POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION

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1 Novi Sad J. Mah. Vol. 32, No. 2, 2002, POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION Hajnalka Péics 1, János Karsai 2 Absrac. We consider he scalar nonauonomous neural delay differenial equaion wih variable delays [ ] d x p j x τ j q i x σ i = 0, d for 0 < T. Using he mehod of characerisic equaions, we give condiions for he exisence of posiive soluions. Our heorems generalize and exend he resuls for simpler cases proved by Chuanxi, Ladas [1] and Győri, Ladas [5]. AMS Mahemaics Subjec Classificaion 2000: 34K15, 34K25 Key words and phrases: neural delay differenial equaion 1 Inroducion Neural delay differenial equaions conain he derivaive of he unknown funcion boh wih and wihou delays. Some new phenomena can appear, hence he heory of neural delay differenial equaions is even more complicaed han he heory of non-neural delay equaions. The oscillaory behavior of he soluions of neural sysems is of imporance in boh he heory and applicaions, such as he moion of radiaing elecrons, populaion growh, he spread of epidemics, in neworks conaining lossless ransmission lines see [2], [5], [6], [7] and he references herein. In our paper we consider he scalar nonauonomous neural delay differenial equaion wih variable delays and coefficiens of he form d 1 x p j x τ j q i x σ i = 0, d for 0 < T, where he nex hypoheses are saisfied: H 1 p j C 1 [[ 0, T, R], τ j C 1 [[ 0, T, R ], j = 1, 2,..., l; 1 Universiy of Novi Sad, Faculy of Civil Engineering, Suboica, Yugoslavia, peics@gf.su.ac.yu 2 Universiy of Szeged, Szeged, Hungary, karsai@dmi.u-szeged.hu

2 96 H. Péics, J. Karsai H 2 q i C[[ 0, T, R], σ i C[[ 0, T, R ], i = 1, 2,..., m. The oscillaory and asympoic behavior of he soluions of non-neural delay differenial equaions wih variable coefficiens and variable delays and also for neural differenial equaions wih consan delays have been sudied in many papers see, for example, [7, 3, 4, 8]. A quie comprehensive reamen of such resuls is given in he monograph [5] by I. Győri and G. Ladas. One of he mos imporan mehods of such invesigaions is he mehod of generalized characerisic equaion, which is based on he idea of finding soluions of linear sysems in he form 2 x = exp αsds. 0 Our main goal is o apply his mehod o equaion 1 o find condiions for he exisence of posiive soluions, and o generalize and exend he resuls proved for special cases of 1. Before formulaing our resuls, we poin o wo characerisic resuls of he recen invesigaions. Chuanxi and Ladas in [1] invesigaed he paricular case of equaion 1 of he form 3 where 4 d [x P x τ] Qx σ = 0 d P C 1 [[ 0,, R], Q C[[ 0,, R], τ 0,, σ [0,, They proved he following resul. Theorem A. Assume ha 4 holds and ha here exiss a posiive number µ such ha 5 P µe µτ P e µτ Q e µσ µ for 0. Then, for every 1 0, equaion 3 has a posiive soluion on [ 1,. The case of variable delays has been considered for equaions of he form 6 ẋ q i x σ i = 0, where, for 0 < T, q i C[[ 0, T, R], σ i C[[ 0, T, R ], i = 1, 2,..., m, by many auhors. Resuls ha give sufficien condiions for he exisence of posiive soluions of equaion 6 on [ 0, T can be found in [5]. Theorem B. Assume ha here exiss a posiive number µ such ha q i e µσi µ

3 Posiive soluions of neural delay differenial equaion 97 for 0 < T. Then for every Φ {ϕ C[[ 1, 0 ], R ] : ϕ 0 > 0, ϕ ϕ 0, 1 0 }, he soluion of equaion 6 hrough 0, Φ remains posiive for 0 < T. 2. Noaions, definiions Define T 1 1 = min 1 j l { inf { τ j} 0 <T } { }, T1 2 = min inf { σ i} 1 i m 0 <T and 1 = min{t 1 1, T 2 1}. A funcion x : [ 1, T R is called a soluion of equaion 1 if x is coninuous on [ 1, T and saisfies equaion 1 on 0, T. An iniial condiion for he soluions of equaion 1 is given in he form 7 x = Φ, 1 0, Φ C 1 [[ 1, 0, R ]. A soluion of he iniial value problem 1 and 7 is a coninuous funcion defined on [ 1, T which coincides wih Φ on [ 1, 0 such ha he difference x l p jxτ j is differeniable and saisfies equaion 1 on 0, T. The unique soluion of he iniial value problem 1 and 7 is denoed by x = xφ and i exiss hroughou he inerval [ 0, T. The coninuous funcion x : [ 1, T R is oscillaory if x has arbirarily large zeros, i.e., for every a 1, here exiss a number c > a such ha xc = 0. Oherwise, x is called nonoscillaory. Rewrie equaion 1 as ẋ [p j 1 τ j ẋ τ j p j x τ j ] q i x σ i = 0. The iniial value problem for his form is as follows. Le Φ be given by 7. A soluion of he iniial value problem 1 and 7 is a coninuous funcion defined on [ 1, T ha coincides wih Φ on [ 1, 0, x being coninuously differeniable and saisfies equaion 1 on 0, T excep a he poins k r, where r = 0 1, k = 0, 1, 2,...

4 98 H. Péics, J. Karsai On he oher hand, if 8 Φ 0 = p j Φ 0 τ j 0 p j 0 1 τ j 0 Φ 0 τ j 0 q i Φ σ i, hen he soluion x is coninuously differeniable for all 1. Consequenly, relaion 8 is necessary and sufficien for he soluion x o have a coninuous derivaive for all 1. In he nex secion we define precisely he generalized characerisic equaion associaed wih he iniial value problem 1 and 7. Using he presenaion 2 we will obain he inegral equaion of he form [ α p j 1 τ j Φh j 9 Φ 0 exp αsds H j which is called characerisic equaion. We will use he following noaions: αh j p j Φh j Φ 0 q i Φg i Φ 0 exp ] G i αsds = 0, h j = min{ 0, τ j }, H j = max{ 0, τ j }, [ 0, T, j = 1, 2,..., l g i = min{ 0, σ i }, G i = max{ 0, σ i }, [ 0, T, i = 1, 2,..., m. Finally, [a] := max0, a and [a] := max0, a denoe he posiive and negaive par of he real number a, respecively. 3. Main Resuls Now we can formulae our main heorem. Theorem 1. Suppose ha H 1 and H 2 hold and le 8 and Φ 0 > 0 be saisfied. Then he following saemens are equivalen: a The iniial value problem 1 and 7 has a posiive soluion on [ 0, T. b The generalized characerisic equaion 9 has a coninuous soluion on [ 0, T. c There exis funcions β, γ C[[ 0, T, R] such ha β γ such ha 10 β δ γ implies β Sδ γ,

5 Posiive soluions of neural delay differenial equaion 99 for every funcion δ C[[ 0, T, R] and 0 < T, where [ Sδ = p j 1 τ j Φh ] j δh j p j Φh j 11 Φ 0 Φ 0 exp δsds q i Φg i exp δsds. Φ 0 H j G i Proof. a b: Le x = xφ be he soluion of he iniial value problem 1 and 7 and suppose ha x > 0 for 0 < T. I will be shown ha he coninuous funcion α defined by α = ẋ x, 0 < T, is a soluion of 9 on [ 0, T. Equaion 1 is equivalen o he form ẋ [p j 1 τ j ẋ τ j p j x τ j ] q i x σ i = 0. By dividing boh sides of he above equaion by x, we obain ha ẋ x [ p j 1 τ j ẋ τ j p j x τ ] j x x q i x σ i x = 0. I follows from he definiion of α ha x = Φ 0 exp αsds, 0 and hence xh j x = exp H j αsds, xg i x = exp G i αsds where j = 1, 2,..., l, i = 1, 2,..., m, and 0 < T. I is obvious for he same values of j, i and ha, x τ j xh j = Φh j, Φ 0 x σ i xg i = Φg i Φ 0.

6 100 H. Péics, J. Karsai I remains o prove ha ẋ τ j ẋh j = Φh j, 0 < T, j = 1, 2,..., l. Φ 0 Observe ha τ j 0 implies h j = 0 and H j = τ j, and hence ẋ τ j ẋh j = ẋh j ẋh j = 1 = Φh j. Φ 0 On he oher hand, τ j < 0 implies h j = τ j and H j = 0, and hence ẋ τ j = ẋh j = Φh j. ẋh j ẋ 0 Φ 0 Using hese equaliies and he definiion of α, we obain ha equaliy 9 holds, and hence he proof of he par a b is complee. b c: If α is a coninuous soluion of 9, hen ake β γ α, 0 < T and he proof is obvious because of he fac ha α = Sα. c a: Firs i mus be shown ha, under hypohesis c, equaion 9 has a coninuous soluion α on [ 0, T, and he funcion x defined by 12 { Φ, 1 < 0 ; x = Φ 0 exp 0 αsds, 0 < T is a posiive soluion of he iniial value problem 1 and 7. The coninuous soluion of equaion 9 will be consruced as he limi of a sequence of funcions {α k } defined by he following successive approximaion. Take any funcion α 0 C[[ 0, T, R] such ha β α 0 γ, 0 < T and se α k1 = Sα k, 0 < T for k = 0, 1, 2,... If follows from he assumpion 10 ha 13 β α k γ, 0 < T, k = 0, 1, 2,..., and clearly α k C[[ 0, T, R]. We show ha he sequence {α k } converges uniformly on any compac subinerval [ 0, T 1 ] of [ 0, T. Se M 1 := max 0 T 1 p j1 τ j Φh j Φ 0, M 2 := max 0 T 1 p j Φh j Φ 0,

7 Posiive soluions of neural delay differenial equaion 101 M 3 := max 0 T 1 q i Φg i Φ 0, L := max 0 T 1 {max { β, γ }}, M := max{m 1, M 2, M 3 }, N 1 := Me LT10, N := max{n 1 L 2, 2LN 1 }. Then from 13 we obain ha max α k L, k = 0, 1, 2,... 0 T 1 Using he mean value heorem we have exp H j = e µ k,j α k sds H j exp H j α k s α k1 s ds, α k1 sds for every j = 1, 2,..., l, k = 0, 1, 2,... and 0 T 1, where µ k,j is beween H j α k sds and H j α k1 sds. Since H j 0 for j = 1, 2,..., l and 0 T 1, µ k,j LT 1 0 and exp α k sds exp α k1 sds H j e LT 1 0 H j 0 α k s α k1 s ds. Similarly, exp G i α k sds exp e LT 1 0 G i 0 α k s α k1 s ds α k1 sds for i = 1, 2,..., m, k = 1, 2,... and 0 T 1. Repeaing he above argumens, we also have α kh j exp α k sds α k1 H j exp α k1 sds H j H j [ α kh j exp α k sds exp α k1 sds] H j H j

8 102 H. Péics, J. Karsai [α kh j α k1 H j ] exp α k1 sds Le LT10 Thus, H j 0 α k s α k1 s ds 2Le LT10. α k1 α k p j1 τ j Φh j Φ 0 α kh j exp α k sds α k1 H j exp H j p j Φh j Φ 0 exp α k sds exp H j q i Φg i Φ 0 exp α k sds exp 2LN 1 N 1 L 2 and now, we can see ha G i 0 α k s α k1 s ds N N α k1 α k N k1 i=0 for k = 0, 1, 2,... and 0 T 1. Since N 0 i i! H j G i H j α k1 sds α k1 sds α k1 sds 0 α k s α k1 s ds, 2L N 0 k. k! N 0 i i=0 i! = e N 0 and N 0 k lim k k! = 0 for 0 T 1, i follows from he Weiersrass crierion ha he sequence defined by k1 α k = α 0 [α j1 α j ] k = 0, 1, 2,..., 0 T 1 j=0 converges uniformly, and hence he limi funcion 14 α = lim k α k is coninuous and solves equaion 9 on [ 0, T 1 ].

9 Posiive soluions of neural delay differenial equaion 103 Finally, he fac ha x defined by 12 is he soluion of he iniial value problem 1 and 7 can be verified by direc subsiuion: ẋ = xα = [ = x p j 1 τ j Φh j Φ 0 exp αsds x = x x = = H j αh j p j Φh j Φ 0 q i Φg i Φ 0 p j 1 τ j ẋ τ j ẋh j ẋh j xh j p j x τ j xh j xh j x x exp xh j x [p j 1 τ j ẋ τ j p j x τ j ] d d [p jx τ j ] q i x σ i for 0 < T. I complees he proof of Theorem 1. G i ] αsds = q i x σ i xg i = xg i x q i x σ i = The above heorem generalizes Theorem in [5], proved for he nonneural equaion 6. Remark. I is clear from he proof ha he posiive soluion of equaion 1 has o saisfy he inequaliy 15 for 0 < T. Φ 0 exp βsds x Φ 0 exp γsds Exisence of Posiive Soluions Using Theorem 1 we formulae condiions for he exisence of posiive soluions. Similar resuls can also be proved for he exisence of negaive soluions. Le F := {Φ C 1 [[ 1, 0 ], R ] : 0 < Φ Φ 0, 0 < Φ Φ 0, 1 0 }. The nex heorem is a common generalizaion of Theorems 1 and 1.

10 104 H. Péics, J. Karsai Theorem 2. Assume ha H 1 and H 2 hold, and here exiss a posiive number µ such ha 16 [ p j 1 τ j µ p j ] e µτj q i e µσi µ for 0 < T. Then, for every Φ F which saisfies he condiion 8, he soluion xφ of equaion 1 remains posiive for 0 < T. Proof. We show ha he condiions of par c in Theorem 1 are saisfied wih β = µ and γ = µ for 0 < T. For any coninuous funcion δ, for which β δ γ, we have µτ j µh j and µσ i µg i H j G i for 0 < T. Then, i follows ha δsds µh j µτ j j = 1, 2,..., l δsds µg i µσ i i = 1, 2,..., m µ [ p j 1 τ j µ p j ] e µτj q i e µσi Sδ [ p j 1 τ j µ p j ] e µτj q i e µσi µ for 0 < T. Therefore, by Theorem 1, he soluion xφ of equaion 1 hrough 0, Φ is posiive on [ 0, T and he proof is complee. Apply his heorem o some special cases. Inroduce he following noaions. Le τ := max τ j, σ := max σ i,,l,m p := p j 1 τ j, r := ṗ j, q := q i. Then, inequaliy 16 follows from he inequaliy 17 pµ re µτ qe µσ µ, which is idenical o 16 for he case of single delays.

11 Posiive soluions of neural delay differenial equaion 105 Now, consider an even more special case. Le p := τ := sup τ, σ := sup σ, [ 0,T [ 0,T sup p, r := sup r, q := sup r [ 0,T [ 0,T [ 0,T be finie. Then, inequaliy 16 follows from he inequaliy 18 pµ re µτ qe µσ µ, which is idenical o 16 for he case of single consan delays and consan coefficiens. In he case τ = σ = λ, we have 19 e µλ µ pµ r q. If 1/ p > 1 and λ < λ 0 for some criical λ 0, hen 18 has a posiive soluion. The criical λ 0 can be found by observing ha he derivaives of he lef and righ sides wih respec o µ are equal for λ 0. Then, λ 0 is he unique soluion of he equaion 2A λ exp A λ A λ 2 = 4 p A λ 2 p A λ A λ 4 p A λ where A = r q. Noe ha in he delay case p = r = 0, we obain he known resul λ 0 = 1/e q. In he following heorem we assume an order of he dominance of delays. This condiion agrees wih several real phenomena. Theorem 3. Assume ha H 1 and H 2 and he following hold for every [ 0, T : 20 0 τ 1 τ 2... τ l, ν 0 σ 1 σ 2... σ m, p j 1 τ j 0, ν p j 0, ν = 1, 2,..., l; ν q i 0 ν = 1, 2,..., m; [p j 1 τ j ] < 1.

12 106 H. Péics, J. Karsai If here exiss a posiive increasing funcion γ C[[ 0, T, R] such ha 25 γ l [ p j] m [q i] 1 l [p, j1 τ j ] hen, for every Φ F which saisfies condiion 8, equaion 1 has a posiive increasing soluion on [ 0, T. This soluion saisfies he inequaliy 26 x Φ 0 exp γsds. 0 Proof. I will be shown ha he saemen c of Theorem 1 is rue wih β = 0 and γ for 0 < T. For any funcion δ [[ 0, T ], R] beween β and γ holds ha Sδ [p j 1 τ j ] γ [ p j ] [q i ] γ [p j 1 τ j ] γ 1 γ. [p j 1 τ j ] Because of he inequaliies H 1 H 2... H l, 0 < T, G 1 G 2... G m, 0 < T, he relaions 22 and 23 yield Sδ { } p j 1 τ Φhj min 1 j l Φ 0 0 exp H l δsds p j Φh j exp Φ 0 [ ] Φg i q i exp Φ 0 0 for 0 < T. H l G m δsds δsds

13 Posiive soluions of neural delay differenial equaion 107 Therefore, he soluion x = xφ of equaion 1 is posiive on [ 0, T. As in he proof of Theorem 1, x can be wrien in he form x = Φ 0 exp αsds for 0 < T, 0 where α is a coninuous soluion of he characerisic equaion 9 such ha 0 α γ for all 0 < T. Hence, x is an increasing soluion of equaion 1, and he proof is complee. Remark. Noe ha condiions show ha smaller delays has o be associaed wih larger coefficiens. Condiions 22 and 23 formulae he same propery for he funcions p j 1 τ j, ṗ j, and q i. For he numbers {a 1, a 2,..., a N }, he condiion can be expanded o n a i 0 for every n = 1, 2,..., N a 1 0, a 1 a 2 0, a 1 a 2 a 3 0,..., N a i 0. For example, his condiion holds if a 1 0 and a 1 N i=2 a i. The case of single delays is sill of imporance. In his case, condiions 20, 21 are empy, 22, 23, and 24 urn o p 1 1 τ 1 < 1, ṗ 1 0, and q 1 0. Finally, 25 becomes p 1 q 1 γ 1 p 1 1 τ 1. For l = m = 1 and consan delays we obain from our resul, as a special case, a heorem of he exisence of posiive soluions, proved in [1] Theorem c. Anoher special case is p j = 0, for j = 1, 2,..., l, [ 0, T. Then, our heorem implies he well known resul for he non-neural equaions proved in [1] Theorem Acknowledgmen This paper was compleed during he firs auhor s visi o he Deparmen of Medical Informaics a he Universiy of Szeged from Ocober 1, o November 30, 2001 under a fellowship Domus Hungarica. The research of J. Karsai is suppored by Hungarian Naional Foundaion for Scienific Research Gran no. T The auhors express heir hanks o Professor Isván Győri for valuable commens and help.

14 108 H. Péics, J. Karsai References [1] Chuanxi, Q., Ladas, G., Exisence of Posiive Soluions for Neural Differenial Equaions, Journal of Applied Mahemaics and Simulaion , [2] Driver, R. D., A Mixed Neural Sysem, Nonlinear Analysis, , [3] El-Morshedy, H. A., Gopalsamy, K., Nonoscillaion, Oscillaion and Convergence of a Class of Neural Equaions, Nonlinear Analysis, , [4] Giang, D. V., Győri, I., Oscillaion of a Linear Neural Delay Differenial Equaion wih Unbounded Time Lag, Differenial Equaions and Dynamical Sysems , [5] Győri, I., Ladas, G., Oscillaion Theory of Delay Differenial Equaions wih Applicaions, Clarendon Press - Oxford, [6] Hale, J., Theory of Funcional Differenial Equaions, Springer-Verlag New York, [7] Kriszin, T., Wu, J., Asympoic Behaviors of Soluions of Scalar Neural Funcional Differenial Equaions, Differenial Equaions and Dynamical Sysems 4 3/41996, [8] Yuecai, F., Yunen, D., Oscillaory and Asympoic Behaviour of Firs Order Differenial Equaion wih Piecewise Consan Deviaing Argumens, Annales of Differenial Equaions, , Received by he ediors February 1, 2002