A maximum principle for second order nonlinear differential inequalities and its applications

Appl. Math. Lett. Vol. 8, No. 4, pp. 91-96, 1995

Pergamon 0893-96S9(95)00055-0

Copynght©1995 Elsevier Scmnce Ltd Printed inGreatBrRain. All rights reserved 0893-9659/95 $9.50 + 0.00

A M a x i m u m Principle for Second Order N o n l i n e a r Differential Inequalities and Its A p p l i c a t i o n s Fu-HSIANG WONG Department of Mathematics and Science National Taipei Teacher's College Hoping E. Rd., Taipei, Taiwan, Republic of China CHEH-CHIH YEH Department of Mathematics National Central University Chung-Li, Taiwan, Republic of China SHIUEH-LING Vu St. John's 8z St. Mary's Institute of Technology Tamsui, Taipei, Taiwan, Republic of China

(Received and accepted October 1994) Abstract--Let

y(t) be a nontrivial solution of the second order differential inequality y(t){(r(t)y'(t))' + f(t, y(t))} < 0.

(El)

We show that the zeros of y(t) are simple; y(t) and yl(t) have at most finite number of zeros on any compact interval [a, b] under suitable conditions on r and f. Using the main result, we establish some nonlinear maximum principles and a nonlinear Levin's comparison theorem, which extend some results of Protter, Weinberger, and Levin. K e y w o r d s - - Z e r o , Second order, Nonlinear differential inequality, Maximum principle, Levin comparison theorem. 1. INTRODUCTION C o n s i d e r t h e function

y(t)

=

{ t9 sin3 -1t ' 0,

t ¢ 0

(1)

t = 0.

I t is clear t h a t y(tn) = y ' ( t , ) = O, where to = 0 a n d t,~ = 1/(nTr) for n = 1, 2 , . . . . T h i s shows t h a t t h e r e exists a n o n t r i v i a l function y(t) such t h a t y a n d y~ have infinitely m a n y zeros on a finite interval. Seeing such facts, we c a n n o t help b u t ask "under w h a t c o n d i t i o n s does e v e r y n o n t r i v i a l s o l u t i o n y(t) of a s e c o n d - o r d e r differential i n e q u a l i t y satisfy y2(t) + yr2(t) ¢ 0 for e v e r y given t; moreover, y a n d y~ always have finite n u m b e r of zeros on a given finite interval." T h e p u r p o s e of t h i s article is to afford a s u i t a b l e criterion to confirm t h e a b o v e - m e n t i o n e d p r o b l e m (which e x t e n d s t h e results in [1]). Using t h e result, we generalize some m a x i m u m p r i n c i p l e s in [2] a n d t h e Levin c o m p a r i s o n t h e o r e m [3] t o n o n l i n e a r cases. For t h e o t h e r r e l a t e d result, we refer t o Sheng a n d A g a r w a l [4]. Typeset by .Ah/eS-TF~ ~LB-4-G

91

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Fu-HSlANG W O N G et al.

2. M A I N R E S U L T In order to discuss our main result, we need the following integral inequality (see, for example, [5] and the book of Agarwal and Lakshmikanthm [6, pp. 12, 19, 47]) which is a generalization of Osgood, Tonelli, Montel and LaSalle.

LEMMA A.

Let

Fc : [0, c] --* [0, c~) be positive and increasing on (0, c) for some c > O,

(C1)

h • LX([a, b]; [0, oe)),

(C2)

y • C([a,b]; [0,el).

(C3)

Then, the inequality y(t) <_

h(s)Fc(y(s)) ds

(

resp: y(t) <

fb

h(s)Fc(y(s)) ds

)

,

t • [a, bl

implies y(t) __ds

<

h(s) ds

F~(s) -<

resp :

h(s) ds

,

t • [a,b].

Jo

In addition, if fo ~ F ds(s)

-

(C4)

oo,

then y(t) = 0 on [a, b]. Given c > 0, we say that a function f : [a, b] × I-c, c] --* R satisfies an "Osgood condition" if there exists a function h satisfying (C2) and a function Fc satisfying (C1), (C4) and

[f(t,y)] <_ h(t)F~(lyl)

on [a,b] × [-c,c].

(2)

REMARK 1. The function Fc can be chosen to be Fc(y) = y (Lipschitz), y l o g ( l / y ) or y(log(1/y)) × (log log(i/y)). Now, we can state and prove the following main result. THEOREM 1. ( M a x i m u m principle) A s s u m e that 1

(c5)

- e Ll([a,b];(O, oo)), r

f • L~o~([a, b] x ~; N), (that is, f is 1ocally Lebesgue-integrable on [a, b] x R)

(c8)

and f satisfies an Osgood condition. I f y ( t ) # 0 in the interval (a,b) with y(a) = 0 or y(b) = 0 and it satisfies

y(t){(r(t)y'(t))' + f(t, y(t))} < 0

a.e. on [a, b],

then y'(a) ~ 0 or y'(b) ¢ O. PROOF. We prove only the case y(a) = O. Assume, on the contrary, t h a t y~(a) = O. CASE

(a).

Suppose that y(t) > 0 on (a,b). It follows from (El) t h a t

r(t)y'(t) < - fa t f ( s , y(s)) ds

a.e. on [a, b],

(El)

Maximum Principle

93

which implies

y(t) < -

f(u,y(u) duds

on [a,b].

(3)

Since y(a) = 0, there exists T1 > a such that 0 < y(t) < c for all t • [a, T1]. It follows from (2) and (3) that

y(t) <_ where m := f : ; ~ . y(t) > 0 on (a, b).

m l I ( s , y ( s ) ) l d s <_

mh(s)Fc(y(s))ds

on [a, T1],

Hence, by Lemma A, y(t) - 0 on [a, T1]. This contradicts the hypothesis

CASE (b). Suppose that y(t) < 0 on (a, b). It follows from (El) that

r(t)y'(t) > -

f ( s , y(s)) ds

which implies

y(t) > -

f ( u , y ( u ) ) du

a.e. on [a, b],

}

ds

on [a, b].

(4)

Since y(a) = 0, there exists T2 > a such that - c <_ y(t) <_ 0 for all t • [0, T2]. It follows from (2) and (4) that

- y ( t ) <_

m]f(s, y(s)) I ds <

mh(s)Fc(]y(s)]) ds --

m h ( s ) F c ( - y ( s ) ) ds

on [a, T2], where m := f : ~--~. Hence, by Lemma A, y(t) =- 0 on [a, T2]. This contradicts the hypothesis y(t) < 0 on (a, b). Thus, by Cases (a) and (b), we complete the proof. COROLLARY 2. Let (C5) and (C6) hold. Then, every nontrivial solution of (El) has at most a finite number of zeros on [a, b]. PROOF. Assume, on infinitely many zeros theorem, we see that contradicts Theorem

the contrary, that there exists a nontrivial solution y(t) of (El) which has on [a, b]. Define Z ( y ) := {t • [a, b] ] y(t) = 0}. By the Solzano-Weierstrass Z ( y ) has an accumulation point, say to. Hence, y(to) = y'(to) = O, which 1. This contradiction completes our proof.

COROLLARY 3. Let (Cs) and

p • Ll([a,b];R)

and f ( t , y ) ~ 0

for (t,y) • [a,b] x { R - {0}}

(c7)

hold. If y(t) is a nontrivial solution of y"(t) + p(t)y'(t) + f(t, y(t)) = O,

(E2)

then y'(t) has at most a finite number of zeros on [a, b]. PROOF. Assume, on the contrary, that there exists a nontrivial solution y(t) of (E2) such that y'(t) has infinitely many zeros on [a, b]. Define

z(y') := {t e [a, b] I y'(t) = 0}. By the Bolzano-Weierstrass theorem, Z ( y ~) has an accumulation point, say to. Hence, y'(to) = y"(to) = O. It follows from (E2) and (C7) that f(to, y(to)) = O. Hence, y(to) = O. That is, (E2) has a solution y(t) satisfying y(to) = y~(to) = O, which contradicts Theorem 1. This contradiction completes our proof.

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Fu-HSIANG W O N G et al.

3. S O M E

APPLICATIONS

Using our maximum principle (Theorem 1), we can generalize some theorems in Chapter 1 of Protter and Weinberger [2], and the Levin comparison theorem [3,7]. THEOREM 4. Let (C5) and (C6) hold. If

for each fixed t • [a, b], f(t, y) is increasing with respect to y,

(Cs)

then, every nonconstant solution y(t ) of (El) has no nonnegative local minimum or nonpositive locaJ maximum in (a, b). PROOF. Without loss of generality, we may assume that y(t) has a nonnegative local minimum in (a, b), say to. It follows from y'(to) = 0 and Theorem 1 that y(to) > 0. Hence, there exists tl • (a, b) such that to < tl and y(t) > y(to) > 0 in (to,tl]. Let z(t) := y(t) - y(to) on [to,tl]. Then, z(t) satisfies y(t) > z(t) > 0 in (to, tl], z(to) = z'(to) = 0 and

z(t){(r(t)z'(t))' + f(t, z(t))} = z(t){(r(t)y'(t))' + f(t, z(t))} <_ z(t){(r(t)y'(t))' + f(t,y(t))} < 0 a.e. on [to, tl], which contradicts Theorem 1. Hence, we complete the proof. Similarly, we have the following. THEOREM 5. Let (C5) and (C6) hold. If for each fixed t 6 [a, b], f(t, y)is decreasing with respect to y,

(C9)

then, every nonconstant solution y(t) of (El) has no nonnegative local maximum or nonpositive local minimum in (a, b). REMARK 2. Theorems 4 and 5 generalize some theorems in Chapter 1 of Protter and Weinberger [2]. THEOREM 6. (Nonlinear Levin's comparison theorem ) Let rl, r2 • Ll([a, b]; (0,co)) satisfy rl(t) _< r2(t) on [a,b],

(Clo)

Pl,P2 • LX([a, b]; ~),

(Cn)

fi • C(]~) n C l ( ( - c ~ , 0 ) U (0, o0)) satisfyyfi(y) > 0 and f[(y) > 0

for all y ~ 0 and i = 1, 2, and f2 satisfy an Osgood condition, f~(y) be decreasing in (0, oo) and increasing in ( - c e , 0),

(C13)

y ( f l ( y ) - f2(y)) > 0 and f (y) > f (y) in R - {0}.

(C14)

(C12)

Suppose that Yl and Y2 are solutions of yl(t){(rl(t)y~(t))' + pl(t)fl(yl(t))} <_0

(E3)

and !

!

(r2(t)Y2(t)) + p2(t)f2(Y2(t))} --- 0 on [a, b], respectively, and assume one of the following conditions holds: (i) yl(t) > 0 on [a,b] and y2(a) >_ yl(a) > O, (ii) yl(t) < 0 on [a,b] and y2(a) _< yl(a) < O.

(E4)

M a x i m u m Principle

g Yl and Y2 satisfy

95

the inequality

f1(v1(~))

+ ffpl(s) ds > - r2(a)y~(a)f2(y2(a))+ ff p2(s)ds

on [a, hi,

(S)

then y2(t) # O, yl(t)y'l(t) < O, ly2(t)l >_ M(t)l and

rl(t)yi(t ) r2(t)y~(t) fl(Yl(t)) > f2(y2(t))

on

[a, b].

(6)

PROOF. Let zl(t) := -rl(t)y~(t) l~(v,(O) on [a, b]. It follows from (E3) that z~(t) --

z2 (t) > pl(t) (rl(t)y~(t))' -t- f~ (_yl_(_(t))~2:÷~ _> pl(t) -t- -f~ (yl(t)) fl(Yl(t)) rl(t ) ~lk~: rl(t) --

(7)

a.e. on [a, b]. This and (5) imply

~at Zl (t) > Zl (a) +

pl(s)

ds > 0

on [a, b].

(8)

Thus, yl(t)y~(t) < 0 on [a,b]. Since y2(a) # 0, there exists t I • (a,b] such that z2(t) :-[-r2(t)y~(t)]/[f2(y2(t))] is continuous and differentiable on [a, tl] and y2(t)y2(a) > 0 in [a, tl]. It follows from (E4) that

4(t) =

' t f~(v2(t))z~(t) = p2(t) + f6(w())

(r2(tM(t))'

> p~(t)

(9)

on [a, tl].

(10)

~,.,

a.e. on [a, tl]. By (5), (8) and (9), we get

z2(t) > z2(a) + fj~ p2(s) ds > -z~(a) -

pl(s) ds >_ - z l ( t )

Now, we claim that zl(t) > z2(t) on [a, tl]. Assume, on the contrary, that there exists t2 E (a, tl) such that zl(t2) = z2(t2) and zl(t) > z2(t) on [a, t2). By (10), we have 0 < Iz2(t)l < zl(t) on [a, t2). Let ds F~(y):=fory#O andi=l,2. ,(a) fi(s)

~v

It is clear that

Fi(y) is decreasing in (0, o0) and increasing in (-oo, 0), where i = 1, 2. Hence, F2(yl(t)) = >_

fa t YIl( S-------~) ds f2(yl(s)) ff

-

y~(s)

ds

f~(~(~)--------)

(by (C14) and Zl > O)

= FI(Yl (t))

f t zl(s) ds

£ ~(~)

> f t z2(s) - A r2(s)

ds

(by (C10))

v2(a) ds = F2(Y2(t)) + J~l(~) f2(s) >_ F2(y2(t)) on [a, t21.

f

Fu-HSIANG WONQ et al.

96

This and (C13), (C14) imply 0 < f~(y2(t)) < f~(yl(t)) < f~(yl(t)) on [a, t2]. Thus, it follows from (5), (7) and (9) t h a t

z2(t2) = z2(a) +

< zl(a) +

fa t2

/?

p2(s) ds +

pl(s) ds +

which contradicts to zl(t2) = z2(t2).

Iz2(t)l <

Zl(t)

on

~a t2 f~(y2(s)) z](s) ds r2(s)

/?

f~(yl(S))z2(s) ds < zl(t2), rl(8 )

Hence, zl(t) > z2(t) on [a, tl].

This and (10) i m p l y

[a, tl].

Next, we claim t h a t the above mentioned tl is b. Assume, on the contrary, t h a t there exists I t ~ 0. Thus, t3 E (a,b) such t h a t y2(t) ¢ 0 in [a, t3) and y2(t3) = 0. Then, by T h e o r e m 1, Y2(3) c¢ -- lim Iz2(t)l < lim zl(t) -- Zl(~3) < c~, which is a contradiction. So, tl = b and we complete the proof of the theorem. REMARK 3. T h e o r e m 6 is a generalization of the Levin comparison t h e o r e m in [3,7,8]. T h e r e are m a n y pairs of functions satisfying conditions (C12), (C13) and (Cla). For example, (fl (y), f2 (y)) -(2y, y); ( 2 s g n ( y ) l n ( l y I + 1), s g n ( y ) l n ( M + 1)); or ( s g n ( y ) M n + y, sgn(y) ln(ly I + 1)) for n = 0,1,2,.... REMARK 4. Using our nonlinear Levin's comparison theorem, we can improve the oscillation criterion in [9]. For the other applications, we refer to the books of Swanson [7] and Agarwal and L a k s h m i k a n t h a m [6].

REFERENCES 1. F.H. Wong, Zeros of solutions of a second order nonlinear differential inequality, Proc. Amer. Math. Soc. 111,497-500 (1991). 2. M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, (1967). 3. A. Yu. Levin, A comparison principle for second order differential equations, Soy. Math. Dokl. 1, 1313-1316 (1960). 4. Q. Sheng and R.P. Agarwal, Generalized maximum principle for higher order differential equations, J. Math. Anal. and Appl. 174, 476-479 (1993). 5. F.H. Wong and C.C. Yeh, LaSalle's inequality and uniqueness theorems, Applicable Anal. 46, 45-58 (1992). 6. R.P. Agarwal and V. Lakshmikantham, Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, World Scientific, Singapore, (1993). 7. C.A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, (1968). 8. C.C. Yeh, Levin's comparison theorem for second order nonlinear differential equations and inequalities, Math. Japonica 36, 703-710 (1991). 9. F.H. Wong and C.C. Yeh, An oscillation criterion for Sturm-Liouville equations with Besicovitch almost-periodic coefficients, Hiroshima Math. J. 21, 521-528 (1991).