On the Periodic Solutions of Certain Fifth Order Nonlinear Vector Differential Equations

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1 On the Periodic Solutions of Certain Fifth Order Nonlinear Vector Differential Equations Melike Karta Department of Mathematics, Faculty of Science and Arts, Agri Ibrahim Cecen University, Agri, Abstract The purpose of this paper is to show that under some sufficient conditions of equation (.) have no periodic solution other than the trivial solution X=. Namely, we improve some well-known periodic results for certain fifth order nonlinear vector differential equations. Keywords: Nonlinear vector differential equation of fifth, Lyapunov function, Periodic solutions.. Introduction Periodic solutions of high order scalar and vector differential equations have emerged in the applied sciences as some practical mechanical problems, physics, chemistry, biology, ecomomics, and control theory. According to observations in the literature, the problems related to the periodic behavior of solutions of a higher order non-linear scalar and vector differential equation have been investigated by many authors. For some related contributors to the subject, we refer to the papers of Li & Duan [], Bereketoğlu [2-3], Ezeilo [4-6], Tejumola [8], Tiryaki [9] and Tunç []. In all of the papers mentioned above, authors used Lyapunov s second (or direct) method []. In this article, based on study of form x i = x i+ ( i,2,3,4 ) x i = a 5 x 5 f 4 (x, x 2, x 3, x 4, x 5 )x 4 f 3 (x 2 )x 3 f 2 (x, x 2, x 3, x 4, x 5 )x 2 f (x ) (f 2 () = f () = ) produced by Li & Duan [5], we investigate that have no periodic solution other than the trivial solution X = of the fifth- order non-linear vector differential equations of the form X (5) + AX (4) + θ(x, X, X, X, X (4) )X + F(X )X + G(X, X, X, X, X (4) )X + H(X) = (.) in the real Euclidean space R n (with the usual norm denoted in what follows by. ), in which X R n ; A is constant n n-symmetric matrix;,f and G are n n-symmetric continuous matrix functions depending, in each case, on the arguments shown; H: R n R n is continuous n-vector function. It is supposed that the function H is continuous and H() =, H(X) for X. Let J H (X) and J F (Y) display the Jacobian matrices corresponding to the functions H(X) and F(Y), respectively, that is, J H (X) = ( h i x j ), J F (Y) = ( f i y j ), (i,j=,2,,n) 38 P a g e

2 in which (x, x 2,, x n ), (y, y 2,, y n ), (f, f 2,,, f n ) and (h, h 2,, h n ), are the components of X, Y, F and H, respectively. The symbol X, Y corresponding to any pair X,Y in R n stands for the usual scalar product n i= x i y i, and the matrix A = (a ij ) is said to be positive definite if and only if the quadratic form X T AX is positive definite, where X R n and X T denotes the traspose of X. Throughout this paper, we consider the following differential systems which are equivalent to the equation (.2) which was attained as usual by setting X = Y, X = Z, X = W, X (4) = U from (.2): X = Y, Y = Z, Z = W, W = U U = AU θ(x, Y, Z, W, U)W F(Y)Z G(X, Y, Z, W, U)Y H(X) (.2) 2. Main Result Our main result is the following theorem. Here we show that equation (.) have no periodic solution other than the trivial solution X =. Theorem 2.. In addition to the basic conditions given above for coefficients A, Θ, F, G and H of (.2) equation, it is assumed that A and J H (X) are symmetric matrices and there is constant a, a positive constant as the following conditions hold: (i) i (A) > a, (i =,2,,n) (ii) λ i (G(X, Y, Z, W, U)) 4 [λ iθ(x, Y, Z, W, U)] 2 >, (i =,2,, n) Then trivial solution X = of (.2) is instability. In prove main result, we use the following lemma. Lemma2.2. Let A be a real symmetric n n-matrix and a ı i (A) a > ( i,2,..., n), (2.) in which a ı, a are constants. Then X, X AX, X a X, X, a ı2 X, X AX, AX a 2 X, X. (2.2) See [2] for proof. Proof. Let (ε, ε 2, ε 3, ε 4, ε 5 ) = (ε (t), ε 2 (t), ε 3 (t), ε 4 (t), ε 5 (t)) be an arbitrary -periodic solution of (.2), that is (ε (t), ε 2 (t), ε 3 (t), ε 4 (t), ε 5 (t)) = (ε (t + α), ε 2 (t) + α, ε 3 (t + α), ε 4 (t + α), ε 5 (t + α)) (2.3) 39 P a g e

3 for some >.It will be shown that, under the conditions in Theorem (2.), ε = ε 2 = ε 3 = ε 4 = ε 5 =. As basic tool the proof of Theorem (2.), we will use Lyapunov function V = V (X, Y, Z, W, U) given as V = Y, U + Z, W AY, W + AZ, Z F(σY)Y, Y dσ H(σX), X dσ (2.4) 2 Consider the function θ(t) = V (ε (t), ε 2 (t), ε 3 (t), ε 4 (t), ε 5 (t)). Since (t) is continuous and, 2, 3, 4, 5 elementary differentiation will show that are periodic in t, (t) is clearly bounded. An V = d V(X, Y, Z, W, U) = W, W + Y, θ(x, Y, Z, W, U)W + Y, F(Y)Z = Y, G(X, Y, Z, W, U)Y + Y, H(X)Y d σf(σy, Y) dσ d H(σX), X dσ (2.5) Now, we recall that d σf(σy, Y) dσ = σ 2 σf(σy)z, Y = F(Y)Z, Y (2.6) and. d H(σX), X dσ = σ H(σX), Y = H(X), Y. (2.7) Substituting (2.5) and (2.6) into (2.4) and taking into account the conditions of the Theorem 2., we obtain V = W, W + Y, θ(x, Y, Z, W, U)W + Y, G(X, Y, Z, W, U)Y = W + θ(x, Y, Z, W, 2 U)Y 2 θ(x, Y, Z, W, U)Y, θ(x, Y, Z, W, U)Y 4 + Y, G(X, Y, Z, W, U)Y 4 P a g e

4 Y, G(X, Y, Z, W, U)Y θ(x, Y, Z, W, U)Y, θ(x, Y, Z, W, U)Y. (2.8) 4 Hence ( t) ;so that (t) is monotone in t, and therefore, being bounded, tends to a limit, say, as t.that is lim ( t).it is readily showed that t ( t ) for all to t. (2.9) from by (2.), (t) = (t+m ) (2.) for any arbitrary fixed t an for arbitrary integer m, and then letting m in the right-hand side of (2.) leads to (2.9). the result (2.7) itself implies that ( t) for all t. Furthermore θ (t) = necessarily implies that ε 2 = for all t.thus, if ε 2 =ε 3 =ε 4 =ε 5 = are written in (.2) system, we can clearly see that it is H( ε ) = H(X ) =, because of H( ) = and H( X) for X under conditions of Theorem (2.) and ε =X.So that ε =X =.It is ε = ε 2 =ε 3 =ε 4 =ε 5 = for all t.since (ε, ε 2, ε 3, ε 4, ε 5 ) is a solution of (.2), it is evident that (ε (t), ε 2 (t), ε 3 (t), ε 4 (t), ε 5 (t)) = (X(t), Y(t), Z(t), W(t), U(t)) = (,,,,) This completes the proof of Theorem (2.). 3.Conclusion The proof of the Theorem2. showed that the equation (.) have no periodic solution other than the trivial solution X =. So, this has helped us to find a new result. 4. References [] Li, W., Duan, K. (2). Instability theorems for some nonlinear differential systems of fifth order. (Chinese) J. Xinjiang Univ. Natur. Sci., 7, -5. [2] Bereketoğlu, H. (992).On the periodic solutions of certain class of seventh- order differential equations. Periodica Mathematica Hungarica., 24, [3] Bereketoğlu, H. (992). On the periodic solutions of certain class of eighth- order differential equations. Commun.Fac. Sci.Univ.Ank.Series A., 4, [4] Ezeilo, J. O. C. (979). Periodic solutions of a certain fourth order differential equations. Atti. Accad. Naz.Lincei CI. Sci. Fis. Mat. Natur. LXVI, P a g e

5 [5] Ezeilo, J. O. C. (983). Uniqueness theorems for periodic solutions of certain fourth and fifth order differential systems. Journal of the Nigerian mathematical society. 2, [6] Ezeilo, J. O. C. (979). Ezeilo, J. O. C. (979). A further instability theorem for a certain fifth-order differential equation. Math. Proc. Cambridge Philos. Soc., 86, [7] Mırsky, L. (99). An Introduction to the linear algebra. Dover Publications, Inc., New York. [8] Tejumola, H.O. (2). Instability and periodic solutions of certain nonlinear differential equations of orders six and seven, in: Ordinary differential equations (Abuja, 2). Proc. Natl. Math. Cent.Abuja Niger.,., Natl. Math. Cent., [9] Tiryaki, A. (99). On the periodic solutions of certain fourth and fifth order differential equations. Pure and Applied Mathematika Sciences., XXXII, -2. [] Tunç, E. (25). On the periodic solutions of certain fourth and fifth order vector differential equations. Mathematical Communıcatıons., [] Lyapunov, A. M. (966). Stability of Motion. Academic Press, New York-London, 23p. [2] Horn, R. A., Johnson, C.R. (994). Topics in Matrix Analysis. Cambridge University Press, Cambridge, UK. 42 P a g e