Result on the Convergence Behavior of Solutions of Certain System of Third-Order Nonlinear Differential Equations

Convergence behaviors of solutions arising from certain system of third-order nonlinear differential equations are studied. Such convergence of solutions corresponding to extreme stability of solutions when P 0 ≠ relates a pair of solutions of the system considered. Using suitable Lyapunov functionals, we prove that the solutions of the nonlinear differential equation are convergent. Result obtained generalizes and improves some known results in the literature. Example is included to illustrate the result.


Introduction
We shall consider here systems of real differential equations of the form which is equivalent to the system where Φ and H are continuous vector functions and Ψ is an n n × -positive definite continuous symmetric matrix function, for the argument displayed explicitly and the dots here as elsewhere stand for differentiation with respect to the independent variable t, t + ∈  ; +  denote the real interval 0 t ≤ < ∞ .n X ∈  and : (1).

( )
JH X are the Jacobian matrices corresponding to the vector functions ( ) ( ) H X respectively exist and are symmetric, positive definite and continuous.
So far in the literature, much attention has been drawn to the boundedness of solutions of ordinary scalar and vector nonlinear differential equations of third order.The book of Reissig et al. [1], the papers by Abou-El-Ela [2], Afuwape [3] [4], Chukwu [5], Ezeilo [6], Ezeilo and Tejumola [7], Meng [8], Omeike [9], Omeike and Afuwape [10], Tiryaki [11], Tunc [12] [13], Tunc and Ates [14], Tunc and Mohammed [15] and the references cited therein have comprehensive treatment of the subject.Throughout the results present in the book of Reissig et al. [1] and the papers mentioned above, Lyapunov's second (direct) method has been used as a basic tool to verify the results established in these works.Equations of the form (1) in which ( )  have been studied by [16] [17].They have obtained some results related to the convergence properties of solutions as well as Afuwape in [18].Very recently, Tunc and Gozen [19] studied the convergence of solution of the equation by extending the result of [17] to the special case ( )  of [17].Also recently, Olutimo [20] studied the equation , where c is a positive constant and obtained some results which guarantee the convergence of the solutions.With respect to our observation in the literature, no work based on (1) was found.The result to be obtained here is different from that in Olutimo [20] and the papers mentioned above.The intuitive idea of convergence of solutions also known as the extreme stability of solutions occurs when the difference between two equilibrium positions tends to zero as time increases infinitely is of practical importance.This intuitive idea is also applicable to nonlinear differential system.The Lyapunov's second method allows us to predict the convergence property of solutions of nonlinear physical system.Result obtained generalizes and improves some known results in the literature.Example is included to illustrate the result.

Definition
Definition 1.1.Any two solutions ( ) If the relations above are true of each other (arbitrary) pair of solutions of (1), we shall describe this saying that all solutions of (1) converge.

Some Preliminary Results
We shall state for completeness, some standard results needed in the proofs of our results.Lemma 1.Let D be a real symmetric n n × matrices.Then for any n X ∈  .
2) The eigenvalues of the sum of Q and D are all real and satisfy where d δ and d ∆ are the least and greatest eigenvalues of D, respectively.Proof of Lemma 3. See [20].Lemma 4. Subject to earlier conditions on Φ and that ( ) See [20].Lemma 5. Subject to earlier conditions on ( ) H X and that ( )

Statement of Results
Throughout the sequel , respectively.Our main result which gives an estimate for the solutions of (1) is the following:  are all symmetric.Jaco- bian matrices ( ) ( ) , JH X J Y Φ exist, positive definite and continuous.Furthermore, there are positive constants α β γ α β γ such that the following conditions are satisfied.
Suppose that ( ) ( )  ( ) , , , , , , for any , , and o ∆ is a finite constant.Then, there exists a finite constant 0 >  such that any two solutions where , µ chosen such that 0 1 µ < < .The following result is immediate from (4).Lemma 6. Assume that, all the hypotheses on matrix ( ) ( ) H X in Theorem 1 are satisfied.Then there exist positive constants 1 D and 2 D such that ( ) ( ) Proof of Lemma 6.In the proof of the lemma, the main tool is the function . This function, after re-arrangement, can be re-written as .
Since matrix J Φ is assumed symmetric and strictly positive definite.Consequently the square root 1 2 J Φ exists which itself is symmetric and non-singular for all .
It is obvious that { } ( ) ( ) Combining all the estimates of 2 V and (11), we have ( ) Thus, it is evident from the terms contained in ( 12) that there exists sufficiently small positive constants 3 D such that ( ) The right half inequality in lemma 6 follows from lemma 1 and 2. Thus, ( )

Proof of Theorem 1
Let ( ) X t be any two solutions of (2), we define ( ).
where V is the function defined in (4) with , , for 3 0 D > and 4 0 D > .The derivative of ( ) W t with respect to t along the solution path and using Lemma 3, 4 and 5, after simplification yields 1 On applying Lemma 1 and 2, we have If we choose µ , such that it satisfies (6), and using (3), we obtain min , , δ δ δ δ = .There exists a constants 6 In view of (14), the above inequality implies ( ) ( ) Let  be now fixed as This completes the proof of Theorem 1.

Conclusions
Analysis of nonlinear systems literary shows that Lyapunov's theory in convergence of solutions is rarely scarce.
The second Lyapunov's method allows predicting the convergence behavior of solutions of sufficiently complicated nonlinear physical system.Example 4.0.1.As a special case of system (2), let us take for 2 n = such that 0 P ≠ is a function of t only and ( ) and JH are symmetric and commute pairwise.That is,