- Email: [email protected]
Existence Theorem for Periodic Solutions of Higher Order Nonlinear Differential Equations
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
216, 481]490 Ž1997.
AY975669
Existence Theorem for Periodic Solutions of Higher Order Nonlinear Differential Equations Zhongdong LiuU and Yiping Mao† Department of Mathematics, Uni¨ ersity of South Carolina, Columbia, South Carolina 29208 Submitted by Zhi¨ ko S. Athanasso¨ Received July 15, 1996
We study the existence of periodic solutions to differential equations of the form X X LŽ x . q g Ž t, x, x , . . . , x Ž m. . s f Ž t . with LŽ x . s x Ž m. q a my 1 x Ž my1. q ??? qa1 x . Q 1997 Academic Press
1. INTRODUCTION The purpose of this paper is to establish the existence of periodic solutions to the nonlinear differential equation x Ž m. q a my 1 x Ž my1. q ??? qa1 xX q g Ž t , x, xX , . . . , x Ž m. . s f Ž t . , Ž 1.1. where a1 , a2 , . . . , a my1 are constants, g : R = R mq 1 ª R is continuous and T-periodic ŽT ) 0. in its first variable, and f Ž t . is a continuous T-periodic function. This and similar types of problems have recently received considerable attention. ŽSee w2, 4, 6, 7, 9]12, 14]18x, etc.. In most known existence results, the nonlinearity g depends at most on the lower order derivatives xX , xY , . . . , and x Ž my1. and, hence, defines a compact nonlinear operator between some appropriate Banach spaces. Therefore, the abstract results used there are not applicable to Ž1.1.. We extend the result of w17x and allow the nonlinearity g to depend on the highest derivative x Ž m.. In our case, the nonlinearity g defines a k-set contractive operator between some * E-mail address: [email protected]. † E-mail address: [email protected]. 481 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
482
LIU AND MAO
Banach spaces. Our method is based on the continuation theory for k-set contractions w7x. As in w17x, the interest of our conditions lies in the possibility of proving an existence theorem for the problem Ž1.1. without needing an assumption on the growth of g for x G 0 or else for x F 0. In w14x and w16x, the authors studied the similar nonlinear periodic boundary value problems and allowed the nonlinearity g to depend on the highest derivative of x Ž t .. However, our conditions on g in this paper are different from theirs. To show the existence of solutions to the considered problems we will use the continuation theory for k-set contractions w7, 10x. Our method in this direction relies on an abstract theorem developed in w16x and a priori bounds on solutions. We will state this abstract theorem in Section 2.
2. ABSTRACT EXISTENCE THEOREMS In this section we will briefly state the part of the abstract continuation theory for k-set contractions that will be used in our study of Eq. Ž1.1.. Let Z be a Banach space. For a bounded subset A ; Z, let GZ Ž A. denote the ŽKuratovski. measure of non-compactness defined by GZ Ž A . s inf d ) 0 : ' a finite number of subsets A i ; A, A s Di A i , diam Ž A i . F d 4 . Ž 2.1. Here, diamŽ A i . denotes the maximum distance between the points in the set A i . Let X and Y be Banach spaces and V a bounded open subset of X. A continuous and bounded map N : V ª Y is called k-set-contractive if for any bounded A ; V we have GY Ž N Ž A . . F kGX Ž A . .
Ž 2.2.
Also, for a continuous and bounded map T : X ª Y we define l Ž T . s sup r G 0 : ; bounded subset A ; X , r GX Ž A . F GY Ž T Ž A . . 4 .
Ž 2.3. Now, let L : X ª Y be a Fredholm operator of index zero, and N : V ª Y be k-set-contractive with k - l Ž L.. Using the approach of Mawhin’s, it was shown by Hetzer w10x that if Lx / Nx for all x g V, then one can associate with the pair Ž L, N . a topological degree DwŽ L, N ., Q x which has most of the important properties of the so-called Leray-Schauder degree. In particular, it has a homotopy invariance property that allows one to prove the following THEOREM 2.1 w16x. Let L : X ª Y be a Fredholm operator of index zero, and y g Y be a fixed point. Suppose that N : V ª Y is k-set-contracti¨ e with
483
PERIODIC SOLUTIONS
k - l Ž L. where V ; X is bounded, open, and symmetric about 0 g V. Suppose further that:
V,
ŽA. Lx / l Nx q l y, for x g V, l g Ž0, 1., and ŽB. w QNŽ x . q Qy, x x ? w QNŽyx . q Qy, x x - 0, for x g KerŽ L. l
where w , x is some bilinear form on Y = X and Q is the projection of Y onto cokerŽ L.. Then there exists x g V such that Lx y Nx s y.
3. MAIN RESULTS Let CT0 denote the linear space of real valued continuous T-periodic functions on R. The linear space CT0 is a Banach space with the usual norm for x g CT0 given by < x < 0 s max t g R < x Ž t .<. Let CTm Ž m G 1. denote the linear space of T-periodic functions with m continuous derivatives. CTm is a Banach space with norm < x < m s max< x Ž i. < 0 : 0 F i F m4 . Let X s CTm and Y s CT0 and let L : X ª Y be given by L Ž x . s x Ž m. q a my 1 x Ž my1. q ??? qa1 xX . It is obvious that L is a bounded linear map. Next define a Žnonlinear. map N : X ª Y by N Ž x . Ž t . s yg Ž t , x Ž t . , xX Ž t . , xY Ž t . , . . . , x Ž m. Ž t . . . Now, the problem Ž1.1. has a solution x Ž t . if and only if Lx y Nx s f for some x g X. We put the following conditions on g and f. They are similar to the ones contained in w17x. ŽH3.1. g : R = R mq 1 ª R is continuous and T-periodic ŽT ) 0. in its first variable. ŽH3.2. There exist measurable functions mq, my : R ª R j "`4 such that
mq Ž t . F lim inf g Ž t , x, x 1 , x 2 , . . . , x m . ,
t g R,
my Ž t . G lim sup g Ž t , x, x 1 , x 2 , . . . , x m . ,
tgR
xªq` xªy`
uniformly for Ž x 1 , x 2 , . . . , x m . g R m . ŽH3.3. There exist constants c1 , c 2 g R, such that g Ž t , x, x 1 , x 2 , . . . , x m . G c1
for x G 0, Ž t , x 1 , x 2 , . . . , x m . g R = R m
484
LIU AND MAO
and g Ž t , x, x 1 , x 2 , . . . , x m . F c2 THEOREM 3.1.
for x F 0, Ž t , x 1 , x 2 , . . . , x m . g R = R m .
Let ŽH3.1. ] ŽH3.3. be satisfied and assume
Ža. The only T-periodic solutions to the equation Lx s 0 are the constants. Žb. There exists a k g w0, 1., such that g Ž t , x, x 1 , . . . , x my1 , p . y g Ž t , x, x 1 , . . . , x my1 , q . F k < p y q < for any Ž t, x, x 1 , . . . , x my1 , p ., Ž t, x, x 1 , . . . , x my1 , q . g R = R mq 1. Žc. There exist positi¨ e constants p 0 , p1 , p 2 , . . . , pm , p such that g Ž t , x, x 1 , x 2 . F g Ž t , x, x 1 , x 2 . q p 0 < x < q p1 < x 1 < q p 2 < x 2 < q ??? qpm < x m < q p, ;Ž t, x, x 1 , x 2 , . . . , x m . g R = R mq 1, or g Ž t , x, x 1 , x 2 . F yg Ž t , x, x 1 , x 2 . q p 0 < x < q p1 < x 1 < q p 2 < x 2 < q ??? qpm < x m < q p ;Ž t, x, x 1 , x 2 , . . . , x m . g R = R mq 1. Žd. There exists T
H0 m
y
T
T
Ž t . dt - H f Ž t . dt - H mq Ž t . dt. 0
0
Then there exists an d ) 0 such that when max p 0 , p1 , p 2 , . . . , pm 4 - d , the problem Ž1.1. has a solution. Notice that if g is non-negative or nonpositive then our key condition Žc. in the above theorem is automatically satisfied. Before proving Theorem 3.1, we need the following lemmas. LEMMA 3.2.
L is a Fredholm map of index 0 and satisfies l Ž L . G 1.
Proof. It is easy to verify that L is a Fredholm map of index 0 due to the condition Ža. of Theorem 3.1. In fact, for y g Y we define QŽ y . s
1
T
H y Ž t . dt. T 0
PERIODIC SOLUTIONS
485
Then ImŽ L. : kerŽ Q .. Applying the L2 theory of Fourier series to the equation Lx s y, we also can see that ImŽ L. = KerŽ Q .. Therefore, ImŽ L. is closed and dim kerŽ L. s codim ImŽ L. s 1. Let A ; X be a bounded subset and let h s GY Ž LŽ A.. ) 0. Given e ) 0, according to the definition, there is a finite number of subsets A i of A such that diam 0 Ž LŽ A i . F qe . Since X is compactly embedded into CTmy 1 and since A i are bounded in X, it follows that there is a finite number of subsets A i j of A i such that diam my 1Ž A i j . - e , and hence, diam mŽ A i j . F h q e q mae , where a s max 1 F iF my1< a i <4 and diam k Ž?. are defined with respect to the norms < ? < m , 0 F k F m. This proves GX Ž A . F h s GY Ž L Ž A . . , that is, l Ž L. G 1. LEMMA 3.3. N : X ª Y is a k-set-contracti¨ e map with k - 1 as gi¨ en in condition Žb. of Theorem 3.1. Proof. Let A ; X be a bounded subset and let h s GX Ž A.. Then for any e ) 0, there is a finite family of subsets A i 4 with A s Di A i and diam mŽ A i . F h q e . Now it follows from the fact that g is uniformly continuous on any compact subset of R = R mq 1 , and from the fact that A and A i are precompact in CTmy 1 with norm < ? < my 1 , that there is a finite family of subsets A i j 4 of A i such that A i s Dj A i j with g Ž t , x Ž t . , xX Ž t . , . . . , x Ž my1. Ž t . , uŽ m. Ž t . . yg Ž t , u Ž t . , uX Ž t . , . . . , uŽ my1. Ž t . , uŽ m. Ž t . . - e for any x, u g A i j . Therefore, for x, u g A i j we have 5 Nx y Nu 5 0 s sup
g Ž t , x, xX , . . . , x Ž my1. , x Ž m. .
0FtF1
yg Ž t , u, uX , . . . , uŽ my1. , uŽ m. . F sup
g Ž t , x, xX , . . . , x Ž my1. , x Ž m. .
0FtF1
yg Ž t , x, xX , . . . , x Ž my1. , uŽ m. . q sup
g Ž t , x, xX , . . . , x Ž my1. , uŽ m. .
0FtF1
yg Ž t , u, uX , . . . , uŽ my1. , uŽ m. . F k 5 x Ž m. y uŽ m. 5 0 q e F kh q Ž k q 1 . e . That is, GY Ž N Ž A . . F kGX Ž A . .
486
LIU AND MAO
The next lemma is from w17x. LEMMA 3.4. Under the assumption Ža. of Theorem 3.1, there is a constant m ) 0 such that my1
Ý
< x Ž i. < 0 q
is1
T
H0
x Ž m. Ž t . dt F m
T
H0
Lx Ž t . dt
for all x g CmT . Proof. See w17x. LEMMA 3.5. There is a number r 0 , such that for each solution x Ž t . to Lx q l Nx s l f, 0 - l - 1, there is a z g w0, T x, with < x Ž z .< F r 0 . Here z may depend on x Ž t . and l. Proof. The proof may be found in w17x but for the sake of completeness we give the proof here. Suppose that for each positive integer n there is a l n g Ž0, 1. and a solution x n of Lx q l n Nx s l n f˜ with x nŽ t . G n for t g w0, T x. Then we would have T
T
H0 Nx Ž t . dt s H0 f Ž t . dt. n
In other words, T
X n
H0 g Ž t , x Ž t . , x Ž t . , . . . , x n
Ž m. n
T
Ž t . . dt s H f Ž t . dt. 0
On the other hand, we also have lim nª` inf g Ž t, x nŽ t ., xXnŽ t ., . . . , x nŽ m. Ž t .. G mq Ž t .. Now, using this and Fatou’s lemma, we get T
T
H0 f Ž t . dt G H0 m
q
Ž t . dt
contradicting condition Žd.. Thus there is a number r 1 such that if x is a ˜ l g Ž0, 1. then there is a number s1 g w0, 1x solution of Lx q l Nx s l f, Ž . with x s1 F r 1. Similarly, by using my and Fatou’s lemma we can show that there must be a number r 2 ) 0 such that for any solution x there is a corresponding value s2 g w0, 1x with x Ž s2 . G yr 2 . By continuity we conclude that for any solution x there is some z x g w0, 1x with < x Ž z x .< F r 0 , r 0 s max r 1 , r 2 4 . LEMMA 3.6. There exist numbers M, d ) 0 such that if conditions of Theorem 3.1 hold and max p 0 , p1 , p 2 , . . . , pm 4 - d then e¨ ery solution x Ž t . of the problem Lx y l Nx s l f , satisfies < x < m F M.
l g Ž 0, 1 .
487
PERIODIC SOLUTIONS
Proof of Lemma 3.6. Let Lx y l Nx s l f for x Ž t . g X, i.e., L Ž x . Ž t . q l g Ž t , x Ž t . , xX Ž t . , . . . , x Ž m. Ž t . . s l f .
Ž 3.1.
Integrating this identity we have T
X
H0 g Ž t , x Ž t . , x Ž t . , . . . , x
Ž m.
T
Ž t . . dt s H f Ž t . dt.
Ž 3.2.
0
Using the condition Žc. of Theorem 3.1, Ž3.1., and Ž3.2. we have T
H0
L Ž x . Ž t . dt F
T
g Ž t , x Ž t . , xX Ž t . , . . . , x Ž m. Ž t . . dt q
H0
F2
T
H0
my1
Ý
f Ž t . dt
f Ž t . dt q p 0 T < x < 0 q p1T < x < 0 q ???
q pmy 1T < x Ž my1. < 0 q pm FT
T
H0
T
H0
x Ž m. Ž t . dt q pT
pi < x i < 0 q p 0 T < x < 0 q q pm
is1
T
H0
x m Ž t . dt q ˜ p,
where ˜ p s pT q 2H0T < f < is a constant. Combining Lemma 3.4 and the above inequality, we get my1
T
Ý Ž 1 y Tm pi . < x i < 0 q Ž 1 y m pm . H
0
is1
x m Ž t . dt F p 0 Tm < x < 0 q m ˜ p. Ž 3.3.
On the other hand, according to Lemma 3.5, there exist z x g w0, T x such that < x Ž z x .< F r 0 . Therefore, < x < 0 s sup
xŽ t.
0FtFT
s sup
xŽ zx . q
0FtFT
t X
Hz x Ž t . dt x
X
F r0 q T < x < 0 . Combining Ž3.3. and Ž3.4., we have my1
Ý Ž 1 y Tm pi . < x i < 0 q Ž 1 y Tm p1 y T 2m p0 . < xX < 0
is2
q Ž 1 y m pm .
T
H0
x m Ž t . dt F m ˜ p q r 0 Tm p 0 .
Ž 3.4.
488
LIU AND MAO
Therefore, there exist constants d ) 0 and M1 ) 0 such that, if pi F d ,
for all i s 0, 1, . . . , m,
then < x Ž i. < 0 F M1 ,
for i s 0, 1, . . . , m y 1.
Moreover, given such a solution of LŽ x .Ž t . q l g Ž t, x, xX , . . . , x Ž m. . s l f Ž t . we have x Ž m. Ž t . F g Ž t , x, xX , . . . , x Ž m. . q f Ž t . q
my1
Ý
< a i < < x Ž i. < 0
is1
F g Ž t , x, xX , . . . , x Ž m. . y g Ž t , x, xX , . . . , 0 . q g Ž t , x, xX , . . . , 0 . q f Ž t . q Ž m y 1 .
max 1FiFmy1
< a i < M1
F k x Ž m. Ž t . q g Ž t , x, xX , . . . , 0 . q f Ž t . q Ž m y 1.
max 1FiFmy1
< a i < M1
F k x Ž m. Ž t . q M2 . Here, M2 is some constant. From this we see that < x Ž m. < 0 F M2rŽ1 y k . so that if we let M s max M1 , M2rŽ1 y k .4 then < x
T
H0 Ž g Ž t , r , 0, . . . , 0. y f Ž t . . dt T
H0 Ž g Ž t , yr , 0, . . . , 0. y f Ž t . . dt.
˜ ) 0 such that if r ) M˜ then By condition Žd., there is a number M T
T
H0 Ž g Ž t , r , 0, . . . , 0. y f Ž t . . dt ? H0 Ž g Ž t , yr , 0, . . . , 0. y f Ž t . . dt - 0.
489
PERIODIC SOLUTIONS
˜ 4 then all of the conditions required in Thus if we pick r ) max M, M Theorem 2.1 hold. It now follows by Theorem 2.1 that there is a function x Ž t . g X, such that Lx y Nx s f . This finishes the proof of Theorem 3.1. EXAMPLE 3.7. Let g Ž t, x, x 1 , . . . , x m . s h1Ž t, x 1 , . . . , x m . eyx q h 2 Ž t, x 1 , . . . , x my 1 . e x , here h1 G 0 and h 2 ) 0 are bounded continuous T-periodic functions in their first variable and h1 satisfies 2
sup
xm
h1 Ž t , x, x 1 , . . . , x m . - 1.
ŽFor example, h1Ž t, x, x 1 , . . . , x m . s differential equation
1 4
sin 2 Ž x m . eyx .. By Theorem 3.1, the 2
x Ž m. q h1 Ž t , xX , . . . , x Ž m. . eyx q h 2 Ž t , xX , . . . , x Ž my1. . e x s f Ž t . 2
has solutions provided H0T f Ž t . dt ) 0. REFERENCES 1. P. B. Bailey, L. F. Shampine, and P. E. Waltman, ‘‘Nonlinear Two-Point Boundary Value Problems,’’ Academic Press, New York, 1968. 2. S. R. Bernfeld and V. Lakshmikantham, ‘‘An Introduction to Nonlinear Boundary Value Problems,’’ Academic Press, New York, 1974. 3. F. E. Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mapping in Banach Spaces, J. Funct. Anal. 3 Ž1969.. 4. E. N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc. 15 Ž1976., 321]328. 5. K. Deimling and V. Lakshmikantham, Periodic sample solutions of second order ODEs, Nonlinear Anal. 10 Ž1986., 403]409. 6. L. H. Erbe, Existence of solutions to boundary value problems for second order differential equations, Nonlinear Anal. 6 Ž1982., 1155]1162. 7. R. E. Gaines and J. L. Mawhin, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin, 1977. 8. A. Granas, R. B. Guenther, and J. W. Lee, Nonlinear boundary value problem for some classes of ordinary differential equations, Rocky Mountain J. Math. 10 Ž1980., 35]58. 9. A. Granas, R. B. Guenther, J. W. Lee, Nonlinear boundary value problem for ordinary differential equations, Diss. Math. 244 Ž1985., 1]128. 10. G. Hetzer, Some remarks on Fq operators and on the coincidence degree for Fredholm equations with noncompact nonlinear perturbations, Ann. Soc. Sci. Bruxelles 89, No. IV Ž1975.. 11. P. Kelevedjiev, Existence of solutions for two-point boundary value problems, Nonlinear Anal. 22, No. 2 Ž1994., 217]224.
490
LIU AND MAO
12. V. Lakshmikantham, J. J. Nieto, and Y. Sun, A existence result about periodic boundary value problems of second order differential equations, Appl. Anal. 40 Ž1991., 1]10. 13. A. C. Lazer, On Schauder’s fixed point theorem and forced second-order nonlinear oscillations, J. Math. Anal. Appl. 21 Ž1968., 421]425. 14. W. V. Petryshyn, ‘‘Approximation-Solvability of Nonlinear Functional and Differential Equations,’’ Dekker, New York, 1993. 15. W. V. Petryshyn and Z. S. Yu, Nonresonance and existence for nonlinear BV problems, Houston J. Math. 9 Ž1982.. 16. W. V. Petryshyn and Z. S. Yu, Existence theorems for higher order nonlinear periodic boundary value problems, Nonlinear Anal. 6, No. 9 Ž1982.. 17. J. Ward, Jr., Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc. 81, No. 3 Ž1981.. 18. S. Umamaheswarman and M. V. S. Suhasini, A global existence theorem for the Y X boundary value problem of y s f Ž x, y, y ., Nonlinear Anal. 10 Ž1986., 679]681.
216, 481]490 Ž1997.
AY975669
Existence Theorem for Periodic Solutions of Higher Order Nonlinear Differential Equations Zhongdong LiuU and Yiping Mao† Department of Mathematics, Uni¨ ersity of South Carolina, Columbia, South Carolina 29208 Submitted by Zhi¨ ko S. Athanasso¨ Received July 15, 1996
We study the existence of periodic solutions to differential equations of the form X X LŽ x . q g Ž t, x, x , . . . , x Ž m. . s f Ž t . with LŽ x . s x Ž m. q a my 1 x Ž my1. q ??? qa1 x . Q 1997 Academic Press
1. INTRODUCTION The purpose of this paper is to establish the existence of periodic solutions to the nonlinear differential equation x Ž m. q a my 1 x Ž my1. q ??? qa1 xX q g Ž t , x, xX , . . . , x Ž m. . s f Ž t . , Ž 1.1. where a1 , a2 , . . . , a my1 are constants, g : R = R mq 1 ª R is continuous and T-periodic ŽT ) 0. in its first variable, and f Ž t . is a continuous T-periodic function. This and similar types of problems have recently received considerable attention. ŽSee w2, 4, 6, 7, 9]12, 14]18x, etc.. In most known existence results, the nonlinearity g depends at most on the lower order derivatives xX , xY , . . . , and x Ž my1. and, hence, defines a compact nonlinear operator between some appropriate Banach spaces. Therefore, the abstract results used there are not applicable to Ž1.1.. We extend the result of w17x and allow the nonlinearity g to depend on the highest derivative x Ž m.. In our case, the nonlinearity g defines a k-set contractive operator between some * E-mail address: [email protected]. † E-mail address: [email protected]. 481 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
482
LIU AND MAO
Banach spaces. Our method is based on the continuation theory for k-set contractions w7x. As in w17x, the interest of our conditions lies in the possibility of proving an existence theorem for the problem Ž1.1. without needing an assumption on the growth of g for x G 0 or else for x F 0. In w14x and w16x, the authors studied the similar nonlinear periodic boundary value problems and allowed the nonlinearity g to depend on the highest derivative of x Ž t .. However, our conditions on g in this paper are different from theirs. To show the existence of solutions to the considered problems we will use the continuation theory for k-set contractions w7, 10x. Our method in this direction relies on an abstract theorem developed in w16x and a priori bounds on solutions. We will state this abstract theorem in Section 2.
2. ABSTRACT EXISTENCE THEOREMS In this section we will briefly state the part of the abstract continuation theory for k-set contractions that will be used in our study of Eq. Ž1.1.. Let Z be a Banach space. For a bounded subset A ; Z, let GZ Ž A. denote the ŽKuratovski. measure of non-compactness defined by GZ Ž A . s inf d ) 0 : ' a finite number of subsets A i ; A, A s Di A i , diam Ž A i . F d 4 . Ž 2.1. Here, diamŽ A i . denotes the maximum distance between the points in the set A i . Let X and Y be Banach spaces and V a bounded open subset of X. A continuous and bounded map N : V ª Y is called k-set-contractive if for any bounded A ; V we have GY Ž N Ž A . . F kGX Ž A . .
Ž 2.2.
Also, for a continuous and bounded map T : X ª Y we define l Ž T . s sup r G 0 : ; bounded subset A ; X , r GX Ž A . F GY Ž T Ž A . . 4 .
Ž 2.3. Now, let L : X ª Y be a Fredholm operator of index zero, and N : V ª Y be k-set-contractive with k - l Ž L.. Using the approach of Mawhin’s, it was shown by Hetzer w10x that if Lx / Nx for all x g V, then one can associate with the pair Ž L, N . a topological degree DwŽ L, N ., Q x which has most of the important properties of the so-called Leray-Schauder degree. In particular, it has a homotopy invariance property that allows one to prove the following THEOREM 2.1 w16x. Let L : X ª Y be a Fredholm operator of index zero, and y g Y be a fixed point. Suppose that N : V ª Y is k-set-contracti¨ e with
483
PERIODIC SOLUTIONS
k - l Ž L. where V ; X is bounded, open, and symmetric about 0 g V. Suppose further that:
V,
ŽA. Lx / l Nx q l y, for x g V, l g Ž0, 1., and ŽB. w QNŽ x . q Qy, x x ? w QNŽyx . q Qy, x x - 0, for x g KerŽ L. l
where w , x is some bilinear form on Y = X and Q is the projection of Y onto cokerŽ L.. Then there exists x g V such that Lx y Nx s y.
3. MAIN RESULTS Let CT0 denote the linear space of real valued continuous T-periodic functions on R. The linear space CT0 is a Banach space with the usual norm for x g CT0 given by < x < 0 s max t g R < x Ž t .<. Let CTm Ž m G 1. denote the linear space of T-periodic functions with m continuous derivatives. CTm is a Banach space with norm < x < m s max< x Ž i. < 0 : 0 F i F m4 . Let X s CTm and Y s CT0 and let L : X ª Y be given by L Ž x . s x Ž m. q a my 1 x Ž my1. q ??? qa1 xX . It is obvious that L is a bounded linear map. Next define a Žnonlinear. map N : X ª Y by N Ž x . Ž t . s yg Ž t , x Ž t . , xX Ž t . , xY Ž t . , . . . , x Ž m. Ž t . . . Now, the problem Ž1.1. has a solution x Ž t . if and only if Lx y Nx s f for some x g X. We put the following conditions on g and f. They are similar to the ones contained in w17x. ŽH3.1. g : R = R mq 1 ª R is continuous and T-periodic ŽT ) 0. in its first variable. ŽH3.2. There exist measurable functions mq, my : R ª R j "`4 such that
mq Ž t . F lim inf g Ž t , x, x 1 , x 2 , . . . , x m . ,
t g R,
my Ž t . G lim sup g Ž t , x, x 1 , x 2 , . . . , x m . ,
tgR
xªq` xªy`
uniformly for Ž x 1 , x 2 , . . . , x m . g R m . ŽH3.3. There exist constants c1 , c 2 g R, such that g Ž t , x, x 1 , x 2 , . . . , x m . G c1
for x G 0, Ž t , x 1 , x 2 , . . . , x m . g R = R m
484
LIU AND MAO
and g Ž t , x, x 1 , x 2 , . . . , x m . F c2 THEOREM 3.1.
for x F 0, Ž t , x 1 , x 2 , . . . , x m . g R = R m .
Let ŽH3.1. ] ŽH3.3. be satisfied and assume
Ža. The only T-periodic solutions to the equation Lx s 0 are the constants. Žb. There exists a k g w0, 1., such that g Ž t , x, x 1 , . . . , x my1 , p . y g Ž t , x, x 1 , . . . , x my1 , q . F k < p y q < for any Ž t, x, x 1 , . . . , x my1 , p ., Ž t, x, x 1 , . . . , x my1 , q . g R = R mq 1. Žc. There exist positi¨ e constants p 0 , p1 , p 2 , . . . , pm , p such that g Ž t , x, x 1 , x 2 . F g Ž t , x, x 1 , x 2 . q p 0 < x < q p1 < x 1 < q p 2 < x 2 < q ??? qpm < x m < q p, ;Ž t, x, x 1 , x 2 , . . . , x m . g R = R mq 1, or g Ž t , x, x 1 , x 2 . F yg Ž t , x, x 1 , x 2 . q p 0 < x < q p1 < x 1 < q p 2 < x 2 < q ??? qpm < x m < q p ;Ž t, x, x 1 , x 2 , . . . , x m . g R = R mq 1. Žd. There exists T
H0 m
y
T
T
Ž t . dt - H f Ž t . dt - H mq Ž t . dt. 0
0
Then there exists an d ) 0 such that when max p 0 , p1 , p 2 , . . . , pm 4 - d , the problem Ž1.1. has a solution. Notice that if g is non-negative or nonpositive then our key condition Žc. in the above theorem is automatically satisfied. Before proving Theorem 3.1, we need the following lemmas. LEMMA 3.2.
L is a Fredholm map of index 0 and satisfies l Ž L . G 1.
Proof. It is easy to verify that L is a Fredholm map of index 0 due to the condition Ža. of Theorem 3.1. In fact, for y g Y we define QŽ y . s
1
T
H y Ž t . dt. T 0
PERIODIC SOLUTIONS
485
Then ImŽ L. : kerŽ Q .. Applying the L2 theory of Fourier series to the equation Lx s y, we also can see that ImŽ L. = KerŽ Q .. Therefore, ImŽ L. is closed and dim kerŽ L. s codim ImŽ L. s 1. Let A ; X be a bounded subset and let h s GY Ž LŽ A.. ) 0. Given e ) 0, according to the definition, there is a finite number of subsets A i of A such that diam 0 Ž LŽ A i . F qe . Since X is compactly embedded into CTmy 1 and since A i are bounded in X, it follows that there is a finite number of subsets A i j of A i such that diam my 1Ž A i j . - e , and hence, diam mŽ A i j . F h q e q mae , where a s max 1 F iF my1< a i <4 and diam k Ž?. are defined with respect to the norms < ? < m , 0 F k F m. This proves GX Ž A . F h s GY Ž L Ž A . . , that is, l Ž L. G 1. LEMMA 3.3. N : X ª Y is a k-set-contracti¨ e map with k - 1 as gi¨ en in condition Žb. of Theorem 3.1. Proof. Let A ; X be a bounded subset and let h s GX Ž A.. Then for any e ) 0, there is a finite family of subsets A i 4 with A s Di A i and diam mŽ A i . F h q e . Now it follows from the fact that g is uniformly continuous on any compact subset of R = R mq 1 , and from the fact that A and A i are precompact in CTmy 1 with norm < ? < my 1 , that there is a finite family of subsets A i j 4 of A i such that A i s Dj A i j with g Ž t , x Ž t . , xX Ž t . , . . . , x Ž my1. Ž t . , uŽ m. Ž t . . yg Ž t , u Ž t . , uX Ž t . , . . . , uŽ my1. Ž t . , uŽ m. Ž t . . - e for any x, u g A i j . Therefore, for x, u g A i j we have 5 Nx y Nu 5 0 s sup
g Ž t , x, xX , . . . , x Ž my1. , x Ž m. .
0FtF1
yg Ž t , u, uX , . . . , uŽ my1. , uŽ m. . F sup
g Ž t , x, xX , . . . , x Ž my1. , x Ž m. .
0FtF1
yg Ž t , x, xX , . . . , x Ž my1. , uŽ m. . q sup
g Ž t , x, xX , . . . , x Ž my1. , uŽ m. .
0FtF1
yg Ž t , u, uX , . . . , uŽ my1. , uŽ m. . F k 5 x Ž m. y uŽ m. 5 0 q e F kh q Ž k q 1 . e . That is, GY Ž N Ž A . . F kGX Ž A . .
486
LIU AND MAO
The next lemma is from w17x. LEMMA 3.4. Under the assumption Ža. of Theorem 3.1, there is a constant m ) 0 such that my1
Ý
< x Ž i. < 0 q
is1
T
H0
x Ž m. Ž t . dt F m
T
H0
Lx Ž t . dt
for all x g CmT . Proof. See w17x. LEMMA 3.5. There is a number r 0 , such that for each solution x Ž t . to Lx q l Nx s l f, 0 - l - 1, there is a z g w0, T x, with < x Ž z .< F r 0 . Here z may depend on x Ž t . and l. Proof. The proof may be found in w17x but for the sake of completeness we give the proof here. Suppose that for each positive integer n there is a l n g Ž0, 1. and a solution x n of Lx q l n Nx s l n f˜ with x nŽ t . G n for t g w0, T x. Then we would have T
T
H0 Nx Ž t . dt s H0 f Ž t . dt. n
In other words, T
X n
H0 g Ž t , x Ž t . , x Ž t . , . . . , x n
Ž m. n
T
Ž t . . dt s H f Ž t . dt. 0
On the other hand, we also have lim nª` inf g Ž t, x nŽ t ., xXnŽ t ., . . . , x nŽ m. Ž t .. G mq Ž t .. Now, using this and Fatou’s lemma, we get T
T
H0 f Ž t . dt G H0 m
q
Ž t . dt
contradicting condition Žd.. Thus there is a number r 1 such that if x is a ˜ l g Ž0, 1. then there is a number s1 g w0, 1x solution of Lx q l Nx s l f, Ž . with x s1 F r 1. Similarly, by using my and Fatou’s lemma we can show that there must be a number r 2 ) 0 such that for any solution x there is a corresponding value s2 g w0, 1x with x Ž s2 . G yr 2 . By continuity we conclude that for any solution x there is some z x g w0, 1x with < x Ž z x .< F r 0 , r 0 s max r 1 , r 2 4 . LEMMA 3.6. There exist numbers M, d ) 0 such that if conditions of Theorem 3.1 hold and max p 0 , p1 , p 2 , . . . , pm 4 - d then e¨ ery solution x Ž t . of the problem Lx y l Nx s l f , satisfies < x < m F M.
l g Ž 0, 1 .
487
PERIODIC SOLUTIONS
Proof of Lemma 3.6. Let Lx y l Nx s l f for x Ž t . g X, i.e., L Ž x . Ž t . q l g Ž t , x Ž t . , xX Ž t . , . . . , x Ž m. Ž t . . s l f .
Ž 3.1.
Integrating this identity we have T
X
H0 g Ž t , x Ž t . , x Ž t . , . . . , x
Ž m.
T
Ž t . . dt s H f Ž t . dt.
Ž 3.2.
0
Using the condition Žc. of Theorem 3.1, Ž3.1., and Ž3.2. we have T
H0
L Ž x . Ž t . dt F
T
g Ž t , x Ž t . , xX Ž t . , . . . , x Ž m. Ž t . . dt q
H0
F2
T
H0
my1
Ý
f Ž t . dt
f Ž t . dt q p 0 T < x < 0 q p1T < x < 0 q ???
q pmy 1T < x Ž my1. < 0 q pm FT
T
H0
T
H0
x Ž m. Ž t . dt q pT
pi < x i < 0 q p 0 T < x < 0 q q pm
is1
T
H0
x m Ž t . dt q ˜ p,
where ˜ p s pT q 2H0T < f < is a constant. Combining Lemma 3.4 and the above inequality, we get my1
T
Ý Ž 1 y Tm pi . < x i < 0 q Ž 1 y m pm . H
0
is1
x m Ž t . dt F p 0 Tm < x < 0 q m ˜ p. Ž 3.3.
On the other hand, according to Lemma 3.5, there exist z x g w0, T x such that < x Ž z x .< F r 0 . Therefore, < x < 0 s sup
xŽ t.
0FtFT
s sup
xŽ zx . q
0FtFT
t X
Hz x Ž t . dt x
X
F r0 q T < x < 0 . Combining Ž3.3. and Ž3.4., we have my1
Ý Ž 1 y Tm pi . < x i < 0 q Ž 1 y Tm p1 y T 2m p0 . < xX < 0
is2
q Ž 1 y m pm .
T
H0
x m Ž t . dt F m ˜ p q r 0 Tm p 0 .
Ž 3.4.
488
LIU AND MAO
Therefore, there exist constants d ) 0 and M1 ) 0 such that, if pi F d ,
for all i s 0, 1, . . . , m,
then < x Ž i. < 0 F M1 ,
for i s 0, 1, . . . , m y 1.
Moreover, given such a solution of LŽ x .Ž t . q l g Ž t, x, xX , . . . , x Ž m. . s l f Ž t . we have x Ž m. Ž t . F g Ž t , x, xX , . . . , x Ž m. . q f Ž t . q
my1
Ý
< a i < < x Ž i. < 0
is1
F g Ž t , x, xX , . . . , x Ž m. . y g Ž t , x, xX , . . . , 0 . q g Ž t , x, xX , . . . , 0 . q f Ž t . q Ž m y 1 .
max 1FiFmy1
< a i < M1
F k x Ž m. Ž t . q g Ž t , x, xX , . . . , 0 . q f Ž t . q Ž m y 1.
max 1FiFmy1
< a i < M1
F k x Ž m. Ž t . q M2 . Here, M2 is some constant. From this we see that < x Ž m. < 0 F M2rŽ1 y k . so that if we let M s max M1 , M2rŽ1 y k .4 then < x
T
H0 Ž g Ž t , r , 0, . . . , 0. y f Ž t . . dt T
H0 Ž g Ž t , yr , 0, . . . , 0. y f Ž t . . dt.
˜ ) 0 such that if r ) M˜ then By condition Žd., there is a number M T
T
H0 Ž g Ž t , r , 0, . . . , 0. y f Ž t . . dt ? H0 Ž g Ž t , yr , 0, . . . , 0. y f Ž t . . dt - 0.
489
PERIODIC SOLUTIONS
˜ 4 then all of the conditions required in Thus if we pick r ) max M, M Theorem 2.1 hold. It now follows by Theorem 2.1 that there is a function x Ž t . g X, such that Lx y Nx s f . This finishes the proof of Theorem 3.1. EXAMPLE 3.7. Let g Ž t, x, x 1 , . . . , x m . s h1Ž t, x 1 , . . . , x m . eyx q h 2 Ž t, x 1 , . . . , x my 1 . e x , here h1 G 0 and h 2 ) 0 are bounded continuous T-periodic functions in their first variable and h1 satisfies 2
sup
xm
h1 Ž t , x, x 1 , . . . , x m . - 1.
ŽFor example, h1Ž t, x, x 1 , . . . , x m . s differential equation
1 4
sin 2 Ž x m . eyx .. By Theorem 3.1, the 2
x Ž m. q h1 Ž t , xX , . . . , x Ž m. . eyx q h 2 Ž t , xX , . . . , x Ž my1. . e x s f Ž t . 2
has solutions provided H0T f Ž t . dt ) 0. REFERENCES 1. P. B. Bailey, L. F. Shampine, and P. E. Waltman, ‘‘Nonlinear Two-Point Boundary Value Problems,’’ Academic Press, New York, 1968. 2. S. R. Bernfeld and V. Lakshmikantham, ‘‘An Introduction to Nonlinear Boundary Value Problems,’’ Academic Press, New York, 1974. 3. F. E. Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mapping in Banach Spaces, J. Funct. Anal. 3 Ž1969.. 4. E. N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc. 15 Ž1976., 321]328. 5. K. Deimling and V. Lakshmikantham, Periodic sample solutions of second order ODEs, Nonlinear Anal. 10 Ž1986., 403]409. 6. L. H. Erbe, Existence of solutions to boundary value problems for second order differential equations, Nonlinear Anal. 6 Ž1982., 1155]1162. 7. R. E. Gaines and J. L. Mawhin, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin, 1977. 8. A. Granas, R. B. Guenther, and J. W. Lee, Nonlinear boundary value problem for some classes of ordinary differential equations, Rocky Mountain J. Math. 10 Ž1980., 35]58. 9. A. Granas, R. B. Guenther, J. W. Lee, Nonlinear boundary value problem for ordinary differential equations, Diss. Math. 244 Ž1985., 1]128. 10. G. Hetzer, Some remarks on Fq operators and on the coincidence degree for Fredholm equations with noncompact nonlinear perturbations, Ann. Soc. Sci. Bruxelles 89, No. IV Ž1975.. 11. P. Kelevedjiev, Existence of solutions for two-point boundary value problems, Nonlinear Anal. 22, No. 2 Ž1994., 217]224.
490
LIU AND MAO
12. V. Lakshmikantham, J. J. Nieto, and Y. Sun, A existence result about periodic boundary value problems of second order differential equations, Appl. Anal. 40 Ž1991., 1]10. 13. A. C. Lazer, On Schauder’s fixed point theorem and forced second-order nonlinear oscillations, J. Math. Anal. Appl. 21 Ž1968., 421]425. 14. W. V. Petryshyn, ‘‘Approximation-Solvability of Nonlinear Functional and Differential Equations,’’ Dekker, New York, 1993. 15. W. V. Petryshyn and Z. S. Yu, Nonresonance and existence for nonlinear BV problems, Houston J. Math. 9 Ž1982.. 16. W. V. Petryshyn and Z. S. Yu, Existence theorems for higher order nonlinear periodic boundary value problems, Nonlinear Anal. 6, No. 9 Ž1982.. 17. J. Ward, Jr., Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc. 81, No. 3 Ž1981.. 18. S. Umamaheswarman and M. V. S. Suhasini, A global existence theorem for the Y X boundary value problem of y s f Ž x, y, y ., Nonlinear Anal. 10 Ž1986., 679]681.