The behavior of solutions of second order delay differential equations

J. Math. Anal. Appl. 332 27) 1278 129 www.elsevier.com/locate/jmaa The behavior of solutions of second order delay differential equations Ali Fuat Yeniçerioğlu Department of Mathematics, The Faculty of Education, Kocaeli University, 4138 Kocaeli, Turkey Received 28 September 26 Available online 8 December 26 Submitted by J. Henderson Abstract In this paper, we study the behavior of solutions of second order delay differential equation y t) = p 1 y t) + p 2 y t τ)+ q 1 yt) + q 2 yt τ), where p 1, p 2, q 1, q 2 are real numbers, τ is positive real number. A basic theorem on the behavior of solutions is established. As a consequence of this theorem, a stability criterion is obtained. 26 Elsevier Inc. All rights reserved. Keywords: Delay differential equation; Characteristic equation; Stability; Trivial solution 1. Introduction and preliminaries Let us consider initial value problem for second order delay differential equation y t) = p 1 y t) + p 2 y t τ)+ q 1 yt) + q 2 yt τ), t, 1) yt) = φt), t, 2) where p 1, p 2, q 1, q 2 are real numbers, τ is positive real number and φt) is a given continuously differentiable initial function on the interval [,]. In many fields of the contemporary science and technology systems with delaying links are often met and the dynamical processes in these are described by systems of delay differential equations [1,4,5]. The delay appears in complicated systems with logical and computing devices, E-mail address: fuatyenicerioglu@kou.edu.tr. 22-247X/$ see front matter 26 Elsevier Inc. All rights reserved. doi:1.116/j.jmaa.26.1.69

A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 1279 where certain time for information processing is needed. The theory of linear delay differential equations has been developed in the fundamental monographs [1,4 6,8]. The equation of form of 1) is of interest in biology in explaining self-balancing of the human body and in robotics in constructing biped robots see [1,12]). These are illustrations of inverted pendulum problems. A typical example is the balancing of a stick see [13]). Equation of form of 1) can be used as test of equations for numerical methods see [7,14]). In [3], it has been established the boundedness under the conditions b>, k>, p> k and q +r p <b,on[, ) of the solution of the second order equation y t) + by t) + qy t r)+ kyt) + pyt r) =, for t, where r>, together with a given continuously differentiable initial function yt) = φt) on [ r, ]. Recently, Cahlon and Schmidt et al. [2] have established the stability criteria for a second order delay differential equation of form 1) with p 1 p 2 and q 1 q 2 <. This paper deals with the stability of the trivial solution for a second order linear delay differential equation with constant delay. An estimate of the solutions is established. The sufficient conditions for the stability, the asymptotic stability and instability of the trivial solution are given. Our results are derived by the use of real roots with an appropriate property) of the corresponding in a sense) characteristic equations. The techniques applied in obtaining our results are originated in a combination of the methods used in [9] and [11]. As usual, a twice continuously differentiable real-valued function y defined on the interval [, ) is said to be a solution of the initial value problem 1) and 2) if y satisfies 1) for all t and 2) for all t. It is known that see, for example, [4]), for any given initial function φ, there exists a unique solution of the initial problem 1) 2) or, more briefly, the solution of 1) 2). If we look for a solution of 1) of the form yt) = e λt for t IR, we see that λ is a root of the characteristic equation λ 2 = p 1 λ + p 2 λe λτ + q 1 + q 2 e λτ. 3) Before closing this section, we will give two well-known definitions see, for example, [5]). The trivial solution of 1) is said to be stable at ) if for every ε>, there exists a number l = lε) > such that, for any initial function φ with φ max φt) <l, t the solution y of 1) 2) satisfies yt) <ε, for all t [, ). Otherwise, the trivial solution of 1) is said to be unstable at ). Moreover, the trivial solution of 1) is called asymptotically stable at ) if it is stable in the above sense and in addition there exists a number l > such that, for any initial function φ with φ <l, the solution y of 1) 2) satisfies lim yt) =.

128 A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 2. Statement of the main results and comments Let now y be the solution of 1) 2). Define xt) = e λ t yt), for t [, ), where λ is a real root of the characteristic equation 3). Then, for every t, we have x t) + 2λ x t) + λ 2 xt) = p 1x t) + p 1 λ xt) + p 2 e λτ x t τ) + p 2 λ e λτ xt τ)+ q 1 xt) + q 2 e λτ xt τ) or [ x t) + 2λ p 1 )xt) p 2 e λ τ xt τ) ] = p 1 λ + q 1 λ 2 ) xt) + p2 λ + q 2 )e λ τ xt τ). 4) Moreover, the initial condition 2) can be equivalently written xt) = e λ t φt), for t [,]. 5) Furthermore, by using the fact that λ is a root of 3) and taking into account 5), we can verify that 4) is equivalent to x t) + 2λ p 1 )xt) p 2 e λτ xt τ) = x ) + 2λ p 1 )x) p 2 e λτ x) + p 1 λ + q 1 λ 2 ) t xs)ds + p 2 λ + q 2 )e λ τ t xs τ)ds, x t) = p 1 2λ )xt) + p 2 e λτ xt τ)+ p 1 λ + q 1 λ 2 ) t xs)ds + p 2 λ + q 2 )e λ τ xs)ds + φ ) + 2λ p 1 )φ) p 2 φ), x t) = p 1 2λ )xt) + p 2 e λτ xt τ)+ p 1 λ + q 1 λ 2 ) t xs)ds + p 2 λ + q 2 )e λ τ xs)ds + Lλ ; φ), t x t) = p 1 2λ )xt) + p 2 e λτ xt τ) p 2 λ + q 2 )e λ τ xs)ds + p 2 λ + q 2 )e λ τ xs)ds + Lλ ; φ),

A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 1281 where Let x t) = p 1 2λ )xt) + p 2 e λ τ xt τ) p 2 λ + q 2 )e λ τ Lλ ; φ) = φ ) + 2λ p 1 )φ) p 2 φ)+ p 2 λ + q 2 )e λ τ β λ p 2 λ + q 2 )τe λ τ + 2λ p 1 p 2 e λ τ and we define zt) = xt) Lλ ; φ), for t. β λ Then we can see that 6) reduces to the following equivalent equation z t) = p 1 2λ )zt) + p 2 e λ τ zt τ) p 2 λ + q 2 )e λ τ t t xs)ds + Lλ ; φ), 6) e λ s φs)ds. 7) zs) ds. 8) If we look for a solution of 8) of the form zt) = e δt for t IR, we see that δ is a root of the second characteristic equation δ = p 1 2λ + p 2 e λ τ e δτ δ 1 1 e δτ) p 2 λ + q 2 )e λ τ. 9) On the other hand, the initial condition 5) can be equivalently written zt) = φt)e λt Lλ ; φ), t [,]. 1) β λ Let Fδ)denote the characteristic function of 9), i.e., Fδ)= δ p 1 + 2λ p 2 e λ τ e δτ + δ 1 1 e δτ) p 2 λ + q 2 )e λ τ. Since δ = is a removable singularity of Fδ), we can regard Fδ)as an entire function with F) = p 2 λ + q 2 )τe λ τ + 2λ p 1 p 2 e λ τ β λ. But, by the definition of β λ, a root of the characteristic equation 9) must become δ. Let now z be the solution of 8) 1) and δ be a real root of the characteristic equation 9). Define for δ vt) = e δ t zt), Then, for every t, we have for all t [, ). v t) = p 1 2λ δ )vt) + p 2 e λ +δ )τ vt τ) p 2 λ + q 2 )e λ τ τ e δ s vt s)ds. 11)

1282 A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 Moreover, the initial condition 1) can be equivalently written vt) = φt)e λ +δ )t e δ t Lλ ; φ), for t [,]. 12) β λ Furthermore, by using the fact that δ is a real root of 9) and taking into account 12), we can verify that 11) is equivalent to t vt) = v) + p 1 2λ δ ) p 2 λ + q 2 )e λ τ τ e δ s vs)ds + p 2 e λ +δ )τ t vt) = φ) Lλ ; φ) β λ + p 1 2λ δ ) p 2 λ + q 2 )e λ τ t vt) = p 1 2λ δ ) + p 2 e λ +δ )τ τ vs)ds e δ s t s s t vu s)du ds, t vs τ)ds vs)ds + p 2 e λ +δ )τ vu)du ds, e δ s e λs φs) Lλ ) ; φ) ds + β λ + φ) Lλ ; φ) p 2 λ + q 2 )e λ τ τ e δ s β λ s e δ u e λu φu) Lλ ) t s ; φ) du+ β λ vs)ds vs)ds vu)du ds, vt) = Rλ,δ ; φ) + δ 1 1 e δ τ ) p 2 λ + q 2 )e λτ p 2 e λ +δ )τ ) t vs)ds + p 2 e λ +δ )τ vs)ds p 2 λ + q 2 )e λ τ vt) = Rλ,δ ; φ) + p 2 λ + q 2 )e λ τ + p 2 e λ +δ )τ τ vs)ds p 2 λ + q 2 )e λ τ τ e δ s t s e δ s ds p 2 e λ +δ )τ τ e δ s t s vu)du ds, ) t vs)ds vu)du ds,

A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 1283 where vt) = Rλ,δ ; φ) p 2 e λ +δ )τ + p 2 λ + q 2 )e λ τ τ e δ s t t vs)ds t s Rλ,δ ; φ) = φ) Lλ ; φ) β λ + p 2 e λ +δ )τ Next, we define p 2 λ + q 2 )e λ τ τ vu)du ds, 13) e δ s s e δ s e λs φs) Lλ ) ; φ) ds β λ e δ u e λu φu) Lλ ) ; φ) du ds. β λ 14) wt) = vt) Rλ,δ ; φ), η λ,δ for t, 15) where η λ,δ 1 + p 2 e λ +δ )τ τ δ 2 1 e δ τ δ τe δ τ ) p 2 λ + q 2 )e λτ. 16) Then we can see that 13) reduces to the following equivalent equation t τ t wt) = p 2 e λ +δ )τ ws)ds + p 2 λ + q 2 )e λ τ e δ s wu)du ds. 17) On the other hand, the initial condition 12) can be equivalently written wt) = φt)e λ +δ )t Lλ ; φ) e δt Rλ,δ ; φ) β λ η λ,δ for t [,]. 18) We have the following basic theorem. Theorem 1. Let λ and δ δ ) be real roots of the characteristic equations 3) and 9). Assume that the roots λ and δ have the following property μ λ,δ p 2 e λ +δ )τ τ + δ 2 1 e δ τ δ τe δ τ ) p 2 λ + q 2 e λτ < 1 19) and β λ p 2 λ + q 2 )τe λ τ + 2λ p 1 p 2 e λ τ. Then, for any φ C[,], IR), the solution y of 1) 2) satisfies e λ +δ )t yt) Lλ ; φ) e δt Rλ,δ ; φ) β λ η Mλ,δ ; φ)μ λ,δ, for all t, λ,δ 2) t s

1284 A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 where Lλ ; φ), Rλ,δ ; φ) and η λ,δ were given in 7), 14) and 16), respectively and Mλ,δ ; φ) = max t e λ +δ )t φt) Lλ ; φ) e δt Rλ,δ ; φ) β. 21) λ η λ,δ Proof. It is easy to see that property 19) guarantees that η λ,δ >. Applying the definitions of x, z, v and w we can obtain that 2) is equivalent to wt) Mλ,δ ; φ)μ λ,δ t. 22) So, we will prove 22). From 18) and 21) it follows that wt) Mλ,δ ; φ), for t [,]. 23) We will show that Mλ,δ ; φ) is a bound of w on the whole interval [, ). Namely wt) Mλ,δ ; φ), for all t [, ). 24) To this end, let us consider an arbitrary number ε>. We claim that wt) <Mλ,δ ; φ) + ε, for every t [, ). 25) Otherwise, by 23), there exists a t > such that wt) <Mλ,δ ; φ) + ε, for t<t and w t ) = Mλ,δ ; φ) + ε. Then using 17), we obtain Mλ,δ ; φ) + ε = w t ) p 2 e λ +δ )τ t t ws) τ ds + p 2 λ + q 2 e λ τ e δ s t t s wu) du ds p 2 e λ +δ )τ τ + δ 2 1 e δ τ δ τe δ τ ) p 2 λ + q 2 e λ τ [ Mλ,δ ; φ) + ε ] < [ Mλ,δ ; φ) + ε ], which, in view of 19), leads to a contradiction. So, our claim is true. Since 25) holds for every ε>, it follows that 24) is always satisfied. By using 24) and 17), we derive wt) t p 2 e λ +δ )τ ws) τ ds + p 2 λ + q 2 e λ τ e δ s t t s wu) du ds p 2 e λ +δ )τ τ + δ 2 1 e δ τ δ τe δ τ ) p 2 λ + q 2 e λ τ Mλ,δ ; φ) = Mλ,δ ; φ)μ λ,δ, for all t. That means 22) holds. Theorem 2. Let λ and δ δ ) be real roots of the characteristic equations 3) and 9). Consider β λ as in Theorem 1. Then, for any φ C[,], IR), the solution y of 1) 2) satisfies lim e λ +δ )t yt) Lλ ; φ) e δ t = Rλ,δ ; φ), β λ η λ,δ where Lλ ; φ), Rλ,δ ; φ) and η λ,δ were given in 7), 14) and 16), respectively.

A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 1285 Proof. By the definitions of x, z, v and w,wehavetoprovethat lim wt) =. 26) In the end of the proof we will establish 26). By using 17) and taking into account 22) and 24), one can show, by an easy induction, that w satisfies wt) μ λ,δ ) n Mλ,δ ; φ), for all t nτ τn=, 1,...). 27) But, 19) guarantees that <μ λ,δ < 1. Thus, from 27) it follows immediately that w tends to zero as t, i.e. 26) holds. The proof of Theorem 2 is completed. Theorem 3. Let λ and δ δ ) be real roots of the characteristic equations 3) and 9) and also the conditions in Theorem 1 β λ and μ λ,δ be provided. Then, for any φ C[,], IR), the solution y of 1) 2) satisfies for all t yt) k λ β λ Nλ,δ ; φ)e λ t [ hλ,δ + + 1 + k λ e δ η λ,δ β λ + h ] λ,δ )μ λ,δ Nλ,δ ; φ)e λ +δ )t, 28) η λ,δ where and η λ,δ was given in 16), k λ = 1 + 2λ p 1 + p 2 e λτ + p 2 λ + q 2 e λτ τ, 29) h λ,δ = 1 + k λ β λ + δ 1 1 e δ τ ) p 2 e λ +δ )τ 1 + k ) λ β λ + δ 2 δ τ + e δτ 1 ) p 2 λ + q 2 e λ τ 1 + k ) λ, 3) β λ e δ = max e δ t 31) t Nλ,δ ; φ) = max max t e λt φt), max t e λ +δ )t φt), max t φ t), max t φt). 32) Moreover, the trivial solution of 1) is stable if λ, λ + δ, it is asymptotically stable if λ <, λ + δ < and it is unstable if δ >, λ + δ >. Proof. By Theorem 1, 2) is satisfied, where Lλ ; φ) and Mλ,δ ; φ) are defined by 7) and 21), respectively. From 2) it follows that e λ +δ )t yt) Lλ ; φ) e δt + Rλ,δ ; φ) + Mλ,δ ; φ)μ λ,δ β λ η. 33) λ,δ Furthermore, by using 29) 32), from 7), 14) and 21), we obtain

1286 A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 Lλ ; φ) φ ) + 2λ p 1 φ) + p2 φ) + p 2 λ + q 2 e λ τ e λ s φs) ds 1 + 2λ p 1 + p 2 e λτ + p 2 λ + q 2 e λτ τ ) Nλ,δ ; φ) = k λ Nλ,δ ; φ), Rλ,δ ; φ) φ) + Lλ ; φ) β λ + p 2 e λ +δ )τ + p 2 λ + q 2 e λ τ τ e δ s s [ 1 + k λ β λ + δ 1 + δ 2 e δ s e λ s φs) + Lλ ) ; φ) ds β λ e δ u e λ u ) φu) Lλ ; φ) + du ds β λ 1 e δ τ ) p 2 e λ +δ )τ δ τ + e δ τ 1 ) p 2 λ + q 2 e λ τ 1 + k ) λ β λ 1 + k λ β λ )] Nλ,δ ; φ) = h λ,δ Nλ,δ ; φ), Mλ,δ ; φ) max e λ +δ )t φt) Lλ ; φ) + max e δ t + Rλ,δ ; φ) t β λ t η λ,δ 1 + k λ e δ β λ + h λ,δ Nλ,δ ; φ). η λ,δ Hence, from 33), we conclude that for all t, e λ +δ )t yt) k λ β λ Nλ,δ ; φ)e δt + h λ,δ Nλ,δ ; φ) η λ,δ + 1 + k λ e δ β λ + h ) λ,δ Nλ,δ ; φ)μ λ,δ η, 34) λ,δ and consequently, 28) holds. Now, let us assume that λ and λ + δ. Define φ max t φt). It follows that φ Nλ,δ ; φ). From 34), it follows that yt) kλ β λ + 1 + k λ e δ β λ for every t. Since k λ β λ > 1, by taking into account the fact that k λ β λ + ) μ λ,δ + 1 + μ λ,δ ) h λ,δ η λ,δ 1 + k ) λ e δ μ λ,δ β λ + 1 + μ λ,δ ) h λ,δ > 1, η λ,δ Nλ,δ ; φ),

A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 1287 we have yt) kλ β λ + 1 + k ) λ e δ μ λ,δ β λ + 1 + μ λ,δ ) h λ,δ Nλ,δ ; φ), η λ,δ for every t [, ), which means that the trivial solution of 1) is stable at ). Next, if λ < and λ + δ <, then 28) guarantees that lim yt) = and so the trivial solution of 1) is asymptotically stable at ). Finally, if δ >, λ +δ >, then the trivial solution of 1) is unstable at ). Otherwise, there exists a number l l1)> such that, for any φ C[,], IR) with φ <l, the solution y of problem 1) 2) satisfies yt) < 1 for all t. 35) Define φ t) = e λ +δ )t e λ t for t [,]. Furthermore, by the definition of Lλ ; φ) and Rλ,δ ; φ), by using 9), we have Lλ ; φ ) = δ p 2 e λ +δ )τ + p 2 e λ τ + p 2 λ + q 2 )e λ τ e δ s ds p 2 λ + q 2 )e λ τ τ = δ + p 1 2λ δ + p 2 e λ τ p 2 λ + q 2 )e λ τ τ β λ, Rλ,δ ; φ ) = 1 + p 2 e λ +δ )τ p 2 λ + q 2 )e λ τ τ e δ s s Let φ C[,], IR) be defined by φ = l 1 φ φ, e δ s e λ s e λ +δ )s e λ s ) + 1 ) ds e δ u e λ u e λ +δ )u e λ u ) + 1 ) du ds = 1 + p 2 e λ +δ )τ τ p 2 λ + q 2 )e λ τ τ e δ s sds η λ,δ >. where l 1 is a number with <l 1 <l. Moreover, let y be the solution of 1) 2). From Theorem 2 it follows that y satisfies

1288 A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 lim e λ +δ )t yt) Lλ ; φ) e δ t β λ = lim e λ +δ )t yt) + l 1 φ e δ t = Rλ,δ ; φ) η λ,δ = l 1/ φ )Rλ,δ ; φ ) = l 1 η λ,δ φ >. But, we have φ =l 1 <land hence from 35) and conditions δ >, λ + δ > it follows that lim e λ +δ )t yt) Lλ ; φ) e δ t β λ This is a contradiction. The proof of Theorem 3 is completed. 3. Examples =. Example 1. Consider y t) = 4y t) + 1 e y t 1 ) 3yt) + 1e 2 y t 1 ), t, 2 yt) = φt), 1 t, 36) 2 where φt) is an arbitrary continuously differentiable initial function on the interval [ 1 2, ]. In this example we apply the characteristic equations 3) and 9). That is, the characteristic equation 3) is λ 2 = 4λ + 1 e λe λ 1 1 2 3 + e e λ 1 2, 37) and we see that λ = 1 is a root of 37). Then, for λ = 1 the characteristic equation 9) is δ = 2 + e 1 2 1+δ). Therefore, δ = δ = 1 is a root, and the conditions of Theorem 3 are satisfied. That is, μ λ,δ = μ 1, 1 = 1 2 < 1 and β λ = β 1 = 2 1 e. Since λ = 1 < and λ + δ = 2 <, the zero solution of 36) is asymptotically stable. Example 2. Consider y t) = e 2 y t) 1 2 y t 1) + yt) yt 1), t, yt) = φt), 1 t, 38) where φt) is an arbitrary continuously differentiable initial function on [ 1, ]. The characteristic equation 3) is λ 2 = e 2 λ 1 2 λe λ + 1 e λ, 39)

A.F. Yeniçerioğlu / J. Math. Anal. Appl. 332 27) 1278 129 1289 and we see easily that λ = is a root of 39). Taking λ =, the characteristic equation 9) is δ 2 = e 2 δ 1 2 e δ δ + 1 e δ. Therefore, we find that δ = δ = 1 is a root. The condition of Theorem 3 is μ λ,δ = μ, 1 > 1. Thus, Theorem 3 is not applicable. But, λ = 1 is another root of 39). Then, for λ = 1we have that δ 2 = e2 ) + 2 δ 1 2 e1 δ δ + 1 2 e 1 2 e1 δ, and we find δ = δ = 1. Corresponding to the roots λ = 1 and δ = 1, the conditions of Theorem 3 are satisfied. Since λ = 1 < and λ + δ =, the zero solution of 38) is stable. Example 3. Consider y t) = 3y t) y t π 2 ) 2yt) + y t π ), t, 2 yt) = φt), π 2 t, 4) where φt) is an arbitrary continuously differentiable initial function on [ π 2, ]. The characteristic equation 3) is λ 2 = 3λ λe λπ 2 2 + e λπ 2, 41) and we see easily that λ = 1 is a root of 41). Taking λ = 1, the characteristic equation 9) is δ = 1 e π 2 δ+1). Therefore, we find that δ = δ =.95351 is a root. Corresponding to the roots λ = 1 and δ =.95351, the conditions of Theorem 3 are satisfied. Since δ > and λ + δ >, the zero solution of 4) is unstable. Acknowledgment The author is grateful to Allaberen Ashyralyev because of his important comments and useful improvements. References [1] R. Bellman, K. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. [2] B. Cahlon, D. Schmidt, Stability criteria for certain second-order delay differential equations with mixed coefficients, J. Comput. Appl. Math. 17 24) 79 12. [3] R.D. Driver, Exponential decay in some linear delay differential equations, Amer. Math. Monthly 85 9) 1978) 757 76. [4] R.D. Driver, Ordinary and Delay Differential Equations, Appl. Math. Sci., vol. 2, Springer, Berlin, 1977. [5] L.E. El sgol ts, S.B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, 1973. [6] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, Berlin, 1993. [7] G.D. Hu, T. Mitsui, Stability of numerical methods for system of neutral delay differential equations, BIT 35 1995) 55 515. [8] V. Kolmanovski, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic, Dordrecht, 1992.

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