order

Sufficiency criteria are established to ensure the asymptotic stability and boundedness of solutions to third-order nonlinear delay differential equations of the form ... x(t) +e(x(t), u x(t), ¨(t))¨x(t) +g(x(t −r), u x(t −r)) + (x(t −r))


Introduction
As is well-known, the area of differential equations is an old but durable subject that remains alive and useful to a wide variety of engineers, scientists, and mathematicians.Now, with over 300 years of history, the subject of differential equations represents a huge body of EJQTDE, 2010 No. 12, p. 1 knowledge including many subfields and a vast array of applications in many disciplines.It is beyond exposition as a whole.It should be noted that principles of differential equations are largely related to the qualitative theory of ordinary differential equations.Qualitative theory refers to the study of behavior of solutions, for example the investigation of stability, instability, boundedness of solutions and etc., without determining explicit formulas for the solutions.Besides, stability and boundedness of solutions are also very important problems in the theory and applications of differential equations.In particular, the stability of motion in dynamical systems is an old but still active area of studies.At the close of the 19th century, three types of stability, Lyapunov stability, Poincare stability and Zhukovskij stability, were established for motion in continuous dynamical systems, i.e., for solutions of differential equations.Among them the Lyapunov stability and Poincare stability are most well-known.It is worth mentioning that if solutions of a differential equation describing a dynamical system or of any differential equation under consideration are known in closed form, one can determine the stability and boundedness properties of the system or the solutions of differential equation, appealing directly the definitions of stability and boundedness.As is well-known, in general, it is also not possible to find the solution of all linear and nonlinear differential equations, except numerically.Moreover, finding of solutions becomes more difficult for delay differential equations rather than the differential equations without delay.Therefore, it is very important to obtain information on the stability and boundedness behavior of solutions to differential equations when there is no analytical expression for solutions.So far, the most efficient tool to the study of stability and boundedness of solutions of a given nonlinear system is provided by Lyapunov's theory [12], that is, the Lyapunov's second (or direct) method.It is also worth mentioning that this theory became an important part of both mathematics and theoretical mechanics in twentieth century.By means of Lyapunov's second method, the stability in the large and boundedness of solutions can be obtained without any prior knowledge of solutions.
That is, the method yields stability and boundedness information directly, without solving the differential equation.The chief characteristic of the method requires the construction of the scalar function and functional for the equation under study.Unfortunately, it is some times very difficult, even impossible, to find a proper Lyapunov function or functional for a EJQTDE, 2010 No. 12, p. 2 given equation.However, within the past forty-five years and so, by using the Lyapunov's [12] second (or direct) method, many good results have been obtained and are still obtaining on the qualitative behaviors of solutions for various third order ordinary non-linear differential equations without delay.In particular, one can refer to the book of Reissig et al. [15] as a survey and the papers of Qian [14], Tunç ([20], [21]) and references quoted in these sources for some publications performed on the topic, which include the differential equations without delay.Besides, it is worth mentioning that, according to our observations, there are only a few papers on the same topics related to certain third order nonlinear differential equations with delay (See, Bereketoglu and Karakoç [1], Chukwu [3], Sadek ([16], [17]), Sinha [18], Tejumola and Tchegnani [19], Tunç [22][23][24][25][26] and Zhu [28]).Perhaps, the possible difficulty raised to this case is due to the construction of Lyapunov functionals for delay differential equations.How to construct those Lyapunov functionals?So far, no author has discussed them.In fact, there is no general method to construct Lyapunov functionals.Clearly, it is also more difficult to construct Lyapunov's functional for higher order differential equations with delay than without delay.Meanwhile, especially, since 1960s many good books, most of them are in Russian literature, have been published on the delay differential equations (see for example the books of Burton [2], Èl'sgol'ts [4], Èl'sgol'ts and Norkin [5], Gopalsamy [6], Hale [7], Hale and Verduyn Lunel [8], Kolmanovskii and Myshkis [9], Kolmanovskii and Nosov [10], Krasovskii [11], Makay [13], Yoshizawa [27] and the references listed in these books).
In the present paper, we take into consideration the following nonlinear differential equation of third order with delay ...
All solutions considered are assumed to be real valued.In addition, it is assumed that the functions e(x(t), y(t), z(t)) , g(x(t − r), y(t − r)) , ψ(x(t − r)) and p(t, x(t), x(t − r), y(t), y(t − r), z(t)) satisfy a Lipschitz condition in x(t), y(t), z(t), x(t − r) and y(t − r) .Then the solution is unique.
It should also be noted that here based of the result of Sinha [18] we establish our asymptotic stability result.Next, in view of the publication dates of the papers mentioned above, (see Sinha [18], Chukwu [3], Zhu [28], Sadek [16], Sadek [17], Bereketoglu and Karakoç [1], it is very interesting that all authors did not make reference to the work that was carried out before their investigation except only the existence of reference [28] in Sadek ([16], [17]) and the reference [3] in [1], respectively.Finally, to the best of our knowledge from the literature, it is not found any boundedness result based on the result of Sinha [18].
Definition 2. (See [2].)A continuous positive definite function W : ) > 0 if s > 0 , and W strictly increasing is a wedge.(We denote wedges by W or W i , where i an integer.) Definition 5. (See [3].)Let V (t, φ) be a continuous functional defined for t ≥ 0 , φ ∈ C H .The derivative of V along solutions of (3) will be denoted by V(3) and is defined by EJQTDE, 2010 No. 12, p. 5 the following relation where x(t 0 , φ) is the solution of (3) with x t 0 (t 0 , φ) = φ .
Theorem 1. (See( [17].)Suppose that there exists a Lyapunov functional V (t, φ) for (3) such that the following conditions are satisfied: where W 1 (r) and W 2 (r) are wedges,) and Then, the zero solution of ( 3) is uniformly stable.Now, we also consider the general autonomous delay differential system where and for H 1 > H , there exists L(H 1 ) > 0 ,with |f (φ)| ≤ L(H 1 ) , when φ ≤ H 1 (see also [26]).It is clear that the general autonomous delay differential system (4) is a special case of system (3), and the following definition and lemma are given. .The derivative of V along solutions of (4) will be denoted by V(4) and is defined by the following relation Lipschitz by condition , and assume that V (0) = 0 and that: ,where W 1 (r) and W 2 (r) are wedges; (ii) V(4) (φ) ≤ 0 for φ ∈ C H . Then the zero solution of ( 4) is uniformly stable.If we define where r is a positive constant; ϕ is a continuous function, ϕ(t, x(t), 0) = 0 and sin(x(t − r)) is a continuously differentiable function, which satisfy the following conditions and ϕ(t, x, y) y ≥ a for all t ≥ 0, x and y(y = 0), where a , b and ε 1 are some positive constants.Equation ( 5) can be rewritten as the following The following Lyapunov functional is defined to verify the stability of trivial solution x = 0 of equation ( 5), where λ is a positive constant which will be determined later, and it is obvious that the term 0 −r t t+s y 2 (θ)dθds is nonnegative.It is clear that the Lyapunov functional V (x t , y t ) in ( 9) is positive definite.Namely, V (0, 0) = 0 , and we also have Making use of the assumption 0 < b ≤ sin x x in (6), it follows that EJQTDE, 2010 No. 12, p. 7 Similarly, in view of the above assumptions, it is readily verified from (9) that Hence, there exist positive constants D 1 and D 2 such that where Finally, evaluating the time derivative of the functional V (x t , y t ) , that is, V = d dt V (x t , y t ), it follows from ( 9) and ( 8) that If we choose λ = L 2 , (10) implies for some constant α > 0 that If we define Z = φ ∈ C H : V(8) (φ) = 0 , then the zero solution of system ( 8) is asymptotically stable, provided that the largest invariant set in Z is Q = {0} .Thus, under the above discussion, one can say that the zero solution of equation ( 5) is asymptotically stable.
EJQTDE, 2010 No. 12, p. 8 Meanwhile, it should be noted that under less restrictive conditions, it can be easily shown that all solutions of delay differential equation are bounded.Therefore, we omit details of related operations.
Our first main result is the following.
Theorem 2. In addition to the basic assumptions imposed on the functions e, g and ψ that appearing in (1), we assume that there are positive constants λ, α, a, b, c, L 1 , L 2 , L 3 and k 1 such that the following conditions hold: (i) e(x, y, z) ≥ a + 2λ α > 0 , ye x (x, y, 0) ≤ 0 and ye z (x, y, z) ≤ 0 for all x, y and z. (iii Then the zero solution of equation ( 1) is asymptotically stable, provided that Proof.Our main tool for the proof of Theorem 2 is the Lyapunov functional V 0 = V 0 (x t , y t , z t ) defined by where EJQTDE, 2010 No. 12, p. 9 and Now, in view of conditions (ii) and (iii) of Theorem 2, it follows from ( 12) that x > k 1 .Similarly, the function V 2 can be expressed as a quadratic form: Therefore, subject to the above discussion, the existence of a continuous function V 0 (x t , y t , z t ) along a solution (x(t), y(t), z(t)) of system (2), we obtain In the light of the hypothesis of Theorem 2 and the mean value theorem (for the derivative), it can be easily obtained the following inequalities for the first four terms in ( 14): e(x, y, z) z − e(x, y, 0) z y z 2 =y z 2 e z (x, y, θz) ≤ 0, where 0 ≤ θ ≤ 1 .Next, the assumption ye x (x, y, 0) ≤ 0 of Theorem 2 shows that EJQTDE, 2010 No. 12, p. 11 Combining the above estimates with that into (14), we get If we take γ = (1+α)(L 1 +L 3 ) 2 and µ = (1+α)L 2
This completes the proof of Theorem 2.
Remark.Our result includes and improves the results of Sadek [16], Sinha [18], Tunç [22] and Zhu [28], which investigated the stability of solutions to third order nonlinear differential equations with delay.Because, equation ( 1) is more general than that considered in the above mentioned papers, and all papers were published without an explanatory example on EJQTDE, 2010 No. 12, p. 13 the stability of solutions.Our assumptions and the Lyapunov functional constructed here are also completely different than that exist in Bereketoglu and Karakoç [1] and Chukwu [3].Now, let p(t, x(t), x(t − r), y(t), y(t − r), z(t)) = 0 .
Our second and last main result is the following.
Theorem 3. Let us assume that assumptions (i)-(iii) of Theorem 2 hold.In addition, we assume that |p(t, x(t), x(t − r), y(t), y(t − r), z(t))| ≤ q(t) for all t, x, x(t − r), y, y(t − r) and z, where q ∈ L 1 (0, ∞) , L 1 (0, ∞) is space of integrable Lebesgue functions.Then, there exists a finite positive constant K such that the solution x(t) of equation ( 1) defined by the initial functions Proof.For the proof of this theorem, as in Theorem 2, we use the Lyapunov functional.
V 0 = V 0 (x t , y t , z t ) given in (11).Obviously, it can be followed from the discussion of Theorem 2 that there exists a positive D 3 such that where D 3 = min 2 −1 δ 1 k 1 , 2 −1 δ 2 .Now, the time derivative of functional V 0 (x t , y t , z t ) along system (2) can be revised as: Clearly, (y 2 + z 2 ) ≤ D −1 3 V 0 (x t , y t , z t ).
Thus, all the assumptions of Theorem 2 hold.We omit details.
and α is a positive constant.If there is a closed subset B ⊂ C H such that the solution remains in B , then α = ∞ .Further, the symbol |. | will denote the norm in ℜ n with |x| = max 1≤i≤n |x i | .Definition 1. (See [2].)Let f (t, 0) = 0 .The zero solution of equation (

Definition 6 .
(See[18].)Let V be a continuous scalar function on C n H , where C n H denotes the set of φ in C n for which φ < H , and C n denotes the spaces of continuous functions mapping from the interval [−r, 0] into ℜ n and for φ ∈ C H , φ = sup −r≤φ≤0 φ(θ)


x, s, 0) − a] sds.Now, aα > 1 .Hence the matrix is positive definite.Making use of the positive definiteness of above matrix and assumption e(x, y, 0) ≥ a > 0 of Theorem 2, we conclude that there exists a positive constant δ 2 such that