BEHAVIOR OF THE SOLUTIONS TO SECOND ORDER LINEAR AUTONOMOUS DELAY DIFFERENTIAL EQUATIONS

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1 Electronic Jornal of Differential Eqation, Vol. 2007(2007, No. 106, pp ISSN: URL: or ftp ejde.math.txtate.ed (login: ftp BEHAVIOR OF THE SOLUTIONS TO SECOND ORDER LINEAR AUTONOMOUS DELAY DIFFERENTIAL EQUATIONS CHRISTOS G. PHILOS, IOANNIS K. PURNARAS Abtract. A wide cla of econd order linear atonomo delay differential eqation with ditribted type delay i conidered. An aymptotic relt, a efl exponential etimate of the oltion, a tability criterion, and a relt on the behavior of the oltion are etablihed. 1. Introdction and preliminarie The theory of delay differential eqation i of both theoretical and practical interet. For the baic theory of delay differential eqation, the reader i referred to the book by Diekmann et al. 2, Driver 7, Hale 12, and Hale and Verdyn Lnel 13. The old bt very intereting aymptotic and tability relt for delay differential eqation de to Driver 5, 6 and to Driver, Saer and Slater 8 gave rie to the pblication of a nmber of article concerning the aymptotic behavior (and, more generally, the behavior and the tability for delay differential eqation, netral delay differential eqation and (netral or non-netral integrodifferential eqation with nbonded delay dring the lat few year. See 1, 3, 4, 9, 10, 14-24; for ome related relt ee 11. Moreover, in the lat few year, a nmber of article dealing with the aymptotic behavior (and, more generally, the behavior and the tability of delay, and netral delay, difference eqation (with dicrete or contino variable and of (netral or non-netral Volterra difference eqation with infinite delay (ee 25 and the reference cited therein appeared in the literatre. Very recently, Yeniçerioğl 26 obtained ome relt on the qalitative behavior of the oltion of a econd order linear atonomo delay differential eqation with a ingle delay. The main idea in 26 i that of tranforming the econd order delay differential eqation into a firt order delay differential eqation, by the e of a real root of the correponding characteritic eqation. The ame idea will be ed in thi paper to obtain ome general relt inclding the relt in 26 a particlar cae. In the preent paper, a wide cla of econd order linear atonomo delay differential eqation with ditribted type delay i conidered. An aymptotic relt for the oltion i obtained. Alo, an etimate of the oltion and a tability 2000 Mathematic Sbject Claification. 34K20, 34K25. Key word and phrae. Delay differential eqation; aymptotic behavior; tability; behavior of oltion; characteritic eqation. c 2007 Texa State Univerity - San Marco. Sbmitted Febrary 16, Pblihed Jly 30,

2 2 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 criterion for the trivial oltion are etablihed. Moreover, a relt on the behavior of the oltion i given. Conider the delay differential eqation x (t x (t dζ( = x(t dη(, (1.1 where r i a poitive real contant, ζ and η are real-valed fnction of bonded variation on the interval, 0, and the integral are Riemann-Stieltje integral. It will be amed that η i not contant on, 0. By a oltion of the delay differential eqation (1.1, we mean a continoly differentiable real-valed fnction x defined on the interval,, which i twice continoly differentiable on 0, and atifie (1.1 for all t 0. Together with the delay differential eqation (1.1, it i ctomary to pecify an initial condition of the form x(t = φ(t for r t 0, (1.2 where the initial fnction φ i a given continoly differentiable real-valed fnction on the initial interval, 0. Eqation (1.1 and (1.2 contitte an initial vale problem (IVP, for hort. It i well-known (ee, for example, Diekmann et al. 2, Driver 7, Hale 12, or Hale and Verdyn Lnel 13 that there exit a niqe oltion x of the delay differential eqation (1.1 which atifie the initial condition (1.2; thi niqe oltion x will be called the oltion of the initial vale problem (1.1 and (1.2 or, more briefly, the oltion of the IVP (1.1 and (1.2. Along with the delay differential eqation (1.1, we aociate it characteritic eqation λ 2 λ e λ dζ( = e λ dη(, (1.3 which i obtained from (1.1 by looking for oltion of the form x(t = e λt for t. For a given real root λ 0 of the characteritic eqation (1.3, we conider the (firt order delay differential eqation z (t 2λ 0 z(t e λ0 z(t dζ( = λ 0 e λ0 z(t d dζ( e λ0 z(t d dη(. (1.4 A oltion of the delay differential eqation (1.4 i a contino real-valed fnction z defined on the interval,, which i continoly differentiable on 0, and atifie (1.4 for all t 0. The characteritic eqation of the delay differential eqation (1.4 i µ 2λ 0 ( = λ 0 e λ0 e (λ0µ dζ( e µ d dζ( e λ0 ( e µ d dη(. (1.5 Thi eqation i obtained from (1.4 by eeking oltion of the form z(t = e µt for t.

3 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 3 For or convenience, we introdce ome notation. For a given real root λ 0 of the characteritic eqation (1.3, we et β(λ 0 = 2λ 0 and, alo, we define K(λ 0 ; φ = φ (0 λ 0 φ(0 1 λ 0 (e λ0 dζ( φ( λ 0 e λ0 e λ0 e λ0 φ(d dη(; in addition, provided that β(λ 0 0, we define Φ(λ 0 ; φ(t = e λ0t φ(t K(λ 0; φ β(λ 0 (e λ0 dη( (1.6 e λ0 φ(d dζ( (1.7 for r t 0. (1.8 We will now give a propoition, which play a crcial role in obtaining or main relt. Propoition 1.1. Let λ 0 be a real root of the characteritic eqation (1.3, and let β(λ 0 and K(λ 0 ; φ be defined by (1.6 and (1.7, repectively. Sppoe that β(λ 0 0, and define Φ(λ 0 ; φ by (1.8. Then a contino real-valed fnction x defined on the interval, i the oltion of the IVP (1.1 and (1.2 if and only if the fnction z defined by z(t = e λ0t x(t K(λ 0; φ β(λ 0 for t (1.9 i the oltion of the delay differential eqation (1.4 which atifie the initial condition z(t = Φ(λ 0 ; φ(t for r t 0. (1.10 Proof. Let x be the oltion of the IVP (1.1 and (1.2, and define y(t = e λ0t x(t for t. (1.11 Then, by taking into accont the fact that λ 0 i a real root of the characteritic eqation (1.3, we get, for every t 0, x (t x (t dζ( = e λ0t { y (t 2λ 0 y (t λ 2 0y(t e λ0 y(t dη( = e λ0t {y (t 2λ 0 y (t λ 0 e λ0 y(t dζ( = e λ0t { y (t 2λ 0 y(t x(t dη( e λ0 y (t λ 0 y(t dζ( e λ0 y (t dζ( λ 2 0y(t e λ0 y(t dη( e λ0 y(t dζ(

4 4 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 λ 0 e λ0 dζ( e λ0 dη( y(t λ 0 e λ0 y(t dζ( e λ0 y(t dη( = e λ0t { y (t 2λ 0 y(t λ 0 e λ0 y(t y(t dζ( e λ0 y(t dζ( e λ0 y(t y(t dη(. Hence, the fact that x i a oltion of the delay differential eqation (1.1 i eqivalent to the fact that y atifie y (t 2λ 0 y(t e λ0 y(t dζ( (1.12 = λ 0 e λ0 y(t y(t dζ( e λ0 y(t y(t dη( for all t 0. On the other hand, x atifie the initial condition (1.2 if and only if y atifie the initial condition y(t = e λ0t φ(t for r t 0. (1.13 Frthermore, we ee that y atifie (1.12 for t 0 if and only if y (t 2λ 0 y(t t = λ 0 e λ0 or, eqivalently, y (t 2λ 0 y(t t e λ0 y(t dζ( t y(d dζ( e λ0 y(d dη( Θ t e λ0 y(t dζ( = λ 0 e λ0 y(t d dζ( e λ0 y(t d dη( Θ for all t 0, where Θ i ome real contant. By ing the initial condition (1.13 and taking into accont the definition of K(λ 0 ; φ by (1.7, we have Θ = y (0 2λ 0 y(0 e λ0 y(dζ( λ 0 e λ0 y(d dζ( = φ (0 λ 0 φ(0 2λ 0 φ(0 φ(dζ( λ 0 e λ0 e λ0 φ(d dζ( = φ (0 λ 0 φ(0 e λ0 y(d dη( e λ0 φ( λ 0 e λ0 e λ0 φ(d dζ( e λ0 φ(d dη(

5 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 5 K(λ 0 ; φ. e λ0 e λ0 φ(d dη( So, the fact that y atifie (1.12 for t 0 i eqivalent to the fact that y atifie y (t 2λ 0 y(t e λ0 y(t dζ( = λ 0 e λ0 y(t d dζ( e λ0 y(t d dη( K(λ 0 ; φ for all t 0. Now, we take into accont the amption β(λ 0 0 and we define z(t = y(t K(λ 0; φ β(λ 0 (1.14 for t. (1.15 Then, becae of the definition of β(λ 0 by (1.6, it i a matter of elementary calclation to how that y atifie (1.14 for t 0 if and only if z atifie (1.4 for all t 0, i.e., if and only if z i a oltion of the delay differential eqation (1.4. Moreover, we ee that the initial condition (1.13 i eqivalently written a follow z(t = e λ0t φ(t K(λ 0; φ β(λ 0 for r t 0. (1.16 We have th proved that x i the oltion of the IVP (1.1 and (1.2 if and only if z i the oltion of the delay differential eqation (1.4 which atifie the initial condition (1.16. By (1.11, we ee that (1.15 coincide with (1.9. Alo, by taking into accont the definition of Φ(λ 0 ; φ by (1.8, we oberve that (1.16 coincide with the initial condition (1.10. The proof of or propoition i complete. Let C(, 0, R be the Banach pace of all contino real-valed fnction on the interval, 0, endowed with the al p-norm ψ = max ψ(t t 0 for ψ C(, 0, R. Moreover, let C 1 (, 0, R be the et of all continoly differentiable real-valed fnction on the interval, 0. Thi et i a Banach pace with the norm ω = max { ω, ω for ω C 1 (, 0, R. A it concern the IVP (1.1 and (1.2 tdied in thi paper, the initial fnction φ belong to C 1 (, 0, R. So, the notation φ ed in Section 3 i defined by { φ = max { φ, φ max max φ(t, max t 0 t 0 φ (t. It will be conidered that the reader i familiar with the notion of tability, niform tability, aymptotic tability, and niform aymptotic tability of the trivial oltion of a linear delay differential ytem. (We chooe to refer to the book by Driver 7. It i known (ee, for example, 7 that, in the cae of atonomo linear delay differential ytem, the trivial oltion i niformly table if and only if it i table (at 0, and the trivial oltion i niformly aymptotically table if and only if it i aymptotically table (at 0.

6 6 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 The btittion x 1 = x, x 2 = x tranform the (econd order linear delay differential eqation (1.1 into the following eqivalent (firt order linear delay differential ytem x 1(t = x 2 (t, x 2(t = x 1 (t dη( x 2 (t dζ(. (1.17 The delay differential ytem (1.17 i conidered in conjnction with the initial condition x 1 (t = φ(t for r t 0, x 2 (t = φ (t for r t 0. (1.18 So, the IVP (1.1 and (1.2 i tranformed into the eqivalent initial vale problem (1.17 and (1.18. On the bai of the above tranformation of the delay differential eqation (1.1 into the eqivalent delay differential ytem (1.17, one can formlate the definition of the notion of the tability, niform tability, aymptotic tability, and niform aymptotic tability of the trivial oltion of (1.1. A the delay differential eqation (1.1 i atonomo, the trivial oltion of (1.1 i niformly table (repectively, niformly aymptotically table if and only if it i table (at 0 (repectively, aymptotically table (at 0. We retrict orelve to giving here the definition of the tability (at 0 and the aymptotic tability (at 0 of the trivial oltion of (1.1. The trivial oltion of the delay differential eqation (1.1 i aid to be table (at 0 if, for each ɛ > 0, there exit a δ δ(ɛ > 0 ch that, for any φ C 1 (, 0, R with φ < δ, the oltion x of the IVP (1.1 and (1.2 atifie max { x(t, x (t < ɛ for all t. Moreover, the trivial oltion of (1.1 i called aymptotically table (at 0 if it i table (at 0 in the above ene and, in addition, there exit a δ 0 > 0 ch that, for any φ C 1 (, 0, R with φ < δ 0, the oltion x of the IVP (1.1 and (1.2 atifie lim max { x(t, t x (t = 0; i.e., lim x(t = lim x (t = 0. t t Let conider the pecial cae of the delay differential eqation x (t = x(t dη(. (1.19 Thi eqation i obtained from (1.1, by conidering that ζ i any contant real -valed fnction on the interval, 0. The characteritic eqation of the delay differential eqation (1.19 i λ 2 = e λ dη(. (1.20 For a given real root λ 0 of the characteritic eqation (1.20, we conider the (firt order delay differential eqation z (t 2λ 0 z(t = e λ0 z(t d dη(. (1.21 The eqation µ 2λ 0 = e λ0 ( e µ d dη( (1.22

7 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 7 i the characteritic eqation of the delay differential eqation (1.21. For a given real root λ 0 of the characteritic eqation (1.20, we define β(λ 0 = 2λ 0 K(λ 0 ; φ = φ (0 λ 0 φ(0 and, provided that β(λ 0 0, (e λ0 dη(, (1.23 e λ0 Φ(λ 0 ; φ(t = e λ0t φ(t K(λ 0 ; φ β(λ 0 e λ0 φ(d dη( (1.24 for r t 0. (1.25 Or relt given in the next ection (Section 25 are formlated a for theorem (Theorem 2.1, 3.1, 4.2, and 4.4, a corollary (Corollary 3.2, and five lemma (Lemma 4.1, 4.3, and Section 2 contain Theorem 2.1, Section 3 i devoted to Theorem 3.1 and Corollary 3.2, Section 4 inclde Lemma 4.1 and 4.3 a well a Theorem 4.2 and 4.4, and Section 5 contain Lemma Theorem 2.1 contitte a baic aymptotic relt for the oltion of the IVP (1.1 and (1.2 a well a for the firt order derivative of thi oltion. Etimate of the oltion of the IVP (1.1 and (1.2 and of the firt order derivative of the oltion are etablihed by Theorem 3.1. Corollary 3.2 i a tability criterion for the trivial oltion of the delay differential eqation (1.1. Lemma 4.1 i an axiliary relt abot the real root of the characteritic eqation (1.5, where λ 0 i a negative real root of the characteritic eqation (1.3. Analogoly, Lemma 4.3 i an axiliary relt concerning the real root of the characteritic eqation (1.22, where λ 0 i a nonzero real root of the characteritic eqation (1.20. Lemma 4.1 and 4.3 are ed in etablihing Theorem 4.2 and 4.4, repectively. Theorem 4.2 i concerned with the behavior of the oltion of the IVP (1.1 and (1.2 and of the firt order derivative of thi oltion. Theorem 4.4 concern the pecial cae of the IVP (1.19 and (1.2, and provide a relt on the behavior of the oltion and of the firt order derivative of thi oltion. For a given real root λ 0 of the characteritic eqation (1.3, Lemma 5.1 etablihe fficient condition for the characteritic eqation (1.5 to have a real root with an appropriate property. Lemma 5.2 i concerned with the real root of (1.5, where λ 0 i a negative real root of (1.3. Finally, Lemma 5.3 concern the real root of the characteritic eqation (1.22, where λ 0 i a nonzero real root of the characteritic eqation (1.20. Theorem 2.1 and 3.1 a well a Corollary 3.2 are obtained, by the e of a real root λ 0 of the characteritic eqation (1.3 and of a real root µ 0 of the characteritic eqation (1.5. Theorem 4.2 i derived, via a negative real root λ 0 of (1.3 and two ditinct real root µ 0 and µ 1 of (1.5. In obtaining Theorem 4.4, a nonzero real root λ 0 of the characteritic eqation (1.20 and two ditinct real root µ 0 and µ 1 of the characteritic eqation (1.22 are ed. Throghot the paper, we need a notation concerning a real-valed fnction θ which i of bonded variation on the interval, 0. By V (θ we will denote the total variation fnction of θ, which i defined on the interval, 0 a follow: V (θ( = 0, and V (θ( i the total variation of θ on, for each in (, 0. Note that the fnction V (θ i nonnegative and increaing on the interval, 0. Moreover, it mt be noted that V (θ i identically zero on, 0 if and only if θ i contant on the interval, 0. So, a η i amed to be not contant on

8 8 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106, 0, the fnction V (η i not identically zero on the interval, 0 (and o it i alway not contant on, 0. It will be conidered that the reader i familiar with the theory of fnction of bonded variation and the theory of Riemann-Stieltje integration. 2. An aymptotic relt Or prpoe in thi ection i to etablih the following theorem. Theorem 2.1. Let λ 0 be a real root of the characteritic eqation (1.3, and let β(λ 0 and K(λ 0 ; φ be defined by (1.6 and (1.7, repectively. Sppoe that β(λ 0 0, and define Φ(λ 0 ; φ by (1.8. Frthermore, let µ 0 be a real root of the characteritic eqation (1.5, and et γ(λ 0, µ 0 = e λ0 (e µ0 λ 0 (e µ0 d dζ( (2.1 e λ0 (e µ0 d dη( and, alo, define L(λ 0, µ 0 ; φ = Φ(λ 0 ; φ(0 { e λ0 e µ0 e µ0 Φ(λ 0 ; φ(d λ 0 e µ0 e µ0v Φ(λ 0 ; φ(vdv d dζ( { e λ0 e µ0 e µ0v Φ(λ 0 ; φ(vdv d dη(. (Note that, becae of β(λ 0 0, we alway have µ 0 0. Ame that e (e λ0 µ0 λ 0 (e µ0 d dv (ζ( e λ0 (e µ0 d dv (η( < 1. (2.2 (2.3 (Thi amption garantee that 1 γ(λ 0, µ 0 > 0. Then the oltion x of the IVP (1.1 and (1.2 atifie lim {e µ0t e λ0t x(t K(λ 0; φ = L(λ 0, µ 0 ; φ (2.4 t β(λ 0 1 γ(λ 0, µ 0 and lim {e µ0t e λ0t x K(λ 0 ; φ (t λ 0 = (λ 0 µ 0 L(λ 0, µ 0 ; φ t β(λ 0 1 γ(λ 0, µ 0. (2.5 Before we prove the above theorem, we will preent ome obervation, which are concerned with a real root λ 0 of the characteritic eqation (1.3 and a real root µ 0 of the characteritic eqation (1.5. We immediately ee that zero i a root of (1.5 if and only if 2λ 0 e λ0 dζ( = λ 0 (e λ0 dζ( (e λ0 dη(; i.e., if and only if β(λ 0 = 0, where β(λ 0 i defined by (1.6. Hence, if we ame that β(λ 0 0, then we alway have µ 0 0.

9 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 9 Let define ρ(λ 0, µ 0 = e λ0 (e µ0 λ 0 e λ0 (e µ0 d dv (η(. (e µ0 d dv (ζ( (2.6 A η i amed to be not contant on, 0, it i clear that ρ(λ 0, µ 0 i poitive. So, (2.3 can eqivalently be written a follow 0 < ρ(λ 0, µ 0 < 1. (2.7 Frthermore, for the real contant γ(λ 0, µ 0 defined by (2.1, we have γ(λ 0, µ 0 e λ0 (e µ0 λ 0 (e µ0 d dζ( e λ0 (e µ0 d dη( 0 e λ0 (e µ0 λ 0 (e µ0 d dv (ζ( e λ0 (e µ0 d dv (η( That i, e λ0 (e µ0 λ 0 e λ0 (e µ0 d dv (η(. (e µ0 d dv (ζ( γ(λ 0, µ 0 ρ(λ 0, µ 0. (2.8 Th, if we ame that (2.3 i atified, i.e., that (2.7 hold, then (2.8 give γ(λ 0, µ 0 < 1. Thi garantee, in particlar, that 1 γ(λ 0, µ 0 > 0. (2.9 Proof of Theorem 2.1. Let x be the oltion of the IVP (1.1 and (1.2. Define the fnction z by (1.9. By Propoition 1.1, the fact that x i the oltion of the IVP (1.1 and (1.2 i eqivalent to the fact that z i the oltion of the delay differential eqation (1.4 which atifie the initial condition (1.10. Set w(t = e µ0t z(t for t. (2.10 Then, ing the fact that µ 0 i a real root of the characteritic eqation (1.5, we obtain, for every t 0, z (t 2λ 0 z(t e λ0 z(t dζ( λ 0 e λ0 z(t d dζ( = e µ0t {w (t (µ 0 2λ 0 w(t e λ0 e (λ0µ0 w(t dζ( z(t d dη(

10 10 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 λ 0 e λ0 = e µ0t {w (t e µ0 w(t d dζ( e λ0 e µ0 w(t d ( e λ0 e µ0 d λ 0 e λ0 = e µ0t (w (t dη( ( e (λ0µ0 dζ( λ 0 e λ0 dη( w(t e µ0 w(t d dζ( e λ0 e µ0 w(t d { λ 0 e λ0 dη( e (λ0µ0 w(t w(t dζ( e µ0 w(t w(t d dζ( { e λ0 e µ0 w(t w(t d e µ0 d dζ( e (λ0µ0 w(t dζ( dη(. So, z i a oltion of the delay differential eqation (1.4 if and only if w atifie for all t 0. atifie w (t = e (λ0µ0 w(t w(t dζ( { λ 0 e λ0 e µ0 w(t w(t d dζ( { e λ0 e µ0 w(t w(t d dη( (2.11 Moreover, z atifie the initial condition (1.10 if and only if w w(t = e µ0t Φ(λ 0 ; φ(t for r t 0. (2.12 Frthermore, we ee that the fact that w atifie (2.11 for t 0 i eqivalent to the fact that w atifie t w(t = e (λ0µ0 w(d dζ( i.e., w(t = t { t λ 0 e λ0 e µ0 { t e λ0 e µ0 e (λ0µ0 t t w(vdv d dζ( w(vdv d dη( Λ, w(t d dζ(

11 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 11 { λ 0 e λ0 e µ0 { e λ0 e µ0 w(t vdv w(t vdv d dζ( d dη( Λ for all t 0, where Λ i ome real nmber. Bt, by taking into accont the initial condition (2.12 and the definition of L(λ 0, µ 0 ; φ by (2.2, we have Λ = w(0 { λ 0 e λ0 = Φ(λ 0 ; φ(0 e (λ0µ0 e µ0 w(d dζ( { e λ0 e µ0 w(vdv e (λ0µ0 w(vdv d dζ( d dη( e µ0 Φ(λ 0 ; φ(d dζ( { λ 0 e λ0 e µ0 e µ0v Φ(λ 0 ; φ(vdv d dζ( = Φ(λ 0 ; φ(0 { e λ0 e µ0 e µ0v Φ(λ 0 ; φ(vdv d dη( { e λ0 e µ0 e µ0 Φ(λ 0 ; φ(d λ 0 e µ0 e µ0v Φ(λ 0 ; φ(vdv d dζ( L(λ 0, µ 0 ; φ. { e λ0 e µ0 e µ0v Φ(λ 0 ; φ(vdv d dη( Th, (2.11 i atified for t 0 if and only if w atifie w(t = e (λ0µ0 w(t d dζ( { λ 0 e λ0 e µ0 { e λ0 e µ0 w(t vdv w(t vdv d dζ( d dη( L(λ 0, µ 0 ; φ (2.13 for all t 0. Next, taking into accont (2.9 (which i a coneqence of the amption (2.3, we define f(t = w(t L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 for t. (2.14

12 12 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 Then, ing the definition of γ(λ 0, µ 0 by (2.1, it i not difficlt to how that the fact that w atifie (2.13 for t 0 i eqivalent to the fact that f atifie f(t = e (λ0µ0 f(t d dζ( { λ 0 e λ0 e µ0 { e λ0 e µ0 f(t vdv f(t vdv d dζ( d dη( (2.15 for all t 0. On the other hand, the initial condition (2.12 take the following eqivalent form f(t = e µ0t Φ(λ 0 ; φ(t L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 for r t 0. (2.16 Now, we hall prove that lim f(t = 0. (2.17 t Define M(λ 0, µ 0 ; φ = max t 0 eµ0t Φ(λ 0 ; φ(t L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0. (2.18 It follow from (2.16 and (2.18 that f(t M(λ 0, µ 0 ; φ for r t 0. (2.19 We will how that M(λ 0, µ 0 ; φ i a bond of the fnction f on the whole interval,, i.e., that f(t M(λ 0, µ 0 ; φ for all t. (2.20 For thi prpoe, we conider an arbitrary poitive real nmber ɛ. We claim that f(t < M(λ 0, µ 0 ; φ ɛ for every t. (2.21 Otherwie, ince (2.19 implie that f(t < M(λ 0, µ 0 ; φ ɛ for t 0, there exit a point t 0 > 0 o that f(t < M(λ 0, µ 0 ; φ ɛ for r t < t 0, and f(t 0 = M(λ 0, µ 0 ; φ ɛ. Then, by taking into accont the definition of ρ(λ 0, µ 0 by (2.6 and ing (2.7 (which i eqivalent to the amption (2.3, from (2.15 we obtain M(λ 0, µ 0 ; φ ɛ = f(t 0 = e (λ0µ0 { λ 0 e λ0 e µ0 f(t 0 d dζ( { e λ0 e µ0 f(t 0 vdv e (λ0µ0 f(t 0 d dζ( f(t 0 vdv d dζ( d dη(

13 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 13 { λ 0 e λ0 e µ0 f(t 0 vdv d dζ( { e λ0 e µ0 f(t 0 vdv d dη( 0 e (λ0µ0 f(t 0 d dv (ζ( 0 λ 0 e λ0 e µ0 f(t 0 vdv d dv (ζ( 0 e λ0 e µ0 f(t 0 vdv d dv (η( e (λ0µ0 f(t 0 d dv (ζ( λ 0 { e λ0 e µ0 { e λ0 e µ0 f(t 0 v dv { ( e (λ0µ0 d dv (ζ( λ 0 { = ( e λ0 e µ0 ( e λ0 e µ0 dv (e (λ0µ0 dv (ζ( λ 0 e λ0 f(t 0 v dv d dv (ζ( d dv (η( dv d dv (ζ( d dv (η( M(λ 0, µ 0 ; φ ɛ (e µ0 d dv (η( e λ0 M(λ 0, µ 0 ; φ ɛ { = e (e λ0 µ0 λ 0 (e µ0 d dv (ζ( e λ0 (e µ0 d dv (η( M(λ 0, µ 0 ; φ ɛ = ρ(λ 0, µ 0 M(λ 0, µ 0 ; φ ɛ < M(λ 0, µ 0 ; φ ɛ. (e µ0 d dv (ζ( We have th arrived at a contradiction, which etablihe or claim, i.e., that (2.21 hold tre. A (2.21 i atified for all real nmber ɛ > 0, it follow that (2.20 i alway flfilled. Frthermore, by ing (2.20, from (2.15 we get, for every t 0, f(t e (λ0µ0 f(t d dζ( λ 0 { e λ0 e µ0 f(t vdv d dζ(

14 14 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 { e λ0 e µ0 f(t vdv d dη( 0 e (λ0µ0 f(t d dv (ζ( 0 λ 0 e λ0 e µ0 f(t vdv d dv (ζ( 0 e λ0 e µ0 f(t vdv d dv (η( e (λ0µ0 f(t d dv (ζ( λ 0 { e λ0 e µ0 { e λ0 e µ0 f(t v dv { ( e (λ0µ0 d dv (ζ( λ 0 { = ( e λ0 e µ0 ( e λ0 e µ0 dv (e (λ0µ0 dv (ζ( λ 0 e λ0 f(t v dv d dv (ζ( d dv (η( dv d dv (ζ( d dv (η( M(λ 0, µ 0 ; φ (e µ0 d dv (η( e λ0 M(λ 0, µ 0 ; φ { = e (e λ0 µ0 λ 0 (e µ0 d dv (ζ( e λ0 (e µ0 d dv (η( M(λ 0, µ 0 ; φ. (e µ0 d dv (ζ( Th, by taking into accont the definition of ρ(λ 0, µ 0 by (2.6, we have f(t ρ(λ 0, µ 0 M(λ 0, µ 0 ; φ for every t 0. (2.22 By ing (2.15 and taking into accont the definition of ρ(λ 0, µ 0 by (2.6 a well a taking into accont (2.20 and (2.22, one can prove, by an eay indction, that the fnction f atifie f(t ρ(λ 0, µ 0 ν M(λ 0, µ 0 ; φ for all t νr r (ν = 0, 1, 2,.... (2.23 Becae of (2.7 (which i eqivalent to the amption (2.3, we have lim ρ(λ 0, µ 0 ν = 0. (2.24 ν In view of (2.24, it follow from (2.23 that lim t f(t = 0, i.e., (2.17 hold tre. Next, we will etablih that lim f (t = 0. (2.25 t

15 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 15 From (2.15 it follow that f atifie f (t = e (λ0µ0 f (t d dζ( { λ 0 e λ0 e µ0 { e λ0 e µ0 f (t vdv for all t 0. Moreover, the initial condition (2.16 give Set f (t vdv d dζ( d dη( (2.26 f (t = e µ0t (Φ(λ 0 ; φ (t µ 0 Φ(λ 0 ; φ(t for r t 0. (2.27 N(λ 0, µ 0 ; φ = max t 0 It follow from (2.27 and (2.28 that e µ 0t (Φ(λ 0 ; φ (t µ 0 Φ(λ 0 ; φ(t. (2.28 f (t N(λ 0, µ 0 ; φ for r t 0. (2.29 By taking into accont the definition of ρ(λ 0, µ 0 by (2.6 and ing (2.29, (2.26 and (2.7, we can follow the ame argment applied previoly in proving (2.20 to conclde that N(λ 0, µ 0 ; φ i a bond of f on the whole interval,, i.e., that f (t N(λ 0, µ 0 ; φ for all t. (2.30 Frthermore, by taking again into accont the definition of ρ(λ 0, µ 0 by (2.6 and ing (2.30 and (2.26, we may apply the ame argment ed above in etablihing (2.22 to obtain f (t ρ(λ 0, µ 0 N(λ 0, µ 0 ; φ for every t 0. (2.31 Taking into accont (2.6 a well a (2.30 and (2.31, one can e (2.26 to how, by indction, that f (t ρ(λ 0, µ 0 ν N(λ 0, µ 0 ; φ for all t νr r (ν = 0, 1, 2,.... (2.32 Becae of (2.24, it follow from (2.32 that lim t f (t = 0. So, (2.25 ha been etablihed. Finally, by (1.9, (2.10 and (2.14, we have f(t = e µ0t e λ0t x(t K(λ 0; φ β(λ 0 L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 for t. (2.33 In view of thi eqality, (2.4 coincide with (2.17. So, the oltion x of the IVP (1.1 and (1.2 atifie (2.4. Frthermore, for t, we define g(t = e µ0t e λ0t x K(λ 0 ; φ (t λ 0 (λ 0 µ 0 L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0. (2.34 Then it i a matter of elementary calclation we check that g(t = f (t (λ 0 µ 0 f(t for all t. (2.35 In view of (2.17 and (2.25, it follow from (2.35 that lim g(t = 0. (2.36 t

16 16 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 By (2.34, we ee that (2.5 coincide with (2.36. Hence, the oltion x of the IVP (1.1 and (1.2 atifie (2.5. The proof of the theorem i complete. 3. An etimate of the oltion. A tability criterion Or relt in thi ection are Theorem 3.1 below and it corollary. Theorem 3.1. Let λ 0 be a real root of the characteritic eqation (1.3, and ppoe that β(λ 0 0, where β(λ 0 i defined by (1.6. Set α(λ 0 = m(λ 0 = max { 1, e λ0r, (3.1 1 λ 0 ( e λ0 dv (ζ( (e λ0 dv (η(. (3.2 Frthermore, let µ 0 be a real root of the characteritic eqation (1.5, and let γ(λ 0, µ 0 and ρ(λ 0, µ 0 be defined by (2.1 and (2.6, repectively. (Note that, becae of β(λ 0 0, we alway have µ 0 0. Alo, et m(µ 0 = max {1, e µ0r. (3.3 Ame that (2.3 hold. (Thi amption garantee that 1 γ(λ 0, µ 0 > 0. Then the oltion x of the IVP (1.1 and (1.2 atifie x(t P (λ 0 e λ0t Q(λ 0, µ 0 e (λ0µ0t φ for all t 0 (3.4 and x (t for all t 0, where and { λ 0 P (λ 0 e λ0t λ 0 µ 0 Q(λ 0, µ 0 R(λ 0, µ 0 e (λ0µ0t φ (3.5 P (λ 0 = 1 λ 0 α(λ 0 m(λ 0, β(λ 0 (3.6 { Q(λ 0, µ 0 = ρ(λ 0, µ 0 m(µ 0 1 ρ(λ 0, µ 0 1 ρ(λ 0, µ 0 m(µ 0 1 γ(λ 0, µ 0 m(λ 0 1 λ 0 α(λ 0 m(λ 0 β(λ 0 (3.7 R(λ 0, µ 0 = ρ(λ 0, µ 0 m(µ 0 (1 λ 0 µ 0 m(λ 0 µ 0 1 λ 0 α(λ 0 m(λ 0. β(λ 0 The contant Q(λ 0, µ 0 i greater than 1. (3.8 Corollary 3.2. Let λ 0 be a real root of the characteritic eqation (1.3, and ppoe that β(λ 0 0, where β(λ 0 i defined by (1.6. Frthermore, let µ 0 be a real root of the characteritic eqation (1.5. (Note that, becae of β(λ 0 0, we alway have µ 0 0. Ame that (2.3 hold. Then the trivial oltion of the delay differential eqation (1.1 i niformly table if λ 0 0 and λ 0 µ 0 0, and it i niformly aymptotically table if λ 0 < 0 and λ 0 µ 0 < 0.

17 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 17 Proof of Theorem 3.1. Firt of all, we oberve that, for any real nmber c, it hold max t 0 e ct = max{1, e cr. So, by taking into accont the definition of m(λ 0 and m(µ 0 by (3.1 and (3.3, repectively, we immediately ee that e λ0t m(λ 0 for r t 0, (3.9 e µ0t m(µ 0 for r t 0. (3.10 Thee ineqalitie will be freqently ed in the eqel. Define K(λ 0 ; φ by (1.7. Then K(λ 0 ; φ φ (0 λ 0 φ(0 φ( λ 0 e λ0 e λ0 φ(d dζ( e λ0 e λ0 φ(d dη( = φ (0 λ 0 φ(0 e λ0 φ( λ 0 e λ0 φ(d e λ0 dζ( e λ0 φ(d e λ0 dη( φ (0 λ 0 φ(0 eλ0 φ( λ 0 e λ0 φ(d eλ0 dv (ζ( e λ0 φ(d eλ0 dv (η( φ (0 λ 0 φ(0 e λ0 φ( λ 0 e λ0 φ( d e λ0 dv (ζ( φ λ 0 e λ0 φ( d e λ0 dv (η( ( e λ0 λ 0 ( e λ0 d e λ0 dv (η( φ. In view of (3.9, we have e λ0 m(λ 0, for every, 0. Th, we obtain ( { K(λ 0 ; φ φ λ 0 (e λ0 dv (η( e λ0 d e λ0 dv (ζ( e λ0 d (m(λ 0 1 λ 0 ( e λ0 dv (ζ( m(λ 0 φ and coneqently, becae of the definition of α(λ 0 by (3.2, we get K(λ 0 ; φ φ λ 0 α(λ 0 m(λ 0 φ.

18 18 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 Thi give K(λ 0 ; φ 1 λ 0 α(λ 0 m(λ 0 φ. (3.11 Conider the fnction Φ(λ 0 ; φ defined by (1.8. Then, by (3.9, we have and o, in view of (3.11, Therefore, Φ(λ 0 ; φ m(λ 0 φ K(λ 0; φ β(λ 0 Φ(λ 0 ; φ m(λ 0 φ 1 λ 0 α(λ 0 m(λ 0 β(λ 0 Φ(λ 0 ; φ φ. m(λ 0 1 λ 0 α(λ 0 m(λ 0 φ. (3.12 β(λ 0 Let conider the contant L(λ 0, µ 0 ; φ defined by (2.2. Then L(λ 0, µ 0 ; φ { Φ(λ 0 ; φ(0 e λ0 e µ0 e µ0 Φ(λ 0 ; φ(d λ 0 e µ0 e µ0v Φ(λ 0 ; φ(vdv d dζ( { e λ0 e µ0 e µ0v Φ(λ 0 ; φ(vdv d dη( Φ(λ 0 ; φ(0 0 e λ0 e µ0 e µ0 Φ(λ 0 ; φ(d λ 0 e µ0 e µ0v Φ(λ 0 ; φ(vdv d dv (ζ( 0 e λ0 e µ0 e µ0v Φ(λ 0 ; φ(vdv d dv (η( { Φ(λ 0 ; φ(0 e λ0 e µ0 e µ0 Φ(λ 0 ; φ( d λ 0 { 1 e µ0 e µ0v Φ(λ 0 ; φ(v dv d dv (ζ( { e λ0 e µ0 e λ0 By (3.10, we have e λ0 e µ0 e µ0 d λ 0 e µ0v Φ(λ 0 ; φ(v dv d dv (η( e µ0 ( ( e µ0 e µ0v dv d dv (η( Φ(λ 0 ; φ. e µ0 d (m(µ 0, e µ0v dv (m(µ 0 for, 0 e µ0v dv d dv (ζ(

19 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 19 for every, 0. Hence, we derive ( { L(λ 0, µ 0 ; φ 1 e (e λ0 µ0 λ 0 e λ0 (e µ0 d dv (η( (e µ0 d dv (ζ( m(µ 0 Φ(λ 0 ; φ. Becae of the definition of ρ(λ 0, µ 0 by (2.6, the lat ineqality i written a L(λ 0, µ 0 ; φ 1 ρ(λ 0, µ 0 m(µ 0 Φ(λ 0 ; φ. A combination of thi ineqality and (3.12 lead to L(λ 0, µ 0 ; φ 1 ρ(λ 0, µ 0 m(µ 0 m(λ 0 1 λ 0 α(λ 0 m(λ 0 φ. β(λ 0 (3.13 Let γ(λ 0, µ 0 be defined by (2.1. Take into accont (2.9 (which i a coneqence of the amption (2.3, and define M(λ 0, µ 0 ; φ by (2.18. Then, by ing (3.10, we have M(λ 0, µ 0 ; φ m(µ 0 Φ(λ 0 ; φ L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0. So, by virte of (3.12 and (3.13, M(λ 0, µ 0 ; φ m(µ 0 1 ρ(λ 0, µ 0 m(µ 0 m(λ 0 1 λ 0 α(λ 0 m(λ 0 φ. 1 γ(λ 0, µ 0 β(λ 0 (3.14 Conider the contant N(λ 0, µ 0 ; φ defined by (2.28. Then, by (3.10, we have N(λ 0, µ 0 ; φ m(µ 0 (Φ(λ 0 ; φ µ 0 Φ(λ 0 ; φ. From the definition of Φ(λ 0 ; φ by (1.18 it follow that (Φ(λ 0 ; φ (t = e λ0t φ (t λ 0 φ(t for r t 0 and coneqently, in view of (3.9, (Φ(λ0 ; φ m(λ0 ( φ λ 0 φ, which give (Φ(λ0 ; φ (1 λ0 m(λ 0 φ. By ing the lat ineqality and ineqality (3.12, we find N(λ 0, µ 0 ; φ m(µ 0 (1 λ 0 µ 0 m(λ 0 µ 0 1 λ 0 α(λ 0 m(λ 0 φ. β(λ 0 (3.15 Now, let x be the oltion of the IVP (1.1 and (1.2, and define the fnction z by (1.9. Alo, we define the fnction w and f by (2.10 and (2.14, repectively. Note that (2.9 (which i a coneqence of the amption (2.3 tate that 1 γ(λ 0, µ 0 > 0. Then, a in the proof of Theorem 2.1, we how that (2.22, (2.31 and (2.33 are atified. Moreover, we conider the fnction g defined by (2.34; the fnction g atifie (2.35. We hall prove that x atifie (3.4 and (3.5, where the contant P (λ 0, Q(λ 0, µ 0 and R(λ 0, µ 0 are defined by (3.6, (3.7 and (3.8, repectively.

20 20 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 From (2.33 it follow that x(t = K(λ 0; φ e λ0t β(λ 0 and coneqently x(t K(λ 0; φ e λ0t β(λ 0 Th, ing (2.22, we obtain x(t K(λ 0; φ e λ0t β(λ 0 f(t L(λ 0, µ 0 ; φ e (λ0µ0t for t 0 1 γ(λ 0, µ 0 f(t L(λ 0, µ 0 ; φ e (λ0µ0t for t 0. 1 γ(λ 0, µ 0 ρ(λ 0, µ 0 M(λ 0, µ 0 ; φ L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 e (λ0µ0t (3.16 for t 0. In view of (3.6, ineqality (3.11 can eqivalently be written a K(λ 0 ; φ β(λ 0 Moreover, by the e of (3.13 and (3.14, we get ρ(λ 0, µ 0 M(λ 0, µ 0 ; φ L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 { ρ(λ 0, µ 0 m(µ 0 1 ρ(λ 0, µ 0 m(µ 0 1 γ(λ 0, µ 0 1 ρ(λ 0, µ 0 m(µ 0 m(λ 0 1 λ 0 α(λ 0 m(λ 0 1 γ(λ 0, µ 0 β(λ 0 { = ρ(λ 0, µ 0 m(µ 0 1 ρ(λ 0, µ 0 1 ρ(λ 0, µ 0 m(µ 0 1 γ(λ 0, µ 0 m(λ 0 1 λ 0 α(λ 0 m(λ 0 φ. β(λ 0 So, becae of (3.7, we have P (λ 0 φ. (3.17 m(λ 0 1 λ 0 α(λ 0 m(λ 0 β(λ 0 φ ρ(λ 0, µ 0 M(λ 0, µ 0 ; φ L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 Q(λ 0, µ 0 φ. (3.18 Uing (3.17 and (3.18, we immediately ee that (3.16 implie (3.4. Hence, (3.4 ha been proved. Next, we ee that (2.34 give x K(λ 0 ; φ (t = λ 0 e λ0t β(λ 0 and hence, by (2.35, we have x K(λ 0 ; φ (t = λ 0 e λ0t β(λ 0 for t 0. Coneqently, x (t λ 0 K(λ 0; φ β(λ 0 { λ 0 µ 0 g(t (λ 0 µ 0 L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 { (λ 0 µ 0 e λ0t f(t L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 e (λ0µ0t for t 0 f(t L(λ 0, µ 0 ; φ f (t e (λ0µ0t 1 γ(λ 0, µ 0 f (t e (λ0µ0t for t 0.

21 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 21 So, in view of (2.22 and (2.31, we derive x (t λ 0 K(λ 0; φ e λ0t β(λ 0 { λ 0 µ 0 ρ(λ 0, µ 0 M(λ 0, µ 0 ; φ L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 ρ(λ 0, µ 0 N(λ 0, µ 0 ; φ e (λ0µ0t for t 0. Bt, becae of (3.8, it follow from (3.15 that (3.19 ρ(λ 0, µ 0 N(λ 0, µ 0 ; φ R(λ 0, µ 0 φ. (3.20 By (3.17, (3.18 and (3.20, we ee that (3.5 can be obtained from (3.19. Th, we have hown that (3.5 hold tre. Finally, we will etablih that the contant Q(λ 0, µ 0 i greater than 1. By (2.8 and (2.9, we have and o, a m(µ 0 1, 0 < 1 γ(λ 0, µ 0 1 γ(λ 0, µ 0 1 ρ(λ 0, µ 0 0 < 1 γ(λ 0, µ 0 1 ρ(λ 0, µ 0 m(µ 0, which enre that 1 ρ(λ 0, µ 0 m(µ γ(λ 0, µ 0 Th, ince ρ(λ 0, µ 0 > 0, we obtain and coneqently 1 ρ(λ 0, µ 0 1 ρ(λ 0, µ 0 m(µ 0 1 γ(λ 0, µ 0 > 1 ρ(λ 0, µ 0 m(µ 0 1 ρ(λ 0, µ 0 1 ρ(λ 0, µ 0 m(µ 0 1 γ(λ 0, µ 0 Moreover, a m(λ 0 1, we have m(λ 0 1 λ 0 α(λ 0 m(λ 0 β(λ 0 > 1. > 1. Hence, it follow from the definition of Q(λ 0, µ 0 by (3.7 that Q(λ 0, µ 0 i alway greater than 1. The proof of the theorem i now complete. Proof of Corollary 3.2. Define m(λ 0, α(λ 0, γ(λ 0, µ 0, ρ(λ 0, µ 0 and m(µ 0 by (3.1, (3.2, (2.1, (2.6 and (3.3, repectively. Note that amption (2.3 garantee that 1 γ(λ 0, µ 0 > 0. Let x be the oltion of the IVP (1.1 and (1.2. By Theorem 3.1, the oltion x atifie (3.4 and (3.5, where P (λ 0, Q(λ 0, µ 0 and R(λ 0, µ 0 are defined by (3.6, (3.7 and (3.8, repectively. The contant Q(λ 0, µ 0 i greater than 1. Ame firt that λ 0 0 and λ 0 µ 0 0. Then (3.4 and (3.5 give x(t P (λ 0 Q(λ 0, µ 0 φ for t 0, x (t λ 0 P (λ 0 λ 0 µ 0 Q(λ 0, µ 0 R(λ 0, µ 0 φ for t 0, repectively. So, if we et S(λ 0, µ 0 = max {P (λ 0 Q(λ 0, µ 0, λ 0 P (λ 0 λ 0 µ 0 Q(λ 0, µ 0 R(λ 0, µ 0,

22 22 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 then we have max{ x(t, x (t S(λ 0, µ 0 φ for every t 0. Since Q(λ 0, µ 0 > 1, we alway have S(λ 0, µ 0 > 1. Th, we obtain max{ x(t, x (t S(λ 0, µ 0 φ for all t. Uing thi ineqality, we can immediately verify that the trivial oltion of (1.1 i table (at 0. Becae of the atonomo character of (1.1, the trivial oltion of (1.1 i niformly table. Next, let ppoe that λ 0 < 0 and λ 0 µ 0 < 0. Then the trivial oltion of (1.1 i table (at 0. Frthermore, we ee that it follow from (3.4 and (3.5 that the oltion x atifie lim x(t = lim t t x (t = 0. Hence, the trivial oltion of (1.1 i aymptotically table (at 0. A (1.1 i atonomo, we conclde that the trivial oltion of (1.1 i niformly aymptotically table. The proof i complete. 4. A relt on the behavior of the oltion We begin thi ection with the following lemma. Lemma 4.1. Sppoe that ζ and η are increaing on, 0. (4.1 Let λ 0 be a negative real root of the characteritic eqation (1.3. Frthermore, let µ 0 be a real root of the characteritic eqation (1.5, and define γ(λ 0, µ 0 by (2.1. Then 1 γ(λ 0, µ 0 > 0 if (1.5 ha another real root le than µ 0, and 1 γ(λ 0, µ 0 < 0 if (1.5 ha another real root greater than µ 0. Before we proceed to the proof of Lemma 4.1, we remark that: if η i increaing on, 0, then, a η i alo amed to be not contant on, 0, we alway have dη( > 0 and o the zero i not a root of the characteritic eqation (1.3. Proof of Lemma 4.1. Conider the real-valed fnction Ω defined by ( Ω(µ = µ 2λ 0 e (λ0µ dζ( λ 0 e λ0 e µ d dζ( We obtain immediately Frthermore, e λ0 ( Ω (µ = 1 Ω (µ = e µ d dη( for µ R. e λ0 (e µ λ 0 (e µ d dζ( e λ0 (e µ d dη( for µ R. e λ0 2 e µ (λ 0 2 e µ d dζ( ( e λ0 2 e µ d dη( for µ R. (4.2 (4.3

23 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 23 So, taking into accont (4.1 and the fact that η i not contant on, 0 and ing the hypothei that λ 0 < 0, we conclde that Ω (µ > 0 for all µ R. (4.4 Now, ame that (1.5 ha another real root µ 1 with µ 1 < µ 0 (repectively, µ 1 > µ 0. From the definition of the fnction Ω by (4.2 it follow that Ω(µ 0 = Ω(µ 1 = 0, and coneqently Rolle Theorem garantee the exitence of a point ξ with µ 1 < ξ < µ 0 (rep., µ 0 < ξ < µ 1 ch that Ω (ξ = 0. Bt, (4.4 implie that Ω i trictly increaing on R and hence, a Ω (ξ = 0, we conclde that Ω i poitive on (ξ, (rep., Ω i negative on (, ξ. Th, we mt have Ω (µ 0 > 0 (rep., Ω (µ 0 < 0. By taking into accont the definition of γ(λ 0, µ 0 by (2.1, from (4.3 we obtain Ω (µ 0 = 1 γ(λ 0, µ 0 and o the proof of the lemma i complete. Now, we will etablih the following theorem. Theorem 4.2. Sppoe that tatement (4.1 i tre. Let λ 0 be a negative real root of the characteritic eqation (1.3, and let β(λ 0 and K(λ 0 ; φ be defined by (1.6 and (1.7, repectively. Sppoe that β(λ 0 0, and define Φ(λ 0 ; φ by (1.8. Frthermore, let µ 0 be a real root of the characteritic eqation (1.5, and let γ(λ 0, µ 0 and L(λ 0, µ 0 ; φ be defined by (2.1 and (2.2, repectively. Alo, let µ 1 be a real root of (1.5 with µ 1 µ 0. (Note that, becae of β(λ 0 0, we alway have µ 0 0 and µ 1 0; moreover, note that Lemma 4.1 garantee that 1 γ(λ 0, µ 0 0. Then the oltion x of the IVP (1.1 and (1.2 atifie C 1 (λ 0, µ 0, µ 1 ; φ e µ1t e λ0t x(t K(λ 0; φ L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 eµ0t (4.5 C 2 (λ 0, µ 0, µ 1 ; φ for all t 0 and D 1 (λ 0, µ 0, µ 1 ; φ e µ1t e λ0t x K(λ 0 ; φ (t λ 0 (λ 0 µ 0 L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 eµ0t D 2 (λ 0, µ 0, µ 1 ; φ for all t 0, (4.6 where C 1 (λ 0, µ 0, µ 1 ; φ = min {e µ1t e λ0t φ(t K(λ 0; φ L(λ 0, µ 0 ; φ t 0 β(λ 0 1 γ(λ 0, µ 0 eµ0t, (4.7 C 2 (λ 0, µ 0, µ 1 ; φ = max {e µ1t e λ0t φ(t K(λ 0; φ L(λ 0, µ 0 ; φ t 0 β(λ 0 1 γ(λ 0, µ 0 eµ0t (4.8 and D 1 (λ 0, µ 0, µ 1 ; φ = min t 0 {e µ1t e λ0t φ K(λ 0 ; φ (t λ 0 (λ 0 µ 0 L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 eµ0t, (4.9

24 24 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 D 2 (λ 0, µ 0, µ 1 ; φ = max {e µ1t e λ0t φ K(λ 0 ; φ (t λ 0 (λ 0 µ 0 L(λ 0, µ 0 ; φ t 0 β(λ 0 1 γ(λ 0, µ 0 eµ0t. (4.10 We ee immediately that ineqalitie (4.5 and (4.6 can eqivalently be written a follow C 1 (λ 0, µ 0, µ 1 ; φe (µ1µ0t e µ0t e λ0t x(t K(λ 0; φ L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 and C 2 (λ 0, µ 0, µ 1 ; φe (µ1µ0t for all t 0 D 1 (λ 0, µ 0, µ 1 ; φe (µ1µ0t e µ0t e λ0t x K(λ 0 ; φ (t λ 0 (λ 0 µ 0 L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 D 2 (λ 0, µ 0, µ 1 ; φe (µ1µ0t for all t 0, repectively. Hence, if µ 1 < µ 0, then the oltion x of the IVP (1.1 and (1.2 atifie (2.4 and (2.5. Alo, we oberve that (4.5 and (4.6 are, repectively, eqivalent to e λ0t C 1 (λ 0, µ 0, µ 1 ; φe µ1t K(λ 0; φ L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 eµ0t x(t e λ0t C 2 (λ 0, µ 0, µ 1 ; φe µ1t K(λ 0; φ L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 eµ0t for all t 0 and e λ0t D 1 (λ 0, µ 0, µ 1 ; φe µ1t K(λ 0 ; φ λ 0 (λ 0 µ 0 L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 eµ0t x (t e λ0t D 2 (λ 0, µ 0, µ 1 ; φe µ1t K(λ 0 ; φ λ 0 (λ 0 µ 0 L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 eµ0t for all t 0. Proof of Theorem 4.2. Let x be the oltion of the IVP (1.1 and (1.2, and conider the fnction z defined by (1.9. Conider alo the fnction w and f defined by (2.10 and (2.14, repectively. Note that, by Lemma 4.1, we necearily have 1 γ(λ 0, µ 0 0. A it ha been hown in the proof of Theorem 2.1, the fact that x atifie (1.1 for t 0 i eqivalent to the fact that f atifie (2.15 for all t 0. Moreover, a in the proof of Theorem 2.1, we ee that f atifie (2.26 for all t 0, and that (2.33 i valid. Frthermore, we conider the fnction g defined by (2.34. A in the proof of Theorem 2.1, eqality (2.35 hold tre. Becae of (2.35, we can e (2.15 and (2.26 to conclde that the fnction g atifie, for all

25 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 25 t 0, g(t = Now, we define e (λ0µ0 g(t d dζ( { λ 0 e λ0 e µ0 { e λ0 e µ0 g(t vdv g(t vdv d dζ( d dη(. (4.11 h(t = e (µ0µ1t f(t for t, (4.12 k(t = e (µ0µ1t g(t for t. (4.13 Then we ee that (2.15 hold for t 0 if and only if h atifie h(t = e (λ0µ0 e (µ0µ1 h(t d dζ( { λ 0 e λ0 e µ0 e (µ0µ1v h(t vdv d dζ( { e λ0 e µ0 e (µ0µ1v h(t vdv d dη( for all t 0, and that (4.11 i flfilled for t 0 if and only if k atifie k(t = e (λ0µ0 e (µ0µ1 k(t d dζ( { λ 0 e λ0 e µ0 e (µ0µ1v k(t vdv d dζ( { e λ0 e µ0 e (µ0µ1v k(t vdv for all t 0. By combining (2.33 and (4.12, we have h(t = e µ1t e λ0t x(t K(λ 0; φ L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 eµ0t d dη( (4.14 (4.15 for t, (4.16 while a combination of (2.34 and (4.13 lead to k(t = e µ1t e λ0t x K(λ 0 ; φ (t λ 0 (λ 0 µ 0 L(λ 0, µ 0 ; φ β(λ 0 1 γ(λ 0, µ 0 eµ0t (4.17 for t. A the oltion x atifie the initial condition (1.2, we can e (4.16 a well a the definition of C 1 (λ 0, µ 0, µ 1 ; φ and C 2 (λ 0, µ 0, µ 1 ; φ by (4.7 and (4.8, repectively, to ee that C 1 (λ 0, µ 0, µ 1 ; φ = min h(t and C 2(λ 0, µ 0, µ 1 ; φ = max h(t. (4.18 t 0 t 0 Moreover, from the initial condition (1.2 we obtain x (t = φ (t for r t 0.

26 26 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 So, by taking into accont (4.17 a well a the definition of D 1 (λ 0, µ 0, µ 1 ; φ and D 2 (λ 0, µ 0, µ 1 ; φ by (4.9 and (4.10, repectively, we have D 1 (λ 0, µ 0, µ 1 ; φ = min k(t and D 2(λ 0, µ 0, µ 1 ; φ = max k(t. (4.19 t 0 t 0 In view of (4.16 and (4.18, the doble ineqality (4.5 can eqivalently written a follow min h( h(t max h( for all t 0. ( Alo, by (4.17 and (4.19, the doble ineqality (4.6 take the following eqivalent form min k( k(t max k( for all t 0. ( All we have to prove i that (4.20 and (4.21 hold. We will e the fact that h atifie (4.14 for all t 0 in order to how that (4.20 i valid. By a imilar way, one can e the fact that k atifie (4.15 for all t 0 to etablih (4.21. So, the proof of (4.21 will be omitted. We retrict orelve to proving that h(t The proof of the ineqality h(t min h( for every t 0. ( max h( for every t 0 0 can be obtained in a imilar way, and o it i omitted. In the ret of the proof we will etablih (4.22. In order to o, we conider an arbitrary real nmber A with A < min 0 h(, i.e., with We will how that h(t > A for r t 0. (4.23 h(t > A for all t 0. (4.24 To thi end, let ame that (4.24 fail to hold. Then, becae of (4.23, there exit a point t 0 > 0 o that h(t > A for r t < t 0, and h(t 0 = A. Th, by ing (4.1 and the fact that η i not contant on, 0 and taking into accont the hypothei that λ 0 < 0, from (4.14 we obtain A = h(t 0 = e (λ0µ0 { λ 0 e λ0 e µ0 e (µ0µ1 h(t 0 d dζ( e (µ0µ1v h(t 0 vdv d dζ( { e λ0 e µ0 e (µ0µ1v h(t 0 vdv ( > A e (λ0µ0 e (µ0µ1 d dζ( { λ 0 e λ0 e µ0 e (µ0µ1v dv d dζ( d dη(

27 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 27 ( = A = = = λ 0 { e λ0 e µ0 ( e (λ0µ0 1 µ 0 µ 1 { e λ0 A µ 0 µ 1 e λ0 { { λ 0 e λ0 A µ 0 µ 1 e µ0 ( 1 e (µ0µ1v dv d dη( 1 e (µ0µ1 dζ( µ 0 µ 1 e µ0 ( 1 µ 0 µ 1 e λ0 (e µ0 e µ1 dζ( (e µ0 e µ1 d dζ( e λ0 (e µ0 e µ1 d { ( e λ0 e µ0 d 1 e (µ0µ1 d dζ( 1 e (µ0µ1 d dη( dη( ( e (λ0µ0 dζ( λ 0 e λ0 dη( ( e (λ0µ1 dζ( λ 0 e λ0 e λ0 ( e µ1 d dη( A µ 0 µ 1 (µ 0 2λ 0 (µ 1 2λ 0 = A. e µ1 d dζ( e µ0 d dζ( We have th arrived at a contradiction and o (4.24 i tre. Since (4.24 i atified for all real nmber A with A < min 0 h(, it follow that (4.22 i alway flfilled. The proof of the theorem i complete. Now, let concentrate on the pecial cae of the delay differential eqation (1.19. In thi cae, the hypothei λ 0 < 0 poed in Lemma 4.1 and Theorem 4.2 can be removed withot damage. More preciely, we have the following relt. Lemma 4.3. Sppoe that η i increaing on, 0. (4.25 Let λ 0 0 be a real root of the characteritic eqation (1.20. Frthermore, let µ 0 be a real root of the characteritic eqation (1.22, and et γ(λ 0, µ 0 = e λ0 (e µ0 d dη(. (4.26 Then 1 γ(λ 0, µ 0 > 0 if (1.22 ha another real root le than µ 0, and 1 γ(λ 0, µ 0 < 0 if (1.22 ha another real root greater than µ 0. Theorem 4.4. Sppoe that (4.25 hold. Let λ 0 0 be a real root of the characteritic eqation (1.20, and let β(λ 0 and K(λ 0 ; φ be defined by (1.23 and (1.24,

28 28 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 repectively. Sppoe that β(λ 0 0, and define Φ(λ 0 ; φ by (1.25. Frthermore, let µ 0 be a real root of the characteritic eqation (1.22, and let γ(λ 0, µ 0 and L(λ 0, µ 0 ; φ be defined by (4.26 and L(λ 0, µ 0 ; φ = Φ(λ 0 ; φ(0 { e λ0 e µ0 e µ0v Φ(λ0 ; φ(vdv d dη(, repectively. Alo, let µ 1 be a real root of (1.22 with µ 1 µ 0. (Note that, becae of β(λ 0 0, we alway have µ 0 0 and µ 1 0; moreover, note that Lemma 4.3 garantee that 1 γ(λ 0, µ 0 0. and Then the oltion x of the IVP (1.19 and (1.2 atifie C 1 (λ 0, µ 0, µ 1 ; φ e µ1t D 1 (λ 0, µ 0, µ 1 ; φ e µ1t where C 1 (λ 0, µ 0, µ 1 ; φ = and C 2 (λ 0, µ 0, µ 1 ; φ = D 1 (λ 0, µ 0, µ 1 ; φ {e µ1t = min t 0 e λ0t x(t K(λ 0 ; φ β(λ 0 C 2 (λ 0, µ 0, µ 1 ; φ for all t 0 e λ0t x (t λ 0 K(λ0 ; φ β(λ 0 D 2 (λ 0, µ 0, µ 1 ; φ for all t 0, min t 0 max t 0 {e µ1t e λ0t φ(t K(λ 0 ; φ β(λ 0 {e µ1t e λ0t φ(t K(λ 0 ; φ β(λ 0 e λ0t φ (t λ 0 K(λ0 ; φ β(λ 0 L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 eµ0t (λ 0 µ 0 L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 eµ0t L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 eµ0t, L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 eµ0t (λ 0 µ 0 L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 eµ0t, D 2 (λ 0, µ 0, µ 1 ; φ {e µ1t = max t 0 e λ0t φ (t λ 0 K(λ0 ; φ β(λ 0 (λ 0 µ 0 L(λ 0, µ 0 ; φ 1 γ(λ 0, µ 0 eµ0t. Note that the obervation preented after the tatement of Theorem 4.2 can alo be formlated in connection with Theorem 4.4, i.e., in the pecial cae of the delay differential eqation (1.19. Proof of Lemma 4.3. The proof will be omitted ince it i imilar to that of Lemma 4.1. We retrict orelve only to noting that, here, we have the real-valed fnction Ω 0 defined by ( Ω 0 (µ = µ 2λ 0 e λ0 e µ d dη( for µ R (4.27

29 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 29 intead of Ω. We note that Ω 0(µ > 0 for all µ R. Thi ineqality hold tre withot the hypothei that λ 0 i negative. Proof of Theorem 4.4. The need for aming, in Theorem 4.2, that the root λ 0 of the characteritic eqation (1.3 i negative i de only to the exitence of the term { λ 0 e λ0 e µ0 e (µ0µ1v h(t vdv d dζ( in (4.14 a well a to the exitence of the term { λ 0 e λ0 e µ0 e (µ0µ1v k(t vdv d dζ( in (4.15. Thee term do not appear in the pecial cae of the delay differential eqation (1.19. In thi pecial cae, h atifie { h(t = e λ0 e µ0 e (µ0µ1v h(t vdv d dη( for all t 0, and k atifie k(t = { e λ0 e µ0 e (µ0µ1v k(t vdv d dη( for all t 0. After thee obervation, we omit the proof of the theorem. 5. Additional Lemma We have already obtained two lemma (Lemma 4.1 and 4.3; Lemma 4.1 i concerned with the real root of the characteritic eqation (1.5, while Lemma 4.3 concern the real root of the characteritic eqation (1.22. Lemma 4.1 and 4.3 have been ed in order to etablih Theorem 4.2 and 4.4, repectively. Here, we will give two lemma abot the real root of (1.5 and a lemma concerning the real root of (1.22. Lemma 5.1. Let λ 0 be a real root of the characteritic eqation (1.3. Ame that e λ0 e (2λ0 1 r λ 0 e (2λ0 1 r d dζ( e λ0 e (2λ0 1 r d dη( < 1 (5.1 r and e (e λ0 (2λ0 1 r λ 0 e λ0 (e (2λ0 1 r d dv (ζ( (e (2λ0 1 r d dv (η( 1. (5.2 Then, in the interval ( 2λ 0 1 r,, the characteritic eqation (1.5 ha a niqe root µ 0 ; thi root atifie (2.3, and the root µ 0 i le than 2λ 0 1 r, provided

30 30 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 that ( e e λ0 2λ 0 1 r λ 0 e λ0 ( 2λ e 0 1 r d dζ( ( 2λ e 0 1 r d dη( > 1r (5.3. Proof. Conider the real-valed fnction Ω defined by (4.2. The derivative Ω of Ω i given by (4.3. It follow from (4.2 that ( Ω 2λ 0 1 r ( = 2λ 0 1 2λ 0 e λ0 e (2λ0 1 r dζ( r λ 0 e λ0 e (2λ0 1 r d dζ( e λ0 = 1 r e λ0 e (2λ0 1 r λ 0 e (2λ0 1 r d dζ( e λ0 e (2λ0 1 r d dη( e (2λ0 1 r d dη( and coneqently, by (5.1, it hold ( Ω 2λ 0 1 < 0. (5.4 r Moreover, from (4.2 we obtain, for µ 2λ 0 1 r, ( Ω(µ = µ 2λ 0 e λ0 e µ λ 0 e µ d dζ( e λ0 ( e µ d dη( ( µ 2λ 0 e λ0 e µ λ 0 e µ d dζ( ( e λ0 e µ d dη( 0 µ 2λ 0 e λ0 e µ λ 0 e µ d dv (ζ( ( e λ0 e µ d dv (η( µ 2λ 0 µ 2λ 0 e λ0 ( e λ0 (e µ λ 0 e µ d dv (η( e λ0 e (2λ0 1 r λ 0 e µ d dv (ζ( e (2λ0 1 r d dv (ζ(

31 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 31 Therefore, e λ0 e (2λ0 1 r d dv (η(. Ω( =. (5.5 Frthermore, ing (4.3, we have, for every µ > 2λ 0 1 r, Ω (µ 1 e λ0 (e µ λ 0 (e µ d dζ( e λ0 (e µ d dη( 0 1 e λ0 (e µ λ 0 (e µ d dv (ζ( e λ0 (e µ d dv (η( 1 > 1 e λ0 (e µ λ 0 e λ0 (e µ d dv (η( e λ0 (e (2λ0 1 r λ 0 e λ0 Coneqently, in view of (5.2, it hold (e µ d dv (ζ( (e (2λ0 1 r d dv (η(. Ω (µ > 0 for all µ > 2λ 0 1 r, (e (2λ0 1 r d dv (ζ( which implie that Ω i trictly increaing on ( 2λ 0 1 r,. By ing thi fact a well a (5.4 and (5.5, we conclde that, in the interval ( 2λ 0 1 r,, the eqation Ω(µ = 0 (which coincide with (1.5 ha a niqe root µ 0. Thi root atifie (2.3. Indeed, by ing again (5.2, we have e (e λ0 µ0 λ 0 (e µ0 d dv (ζ( < 1. e λ0 (e µ0 d dv (η( e λ0 (e (2λ0 1 r λ 0 e λ0 (e (2λ0 1 r d dv (η( (e (2λ0 1 r d dv (ζ( Finally, let ame that (5.3 hold. Then it follow from (4.2 that ( Ω 2λ 0 1 r

32 32 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 = ( 2λ 0 1 2λ 0 r λ 0 e λ0 = 1 ( r e e λ0 e λ0 ( 2λ e 0 1 r 2λ 0 1 r ( e λ0 2λ e 0 1 r dζ( d dζ( e λ0 ( 2λ λ 0 e 0 1 r d dζ( ( 2λ e 0 1 r d dη( > 0. ( 2λ e 0 1 r d dη( A Ω ( 2λ 0 1 r > 0, we ee that µ0 mt be le than 2λ 0 1 r. Thi complete the proof of the lemma. Lemma 5.1 can be applied to the pecial cae of the characteritic eqation (1.22, where λ 0 i a real root of the characteritic eqation (1.20. In thi particlar cae, condition (5.1, (5.2 and (5.3 become e λ0 e (2λ0 1 r d dη( < 1 r, (5.6 e λ0 (e (2λ0 1 r d dv (η( 1, (5.7 e λ0 ( 2λ e 0 1 r d dη( > 1 r, (5.8 repectively. It i remarkable that condition (5.6, (5.7 and (5.8 are atified if the following tronger condition hold: e λ0 e (2λ0 1 r d dv (η( < 1 r. (5.9 In fact, we have e λ0 and e λ0 e (2λ0 1 r d dη( r e λ0 ( 2λ e 0 1 r d e λ0 e λ0 e λ0 (e (2λ0 1 r d dv (η( dη( e (2λ0 1 r d dv (η( e λ0 e (2λ0 1 r d dη( e (2λ0 1 r d dv (η(, e λ0 e λ0 ( 2λ e 0 1 r d dη( ( 2λ e 0 1 r d dv (η( e (2λ0 1 r d dv (η(.

33 EJDE-2007/106 BEHAVIOR OF THE SOLUTIONS 33 So, condition (5.9 implie each one of the condition (5.6, (5.7 and (5.8. Lemma 5.2. Sppoe that tatement (4.1 i tre. Let λ 0 be a negative real root of the characteritic eqation (1.3. Then we have: (I In the interval 2λ 0,, the characteritic eqation (1.5 ha no root. (II Ame that (5.1 hold. Then: (i µ = 2λ 0 1 r i not a root of the characteritic eqation (1.5. (ii In the interval ( 2λ 0 1 r, 2λ 0, (1.5 ha a niqe root. (iii In the interval (, 2λ 0 1 r, (1.5 ha a niqe root. Proof. (I Let µ be a real root of the characteritic eqation (1.5. By taking into accont (4.1 a well a the fact that η i not contant on, 0 and ing the hypothei that λ 0 < 0, we can immediately ee that ( ( e (λ0eµ dζ( λ 0 e λ0 e eµ d dζ( e λ0 e eµ d dη( < 0. Hence, from (1.5 it follow that µ 2λ 0 < 0, i.e., µ < 2λ 0. We have th proved that every real root of (1.5 i alway le than 2λ 0. (II Conider the real-valed fnction Ω defined by (4.2. A in the proof of Lemma 4.1, we ee that (4.4 hold and coneqently Ω i convex on R. (5.10 Next, we oberve that, a in the proof of Lemma 5.1, amption (5.1 mean that (5.4 hold tre. Ineqality (5.4 implie, in particlar, that µ = 2λ 0 1 r i not a root of the characteritic eqation (1.5. From (4.2 we obtain ( Ω(2λ 0 = e λ0 dζ( λ 0 e λ0 e 2λ0 d dζ( ( e λ0 e 2λ0 d dη(. So, by ing (4.1 and taking into accont the fact that η i not contant on, 0 and that λ 0 i negative, we conclde that Ω(2λ 0 > 0. (5.11 Frthermore, a λ 0 < 0 and ζ i increaing on, 0, from (4.2 we get ( Ω(µ µ 2λ 0 e λ0 e µ d dη( for µ R. Uing thi ineqality and the fact that η i increaing and not contant on, 0, it i not difficlt to how that Ω( =. (5.12 From (5.10, (5.4 and (5.11 it follow that, in the interval ( 2λ 0 1 r, 2λ 0, the characteritic eqation (1.5 ha a niqe root. Moreover, (5.10, (5.4 and (5.12 garantee that, in the interval (, 2λ 0 1 r, (1.5 ha alo a niqe root. The proof of the lemma i complete.

34 34 CH. G. PHILOS, I. K. PURNARAS EJDE-2007/106 If omeone read careflly the proof of Lemma 5.2, he/he may ee that the amption that the root λ 0 of the characteritic eqation (1.3 i negative i a neceity becae of the preence of the term ( λ 0 e λ0 e µ d dζ( in the characteritic eqation (1.5. A thi term doe not exit in the characteritic eqation (1.22, ing the fnction Ω 0 defined by ( Ω 0 (µ = µ 2λ 0 e λ0 e µ d dη( for µ R intead of the fnction Ω and following the tep of the proof of Lemma 5.2, we can prove the next lemma valid in the pecial cae of the characteritic eqation (1.22, withot the retriction that the root λ 0 of the characteritic eqation (1.20 i necearily negative. Lemma 5.3. Sppoe that (4.25 hold. Let λ 0 0 be a real root of the characteritic eqation (1.20. Then we have: (I In the interval 2λ 0,, the characteritic eqation (1.20 ha no root. (II Ame that (5.6 hold. Then: (i µ = 2λ 0 1 r i not a root of the characteritic eqation (1.22. (ii In the interval ( 2λ 0 1 r, 2λ 0, (1.22 ha a niqe root. (iii In the interval (, 2λ 0 1 r, (1.22 ha a niqe root. Reference 1 O. Arino and M. Pitk, More on linear differential ytem with mall delay, J. Differential Eqation 170 (2001, O. Diekmann, S. A. van Gil, S. M. Verdyn Lnel and H.-O. Walther, Delay Eqation: Fnctional-, Complex-, and Nonlinear Analyi, Springer-Verlag, New York, J. G. Dix, Ch. G. Philo and I. K. Prnara, An aymptotic property of oltion to linear nonatonomo delay differential eqation, Electron. J. Differential Eqation 2005 (2005, No. 10, pp J. G. Dix, Ch. G. Philo and I. K. Prnara, Aymptotic propertie of oltion to linear non-atonomo netral differential eqation, J. Math. Anal. Appl. 318 (2006, R. D. Driver, Some harmle delay, Delay and Fnctional Differential Eqation and Their Application, Academic Pre, New York, 1972, pp R. D. Driver, Linear differential ytem with mall delay, J. Differential Eqation 21 (1976, R. D. Driver, Ordinary and Delay Differential Eqation, Springer-Verlag, New York, R. D. Driver, D. W. Saer and M. L. Slater, The eqation x (t = ax(t bx(t τ with mall delay, Amer. Math. Monthly 80 (1973, M. V. S. Fraon and S. M. Verdyn Lnel, Large time behavior of linear fnctional differential eqation, Integral Eqation Operator Theory 47 (2003, J. R. Graef and C. Qian, Aymptotic behavior of forced delay eqation with periodic coefficient, Commn. Appl. Anal. 2 (1998, I. Györi, Invariant cone of poitive initial fnction for delay differential eqation, Appl. Anal. 35 (1990, J. K. Hale, Theory of Fnctional Differential Eqation, Springer-Verlag, New York, J. K. Hale and S. M. Verdyn Lnel, Introdction to Fnctional Differential Eqation, Springer-Verlag, New York, I.-G. E. Kordoni, N. T. Niyianni and Ch. G. Philo, On the behavior of the oltion of calar firt order linear atonomo netral delay differential eqation, Arch. Math. (Bael 71 (1998,