International Journal of Mathematical Archive-5(6), 2014, Available online through ISSN

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1 Inenaional Jonal o Mahemaical Achive-6, 0, 09-8 Availale online hogh ISSN 9 06 EXISENCE OF NONOSCILLAORY SOLUIONS OF A CLASS OF NONLINEAR NEURAL DELAY DIFFERENIAL EQUAIONS OF HIRD ORDER K V V Sehagii Rao Depamen o Mahemaic, Kakaia Inie o echnolog Science, Waangal, India. P V H S Sai Kma* Depamen o Mahemaic, Swana Bhaahi College o Engineeing, Maddlapalli, Kodad Co Road, Khammam, India. Received on: 9-06-; Revied & Acceped on: ABSRAC he condiion o he eience o nonocillao olion o nonlinea neal dela dieenial eqaion wih poiive negaive coeicien o he om d d d p τ G σ G σ 0 ae died d d d nde he ampion ha o vaio ange o p. 0 AMS Claiicaion: K0, K0, C0. ddd < ; i,. i Ke wod: Ocillaion, Nonocillaion, Neal dieenial eqaion, hid ode.. INRODUCION In hi pape we conide he neal dela dieenial eqaion NDDE o hid ode d d d p τ G σ G σ 0 d d d τ, σ, σ ae poiive nme C [ 0,,[0,,, [0,,[0, C G wih G i > 0 o 0 ; i,. whee p C[ 0,, R, R, G C R,, R When p 0, 0 he aove eqaion edce o he hid ode dela dieenial eqaion d d d G σ 0 d d d I, G hi eqaion ecome d d d σ 0 d d d Coeponding aho: P V H S Sai Kma* Depamen o Mahemaic, Swana Bhaahi College o Engineeing, Maddlapalli, Kodad Co Road, Khammam, India. Inenaional Jonal o Mahemaical Achive- 6, Jne 0 09

2 K V V Sehagii Rao P V H S Sai Kma*/ Eience o Nonocillao olion o a cla o nonlinea Neal Dela Dieenial Eqaion o hid Ode/ IJMA- 6, Jne-0. he el on he nonocillao ehavio o ae volmino. Some el on he ocillao ehavio o olion o he eqaion ae diced in [8]. he aho peened nonocillao o he olion o he i ode neal dieenial eqaion in [7]. We oeve ha mch le i availale egading he eience o nonocillao olion o nonlinea neal dela dieenial eqaion wih poiive negaive coeicien. We ecall he ollowing: Le C[ 0,, R 0 0, ρ ma τ, σ, σ e he given ncion le 0 e he given conan. B a olion o he eqaion we mean a ncion C[ 0 ρ,, R in he ene ha oh [ p τ ] [ p τ ]'' ae coninol dieeniale o 0 ; Sp : 0 >, aiie he eqaion o 0 ρ, ] ρ whee { } { } 0 [ p τ ]' 0 [ 0 I hee ei a eqence o poin{ j } ch ha j 0, hen he olion i called ocillao. I all he olion o eqaion ae ocillao, hen eqaion i called ocillao; ohewie i i called nonocillao.. MAIN RESULS We need he ollowing in o dicion H : ; < ddd 0 0 H : G > o 0 H : vg v o v 0 H : ddd < ; 0 0 > When p - we ee ha eqaion i edced o d d d τ G σ G σ 0 d d d heoem.: Sppoe G hold, hen he eqaion poee a nonocillao olion. Poo: We can chooe icienl lage ch ha M ddd < 0 whee on M ma{ ; G },. 0 L ; G ae Lipchiizian on he ineval o he pe [a,], 0 < a < < H - H M ma{ G } M ddd < L; 0 L, L ae Lipchiz conan o G G epecivel Le XBC [ 0,, R e he pace o all onded eal valed conino ncion deined, Cleal, X i a Banach pace. p { : }, wih pemm nom 0, IJMA. All Righ Reeved 0

3 Le S K V V Sehagii Rao P V H S Sai Kma*/ Eience o Nonocillao olion o a cla o nonlinea Neal Dela Dieenial Eqaion o hid Ode/ IJMA- 6, Jne-0. X : ; 0 I i ea o vei ha S i a cloed onded conve e o X. Fo S, ρ, whee ρ τ G σ ddd 0 G σ ddd Indeed, < < > 0 Hence 0, implie ha S h : S S. Fo, S. conide, whee ρ τ 0 σ G ddd G σ ddd τ 0 σ G ddd σ G ddd 0, IJMA. All Righ Reeved

4 K V V Sehagii Rao P V H S Sai Kma*/ Eience o Nonocillao olion o a cla o nonlinea Neal Dela Dieenial Eqaion o hid Ode/ IJMA- 6, Jne ha i, h < 0. Hence i a conacion. B he Banach Conacion heoem, ha niqe ied poin in S. hen i a olion o eqaion in he ineval 0, hi i a nonocillao olion o eqaion. Remak.: I i ea o ee ha he olion o eqaion oained in he aove heoem i poiive onded, h he condiion o he heoem gaanee he eience o poiive onded olion. Remak.: When p eqaion can e wien a d d d τ G σ G σ 0 d d d We can alo pove he eience o a poiive olion which i onded o he eqaion Le p, p need no e o one ign hi ineval heoem.: Le -< p <. Ame hag i, i, i a Lipchizian on he ineval o he om [a, ], 0<a<<, H - H hold. hen he eqaion admi a nonocillao olion. Poo: We divide he poo ino hee cae. Cae - i: 0 < p < I i poile o chooe a poiive nme ch ha M ddd < M ddd < 0 whee on 0 M ma{ ; },. L ; G M ma{ ; } L G L, L ae Lipchiz conan o G G epecivel, We again conide, X BC [ 0,, R he Banach pace o all onded eal valed conino ncion 0 wih pemm nom, Deine S X : ; 0 p { : } 0, IJMA. All Righ Reeved

5 K V V Sehagii Rao P V H S Sai Kma*/ Eience o Nonocillao olion o a cla o nonlinea Neal Dela Dieenial Eqaion o hid Ode/ IJMA- 6, Jne-0. Hence S i a complee meic pace whee he meic i indced he nom on X. Fo S we deine, ρ, whee ρ p τ G σ ddd G σ ddd Conide < G ddd σ < < G ddd σ 0 M 0 ddd Hence, 0, h S ; i.e. : S S Alo o, S whee τ σ p G ddd ρ G σ ddd τ σ p G ddd G ddd σ 0, IJMA. All Righ Reeved

6 K V V Sehagii Rao P V H S Sai Kma*/ Eience o Nonocillao olion o a cla o nonlinea Neal Dela Dieenial Eqaion o hid Ode/ IJMA- 6, Jne-0. 0, IJMA. All Righ Reeved h 0 7 Hence i a conacion. B Conacion mapping pinciple ha niqe ied poin S.h i a nonocillao olion o eqaion. Cae - ii: Sppoe - < p < 0. I i poile o chooe a poiive nme ch ha < ddd M ddd M < 0 whee M ma{ } ; G L ; M ma{ } ; G L L, L ae Lipchiz conan o G G epecivel on, 0. We chooe he pace X a in cae i wih he ame nom Deine S ; 0 : X Hence S i a complee meic pace whee he meic i indced he nom on X. Fo S we deine, whee ddd G ddd G p whee ρ σ σ τ ρ ρ ;, Conide ddd G σ ddd M <

7 K V V Sehagii Rao P V H S Sai Kma*/ Eience o Nonocillao olion o a cla o nonlinea Neal Dela Dieenial Eqaion o hid Ode/ IJMA- 6, Jne-0. > G σ M > ddd > 0 0 Hence, S : S S ddd Alo o, S p G ddd τ σ G σ ddd τ σ G ddd σ p G ddd whee ρ h 7 0 Hence i a conacion. B Conacion mapping pinciple ha niqe ied poin S. h i a nonocillao olion. Cae - iii: Sppoe < p < ch ha < 0 > 0 <. A in he ealie cae i i poile o chooe a poiive nme ch ha M ddd < whee M, M, M ddd < L, L ae a in cae II on he ineval o he om, 0 0 0, IJMA. All Righ Reeved

8 Le S K V V Sehagii Rao P V H S Sai Kma*/ Eience o Nonocillao olion o a cla o nonlinea Neal Dela Dieenial Eqaion o hid Ode/ IJMA- 6, Jne-0. X : ; 0 whee X i he ame e a in cae ii. Fo S we deine, ρ, whee ρ p τ G σ ddd G σ ddd ; whee ρ G σ ddd M ddd < > G σ ddd > M ddd > 0 0 Hence, S S S Alo o, ε S, :. p G ddd τ σ G σ ddd τ σ G ddd σ p G ddd 0, IJMA. All Righ Reeved 6

9 K V V Sehagii Rao P V H S Sai Kma*/ Eience o Nonocillao olion o a cla o nonlinea Neal Dela Dieenial Eqaion o hid Ode/ IJMA- 6, Jne-0. whee ρ B < h 0 0 Hence i a conacion. Again, Conacion pinciple ha niqe ied poin S. h i a nonocillao olion o eqaion which i onded. SUMMARY We can pove he el which gaanee he eience o nonocillao olion which ae onded o he eqaion when p.howeve ollowing he line o hee el we can pove he eience o nonocillao olion o ohe ange o p a well. Alo we ee he dominance o pon i no oo eicive in hi wok. REFERENCES [] D.D. Bainov, D.P. Mihev, Ocillaion heo o Neal Dieenial Eqaion wih Dela, Hilge, Biol 99. [] L.H Ee, Q.K.Kong, B.G.Zhang, Ocillaion heo o Fncional Dieenial Eqaion, Macel Dekke, New Yok 99. [] I Goi, G. Lada, Ocillaion heo o Dela Dieenial Eqaion wih Applicaion, Ood Univ.Pe, London 99. [] J.K.Hale, heo o Fncional Dieenial Eqaion, Spinge-Velag, New Yok 977. [] Y.Li, W. Ge, Ampoic ehavio o ceain dela dieenial eqaion wih ocing em, J.Mah.Appl , , IJMA. All Righ Reeved 7

10 K V V Sehagii Rao P V H S Sai Kma*/ Eience o Nonocillao olion o a cla o nonlinea Neal Dela Dieenial Eqaion o hid Ode/ IJMA- 6, Jne-0. [6] J.Shen,. ang, New nonocillaion cieia o dela dieenial eqaion, J.Mah.Anal.Appl, 90 00, -9. [7] Shiping L Weigao Ge, On Eience o Peiodic Solion o Neal Dieenial Eqaion, Nonlinea Anali,, 00, [8] K.V.V. Sehagii Rao P.V.H.S Sai Kma, On he Ocillaion o hid Ode Linea Neal Dela Dieenial Eqaion, Inenaional Jonal o Mahemaic end echnolog, 0, Soce o ppo: Nil, Conlic o inee: None Declaed [Cop igh 0 hi i an Open Acce aicle diied nde he em o he Inenaional Jonal o Mahemaical Achive IJMA, which pemi neiced e, diiion, epodcion in an medim, povided he oiginal wok i popel cied.] 0, IJMA. All Righ Reeved 8

Degree of Approximation of a Class of Function by (C, 1) (E, q) Means of Fourier Series

Degree of Approximation of a Class of Function by (C, 1) (E, q) Means of Fourier Series IAENG Inenaional Jounal of Applied Mahemaic, 4:, IJAM_4 7 Degee of Appoximaion of a Cla of Funcion by C, E, q Mean of Fouie Seie Hae Kihna Nigam and Kuum Shama Abac In hi pape, fo he fi ime, we inoduce