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AFRIKA MATEMATIKA

Journal of the African Mathematical Union

Journal de l 'Union Mathematique Africaine

Serie 3, vol. 7 (1997)

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Afrika matcmatika serie 3, vol. 7 (1997)

OSCILLATING PERIODIC SOLUTIONS OF SOME

NON-LINEAR DEFERENTIAL EQUATIONS

BY

S .A. lYASE

DEPARTMENT OF MATHS, STATISTICS AND

COMPUTER SCIENCE UNIVERSITY OF ABUJA

PMB 117 ABUJA, FCT, NIGERIA

WEST AFRICA

1 INTRODUCTION

We consider here ordinary differential equations of the form

u (m) = .Xu + p(t)g(u) - e /(t) {1.1)

where .X and e are real parameters, P,/ EC ([o, + oo], R) g EC (R,R) . Our

purpose is tc• obtain existence of w - periodic solutions of (1.1). We shall first consider a periodic existence boundary value problem (PBVP) associated with

(1.1) and obtain existence of solutions by employing the alternative method for

non-linear problems at resonance [1]. We shall then relate these solut ions t o thew

- periodic solutions of (1.1).

Our results generalize the corresponding results of [4] in several ways. For example we obtain our results for any .X E R and for arbitrary m. In the case

>. = - 1, / U) = coswt and p(t) = - p., in.equation {1.1) Ezeilo [2] has obtained

existence results using the Leray - Schauder technique.

2 PRELIMINARIES

Let X be a real Hilbert Space with inner product (•,. ) and norm 11 ·11· Let L : D(L) C X --. X be a linear operator wit h finite d imentional kernel Xo and

.

•

OSCD...LATING PERIODIC SOLUTIONS •

u (m) = ~u + p(t)g(u)- e/(t )

u(i)(o) = u (i)(w), (i = 0, 1,2 ... m- 1)

2.5

in which P, f are continuous functions on fo,w]. Our procedure is to shift all

considerations away from (2.5) to the mo~ified PBVP

u(m) = ~<Pa(u) + p(t)g(¢a(u)) - e/(t)

u(i)(o) = u (i)(w), (i = 0,1,2 ... m- 1)

2.6

and to show that the equation (2.6) has an w-periodic solution fot all arbitrary A I

and e. The existence of periodic solution of (2.5) will then follow in view of (2.4) after it is verified that every w- per\odic solution u(t) of (~.6) satisfies lu(t)l $ a:.

3 EXISTENCE OF PERIODIC SOLUTIONS

OF 2 .6

We shall prove the following

Lemma 3.1

Let m ;::: 1. Then the PBVP

u(m) = 0

u(o) = u(i)(w), (i = 0.1,2, ... m-l)

has non- trivial periodic solutions.

Proof t!(m ) = 0

implies that m-1

u (t) = L bJt''"'-J + bm J=l

for some b, i <;: R.

Now m- •

(i)tt) " ·tl ")b tm-j-i u ~ = ~ l. m- J i

J=l

IS

OSCll..LATING PERIODIC: SOLUTIONS

as follows 11W P:r =- :r(s)ds w 0

d H · f d I ·r (m) (i) ( ) (i)'( ) · 0 1 2 1 an :r1 = Y1 1 an on y 1 y1 = :r1 , y1 o = y 1 w , t = , , , .. . m -

Clearly dom(Nor) = X since <Po: is bounded. Thus the PBVP (2.6) may be

translated into the operator equation

Lu = N"'u (3.2)

For u E X, seL u = u0 + u1 where u0 = Pu and u1 = (I - P)u. Thus (2.6) is

equivalent to the coupled system of equations

u1 = H(I- P)N(u0 + u1) (3.3)

and

PNor(tt.o + u1) = 0

Since <Po: is bounded there exists a constant· J > 0 such that IINo:ull :::; J for all

uE X. Given U 0 E X 0 we can find a constant c :> o independent of u 0 E X 0 such that

max Ju1(t)l :::; c for any solution u1 of (3.3) . te[o,w)

Now choose R > o > 0 large enough so that

lu1(t)l :S R - o (3.4)

Since u0 =spun{!} we can see that if u 0 E X 0 and lluoll = R then either u0 = R

oru0 = -R. If u 0 = R then (2.4) and hypothesis (i) implies

(No:(u0 + u1) , u0 ) = R J:'[A¢o:(u) + p(t)g(¢o:(u)) - c/(t)]dt

= R J:'[Ao + p(t)g(o) '-- c/(t)]dt

= Rw[Ao +Mpg(o) - cMtJ ?: 0

Similarly if u ., = -R we again obtain that (No:(u0 + u 1).u0 > 0. Hence the hypothesis of theorem is satisfied and the proof is complete. Lemma 3.3

Assume that

11"' M1 =- f(t)dt = 0 and g(O) = 0 w 0 .

Then for any ()' > 0 and for any of the following choices of A, p and g given by

17

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~

OSCD..LATING PERIODIC SOLI.JilONS

where Ya = max lg(u) l - a < u<a

then the equation (1.1) has a solution which is both oscillatory and w-periodic.

Proof

From Lemma 3.3, the modified PBVP (2.6) has a solution u on [o,w] and

u(17) = 0 for some '7 E [o,w]. Let u(t) be any w-periodic solution of (2.6) then

lu(m)(t)l :5 1-XIo + IPiga + lellfl 4.2

and thus, since

lu(m-k)(t) l = 0 at some .t(k = 1, 2, ... m - 1)

we will derive successively from (4.2) that

lu(m- kl(t)l :5 wk[i -XIo + g,.MIPI + le i'MI!il ·

In particular

lu'(t)l :5 wm-1[1-XIo + g,.MiPI + leiMi!ll

Hence,

u(t) = u(17) + J: u'(s)ds

and

lu(t)l :5 wm[I-XIo + 9aMIPi + leiMI/1] :5 o

Since the existence of a positive constant o such that (4.1) is satisfied is given, then the solution obtained above for the PBVP (2.6) is a solution of the PBVP

(2.5) . This solut ion may then be extended periodically on [o, oo] having at least

one zero on (o, w], one zero on (w, 2Lv] and so on. Thus there exist s arbitrarily large

zeros for u . Also from the remark, u changes sign on [o, w]. [w, 2w] and so on.

T his completes the proof.

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BREVES INFORMATIONS SUR LE JOURNAL

Afrika Matematika est le Journal de I' Union /'vfathematique Africaine, avec des referees intemationaux. Sa publication demarra en 1978. 11 a depuis offert un debouche a Ia recherche mathematique faite 1:n Afrique. Afrika Matematika publie des articles de recherche, sans restriction de longur:.Jr, dans tous les domaines des Mathematiques et de leurs applications. II publie egalement de:; articles de synthese. Les conferences et seminaires organises sous les auspices de !'Union Mathematique Africaine soot publies dans des editions speciales du journal.

I Nous invitons les mathematiciens de tous pays (Afrique, USA, Europe, Asie, etc ... ) a

publier des articles dans Afrika Matematika. Tousles manuscrits doivent etre Soumis pour publication a tout m•:mbre du Comite de Redaction. lis doivent etre tapes seton les normes intemationales (consulter un modele). Le manuscrit devra etre dispose comme suit :

( 1) Titre de I' article ; (2) Nom de I' Auteur : (3) Adresse de l'Auteur ; (4) Resume en anglais; (5) Introduction suivie du contenu de l'article; (6) Remerciements (le cas echeant), (7) Bibliographie.

, Tousles papicrs acceptes pour publication dans Afrika Matematika doivent etre traites, si possible, en TEX IIU equivalent.

Les auteurs n:cevront des copies de Ia version finale de leurs articles acceptes. Les langues otlicielles de Afrika Matematika sont I'anglais et le franyais. Le montant des frais d'abonnement pour 2 volumes, y compris les frais d'envoi par

avion est:

Afrique : SO US$. Autre continent : 75 US $ .

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REMER ClEMENTS

L'Union Mathematique Africaine exprime sa gratitude au International Centre forTheoretical Physics (ICTP), Trieste, Italie, et a I'UNESCO pour leur soutien financier tors de Ia realisation de ce 'IO)ume.