- Email: [email protected]
Generalized Averages for Solutions of Nonlinear Operators
Journal of Mathematical Analysis and Applications 250, 196᎐203 Ž2000. doi:10.1006rjmaa.2000.7077, available online at http:rrwww.idealibrary.com on
Generalized Averages for Solutions of Nonlinear Operators Xiaojing Yang Department of Mathematics, Tsinghua Uni¨ ersity, Beijing, People’s Republic of China Submitted by L. Debnath Received May 4, 2000
1. INTRODUCTION Recently, Korman and Li w1x proved the following theorem. THEOREM A. Consider the two point Dirichlet problem u⬙ q f Ž u . s 0,
0 - x - T,
u Ž 0 . s u Ž T . s 0.
Ž 1.1.
Assume f g C 1 Ž Rq ., and let uŽ x . be a positi¨ e solution of Ž1.1., uŽ x . ) 0, x g Ž0, T .; we ha¨ e f Ž u Ž x . . ) 0.
Ž 1.2.
Define LŽ u. s
1r2
u
žH
f Ž s . ds
0
/
,
g Ž u . s L⬘ Ž u . .
Ž 1.3.
Then T
H0
g Ž u Ž x . . dx s r'2 .
Ž 1.4.
Theorem A is a generalization of the following boundary value problem: u⬙ q u 3 s 0,
0-x-T
u Ž 0 . s u Ž T . s 0.
Ž 1.5. Ž 1.6.
The problem Ž1.5. ᎐ Ž1.6. has a unique positive solution uŽ x . ) 0, x g Ž0, T . w1, 2x. Then for any T ) 0 this solution satisfies the following average 196 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
SOLUTIONS OF NONLINEAR OPERATORS
197
condition: T
H0
u Ž x . dx s
'2
.
Ž 1.7.
This fact is observed in w2x and is proved in w1x by setting y s H0x uŽ s . ds, hŽ y . s u 2 ; then Ž1.1. is transformed to h⬙ q 2 h s 0,
0 - y - R,
h Ž 0. s h Ž R . s 0
Ž 1.8.
with Rs
T
H0
u Ž s . ds.
Ž 1.9.
A solution of Ž1.8. is hŽ y . s C sin '2 y; C is a constant, hŽ R . s 0, and the positivity of h implies that '2 R s . Then Ž1.9. implies Ž1.7.. By using the same ideal, let hŽ y . s LŽ uŽ y .., y s H0x g Ž uŽ s .. ds; Ž1.1. is transformed to L⬘ Ž u . Ž h⬙ q 2 h . s 0.
Ž 1.10.
Then L⬘Ž u. ) 0, x g Ž0, T ., Ž1.10. becomes Ž1.8., and the results of Theorem A follow. The method of Theorem A is clear now; by using variables transformation Ž1.3., hŽ y . s LŽ uŽ y .., y s H0x g Ž uŽ s .. ds. Ž1.1. is transformed into a simple linear equation Ž1.8.; then the result follows. Inspired by this ideal, we consider in this paper the more general problem
Ž Ž u⬘ . . ⬘ q f Ž u . s 0,
u Ž 0 . s u Ž T . s 0,
Ž 1.11.
where g C 1 Ž R, R ., is strictly increasing, Ž R . s R, and Ž0. s 0. An interesting example is the well-known p-Laplacian: Ž x . s p Ž x . s < x < py 2 x, where p ) 1 is a constant. By using suitable variable transformation, Ž1.11. can be transformed into a kind of simpler form; then by applying known eigenvalue problem results, we obtain a generalized average result for positive solutions of Ž1.11..
2. MAIN RESULTS LEMMA 2.1. Let g C 1 Ž R, R .; is strictly increasing, Ž R . s R, and Ž0. s 0. If the boundary ¨ alue problem ŽBVP.
Ž Ž u⬘ . . q Ž u . s 0, u Ž 0. s u Ž T . s 0
Ž 2.1. Ž 2.2.
198
XIAOJING YANG
has nonzero solutions, then ) 0, and is called an eigen¨ alue of Ž2.1. ᎐ Ž2.2. with eigenfunction uŽ x .. Let 1 s inf N Ž2.1. ᎐ Ž2.2. has nonzero solutions4 : then 1 ) 0 and
1 s inf
¡H
T
~
u/0
0
¦
Ž u⬘ Ž s . . u⬘ Ž s . ds
¢H
T
0
Ž u Ž s . . u Ž s . ds
¥.
Ž 2.3.
§
Proof. Multiplying Ž2.1. by u, integrating by parts over w0, T x, and then using Ž2.2. we obtain T
s
Ž u⬘ Ž s . . u⬘ Ž ds . ds
H0
T
H0
.
Ž u Ž s . . u Ž s . ds
Now the definitions of and uŽ x . / 0 imply ) 0 and Ž2.3.. THEOREM 2.2. Suppose satisfies the assumptions of Lemma 2.1; Ž2.1. has at least one positi¨ e solution and all positi¨ e solutions of Ž2.1. corresponding to the same eigen¨ alue 1 ) 0. Let LŽ u. and g Ž u. be defined as L Ž u . s ⌽ry1 Ž F Ž u . . ,
g Ž u . s L⬘ Ž u . ,
Ž 2.4.
where ⌽ry1 is the right in¨ erse of ⌽, that is, the in¨ erse of ⌽ restricted to w0, ⬁., x
⌽Ž x. s
H0 1
H0
Ž s . ds ,
Ž s . ds
FŽ x. s
x
H0 f Ž s . ds,
f g C 1 Ž Rq . ,
f Ž u. ) 0
if u ) 0. Then for any positi¨ e solution of Ž1.11., if it exists, we ha¨ e T
H0
g Ž u Ž s . . ds s R
and
1 s
1 1
H0
Ž s . ds
.
Ž 2.5.
199
SOLUTIONS OF NONLINEAR OPERATORS
Proof. From the definition of L and g, we have
H0L u Ž s . ds Ž .
F Ž u. s ⌽ Ž LŽ u. . s
1
H0
.
Ž 2.6.
Ž s . ds
Define a time variable transformation
s
t
t g Ž 0, T . ;
H0 g Ž u Ž s . . ds,
Ž 2.7.
then u⬘ s
du dt
du d
s
du
s
d dt
d
g Ž u.
and u⬙ s
d2 u
d2 u
s
dt 2
d 2
2
du
g q 2
g ⭈ g ⬘.
ž / d
Ž 2.8.
From Ž2.4. and Ž2.6. we obtain
Ž L Ž u . . L⬘ Ž u .
f Ž u . s F⬘ Ž u . s
1
H0
Ž L. g
s
Ž s . ds
1
H0
.
Ž 2.9.
Ž s . ds
Substituting Ž2.8. and Ž2.9. into Ž1.11., we obtain g ⭈ ⬘ g
du
ž /ž d
g
d2 u d
2
q g⬘
Ž L.
2
du
ž // d
q
1
H0
Ž s . ds
s 0.
Ž 2.10.
Since g Ž u. / 0 for u / 0, we have
⬘ g
du
ž /ž d
g
d2 u d
2
q g⬘
du
Ž LŽ u. .
2
q
ž // d
1
H0
Ž s . ds
Now define hŽ . s LŽ uŽ ..; then we have dh d d2 h d 2
s L⬘ Ž u . s L⬙ Ž u . sg
d2 u d 2
dh d
sg
du
du d
2
q L⬘ Ž u .
ž / ž / d
q g⬘
du d
2
.
d2 u d 2
s 0.
Ž 2.11.
200
XIAOJING YANG
Hence Ž2.11. can be written as 1
Ž Ž h⬘ . . ⬘ q
1
Ž s . ds
H0
Ž h. s 0
Ž 2.12.
with hŽ0. s hŽ R . s 0, where Rs
T
H0
g Ž u Ž s . . ds.
Since uŽ t . is a positive solution of Ž1.11., then by comparing Ž2.1. with Ž2.12. and by using the assumptions of Theorem 2.2, we have 1
1 s
1
H0 COROLLARY 2.3. then
.
Ž s . ds
Let Ž u. s p Ž u. s < u < py 2 u with p ) 1 a constant;
⌽ Ž u. s F Ž u. s
u
H0
u
H0
p Ž s . ds s
f Ž s . ds s
up p
u G 0.
,
⌽ Ž LŽ u. . ⌽ Ž 1.
s L p Ž u. ,
so LŽ u. s F 1r p Ž u., and g Ž u . s L⬘ Ž u . s
1 p
F Ž1r p.y1 Ž u . f Ž u . .
Let hŽ . s LŽ uŽ . . ,
s
t
H0 g Ž u Ž s . . ds;
then Ž1.11. can be written as
Ž p Ž h⬘. . ⬘ q pp Ž h . s 0,
Ž 2.13.
with h Ž 0 . s h Ž R . s 0,
Rs
T
H0
g Ž u Ž s . . ds.
Ž 2.14.
201
SOLUTIONS OF NONLINEAR OPERATORS
Therefore 1 s p, by the well-known result of w3x,
1 s
p
p
ž /
Ž 2.15.
R
with
p s
2
Ž 2.16.
Ž p sin Ž rp . .
and h Ž . s c sin p
p
ž / R
,
where c is any nonzero constant and sin p Ž t . is the unique 2p periodic function w implicitly defined by
H0w t
ds
Ž .
Ž 1 y s pr Ž p y 1 . .
1rp
s t.
Ž 2.17.
By 1 s p and Ž2.15., we get Rs
2 p
Ž1qp.r p
Ž 2.18.
sin Ž rp .
and T
H0
g Ž u Ž s . . ds s
2 p
Ž1qp.r p
Ž 2.19.
sin Ž rp .
Let p s 2 in Corollary 2.3; then R s r '2 in Ž2.18. ᎐ Ž2.19. and hŽ . s c sin '2 ; hence Corollary 2.3 is a generalization of w1, Theorem Ax. By using the similar method used in w1x we can prove the following lemma Žthe proof is omitted.. LEMMA 2.4. Any two positi¨ e solutions uŽ t . and ¨ Ž t . of Ž1.11. are strictly ordered; i.e., we may assume that uŽ t . - ¨ Ž t . ,
t g Ž 0, T . .
Ž 2.20.
THEOREM 2.5. Assume that f g C 1 Ž Rq. and f Ž u. ) 0 for u ) 0 and either GŽ u. ) 0 or GŽ u. - 0 for all u ) 0, where GŽ u. s f ⬘Ž u. 2 Ž LŽ u. . y ⬘Ž LŽ u. . f 2 Ž u.
1
H0
Ž s . ds.
Ž 2.21.
202
XIAOJING YANG
Then the problem Ž1.11. has at most one positive solution; especially, if Žu. s p Žu., then G Ž u . s Gp Ž u . s f ⬘ Ž u . F Ž u . y
py1 p
f 2 Ž u. .
Ž 2.22.
Proof. Conditions Ž2.21. and Ž2.22. imply that g ⬘ s L⬙ s G ) 0 Žor - 0., and hence the function g is either strictly increasing or strictly decreasing; the result now follows from Ž2.5. and Lemma 2.4. Assume f g C 2 Ž Rq. with f Ž u. ) 0 for u ) 0 and let
THEOREM 2.6.
H Ž u. s f ⬙ Ž u. 3 Ž LŽ u. . y ⬘Ž LŽ u. . 2 Ž LŽ u. . g Ž u. f ⬘Ž u. 1
y
H0
Ž s . ds f 2 Ž u . g Ž u . Ž L Ž u . . q2 f ⬘ Ž u . ⬘ Ž L Ž u . . Ž L Ž u . . y3 g Ž u . f Ž u . Ž ⬘ Ž L Ž u . . .
2
,
Ž 2.23.
especially if Ž u. s p Ž u., H Ž u. s H p Ž u., Hp Ž u . s f ⬙ Ž u . F 2 Ž u . q y
3 Ž p y 1. p
Ž p y 1. Ž 2 p y 1. p2
f 3 Ž u.
f ⬘Ž u. f Ž u. F Ž u. .
Ž 2.24.
If H Ž u. ) 0 for u ) 0, then the problem Ž1.11. admits at most two positive solutions. Proof. By a direct computation we see Ž2.23. and Ž2.24. are equivalent to g ⬙ Ž u . ) 0,
u)0
Ž 2.25.
If there were three solutions u, ¨ , and w, by Lemma 2.4, we may assume that uŽ t . - ¨ Ž t . - w Ž t . ,
t g Ž 0, T . .
Set pŽ t . s ¨ Ž t . y uŽ t ., q Ž t . s w Ž t . y uŽ t .; then 0 - pŽ t . - q Ž t . ,
t g Ž 0, T . .
Ž 2.26.
Writing the formula Ž2.5. at u and ¨ , respectively, and then subtracting, we have Žby using the mean value theorem. T
1
H0 H0
g ⬘ Ž ¨ Ž t . q Ž 1 y . u Ž t . . p Ž t . d dt s 0.
Ž 2.27.
SOLUTIONS OF NONLINEAR OPERATORS
203
Similarly T
1
H0 H0
g ⬘ Ž w Ž t . q Ž 1 y . u Ž t . . q Ž t . d dt s 0.
Ž 2.28.
In view of the above inequalities, the integrand in Ž2.28. is pointwise greater than the one in Ž2.27., a contradiction. Remark. Let p s 2 in Ž2.24.; we have H2 Ž u . s f ⬙ Ž u . F 2 Ž u . q 34 f 3 Ž u . y 32 f ⬘ Ž u . f Ž u . F Ž u . , which is the result of w1, Theorem 2.3x. Hence our results above are a generalization of the results in w1x.
REFERENCES 1. P. Korman and Y. Li, Generalized averages for solutions of two-point Dirichlet problems, J. Math. Anal. Appl. 239 Ž1999., 478᎐484. 2. P. Korman, Problem 97-8, SIAM Re¨ . 39 Ž1997., 318. 3. M. Del Pino, M. Elgueta, and R. Manasevich, A homotopic deformation along p of a Leray᎐Schauder degree result and existence for Ž< u⬘ < py 2 u⬘.⬘ q f Ž t, u. s 0, uŽ0. s uŽT . s 0, p ) 1, J. Differential Equations 80 Ž1989., 1᎐13.
Generalized Averages for Solutions of Nonlinear Operators Xiaojing Yang Department of Mathematics, Tsinghua Uni¨ ersity, Beijing, People’s Republic of China Submitted by L. Debnath Received May 4, 2000
1. INTRODUCTION Recently, Korman and Li w1x proved the following theorem. THEOREM A. Consider the two point Dirichlet problem u⬙ q f Ž u . s 0,
0 - x - T,
u Ž 0 . s u Ž T . s 0.
Ž 1.1.
Assume f g C 1 Ž Rq ., and let uŽ x . be a positi¨ e solution of Ž1.1., uŽ x . ) 0, x g Ž0, T .; we ha¨ e f Ž u Ž x . . ) 0.
Ž 1.2.
Define LŽ u. s
1r2
u
žH
f Ž s . ds
0
/
,
g Ž u . s L⬘ Ž u . .
Ž 1.3.
Then T
H0
g Ž u Ž x . . dx s r'2 .
Ž 1.4.
Theorem A is a generalization of the following boundary value problem: u⬙ q u 3 s 0,
0-x-T
u Ž 0 . s u Ž T . s 0.
Ž 1.5. Ž 1.6.
The problem Ž1.5. ᎐ Ž1.6. has a unique positive solution uŽ x . ) 0, x g Ž0, T . w1, 2x. Then for any T ) 0 this solution satisfies the following average 196 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
SOLUTIONS OF NONLINEAR OPERATORS
197
condition: T
H0
u Ž x . dx s
'2
.
Ž 1.7.
This fact is observed in w2x and is proved in w1x by setting y s H0x uŽ s . ds, hŽ y . s u 2 ; then Ž1.1. is transformed to h⬙ q 2 h s 0,
0 - y - R,
h Ž 0. s h Ž R . s 0
Ž 1.8.
with Rs
T
H0
u Ž s . ds.
Ž 1.9.
A solution of Ž1.8. is hŽ y . s C sin '2 y; C is a constant, hŽ R . s 0, and the positivity of h implies that '2 R s . Then Ž1.9. implies Ž1.7.. By using the same ideal, let hŽ y . s LŽ uŽ y .., y s H0x g Ž uŽ s .. ds; Ž1.1. is transformed to L⬘ Ž u . Ž h⬙ q 2 h . s 0.
Ž 1.10.
Then L⬘Ž u. ) 0, x g Ž0, T ., Ž1.10. becomes Ž1.8., and the results of Theorem A follow. The method of Theorem A is clear now; by using variables transformation Ž1.3., hŽ y . s LŽ uŽ y .., y s H0x g Ž uŽ s .. ds. Ž1.1. is transformed into a simple linear equation Ž1.8.; then the result follows. Inspired by this ideal, we consider in this paper the more general problem
Ž Ž u⬘ . . ⬘ q f Ž u . s 0,
u Ž 0 . s u Ž T . s 0,
Ž 1.11.
where g C 1 Ž R, R ., is strictly increasing, Ž R . s R, and Ž0. s 0. An interesting example is the well-known p-Laplacian: Ž x . s p Ž x . s < x < py 2 x, where p ) 1 is a constant. By using suitable variable transformation, Ž1.11. can be transformed into a kind of simpler form; then by applying known eigenvalue problem results, we obtain a generalized average result for positive solutions of Ž1.11..
2. MAIN RESULTS LEMMA 2.1. Let g C 1 Ž R, R .; is strictly increasing, Ž R . s R, and Ž0. s 0. If the boundary ¨ alue problem ŽBVP.
Ž Ž u⬘ . . q Ž u . s 0, u Ž 0. s u Ž T . s 0
Ž 2.1. Ž 2.2.
198
XIAOJING YANG
has nonzero solutions, then ) 0, and is called an eigen¨ alue of Ž2.1. ᎐ Ž2.2. with eigenfunction uŽ x .. Let 1 s inf N Ž2.1. ᎐ Ž2.2. has nonzero solutions4 : then 1 ) 0 and
1 s inf
¡H
T
~
u/0
0
¦
Ž u⬘ Ž s . . u⬘ Ž s . ds
¢H
T
0
Ž u Ž s . . u Ž s . ds
¥.
Ž 2.3.
§
Proof. Multiplying Ž2.1. by u, integrating by parts over w0, T x, and then using Ž2.2. we obtain T
s
Ž u⬘ Ž s . . u⬘ Ž ds . ds
H0
T
H0
.
Ž u Ž s . . u Ž s . ds
Now the definitions of and uŽ x . / 0 imply ) 0 and Ž2.3.. THEOREM 2.2. Suppose satisfies the assumptions of Lemma 2.1; Ž2.1. has at least one positi¨ e solution and all positi¨ e solutions of Ž2.1. corresponding to the same eigen¨ alue 1 ) 0. Let LŽ u. and g Ž u. be defined as L Ž u . s ⌽ry1 Ž F Ž u . . ,
g Ž u . s L⬘ Ž u . ,
Ž 2.4.
where ⌽ry1 is the right in¨ erse of ⌽, that is, the in¨ erse of ⌽ restricted to w0, ⬁., x
⌽Ž x. s
H0 1
H0
Ž s . ds ,
Ž s . ds
FŽ x. s
x
H0 f Ž s . ds,
f g C 1 Ž Rq . ,
f Ž u. ) 0
if u ) 0. Then for any positi¨ e solution of Ž1.11., if it exists, we ha¨ e T
H0
g Ž u Ž s . . ds s R
and
1 s
1 1
H0
Ž s . ds
.
Ž 2.5.
199
SOLUTIONS OF NONLINEAR OPERATORS
Proof. From the definition of L and g, we have
H0L u Ž s . ds Ž .
F Ž u. s ⌽ Ž LŽ u. . s
1
H0
.
Ž 2.6.
Ž s . ds
Define a time variable transformation
s
t
t g Ž 0, T . ;
H0 g Ž u Ž s . . ds,
Ž 2.7.
then u⬘ s
du dt
du d
s
du
s
d dt
d
g Ž u.
and u⬙ s
d2 u
d2 u
s
dt 2
d 2
2
du
g q 2
g ⭈ g ⬘.
ž / d
Ž 2.8.
From Ž2.4. and Ž2.6. we obtain
Ž L Ž u . . L⬘ Ž u .
f Ž u . s F⬘ Ž u . s
1
H0
Ž L. g
s
Ž s . ds
1
H0
.
Ž 2.9.
Ž s . ds
Substituting Ž2.8. and Ž2.9. into Ž1.11., we obtain g ⭈ ⬘ g
du
ž /ž d
g
d2 u d
2
q g⬘
Ž L.
2
du
ž // d
q
1
H0
Ž s . ds
s 0.
Ž 2.10.
Since g Ž u. / 0 for u / 0, we have
⬘ g
du
ž /ž d
g
d2 u d
2
q g⬘
du
Ž LŽ u. .
2
q
ž // d
1
H0
Ž s . ds
Now define hŽ . s LŽ uŽ ..; then we have dh d d2 h d 2
s L⬘ Ž u . s L⬙ Ž u . sg
d2 u d 2
dh d
sg
du
du d
2
q L⬘ Ž u .
ž / ž / d
q g⬘
du d
2
.
d2 u d 2
s 0.
Ž 2.11.
200
XIAOJING YANG
Hence Ž2.11. can be written as 1
Ž Ž h⬘ . . ⬘ q
1
Ž s . ds
H0
Ž h. s 0
Ž 2.12.
with hŽ0. s hŽ R . s 0, where Rs
T
H0
g Ž u Ž s . . ds.
Since uŽ t . is a positive solution of Ž1.11., then by comparing Ž2.1. with Ž2.12. and by using the assumptions of Theorem 2.2, we have 1
1 s
1
H0 COROLLARY 2.3. then
.
Ž s . ds
Let Ž u. s p Ž u. s < u < py 2 u with p ) 1 a constant;
⌽ Ž u. s F Ž u. s
u
H0
u
H0
p Ž s . ds s
f Ž s . ds s
up p
u G 0.
,
⌽ Ž LŽ u. . ⌽ Ž 1.
s L p Ž u. ,
so LŽ u. s F 1r p Ž u., and g Ž u . s L⬘ Ž u . s
1 p
F Ž1r p.y1 Ž u . f Ž u . .
Let hŽ . s LŽ uŽ . . ,
s
t
H0 g Ž u Ž s . . ds;
then Ž1.11. can be written as
Ž p Ž h⬘. . ⬘ q pp Ž h . s 0,
Ž 2.13.
with h Ž 0 . s h Ž R . s 0,
Rs
T
H0
g Ž u Ž s . . ds.
Ž 2.14.
201
SOLUTIONS OF NONLINEAR OPERATORS
Therefore 1 s p, by the well-known result of w3x,
1 s
p
p
ž /
Ž 2.15.
R
with
p s
2
Ž 2.16.
Ž p sin Ž rp . .
and h Ž . s c sin p
p
ž / R
,
where c is any nonzero constant and sin p Ž t . is the unique 2p periodic function w implicitly defined by
H0w t
ds
Ž .
Ž 1 y s pr Ž p y 1 . .
1rp
s t.
Ž 2.17.
By 1 s p and Ž2.15., we get Rs
2 p
Ž1qp.r p
Ž 2.18.
sin Ž rp .
and T
H0
g Ž u Ž s . . ds s
2 p
Ž1qp.r p
Ž 2.19.
sin Ž rp .
Let p s 2 in Corollary 2.3; then R s r '2 in Ž2.18. ᎐ Ž2.19. and hŽ . s c sin '2 ; hence Corollary 2.3 is a generalization of w1, Theorem Ax. By using the similar method used in w1x we can prove the following lemma Žthe proof is omitted.. LEMMA 2.4. Any two positi¨ e solutions uŽ t . and ¨ Ž t . of Ž1.11. are strictly ordered; i.e., we may assume that uŽ t . - ¨ Ž t . ,
t g Ž 0, T . .
Ž 2.20.
THEOREM 2.5. Assume that f g C 1 Ž Rq. and f Ž u. ) 0 for u ) 0 and either GŽ u. ) 0 or GŽ u. - 0 for all u ) 0, where GŽ u. s f ⬘Ž u. 2 Ž LŽ u. . y ⬘Ž LŽ u. . f 2 Ž u.
1
H0
Ž s . ds.
Ž 2.21.
202
XIAOJING YANG
Then the problem Ž1.11. has at most one positive solution; especially, if Žu. s p Žu., then G Ž u . s Gp Ž u . s f ⬘ Ž u . F Ž u . y
py1 p
f 2 Ž u. .
Ž 2.22.
Proof. Conditions Ž2.21. and Ž2.22. imply that g ⬘ s L⬙ s G ) 0 Žor - 0., and hence the function g is either strictly increasing or strictly decreasing; the result now follows from Ž2.5. and Lemma 2.4. Assume f g C 2 Ž Rq. with f Ž u. ) 0 for u ) 0 and let
THEOREM 2.6.
H Ž u. s f ⬙ Ž u. 3 Ž LŽ u. . y ⬘Ž LŽ u. . 2 Ž LŽ u. . g Ž u. f ⬘Ž u. 1
y
H0
Ž s . ds f 2 Ž u . g Ž u . Ž L Ž u . . q2 f ⬘ Ž u . ⬘ Ž L Ž u . . Ž L Ž u . . y3 g Ž u . f Ž u . Ž ⬘ Ž L Ž u . . .
2
,
Ž 2.23.
especially if Ž u. s p Ž u., H Ž u. s H p Ž u., Hp Ž u . s f ⬙ Ž u . F 2 Ž u . q y
3 Ž p y 1. p
Ž p y 1. Ž 2 p y 1. p2
f 3 Ž u.
f ⬘Ž u. f Ž u. F Ž u. .
Ž 2.24.
If H Ž u. ) 0 for u ) 0, then the problem Ž1.11. admits at most two positive solutions. Proof. By a direct computation we see Ž2.23. and Ž2.24. are equivalent to g ⬙ Ž u . ) 0,
u)0
Ž 2.25.
If there were three solutions u, ¨ , and w, by Lemma 2.4, we may assume that uŽ t . - ¨ Ž t . - w Ž t . ,
t g Ž 0, T . .
Set pŽ t . s ¨ Ž t . y uŽ t ., q Ž t . s w Ž t . y uŽ t .; then 0 - pŽ t . - q Ž t . ,
t g Ž 0, T . .
Ž 2.26.
Writing the formula Ž2.5. at u and ¨ , respectively, and then subtracting, we have Žby using the mean value theorem. T
1
H0 H0
g ⬘ Ž ¨ Ž t . q Ž 1 y . u Ž t . . p Ž t . d dt s 0.
Ž 2.27.
SOLUTIONS OF NONLINEAR OPERATORS
203
Similarly T
1
H0 H0
g ⬘ Ž w Ž t . q Ž 1 y . u Ž t . . q Ž t . d dt s 0.
Ž 2.28.
In view of the above inequalities, the integrand in Ž2.28. is pointwise greater than the one in Ž2.27., a contradiction. Remark. Let p s 2 in Ž2.24.; we have H2 Ž u . s f ⬙ Ž u . F 2 Ž u . q 34 f 3 Ž u . y 32 f ⬘ Ž u . f Ž u . F Ž u . , which is the result of w1, Theorem 2.3x. Hence our results above are a generalization of the results in w1x.
REFERENCES 1. P. Korman and Y. Li, Generalized averages for solutions of two-point Dirichlet problems, J. Math. Anal. Appl. 239 Ž1999., 478᎐484. 2. P. Korman, Problem 97-8, SIAM Re¨ . 39 Ž1997., 318. 3. M. Del Pino, M. Elgueta, and R. Manasevich, A homotopic deformation along p of a Leray᎐Schauder degree result and existence for Ž< u⬘ < py 2 u⬘.⬘ q f Ž t, u. s 0, uŽ0. s uŽT . s 0, p ) 1, J. Differential Equations 80 Ž1989., 1᎐13.