. ISSN (print), (online) International Journal of Nonlinear Science Vol.6(2008) No.3,pp

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1 . ISSN (print), (online) International Journal of Nonlinear Science Vol.6(8) No.3,pp A Bouneness Criterion for Fourth Orer Nonlinear Orinary Differential Equations with Delay Cemil Tunç Department of Mathematics, Faculty of Arts an Sciences Yüüncü Yıl University, 658, Van, TURKEY. (Receive 6 September 7, March accepte 8) Abstract: In this paper, by using Lyapunov s secon metho, we stuy a certain fourth orer nonlinear orinary elay ifferential equation an obtain some sufficient conitions for the bouneness of solutions of this equation. Keywors: fourth orer nonlinear elay ifferential equation; bouneness; lyapunov functional AMS Classification number: 34K. 1 Introuction Especially, since 196s many goo books, most of them are in Russian literature, ealt with elay ifferential equations. For a comprehensive treatment of subject we refer the reaer to the book by Burton [1], Èl sgol ts [], Èl sgol ts an Norkin [3], Hale [4], Hale an Veruyn Lunel [5], Kolmanovskii an Myshkis [6], Kolmanovskii an Nosov [7], Krasovskii [8], Yoshiawa [13] an the references citie in these books. With respect to our observations from the literature, it is only foun two works achieve on the bouneness of solutions of fourth orer nonlinear elay ifferential equations. These works can be summarie as follows: First, in 1989, Okoronkwo [1] consiere the fourth-orer nonlinear elay ifferential equation of the form x (4) (t) + f(x (t))x (t) + α x (t) + β x (t h) + g(x (t h)) + α 4 x(t) + β 4 x(t h) = p(t). Subject specifie conitions impose on the functions f, g, p an the constants α, α 4, β, β 4 appeare in this equation, he establishe some sufficient conitions that guarantee the bouneness of the solutions of the equation. Later, in, Tejumola&Tchegnani [11] took into consieration the fourth-orer nonlinear elay ifferential equation x (4) (t) + ϕ(t, x(t), x (t), x (t), x (t))x (t) + ψ(t, x (t τ), x (t τ)) + χ(t, x(t τ), x (t τ)) +h(x(t τ)) = p (t, x(t), x (t), x (t), x (t), x(t τ), x (t τ), x (t τ)). They prove a result [11, Theorem.4] on the uniformly boune an uniformly ultimately boune of solutions of this equation. In this paper we are concerne with the fourth orer nonlinear elay ifferential equations of the type x (4) (t) + ϕ(x (t))x (t) + h(x (t r)) + φ(x (t r)) + f(x(t r)) = p(t, x(t), x (t), x (t), x (t), x(t r), x (t r), x (t r)) (1) Corresponing author. aress: cemtunc@yahoo.com Copyright c Worl Acaemic Press, Worl Acaemic Union IJNS /18

2 196 International Journal of Nonlinear Science,Vol.6(8),No.3,pp in which ϕ, h, φ, f an p epen only on the variables isplaye explicitly an r is a positive constant, fixe elay; the primes in equation (1) enote ifferentiation with respect to t, t [, ). It is assume as basic that the functions ϕ, h, φ, f an p are continuous in their respective arguments an satisfy a Lipschit conition in x(t), x (t), x (t), x (t), x(t r), x (t r) an x (t r); h() = φ() = f() = an the erivatives φ x φ (x ) an f x f (x) exist an are also continuous. Equation (1) can be transforme into an equivalent system of the form u = ϕ()u h() φ(y) f(x) + + x = y, y =, = u, h ((s))u(s)s + φ (y(s))(s)s f (x(s))y(s)s + p(t, x(t), y(t), (t), u(t), x(t r), y(t r), (t r)) where x(t), y(t), (t) an u(t) are respectively abbreviate as x, y, an u throughout the paper. All solutions consiere are also assume to be real value. It shoul be note in proving the main result of this paper we make use of Lyapunov s secon metho [9] as in Okoronkwo [1] an Tejumola an Tchegnani [11]. Our assumptions an the Lyapunov functional use here will be completely ifferent than that in Okoronkwo [1] an Tejumola &Tchegnani [11]. Preliminaries In orer to reach our main result, we give some important basic information for the general non-autonomous elay ifferential system (see also Èl sgol ts [], Èl sgol ts an Norkin [3], Hale [4], Hale an Veruyn Lunel [5], Kolmanovskii an Myshkis [6], Kolmanovskii an Nosov [7], Krasovskii [8] ). Now, we consier the general non-autonomous elay ifferential system ẋ = f(t, x t ), x t (θ) = x(t + θ), θ, t, (3) where f : [, ) C H R n is a continuous mapping, f(t, ) =, an we suppose that f takes close boune sets into boune sets of R n. Here (C,. ) is the Banach space of continuous function φ : [, ] R n with supremum norm, r >,C H is the open H -ball in C ; C H := {φ (C [, ], R n ) : φ < H}. Stanar existence theory, see Burton [1, pp.31], shows that if φ C H an t, then there is at least one continuous solution x(t, t, φ) such that on [t, t + α) satisfying equation (3) for t > t, x t (s, t, φ) = φ t (s) an α is a positive constant. If there is a close subset B C H such that the solution remains in B, then α =. Further, the symbol. will enote the norm in R n with x = max 1 i n x i. Definition 1 (See [1, pp.3].) A continuous function W : [, ) [, ) with W () =, W (s) > if s >, an W strictly increasing is a wege. (We enote weges by W or W i, where i an integer.) Definition (See [1, pp. 6].) Let V (t, φ) be a continuous functional efine for t, φ C H. The erivative of V along solutions of (3) will be enote by V (3) an is efine by the following relation V (3) (t, φ) = lim sup h where x(t, φ) is the solution of (3) with x t (t, φ) = φ. V (t + h, x t+h (t, φ)) V (t, x t (t, φ)), h Definition 3 (See [13, pp.184].) A function x(t, φ) is sai to be a solution of (3) with the initial conition φ C H at t = t, t, if there is a constant A > such that x(t, φ) is a function from [t h, t + A] into R n with the properties: (i) x t (t, φ) C H for t t < t + A, (ii) x t (t, φ) = φ, (iii) x(t, φ) satisfies (3) for t t < t + A. () IJNS for contribution: eitor@nonlinearscience.org.uk

3 Cemil Tunç: A Bouneness Criterion for Fourth Orer Nonlinear Orinary 197 Theorem 1 (See [13, pp.184].) If f(t, φ) in (3) is continuous in t, φ, for every φ C H1, H 1 < H, an t, t < c, where c is a positive constant, then there exist a solution of (3) with initial value φ at t = t, an this solution has a continuous erivative for t > t. 3 Main result First, we introuce the following notations: ϕ 1 () = 1 ϕ(τ)τ, ϕ(), = an φ 1 (y) = { φ(y) y, y φ (), y =. Our main result is the following theorem. Theorem In aition to the basic assumptions impose on ϕ, h, φ, f an p, we assume the following conitions are satisfie: (i) There are positive constants α 1, α, α 3, α 4,, L, 1,, 3 an ε such that α 1 α α 3 α 3 φ (y) α 1 α 4 ϕ() > for all y an, in which ε α 1 α 3 D, D = α 1α + α α 3 α 4 ; (ii) < α 4 α 1 4α 3 < f (x) α 4 for all x; (iii) φ (y) α 3 an φ 1 (y) α 3 < ε (iv) h() α α 3 8α 4 (v) ϕ() α 1, ϕ 1 () ϕ() < 8α 3 α4 α 1 α 3 for all y; α 1 for all, ( ) an h () L for all ; for all ; α 1 α 3 (vi) p(t, x(t), y(t), (t), u(t), x(t r), y(t r), (t r)) q(t), where max q(t) < an q L 1 (, ), L 1 (, ) is space of integrable Lebesgue functions. Then, there exists a finite positive constant K such that the solution x(t) of equation (1) efine by the initial functions x(t) = φ(t), x (t) = φ (t), x (t) = φ (t), x (t) = φ (t) satisfies the inequalities x(t) K, x (t) K, x (t) K, x (t) K for all t t, where φ C 3 ([t r, t ], R), provie that { εα 3 r < min ( α 4 + L + α 1 α + λ), 8α 1 α 3 (α 1 α + L + α 4 + µ), } εα 1 ( 1 α 1 α + 1 L + 1 α 4 + ρ) with λ = α 4 ( ) >, µ = α 1α ( ) > an ρ = L ( ) >. Remark 3 Making use of conitions(i),(iii) an (v) of Theorem we obtain that ϕ() < α α 3 α 4, φ (y) < α 1 α. IJNS homepage:

4 198 International Journal of Nonlinear Science,Vol.6(8),No.3,pp Proof. Now, to verify Theorem, we introuce the Lyapunov functional x y V (x t, y t, t, u t ) = f(ξ)ξ + α y 1 α 4 y + φ(η)η + 1 h(ζ)ζ + ϕ(τ)ττ + 1 u + f(x)y + 1 f(x) + 1 φ(y) + y ϕ(τ)τ + yu + u + λ y (θ)θs (4) +µ (θ)θs+ρ u (θ)θs where 1 = ε + 1, = ε + α 4, α 1 α 3 an λ, µ an µ are some positive constants which will be etermine later in the proof. In view of the assumptions of Theorem, one can easily obtain that ( ) ( ) ( ) V ε α 4 α 1 4α 3 x + α4 y + 8α 1 α + εu 3 4α 1 α 3 +λ y (θ)θs+µ (θ)θs+ρ u (θ)θs D 1 x + D y + D 3 + D 4 u +λ y (θ)θs (5) +µ (θ)θs+ρ u (θ)θs D 1 x + D y + D 3 + D 4 u ( D 5 x + y + + u ), ( ) where D 1 = 1 ε α 4 α 1 4α 3, D = α 4, D 8α 1 α 3 = 3 16α 1 α, D 4 = ε an D 5 = min {D 1, D, D 3, D 4 }. 3 (See, also, for the etails of the operations to Tunç [1]). Now, ifferentiating the functional V = V (x t, y t, t, u t ) in (4), we have [ ] [ ] t V (x t, y t, t, u t ) = [α 4 f (x)]. y + 1 h() 1 φ (y) ϕ 1 () [ ] [α 4 f (x)] [ 1 ϕ() 1] u φ(y) y α 4 y [ h() α ] y 1 [α 4 f (x)] y +( 1 u + + y)p(t, x(t), y(t), (t), u(t), x(t r), y(t r), (t r)) +( 1 u + + y) h ((s))u(s)s + ( 1 u + + y) φ (y(s))(s)s (6) +( 1 u + + y) +µ r µ f (x(s))y(s)s + λy r λ (s)s + ρu r ρ u (s)s. y (s)s IJNS for contribution: eitor@nonlinearscience.org.uk

5 Cemil Tunç: A Bouneness Criterion for Fourth Orer Nonlinear Orinary 199 By following some similar lines taken place in Tunç [1], one can easily obtain that t V (x t, y t, t, u t ) ( ) ( ) εα 3 y 8α 1 α 3 (εα 1 )u + ( 1 u + + y) +( 1 u + + y) φ (y(s))(s)s + ( 1 u + + y) h ((s))u(s)s f (x(s))y(s)s +( 1 u + + y)p(t, x(t), y(t), (t), u(t), x(t r), y(t r), (t r)) +λy r λ y (s)s + µ r µ (s)s + ρu r ρ u (s)s. Now, in view of the assumptions f (x) α 4, φ (y) α 1 α, h () L an ab a + b, it follows the following inequalities for some terms containe in (6): ( 1 u + + y) h ((s))u(s)s 1L ru (t) + L r (t) + L ry (t) ( 1 u + + y) + L ( ) u (s)s, φ (y(s))(s)s 1α 1 α ru (t) + α 1α r (t) + α 1 α ry (t) an ( 1 u + + y) + α 1α ( ) (s)s f (x(s))y(s)s 1α 4 ru (t) + α 4 r (t) + α 4 ry (t) Substituting these estimates into (6) we get + α 4 ( ) y (s)s. t V (x t, y t, t, u t ) [ εα 3 1 ( α 4 + L + α 1 α + λ)r ] y ( ) 8α 1 α 3 1 (α 1α + L + α 4 + µ)r +( 1 u + + y)p(t, x(t), y(t), (t), u(t), x(t r), y(t r), (t r)) ( εα 1 1 ( 1α 1 α + 1 L + 1 α 4 + ρ)r ) u + [ α 4 ( ) λ ] t + [ α 1 α ( ) µ ] + [ L ( ) ρ ] t y (s)s (s)s u (s)s. IJNS homepage:

6 International Journal of Nonlinear Science,Vol.6(8),No.3,pp an Let us choose Hence λ = α 4 ( ) >, µ = α 1α ( ) > ρ = L ( ) >. t V (x t, y t, t, u t ) [ εα 3 1 ( α 4 + L + α 1 α + λ)r ] y Now, in fact, we can obtain t V (x t, y t, t, u t ) τ(y + + u ) ( ) 8α 1 α 3 1 (α 1α + L + α 4 + µ)r ( εα 1 1 ( 1α 1 α + 1 L + 1 α 4 + ρ)r ) u +( 1 u + + y)p(t, x(t), y(t), (t), u(t), x(t r), y(t r), (t r)). + 1 u + + y p(t, x(t), y(t), (t), u(t), x(t r), y(t r), (t r)) for some constant τ > provie that { εα 3 r < min ( α 4 + L + α 1 α + λ), 8α 1 α 3 (α 1 α + L + α 4 + µ), } εα 1 ( 1 α 1 α + 1 L + 1 α 4 + ρ) Therefore t V (x t, y t, t, u t ) 1 u + + y p(t, x(t), y(t), (t), u(t), x(t r), y(t r), (t r)) D 6 ( y + + u ) q(t) for a constant D 6 > by (vi), where D 6 = max { 1, 1, }. Now, making use of the inequalities y < 1 + y, < 1 + an u < 1 + u, it is clear that t V (x ( t, y t, t, u t ) D y + + u ) q(t). By (5), we also have ( y + + u ) ( x + y + + u ) D5 1 V (x t, y t, t, u t ). Hence t V (x t, y t, t, u t ) D 6 ( 3 + D 1 5 V (x t, y t, t, u t ) ) q(t) = 3D 6 q(t) + D 6 D 1 5 V (x t, y t, t, u t )q(t). Now, integrating the last inequality from to t, using the assumption q L 1 (, ) an Gronwall- Rei-Bellman inequality, we obtain V (x t, y t, t, u t ) V (x, y,, u ) + 3D 6 A + D 6 D5 1 (V (x s, y s, s, u s )) q(s)s ( (V (x, y,, u ) + 3D 6 A) exp D 6 D5 1 ) q(s)s (V (x, y,, u ) + 3D 6 A) exp ( D 6 D 1 5 A) = K 1 <, (7) IJNS for contribution: eitor@nonlinearscience.org.uk

7 Cemil Tunç: A Bouneness Criterion for Fourth Orer Nonlinear Orinary 1 where K 1 > is a constant, K 1 = (V (x, y,, u ) + 3D 6 A) exp ( D 6 D5 1 A) = K 1 < an A = q(s)s. Now, the inequalities (5) an (7) together yiels that x (t) + y (t) + (t) + u (t) D 1 5 V (x t, y t, t, u t ) K, where K = K 1 D5 1. Thus, we can conclue that for all t t. That is, for all t t. The proof of Theorem is complete. References x(t) K, y(t) K, (t) K, u(t) K x(t) K, x (t) K, x (t) K, x (t) K [1] T.A.Burton: Stability an perioic solutions of orinary an functional ifferential equations. Acaemic Press, Orlano(1985). [] L. È.Èl sgol ts: Introuction to the theory of ifferential equations with eviating arguments. Translate from the Russian by Robert J. McLaughlin Holen-Day, Inc., San Francisco, Calif.-Lonon- Amsteram(1966). [3] L. È. Èl sgol ts, S. B. Norkin: Introuction to the theory an application of ifferential equations with eviating arguments. Translate from the Russian by John L. Casti. Mathematics in Science an Engineering, Vol. 15. Acaemic Press [A Subsiiary of Harcourt Brace Jovanovich, Publishers], New York-Lonon(1973). [4] J. Hale: Theory of functional ifferential equations. Springer-Verlag, New York-Heielberg(1977). [5] J. Hale, S. M. Veruyn Lunel: Introuction to functional-ifferential equations. Applie Mathematical Sciences, 99. Springer-Verlag, New York(1993). [6] V. Kolmanovskii, A. Myshkis: Introuction to the theory an applications of functional ifferential equations. Kluwer Acaemic Publishers, Dorrecht(1999). [7] V. B. Kolmanovskii, V. R. Nosov: Stability of functional-ifferential equations. Mathematics in Science an Engineering, Acaemic Press, Inc. [Harcourt Brace Jovanovich, Publishers], Lonon(1986). [8] N. N. Krasovskii: Stability of motion. Applications of Lyapunov s secon metho to ifferential systems an equations with elay. Translate by J. L. Brenner Stanfor University Press, Stanfor, Calif (1963). [9] A.M.Lyapunov: Stability of Motion. Acaemic Press, Lonon(1966). [1] O. E. Okoronkwo: On stability an bouneness of solutions of a certain fourth-orer elay ifferential equation, Internat. J. Math. Math. Sci. 1(3): (1989). [11] H. O. Tejumola, B. Tchegnani: Stability, bouneness an existence of perioic solutions of some thir an fourth orer nonlinear elay ifferential equations. J. Nigerian Math. Soc. 19: 9-19 (). [1] C.Tunç: On the stability of solutions of certain fourth-orer elay ifferential equations.applie Mathematics an Mechanics (English Eition). 7(8): (6). [13] T.Yoshiawa: Stability theory by Liapunov s secon metho. The Mathematical Society of Japan,Tokyo(1966). IJNS homepage: