Oscillations of the solutions of nonlinear hyperbolic equations of neutral type

Publicacions Ma emá iques, Vol 36 (1992), 3-18 . Abs ac OSCILLATIONS OF THE SOLUTIONS OF NONLINEAR HYPERBOLIC EQUATIONS OF NEUTRAL TYPE D .P . MISHEV AND D .D . BAINOV In his pape nonlinea hype bolic equa ions o neu al ypé o he o m 2 (~ 2 [ u ( X ' ) + a( )u(x' - T)] - [Au(x, ) + 11( )AU(x, - u)] + c(x, , u) = (X, ),  (x, ) E 9 x (0, oo) - G, a e conside ed, we e , a = cons > 0, wi h bounda y condi ions an +7(x, )u = g(x, ),  (x, ) E as x [0, oo) o u = 0,  (x, ) E asl x [O,oo) . Unde ce ain cons ain s on he coe icien s o he equa ion and he bounda y condi ions, su icien condi ions o oscilla ion o he solu ions o he p oblems conside ed a e ob ained . 1 . In oduc ion In he las ew yea s esul s ela ed o he oscilla o y p ope ies and asymp o ic beha iou o he solu ions o some classes o hype bolic equa- ions we e published . We shall men ion especially he wo k o K . K ei h, T . Kusano and N . Yoshida [4] in which su icien condi ions o oscilla ion o he solu ions o he nonlinea hype bolic equa ion u - Au + c(x, , u) = (x, ) conside ed in a cylind ical domain a e ob ained . Oscilla o y p ope ies o he solu ions o hype bolic di e en ial equa ions wi h a de ia ing a - gumen we e in es iga ed in he wo ks o D . Geo giou, K . K ei h [2], D . Geo giou [3] . Hype bolic di e en ial equa ions wi h maxima we e in es- iga ed in he wo k o D . Mishe [6] and some condi ions o oscilla ion o he solu ions o hype bolic equa ions o neu al ype we e ob ained by D . Mishe and D . Baino in [7], [8] . The p esen in es iga ion is suppo ed by he Minis y o Cul u e, Science and Edu- ca ion o People's Republic o Bulga ia unde G an 61 . 4  D .P . MISHE , D .D . BAINOV 2 . P elimina y no es In he p esen pape su icien condi ions o oscilla ion o he solu ions o nonlinea hype bolic equa ions o neu al ype o he o m a (1)  e a [u(x, ) + A( )u(x, - T)] - [DU(x, ) + p,( )DU(x, - Q)]+ + c(x, , u) = (x, ),  (x, ) E SZ x (0, oo) - G, n a e ob ained, whe e T, Q = cons > 0, niu(x, ) _  u ., ., (x, ) and 9 is -i a bounded domain in R'n wi h a piecewise smoo h bounda y . Conside bounda y condi ions o he o in (2)  ~n+ ~(x, )u = g(x, ),  (x, ) E &SZ x [0, 00) (3)  u = 0,  (x, ) E áQ x [0, 00) We shall say ha condi ions (H) a e sa is ied i he ollowing condi ions hold : H1 . ~( ) E C a([0, 00) ; [0, 00 )), p, ( ) E C([0, 00) ; R), H2 .  c(x, , u) E C(G x R ; R), 113 .  c(x, , -u) = -c(x, , u),  (x, , u) E G x (0, oo), 114 .  c(x, , u) > p( ) - h(u),  (x, , u) E G x . (0, oo), whe e p( ) is a con inuous and posi i e unc ion in he in e al (0, 00) and h(u) is a con inuous, posi i e and con ex unc ion in he same in e al ( 0 , 00 ) . H5 .  (x, ) E C(G ; IEB) 116 .  g (x, ) E C (8Q x [0, oc) ; IR) H7 . - y(x, ,) E C(09 x [0, oc) ; [0, o0)) . De ini ion 1 .  The solu ion u(x, ) E C a (G) n Cl(G) o p oblem (1), (2) ((1, (3)) is said o oscilla e in he domain G i o any posi i e numbe p, he e exis s a poin (xo, o) E 9 x [p,, 0o), such ha he equali y u(xo, o) = 0 holds . In he subsequen heo ems su icien condi ions o oscilla ion o he solu ions o p obem (1), (2) and (1),, (3) in he domain G a e ob ained . We shall no e ha in he wo k o K . K ei h, T . Kusano, N . Yoshida [4] condi ions a e ob ained o he oscilla ion o he solu ions only o p oblem (1), (2) in he case when A ( ) -- 0, p( ) -- 0 and -y(x, ) - 0 . NONLINEAR HYPERBOLIC EQUATIONS OF NEUTRAL TYPE  5 In oduce he ollowing no a ion : whe e 191 = sz dx . Wi h any solu ion u(x, ) E C 2 (G) 1 C l (G) o p oblem (1), (2) we associa e he unc ion Lemma 1 . Le , condi ions (H) hold and le u(x, ) be a posi i e solu- ion o p oblem (1), (2) in he domain G . Then he unc ion ( ) de ined by (5) sa is ies he di e en ial inequali y o neu al ype 2 (6)  d 2 [ ( ) + A( ) ( - , )] + p( )h( ( )) < G( ) + p( )G( - a) + F( ), _> o, whe e o is a su icien ly la ge posi i e numbe . P oo£- Le u(x, ) be a posi i e solu ion in he domain G o p oblem (1), (2) and o = max{T, a} . Then u(x, - - ) > 0and u(x, - a) > 0 o (x, ) E 52 x [ o, oc) . We in eg a e bo h sides o equa ion (1) wi h espec o x o e he domain 9 and ob ain o ? o : ( ) = 191 1  - 1~u (x, ) dx,  > 0 . j 2  u(x, ) dx + A ( )  u(x, - - ) dx J - ~~ áu(x, ) dx+ + p, ( )  Du(x, - a) dx] +  c(x, , u) dx =  , ( .x, ) dx . in  9 Rom G een's o mula and condi ion H7 i ollows ha Du(x, ) dx = ao án ds = ~~ [g(x, ) - -y(x, )u] ds < ~~ g(x, ) ds (9)  áu(x, - a) dx =  &u (x, - a) ds = 9  lasa 8n = ~~[g(x, - a) - 7(x, - a) - u(x, - a)] ds <_< ~ sa g(x ' - a) ds (4) F ( ) 1 = (x, ) dx, > 0, (5) G ( ) = IS2I ~~ g (x, ) ds, > 0, 6  D .P . Mis¡¡ , D .D . BAINOV Mo eo e , om condi ion H4 and Jensen's inequali y i ollows ha (10)  c(x, , u) dx -> p( ) h(u(x, )) dx >_ s  - >_ p( )h ( o u(x, ) dx  dx) -1/  dx = p( ) - h( ( )) - 1521 Using (8)-(10) and condi ion H1, om (7) we ob ain z d 2 [ ( ) + ( ) ( - , )] <_ G( ) + c( )G( - ) + F( ) - p( )  h( ( )), which p o es Lemma 1 . 3 . Main esul s Theo em 1 . Le condi ions (H) hold and le he di e en ial inequal- i ies o neu al ype (11) 2 d 2 [ ( ) + A( ) ( - - )] + p( ) - h( ( )) <_- G( ) + u( )G( - o,) + F( ) (12) z d 2 [ ( ) + A( ) ( - T)] + p( ) - h( ( )) <_ G( ) - h( ) - G( - Q) - F( ) ha e no e en ually posi i e solu ions . Then each solu ion u(x, ) o p ob- lem (1), (2) oscilla es in he domain G . P oó% Le p > 0 be a posi i e numbe . Suppose ha he asse ion o he heo em is no ue and le u(x, ) be a solu ion o p oblem (1), (2) wi hou ze oes in he domain G, = 9 x [p,, oo) . I u(x, ) > 0 o (x, ) E G p , hen om Lemma 1 i ollows ha he unc ion ( ) de ined by (5) is a posi i e solu ion o inequali y (11) o >-- o + ju, Le i is an e en ually posi i e solu ion o (11) which con adic s he assump ion o he heo em . I u(x, ) < 0 o (x, ) E G,,, hen he unc ion -u(x, ) is a posi i e solu ion o he p oblem 2 á 2 [u + A( )u(x, - T)] - [Du +, ( )DU(x, - Q)]+ + c(x, , u) =- (x, ), ( .x, ) E G ón + -y(x, )u =- g(x, ),  (x, ) E óg x [0, oo) . NONLINEAR HYPERBOLIC EQUATIONS OF NEUTRAL TYPE  7 F~om Lemma 1 i ollows ha he unc ion Tg-'T ez(-u(x, )) dx is a posi- i e solu ion o inequali y (12) o _>- o + c which also con adic s he assump ion o he heo em . Thus Theo em 1is p o ed . Now we shall in es iga ge he oscilla o y p ope ies o he solu ions o p oblem (1), (3) . Conside in he domain 9 l e ollowing Di ichle p oblem : DU + caU = 0 in S2 UlaQ = 0 whe e a = cons . I is well known [1] ha he smalles eigen alue ao is posi i e and he co esponding eigen unc ion W(x) can be chosen o sa is y he inequali y W(x) > 0 o x E 9 . Wi h any solu ion u(x, ) E C 2 (G) 1 Cl (G) o p oble i (1), (3) we associa e he unc ion -I (13)  w ( )  u(x, )W(x) dx  w(x) dx)  ,  > 0 sz  sz We shall no e ha a simila a e aging was i s used by N . Yoshida in he wo k [10] . Lemma 2 . Le condi ions Hl-H6 hold and le u(x, ) be a posi i e solu ion in he domain G o p oblem (1), (3) . Then he unc ion w( ) de zned by (13) sa is aes he di e en ial inequali y o neu al ype 2 (14)  T 2 [w( ) +, ( )w( - T)] + aol ( ) + aoh( ) - w( - o-)+ + p( ) - h(w( )) :5 J , (x, )w(x) dx  ~~ W (x) dx) - '  --> o, sz  sz  - whe e o is a su cien ly la ge posi i e numbe . P oo .. Le u(x, ) be a posi i e solu ion in he , domain G o p oblem (1), (3) and o = max{T, u} .  Then u(x, - T) > 0 and u(x, - a) > 0 o (x, ) E 9 x ( o, oo) .  Mul iply bo h sides o equa ion (1) by he eigen unc ion W(x) o he Di ichle p oblem and in eg a e wi h espec o .x o e he domain 9 . Fo -> o we ob ain (15) d 2 U ( x, ) w(x) dx + A ( )  u(x, - - )cp(x) dx] - d - ~~ Du(x, )W(x) dx + p( ) n  2 nu(x, - u)p(x) dx] + + .~s c(x, , u) W(x) dx = .~ (x, ) ~o(x) dx . 8  D .P . Misü , D .D . BAINOV F o n G ee 's o iula i ollows ha (16)  L Du(x, )W(x) dx = i u(x, ) AW(x) dx = s  in =- ao -  u(x, )W(x) dx = -aow( ) - 1 W(x) dx sz  sz (17)  L Du(x, - u)cp(x) dx = 1 u(x, - o,)Ocp(x) dx = sz  sz =- ao u(x, - u)W(x) dx = -cxow( - a) - ~p(x) dx, sz  sz whe e ao is he smalles eigen alue . Mo eo e , om condi ion H4 and Jenscn's inequali y i ollows ha (18)  ,ISZ c(x, , u) W(x) dx ~ p( ) 19 h(u)~o(x) dx _1 p( ) - 1¿  u(x, ) W(x) dx - C W(x) dx / l  ~, co(x) dx = S2  S2  S2 Using (16)-(18) a ad condi ion H1, om (15) we ob ain 2 2 ~1A1( ) + ñ( )1l1( - T) 1 < -00 [ID ( ) + /b( )1U( - u)~- _ p( ) . h(w( )) + ig (x' )~o(x) dx - (I 2 W(x) dx) -1 which comple es he p oo o Lemma 2 . In oduce he no a ion = p( ) - h(w( )) - ~, cp(x) dx sz (19)  Fi ( ) _  (x, )~(x) dx  ~ W(x) dx) -1 ,  > 0 . sz  Sa Analogously o Theo em 1 he ollowing heo em is p o ed . Theo em 2 . Le , condi ions Hl-H5 hold and le he di e en ial in- equali ies o neu al, ype z (20)  22 ~1U( ) + ~( )1U( - T)) + a o [w( ) + ~( )w( - ~) 1+ + p( ) - h(w( )) <_ Fl( ),  ? o, NONLINBAR HYPERBOLIC EQUATIONS O NEUTRAL TYP13  9 2 (21)  d 2 [w( ) + ( )W( - T)] + c1!o [w( ) + P( )w( - o,)]+ + p( ) - h(w( )) <- - FI ( ),  ? o ha e no e en ually posi i e solu ions . Then each solu ion u(x, ) o p ob- lem (1), (3) oscillla es in he domain G . F om he heo ems p o ed abo e i ollows ha he inding o su iicien condi ions o oscilla ion o he solu ions o equa ion (1) in he domain G is educed o he in es iga ion o he oscilla o y p ope ies o di e en ial inequali ies o neu al ype o he o m 2 (22)  d 2 [x( ) + ( )x( - T)] + go ( )x( ) + q( )x( - o  )+ We shall say ha condi ion (A) a e sa is ice i lic 'ollowing condi ions hold : A1 . ~ ( ) E C 2 ([ o, 00) ; [0, 00 )), A2 . qo( ), q( ) E C([ o, co) ;[ 0 , oo)), A3 . p( ) E C([ o, oo) ; [0, oo)), A4 . h(u) E C(R ; R), h(u) > 0 o u > 0, A5 . H ( ) E C([ o , oo) ; H) . + p( ) - h(x( )) <_ H( ),  >- 11 Theo em 3 . Le condi ions (A) hold as well as he condi ion (23)  lim in  1 -00 - 1 o i >- o . Then he di 'e en ial inequali y (22) has no e en ualllly posi- i e solu ions . P oo :: Suppose ha his is no ue and le x( ) be a posi i e solu ion o inequali y (22) de ined in he in e al [ 1,oo), whe e 1 >__ o . Then in i ue o condi ions A2-A4 wc ob ain o , >- 2 ( 2 >- 1 + nax{a,T}) 2 d 2 [x( ) + ( )x( - T)] <_ H( ) - go( )x( ) - q( )x( - o-)- - p( ) - h(x( )) <- H( ) . We in eg a e wice he abo e inequali y o e he in e al [ 2, .], . > 2 and ob ain ( - s) - H(s) ds = - n  1 x ( ) + A( )x( - T) < C1 + C2 ( - 2) +'12' [£  H(s) ds ] dp, Z 10  D .P . MISHEV, D .D . BAINOV whe é Cl, C2 = cons . Since (24)  x( ) + ~( )x( - T)  C  + C2 +  1  J ~( - s)H(s) ds - 2  - - 2  - 2 , ha  1  e H (s) ds J dp =  ( - s)H(s) ds, 2 2  2 di iding bo h sides o las inequali y by - 2 > 0, we ob ain Then o -> oo om (24), making use o condi ion (23), we ob ain x( ) +, ( )x( - T) _ (25)  l m i0  -  - 2 On he o he hand, using condi ion A1 and he ac ha x( ) > 0, x( - T) > 0 o _> 2, we ob ain ha li a in  1  [x( ) + A( )X( - T)J >- 0, -00 - op  - which con adic s equali y (25) . This comple es he p oo o Theo em 3 . Thé ollowing su icien condi ion o oscilla ion o he solu ions o p oblem (1), (2) is a co olla y o Theo em 1 and Theo em 3 . Theo em 4 . Le condi ions (H) hold as well as he condi ions o any_su cien ly la ge numbe e, whe e he unc ions G( ) and F( ) a e de ined by (4) .  Then each solu ion u(x, ) Q p oblem (I), (2) oscil- la es in he do nain G . The . ollowing su ñcien condi ion o oscilla ion o he solu ions o p oblem (1), (3) is a co olla y o Theo em 2 and Theo em 3 . (26) lim in £ (1 - ') (G(s) + (s)G(s - a) + F(s)) ds = -oo, (27) limsu ) ~ 0 (1 - ) (G(s) + p(s)G(s - a) + F(s)) ds = +oo NONLINEAR HYPERBOLIC EQUATIONS O NEUTRAL TYPE  11 Theo em 5 . Le condi ions Hl-H5 hold as well as he condi ions (28)  M( ) ? 0 o > 0 (29)  lim in £ (1 - ) Fi (s) ds = -oo -oo (30)  lim sup J (1 - S) F (s) ds = +oo --- oo   o any su icien ly la ge numbe o, whe e he unc ion FI ( ) is de ined by (19) . Then each solu ion u(x, ) o P oblem (1), (i) oscilla es in he domain G . Example 1 . Conside he equa ion (31)  u + u (x, - 7 ) - u . x +u = 2e cos x(sin . + cos í - e' - cos ), (x, ) E (0, 2 ) x (0, oo) - G, and he bounda y condi ions (32)  -u x (0, ) = 0,  u y ( 7 , ) ,_ -e - sin ,  ? 0 A s aigh o wa d e i ica ion shows ha he unc ions c(x, , u) = u, (x, ) = 2e - cos x - (sin + cos , - e' cos ), g(0, ) = 0, g ( 2 , ) _-e  sin , A( ) - 1, p( ) - 0, -Y (x, ) - 0 sa is y condi ions (H) . Mo eo e , om (4) we ob ain ha By s aigh o wa d calcula ions we ind ha I( ) .= ~ (1 - ) (G(s) + p(s)G(s - a) + F(s)) ds o = e - ( i ) -I - (2 sin - 2e - ' sin - cos ) + C, 1 8  D .P . MISHEV, D .D . BAINOV Re e en es 1 .  V .S . VLADIMIROV, "Equa ions o Ma hema ical Physics," Moscow, Nauka, 1981 (in Russian) . 2 .  D . GEoRGiou, K . KREITH, Func ional cha ac e is ic ini ial alue p oblems, J . Ma h . Anal . Appl . 10 7 (1985), 414-424 . 3 .  D . GEORGIOU, "Ex ema solu ions o une ional hype bolic ini ial alue p oblems, Di e en ial equa ions : quali a i e heo y," ol . I, 11 (Szeged, 1984), Colloq . Ma h . Soc . János Bolyai 47, No h-Holland, 1987 . 4 .  K . KREITII, T . KUSANO, N . YOSHIDA, Oscilla ion p ope ies o no llinea hype bolic equa ions, Siam J . Ma h . Anal . 15, 3 (1984), 570-578 . 5 .  T . KUSANO, M . NAITO, Oscilla ion c i e ia o a class o pe u bed Scl ódinge equa ions, Canad . Ma h . Bull . 25, 1 (1982), 71-77 . 6 .  D .P . MISHEV, Oscilla o y p ope ies o he solu ions o hype - bolic di e en ial equa ions wi h "maximun ", Hi oshi na Ma h . J . 16 (1986), 77-83 . 7 .  D .P . MISHEV, D .D . BAINOV, Oscilla ion p ope ies o he solu- ions o a class o hype bolic equa ions o neu a' ype, Funkcialaj Ek acioj 29 (1986), 213-218 . 8 .  D .P . MISHEV, D .D . BAINOV, "Oscilla ion p ope ies o he solu- ions o hype bolic equa ions o neu al ype, Di e en ial equa ions : quali a i e heo y," ol . 1, 11 (Szeged, 1984), 771-780, Colloc . Ma h . Soc . János Bolyai 47, No h-Holland, 1987 . 9 .  D .P . MISHEV, Oscilla ion o he solu ions o non-linea pa abolic equa ions o neu al ype ( o appea ) . 10 .  N . YOSHIDA, Oscilla ion o nonlinca pa abolic equa ions wi h une- ional a gunien s, Hi oshima Ma h . J . 16, 2 (1986), 305-314 . 11 .  A .I . ZAHARmV, D .D . BAINOV, Oscilla ing p ope ies o he solu- ions o a class o neu al ype une ional di e en ial equa ions, Bull . Aus al . Ma h . Soe . 22, 3 (1980), 365-372 . P .O . Box 45 1504 So ia BULGARIA Rebu el 23 d'Agos de 1990