The bounded solutions to nonlinear fifth-order differential equations with delay

The bounded solutions to nonlinear fifth-order differential equations with delay

Cemil Tunç

Department of Mathematics, Faculty of Arts and Sciences, Yüzüncü Yıl University, 65080, Van-Turkey E-mail: [e-mail]

ABSTRACT

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.

1 Introduction

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 [1], Chukwu [3], Sinha [14], Tunç [15, 16, 17], Yuan Hong [21] 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 [12] 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 [2], Èl'sgol'ts[4], Èl'sgol'ts and Norkin [5], Gopalsamy [6], Hale [7], Hale and Verduyn Lunel[8], Kolmanovskii and Myshkis [9], Kolmanovskii and Nosov [10], Krasovskii [11], Makay [13] and Yoshizawa [20] 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 ƒ₁, ƒ₅ 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]).

2 Preliminaries

In order to reach our main result, we will give some basic information for the general non-autonomous delay differential system, see also Burton [2], Èl'sgol'ts [4], Èl'sgol'ts and Norkin [5], Gopalsamy [6], Hale [7], Hale and Verduyn Lunel [8], Kolmanovskii and Myshkis [9], Kolmanovskii and Nosov [10], Krasovskii [11], Makay [13] and Yoshizawa [20]. Now, we consider the general non-autonomous delay differential system

where ƒ: [0, ) × C_H

is a continuous mapping, ƒ(t, 0) = 0, and we suppose that ƒ 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; C_His the open H-ball in C;

.

Standard existence theory, see Burton [2], shows that if Φ C_Hand 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, x_t(t, Φ) = Φ and α is a positive constant. If there is a closed subset B C_Hsuch that the solution remains in B, then α =. Further, the symbol |.| will denote the norm in with |x|= max_1<i<n |x_i|.

Definition 1 (See [2]). 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 W_i, where i an integer.)

Definition 2 (See [2]). 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 [2]). Let V (t, Φ) be a continuous functional defined for t > 0, Φ C_H. 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 x_t0 (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:

and

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, ); ƒ is continuously differentiable satisfying the conditions: x^-1 ƒ(x) > α₂,(x 0), and |ƒ'(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, x_t , y_t) 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

we find

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 ƒ₁, ƒ₅and 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< ƒ₁(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 ε₁satisfies the inequality

.

(iii) ƒ₅(0) = 0, ƒ₅(x) 0 if x 0,x^-1 ƒ₅(x) > α(x 0), and ƒ'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 ΦC⁴ ([t₀ - r, t₀], f), provided that

.

Proof. For the proof, we introduce the Liapunov functional V₀ = V₀ (t, x_t , y_t, z_t,w_t , u_t):

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

where

The assumptions ƒ₅(0) = 0, ƒ₅(x) sgn x > 0, x^-1 ƒ₅(x) > α (x 0), ƒ'₅(x) < α₅ and (5) imply that

and

provided that

,

which we now assume. Now, the estimates related to V₁ and V₂ yield

Following a similar way as that of Tunç [18], 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

and

By the assumption ƒ'₅(x) < α₅ and inequality 2 |uν|< u² + ν², we obtain the following:

and

.

Making use of these inequalities, we get

where

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

where

.

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,

,

and

.

Now, the inequality (7) and the last inequality together give that

,

where K = K₁D₈^-1. This fact completes the proof of theorem.

Received: 18/IX/08.

Accepted: 09/II/09.

#CAM-23/08.

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Publication Dates

• Publication in this collection
  08 July 2009
• Date of issue
  2009

History

• Received
  18 Sept 2008
• Accepted
  09 Feb 2009