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The Method of Lower and Upper Solutions for Fourth-Order Two-Point Boundary Value Problems
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
215, 415]422 Ž1997.
AY975639
The Method of Lower and Upper Solutions for Fourth-Order Two-Point Boundary Value Problems Ma Ruyun, Zhang Jihui, and Fu Shengmao Department of Mathematics, Northwest Normal Uni¨ ersity, Lanzhou, 730070, Gansu, People’s Republic of China Submitted by Harlan W. Stech Received November 7, 1995
In this paper we develop the monotone method in the presence of lower and upper solutions for the problem uŽ I V . s f Ž x, u Ž x . , u0 Ž x . . , u Ž 0 . s u Ž 1 . s u0 Ž 0 . s u0 Ž 1 . s 0, where f : w0, 1x = R = R ª R is continuous.
Q 1997 Academic Press
1. INTRODUCTION In this paper we are mainly concerned with existence for a fourth-order boundary value problem of the form uŽ I V . Ž x . s f Ž x, u Ž x . , u0 Ž x . . ,
Ž 1.1.
u Ž 0 . s u Ž 1 . s u0 Ž 0 . s u0 Ž 1 . s 0,
Ž 1.2.
where f : w0, 1x = R = R ª R is continuous. This problem has been studied by several authors. In w1x, Aftabizadeh showed the existence of a solution to Ž1.1. ] Ž1.2. under the restriction that f is a bounded function. In w10, Theorem 1x, Yang extended these results, letting f satisfy a growth condition of the form f Ž x, u, ¨ . F a < u < q b < ¨ < q c,
Ž 1.3.
415 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
416
RUYUN, JIHUI, AND SHENGMAO
where a, b, and c are positive constants such that a
p
4
q
b
p2
- 1.
Ž 1.4.
In w6x Del Pino and Manasevich extended this work of Yang. They showed the existence for Ž1.1. ] Ž1.2. under a growth condition of the form f Ž x, u, ¨ . y Ž a u y b ¨ . F a < u < q b < ¨ < q c
Ž 1.5.
and a nonresonance condition involving a two-parameter linear eigenvalue problem. The result of Yang w18x was also improved by Coster, Fabry, and Munyamarere w4x. Other existence results for Ž1.1. ] Ž1.2. can be found in w8]10, 12, 13, 17x. All of those results are based upon the Leray]Schauder continuation method and topological degree. The upper and lower solution method has been studied for the fourthorder problem with boundary condition Ž1.2. and others by several authors; see Agarwal w2x, Cabada w3x, De Coster and Sanchez w5x, Dunninger w7x, Korman w11x, Sadyrbaev w15x, and Schroder w16x for references along this line. But all of these authors consider only an equation of the form uŽ I V . Ž x . s f Ž x, u Ž x . . , with diverse kind of boundary conditions. It is the purpose of this paper to develop the monotone method in the presence of a lower and an upper solution for the problem Ž1.1. ] Ž1.2.. We show the existence of solutions between a lower solution b and an upper solution a without any growth restriction on f.
2. MAXIMUM PRINCIPLE To prove the validity of the monotone method for the fourth-order two-point boundary value problem Ž1.1. ] Ž1.2., we present a maximum principle for the operator L: F ª C w 0, 1 x defined by Lu s uŽ I V ., where u g F and F s u g C 4 w 0, 1 x u0 Ž 0 . F 0, u0 Ž 1 . F 0, u Ž 0 . G 0, u Ž 1 . G 0 4 .
METHOD OF LOWER AND UPPER SOLUTIONS
LEMMA 2.1.
417
If u g F satisfies Lu G 0, then u G 0 in w0, 1x.
Proof. Let uŽ I V . Ž x . s s Ž x . , u Ž 0 . s a, u0 Ž 0 . s c,
Ž 2.1.
u Ž 1 . s b. u0 Ž 1 . s d.
Ž 2.2.
Then a G 0, b G 0, c F 0, d F 0, and s G 0.
Ž 2.3.
It is easy to check that such a u can be given by the expression uŽ x . s
1
1
H0 H0 G Ž x, s . G Ž s, t . s Ž t . dt
ds q R Ž x . ,
Ž 2.4.
where RŽ x . s
dyc 6
c
x3 q
2
x2 q b y a y
ž
c 3
y
d 6
/
x q a,
Ž 2.5.
and where G: w0, 1x = w0, 1x ª R denotes the Green’s function for the boundary value problem yu0 s h,
Ž 2.6.
u Ž 0 . s u Ž 1 . s 0,
Ž 2.7.
and is explicitly given by G Ž x, s . s
½
x Ž1 y s. , sŽ1 y x . ,
for x F s, for x ) s.
Ž 2.8.
From Ž2.5., we get R0 Ž x . s dx q c Ž 1 y x . .
Ž 2.9.
By Ž2.9. and Ž2.3., we have that R0 Ž x . F 0
for x g w 0, 1 x .
Ž 2.10.
This together with the fact that RŽ0. s a G 0 and RŽ1. s b G 0 implies RŽ x . G 0
for x g w 0, 1 x .
Ž 2.11.
By combining Ž2.4. and Ž2.11. and using Ž2.3. and the fact G G 0 in w0, 1x = w0, 1x, we obtain the result that u G 0.
418
RUYUN, JIHUI, AND SHENGMAO
3. THE MONOTONE METHOD In this section we develop the monotone method for the fourth-order two-point boundary value problem Ž1.1. ] Ž1.2.. DEFINITION 3.1. Letting a g C 4 w0, 1x, we say a is an upper solution for the problem Ž1.1. ] Ž1.2. if a satisfies
a Ž I V . G f Ž x, a Ž x . , a 0 Ž x . . ,
for x g Ž 0, 1 .
a Ž 0 . G 0,
a Ž 1 . G 0,
a 0 Ž 0 . F 0,
a 0 Ž 1 . F 0.
DEFINITION 3.2. Letting b g C 4 w0, 1x, we say b is a lower solution for the problem Ž1.1. ] Ž1.2. if b satisfies
b Ž I V . F f Ž x, b Ž x . , b 0 Ž x . . ,
for x g Ž 0, 1 .
b Ž 0 . F 0,
b Ž 1 . F 0,
b 0 Ž 0 . G 0,
b 0 Ž 1 . G 0.
THEOREM 3.1. If there exist a and b , upper and lower solutions, respecti¨ ely, for the problem Ž1.1. ] Ž1.2. which satisfy
bFa
and
b0 G a0 ,
Ž 3.1.
and if f : w0, 1x = R = R ª R is continuous and satisfies f Ž x, u 2 , ¨ . y f Ž x, u1 , ¨ . G 0, for b Ž x . F u1 F u 2 F a Ž x . , ¨ g R , and x g w 0, 1 x Ž 3.2. f Ž x, u, ¨ 2 . y f Ž x, u, ¨ 1 . F 0, for a 0 Ž x . F ¨ 1 F ¨ 2 F b 0 Ž x . , u g R, and x g w 0, 1 x
Ž 3.3. then there exist two monotone sequences a n4 and bn4 , nonincreasing and nondecreasing, respecti¨ ely, with a 0 s a and b 0 s b , which con¨ erge uniformly to the extremal solutions in w b , a x of the problem Ž1.1. ] Ž1.2.. Proof. We consider the problem uŽ I V . s f Ž x, h Ž x . , h 0 Ž x . . ,
for x g Ž 0, 1 .
u Ž 0 . s u Ž 1 . s u0 Ž 0 . s u0 Ž 1 . s 0, with h g C 2 w0, 1x.
Ž 3.4. Ž 3.5.
METHOD OF LOWER AND UPPER SOLUTIONS
419
According to w10, Theorem 2x, the problem Ž3.4. ] Ž3.5. has a unique solution u. Define T : C 2 w0, 1x ª C 4 w0, 1x by Th s u.
Ž 3.6.
Now, we divide the proof into three steps. Step 1. We show TC : C.
Ž 3.7.
Here C s h g C 2 w0, 1x ¬ b F h F a , b 0 G h 0 G a 0 4 is a nonempty bounded closed subset in C 2 w0, 1x. In fact, for z g C, set w s Tz . From the definitions of a , b , and C, we have that
Ž a y w.
ŽIV .
Ž x . G f Ž x, a Ž x . , a 0 Ž x . . y f Ž x, z Ž x . , z 0 Ž x . . G 0 Ž 3.8. Ž a y w . Ž 0 . G 0,
Ž a y w . Ž 1 . G 0,
Ž 3.9.
Ž a y w . 0 Ž 0 . F 0,
Ž a y w . 0 Ž 1 . F 0.
Ž 3.10.
Using Lemma 2.1 we obtain that a G w. Analogously we can prove that w G b. Setting zŽ x. s Ž a y w.0 Ž x. ,
x g w 0, 1 x
Ž 3.11.
from Ž3.8. and Ž3.10. we obtain z0 Ž x . G 0 z Ž 0 . F 0,
Ž 3.12.
z Ž 1 . F 0,
Ž 3.13.
for x g w 0, 1 x .
Ž 3.14.
and Ž3.12. ] Ž3.13. yields zŽ x. F 0
Hence, a 0 Ž x . F w0 Ž x . for x g w0, 1x. Analogously we can prove that w0 Ž x . F b 0 Ž x . for x g w0, 1x. Thus, Ž3.7. holds. Step 2. Let u1 s Th1 , u 2 s Th 2 , where h1 , h 2 g C satisfy a F h1 F h 2 F b and a 0 G h1Y G hY2 G b 0. We show u1 F u 2 ,
uY1 G uY2
Ž 3.15.
420
RUYUN, JIHUI, AND SHENGMAO
In fact,
Ž u 2 y u1 .
ŽIV .
Ž x . s f Ž x, h2 Ž x . , hY2 Ž x . . y f Ž x, h1 Ž x . , h1Y Ž x . . G 0 Ž 3.16. Ž u 2 y u1 . Ž 0 . s Ž u 2 y u1 . Ž 1 . s 0,
Ž 3.17.
Ž u 2 y u1 . 0 Ž 0 . s Ž u 2 y u1 . 0 Ž 1 . s 0.
Ž 3.18.
Using Lemma 2.1, we get that u1 F u 2 . Setting ¨ s uY2 y uY1 , from Ž3.16. and Ž3.18., we have ¨0 Ž x. G 0
Ž 3.19.
¨ Ž 0 . s ¨ Ž 1 . s 0.
Ž 3.20.
This yields ¨ F 0 in w0, 1x. That is, uY1 G uY2 . Thus, Ž3.15. holds. Step 3. The sequences a n4 and bn4 are obtained by recurrence: a 0 s a , b 0 s b , a n s Ta ny1 , bn s Tbny1; n s 1, 2, . . . . From the results of Step 1 and Step 2, we know
b s b 0 F b 1 F ??? F bn F ??? F a n F ??? F a 1 F a 0 s a , b0 s
b 0Y
G
b 1Y
G ??? G
bnY
a nY
G ??? G
G ??? G
a 1Y
G
a 0Y
Ž 3.21.
s a 0 . Ž 3.22.
Moreover from the definition of T Žsee Ž3.6.., we have Y a nZX Ž x . s f Ž x, a ny1 Ž x . , a ny1 Ž x. .,
Ž 3.23.
a nY
Ž 3.24.
a n Ž 0. y a n Ž 1. s
Ž 0. y
a nY
Ž 1. s 0
and Y bnZX Ž x . s f Ž x, bny1 Ž x . , bny1 Ž x. .,
Ž 3.25.
bnY
Ž 3.26.
bn Ž 0 . y bn Ž 1 . s
Ž 0. y
bnY
Ž 1 . s 0.
From Ž3.21., Ž3.22., and Ž3.23., we have that there exists Ma ) 0 depending only on a Žbut not on n or x . such that
a nZX Ž x . F Ma ,
for all x g w 0, 1 x .
Ž 3.27.
Using the boundary condition Ž3.24. we get that for each n g N, there exists j n g Ž0, 1. such that
a nZ Ž j n . s 0.
Ž 3.28.
METHOD OF LOWER AND UPPER SOLUTIONS
421
This together with Ž3.27. yields
a nZ Ž x . s a nZ Ž j n . q
x
Hj a
ZX n
Ž s . ds F Ma .
Ž 3.29.
n
By combining Ž3.22. and Ž3.24., we can easily get that there is Ca ) 0 depending only on a Žbut not on n or x . such that
a nX Ž x . F Ca .
Ž 3.30.
Thus, from Ž3.21., Ž3.22., Ž3.29., and Ž3.30., we know that a n4 is bounded in C 3 w0, 1x. Similarly, bn4 is bounded in C 3w0, 1x. Now, by using the fact that a n4 and bn4 are bounded in C 3 w0, 1x, we can conclude that a n4 , bn4 converge uniformly to the extremal solutions in w0, 1x of the problem Ž1.1. ] Ž1.2.. Remark 3.2. As an example we mention the boundary value problem uŽ4. s u m y
1
p
10
5 Ž u0 . q sin p x
u Ž 0 . s u Ž 1 . s u0 Ž 0 . s u0 Ž 1 . s 0,
Ž 3.31. Ž 3.32.
where m g N. It is clear that the results of w1]13, 15]18x do not apply to the example. On the other hand, it is easy to check that a s sin p x, b s 0 are upper and lower solutions of Ž3.31. ] Ž3.32., respectively. Clearly, all assumptions of Theorem 3.1 are fulfilled. So the problem Ž3.31. ] Ž3.32. has at least one solution u, which satisfies 0 F u F sin p x,
and
yp 2 sin p x F u0 F 0.
Ž 3.33.
Moreover, by combining Ž3.33. and Ž3.31. and using Ž3.32. and w14, Lemma 1x, we obtain uŽ x . ) 0 for x g Ž0, 1..
ACKNOWLEDGMENT The authors thank the referees for useful suggestions.
REFERENCES 1. A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl. 116 Ž1986., 415]426. 2. R. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations 2 Ž1989., 91]110.
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RUYUN, JIHUI, AND SHENGMAO
3. A. Cabada, The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems, J. Math. Anal. Appl. 185 Ž1994., 302]320. 4. C. De Coster, C. Fabry, and F. Munyamarere, Nonresonance conditions for fourth-order nonlinear boundary value problems, Internat. J. Math. Math. Sci. 17 Ž1994., 725]740. 5. C. De Coster and L. Sanchez, Upper and lower solutions, Ambrosetti]Prodi problem and positive solutions for fourth order O.D.E., Ri¨ . Mat. Pura Appl. 14 Ž1994., 1129]1138. 6. M. A. Del Pino and R. F. Manasevich, Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc. 112 Ž1991., 81]86. 7. D. Dunninger, Existence of positive solutions for fourth-order nonlinear problems, Boll. Un. Mat. Ital. 7 Ž1987., 1129]1138. 8. C. P. Gupta, Existence and uniqueness results for some fourth order fully quasilinear boundary value problem, Appl. Anal. 36 Ž1990., 175]169. 9. C. P. Gupta, Existence and uniqueness results for the bending of an elastic beam equation at resonance, J. Math. Appl. Appl. 135 Ž1988., 208]225. 10. C. P. Gupta, Existence and uniqueness theorem for a bending of an elastic beam equation, Appl. Anal. 26 Ž1988., 289]304. 11. P. Korman, A maximum principle for fourth-order ordinary differential equations, Appl. Anal. 33 Ž1989., 267]273. 12. R. Y. Ma, Some multiplicity results for an elastic beam equation at resonance, Appl. Math. Mech. (English ed.) 14 Ž1993., 193]200. 13. R. Y. Ma and H. Y. Wang, On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal. 59 Ž1995., 225]231. 14. F. I. Njoku, Positive solutions for two-point BVP’s: Existence and multiplicity results, Nonlinear Anal. 13 Ž1989., 1329]1338. 15. F. Sadyrbaev, Two-point boundary value problem for fourth-order, Acta Uni¨ . Lat¨ iensis 553 Ž1990., 84]91. 16. J. Schroder, Fourth-order two-point boundary value problems; estimates by two side bounds, Nonlinear Anal. 8 Ž1984., 107]114. 17. R. A. Usmani, A uniqueness theorem for a boundary value problem, Proc. Amer. Math. Soc. 77 Ž1979., 327]335. 18. Y. Yang, Fourth-order two-point boundary value problem, Proc. Amer. Math. Soc. 104 Ž1988., 175]180.
215, 415]422 Ž1997.
AY975639
The Method of Lower and Upper Solutions for Fourth-Order Two-Point Boundary Value Problems Ma Ruyun, Zhang Jihui, and Fu Shengmao Department of Mathematics, Northwest Normal Uni¨ ersity, Lanzhou, 730070, Gansu, People’s Republic of China Submitted by Harlan W. Stech Received November 7, 1995
In this paper we develop the monotone method in the presence of lower and upper solutions for the problem uŽ I V . s f Ž x, u Ž x . , u0 Ž x . . , u Ž 0 . s u Ž 1 . s u0 Ž 0 . s u0 Ž 1 . s 0, where f : w0, 1x = R = R ª R is continuous.
Q 1997 Academic Press
1. INTRODUCTION In this paper we are mainly concerned with existence for a fourth-order boundary value problem of the form uŽ I V . Ž x . s f Ž x, u Ž x . , u0 Ž x . . ,
Ž 1.1.
u Ž 0 . s u Ž 1 . s u0 Ž 0 . s u0 Ž 1 . s 0,
Ž 1.2.
where f : w0, 1x = R = R ª R is continuous. This problem has been studied by several authors. In w1x, Aftabizadeh showed the existence of a solution to Ž1.1. ] Ž1.2. under the restriction that f is a bounded function. In w10, Theorem 1x, Yang extended these results, letting f satisfy a growth condition of the form f Ž x, u, ¨ . F a < u < q b < ¨ < q c,
Ž 1.3.
415 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
416
RUYUN, JIHUI, AND SHENGMAO
where a, b, and c are positive constants such that a
p
4
q
b
p2
- 1.
Ž 1.4.
In w6x Del Pino and Manasevich extended this work of Yang. They showed the existence for Ž1.1. ] Ž1.2. under a growth condition of the form f Ž x, u, ¨ . y Ž a u y b ¨ . F a < u < q b < ¨ < q c
Ž 1.5.
and a nonresonance condition involving a two-parameter linear eigenvalue problem. The result of Yang w18x was also improved by Coster, Fabry, and Munyamarere w4x. Other existence results for Ž1.1. ] Ž1.2. can be found in w8]10, 12, 13, 17x. All of those results are based upon the Leray]Schauder continuation method and topological degree. The upper and lower solution method has been studied for the fourthorder problem with boundary condition Ž1.2. and others by several authors; see Agarwal w2x, Cabada w3x, De Coster and Sanchez w5x, Dunninger w7x, Korman w11x, Sadyrbaev w15x, and Schroder w16x for references along this line. But all of these authors consider only an equation of the form uŽ I V . Ž x . s f Ž x, u Ž x . . , with diverse kind of boundary conditions. It is the purpose of this paper to develop the monotone method in the presence of a lower and an upper solution for the problem Ž1.1. ] Ž1.2.. We show the existence of solutions between a lower solution b and an upper solution a without any growth restriction on f.
2. MAXIMUM PRINCIPLE To prove the validity of the monotone method for the fourth-order two-point boundary value problem Ž1.1. ] Ž1.2., we present a maximum principle for the operator L: F ª C w 0, 1 x defined by Lu s uŽ I V ., where u g F and F s u g C 4 w 0, 1 x u0 Ž 0 . F 0, u0 Ž 1 . F 0, u Ž 0 . G 0, u Ž 1 . G 0 4 .
METHOD OF LOWER AND UPPER SOLUTIONS
LEMMA 2.1.
417
If u g F satisfies Lu G 0, then u G 0 in w0, 1x.
Proof. Let uŽ I V . Ž x . s s Ž x . , u Ž 0 . s a, u0 Ž 0 . s c,
Ž 2.1.
u Ž 1 . s b. u0 Ž 1 . s d.
Ž 2.2.
Then a G 0, b G 0, c F 0, d F 0, and s G 0.
Ž 2.3.
It is easy to check that such a u can be given by the expression uŽ x . s
1
1
H0 H0 G Ž x, s . G Ž s, t . s Ž t . dt
ds q R Ž x . ,
Ž 2.4.
where RŽ x . s
dyc 6
c
x3 q
2
x2 q b y a y
ž
c 3
y
d 6
/
x q a,
Ž 2.5.
and where G: w0, 1x = w0, 1x ª R denotes the Green’s function for the boundary value problem yu0 s h,
Ž 2.6.
u Ž 0 . s u Ž 1 . s 0,
Ž 2.7.
and is explicitly given by G Ž x, s . s
½
x Ž1 y s. , sŽ1 y x . ,
for x F s, for x ) s.
Ž 2.8.
From Ž2.5., we get R0 Ž x . s dx q c Ž 1 y x . .
Ž 2.9.
By Ž2.9. and Ž2.3., we have that R0 Ž x . F 0
for x g w 0, 1 x .
Ž 2.10.
This together with the fact that RŽ0. s a G 0 and RŽ1. s b G 0 implies RŽ x . G 0
for x g w 0, 1 x .
Ž 2.11.
By combining Ž2.4. and Ž2.11. and using Ž2.3. and the fact G G 0 in w0, 1x = w0, 1x, we obtain the result that u G 0.
418
RUYUN, JIHUI, AND SHENGMAO
3. THE MONOTONE METHOD In this section we develop the monotone method for the fourth-order two-point boundary value problem Ž1.1. ] Ž1.2.. DEFINITION 3.1. Letting a g C 4 w0, 1x, we say a is an upper solution for the problem Ž1.1. ] Ž1.2. if a satisfies
a Ž I V . G f Ž x, a Ž x . , a 0 Ž x . . ,
for x g Ž 0, 1 .
a Ž 0 . G 0,
a Ž 1 . G 0,
a 0 Ž 0 . F 0,
a 0 Ž 1 . F 0.
DEFINITION 3.2. Letting b g C 4 w0, 1x, we say b is a lower solution for the problem Ž1.1. ] Ž1.2. if b satisfies
b Ž I V . F f Ž x, b Ž x . , b 0 Ž x . . ,
for x g Ž 0, 1 .
b Ž 0 . F 0,
b Ž 1 . F 0,
b 0 Ž 0 . G 0,
b 0 Ž 1 . G 0.
THEOREM 3.1. If there exist a and b , upper and lower solutions, respecti¨ ely, for the problem Ž1.1. ] Ž1.2. which satisfy
bFa
and
b0 G a0 ,
Ž 3.1.
and if f : w0, 1x = R = R ª R is continuous and satisfies f Ž x, u 2 , ¨ . y f Ž x, u1 , ¨ . G 0, for b Ž x . F u1 F u 2 F a Ž x . , ¨ g R , and x g w 0, 1 x Ž 3.2. f Ž x, u, ¨ 2 . y f Ž x, u, ¨ 1 . F 0, for a 0 Ž x . F ¨ 1 F ¨ 2 F b 0 Ž x . , u g R, and x g w 0, 1 x
Ž 3.3. then there exist two monotone sequences a n4 and bn4 , nonincreasing and nondecreasing, respecti¨ ely, with a 0 s a and b 0 s b , which con¨ erge uniformly to the extremal solutions in w b , a x of the problem Ž1.1. ] Ž1.2.. Proof. We consider the problem uŽ I V . s f Ž x, h Ž x . , h 0 Ž x . . ,
for x g Ž 0, 1 .
u Ž 0 . s u Ž 1 . s u0 Ž 0 . s u0 Ž 1 . s 0, with h g C 2 w0, 1x.
Ž 3.4. Ž 3.5.
METHOD OF LOWER AND UPPER SOLUTIONS
419
According to w10, Theorem 2x, the problem Ž3.4. ] Ž3.5. has a unique solution u. Define T : C 2 w0, 1x ª C 4 w0, 1x by Th s u.
Ž 3.6.
Now, we divide the proof into three steps. Step 1. We show TC : C.
Ž 3.7.
Here C s h g C 2 w0, 1x ¬ b F h F a , b 0 G h 0 G a 0 4 is a nonempty bounded closed subset in C 2 w0, 1x. In fact, for z g C, set w s Tz . From the definitions of a , b , and C, we have that
Ž a y w.
ŽIV .
Ž x . G f Ž x, a Ž x . , a 0 Ž x . . y f Ž x, z Ž x . , z 0 Ž x . . G 0 Ž 3.8. Ž a y w . Ž 0 . G 0,
Ž a y w . Ž 1 . G 0,
Ž 3.9.
Ž a y w . 0 Ž 0 . F 0,
Ž a y w . 0 Ž 1 . F 0.
Ž 3.10.
Using Lemma 2.1 we obtain that a G w. Analogously we can prove that w G b. Setting zŽ x. s Ž a y w.0 Ž x. ,
x g w 0, 1 x
Ž 3.11.
from Ž3.8. and Ž3.10. we obtain z0 Ž x . G 0 z Ž 0 . F 0,
Ž 3.12.
z Ž 1 . F 0,
Ž 3.13.
for x g w 0, 1 x .
Ž 3.14.
and Ž3.12. ] Ž3.13. yields zŽ x. F 0
Hence, a 0 Ž x . F w0 Ž x . for x g w0, 1x. Analogously we can prove that w0 Ž x . F b 0 Ž x . for x g w0, 1x. Thus, Ž3.7. holds. Step 2. Let u1 s Th1 , u 2 s Th 2 , where h1 , h 2 g C satisfy a F h1 F h 2 F b and a 0 G h1Y G hY2 G b 0. We show u1 F u 2 ,
uY1 G uY2
Ž 3.15.
420
RUYUN, JIHUI, AND SHENGMAO
In fact,
Ž u 2 y u1 .
ŽIV .
Ž x . s f Ž x, h2 Ž x . , hY2 Ž x . . y f Ž x, h1 Ž x . , h1Y Ž x . . G 0 Ž 3.16. Ž u 2 y u1 . Ž 0 . s Ž u 2 y u1 . Ž 1 . s 0,
Ž 3.17.
Ž u 2 y u1 . 0 Ž 0 . s Ž u 2 y u1 . 0 Ž 1 . s 0.
Ž 3.18.
Using Lemma 2.1, we get that u1 F u 2 . Setting ¨ s uY2 y uY1 , from Ž3.16. and Ž3.18., we have ¨0 Ž x. G 0
Ž 3.19.
¨ Ž 0 . s ¨ Ž 1 . s 0.
Ž 3.20.
This yields ¨ F 0 in w0, 1x. That is, uY1 G uY2 . Thus, Ž3.15. holds. Step 3. The sequences a n4 and bn4 are obtained by recurrence: a 0 s a , b 0 s b , a n s Ta ny1 , bn s Tbny1; n s 1, 2, . . . . From the results of Step 1 and Step 2, we know
b s b 0 F b 1 F ??? F bn F ??? F a n F ??? F a 1 F a 0 s a , b0 s
b 0Y
G
b 1Y
G ??? G
bnY
a nY
G ??? G
G ??? G
a 1Y
G
a 0Y
Ž 3.21.
s a 0 . Ž 3.22.
Moreover from the definition of T Žsee Ž3.6.., we have Y a nZX Ž x . s f Ž x, a ny1 Ž x . , a ny1 Ž x. .,
Ž 3.23.
a nY
Ž 3.24.
a n Ž 0. y a n Ž 1. s
Ž 0. y
a nY
Ž 1. s 0
and Y bnZX Ž x . s f Ž x, bny1 Ž x . , bny1 Ž x. .,
Ž 3.25.
bnY
Ž 3.26.
bn Ž 0 . y bn Ž 1 . s
Ž 0. y
bnY
Ž 1 . s 0.
From Ž3.21., Ž3.22., and Ž3.23., we have that there exists Ma ) 0 depending only on a Žbut not on n or x . such that
a nZX Ž x . F Ma ,
for all x g w 0, 1 x .
Ž 3.27.
Using the boundary condition Ž3.24. we get that for each n g N, there exists j n g Ž0, 1. such that
a nZ Ž j n . s 0.
Ž 3.28.
METHOD OF LOWER AND UPPER SOLUTIONS
421
This together with Ž3.27. yields
a nZ Ž x . s a nZ Ž j n . q
x
Hj a
ZX n
Ž s . ds F Ma .
Ž 3.29.
n
By combining Ž3.22. and Ž3.24., we can easily get that there is Ca ) 0 depending only on a Žbut not on n or x . such that
a nX Ž x . F Ca .
Ž 3.30.
Thus, from Ž3.21., Ž3.22., Ž3.29., and Ž3.30., we know that a n4 is bounded in C 3 w0, 1x. Similarly, bn4 is bounded in C 3w0, 1x. Now, by using the fact that a n4 and bn4 are bounded in C 3 w0, 1x, we can conclude that a n4 , bn4 converge uniformly to the extremal solutions in w0, 1x of the problem Ž1.1. ] Ž1.2.. Remark 3.2. As an example we mention the boundary value problem uŽ4. s u m y
1
p
10
5 Ž u0 . q sin p x
u Ž 0 . s u Ž 1 . s u0 Ž 0 . s u0 Ž 1 . s 0,
Ž 3.31. Ž 3.32.
where m g N. It is clear that the results of w1]13, 15]18x do not apply to the example. On the other hand, it is easy to check that a s sin p x, b s 0 are upper and lower solutions of Ž3.31. ] Ž3.32., respectively. Clearly, all assumptions of Theorem 3.1 are fulfilled. So the problem Ž3.31. ] Ž3.32. has at least one solution u, which satisfies 0 F u F sin p x,
and
yp 2 sin p x F u0 F 0.
Ž 3.33.
Moreover, by combining Ž3.33. and Ž3.31. and using Ž3.32. and w14, Lemma 1x, we obtain uŽ x . ) 0 for x g Ž0, 1..
ACKNOWLEDGMENT The authors thank the referees for useful suggestions.
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