Oscillations and Nonoscillations in Mixed Differential Equations with Monotonic Delays and Advances

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1 Advances in Dynamical Systems Applications ISSN , Volume 4, Number 1, pp ( Oscillations Nonoscillations in Mixed Differential Equations with Monotonic Delays Advances Sra Pinelas Universidade dos Açores Departamento de Matemática R. Mãe de Deus Ponta Delgada, Portugal Abstract In this work we study the oscillatory behavior of the delay differential equation of mixed type x (t r(θ dν(θ + x (t + τ(θ dη(θ. Some criteria are obtained in order to guarantee that all solutions are oscillatory. The existence of nonoscillatory solutions is also considered. AMS Subject Classifications: 34K11. Keywords: ODE, oscillation, nonoscillation, advanced, delayed. 1 Introduction The aim of this work is to study the oscillatory behavior of the differential equation of mixed type x (t r(θ dν(θ + x (t + τ(θ dη(θ, (1.1 where x(t R, r(θ τ(θ are nonnegative real continuous functions on [, 0]. The advance τ(θ will be assumed such that τ(θ 0 > τ(θ θ θ 0, Received May 8, 2008; Accepted August 25, 2008 Communicated by Elena Braverman

2 108 Sra Pinelas where τ(θ 0 = max {τ(θ : θ [, 0]}. Both ν(θ η(θ are real functions of bounded variation on [, 0], η(θ is atomic at θ 0, that is, such that η(θ + 0 η(θ 0 0. (1.2 The equation (1.1 represents a wide class of linear functional differential equations of mixed type. The same equation is considered by Krisztin [5] in basis of some mathematical applications appearing in the literature, such as in [1] [6] (see also [2]. When η(θ is the null function, one obtains a retarded functional differential equation whose oscillatory behavior is studied in [3, 4]. As usual, we will say that a solution x of (2.1 oscillates if it has arbitrary large zeros. Notice that for equations, this definition coincides with the nonexistence of an invariant cone as is considered an oscillatory solution in [5]. When all solutions oscillate, (2.1 will be called oscillatory. By [5, Corollary 5], under the condition (1.2, the oscillatory behavior of (2.1 can be studied through the analysis of the real zeros of the characteristic equation Letting = = exp ( r(θ dν(θ + exp ( r(θ dν(θ + exp (τ(θ dη(θ. (1.3 exp (τ(θ dη(θ, the equation (1.1 is oscillatory if only if for every real. So, if either or >, R (1.4 <, R (1.5 then we can conclude that equation (1.1 is oscillatory. In case of having = for some real, there exists at least one nonoscillatory solution. 2 Oscillations Nonoscillations independent of Delays Advances For ν(θ η(θ monotonous on [, 0], some general criteria for oscillations can be obtained independently of either the delays or the advances. Theorem 2.1. (i Let ν(θ η(θ be increasing functions on [, 0]. If τ(θdη(θ > e, (2.1 then equation (1.1 is oscillatory independently of the delays.

3 Mixed Differential Equations 109 (ii If ν(θ η(θ are decreasing functions on [, 0] then (1.1 is oscillatory independently of the advances. r(θdν(θ < e, (2.2 Proof. We first prove (i. Since > 0 for every R, one has > for every 0. For > 0 we have = exp ( r(θ 0 r(θdν(θ + r(θ exp (τ(θ τ(θdη(θ, τ(θ as exp( u u > 0 exp u u > e for every u > 0, we obtain > e τ(θdη(θ > 1. Therefore (1.4 is verified. Now we show (ii. We have < 0 for every R so < for every 0. Through the same arguments one has > e r(θdν(θ > 1 for every < 0, which implies (1.5. Remark 2.2. Denoting m r = min {r(θ : θ [, 0]}, m τ = min {τ(θ : θ [, 0]} considering the differences ν = ν(0 ν(, η = η(0 η(, in the case (i of the Theorem 2.1, the assumption (2.1 is fulfilled whenever m τ η > e. In the case (ii of the same theorem, also condition (2.2 is satisfied if m r ν < e. Example 2.3. By Theorem 2.1, for every delay function r(θ any increasing function ν(θ, the equation x (t r(θ dν(θ + x (t + (θ + 2 dθ is oscillatory since (θ + 2dθ = 1.5 > e.

4 110 Sra Pinelas Example 2.4. Analogously for every advance function τ(θ, the equation x (t (θ + 3 d( θ + where η(θ is a decreasing function, is oscillatory since (θ + 3d( θ = 2.5 < e. x (t + τ(θ dη(θ, In case that the functions ν(θ η(θ have an opposite type of monotonicity, one obtains nonoscillations, as is stated in the following theorem. Theorem 2.5. (i If ν(θ is decreasing η(θ is increasing, then (1.1 is nonoscillatory independently of the delays advances. (ii If ν(θ is increasing η(θ is decreasing, then (1.1 is nonoscillatory independently of the delays advances. Proof. We prove only (i as (ii can be obtained analogously. Let ν(θ be decreasing η(θ increasing on [, 0]. For > 0 one has exp ( r(θ dν(θ + exp (m τ η 0 exp ( r(θ dν(θ exp ( m r dν(θ. (2.3 Then + as +. On the other h, for < 0, we have exp ( m r ν + 0 exp (τ(θ dη(θ exp (m τ exp (τ(θ dη(θ dη(θ, (2.4 which imply that as. So there is at least one 0 R such that M( 0 = 0 so (1.1 is nonoscillatory independently of the delays advances. Example 2.6. The equation x (t r(θ d(1 θ + is nonoscillatory for every delays advances. x (t + τ(θ dθ

5 Mixed Differential Equations Monotonic Differentiable Delays Advances In this section we analyze the oscillatory behavior of (1.1 for the relevant case where both delays advances are monotonous. So in the remainder of this section, the delays advances will be assumed monotonous differentiable functions on [, 0]. For the sake of simplicity we will normalize the functions ν(θ η(θ in the following way: a if r(θ is increasing, then ν(0 = 0, b if r(θ is decreasing, then ν( = 0, c if τ(θ is increasing, then η(0 = 0, d if τ(θ is decreasing, then η( = 0. With regard of condition (1.4 we obtain the following theorem. Theorem 3.1. Let { δ = min e ( mτ mr ν mr+mτ m τ η ( } mτ e + 1 η, (1 + ln ( m r ν. m r m r If r (θν(θ 0 τ (θη(θ 0 for every θ [, 0] (3.1 ν(θd ln r(θ + then the equation (1.1 is oscillatory. η(θd ln τ(θ < δ, (3.2 Proof. Assumptions a, b, c d jointly with (3.1 imply that ν 0 η 0. However the value δ defined above has no meaning when either ν or η are equal to zero. This way, implicitly we are assuming that ν < 0 η < 0. One easily notices that M(0 = ν + η. So M(0 < 0. Let 0. Integrating by parts both integrals of (1.3 we have = exp ( r(0 ν(0 exp ( r( ν( + exp (τ(0 η(0 exp (τ( η( + exp ( r(θ ν(θdr(θ By (3.1 the inequality u exp( u e, we obtain exp ( r(0 ν(0 exp ( r( ν( exp (τ(θ η(θdτ(θ. (3.3 + exp (τ(0 η(0 exp (τ( η( ( +e ν(θd ln r(θ + η(θd ln τ(θ.

6 112 Sra Pinelas For r(θ increasing τ(θ increasing, one has by a c exp ( r( ν( exp (τ( η( ( +e ν(θd ln r(θ + η(θd ln τ(θ. For r(θ increasing τ(θ decreasing, by a d we have exp ( r( ν( + exp (τ(0 η(0 ( +e ν(θd ln r(θ + η(θd ln τ(θ. For r(θ decreasing τ(θ increasing, we have by b c exp ( r(0 ν(0 exp (τ( η( ( +e ν(θd ln r(θ + η(θd ln τ(θ. For r(θ decreasing τ(θ decreasing, by b d one has that exp ( r(0 ν(0 + exp (τ(0 η(0 ( +e ν(θd ln r(θ + η(θd ln τ(θ. So, for all possible cases one obtains Let > 0. Then exp ( m r ν + exp (m τ η ( +e ν(θd ln r(θ + η(θd ln τ(θ. < exp ( m r ν + exp (m τ η ( +e ν(θd ln r(θ + η(θd ln τ(θ. The function f : R R given by has an absolute maximum at f( = exp ( m r ν + exp (m τ η ( +e ν(θd ln r(θ + η(θd ln τ(θ 0 = 1 m r + m τ ln m r ν m τ η,

7 Mixed Differential Equations 113 in view of (3.2 f( 0 = Therefore, for every > 0, For < 0, one has ( mτ mr ν mr+mτ m τ η ( +e ( mτ + 1 η m r ν(θd ln r(θ + < 0. exp ( m r ν ( +e ν(θd ln r(θ + The function g : R R given by g( = exp ( m r ν ( +e ν(θd ln r(θ + attains its absolute maximum at by (3.2 it is Thus, also 1 = 1 m r ln ( m r ν, g( 1 = 1 m r 1 m r ln ( m r ν ( +e ν(θd ln r(θ + < 0 for every < 0. Hence (1.5 holds (1.1 is oscillatory. Example 3.2. By Theorem 3.1 the equation x (t (θ + 2 d ( 12 θ + is oscillatory. In fact, for θ [, 0], we have η(θd ln τ(θ < 0. η(θd ln τ(θ. η(θd ln τ(θ η(θd ln τ(θ < 0. x (t + (4 θ d( 4θ 4 r(θ increasing, r (θν(θ = 1 θ 0, with ν(0 = 0 2 τ(θ decreasing, τ (θη(θ = 4θ + 4 0, with η( = 0

8 114 Sra Pinelas since ν(θd ln r(θ + e Theorem 3.3. If ( mτ mr ν mr+mτ m τ η η(θd ln τ(θ = + ( ( mτ η = 4e m r 2 ( 12 θ d ln(θ + 2 ( 4θ 4d ln(4 θ < δ 1 16 e m r (1 + ln ( m r ν ( r (θν(θ 0 τ (θη(θ 0 for every θ [, 0] ( em r ν < then the equation (1.1 is oscillatory. ν(θdr(θ η(θdτ(θ < 1 em τ η, (3.5 Proof. As in Theorem 3.1 one has ν 0 η 0. For = 0, we have M(0 = ν + η < 0 since by (3.5 one cannot have both ν η equal to zero. Notice that for every u > 0 we have exp( u < 1, exp(u > 1, exp( u u Let > 0. By (3.3 (3.6 we have, for τ(θ increasing for τ(θ decreasing < eτ(η( + ν(θdr(θ > 0 exp u u > e. (3.6 η(θdτ(θ, Therefore < eτ(0η(0 + < em τ η + ν(θdr(θ ν(θdr(θ η(θdτ(θ. η(θdτ(θ < 1

9 Mixed Differential Equations 115 consequently < for every > 0. If < 0, then by (3.3 (3.6 we have for r(θ increasing for r(θ decreasing. Thus also > er(ν( + > er(0ν(0 + > em r ν + ν(θdr(θ ν(θdr(θ ν(θdr(θ < for every < 0. Then (1.5 is verified (1.1 is oscillatory. We illustrate this theorem through the following example. η(θdτ(θ η(θdτ(θ η(θdτ(θ > 1 Example 3.4. Through Theorem 3.3 one can show that the equation x (t (7 2θ d( θ 1 + is oscillatory. As a matter of fact, for θ [, 0] one has r (θ = 2, ν( = 0, τ (θ =, η( = 0 1 em τ η , 1 + em r ν 8.028, x (t + ( θ + 1 d( 3θ 3 while ( θ 1 d(7 2θ ( 3θ 3d( θ + 1 = 0.5. Remark 3.5. Notice that Theorem 3.1 cannot be applied to the equation of Example 3.4 since ν(θd ln r(θ + η(θd ln τ(θ e m r (1 + ln ( m r ν

10 116 Sra Pinelas On the other h, Theorem 3.3 cannot be used in the equation of Example 3.2 since ν(θdr(θ η(θdτ(θ em r ν Similar results can be obtained through the use of condition (1.4, as is stated in the two following theorems. Theorem 3.6. Let { ε = max e ( mr mτ η mr+mτ m r ν ( } mr + 1 ν, e (1 + ln (m τ η. m τ m τ If r (θν(θ 0 τ (θη(θ 0 for every θ [, 0] (3.7 ν(θd ln r(θ + then the equation (1.1 is oscillatory. η(θd ln τ(θ > ε, (3.8 Proof. We will follow closely the proof of Theorem 3.1. Here the assumptions a, b, c, d (3.7 imply that ν 0 η 0. As the value ε above has no meaning when either ν = 0 or η = 0, we implicitly are assuming that ν η are both positive real numbers. For = 0 one has then M(0 = ν + η > 0. Let 0. Integrating again by parts both integrals which define the function, the assumptions (3.7 the inequality u exp( u 1 imply that exp ( r(0 ν(0 exp ( r( ν( + exp (τ(0 η(0 exp (τ( η( ( +e ν(θd ln r(θ + η(θd ln τ(θ. The arguments used in proof of Theorem 3.1 enable us to conclude that exp ( m r ν + exp (m τ η ( +e ν(θd ln r(θ + η(θd ln τ(θ exp ( m r ν + exp (m τ η ( +e ν(θd ln r(θ + η(θd ln τ(θ.

11 Mixed Differential Equations 117 Letting < 0, analogously to the proof of Theorem 3.1 we obtain > exp ( m r ν + exp (m τ η ( +e ν(θd ln r(θ + η(θd ln τ(θ ( mτ η m r ν For > 0, we have in the same way mr mr+mτ ( mr m τ + 1 ( +e ν(θd ln r(θ + exp (m τ η ( +e ν(θd ln r(θ + 1 m τ + 1 m τ ln (m τ η ( +e ν(θd ln r(θ + ν η(θd ln τ(θ > 0. η(θd ln τ(θ η(θd ln τ(θ > 0. Hence (1.1 is oscillatory. > 0 for every R Example 3.7. By Theorem 3.6 the equation x (t (2θ + 4 d ( θ 2 + x(t + θ + 3dθ is oscillatory. In fact, for every θ [, 0] we have r(θ increasing, r (θν(θ = 2θ 2 0, with ν(0 = 0, τ(θ increasing, τ (θη(θ = θ 0, with η(0 = 0. Moreover ν(θd ln r(θ + η(θd ln τ(θ = ( θ 2 d ln(2θ θd ln(θ > ε

12 118 Sra Pinelas since e ( mr mτ η mr+mτ m r ν ( ( 2 ( mr ν = e m τ e (1 + ln (m τ η = e (1 + ln m τ 2 Theorem 3.8. If r (θν(θ 0 τ (θη(θ 0, for every θ [, 0] (3.9 1 em τ η < then the equation (1.1 is oscillatory. ν(θdr(θ η(θdτ(θ < 1 + em r ν, (3.10 Proof. As before, one has ν 0 η 0, (3.10 implies that either ν > 0 or η > 0. Therefore M(0 = ν + η > 0. Let > 0. By (3.3 (3.6 we have, for τ(θ increasing > eτ(η( + ν(θdr(θ η(θdτ(θ > eτ(0η(0 + for τ(θ decreasing. So, for every real > 0, ν(θdr(θ η(θdτ(θ consequently > em τ η + ν(θdr(θ >. η(θdτ(θ > 1 If < 0, by (3.3 (3.6 we have < er(ν( + ν(θdr(θ η(θdτ(θ when r(θ is increasing, < er(0ν(0 + ν(θdr(θ η(θdτ(θ

13 Mixed Differential Equations 119 for r(θ decreasing. So, for every real < 0, < em r ν + consequently >. Example 3.9. Let ν(θdr(θ x (t (2 θ d(θ By Theorem 3.8 the equation is oscillatory since η(θdτ(θ < 1 x (t + (2θ + 3 dθ. (θ + 1d(2 θ θd(2θ + 3 = em τ η = 1 e.7183, 1 + em r ν = 1 + 2e Nonoscillations appear in situations as given in the following theorem. Theorem (i If for every θ [, 0] then (1.1 is nonoscillatory. (ii If for every θ [, 0] then (1.1 is nonoscillatory. r (θν(θ 0 τ (θη(θ 0, r (θν(θ 0 τ (θη(θ 0, Proof. We will prove only (i as analogous arguments enable to obtain (ii. Let then r (θν(θ 0 τ (θη(θ 0 for every θ [, 0]. Integrating by parts the first integral in (1.3 we have exp ( r(θ dν(θ = exp ( r( ν( + Therefore with < 0 we obtain for r(θ increasing ( exp ( r(θ dν(θ exp ( r( ν( exp ( r(θ ν(θdr(θ. ν(θr (θdθ

14 120 Sra Pinelas for r(θ decreasing ( exp ( r(θ dν(θ exp ( r(0 ν(0 + ν(θr(θdθ. As both right-h parts of these inequalities tend to + as, by (2.4 we have = + for every monotonous delay function r(θ. Analogously, lim integrating by parts the second integral in (1.3 exp (τ(θ dη(θ = exp (τ( η( η(θ exp (τ(θ dτ(θ with > 0, we have for τ(θ increasing ( exp (τ(θ dη(θ exp (τ( η( + η(θτ (θdθ for τ(θ decreasing ( exp (τ(θ dη(θ = exp (τ(0 η(0 η(θτ (θdθ. This enables to conclude that, as +, exp (τ(θ dη(θ exponentially, for every monotonous advance function τ(θ. In view of (2.3, the same conclusion holds for the function. Thus there exists a 0 R such that M( 0 = 0 consequently (1.1 is nonoscillatory. Example Through (i of Theorem 3.10 one easily sees that the equation x (t (2 θ d (( θ 1(2θ 1 + x (t + (θ + 3 d (θ( θ 1 is nonoscillatory. Notice that in this example we cannot use Theorem 2.5 since ν(θ η(θ are not monotonic functions on [, 0]. References [1] Henjin Chi, Jonathan Bell, Brian Hassard. Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory. J. Math. Biol., 24(5: , 1986.

15 Mixed Differential Equations 121 [2] Kenneth L. Cooke Joseph Wiener. An equation alternately of retarded advanced type. Proc. Amer. Math. Soc., 99(4: , [3] José M. Ferreira. Oscillations nonoscillations in retarded equations. Port. Math. (N.S., 58(2: , [4] José M. Ferreira. On the stability oscillatory behavior of a retarded functional equation. In Differential equations dynamical systems (Lisbon, 2000, volume 31 of Fields Inst. Commun., pages Amer. Math. Soc., Providence, RI, [5] Tibor Krisztin. Nonoscillation for functional differential equations of mixed type. J. Math. Anal. Appl., 245(2: , [6] Aldo Rustichini. Hopf bifurcation for functional-differential equations of mixed type. J. Dynam. Differential Equations, 1(2: , 1989.