Stability and boundedness in di ff erential systems of third order with variable delay

In this paper, we consider a non-linear system of differential equations of third order with variable delay. We discuss the globally asymptotic stability/uniformly stability, boundedness and uniformly boundedness of solutions for the considered system. The technique of proofs involves defining an appropriate Lyapunov functional. The obtained results include and improve the results in literature.

Besides, it is well known that differential equations of third order play extremely important and useful roles in many scientific areas such as atomic energy, biology, chemistry, control theory, economy, engineering, information theory, mechanics, medicine, physics, etc.. Indeed, we can find applications such as nonlinear oscillations in Afuwape et al. [51], Andres [52], Fridedrichs [53], physical applications in Animalu and Ezeilo [54], non-resonant oscillations in Ezeilo and Onyia [55], prototypical examples of complex dynamical systems in a high-dimensional phase space, displacement in a mechanical system, velocity, acceleration in Chlouverakis and Sprott [56], Eichhorn et al. [57], Linz [58], the biological model and other models in Cronin-Scanlon [59], problems in biomathematics in Chlouverakis and Sprott [56], electronic theory in Rauch [60], and etc.Further, we refer the readers to the book of Smith [61] for some important applications of delay differential equation in sciences, biomathematics, engineering, and etc..In 2015, Omeike [21] investigated the stability and boundedness of nonlinear differential system of third order with variable delay, τ (t) : In this paper, we consider nonlinear differential system of the third order with variable delay, τ (t) : where are continuous differentiable functions with G(0) = H(0) = 0 such that the Jacobian matrices J G (X 0 ) and J H (X) exist and are symmetric and continuous, that is, ´, (i, j = 1, 2, ..., n), exist and are symmetric and continuous, where (x 1 , x 2 , ..., x n ), (x It is more convenient to consider not Eq.(1.2) itself, but rather the system X The continuity of the functions τ, H, G and F is a sufficient condition to guarantee the existence of solutions of Eq. (1.2).Besides, we assume that the functions H, G and F satisfy a Lipschitz condition with respect to their respective arguments, like X, X 0 and X 00 .In this case, the uniqueness of solutions of Eq. (1.2) is guaranteed.
The motivation of this paper comes from the results established by Omeike [21], the mentioned books, papers and theirs references.The main purpose of this paper is to get new the globally asymptotic stability /uniformly stability, boundedness and uniformly boundedness results in Eq. (1.2) by defining a suitable new Lyapunov functional.By this paper, we extend and improve the stability and boundedness results of Omeike [21], and we give additional two results to that in Omeike [21], like uniformly stability and uniformly boundedness results.It follows that if we choose G(X 0 (t − τ (t))) = BX 0 (t), B is an n × n-constant symmetric matrix, and F (t, X, X 0 , X 00 ) = P (t) in Eq. (1.2), then Eq. (1.2) reduces to Eq. (1.1), which is discussed by Omeike [21].This means that instead of the linear term BX 0 (t) in Eq. (1.1), we take the non-linear term G(X which includes a variable delay τ (t), and we also take a nonlinear generalization of the term P (t) in Eq. (1.1) like F (t, X, X 0 , X 00 ).Probably, these cases seem as similarity of Eq. (1.1) and Eq.(1.2).
However, till now, throughout all the papers published in the literature, no author discussed the stability and boundedness of the solutions when we take the possible second term in Eq. (1.1), G(X 0 ) or BX 0 (t) as a non-linear term with a deviating argument like G(X 0 (t − τ (t))).To the best of our knowledge, it is not easy to discuss the topic for Eq.(1.2).The possible reason is that the construction or definition of a suitable Lyapunov function or functional for higher differential systems remains as an open problem in the literature by this time.This case is more difficult for the functional differential systems of higher order with variable delay.The choice of the second term in Eq.(1.2) as G(X ) is important and the discussion of the problems for this case is some hard.This means that, in view of the whole mentioned discussion, it is worth to discuss the globally asymptotic stability /uniformly stability, boundedness and uniformly boundedness to Eq. (1.2).
Besides, this paper may be useful for researchers working on the qualitative behaviors of solutions of third differential equations and completes that in the literature.These cases show the novelty and originality of the present paper.
Consider general delay differential system Lemma 1 (Hale [62]).Suppose f (0) = 0. Let V be a continuous functional defined on C H = C with V (0) = 0 and let u(s) be a function, non-negative and continuous for 0 If we define Z = {ϕ ∈ C H : V (ϕ) = 0}, then the zero solution of ẋ = f (x t ) is asymptotically stable, provided that the largest invariant set in Z is O ¯= 0. Besides, we consider the general non-autonomous delay differential system ẋ = g(t, x t ), Theorem A(Yoshizawa [63 pp.191]).Assume that there is a Lyapunov functional V 0 (t, x) for ẋ = g(t, x t ), and wedges satisfying; , (where W 1 (r) and W 2 (r) are wedges), Then the zero solution of ẋ = g(t, x t ) is uniformly stable.
Lemma 2. (Bellman [64, pp.288]).Let M be a real symmetric n × n -matrix.Then for any where δ M and ∆ M are, respectively, positive and simple, the least and greatest eigenvalues of the matrix M .

Stability
We introduce some basic assumptions needed in the proofs.(A1) A is an n × n-symmetric constant matrix, where δ a and 4 a are positive constants.(A2) H(0) = 0, J H exists and is an n × n-symmetric matrix, H(X 1 ) 6 = 0, when X 1 6 = 0, where δ h and 4 h are positive constants.(A3) G(0) = 0, J G exists and is an n × n-symmetric matrix, where δ b and 4 b are positive constants, and the matrices J G and J H commute with each other.
Let F (.) ≡ 0. The stability result of this paper is given by the following theorem.
where β, λ and η are positive constants, the constants λ and η will be determined later in the proof.
From assumption (A 3 ), Lemma 2 and We obtain 2 Similarly, it follows from Integrating, from σ 1 = 0 to σ 1 = 1, the both sides of the last estimates, respectively, we get It can also be seen that 2 Further, it is obvious that Integrating the both sides of the last equality from σ 2 = 0 to σ 2 = 1, we obtain Then, we have hH(X 1 ), H(X 1 )i = 2 From (2.1), the above discussion, Lemma 2 and the assumptions of Theorem 1, we can obtain Stability and boundedness in differential systems of third order ... 325 It is clear all the coefficients in the last inequality are positive and hence there exists a positive constant K such that and For the time derivative the functional W 0 by a straightforward calculation from (1.3) and (2.1), we obtain The assumptions of Theorem 1, 0 ≤ τ (t) ≤ τ 0 and τ On gathering the obtained inequalities into Ẇ0 (t), we arrive at for some positive constants K 1 and K 2 .In addition, we can easily see that Consider the set defined by When we apply the LaSalle's invariance principle, we observe that (X 1 , X 2 , X 3 ) ∈ Q implies that X 2 = X 3 = 0 and hence X 1 = µ, (µ 6 = 0 is a constant vector).From the last estimate and system (1.3), we have H(µ) = 0, which necessarily implies that µ = 0 since H(0) = 0. Therefore In fact, this result implies that the largest invariant set contained in Q is (0, 0, 0) ∈ Q.By Lemma 1, we conclude that the zero solution of system (1.3) is asymptotically stable.Hence, the zero solution of Eq. (1.2) is the globally asymptotic stable.This completes the proof of Theorem 1.
Proof.To prove Theorem 2, our main tool is the functional W 0 given by (2.1).It is clear from the proof of Theorem 1 that the functional W 0 and its time derivative satisfy the assumptions of Theorem A, except W 2 (kφk); Besides, subject to the assumptions of Theorem 2, it can be easily obtained that W 0 ≤ W 2 (kφk) We omit the detail of the proof.This completes the proof of Theorem 2.

Boundedness
Let F (.) 6 = 0.The boundedness results of this paper are given by the following theorems.Theorem 3. We assume that all the assumptions of Theorem 1 hold, except F (.) ≡ 0. Further, we suppose that there exists a non-negative and continuous function P = P (t) such that kF (t, X 1 , X 2 , X 3 )k ≤ P (t) for all t ≥ 0, max P(t) < ∞ and P ∈ L 1 (0, ∞), where L 1 (0, ∞) denotes the space of Lebesgue integrable functions.If then there exists a constant M > 0 such that any solution (X 1 (t), X 2 (t), X 3 (t)) of system (1.3) determined by Proof.Let F (.) = F (t, X 1 , X 2 , X 3 ).For the case of F (.) 6 = 0, it can be concluded that ≤ 2K 3 P (t) + K −1 K 3 W 0 (t)P (t), by the assumptions of Theorem 3 and (2.2), where The integration of both sides of the last inequality, between 0 to t, (t ≥ 0), leads that W 0 (t) ≤ W 0 (0) + 2K 3 By the estimate kX 1 k 2 + kX 2 k 2 + kX 3 k 2 ≤ K −1 W 0 and the assumption P ∈ L 1 (0, ∞), we can conclude that all solutions of system (1.3) are bounded.This completes the proof of Theorem 3. Proof.To complete the poof of Theorem 4, the main tool is the functional W 0 given by (2.1).When we benefit from the functional W 0 and the assumptions of Theorem 4, we can easily complete the poof of Theorem 4. Therefore, we omit the details of the proof.

Conclusion
We consider a functional differential system of third order with variable delay.We investigate the globally asymptotic stability/uniformly stability/boundedness/ uniformly boundedness of solutions.The technique of proofs involves defining an appropriate Lyapunov functional.Our results include, improve and complete some recent results in the literature.
a continuous function and the primes in Eq. (1.2) indicate differentiation with respect to t, t ≥ t 0 ≥ 0.