Nonexistence of Nontrivial Periodic Solutions to a Class of Nonlinear Differential Equations of Eighth Order

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1 BULLETIN of the Malaysian Mathematical Sciences Society Bull. Malays. Math. Sci. Soc. () 3(3) (9), Nonexistence of Nontrivial Periodic Solutions to a Class of Nonlinear Differential Equations of Eighth Order Cemil Tunç Department of Mathematics, Faculty of Arts and Sciences, Yüzüncü Yıl University, 658, Van - Turkey cemtunc@yahoo.com Abstract. By constructing a Lyapunov function, a new result is given, which guarantees the non-existence of nontrivial periodic solutions to nonlinear vector differential equation of eighth order: X (8) +AX (7) +BX (6) +CX (5) +DX (4) +E X... +F (Ẋ)Ẍ +G(X)Ẋ +H(X) =. An example is also established for the illustrations of topic. By this way, our findings raise a new result for the nonexistence of nontrivial periodic solutions related to this nonlinear vector differential equation of eighth order. Mathematics Subject Classification: 34D5, 34D Key words and phrases: Nonlinear differential equation, eighth order, periodic solution, Lyapunov s second (or direct) method. 1. Introduction By now, in the relevant literature, the instability of solutions for various nonlinear scalar and vector differential equations of eighth order has been considered only by a few authors, see, Tunç [3], Tunç and Tunç [4] and the references registered in these papers. In addition, in 1996, Iyase [] proved a result on the nonexistence of nontrivial periodic solutions to nonlinear eighth order scalar differential equation: (1.1) x (8) + a 1 x (7) + a x (6) + a 3 x (5) + a 4 x (4)... + a 5 x + f 6 (ẋ)ẍ + f 7 (x)ẋ + f 8 (x) =. Now, for basic information, we consider the linear constant coefficient differential equation of eighth order: (1.) x (8) + a 1 x (7) + a x (6) + a 3 x (5) + a 4 x (4)... + a 5 x + a 6 ẍ + a 7 ẋ + a 8 x =. It is well-known that the auxiliary equation of (1.) is given by: ψ(λ) λ 8 + a 1 λ 7 + a λ 6 + a 3 λ 5 + a 4 λ 4 + a 5 λ 3 + a 6 λ + a 7 λ + a 8 =. If β is an arbitrary real number, then the real part of ψ(iβ) is given by φ(β) = β 8 a β 6 + a 4 β 4 a 6 β + a 8. Received: July 1, 8; Revised: June 1, 9.

2 38 C. Tunç It is also well-known that if a, a 4, a 6, a 8 > in which case φ(β) >, then the auxiliary equation cannot have any purely imaginary root whatever. It therefore follows from general theory that equation (1.) does not have a periodic solution except x =. An analogous consideration of the imaginary part of ψ(iβ) also leads to conditions on a 1, a 3, a 5 and a 7 for the nonexistence of any periodic solution of (1.) other than x =. Now, in this paper, we consider instead of scalar nonlinear differential equation given by (1.1), its nonlinear vector differential equation form: (1.3) X (8) +AX (7) +BX (6) +CX (5) +DX (4) +E X... +F (Ẋ)Ẍ +G(X)Ẋ +H(X) = in the real Euclidean space R n (with the usual norm denoted in what follows by. ), where A, B, C, D and E are constant n n- symmetric matrices; F and G are continuous n n- symmetric matrix functions depending in each case on the arguments shown; H : R n R n, H is continuous and H() =. Throughout this paper, instead of equation (1.3), we consider its equivalent differential system: (1.4) Ẋ = Y, Ẏ = Z, Ż = S, Ṡ = T, T = U, U = V, V = W, Ẇ = AW BV CU DT ES F (Y )Z G(X)Y H(X), which was obtained as usual by setting Ẋ = Y,Ẍ = Z, X = S, X(4) = T,X (5) = U, X (6) = V, X (7) = W from (1.3). Let J G (X) and J F (Y ) denote the Jacobian matrices corresponding to the matrix functions G(X) and F (Y ), that is, ( gi J G (X) = x j ), J F (Y ) = ( fi y j ), (i, j = 1,,..., n), where (x 1, x,..., x n ), (y 1, y,..., y n ), (g 1, g,..., g n ) and (f 1, f,..., f n ) are the components of X, Y, G and F, respectively. In addition, it is assumed that the Jacobian matrices J G (X) and J F (Y ) exist and are continuous. The symbol X, Y corresponding to any pair X, Y in R n stands for the usual scalar product n x iy i and λ i (A), (i = 1,,..., n), are the eigenvalues of the n n-matrix A. It is also well-known that a real symmetric matrix A = (a ij ), (i, j = 1,,..., n), 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 transpose of X. Obviously, for the case n = 1, equation (1.3) reduces to equation (1.1). It should also be noted that according to our observations in the literature, it is not founded any paper on the nonexistence of nontrivial periodic solutions of nonlinear vector differential equations of eighth order. The present study is the first attempt to obtain sufficient conditions related to the nonexistence of nontrivial periodic solutions of nonlinear vector differential equations of eighth order. The motivation for the present study has come, especially, from the paper of Iyase [], Tunç [5], Tunç [6] and the papers mentioned above. Our aim is to accomplish the result of Iyase [Theorem 1, ] to the nonlinear vector differential equation of eighth order, (1.3). We shall use the following well-known algebraic result to prove our main result.

3 Nonexistence of Nontrivial Periodic Solutions to Nonlinear Differential Equations 39 Lemma 1.1. [1] Let A be a real symmetric n n matrix and a λ i (A) a > (i = 1,,..., n), where a, a are constants. Then a X, X AX, X a X, X and a X, X AX, AX a X, X.. Main result The main result is the following theorem. Theorem.1. Suppose that there are constants a 1, a, a 4 and n n- symmetric constant matrices A, B, D, n n- symmetric matrix functions F, G and n-vector function H such that the following conditions hold: λ i (A) a 1 >, λ i (B) a, λ i (D) a 4, λ i (F (Y )) for all Y R n, H() =, n H(X) if X, x i h i (X) > for all X R n, where H(X) = (h 1 (X),..., h n (X)), (i = 1,,..., n). Then equation (1.3) has no non-trivial periodic solutions. Note that there are no restrictions on C, E and G. Proof. To prove the theorem, we introduce a Lyapunov function defined as follows: (.1) V 1 = V 1 (X, Y, Z, S, T, U, V, W ) V 1 = X, W X, AV X, BU X, CT X, DS X, EZ + Y, V + Y, AU + Y, BT + Y, CS + Y, DZ + 1 Y, EY Z, U Z, AT Z, BS 1 Z, AZ + S, T S, AS F (σy )Y, X dσ σg(σx)x, X dσ. Clearly, we have V 1 (,,,,,,, ) =. Next, from the assumptions of theorem and the above lemma it follows that: V 1 (,,, ε,,,, ) = 1 ε, Aε 1 ε, a 1ε = 1 a 1 ε > for all arbitrary ε R n, ε. Thus, in every neighborhood of (,,,,,,, ), there exists a point (ξ, η, ζ, µ, τ, ω, ρ, λ) such that V 1 (ξ, η, ζ, µ, τ, ω, ρ, λ) > for arbitrary ξ, η, ζ, µ, τ, ω, ρ, λ in R n. Now, let (X, Y, Z, S, T, U, V, W ) = (X(t), Y (t), Z(t), S(t), T (t), U(t), V (t), W (t))

4 31 C. Tunç be an arbitrary periodic solution of system (1.4). function (.1) along this solution, we obtain Differentiating the Lyapunov (.) Recall that (.3) and (.4) d dt V 1 = d dt V 1(X, Y, Z, S, T, U, V, W ) d dt = T, T S, BS + Z, DZ + X, H(X) + G(X)Y, X + F (Y )X, Z d dt F (σy )Y, X dσ d dt σg(σx)x, X dσ = F (σy )Y, X dσ = d dt σg(σx)x, X dσ. σg(σx)y, X dσ + = σ G(σX)Y, X 1 = G(X)Y, X = + F (σy )X, Y dσ F (σy )X, Z dσ + F (σy )Y, Y dσ = σ F (σy )X, Z 1 + = F (Y )X, Z + σ σg(σx)y, X dσ σ σ F (σy )X, Z dσ σ F (σy )Y, Y dσ F (σy )Y, Y dσ. Now, combining estimates (.3) and (.4) into (.), we get V 1 = T, T S, BS + Z, DZ + X, H(X) Since λ i (B) a, λ i (D) a 4, F (σy )Y, Y dσ. n x i h i (X) > and λ i (F (Y )), then it is clear that V 1 (t) for all t, which implies that V 1 (t) is monotone in t. Since V 1 is continuous and X, Y, Z, S, T, U, V, W are periodic in t, V 1 (t) is bounded. Hence lim t V 1 (t) = V (constant). By using the fact V 1 (t) = V 1 (t + mω) for arbitrary fixed t and arbitrary m, ω, we have V 1 (t) = V for all t. Thus V 1 = for all t. This case necessarily implies

5 Nonexistence of Nontrivial Periodic Solutions to Nonlinear Differential Equations 311 that Y = for all t, and therefore also that X = ξ (a constant vector), Z =... Ẏ =, S = Ÿ =, T = Y =, U = Y (4) =, V = Y (5) =, W = Y (6) =, Y (7) = Ẇ = for t. Substituting the estimates X = ξ, Y = Z = S = T = U = V = W = in (1.4), it follows that H(ξ) = which necessarily implies that ξ = because of H() = and H(X) when X. Hence for all t. X = Y = Z = S = T = U = V = W = Example.1. As a special case of system (1.4), for the case n =, let us choose A, B, D, F and H as follows: [ ] [ ] [ ] A =, B =, D =, 1 3 [ ] [ ] x 3 H(X) = 1 + x 5 1 y x 3 + x 5, F (Y ) = 1 y y1 y. Then, respectively, we have λ 1 (A) =, λ (A) = 4, λ 1 (B) =, λ (B) =, λ 1 (D) =, λ (D) = 6, x i F i (x) = x x x 4 + x 6 > for all x 1 and x, and λ 1 (F (Y )) = λ (F (Y )) = y 1 y. Thus, all the conditions of Theorem.1 hold. References [1] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Corrected reprint of the 1991 original, Cambridge Univ. Press, Cambridge, [] S. A. Iyase, Periodic solutions of certain eighth order differential equations, J. Nigerian Math. Soc. 14/15 (1995/96), [3] C. Tunç, Instability of solutions of a certain non-autonomous vector differential equation of eighth-order, Ann. Differential Equations (6), no. 1, 7 1. [4] Dzh. Tundzh and E. Tundzh, Differ. Uravn. 4 (6), no. 1, , 143; translation in Differ. Equ. 4 (6), no. 1, [5] C. Tunç, About uniform boundedness and convergence of solutions of certain non-linear differential equations of fifth-order, Bull. Malays. Math. Sci. Soc. () 3 (7), no. 1, 1 1. [6] C. Tunç, On the stability and boundedness of solutions of nonlinear vector differential equations of third order, Nonlinear Anal. 7 (9), no. 6, [7] C. Tunç, On the existence of periodic solutions to a certain fourth-order nonlinear differential equation, Ann. Differential Equations 5 (9), no. 1, 8 1.