A Necessary and Sufficient Condition for the Solutions of a Functional Differential Equation to Be Oscillatory or Tend to Zero

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1 JOURNAL OF MAEMAICAL ANALYSIS AND APPLICAIONS 24, ARICLE NO. AY A Necessary and Sufficien Condiion for he Soluions of a Funcional Differenial Equaion o Be Oscillaory or end o Zero Piambar Das* Deparmen of Mahemaics, I.G.I.., Sarang, alcher , India and N. Misra Deparmen of Mahemaics, Berhampur Uniersiy, Berhampur-76 7, India Submied by V. Lakshmikanham Received January 17, 1995 I is shown ha every soluion of he nonhomogeneous funcional differenial equaion d Ž xž. pxž.. QŽ. GŽ xž.. fž., d. where f, Q C,,,,,,,p1, G: R R such ha xg x for x, G is nondecreasing, Lipschizian, and saisfy a sublinear condiion k dx GŽ x. and f s ds, is eiher oscillaory or ends o zero asympoically if and only if QŽ s. ds Academic Press * Research suppored by he Naional Board for igher Mahemaics, Deparmen of Aomic Energy, Governmen of India X97 $25. Copyrigh 1997 by Academic Press All righs of reproducion in any form reserved. 78

2 FUNCIONAL DIFFERENIAL EQUAIONS INRODUCION In his paper we consider he nonhomogeneous neural delay differenial equaion d Ž xž. pxž.. QŽ. GŽ xž.. fž., Ž 1.1. d wih he assumpions ha f, Q CŽ,., Ž,..,, Ž,., G: R R such ha xgž x. for x, G is nondecreasing, Lipschizian, and saisfy K dx GŽ x. for every K, and prove ha if p 1, every soluion of 1.1 is eiher oscillaory or ends o zero asympoically if and only if QŽ s. ds. Se max, 4. By a soluion of Ž 1.1. wih iniial funcion CŽ,, R., we mean a funcion x CŽ,., R. such ha x Ž. Ž Ž.. Ž1. for, x px C Ž,., R., and x saisfies Ž A soluion of Ž 1.1. is said o be coninuable if i exiss in a half-line,. for some. Such a coninuable soluion x Ž. of Ž 1.1. is called oscillaory if x Ž. has zeros for arbirarily large. Oherwise, x Ž. is called nonoscillaory. Equaion Ž 1.1. is said o be generalised-sublinear if G saisfies K dx GŽ x. for every posiive real K, which includes he case G x x,1. Similarly, Eq. Ž 1.1. is said o be generalised-superlinear in case dx K GŽ x. for every posiive real K, which includes he case G x x, 1.

3 8 DAS AND MISRA here is absoluely no resul dealing wih a necessary and sufficien condiion for oscillaionnonoscillaion of Eq. Ž 1.1., excep a few wih fž., which deals wih such condiions where he coefficien funcions are real consans Žsee 7, 8. and he oscillaory behaviour of such equaions are mainly characerised by he exisence of no real roos of an associaed characerisic equaion which iself leads o a research problem. In 4 he auhor considered Eq. Ž 1.1. and obained a necessary and sufficien condiion for oscillaion of all soluions in which he coefficiens of he equaion are assumed o be he periodic funcions and fž.. In 2, 3 equaions of he ype and xž. QŽ. x Ž. x Ž n. Ž. QŽ. GŽ xž.. were considered and necessary and sufficien condiions, in erms of coefficiens, were obained for all soluions of he prescribed equaions which were oscillaory. Bu he simple example xž. x Ž. Ž e Ž. e., Ž. which admis a nonoscillaory soluion x e in which he associaed homogeneous equaion has only oscillaory soluions for 1e Žsee. 2, moivaed us o sudy oscillaion along wih asympoic behavior of soluions of Eq. Ž We obained he following resul. 2. MAIN RESULS EOREM 1. Suppose ha Q CŽ,.Ž,,.., G: RR such ha xgž x., x, G is nondecreasing, and saisfy K dx GŽ x. If p 1,,,, and for every K. Ž 2.1. fž x. dx, Ž 2.2. hen a necessary and sufficien condiion for a soluion of Ž xž. pxž.. QŽ. GŽ xž.. fž. Ž 2.3.

4 FUNCIONAL DIFFERENIAL EQUAIONS 81 which is eiher oscillaory or ends o zero asympoically is ha QŽ s. ds. Ž 2.4. Proof. Suppose ha Ž 2.4. holds. Le x be a soluion of Ž If x is oscillaory, here is nohing o prove. Assume ha x Ž. is nonoscillaory, ha is, x Ž. or x Ž. for for some. We shall show ha x Ž. as. If x Ž. for, from Ž 2.3. i follows ha zž. for where z Ž. x Ž. pxž.. Consequenly z Ž. orz Ž. for 1. If he former holds hen obviously Seing xž. pxž., 1. Ž 2.5. max xž., 11, we see ha for every, here exiss an ineger N such ha p n, n N. Se N. Now implies ha 1 xž. pxž. p 2 xž 2. p N xž N.. ence by definiion x Ž. as. If he laer holds, ha is, z Ž. for hen zž. and consequenly he inequaliy z Ž. z 1 x pxž. x, and G is nondecreasing implies ha zž. QŽ. GŽ zž.. zž. QŽ. GŽ xž.. fž.. Dividing he above inequaliy by G z and using he fac ha G z we obain zž. G zž. QŽ.. Inegraing he above inequaliy from o and using 2.1 and 2.4 on he resuling inequaliy we reach a conradicion.

5 82 DAS AND MISRA Nex, suppose ha x Ž.,. We shall show ha x Ž. as. Se YŽ. xž. pxž. fž s. ds. In view of 2.3, we verify ha Y saisfies YŽ. QŽ. GŽ xž... Ž 2.6. ence YŽ. evenually. Consequenly Y Ž. ory Ž. for large, say for 1. If Y Ž., hen YŽ. concludes ha Y Ž. is bounded. Inegraing Ž 2.6. from o Ž. and using he boundedness of Y we obain 1 QŽ. G xž. d. Ž 2.7. Again Y, 1 implies ha and consequenly, which furher implies ha 1 xž. fž s. ds, ž / G xž. G fž s. ds, QŽ. G xž. d. his is a conradicion o Ž On he oher hand, if Y Ž., where 1 hen xž. pxž. fž s. ds pxž. fž s. ds pxž. M, Ž 2.8. M fž s. ds.

6 FUNCIONAL DIFFERENIAL EQUAIONS 83 Replacing by, 2,...,n successively in 2.8 hen using he resuling inequaliies we obain ha is, xž. pxž. M pž pxž 2. M. Mp 2 xž 2. pm M 2 p pxž 3. M PM M p 3 xž 3. p 2 MPM M p n xž n. p n1 Mp n2 M PM M, n1 n i xž. p xž n. M ž Ý p /. i he convergence of he geomeric series Ý is bounded. We may see ha lim inf x. Because, oherwise lim inf x Ž. implies he exisence of a such ha x Ž 1. 2 for. Inegraing Ž 2.6. from o, we obain i p leads o he fac ha x Ž. i Ž. YŽ s. ds QŽ s. G xž s. ds, which furher implies Y Ž. as. Consequenly, x Ž. bounded. his is a conradicion. ence lim inf x Ž.. Le lim sup xž.. is un- If we shall reach he following conradicion. By definiion, here exiss a divergen sequence of reals ² : such ha x n n1 n as n. Since x Ž. is bounded, he bounded sequence of real numbers ² x Ž n.: n1 admis a convergen subsequence, say ² x : n k which con- verges o for some,. Now seing z Ž. x Ž. pxž. 1 1 we see ha lim z lim x px k nk nk nk k lim x p lim x k nk k p 1p nk

7 84 DAS AND MISRA his shows ha lim sup z Ž. 1p. For our convenience ake lim sup zž.. Again lim inf x Ž. implies, by definiion, ha here exiss a diver- gen sequence ² : of reals such ha s Ž. n n1 n as n. Conse- quenly, he bounded sequence of real numbers ² x Ž.: n admis a convergen subsequence ² x Ž.: n k k1 which converges o some real number where. ence lim z lim x px. k nk nk nk k p. his shows ha lim inf z Ž.. Le us suppose ha lim inf zž., where is some nonposiive real number. Again from he definiion of limi supremum, here exis wo divergen sequences of reals ² s : n n1 and ² s: n1 such ha zžs. and zž s. n n as n. Wihou any loss of generaliy, assume ha s s n n for every index n. ake N large enough such ha zž sn. zž sn., nn. Ž Since x Ž. is bounded implies ha z Ž. is bounded, inegraing Ž 2.3. from o and using Ž 2.2. we obain QŽ s. G xž s. ds. Ž From Ž 2.2. and Ž i follows ha here exiss such ha QŽ s. GŽ xž s.. ds 8 and f s ds. 8

8 FUNCIONAL DIFFERENIAL EQUAIONS 85 Le s, s for n N N. Inegraing Ž 2.3. from s o s for n N we see ha n n n n zž sn. zž sn. 2 sn sn QŽ s. G xž s. ds fž s. ds sn, which is a conradicion. his concludes ha lim sup xž.. Ž. ere x as. his complees he proof. Necessary Par. o he conrary, suppose ha Choose large enough so ha QŽ s. ds. Ž 1p. Ž 1p. fž s. ds and K QŽ s. ds, 1 5 where K is he Lipschiz consan of G in Ž 1 p. 1, 1. Le C,. b denoe he Banach space of all funcions f CŽ,.Ž,,.. associaed wih he sup-norm on,.. Consider he operaor R: S S defined by 1p R yž. pyž. QŽ s. G yž s. ds fž s. ds, sn 5. where S is he se of all funcions g C, such ha b, 1 p g 1. 1

9 86 DAS AND MISRA For any g S, i is easy o see ha and 1 p 1 p RŽ gž.. p Ž 1 p. 1 p 1 p 1 p RŽ gž.. p Ž. his shows ha R g S and hence R maps S o S. Now for any g, h S we have R R sup R gž. R hž. g h psup gž. hž. sup QŽ s. G gž s. G hž s. ds p sup gž. hž. Ksup gž. hž. QŽ s. ds ž / pgh K Q s ds gh ž / 1p p gh. 5 his furher shows ha R is a concenraion mapping and hence admis a fixed poin say w Ž. in S, ha is, 1p wž. pwž. QŽ s. G wž s. ds fž s. ds which is equivalen o 5 Ž wž. pwž.. QŽ. GŽ wž.. fž.. ence w is he required soluion wih he propery ha Ž 11.Ž 1 p. w Ž. 1 which is neiher oscillaory nor ends o zero asympoically. his complees he proof of he heorem.

10 FUNCIONAL DIFFERENIAL EQUAIONS 87 EXAMPLE EXAMPLES AND COUNEREXAMPLES Consider he differenial equaion 1 13 Ž13.Ž. x 2x x e 1e 2 e, Ž where 1 e 2.. his equaion saisfies he hypohesis of heorem 1. Ž. Clearly, x e is a soluion of his equaion. I is impossible o exend heorem 1 for p 1. he following example jusifies our claim. EXAMPLE 2. Le, Ž,.. Consider he equaion Ž xž. pxž.. e Ž e 2 pe 1. xž. e, Ž. where p e. his equaion admis a soluion x e oscillaory nor ends o zero asympoically. which is neiher ACKNOWLEDGMEN he auhors are hankful o he referee for suggesing some modificaions. REFERENCES 1. R. S. Dahiya and O. Akinyele, Oscillaion heorems of N h order funcional differenial equaions wih forcing erm, J. Mah. Anal. Appl. 19 Ž 1985., P. Das, A noe on a paper of Shreve, J. Mah. Phys. Sci. 27 Ž 1993., P. Das, Oscillaion of odd order delay differenial equaions, Proc. Indian Acad. Sci. 13 Ž 1993., P. Das, Necessary and sufficien condiion for oscillaion of neural equaions wih periodic coefficiens, Czechosloak Mah. J. 44 Ž 1994., R. D. Driver, Ordinary and Funcional Differenial Equaions, Springer-Verlag, New York, A. G. Karsaos, Recen resuls on oscillaion of soluion of forced and perurbed nonlinear differenial equaions of even order, in Sabiliy of Dynamical Sysems, NSF-CBMS Conf. Mississippi Sae Universiy, Mississippi, M. R. S. Kulenovic, G. Ladas, and A. Meimaridou, Necessary and sufficien condiion for oscillaion of neural differenial equaions, J. Ausral. Mah. Soc. Ser. B 28 Ž 1987., G. Ladas, Y. G. Sficas, and I. P. Savroulakis, Necessary and sufficien condiions for oscillaion of higher order delay differenial equaions, rans. Amer. Mah. Soc. 285 Ž 1984., 819.