On the stability and boundedness of certain third order non-autonomous di ff erential equations of retarded type

In this paper, based on the Lyapunov-Krasovskii functional approach, we obtain sufficient conditions which guarantee stability, uniformly stability, boundedness and uniformly boundedness of solutions of certain third order nonautonomous differential equations of retarded type. Our results complement and improve some recent ones. Subjclass : [2010]34K20.

Differential equations of the type (1.1) and (1.2) have been shown to be useful in modeling many phenomena in various fields of science and engineering and in more recent years to problems in biomathematics (see, for example, Cronin-Scanlon [8] and Smith [19]).One special case of nonlinear differential equations of third order is what is known as the jerky dynamics equation x 000 (t) + k 1 (x(t), x 0 (t))x 00 (t) + k 2 (x(t), x 0 (t), x 00 (t)) = 0 that has gained some attention in the literature (see, Chlouverakis and Sprott [7], Eichhorn et al. [9], Elhadj and Sprott [10] and Linz [14]).Besides, qualitative properties of solutions of third order differential equations such as stability, instability, boundedness, oscillation, and periodicity of solutions have been studied by many authors; in this regard, we refer the reader to the monograph by Reissig et al. [17], and the recent papers of Adams et al. [1], Ademola and Arawomo [2], Afuwape and Adesina [3], Bai and Guo [5], Ogundare and Okecha [15], Rauch [16], Sadek [18], Tunç ([20]- [27]), Zhang and Yu [29] and the references cited therein.However, to the best of our knowledge, there exist few results on the mentioned qualitative behaviors of solutions for the non-autonomous third order differential equations of retarded type (see, the references of this paper).Motivated by the above discussions, the main purpose of this paper is to give some sufficient conditions for the stability, uniformly stability, boundedness and uniformly boundedness of solutions of equation (1.1).Our results complement and improve some recent ones.
One tool to be used here is the stability and boundedness theorems.
Let us consider non-autonomous delay differential equation ), is a continuous mapping, F (t, 0) = 0, and we suppose that F takes closed bounded sets into bounded sets of < n .
First, we will give some basic definitions.[6]) A continuous functions W : < + → < + with W (0) = 0, W (s) > 0 if s > 0, and W strictly increasing is a wedge.(We denote wedges by W or W i , where i an integer.) The following theorems are basic tools for our results.
Assume that: Define the function H(x, y) by and Ω 0 is a domain of the two dimensional Euclidean space < 2 .
Then, the function H(x, y) = Lx 2 + 2Mxy + Ny 2 is positive definite and decrescent, where Proof.By noting the assumptions of Lemma 2.1, it follows that Then, we can conclude that where K = min{[inf L(x, y)] for all x, y ∈ Ω 0 , N}, K > 0. This means that H(x, y) is positive definite.It is also clear that the quadratic form H(x, y) can be rearranged as Let λ 1 (x, y) and λ 2 (x, y) denote the characteristic roots of the matrix T (x, y).Then, it is clear that where K = sup[λ 2 1 (x, y) + λ 2 2 (x, y)] for all x, y ∈ Ω 0 , and K > 0. Thus, the function H(x, y) = Lx 2 + 2Mxy + Ny 2 is decrescent.This completes the proof of Lemma 2.1.2 Theorem 2.1.Assume that p(t) ≡ 0, conditions (C1) − (C7) hold, and Then, the zero solution of equation (2.1) is stable.
Consider the terms It is clear that ∂x , g y = ∂g ∂y .In view of the assumption (C4), it follows that y y R 0 g x (x, u)du ≥ 0.