Print version ISSN 2238-3603
On-line version ISSN 1807-0302
Comput. Appl. Math. vol.28 no.2 São Carlos 2009
The bounded solutions to nonlinear fifth-order differential equations with delay
Department of Mathematics, Faculty of Arts and Sciences, Yüzüncü Yıl University, 65080, Van-Turkey E-mail: firstname.lastname@example.org
In this paper, we improve some boundedness results, which have been obtained with respect to nonlinear differential equations of fifth order without delay, to a certain functional differential equation with constant delay. We give an illustrative example and also verify our main result by means of Liaponov tecnique.
Mathematical subject classification: 34K20.
Key words: boundedness, Liapunov functional, nonlinear differential equations of fifth.
Among the scores of articles on the qualitative theory of differential equations, the number of articles on boundedness of solutions to nonlinear fifth order differential equations with delay is significantly less than those on differential equations without delay. For those contributions on the boundedness of solutions of nonlinear fifth order differential equations without delay, one can refer to the papers of Abou-El Ela and Sadek , Chukwu , Sinha , Tunç [15, 16, 17], Yuan Hong  and some other references thereof. For those regarding the boundedness of solutions of nonlinear fifth order differential equations with delay, wecite Tunç ([18, 19]). All of the aforementioned contributions have focused onthe Liapunov's second (direct) method  utilizing Liapunov functions and functionals.
It is also worth mentioning that the construction of Liapunov functions and functionals for higher order nonlinear differential equations still remains as a general problem. In fact, the construction of Liapunov functional for delay differential equations of higher orders is more difficult than the derivation of Liapunov function for differential equations without delay. Since 1960 many excellent books, most of them in Russian, have been published on the qualitative behaviors of delay differential equations. See, for example, Burton , Èl'sgol'ts, Èl'sgol'ts and Norkin , Gopalsamy , Hale , Hale and Verduyn Lunel, Kolmanovskii and Myshkis , Kolmanovskii and Nosov , Krasovskii , Makay  and Yoshizawa  and the references listed in these books.
In this paper, we consider nonlinear fifth order delay differential equation
which is equivalent to the system
where 1, 5 and p are continuous functions for the arguments displayed explicitly in equation (1); r is a positive constant , that is, r is a constant delay; α2, α3 and α4 are some positive constants. This fact guarantees the existence of the solution of delay differential equation (1) (see Èl'sgol'ts [4, p. 14]). It is assumed that the derivative exists and is continuous for all x and all solutions considered are assumed to be real valued. In addition, we assume that the right-hand side of system (2) satisfies a Lipschitz condition in x(t), y(t), z(t), w(t), u(t), x(t - r), y(t - r), z(t - r), w(t - r) and u (t - r). Then the solution is unique (see Èl'sgol'ts [4, p. 15]). Throughout the paper x(t), y(t), z(t), w(t) and u(t) are also abbreviated as x, y, z, w and u, respectively. It should be noted that the equation considered here, (1), is completely different than that investigated by Tunç ([18, 19]).
In order to reach our main result, we will give some basic information for the general non-autonomous delay differential system, see also Burton , Èl'sgol'ts , Èl'sgol'ts and Norkin , Gopalsamy , Hale , Hale and Verduyn Lunel , Kolmanovskii and Myshkis , Kolmanovskii and Nosov , Krasovskii , Makay  and Yoshizawa . Now, we consider the general non-autonomous delay differential system
where : [0, ) × CH is a continuous mapping, (t, 0) = 0, and we suppose that takes closed bounded sets into bounded sets of . Here (C, .) is the Banach space of continuous function Φ : [ -r, 0] with supremum norm, r > 0; CH is the open H-ball in C;
Standard existence theory, see Burton , shows that if Φ CH and t > 0, then there is at least one continuous solution x(t, t0, Φ) such that on [t0, t0 + α) satisfying equation (3) for t > t0, xt (t, Φ) = Φ and α is a positive constant. If there is a closed subset B CH such that the solution remains in B, then α =. Further, the symbol |.| will denote the norm in with |x|= max1<i<n |xi|.
Definition 1 (See ). A continuous function W : [0, ) [0, ) with W (0) = 0, W (s) > 0 if s > 0, and W strictly increasing is a wedge. (We denote wedges by W or Wi , where i an integer.)
Definition 2 (See ). Let D be an open set in with 0 D. A function V : [0, ) × D [0, ) is called positive definite if V (t, 0) = 0 and if there is a wedge W1 with V (t, x) > W1(|x|), and is called decrescent if there is a wedge W2 with V (t, x) < W2(|x|).
Definition 3 (See ). Let V (t, Φ) be a continuous functional defined for t > 0, Φ CH . The derivative of V along solutions of (3) will be denoted by and is defined by the following relation
where x(t0, Φ) is the solution of (3) with xt0 (t0, Φ) = Φ.
Example. Let us consider the nonlinear second order delay differential equation:
which is equivalent to the system
We assume that the functions and p are continuous and satisfy the following:
for all t, t [0, ), x, x(t - r), y and y(t - r), where r is a positive constant, constant delay, which will be determined later; max q(t)< and q L1(0, ); is continuously differentiable satisfying the conditions: x-1 (x) > α2,(x 0), and |'(x)| < L for all x; α1,α2 and L are some positive constants. In particular, let us take
Then, it follows that
If we choose
then we have
that is, q L1(0, ). Now, we introduce the Liapunov functional
where λ is a positive constant which will be determined later. It is clear that the functional V (t, xt , yt ) is positive definite:
for all x( 0) and y. Along a trajectory of the equation, we have
Hence, we get
In view of afore mentioned assumptions, the inequality 2 |uν|< u2 + ν2 and the fact
If we choose , then we have
Therefore, it follows that
for some constant α > 0 provided that r < . Thus, we get
Now, integrating this inequality from 0 to t and using the fact L1(0, ), we have
Therefore, one can conclude that
for all t > 0. That is,
for all t > 0. This fact shows that all solutions of the equation considered are bounded.
3 Main result
We establish the following result.
Theorem. In addition to the basic assumptions imposed on the functions 1, 5 and p appearing in equation (1), we assume that there are positive constants α, α1, α2, α3, α4, ε, ε0, ε1 , δ and λ such that the following conditions hold:
(ii) ε0 < 1(t, x(t - r), y(t - r), z(t - r), w(t - r), u(t - r)) - α1 < ε1 for all t, x(t - r), y(t - r), z(t - r), w(t - r) and u(t - r), where the constant ε1 satisfies the inequality
(iii) 5(0) = 0, 5(x) 0 if x 0,x-1 5(x) > α(x 0), and '5(x) < α5 for all x.
(iv) |p(t, x(t - r), y(t - r), z(t - r), w(t - r), u(t - r))|< q(T) for all t, x(t - r), y(t - r), z(t - r), w(t - r) and u(t - r), where max q(T)< and q L1(0, ).
Then, there exists a finite positive constant K such that the solution x(t) of equation (1) defined by the initial functions
satisfies the inequalities
for all t > t0, where Φ C4 ([t0 - r, t0], f), provided that
Proof. For the proof, we introduce the Liapunov functional V0 = V0 (t, xt , yt, zt ,wt , ut ):
where λ, µ and ρ are some positive constants which will be determined later in the proof and δ is also a positive constant satisfying
Now, in view of (4), it follows that
The assumptions 5(0) = 0, 5(x) sgn x > 0, x-1 5(x) > α (x 0), '5(x) < α5 and (5) imply that
which we now assume. Now, the estimates related to V1 and V2 yield
Following a similar way as that of Tunç , one can easily conclude from (6) that
for a sufficiently small positive constant D8. Now, let
be a solution of system (2). Then, a direct computation along this solution shows that
In view of the assumptions of theorem and expression (5), we have that
By the assumption '5(x) < α5 and inequality 2 |uν|< u2 + ν2, we obtain the following:
Making use of these inequalities, we get
Now, subject to the assumptions (ii) and (iii) of theorem, one easily finds that
V4 > 0, V5 > 0, V6 > 0, V7 > 0, V8 > 0,
respectively. Gathering the above discussion into (8) and making use of the assumption (ii), it follows that
If we let
then subject to the assumption (i) of theorem the inequality (9) implies that
Thus, one easily obtains that
for some constant τ > 0 provided that
Clearly, we have
Using the fact |y| < 1 + y2, |z| < 1 + z2, |w| < 1 + w2 and |u| < 1 + u2,we see that
In view of (7), it follows that
Finally, integrating this inequality from t0, (t0 > 0), to t and using the assumption q L1(0, ) and Gronwall-Reid-Bellman inequality, we conclude that
where K1 > 0 is a constant,
Now, the inequality (7) and the last inequality together give that
where K = K1 D8-1. This fact completes the proof of theorem.
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