_𝑡−𝜏1𝑋₂(𝑠)²𝜆𝑑𝑠+⋯+_𝑛−12√𝑛𝛼_𝑛^𝑡_{𝑡−𝜏𝑛}𝑋₂(𝑠)²𝑑𝑠. (32)
Let 𝜆_𝑖=(1/2)√𝑛𝛼_𝑖. Then, using the assumptions of the theorem and the estimate 𝜏=max𝜏_𝑖,we have 𝑉̇(⋅)≥𝑋₅²−𝑎₂𝑋₄+2⁻¹𝑎₂⁻¹Ψ₄(⋅)𝑋₂²+14𝑎₂Ψ₄(⋅)𝑋₂,Ψ₄(⋅)𝑋₂−Ψ₆(⋅)𝑋₂,𝑋₂−√𝑛𝛼₁𝜏₁𝑋₂²−⋯−√𝑛𝛼_𝑛𝜏_𝑛𝑋₂²≥𝛿−√𝑛𝛼₁+⋯+𝛼_𝑛𝜏𝑋₂². (33)
If 𝜏<𝛿/√𝑛(𝛼₁+⋯+𝛼_𝑛), then, for some positive constant 𝑘, we have 𝑉̇(⋅)≥𝑘𝑋₂²>0. (34)
Finally, 𝑉̇(⋅)=0 for all 𝑡≥0 necessarily implies that 𝑋₂=0. Hence, it follows that 𝑋₁=𝜉₁(constantvector),𝑋₂=𝑋^′𝑋=0,₃=𝑋^′′=0,𝑋₄=𝑋^′′′=0,𝑋₅=𝑋⁽⁴⁾𝑋=0,₆=𝑋⁽⁵⁾=0,𝑋₇=𝑋⁽⁶⁾=0, (35) for all 𝑡≥0 so that 𝑋₁=𝜉₁(constantvector),𝑋₂=𝑋₃=𝑋₄=𝑋₅=𝑋₆=𝑋₇=0,(𝑡≥0). (36) Therefore, the estimates (𝑑/𝑑𝑡)𝑉(⋅)=0 and (6) imply 𝑋₁=𝑋₂=𝑋₃=𝑋₄=𝑋₅=𝑋₆=𝑋₇=0 since 𝐻_𝑖(𝜉₁)=0 if and only if 𝜉₁=0. It now follows that functional 𝑉 thus has all the requisite Krasovskii [9] properties subject to the conditions of the theorem. Hence, we can conclude that the zero solution of (5) is unstable.
The proof of the theorem is completed.