Result on the Convergence Behavior of Solutions of Certain System of Third-Order Nonlinear Differential Equations ()
Received 30 November 2015; accepted 4 March 2016; published 7 March 2016
1. Introduction
We shall consider here systems of real differential equations of the form
(1)
which is equivalent to the system
(2)
where and H are continuous vector functions and is an -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,; denote the real interval. and in Equation (1)., are the Jacobian matrices corresponding to the vector functions and respectively exist and are symmetric, positive definite and continuous.
by extending the result of [17] to the special case of [17] . Also recently, Olutimo [20] studied the equation
a variant of (1), 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, of (1) are said to converge if
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.
2. 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 matrices. Then for any.
where and are the least and greatest eigenvalues of D, respectively.
Proof of Lemma 1. See [3] [7] .
Lemma 2. Let be real symmetric commuting matrices. Then,
1) The eigenvalues of the product matrix are all real and satisfy
2) The eigenvalues of the sum of Q and D are all real and satisfy
where and are respectively the eigenvalues of Q and D.
Proof of Lemma 2. See [3] [7] .
Lemma 3. Subject to earlier conditions on the following is true
where and are the least and greatest eigenvalues of D, respectively.
Proof of Lemma 3. See [20] .
Lemma 4. Subject to earlier conditions on and that, then
1)
2)
Proof of Lemma 4. See [20] .
Lemma 5. Subject to earlier conditions on and that, then
1)
2)
Proof of Lemma 5. See [3] [7] [11] .
3. Statement of Results
Throughout the sequel are the Jacobian matrices corresponding to the vector
functions, respectively.
Our main result which gives an estimate for the solutions of (1) is the following:
Theorem 1. Assume that and, for all in are all symmetric. Jacobian matrices exist, positive definite and continuous. Furthermore, there are positive constants such that the following conditions are satisfied.
Suppose that and that
1) The continuous matrices, and are symmetric, associative and commute pairwise. Then eigenvalues of, of and of , satisfy
2) P satisfies
(3)
for any (i = 1, 2) in, and is a finite constant. Then, there exists a finite constant such that any two solutions of (2) necessarily converge if.
Our main tool in the proof of the result is the function defined for any in by
(4)
where
and is a fixed constant chosen such that
(5)
(6)
chosen such that.
The following result is immediate from (4).
Lemma 6. Assume that, all the hypotheses on matrix and vectors and in Theorem 1 are satisfied. Then there exist positive constants and such that
(7)
Proof of Lemma 6. In the proof of the lemma, the main tool is the function in (4).
This function, after re-arrangement, can be re-written as
Since
And
we have that
Since matrix is assumed symmetric and strictly positive definite. Consequently the square root exists which itself is symmetric and non-singular for all Therefore, we have
(8)
where stands for.
Thus,
(9)
From (9), the term
(10)
Since
by integrating both sides from to and because, then we obtain
But from
integrating both sides from to and because, we find
Hence, (10) becomes
combining the estimate for in (9), we have
By hypothesis (1) of Theorem 1 and lemmas 1 and 2, we have
where and by (5).
Similarly, after re-arrangement becomes
(11)
It is obvious that
also,
and
Combining all the estimates of and (11), we have
Now, combining and we must have
that is,
(12)
Thus, it is evident from the terms contained in (12) that there exists sufficiently small positive constants such that
where
The right half inequality in lemma 6 follows from lemma 1 and 2.
Thus,
where
Hence,
(13)
4. Proof of Theorem 1
Let, be any two solutions of (2), we define
By
where V is the function defined in (4) with replaced by respectively.
By lemma 6, (13) becomes
(14)
for and.
The derivative of with respect to t along the solution path and using Lemma 3, 4 and 5, after simplification yields
where, , and .
Using the fact that
and
where
Following (8),
and
Thus,
Note that
and
We have;
On applying Lemma 1 and 2, we have
If we choose, such that it satisfies (6), and using (3), we obtain
where
Thus,
with.
There exists a constants such that
In view of (14), the above inequality implies
(15)
Let be now fixed as. Thus, last part of the theorem is immediate, provided and on integrating (15) between and t, we have
which implies that
Thus, by (14), it shows that
From system (1) this implies that
This completes the proof of Theorem 1.
5. 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 such that is a function of t only and
Thus,
Clearly, and are symmetric and commute pairwise. That is,
and
Then, by easy calculation, we obtain eigenvalues of the matrices and as follows
It is obvious that, , , , and.
If we choose, we must have that
Thus, all the conditions of Theorem 1 are satisfied. Therefore, all solutions of (1) converge since (5) and (6) hold.