Qualitative Behaviors of Functional Differential Equations of Third Order with Multiple Deviating Arguments

and Applied Analysis 3 i gi 0 g 0 h 0 0, a t ≥ 2α a, βi ≤ bi t ≤ Bi, 0 < c1 ≤ h′ x ≤ c, αb − c > δ, gi ( y ) y ≥ bi, g ( y ) y ≥ b, y / 0 ) , ∣g ′ i ( y )∣∣ ≤ Li, 2.1 ii ∑n i 1{αbibi t } − c y2 ≥ 2−1αa′ t y2 ∑n i 1 b ′ i t ∫y 0 gi η dη.


Introduction
In this paper, we consider nonautonomous differential equation of the third order with constant multiple deviating arguments, τ, τ i , i 1, 2, . . . , n as follows: x t a t x t where τ and τ i , i 1, 2, . . . , n , are positive constants, that is, fixed delays; the functions a, b i , g, g i , h, and p are continuous for their all respective arguments and the primes in 1.1 denote differentiation with respect to t, t ∈ 0, ∞ . It is also assumed that the derivatives a t ≡ d/dt a t , b i t ≡ d/dt b i t , g i y ≡ d/dy g i y , and h x ≡ d/dx h x exist and are continuous; throughout the paper x t , y t , and z t are abbreviated as x, y, and z, respectively. Finally, the existence and uniqueness of solutions of 1.1 are assumed and all solutions considered are supposed to be real valued.
To the best of our knowledge from the literature, in the last five decades, there has been much attention paid to the discussion of stability and boundedness of solutions of nonlinear differential equations of the third order without a deviating argument. For a comprehensive treatment of the subject, we refer the readers to the book of Reissig  It should be noted that throughout all the above mentioned papers, Lyapunov's functions or the Lyapunov-Krasovskii functionals have been used as a basic tool to prove the results established there. It is also worth mentioning that the most effective method to study the stability and boundedness of solutions of nonlinear differential equations of higher orders without or with a deviating argument in the literature is still the Lyapunov's direct method, despite its use for a past long period by now.
The motivation for this paper comes from the above mentioned papers and the books. Our results improve and include the results existing in the literature. This work makes also a contribution to the existing studies made in the literature.

Main Results
Let p · 0. Our first result is given by the following theorem.
Theorem 2.1. In addition to the basic assumptions imposed to the functions a t , b t , g, g i , and h appearing in 1.1 , we assume that there exist positive constants a, α, β, δ, b, b i , B i , c, c 1 , and L i such that the following conditions hold: where λ and λ i are some positive constants to be chosen later.
Using the assumptions

2.5
On the other hand, it is obvious that

2.7
Hence, we can obtain some positive constants D j , j 1, 2, 3 , such that where

2.9
Using the assumptions of Theorem 2.1 and the estimate 2|mn| ≤ m 2 n 2 , we get
Let p · 0. Our second main result is given by the following theorem.
Abstract and Applied Analysis 7 Theorem 2.2. In addition to all the assumptions of Theorem 2.1, we assume that the condition p · ≤ q t 2.16 holds, where |q| ∈ L 1 0, ∞ . If then, there exists a finite positive constant K such that the solution x t of 1.1 defined by the initial function Proof. Under the assumptions of Theorem 2.2, the time derivative of the Lyapunov-Krasovskii functional V · satisfies Using the estimates |m| < 1 m 2 and D 4 x 2 y 2 z 2 ≤ V · , it follows that
Integrating the above estimate from 0 to t, using the assumption |q| ∈ L 1 0, ∞ and the Gronwall-Bellman inequality see Gronwall 65 and Mitrinović 66 , we can conclude that all solutions of 1.1 are bounded.

Abstract and Applied Analysis
Example 2.3. Consider the nonlinear differential equation of the third order with two deviating arguments as follows:

2.22
Writing 2.22 as a system of first order equations, we obtain x y, y z,

2.24
In view of the above discussion, it follows that αb − c 4 > 0,