Existence of Periodic Solutions to Nonlinear Differential Equations of Third Order with Multiple Deviating Arguments

It is known that functional differential equations, in particular, that delay differential equations can be used as models to describe many physical, biological systems, and so forth. In reality, many actual systems have the property aftereffect, that is, the future states depend not only on the present, but also on the past history, and after effect is also known to occur in mechanics, control theory, physics, chemistry, biology, medicine, economics, atomic energy, information theory, and so forth Burton 1 , Kolmanovskii and Myshkis 2 . Therefore, it is important to investigate the qualitative behaviors of functional differential equations. In 1978, using the known theorem of Yoshizawa 3, Theorem 37.2 , Chukwu 4 found certain sufficient conditions that guarantee the existence of a periodic solution to nonlinearlinear differential of the third order with the constant deviating argument h >0 :


Introduction
It is known that functional differential equations, in particular, that delay differential equations can be used as models to describe many physical, biological systems, and so forth.In reality, many actual systems have the property aftereffect, that is, the future states depend not only on the present, but also on the past history, and after effect is also known to occur in mechanics, control theory, physics, chemistry, biology, medicine, economics, atomic energy, information theory, and so forth Burton 1 , Kolmanovskii and Myshkis 2 .Therefore, it is important to investigate the qualitative behaviors of functional differential equations.
In 1978, using the known theorem of Yoshizawa 3, Theorem 37.2 , Chukwu 4 found certain sufficient conditions that guarantee the existence of a periodic solution to nonlinearlinear differential of the third order with the constant deviating argument h >0 : x f x, x , x x g x t − h , x t − h i x t − h p t, x, x , x t − h , x t − h , x .

International Journal of Differential Equations
Later, in 1992, Zhu 5 considered the nonlinear differential equation of the third order with the constant deviating argument r >0 : x ax φ x t − r f x p t , 1.2 and he discussed the existence of periodic solutions for this equation when p t is a periodic function of period T , T > 0.
In 2000, Tejumola and Tchegnani 6 considered the nonlinear differential equation of the third order with the constant deviating argument τ >0 :

1.3
The authors established certain sufficient conditions on the existence of periodic of solutions of this equation.
In 2010, Tunc ¸ 7 established certain sufficient conditions for the existence of a periodic solution for the nonlinear differential equation of the third order with the constant deviating argument r >0 : However, a review to date of the literature indicates that the existence of periodic solutions to the nonlinear differential equation of the third order with multiple deviating arguments has not been investigated.The paper considers the nonlinear differential equation of the third order with multiple constant deviating arguments τ i , i 1, 2, . . ., n :

1.5
The equation 1.5 is stated in system form as follows: The motivation for this paper is a result of the research mentioned regarding ordinary differential equations with a deviating argument.Our aim is to achieve the results established in 5, 7 to 1.5 with multiple deviating arguments.Our results generalize the results established on the existence of periodic solution in 5, 7 .This paper is the first known publication regarding the existence of periodic solution for differential equations of the third order with multiple deviating arguments.
In order to reach our main result, this paper offers fundamental information regarding the general nonautonomous delay periodic differential system.Consider the delay periodic system: where F : 0, ∞ ×C H → n is a continuous mapping, F t T, ϕ F t, ϕ for all ϕ ∈ C and for some constant T > 0. We assume that F takes closed bounded sets into bounded sets of n .Here C, • is the Banach space of continuous function φ : −r, 0 → n with supremum norm, r > 0; for H > 0, we define Theorem 1.1.Suppose that F t, ϕ ∈ C 0 ϕ and F t, ϕ is periodic in t of period T, T ≥ r, and consequently for any α > 0 there exists an may be large), and that V t, ϕ satisfies the following conditions.
i Continuous increasing functions a s and b s exist, satisfying a s > 0, b s > 0 for s ≥ H and a s → ∞ as s → ∞, such that ii A continuous and positive function w s exists such that where γ * > 0 is a constant which is determined in the following way.
Using the condition on V t, ϕ , constants α > 0, β > 0, and Under these conditions, a periodic solution of 1.7 of period T exists.In particular, the relation rL γ * < H 1 − H is always satisfied if r is sufficiently small (see Yoshizawa [3]).

Main Result
The main result is the following theorem: let τ max 1≤i≤n τ i .
Theorem 2.1.Suppose that positive constants a, b i , c, m, δ, L i , and τ exist such that the following conditions hold: Proof.Define a Lyapunov functional V V x t , y t , z t by: where M, M > 1 , and γ i are certain positive constants; the constants γ i will be determined later in the proof.

2.5
Using the assumptions of Theorem 2.1, have where It is also clear that the function V 2 is continuous and satisfies In view of 2.3 , 2.7 , 2.8 , and the assumptions of Theorem 2.1, it can be shown that V satisfies the condition i of Theorem 1.1.
Using a basic calculation, the time derivative of V 1 along solutions of

2.11
Using the assumptions sup{f x } c > 0, ψ y ≥ a and ab − c > 0, and the estimation μ ab c /2b, it follows that μm y m|z|.

2.13
An easy calculation from V 2 x, y, z and 1.6 leads to using the assumptions of Theorem 2.1.First, we consider V in the domain max{|y| − K, |z| − M} ≥ 0, where the constants K and M are large enough, which will be determined later.We have to discuss the following two cases.Case 1 0 |y| ≥ K ≥ 1, and x, z are arbitrary .In this case, it follows that:

2.17
We now consider the term and define h ab 3c 2 ab c < 1.

2.19
Then, there exists a constant

2.21
where then the above estimate implies V x t , y t , z t ≤ −δ 1 y 2 z 2 μm a δ m 1 y m N 1 τ |z| 2.24 for a positive constant δ 1 .Let

2.35
Subject to the evidence thus far, we can conclude that there exists a positive constant R, which is large enough, such that V x t , y t , z t ≤ −w u for u 2 ≥ R 2 , 2.36 where u x 2 y 2 z 2 1/2 .Thus, the Lyapunov functional V x t , y t , z t satisfies all the assumptions of Theorem 1.1.The proof for Theorem 2.1 is complete.

2.37
which is a special case of 1.5 .
• is periodic in t of period T , T ≥ τ i , the derivatives g i y ≡ d/dy g i y exist and are also continuous; throughout what follows x t , y t , and z t are abbreviated as x,y, and z, respectively.