NONLINEAR OSCILLATIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS OF EULER TYPE

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Numbe 1, Octobe 1996 NONLINEAR OSCILLATIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS OF EULER TYPE JITSURO SUGIE AND TADAYUKI HARA (Communicated by Hal L. Smith) Dedicated to Pofesso Junji Kato on the occasion of his 6th bithday Abstact. We conside the nonlinea equation t 2 x + g(x) =,wheeg(x) satisfies xg(x) > fox, but is not assumed to be sublinea o supelinea. We discuss whethe all nontivial solutions of the equation ae oscillatoy o nonoscillatoy. Ou esults can be applied even to the case g(x) 1 as x x 4, which is most difficult. 1. Intoduction and statement of esults We conside the oscillation poblem fo the second ode nonlinea diffeential equation (1.1) t 2 x + g(x) =, t >, whee g(x) is locally Lipschitz continuous on R and (1.2) xg(x) > if x. A nontivial solution of (1.1) is said to be oscillatoy if it has abitaily lage zeos. Othewise, the solution is said to be nonoscillatoy. In the theoy of oscillations, the numbe 1 4 vey often appeas as a citical value. The following esult is a good illustation of this fact: all nontivial solutions of Eule s equation (1.3) t 2 x + λx = ae oscillatoy if and only if λ> 1 4. Othe examples ae found in [3], [6] and the efeences cited theein. Because of Stum s sepaation theoem, the solutions of second ode linea diffeential equations ae eithe all oscillatoy o all nonoscillatoy, but cannot be both. Thus, we can classify second ode linea diffeential equations into the two types. Howeve, the oscillation poblem fo (1.1) is not so easy, because g(x) is nonlinea. Judging fom the oscillation esult fo Eule s equation (1.3), we see that all nontivial solutions of (1.1) have a tendency to be oscillatoy as g(x) gows lage Received by the editos Apil 6, Mathematics Subject Classification. Pimay 34C1, 34C15; Seconday 34A12, 7K5. Key wods and phases. Oscillation, nonlinea diffeential equations, Liénad system, global phase potait. The fist autho was suppoted in pat by Gant-in-Aid fo Scientific Reseach c 1996 Ameican Mathematical Society

2 3174 JITSURO SUGIE AND TADAYUKI HARA in some sense; and we must conside the case g(x) (1.4) x 1 as x 4 to solve completely the oscillation poblem fo (1.1). The pupose of this pape is to give ou answe to this delicate poblem. Ou main esults ae stated in the following: Theoem 1.1. Let λ>. Then all nontivial solutions of (1.1) ae oscillatoy if g(x) (1.5) x λ log x fo x >Rwith a sufficienlty lage R>. Theoem 1.2. Suppose that thee exists a λ with <λ< 1 4 such that g(x) x 1 ( ) 2 λ (1.6) 4 + log x fo x>ro x< Rwith a sufficiently lage R>. Then all nontivial solutions of (1.1) ae nonoscillatoy. Remak 1.1. We note that condition (1.4) is satisfied in eithe case g(x) = 1 4 x+ o g(x) = 1 ( λ 4 x+ log x fo x sufficiently lage. λx log x with λ> ) 2 x with <λ< Global existence of solutions In this section we will show that each solution of (1.1) exists in the futue. To see this, we conside the following system x = y (2.1) y = g(x) t 2 equivalent to (1.1). Poposition 2.1. Assume (1.2). Then evey solution of (2.1) exists in the futue. To pove Poposition 2.1, we supply a continuation esult fo the geneal system x = F 1 (t, x, y) (2.2) y = F 2 (t, x, y) in which F 1 :[, ) R m R n R m and F 2 :[, ) R m R n R n with both functions being continuous. A continuous function V :[, ) R m R n Ris called a Liapunov function fo (2.2) if V (t, x, y) is locally Lipschitz in (x, y). We define V (2.2) (t, x, y) = lim sup h + 1 { V (t + h, x + hf1 (t, x, y),y+hf 2 (t, x, y)) V (t, x, y) }. h

3 NONLINEAR OSCILLATIONS OF SECOND ORDER ODE 3175 We also say that a continuous scala function φ :[, ) R Ris of the class G if, fo any t, u R, the maximal solution u(t, t,u ) of the equation u = φ(t, u) exists in the futue. In [2], [4], [5], the continuation poblem is discussed by means of two Liapunov functions fo (2.2): Poposition 2.2. Let V : [, ) R m R n R be a Liapunov function with the popety that V (t, x, y) as y unifomly in x R m (2.3) fo each fixed t; (2.4) V (2.2) (t, x, y) φ(t, V (t, x, y)) fo some φ G. Suppose that fo each K > and T > thee exists a Liapunov function W : [,T] R m SK n R,Sn K ={y Rn : y K}which satisfies (2.5) W (t, x, y) as x unifomly in y SK n fo each fixed t; (2.6) Ẇ (2.2) (t, x, y) ψ(t, W (t, x, y)) fo some ψ G. Then evey solution of (2.2) exists in the futue. Using Poposition 2.2, we can pove Poposition 2.1. Poof of Poposition 2.1. Let G(x) = x g(ξ)dξ. Then we have G(x) fo x R by (1.2). Hence, condition (2.3) is satisfied when V (t, x, y) = 1 2 y2 + G(x) t 2. Define anothe Liapunov function W (t, x, y) = x. Clealy, this satisfies condition (2.5). We obtain and V (2.1) (t, x, y) = 2 t 3G(x) Ẇ (2.1) (t, x, y) y K on SK. 1 Since φ(t, u) = and ψ(t, u) = K belong to G, conditions (2.4) and (2.6) ae satisfied. Thus, by Poposition 2.2 all solutions of (2.1) ae continuable in the futue. Remak 2.1. Poposition 2.1 shows that each solution of (1.1) and its deivative exist in the futue. Remak 2.2. The conclusion of Poposition 2.1 is still tue if (1.2) is eplaced by the moe geneal condition G(x) d fo x R, whee d is an abitay constant.

4 3176 JITSURO SUGIE AND TADAYUKI HARA 3. Some lemmas The change of vaiable t = e s educes (1.1) to the equation ẍ ẋ + g(x) =, s R, whee = d ds. This equation is equivalent to the system ẋ = y + x (3.1) ẏ = g(x) which is of Liénad type. Note that evey solution of (3.1) exists in the futue by Poposition 2.1. In this section we give some esults on the asymptotic behavio of tajectoies of (3.1). We wite γ + (P ) fo the positive semitajectoy of (3.1) stating at a point P R 2. Lemma 3.1. Fo each point P =(p, p) with p>, the positive semitajectoy γ + (P ) cosses the negative y-axis. Poof. Suppose that thee exists a point P =(p, p) withp> such that γ + (P ) does not intesect the negative y-axis. Let (x(s),y(s)) be a solution of (3.1) definedonaninteval[s, )with(x(s ),y(s )) = P. Then the solution (x(s),y(s)) coesponds to γ + (P ). Taking into account the vecto field of (3.1), we see that <x(s) x(s ) fo s s and y(s) as s. Hence, it follows fom the fist equation of (3.1) that ẋ(s) as s, and theefoe, thee exists an s 1 >s such that ẋ(s) 1 fo s s 1. Integation of the above leads to x(s 1 ) <x(s) x(s 1 ) s 1 s as s. This is a contadiction and completes the poof. Similaly, tuning ou attention to the left half-plane, we have the following esult. Lemma 3.2. Fo each point P =( p, p) with p>, the positive semitajectoy γ + (P ) cosses the positive y-axis. We intoduce hee a new impotant concept which is useful in the theoy of oscillations. We say that system (3.1) has popety (X + ) in the ight half-plane (esp., left half-plane) if, fo evey point P in the egion { (x, y) : x and y> x }( esp., { (x, y) : x andy< x }), the positive semitajectoy γ + (P ) cosses the cuve y = x. In [1] the authos went into details about popety (X + ) and gave some necessay conditions and some sufficient conditions fo popety (X + ). We state below special cases of those esults. Let G( ) = g(ξ)dξ and G( ) = g(ξ)dξ.

5 NONLINEAR OSCILLATIONS OF SECOND ORDER ODE 3177 Lemma 3.3 ([1, Theoem 4.1]). Assume G( ) < (esp., G( ) < ). Then system (3.1) fails to have popety (X + ) in the ight half-plane (esp., left halfplane). Lemma 3.4 ([1, Theoem 5.4]). Assume G( ) = (esp., G( ) = ). Then system (3.1) fails to have popety (X + ) in the ight half-plane (esp., left halfplane) if x 2 ( ) (3.2) 2G(x) h 2G(x) fo x sufficiently lage, whee h() is a continunous function on [, ) such that fo sufficiently lage (3.3) h() is noninceasing and nonnegative; (3.4) ( ) 2 h(ξ) ξ 2 dξ 1 4 h(). Lemma 3.5 ([1, Theoem 5.2]). Assume G( ) = (esp., G( ) = ). Then system (3.1) has popety (X + ) in the ight half-plane (esp., left half-plane) if x 2 ( ) (3.5) 2G(x) h 2G(x) fo x sufficiently lage, whee h() is a continunous function on [, ) with h() is noninceasing, nonnegative (3.6) and is not geate than 2 fo sufficiently lage; (3.7) h() d = Poof of the theoems Poof of Theoem 1.1. As shown in Section 2, each solution of (1.1) exists in the futue. Suppose that system (3.1) which is equivalent to (1.1) has popety (X + ) in the ight and left half-plane. Then it follows fom Lemmas 3.1 and 3.2 that evey solution of (3.1) keeps on otating aound the oigin except the zeo solution. Hence, all nontivial solutions of (1.1) ae oscillatoy. Thus, to pove Theoem 1.1, it is enough to show that system (3.1) has popety (X + ) in the ight and left half-plane. We will demonstate this fact by means of Lemma 3.5. Note that (1.5) implies G(± ) =. Let <ν<λand h() = ν log fo sufficienlty lage. Then it is clea that conditions (3.6) and (3.7) ae satisfied. We next define continuous functions k(x),k(x)andl(x)onrby k(x) = λx log x, K(x)= k(ξ)dξ and L(x) = λx2 2log x fo x sufficiently lage, espectively. Then we have K(x) L(x) M fo some M> x

6 3178 JITSURO SUGIE AND TADAYUKI HARA and by (1.5) G(x) 1 8 x2 + K(x) N fo some N>. Since xk(x) is inceasing fo x sufficiently lage, we get ( K 2u νu ) νu2 log u 2log u K(u) νu2 2log u L(u) M νu2 2log u (λ ν)u2 M 2log u which tends to as u. Hence, fo u sufficiently lage 1 2 u2 1 ( 2 u2 + K 2u νu ) νu2 log u 2log u N + ν2 u 2 8(log u ) 2 = 1 ( 2u νu ) 2 + K( 2u νu ) N 8 log u log u ( G 2u νu ), log u namely, { 1 G(2u h(u)) if u > 2 u2 G(2u + h( u)) if u <. Letting { 2G(x) if x> u = 2G(x) if x<, we have x 2 ( ) 2G(x) h 2G(x) fo x sufficiently lage, that is, condition (3.5) is also satisfied. Thus, by Lemma 3.5 system (3.1) has popety (X + ) in the ight and left half-plane. The poof is complete. To pove Theoem 1.2, we need Lemmas 4.1 and 4.2 below. Lemma 4.1. Evey solution of (3.1) is unbounded except the zeo solution. Poof. By way of contadiction, we suppose that thee exists a bounded solution (x(s),y(s)) of (3.1) initiating at s = s >, that is, (4.1) x(s) + y(s) B fo s s with a B>. Then the solution (x(s),y(s)) cicles clockwise aound the oigin. Let s i >s and p i > with (x(s i ),y(s i )) = (,p i ) fo i =1, 2. Define a Liapunov function V (x, y) = 1 2 y2 +G(x)

7 NONLINEAR OSCILLATIONS OF SECOND ORDER ODE 3179 so that V (3.1) (x, y) =xg(x) > if x. If s 1 <s 2,thenp 1 <p 2.Infact,wehave 1 2 p2 1 =V(x(s 1),y(s 1 )) <V(x(s 2 ),y(s 2 )) = 1 2 p2 2. Hence, thee exists a simple closed cuve C suounding the oigin such that (4.2) dist { (x(s),y(s)),c } as s. Let ε > be sufficiently small and define l =max { g(x):ε x B }, m =min { xg(x) :ε x B }. By (4.2) the solution (x(s),y(s)) does not stay in { (x, y) : x <ε }. Hence, thee exist sequences {τ n } and {σ n } with s <τ n <σ n <τ n+1 and τ n as n such that and We have x(τ n )=x(σ n )=ε, y(τ n ) >ε, y(σ n ) < ε x(s) >ε fo τ n <s<σ n. σn 2ε >y(σ n ) y(τ n )= g(x(s))ds l(σ n τ n ), τ n and theefoe, fo s>σ n s n V(x(s),y(s)) V (x(s ),y(s )) = x(σ)g(x(σ))dσ s k=1 n m (σ k τ k ) 2mε n l k=1 σk τ k x(s)g(x(s))ds which tends to as n. This is a contadiction to (4.1) and completes the poof. Let V (x, y) = 1 2 y2 +G(x) and conside the cuve V (x, y) =V(x,y ), whee x >. Then thee exist two points of intesection of the cuve with the line y = x. In fact, the equation V (x, x) =V(x,y ) has exactly two oots because V (x, x) is inceasing fo x> and deceasing fo x<, and V (, ) =. Let ( a, a) and(b, b) be the intesection points, whee a>andb>. Define S = { (x, y) : a x c and V (x, y) V (x,y ) }

8 318 JITSURO SUGIE AND TADAYUKI HARA in which c =max{b, x }. Then it is clea that S is a bounded set. Lemma 4.1 shows that evey solution of (3.1) stating in S \{}does not emain in S. Take note of the vecto field of (3.1) and the fact that V (3.1) (x, y) =xg(x) > if x. Then we also see that evey solution of (3.1) stating in S c, the complement of S in R 2,staysinS c fo all futue time. Thus, we have Lemma 4.2. Evey solution of (3.1) stating in S\{} entes S c which is a positive invaiant set with espect to (3.1). Poof of Theoem 1.2. We pove only the case that condition (1.6) is satisfied fo x>r, because the othe case is caied out in the same way. Fist, we will show that system (3.1) fails to have popety (X + )intheight half-plane. If G( ) <, then this fact is clea because of Lemma 3.3. Suppose that G( ) =. To use Lemma 3.4, we will check that conditions (3.2) (3.4) hold. Let h() = 4(log ) 2 fo sufficiently lage. Then h() ( h(ξ) ξ 2 dξ is noninceasing and nonnegative; and we have ) 2 = 16(log ) 2 = 1 4 h(), that is, conditions (3.3) and (3.4) ae satisfied. Define continuous functions k(x) and L(x) onrby ( ) 2 ( ) 2 λ νx k(x) = x and L(x) = log x log x fo x>rwith λ 2 < 2ν 2 < 1 16.Then K(x) x k(ξ)dξ is inceasing fo x>r, and thee exist constants M>andN> such that and L(x)+M K(x) G(x) 1 8 x2 + K(x)+N fo x>. Hence, we obtain 1 2 uh(u)+ 1 8 (h(u))2 + K(2u h(u)) u2 8(log u) 2 + u 2 128(log u) 4 + K(2u) u2 8(log u) 2 + u 2 128(log u) 4 + L(2u)+M (1 32ν2 )u 2 u 2 8(log u) (log u) 4 + M as u,

9 NONLINEAR OSCILLATIONS OF SECOND ORDER ODE 3181 and theefoe, fo u sufficiently lage 1 2 u2 1 2 u2 1 2 uh(u)+ 1 8 (h(u))2 + K(2u h(u)) + N = 1 8 (2u h(u))2 + K(2u h(u)) + N G(2u h(u)). Let u = 2G(x). Then we have x 2 ( ) 2G(x) h 2G(x) fo x sufficiently lage. Thus, condition (3.2) is also satisfied, and so system (3.1) fails to have popety (X + ) in the ight half-plane by Lemma 3.3. Hence, thee exists a point P (x,y )withx andy > x such that γ + (P ) uns to infinity without intesecting the cuve y = x. We hee suppose that (1.1) has an oscillatoy solution. Let γ + (Q) bethepositive semitajectoy which coesponds to the oscillatoy solution of (1.1). By vitue of Lemma 4.2, we see that γ + (Q) eventually goes aound the set S infinitely many times. Hence, it cosses the half-line { } (x, y) : x = x and y > y at a point P 1 (x,y 1 )withy 1 >y. Fom the uniqueness of solutions fo the initial value poblem, it tuns out that (i) γ + (Q) coincideswithγ + (P 1 ) except fo the ac QP 1. (ii) γ + (P 1 ) lies above γ + (P ). Hence, γ + (Q) uns to infinity without cossing the cuve y = x. This contadicts the fact that γ + (Q) cicles the set S. The poof is now complete. Refeences [1] T. Haa and J. Sugie, When all tajectoies in the Liénad plane coss the vetical isocline?, Nonlin. Diff. Eq. Appl. 2 (1995), [2] T. Haa, T. Yoneyama and J. Sugie, Continuation esults fo diffeential equations by two Liapunov functions, Ann. Mat. Pua Appl. 133 (1983), MR 85k:349 [3] E. Hille, Non-oscillation theoems, Tan. Ame. Math. Soc. 64 (1948), MR 1:376 [4] J. Sugie, Continuation esults fo diffeential equations without uniqueness by two Liapunov functions, Poc. Japan Acad. Math. Sci. Se. A6(1984), MR 86a:346 [5], Global existence and boundedness of solutions of diffeential equations, doctoal dissetation, Tôhoku Univesity, 199. [6] C. A. Swanson, Compaison and oscillation theoy of linea diffeential equations, Academic Pess, New Yok and London, MR 57:3515 Depatment of Mathematics, Faculty of Science, Shinshu Univesity, Matsumoto 39, Japan Cuent addess: Depatment of Mathematics and Compute Science, Shimane Univesity, Matsue 69, Japan addess: jsugie@botan.shimane-u.ac.jp Depatment of Mathematical Sciences, Univesity of Osaka Pefectue, Sakai 593, Japan addess: haa@ms.osakafu-u.ac.jp

ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS

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