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Differential Equations for Which the Cross-Ratio of Four Solutions Is Weakly Monotone
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
201, 588]599 Ž1996.
0275
Differential Equations for Which the Cross-Ratio of Four Solutions Is Weakly Monotone Kurt Munk Andersen and Allan Sandqvist Mathematical Institute, Technical Uni¨ ersity of Denmark, Building 303, DK-2800 Lyngby, Denmark Submitted by Jack K. Hale Received October 30, 1995
INTRODUCTION Consider the first order differential equation
˙x s f Ž t , x . ,
Ž t , x . g I = R,
Ž 1.
where I 9 R is an open interval and f Ž t, x ., Ž t, x . g I = R a continuous function Žit is not assumed that there locally is uniqueness of solutions of Eq. Ž1... If w i Ž t ., t g J, i s 1, 2, 3, 4 are four distinct solutions of Eq. Ž1. on the same interval J 9 I, their cross-ratio RŽ t ., t g J is defined by RŽ t . s
w 3 Ž t . y w 1Ž t . w4 Ž t . y w 1 Ž t .
:
w3Ž t . y w2 Ž t . w4 Ž t . y w 2 Ž t .
,
t g J.
Ž 2.
It is a well-known fact Žsee, e.g., w4, p. 274x. that the cross-ratio Ž2. is constant for all choices of the four solutions if Eq. Ž1. is a Riccati equation, i.e., an equation of the form
˙x s a2 Ž t . x 2 q a1 Ž t . x q a0 Ž t . ,
t g I,
Ž 3.
where a i Ž t ., t g I, i s 0, 1, 2 are continuous functions. It turns out that the converse statement is also true. This can be proved elementary Žsee Remark 2 after Theorem 3.. Now, instead of the explicit form Ž3., a Riccati equation can be characterized as an equation Ž1., for which the derivative f xX Ž t, x . exists and is continuous for all Ž t, x . g I = R and is linear in x for any fixed t g I. Thus, the cross-ratio Ž2. is constant for all choices of the 588 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
MONOTONE CROSS-RATIO
589
four solutions of Eq. Ž1. if and only if the derivative f xX Ž t, x . exists and is continuous for all Ž t, x . g I = R and is linear in x for any fixed t g I. In Section 2 it is proved that the cross-ratio Ž2. is weakly increasing Ž weakly decreasing . for all choices of the four solutions of Eq. Ž1. if and only if the derivative f xX Ž t, x . exists and is continuous for all Ž t, x . g I = R and is weakly convex Žweakly concave. in x for any fixed t g I. The ‘‘if’’ part of the mentioned result in Section 2 is already proved in w1, Theorem 1x. The ‘‘only if’’ part is based on a connection between the cross-ratio Ž2. and the second divided difference of the function x ª f Ž t, x ., x g R Ž t g I fixed. and on a theorem of Bullen on 2-convexity. These two ingredients are presented in Section 1. The two Appendices A and B contain proofs of some auxiliary theorems on real functions and sequences of real functions.
1. DIVIDED DIFFERENCES AND n-CONVEXITY Let F Ž x ., x g R be a real function. The nth divided difference F w x 1 , x 2 , . . . , x n x on n distinct real numbers x 1 , x 2 , . . . , x n of the function F Ž x . is defined recursively by F w x 1 x s F Ž x 1 . and F w x1 , x 2 , . . . , x n x s
F w x 1 , . . . , x ny1 x y F w x 2 , . . . , x n x x1 y x n
Ž 4.
if n G 2. The function F Ž x ., x g R is called n-convex Ž n-concave. if and only if F w x 1 , x 2 , . . . , x nq2 x G 0 ŽF 0. for all choices of the real numbers x 1 , x 2 , . . . , x nq2 . In this definition we may impose the restriction x 1 x 2 -???- x nq2 Žsee w2, Lemma 1x.. It is seen that a 1-convex Ž1-concave. function is what is usually called a weakly convex Žweakly concave. function. For later use we state two results of Bullen on 2-convex functions. THEOREM 1. If F Ž x ., x g R, is 2-con¨ ex, then F9Ž x . exists for all x g R and is a weakly con¨ ex Ž and hence continuous. function. For a proof, see w3, Theorem 7 and Corollary 15x. Note that there is a difference in notation: n-convexity in w3x corresponds with what we here call Ž n y 1.-convexity. Before stating the other result of Bullen we introduce the following notation. By P2 Ž x; x 1 , x 2 , x 3 ; F . is denoted the uniquely determined polynomial of degree at most two having the same values as the function F Ž x ., x g R in the points x 1 , x 2 , and x 3 Žthis polynomial is usually called the Lagrange interpolating polynomial..
590
ANDERSEN AND SANDQVIST
THEOREM 2. The function F Ž x ., x g R, is 2-con¨ ex if and only if for any x 1 , x 2 , x 3 g R with x 1 - x 2 - x 3 there hold x F x 1 k x 2 F x F x 3 « F Ž x . F P2 Ž x ; x 1 , x 2 , x 3 ; F . and x 1 F x F x 2 k x 3 F x « F Ž x . G P2 Ž x ; x 1 , x 2 , x 3 ; F . .
Ž 5.
For a proof see w3, Theorem 5x. Condition Ž5. expresses that the graph of F lies alternately below and abo¨ e the cur¨ e y s P2 Ž x; x 1 , x 2 , x 3 ; F . in the four inter¨ als, into which x 1 , x 2 , and x 3 subdi¨ ide R. LEMMA 1.
For any four distinct real numbers x 1 , x 2 , x 3 , x 4 there holds
Ž x 2 y x1 . Ž x 4 y x 3 . F w x1 , x 2 , x 3 , x 4 x F Ž x 3 . y F Ž x1 .
s
x 3 y x1 y
F Ž x4 . y F Ž x2 .
q
F Ž x 4 . y F Ž x1 . x 4 y x1
x4 y x2 F Ž x3 . y F Ž x2 .
y
.
x3 y x2
Ž 6.
Proof. From w3, Ž8. ] Ž10.x it follows that F w x1 , x 2 , x 3 , x 4 x s
4
F Ž xk .
Ý
.
4
ks1
Ž 7.
Ł Ž xk y x j . js1 j/k
The right hand side of Ž6. can be written F Ž x1 .
y1 x 3 y x1
q
qF Ž x 3 . s F Ž x1 .
1 x 4 y x1 1
x 3 y x1
q F Ž x2 .
q
y1 x3 y x2
x3 y x4
Ž x1 y x 3 . Ž x1 y x 4 .
q F Ž x3 .
y1 x4 y x2
q F Ž x4 .
q F Ž x2 .
x1 y x 2
Ž x 3 y x1 . Ž x 3 y x 2 .
q
x3 y x2 1
x4 y x2
q
y1 x 4 y x1
x4 y x3
Ž x2 y x4 . Ž x2 y x3 .
q F Ž x4 .
s Ž x 2 y x1 . Ž x 4 y x 3 . F w x1 , x 2 , x 3 , x 4 x . where the last equality sign comes from Ž7..
1
x 2 y x1
Ž x 4 y x 2 . Ž x 4 y x1 .
MONOTONE CROSS-RATIO
591
2. ON THE CROSS-RATIO OF FOUR SOLUTIONS OF EQ. Ž1. Let f t Ž x ., x g R denote the function defined by x ª f Ž t, x . for t g I fixed. THEOREM 3. The function f Ž t, x ., Ž t, x . g I = R on the right hand side of Eq. Ž1. has a continuous deri¨ ati¨ e f xX Ž t, x ., Ž t, x . g I = R which is weakly con¨ ex w weakly conca¨ e x in x for any fixed t g I if and only if for any four solutions w i Ž t ., t g J, i s 1, 2, 3, 4 of Eq. Ž1. on some common inter¨ al J 9 I such that w 1Ž t . - w 2 Ž t . - w 3 Ž t . - w4Ž t . for all t g J, the cross-ratio RŽ t . s
w 3 Ž t . y w 1Ž t .
:
w4 Ž t . y w 1 Ž t .
w3Ž t . y w2 Ž t . w4 Ž t . y w 2 Ž t .
tgJ
,
Ž 8.
is weakly increasing w weakly decreasing x. Proof. The ‘‘only if ’’ part follows from w1, Theorem 1x. Below we prove the ‘‘if ’’ part in the weakly convexrweakly increasing case. The weakly concaverweakly decreasing case is treated analogously. From Ž8. we get by differentiation, using Ž6. and suppressing the variable t under the w i-signs R9 Ž t . RŽ t .
s s
d dt
log R Ž t . s
w ˙3 y w˙1 w3 y w1
f Ž t , w3 . y f Ž t , w1 .
w3 y w1 y
q
f Ž t , w4 . y f Ž t , w 1 .
w4 y w 1
q
w ˙4 y w˙2 w4 y w 2
y
w ˙4 y w˙1 w4 y w 1
y
w ˙3 y w˙2 w3 y w2
f Ž t , w4 . y f Ž t , w 2 .
w4 y w 2 y
f Ž t , w3 . y f Ž t , w2 .
w3 y w2
s Ž w 2 y w 1 . Ž w4 y w 3 . f t w w 1 , w 2 , w 3 , w4 x ,
t g J.
Ž 9.
Suppose that RŽ t ., t g J is weakly increasing for any choice of the four solutions. From Ž9. we infer that f t w w 1Ž t ., w 2 Ž t ., w 3 Ž t ., w4Ž t .x G 0, t g J for any choice of the four solutions. Let t 0 g I and x i g R, i s 1, 2, 3, 4, x 1 - x 2 - x 3 - x 4 be arbitrary. Let w i Ž t . denote a solution of Eq. Ž1. through the point Ž t 0 , x i ., i s 1, 2, 3, 4. These four solutions can be chosen such that they are defined on some common interval J around t 0 , and w 1Ž t . - w 2 Ž t . - w 3 Ž t . - w4Ž t . for all t g J. Hence f t w w 1Ž t ., w 2 Ž t ., w 3 Ž t ., w4Ž t .x G 0 for all t g J, in particular for t s t 0 . This gives f t 0 w x 1 , x 2 , x 3 , x 4 x G 0. Hence f t 0Ž x ., x g R is 2-convex by definition. Theorem 1 implies that f xX Ž t 0 , x ., x g R exists and is weakly convex Žand
592
ANDERSEN AND SANDQVIST
hence continuous. }for any fixed t 0 g I. From Theorem A in Appendix A it follows that f xX Ž t, x ., Ž t, x . g I = R is a continuous function. COROLLARY. Equation Ž1. is a Riccati equation if and only if for any four solutions w i Ž t ., t g J of Eq. Ž1. on some common inter¨ al J 9 I such that w 1Ž t . - w 2 Ž t . - w 3 Ž t . - w4Ž t . for all t g J the cross-ratio Ž8. is constant. Remark 1. From Theorem 3 it follows that there locally is uniqueness of solutions of Eq. Ž1., if the cross-ratio of any four solutions is weakly increasing wweakly decreasingx. This can be proved directly as follows in the weakly increasing case. The weakly decreasing case is treated similarly. If there is not locally uniqueness, then there exist a point Ž t 0 , x 0 . g I = R and two solutions x 1Ž t . and x 2 Ž t . of Eq. Ž1., both defined in some interval x t 0 y d , t 0 q d w , such that x 1Ž t 0 . s x 2 Ž t 0 . s x 0 and either x 1Ž t . x 2 Ž t . for all t g x t 0 y d , t 0 w or x 1Ž t . - x 2 Ž t . for all t g x t 0 , t 0 q d w . Consider the first case. We may choose two other solutions w 3 Ž t . and w4Ž t . of Eq. Ž1. such that all four solutions are defined on some interval xt 0 y d 1 , t 0 x, 0 - d 1 F d and x 1Ž t . - x 2 Ž t . - w 3 Ž t . - w4Ž t . for all t g xt 0 y d 1 , t 0 w , and x 0 - w 3 Ž t 0 . - w4 Ž t 0 .. Then RŽ t . s
w 3 Ž t . y x1Ž t .
w3 Ž t . y x2 Ž t .
:
w4 Ž t . y x 1 Ž t .
w4 Ž t . y x 2 Ž t .
ª1
as t ª t 0 y ,
whence RŽ t . F 1 for all t g x t 0 y d 1 , t 0 w , i.e., w w 3 Ž t . y x 1Ž t .x w w4 Ž t . y x 2 Ž t .x F w w4Ž t . y x 1Ž t .x w w 3 Ž t . y x 2 Ž t .x for all t g x t 0 y d 1 , t 0 w . This gives the contradiction w4 Ž t . F w 3 Ž t . for all t g x t 0 y d 1 , t 0 w . Consider next the case x 1Ž t . - x 2 Ž t . for all t g x t 0 , t 0 q d w . Choose two other solutions c 1Ž t . and c4Ž t . of Eq. Ž1. such that all four solutions are defined on some interval w t 0 , t 0 q d 1w , 0 - d 1 F d and c 1Ž t . - x 1Ž t . - x 2 Ž t . - c4 Ž t . for all t g x t 0 , t 0 q d 1w , and c 1Ž t 0 . - x 0 - c4Ž t 0 .. Then RŽ t . s
x 2 Ž t . y c 1Ž t .
c4 Ž t . y c 1 Ž t .
:
x 2 Ž t . y x1Ž t .
c4 Ž t . y x 1 Ž t .
ª q`
as t ª t 0 q .
This contradicts that RŽ t ., t g x t 0 , t 0 q d 1w is weakly increasing. Remark 2. The ‘‘if ’’ part of the corollary can be proved elementary as follows. Let in Ž8. the solutions w 1Ž t ., w 3 Ž t ., and w4Ž t . be fixed and w 2 Ž t . s x Ž t . be arbitrary. When RŽ t . is constant we get
w ˙3 y w˙1 w3 y w1
q
w ˙4 y ˙x w4 y x
y
w ˙4 y w˙1 w4 y w 1
y
w ˙3 y ˙x w3 y x
s 0,
t g J.
MONOTONE CROSS-RATIO
593
Solving for ˙ x we obtain an equation on the form Ž3. on the interval J. This means that f Ž t, x . is a polynomial in x of degree at most two in the domain w 1Ž t . - x - w 3 Ž t ., t g J. Consequently to any point Ž t, x . g I = R there exists a neighborhood in which f xŽ nn. Ž t, x ., n s 1, 2, 3 exist and are continuous and f xZ3 Ž t, x . is identically zero. Hence these statements on the derivatives hold for all Ž t, x . g I = R. It follows that Eq. Ž1. has the form Ž3., i.e., it is a Riccati equation. Remark 3. If Eq. Ž1. is autonomous}say f Ž t, x . s F Ž x . }then f t 0 s F for any t 0 g I. From the proof of Theorem 3 it follows that F Ž x ., x g R is 2-convex provided that the cross-ratios are weakly increasing, which by the theorem itself will hold if F9Ž x ., x g R exists and is weakly convex. Hence, the converse statement of Theorem 1 is also true. Remark 4. From Remark 3 and Theorem A in Appendix A it follows that a continuous function f Ž t, x ., Ž t, x . g I = R has a continuous derivative f xX Ž t, x ., Ž t, x . g I = R which is weakly convex wweakly concavex in x for any fixed t g I if and only if it is 2-convex w2-concavex in x for any fixed t g I. Hence, Theorem 3 can also be formulated as a necessary and sufficient condition for the function f Ž t, x ., Ž t, x . g I = R to be 2-convex w2-concavex in x for any fixed t g I. In the remaining part of this section we assume that f Ž t, x ., Ž t, x . g I = R is 2-convex in x for any fixed t g I. Let w 1Ž t ., w 2 Ž t ., w 3 Ž t ., t g J denote three solutions of Eq. Ž1. on the same interval J 9 I and suppose that w 1Ž t . - w 2 Ž t . - w 3 Ž t . for all t g J. If x Ž t ., t g J is any solution of Eq. Ž1. on the same interval J, according to Ž5. in Theorem 2 there hold x Ž t . - w 1Ž t . k w 2 Ž t . - x Ž t . - w 3 Ž t . «˙ x Ž t . F P2 Ž x Ž t . ; w 1 Ž t . , w 2 Ž t . , w 3 Ž t . ; f t .
Ž 10 .
and
w 1Ž t . - x Ž t . - w 2 Ž t . k w 3 Ž t . - x Ž t . «˙ x Ž t . G P2 Ž x Ž t . ; w 1 Ž t . , w 2 Ž t . , w 3 Ž t . ; f t .
Ž 11 .
for all t g J. We may write P2 Ž x ; w 1 Ž t . , w 2 Ž t . , w 3 Ž t . ; f t . s a Ž t . x 2 q b Ž t . x q g Ž t . ,
t g J,
where a Ž t ., b Ž t ., g Ž t ., t g I are continuous functions. The two implications Ž10. and Ž11. indicate that the solution curves of Eq. Ž1. in the strip t g J can be compared to those of the Riccati equation
˙x s a Ž t . x 2 q b Ž t . x q g Ž t . ,
t g J,
Ž 12 .
594
ANDERSEN AND SANDQVIST
having x s w i Ž t ., t g J, i s 1, 2, 3 as solutions. From this it is easy to derive a stability result. Suppose namely that J s I > w0, 1x and that w i Ž t ., t g I, i s 1, 2, 3 are closed, i.e., w i Ž0. s w i Ž1., i s 1, 2, 3. It is well known that then any on the interval I defined solution of the corresponding Riccati equation Ž12. is also closed Žsee, e.g., w1, Lemma 2x.. From Ž10. and Ž11. it follows that w 1Ž t . and w 3 Ž t . are unstable and w 2 Ž t . is stable}unless Eq. Ž1. and Eq. Ž12. coincide. This is also proved in w2, Theorem 9x.
APPENDIX A THEOREM A. Let f Ž t, x ., Ž t, x . g I = R be a continuous function, where I 9 R is an open inter¨ al. Assume that the deri¨ ati¨ e f xX Ž t, x . exists for all Ž t, x . g I = R and that f xX Ž t, x . is weakly con¨ ex Ž and hence continuous. in x for any fixed t g I. Then f xX Ž t, x ., Ž t, x . g R is a continuous function. Proof. Let Ž t 0 , x 0 . g I = R be arbitrary and let ŽŽ t n , x n .. be some sequence in I = R converging to Ž t 0 , x 0 . as n ª q`. We prove that f xX Ž t n , x n . ª f xX Ž t 0 , x 0 . as n ª q`. From this the desired property follows. Choose the interval w a, b x and the number d ) 0 such that x 0 g x a, bw and w t 0 y d , t 0 q d x ; I. Without loss of generality we may assume that Ž t n , x n . g w t 0 y d , t 0 q d x = w a, b x for all n g N. Define the functions g n Ž x . s f Ž tn , x . ,
g Ž x . s f Ž t 0 , x . , x g w a, b x , n g N.
These functions are all C 1 and their derivatives are weakly convex. Our aim is to prove that g XnŽ x n . ª g 9Ž x 0 . as n ª q`. First we prove that g nŽ x . ª g Ž x . as n ª q`, uniformly for x g w a, b x. Let « ) 0 be arbitrary. Since f Ž t, x . is uniformly continuous for Ž t, x . g w t 0 y d , t 0 q d x = w a, b x, an h ) 0 can be chosen such that < f Ž t1 , x 1 . y f Ž t 2 , x 2 . < F «
if 5 Ž t 1 y t 2 , x 1 y x 2 . 5 F h
Ž A1.
provided that t 1 , t 2 g w t 0 y d , t 0 q d x and x 1 , x 2 g w a, b x. We may choose N g N such that < t 0 y t n < F h for all n G N, whence by ŽA1. < g Ž x . y g n Ž x . < s < f Ž t0 , x . y f Ž tn , x . < F « for all x g w a, b x and all n G N. This implies the uniform convergence of Ž g nŽ x .., x g w a, b x with limit g Ž x ., x g w a, b x. From Theorem B2 in Appendix B it follows that g XnŽ x . ª g 9Ž x . as n ª q`, uniformly for x g w a, b x. Again, let « ) 0 be arbitrary and choose N g N such that < g 9Ž x . y g XnŽ x .< F «r2 for all x g w a, b x and all n G N. Moreover, choose h ) 0 such that < g 9Ž x . y g 9Ž x 0 .< F «r2 if < x y x 0 < F h , x g w a, b x. Finally, choose N1 g N, N1 G N such that < x n y x 0 < F
MONOTONE CROSS-RATIO
595
h for all n G N1. From this we get < g 9 Ž x 0 . y g Xn Ž x n . < F < g 9 Ž x 0 . y g 9 Ž x n . < q < g 9 Ž x n . y g Xn Ž x n . < F
« 2
q
« 2
s«
for all n G N1 ,
i.e., g XnŽ x n . ª g 9Ž x 0 . as n ª q`. Remark. From the proof and Theorem B1 in Appendix B it follows that the statement in Theorem A remains true if instead of the derivative f xX Ž t, x . the function f Ž t, x . itself is weakly convex in x for any fixed t g I Žand f xX Ž t, x . is continuous in x for any fixed t g I .. Moreover, in both cases weakly con¨ ex may be replaced by weakly conca¨ e.
APPENDIX B In this appendix we prove the result on uniformly convergent sequences of C 1-functions with weakly convex derivatives needed in the proof of Theorem A in Appendix A. The proof is based on a preliminary theorem, which is interesting in itself. THEOREM B1. Let Ž g nŽ x .., x g w a, b x denote a sequence of weakly con¨ ex differentiable functions. Assume that g nŽ x . ª g Ž x . as n ª q` uniformly for x g w a, b x, where g Ž x ., x g w a, b x is a C 1-function. Then g XnŽ x . ª g 9Ž x . as n ª q`, uniformly for x g w a, b x. Proof. Suppose that g XnŽ x . does not converge uniformly for x g w a, b x to g 9Ž x . as n ª q`. Then there exist an « 0 ) 0 and sequences Ž x n . on w a, b x and Ž pn . on N such that pn ª q` as n ª q` and < g 9Ž x n . y g Xp nŽ x n .< G « 0 for all n g N. Passing}if necessary}to a subsequence we may suppose that x n ª x 0 as n ª q` for some x 0 g w a, b x. Suppose that x 0 g x a, bw and let h ) 0 be arbitrary such that w x 0 y h, x 0 q h x ; x a, bw Žif x s a or x s b the following reasoning holds with minor modifications.. Then x n g w x 0 y h, x 0 q h x if n is large enough, say n G N1. Hence, for any « ) 0 we get g p nŽ x n q h . y g p nŽ x n . y g Ž x 0 q h . y g Ž x 0 . F g Ž x 0 . y g p nŽ x n . q g Ž x 0 q h . y g p nŽ x n q h . F g Ž x 0 . y g Ž x n . q g Ž x n . y g p nŽ x n . q g Ž x 0 q h . y g Ž x n q h . q g Ž x n q h . y g p nŽ x n q h . F « q « q « q « s 4« ,
596
ANDERSEN AND SANDQVIST
provided that n G N1 and}besides}n is large enough. This follows from the facts: < g Ž x 1 . y g Ž x 2 .< F « if < x 1 y x 2 < is small enough Žuniform continuity. and < g Ž x . y g nŽ x .< F « for all x g w a, b x if n is large enough Žuniform convergence.. We infer that g p nŽ x n q h . y g p nŽ x n . ª g Ž x 0 q h . y g Ž x 0 .
as n ª q`. Ž B1 .
In the same way it is proved that g p nŽ x n . y g p nŽ x n y h . ª g Ž x 0 . y g Ž x 0 y h .
as n ª q`. Ž B2 .
From the convexity assumption we infer that 1 h
g p nŽ x n . y g p nŽ x n y h . F g Xp nŽ x n . F
1 h
g p nŽ x n q h . y g p nŽ x 0 . .
Hence either 1 h
g p nŽ x n . y g p nŽ x n y h . F g 9 Ž x n . y « 0
Ž B3 .
g p nŽ x n q h . y g p nŽ x 0 . G g 9 Ž x n . q « 0
Ž B4 .
or 1 h
holds for infinitely many values of n since < g 9Ž x n . y g p nŽ x n .< G « 0 for all n g N. Via ŽB1. or ŽB2. we get from ŽB3. or ŽB4. passing to the limit as n ª q` that either 1 h
g Ž x0 . y g Ž x0 y h. F g 9Ž x0 . y «
or 1 h
g Ž x0 q h. y g Ž x0 . G g 9Ž x0 . q « .
Passing to the limit as h ª 0 q we arrive at the contradiction g 9Ž x 0 . F g 9Ž x 0 . y « 0 k g 9Ž x 0 . G g 9Ž x 0 . q « 0 . Remark. Obviously, the theorem remains true if the functions g nŽ x ., x g w a, b x are all weakly concave instead of all weakly convex. THEOREM B2. Let Ž g nŽ x .., x g w a, b x denote a sequence of C 1-functions for which g XnŽ x ., x g w a, b x are weakly con¨ ex for all n g N. Assume that g nŽ x . ª g Ž x . as n ª q` uniformly for x g w a, b x, where g Ž x ., x g w a, b x is
MONOTONE CROSS-RATIO
597
a C 1-function with g 9Ž x ., x g w a, b x weakly con¨ ex. Then g XnŽ x . ª g 9Ž x . as n ª q` uniformly for x g w a, b x. Proof. We may assume that all functions g 9Ž x ., g XnŽ x ., x g w a, b x are strictly convex. Otherwise we could pass to the functions h nŽ x . s g nŽ x . q x 3 and hŽ x . s g Ž x . q x 3. A strictly convex function is either strictly increasing, strictly decreasing, or it has a unique minimum point, to the left Žright. of which the function is strictly decreasing Žstrictly increasing .. Consequently, a function with a strictly convex derivative is either strictly convex, strictly concave, or it has a unique inflexion point, to the left Žright. of which the function is strictly concave Žstrictly convex.. If there is an inflexion point, this point is the unique minimum point of the derivative. The sequence Ž g nŽ x .. consists of three subsequences Ž g q nŽ x .., Ž g r nŽ x .., and Ž g s nŽ x .., where one or two may be finite or empty: g q nŽ x ., x g w a, b x has an inflexion point, g r nŽ x ., x g w a, b x is strictly convex, and g s nŽ x ., x g w a, b x is strictly concave, respectively, for all n. Theorem B1 and its counterpart with concave functions apply to the subsequence Ž g r nŽ x .. and the subsequence Ž g s nŽ x .., respectively, if the subsequence is not finite or empty. Consequently, the theorem is proved if we prove that g Xq nŽ x . ª g 9Ž x . as n ª q` uniformly for x g w a, b x in the case where the subsequence Ž g q n Ž x .. is not finite or empty. Hence, the theorem is proved if we prove it with the additional assumption that all the functions g nŽ x ., x g w a, b x have an inflexion point. Let g nŽ x ., x g w a, b x have the inflexion point x n g x a, bw , n g N. There are two cases according to the existence or nonexistence of an inflexion point of the function g Ž x ., x g w a, b x. Case 18. Suppose that g Ž x ., x g w a, b x has an inflexion point x 0 g xa, bw . Let Ž x p . be any subsequence of Ž x n .. This subsequence in w a, b x has n a convergent subsequence Ž x pXn . with limit, say y 0 g w a, b x. Since g nŽ x . is strictly convex Žstrictly concave. for x n F x F b Ž a F x F x n ., it follows by a standard argument that g Ž x . is weakly convex Žweakly concave. for y 0 F x F b Ž a F x F y 0 .. Hence, there must hold y 0 s x 0 . In other words, x pXn ª x 0 as n ª q`. By a standard argument we infer that x n ª x 0 as n ª q`. Let h ) 0 be arbitrary such that w x 0 y 2 h, x 0 q 2 h x ; x a, bw . Then < x 0 y x n < - h for}say}n G N1. From Theorem B1 and its counterpart with concave functions it follows that g XnŽ x . ª g 9Ž x . as n ª q`, uniformly Žand hence pointwise. for x g w a, x 0 y h x j w x 0 q h, b x. From the convexity assumption we infer that the curve y s g XnŽ x . in the interval w a, x 0 q h x lies above the line through the two points Ž x 0 q h, g XnŽ x 0 q h..
598
ANDERSEN AND SANDQVIST
and Ž x 0 q 2 h, g XnŽ x 0 q 2 h... Hence g Xn Ž x 0 q h . q
1 h
g Xn Ž x 0 q 2 h . y g Xn Ž x 0 q h . Ž x n y x 0 y h .
- g Xn Ž x n . - g Xn Ž x 0 q h .
Ž B5 .
for n G N. Passing to the limit as n ª q` we get g 9Ž x0 q h. yŽ g 9Ž x0 q 2 h. y g 9Ž x0 q h. . F lim inf g Xn Ž x n . F lim sup g Xn Ž x n . F g 9 Ž x 0 q h . . Via h ª 0 q and the continuity of g 9Ž x ., x g w a, b x we get lim nªq` g XnŽ x n . s g 9Ž x 0 .. We are now in a position to conclude the proof in this case. Let « ) 0 be given. By continuity d ) 0 can be chosen such that < g 9Ž x . y g 9Ž x 0 .< F «r3 if < x y x 0 < F d , x g w a, b x. Moreover < g XnŽ x n . y g 9Ž x 0 .< F «r3 and < x n y x 0 < F d if}say}n G N. Hence, g 9Ž x0 . y
« 3
F g Xn Ž x n . F g Xn Ž x . F max g Xn Ž x 0 y d . , g Xn Ž x 0 q d . 4 , Ž B6 .
if n G N and < x y x 0 < F d . From Theorem B1 and its counterpart with concave functions it follows that g XnŽ x . ª g 9Ž x . as n ª q`, uniformly for x g w a, x 0 y d x j w x 0 q d , b x. Hence, there exists a number N1 G N such that < g XnŽ x . y g 9Ž x .< F «r3 for all x g w a, x 0 y d x j w x 0 q d , b x, provided that n G N1. In particular, max g XnŽ x 0 y d ., g XnŽ x 0 q d .4 F max g 9Ž x 0 y d ., g 9Ž x 0 q d .4 q «r3 F g 9Ž x 0 . q Ž2r3. « if n G N1. Then, by ŽB6. it follows that < g XnŽ x . y g 9Ž x 0 .< F Ž2r3. « if n G N1 and < x y x 0 < - d , whence < g Xn Ž x . y g 9 Ž x . < F < g Xn Ž x . y g 9 Ž x 0 . < q < g 9 Ž x 0 . y g 9 Ž x . < F « if n G N1 and x g w x 0 y d , x 0 q d x. Consequently, < g XnŽ x . y g 9Ž x .< F « for all x g w a, b x if n G N1. This proves that g XnŽ x . ª g 9Ž x . as n ª q` uniformly for x g w a, b x. Case 28. Suppose that g Ž x ., x g w a, b x is either strictly convex or strictly concave. By a variant of the argument in Case 18 it follows that}respectively}x n ª a as n ª q` or x n ª b as n ª q`. In the convex case, the proof in Case 18 holds if x 0 , x 0 y 2 h and x 0 y d are substituted by a. In the concave case the proof in Case 18 holds if x 0 ,
MONOTONE CROSS-RATIO
599
x 0 q 2 h and x 0 q d are substituted by b, and if the inequality ŽB5. is replaced by g Xn Ž b y h . y
1 h
g Xn Ž b y 2 h . y g Xn Ž b y h . Ž x n y b q h .
- g Xn Ž x n . - g Xn Ž b . . This completes the proof of the theorem. Remark. Theorem B2 remains valid if all the functions are weakly concave instead of weakly convex.
REFERENCES 1. K. M. Andersen and A. Sandqvist, On the cross-ratio of four solutions of a first order ordinary differential equation, J. Differential Equations 108 Ž1994., 89]99. 2. K. M. Andersen and A. Sandqvist, On the characteristic exponent function for a first order ordinary differential equation, Differential Integral Equations 8 Ž1995., 717]728. 3. P. S. Bullen, A criterion for n-convexity, Pacific J. Math. 36 Ž1971., 81]98. 4. E. Hille, ‘‘Lectures on Ordinary Differential Equations,’’ Addison]Wesley, Reading, MA, 1969.
201, 588]599 Ž1996.
0275
Differential Equations for Which the Cross-Ratio of Four Solutions Is Weakly Monotone Kurt Munk Andersen and Allan Sandqvist Mathematical Institute, Technical Uni¨ ersity of Denmark, Building 303, DK-2800 Lyngby, Denmark Submitted by Jack K. Hale Received October 30, 1995
INTRODUCTION Consider the first order differential equation
˙x s f Ž t , x . ,
Ž t , x . g I = R,
Ž 1.
where I 9 R is an open interval and f Ž t, x ., Ž t, x . g I = R a continuous function Žit is not assumed that there locally is uniqueness of solutions of Eq. Ž1... If w i Ž t ., t g J, i s 1, 2, 3, 4 are four distinct solutions of Eq. Ž1. on the same interval J 9 I, their cross-ratio RŽ t ., t g J is defined by RŽ t . s
w 3 Ž t . y w 1Ž t . w4 Ž t . y w 1 Ž t .
:
w3Ž t . y w2 Ž t . w4 Ž t . y w 2 Ž t .
,
t g J.
Ž 2.
It is a well-known fact Žsee, e.g., w4, p. 274x. that the cross-ratio Ž2. is constant for all choices of the four solutions if Eq. Ž1. is a Riccati equation, i.e., an equation of the form
˙x s a2 Ž t . x 2 q a1 Ž t . x q a0 Ž t . ,
t g I,
Ž 3.
where a i Ž t ., t g I, i s 0, 1, 2 are continuous functions. It turns out that the converse statement is also true. This can be proved elementary Žsee Remark 2 after Theorem 3.. Now, instead of the explicit form Ž3., a Riccati equation can be characterized as an equation Ž1., for which the derivative f xX Ž t, x . exists and is continuous for all Ž t, x . g I = R and is linear in x for any fixed t g I. Thus, the cross-ratio Ž2. is constant for all choices of the 588 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
MONOTONE CROSS-RATIO
589
four solutions of Eq. Ž1. if and only if the derivative f xX Ž t, x . exists and is continuous for all Ž t, x . g I = R and is linear in x for any fixed t g I. In Section 2 it is proved that the cross-ratio Ž2. is weakly increasing Ž weakly decreasing . for all choices of the four solutions of Eq. Ž1. if and only if the derivative f xX Ž t, x . exists and is continuous for all Ž t, x . g I = R and is weakly convex Žweakly concave. in x for any fixed t g I. The ‘‘if’’ part of the mentioned result in Section 2 is already proved in w1, Theorem 1x. The ‘‘only if’’ part is based on a connection between the cross-ratio Ž2. and the second divided difference of the function x ª f Ž t, x ., x g R Ž t g I fixed. and on a theorem of Bullen on 2-convexity. These two ingredients are presented in Section 1. The two Appendices A and B contain proofs of some auxiliary theorems on real functions and sequences of real functions.
1. DIVIDED DIFFERENCES AND n-CONVEXITY Let F Ž x ., x g R be a real function. The nth divided difference F w x 1 , x 2 , . . . , x n x on n distinct real numbers x 1 , x 2 , . . . , x n of the function F Ž x . is defined recursively by F w x 1 x s F Ž x 1 . and F w x1 , x 2 , . . . , x n x s
F w x 1 , . . . , x ny1 x y F w x 2 , . . . , x n x x1 y x n
Ž 4.
if n G 2. The function F Ž x ., x g R is called n-convex Ž n-concave. if and only if F w x 1 , x 2 , . . . , x nq2 x G 0 ŽF 0. for all choices of the real numbers x 1 , x 2 , . . . , x nq2 . In this definition we may impose the restriction x 1 x 2 -???- x nq2 Žsee w2, Lemma 1x.. It is seen that a 1-convex Ž1-concave. function is what is usually called a weakly convex Žweakly concave. function. For later use we state two results of Bullen on 2-convex functions. THEOREM 1. If F Ž x ., x g R, is 2-con¨ ex, then F9Ž x . exists for all x g R and is a weakly con¨ ex Ž and hence continuous. function. For a proof, see w3, Theorem 7 and Corollary 15x. Note that there is a difference in notation: n-convexity in w3x corresponds with what we here call Ž n y 1.-convexity. Before stating the other result of Bullen we introduce the following notation. By P2 Ž x; x 1 , x 2 , x 3 ; F . is denoted the uniquely determined polynomial of degree at most two having the same values as the function F Ž x ., x g R in the points x 1 , x 2 , and x 3 Žthis polynomial is usually called the Lagrange interpolating polynomial..
590
ANDERSEN AND SANDQVIST
THEOREM 2. The function F Ž x ., x g R, is 2-con¨ ex if and only if for any x 1 , x 2 , x 3 g R with x 1 - x 2 - x 3 there hold x F x 1 k x 2 F x F x 3 « F Ž x . F P2 Ž x ; x 1 , x 2 , x 3 ; F . and x 1 F x F x 2 k x 3 F x « F Ž x . G P2 Ž x ; x 1 , x 2 , x 3 ; F . .
Ž 5.
For a proof see w3, Theorem 5x. Condition Ž5. expresses that the graph of F lies alternately below and abo¨ e the cur¨ e y s P2 Ž x; x 1 , x 2 , x 3 ; F . in the four inter¨ als, into which x 1 , x 2 , and x 3 subdi¨ ide R. LEMMA 1.
For any four distinct real numbers x 1 , x 2 , x 3 , x 4 there holds
Ž x 2 y x1 . Ž x 4 y x 3 . F w x1 , x 2 , x 3 , x 4 x F Ž x 3 . y F Ž x1 .
s
x 3 y x1 y
F Ž x4 . y F Ž x2 .
q
F Ž x 4 . y F Ž x1 . x 4 y x1
x4 y x2 F Ž x3 . y F Ž x2 .
y
.
x3 y x2
Ž 6.
Proof. From w3, Ž8. ] Ž10.x it follows that F w x1 , x 2 , x 3 , x 4 x s
4
F Ž xk .
Ý
.
4
ks1
Ž 7.
Ł Ž xk y x j . js1 j/k
The right hand side of Ž6. can be written F Ž x1 .
y1 x 3 y x1
q
qF Ž x 3 . s F Ž x1 .
1 x 4 y x1 1
x 3 y x1
q F Ž x2 .
q
y1 x3 y x2
x3 y x4
Ž x1 y x 3 . Ž x1 y x 4 .
q F Ž x3 .
y1 x4 y x2
q F Ž x4 .
q F Ž x2 .
x1 y x 2
Ž x 3 y x1 . Ž x 3 y x 2 .
q
x3 y x2 1
x4 y x2
q
y1 x 4 y x1
x4 y x3
Ž x2 y x4 . Ž x2 y x3 .
q F Ž x4 .
s Ž x 2 y x1 . Ž x 4 y x 3 . F w x1 , x 2 , x 3 , x 4 x . where the last equality sign comes from Ž7..
1
x 2 y x1
Ž x 4 y x 2 . Ž x 4 y x1 .
MONOTONE CROSS-RATIO
591
2. ON THE CROSS-RATIO OF FOUR SOLUTIONS OF EQ. Ž1. Let f t Ž x ., x g R denote the function defined by x ª f Ž t, x . for t g I fixed. THEOREM 3. The function f Ž t, x ., Ž t, x . g I = R on the right hand side of Eq. Ž1. has a continuous deri¨ ati¨ e f xX Ž t, x ., Ž t, x . g I = R which is weakly con¨ ex w weakly conca¨ e x in x for any fixed t g I if and only if for any four solutions w i Ž t ., t g J, i s 1, 2, 3, 4 of Eq. Ž1. on some common inter¨ al J 9 I such that w 1Ž t . - w 2 Ž t . - w 3 Ž t . - w4Ž t . for all t g J, the cross-ratio RŽ t . s
w 3 Ž t . y w 1Ž t .
:
w4 Ž t . y w 1 Ž t .
w3Ž t . y w2 Ž t . w4 Ž t . y w 2 Ž t .
tgJ
,
Ž 8.
is weakly increasing w weakly decreasing x. Proof. The ‘‘only if ’’ part follows from w1, Theorem 1x. Below we prove the ‘‘if ’’ part in the weakly convexrweakly increasing case. The weakly concaverweakly decreasing case is treated analogously. From Ž8. we get by differentiation, using Ž6. and suppressing the variable t under the w i-signs R9 Ž t . RŽ t .
s s
d dt
log R Ž t . s
w ˙3 y w˙1 w3 y w1
f Ž t , w3 . y f Ž t , w1 .
w3 y w1 y
q
f Ž t , w4 . y f Ž t , w 1 .
w4 y w 1
q
w ˙4 y w˙2 w4 y w 2
y
w ˙4 y w˙1 w4 y w 1
y
w ˙3 y w˙2 w3 y w2
f Ž t , w4 . y f Ž t , w 2 .
w4 y w 2 y
f Ž t , w3 . y f Ž t , w2 .
w3 y w2
s Ž w 2 y w 1 . Ž w4 y w 3 . f t w w 1 , w 2 , w 3 , w4 x ,
t g J.
Ž 9.
Suppose that RŽ t ., t g J is weakly increasing for any choice of the four solutions. From Ž9. we infer that f t w w 1Ž t ., w 2 Ž t ., w 3 Ž t ., w4Ž t .x G 0, t g J for any choice of the four solutions. Let t 0 g I and x i g R, i s 1, 2, 3, 4, x 1 - x 2 - x 3 - x 4 be arbitrary. Let w i Ž t . denote a solution of Eq. Ž1. through the point Ž t 0 , x i ., i s 1, 2, 3, 4. These four solutions can be chosen such that they are defined on some common interval J around t 0 , and w 1Ž t . - w 2 Ž t . - w 3 Ž t . - w4Ž t . for all t g J. Hence f t w w 1Ž t ., w 2 Ž t ., w 3 Ž t ., w4Ž t .x G 0 for all t g J, in particular for t s t 0 . This gives f t 0 w x 1 , x 2 , x 3 , x 4 x G 0. Hence f t 0Ž x ., x g R is 2-convex by definition. Theorem 1 implies that f xX Ž t 0 , x ., x g R exists and is weakly convex Žand
592
ANDERSEN AND SANDQVIST
hence continuous. }for any fixed t 0 g I. From Theorem A in Appendix A it follows that f xX Ž t, x ., Ž t, x . g I = R is a continuous function. COROLLARY. Equation Ž1. is a Riccati equation if and only if for any four solutions w i Ž t ., t g J of Eq. Ž1. on some common inter¨ al J 9 I such that w 1Ž t . - w 2 Ž t . - w 3 Ž t . - w4Ž t . for all t g J the cross-ratio Ž8. is constant. Remark 1. From Theorem 3 it follows that there locally is uniqueness of solutions of Eq. Ž1., if the cross-ratio of any four solutions is weakly increasing wweakly decreasingx. This can be proved directly as follows in the weakly increasing case. The weakly decreasing case is treated similarly. If there is not locally uniqueness, then there exist a point Ž t 0 , x 0 . g I = R and two solutions x 1Ž t . and x 2 Ž t . of Eq. Ž1., both defined in some interval x t 0 y d , t 0 q d w , such that x 1Ž t 0 . s x 2 Ž t 0 . s x 0 and either x 1Ž t . x 2 Ž t . for all t g x t 0 y d , t 0 w or x 1Ž t . - x 2 Ž t . for all t g x t 0 , t 0 q d w . Consider the first case. We may choose two other solutions w 3 Ž t . and w4Ž t . of Eq. Ž1. such that all four solutions are defined on some interval xt 0 y d 1 , t 0 x, 0 - d 1 F d and x 1Ž t . - x 2 Ž t . - w 3 Ž t . - w4Ž t . for all t g xt 0 y d 1 , t 0 w , and x 0 - w 3 Ž t 0 . - w4 Ž t 0 .. Then RŽ t . s
w 3 Ž t . y x1Ž t .
w3 Ž t . y x2 Ž t .
:
w4 Ž t . y x 1 Ž t .
w4 Ž t . y x 2 Ž t .
ª1
as t ª t 0 y ,
whence RŽ t . F 1 for all t g x t 0 y d 1 , t 0 w , i.e., w w 3 Ž t . y x 1Ž t .x w w4 Ž t . y x 2 Ž t .x F w w4Ž t . y x 1Ž t .x w w 3 Ž t . y x 2 Ž t .x for all t g x t 0 y d 1 , t 0 w . This gives the contradiction w4 Ž t . F w 3 Ž t . for all t g x t 0 y d 1 , t 0 w . Consider next the case x 1Ž t . - x 2 Ž t . for all t g x t 0 , t 0 q d w . Choose two other solutions c 1Ž t . and c4Ž t . of Eq. Ž1. such that all four solutions are defined on some interval w t 0 , t 0 q d 1w , 0 - d 1 F d and c 1Ž t . - x 1Ž t . - x 2 Ž t . - c4 Ž t . for all t g x t 0 , t 0 q d 1w , and c 1Ž t 0 . - x 0 - c4Ž t 0 .. Then RŽ t . s
x 2 Ž t . y c 1Ž t .
c4 Ž t . y c 1 Ž t .
:
x 2 Ž t . y x1Ž t .
c4 Ž t . y x 1 Ž t .
ª q`
as t ª t 0 q .
This contradicts that RŽ t ., t g x t 0 , t 0 q d 1w is weakly increasing. Remark 2. The ‘‘if ’’ part of the corollary can be proved elementary as follows. Let in Ž8. the solutions w 1Ž t ., w 3 Ž t ., and w4Ž t . be fixed and w 2 Ž t . s x Ž t . be arbitrary. When RŽ t . is constant we get
w ˙3 y w˙1 w3 y w1
q
w ˙4 y ˙x w4 y x
y
w ˙4 y w˙1 w4 y w 1
y
w ˙3 y ˙x w3 y x
s 0,
t g J.
MONOTONE CROSS-RATIO
593
Solving for ˙ x we obtain an equation on the form Ž3. on the interval J. This means that f Ž t, x . is a polynomial in x of degree at most two in the domain w 1Ž t . - x - w 3 Ž t ., t g J. Consequently to any point Ž t, x . g I = R there exists a neighborhood in which f xŽ nn. Ž t, x ., n s 1, 2, 3 exist and are continuous and f xZ3 Ž t, x . is identically zero. Hence these statements on the derivatives hold for all Ž t, x . g I = R. It follows that Eq. Ž1. has the form Ž3., i.e., it is a Riccati equation. Remark 3. If Eq. Ž1. is autonomous}say f Ž t, x . s F Ž x . }then f t 0 s F for any t 0 g I. From the proof of Theorem 3 it follows that F Ž x ., x g R is 2-convex provided that the cross-ratios are weakly increasing, which by the theorem itself will hold if F9Ž x ., x g R exists and is weakly convex. Hence, the converse statement of Theorem 1 is also true. Remark 4. From Remark 3 and Theorem A in Appendix A it follows that a continuous function f Ž t, x ., Ž t, x . g I = R has a continuous derivative f xX Ž t, x ., Ž t, x . g I = R which is weakly convex wweakly concavex in x for any fixed t g I if and only if it is 2-convex w2-concavex in x for any fixed t g I. Hence, Theorem 3 can also be formulated as a necessary and sufficient condition for the function f Ž t, x ., Ž t, x . g I = R to be 2-convex w2-concavex in x for any fixed t g I. In the remaining part of this section we assume that f Ž t, x ., Ž t, x . g I = R is 2-convex in x for any fixed t g I. Let w 1Ž t ., w 2 Ž t ., w 3 Ž t ., t g J denote three solutions of Eq. Ž1. on the same interval J 9 I and suppose that w 1Ž t . - w 2 Ž t . - w 3 Ž t . for all t g J. If x Ž t ., t g J is any solution of Eq. Ž1. on the same interval J, according to Ž5. in Theorem 2 there hold x Ž t . - w 1Ž t . k w 2 Ž t . - x Ž t . - w 3 Ž t . «˙ x Ž t . F P2 Ž x Ž t . ; w 1 Ž t . , w 2 Ž t . , w 3 Ž t . ; f t .
Ž 10 .
and
w 1Ž t . - x Ž t . - w 2 Ž t . k w 3 Ž t . - x Ž t . «˙ x Ž t . G P2 Ž x Ž t . ; w 1 Ž t . , w 2 Ž t . , w 3 Ž t . ; f t .
Ž 11 .
for all t g J. We may write P2 Ž x ; w 1 Ž t . , w 2 Ž t . , w 3 Ž t . ; f t . s a Ž t . x 2 q b Ž t . x q g Ž t . ,
t g J,
where a Ž t ., b Ž t ., g Ž t ., t g I are continuous functions. The two implications Ž10. and Ž11. indicate that the solution curves of Eq. Ž1. in the strip t g J can be compared to those of the Riccati equation
˙x s a Ž t . x 2 q b Ž t . x q g Ž t . ,
t g J,
Ž 12 .
594
ANDERSEN AND SANDQVIST
having x s w i Ž t ., t g J, i s 1, 2, 3 as solutions. From this it is easy to derive a stability result. Suppose namely that J s I > w0, 1x and that w i Ž t ., t g I, i s 1, 2, 3 are closed, i.e., w i Ž0. s w i Ž1., i s 1, 2, 3. It is well known that then any on the interval I defined solution of the corresponding Riccati equation Ž12. is also closed Žsee, e.g., w1, Lemma 2x.. From Ž10. and Ž11. it follows that w 1Ž t . and w 3 Ž t . are unstable and w 2 Ž t . is stable}unless Eq. Ž1. and Eq. Ž12. coincide. This is also proved in w2, Theorem 9x.
APPENDIX A THEOREM A. Let f Ž t, x ., Ž t, x . g I = R be a continuous function, where I 9 R is an open inter¨ al. Assume that the deri¨ ati¨ e f xX Ž t, x . exists for all Ž t, x . g I = R and that f xX Ž t, x . is weakly con¨ ex Ž and hence continuous. in x for any fixed t g I. Then f xX Ž t, x ., Ž t, x . g R is a continuous function. Proof. Let Ž t 0 , x 0 . g I = R be arbitrary and let ŽŽ t n , x n .. be some sequence in I = R converging to Ž t 0 , x 0 . as n ª q`. We prove that f xX Ž t n , x n . ª f xX Ž t 0 , x 0 . as n ª q`. From this the desired property follows. Choose the interval w a, b x and the number d ) 0 such that x 0 g x a, bw and w t 0 y d , t 0 q d x ; I. Without loss of generality we may assume that Ž t n , x n . g w t 0 y d , t 0 q d x = w a, b x for all n g N. Define the functions g n Ž x . s f Ž tn , x . ,
g Ž x . s f Ž t 0 , x . , x g w a, b x , n g N.
These functions are all C 1 and their derivatives are weakly convex. Our aim is to prove that g XnŽ x n . ª g 9Ž x 0 . as n ª q`. First we prove that g nŽ x . ª g Ž x . as n ª q`, uniformly for x g w a, b x. Let « ) 0 be arbitrary. Since f Ž t, x . is uniformly continuous for Ž t, x . g w t 0 y d , t 0 q d x = w a, b x, an h ) 0 can be chosen such that < f Ž t1 , x 1 . y f Ž t 2 , x 2 . < F «
if 5 Ž t 1 y t 2 , x 1 y x 2 . 5 F h
Ž A1.
provided that t 1 , t 2 g w t 0 y d , t 0 q d x and x 1 , x 2 g w a, b x. We may choose N g N such that < t 0 y t n < F h for all n G N, whence by ŽA1. < g Ž x . y g n Ž x . < s < f Ž t0 , x . y f Ž tn , x . < F « for all x g w a, b x and all n G N. This implies the uniform convergence of Ž g nŽ x .., x g w a, b x with limit g Ž x ., x g w a, b x. From Theorem B2 in Appendix B it follows that g XnŽ x . ª g 9Ž x . as n ª q`, uniformly for x g w a, b x. Again, let « ) 0 be arbitrary and choose N g N such that < g 9Ž x . y g XnŽ x .< F «r2 for all x g w a, b x and all n G N. Moreover, choose h ) 0 such that < g 9Ž x . y g 9Ž x 0 .< F «r2 if < x y x 0 < F h , x g w a, b x. Finally, choose N1 g N, N1 G N such that < x n y x 0 < F
MONOTONE CROSS-RATIO
595
h for all n G N1. From this we get < g 9 Ž x 0 . y g Xn Ž x n . < F < g 9 Ž x 0 . y g 9 Ž x n . < q < g 9 Ž x n . y g Xn Ž x n . < F
« 2
q
« 2
s«
for all n G N1 ,
i.e., g XnŽ x n . ª g 9Ž x 0 . as n ª q`. Remark. From the proof and Theorem B1 in Appendix B it follows that the statement in Theorem A remains true if instead of the derivative f xX Ž t, x . the function f Ž t, x . itself is weakly convex in x for any fixed t g I Žand f xX Ž t, x . is continuous in x for any fixed t g I .. Moreover, in both cases weakly con¨ ex may be replaced by weakly conca¨ e.
APPENDIX B In this appendix we prove the result on uniformly convergent sequences of C 1-functions with weakly convex derivatives needed in the proof of Theorem A in Appendix A. The proof is based on a preliminary theorem, which is interesting in itself. THEOREM B1. Let Ž g nŽ x .., x g w a, b x denote a sequence of weakly con¨ ex differentiable functions. Assume that g nŽ x . ª g Ž x . as n ª q` uniformly for x g w a, b x, where g Ž x ., x g w a, b x is a C 1-function. Then g XnŽ x . ª g 9Ž x . as n ª q`, uniformly for x g w a, b x. Proof. Suppose that g XnŽ x . does not converge uniformly for x g w a, b x to g 9Ž x . as n ª q`. Then there exist an « 0 ) 0 and sequences Ž x n . on w a, b x and Ž pn . on N such that pn ª q` as n ª q` and < g 9Ž x n . y g Xp nŽ x n .< G « 0 for all n g N. Passing}if necessary}to a subsequence we may suppose that x n ª x 0 as n ª q` for some x 0 g w a, b x. Suppose that x 0 g x a, bw and let h ) 0 be arbitrary such that w x 0 y h, x 0 q h x ; x a, bw Žif x s a or x s b the following reasoning holds with minor modifications.. Then x n g w x 0 y h, x 0 q h x if n is large enough, say n G N1. Hence, for any « ) 0 we get g p nŽ x n q h . y g p nŽ x n . y g Ž x 0 q h . y g Ž x 0 . F g Ž x 0 . y g p nŽ x n . q g Ž x 0 q h . y g p nŽ x n q h . F g Ž x 0 . y g Ž x n . q g Ž x n . y g p nŽ x n . q g Ž x 0 q h . y g Ž x n q h . q g Ž x n q h . y g p nŽ x n q h . F « q « q « q « s 4« ,
596
ANDERSEN AND SANDQVIST
provided that n G N1 and}besides}n is large enough. This follows from the facts: < g Ž x 1 . y g Ž x 2 .< F « if < x 1 y x 2 < is small enough Žuniform continuity. and < g Ž x . y g nŽ x .< F « for all x g w a, b x if n is large enough Žuniform convergence.. We infer that g p nŽ x n q h . y g p nŽ x n . ª g Ž x 0 q h . y g Ž x 0 .
as n ª q`. Ž B1 .
In the same way it is proved that g p nŽ x n . y g p nŽ x n y h . ª g Ž x 0 . y g Ž x 0 y h .
as n ª q`. Ž B2 .
From the convexity assumption we infer that 1 h
g p nŽ x n . y g p nŽ x n y h . F g Xp nŽ x n . F
1 h
g p nŽ x n q h . y g p nŽ x 0 . .
Hence either 1 h
g p nŽ x n . y g p nŽ x n y h . F g 9 Ž x n . y « 0
Ž B3 .
g p nŽ x n q h . y g p nŽ x 0 . G g 9 Ž x n . q « 0
Ž B4 .
or 1 h
holds for infinitely many values of n since < g 9Ž x n . y g p nŽ x n .< G « 0 for all n g N. Via ŽB1. or ŽB2. we get from ŽB3. or ŽB4. passing to the limit as n ª q` that either 1 h
g Ž x0 . y g Ž x0 y h. F g 9Ž x0 . y «
or 1 h
g Ž x0 q h. y g Ž x0 . G g 9Ž x0 . q « .
Passing to the limit as h ª 0 q we arrive at the contradiction g 9Ž x 0 . F g 9Ž x 0 . y « 0 k g 9Ž x 0 . G g 9Ž x 0 . q « 0 . Remark. Obviously, the theorem remains true if the functions g nŽ x ., x g w a, b x are all weakly concave instead of all weakly convex. THEOREM B2. Let Ž g nŽ x .., x g w a, b x denote a sequence of C 1-functions for which g XnŽ x ., x g w a, b x are weakly con¨ ex for all n g N. Assume that g nŽ x . ª g Ž x . as n ª q` uniformly for x g w a, b x, where g Ž x ., x g w a, b x is
MONOTONE CROSS-RATIO
597
a C 1-function with g 9Ž x ., x g w a, b x weakly con¨ ex. Then g XnŽ x . ª g 9Ž x . as n ª q` uniformly for x g w a, b x. Proof. We may assume that all functions g 9Ž x ., g XnŽ x ., x g w a, b x are strictly convex. Otherwise we could pass to the functions h nŽ x . s g nŽ x . q x 3 and hŽ x . s g Ž x . q x 3. A strictly convex function is either strictly increasing, strictly decreasing, or it has a unique minimum point, to the left Žright. of which the function is strictly decreasing Žstrictly increasing .. Consequently, a function with a strictly convex derivative is either strictly convex, strictly concave, or it has a unique inflexion point, to the left Žright. of which the function is strictly concave Žstrictly convex.. If there is an inflexion point, this point is the unique minimum point of the derivative. The sequence Ž g nŽ x .. consists of three subsequences Ž g q nŽ x .., Ž g r nŽ x .., and Ž g s nŽ x .., where one or two may be finite or empty: g q nŽ x ., x g w a, b x has an inflexion point, g r nŽ x ., x g w a, b x is strictly convex, and g s nŽ x ., x g w a, b x is strictly concave, respectively, for all n. Theorem B1 and its counterpart with concave functions apply to the subsequence Ž g r nŽ x .. and the subsequence Ž g s nŽ x .., respectively, if the subsequence is not finite or empty. Consequently, the theorem is proved if we prove that g Xq nŽ x . ª g 9Ž x . as n ª q` uniformly for x g w a, b x in the case where the subsequence Ž g q n Ž x .. is not finite or empty. Hence, the theorem is proved if we prove it with the additional assumption that all the functions g nŽ x ., x g w a, b x have an inflexion point. Let g nŽ x ., x g w a, b x have the inflexion point x n g x a, bw , n g N. There are two cases according to the existence or nonexistence of an inflexion point of the function g Ž x ., x g w a, b x. Case 18. Suppose that g Ž x ., x g w a, b x has an inflexion point x 0 g xa, bw . Let Ž x p . be any subsequence of Ž x n .. This subsequence in w a, b x has n a convergent subsequence Ž x pXn . with limit, say y 0 g w a, b x. Since g nŽ x . is strictly convex Žstrictly concave. for x n F x F b Ž a F x F x n ., it follows by a standard argument that g Ž x . is weakly convex Žweakly concave. for y 0 F x F b Ž a F x F y 0 .. Hence, there must hold y 0 s x 0 . In other words, x pXn ª x 0 as n ª q`. By a standard argument we infer that x n ª x 0 as n ª q`. Let h ) 0 be arbitrary such that w x 0 y 2 h, x 0 q 2 h x ; x a, bw . Then < x 0 y x n < - h for}say}n G N1. From Theorem B1 and its counterpart with concave functions it follows that g XnŽ x . ª g 9Ž x . as n ª q`, uniformly Žand hence pointwise. for x g w a, x 0 y h x j w x 0 q h, b x. From the convexity assumption we infer that the curve y s g XnŽ x . in the interval w a, x 0 q h x lies above the line through the two points Ž x 0 q h, g XnŽ x 0 q h..
598
ANDERSEN AND SANDQVIST
and Ž x 0 q 2 h, g XnŽ x 0 q 2 h... Hence g Xn Ž x 0 q h . q
1 h
g Xn Ž x 0 q 2 h . y g Xn Ž x 0 q h . Ž x n y x 0 y h .
- g Xn Ž x n . - g Xn Ž x 0 q h .
Ž B5 .
for n G N. Passing to the limit as n ª q` we get g 9Ž x0 q h. yŽ g 9Ž x0 q 2 h. y g 9Ž x0 q h. . F lim inf g Xn Ž x n . F lim sup g Xn Ž x n . F g 9 Ž x 0 q h . . Via h ª 0 q and the continuity of g 9Ž x ., x g w a, b x we get lim nªq` g XnŽ x n . s g 9Ž x 0 .. We are now in a position to conclude the proof in this case. Let « ) 0 be given. By continuity d ) 0 can be chosen such that < g 9Ž x . y g 9Ž x 0 .< F «r3 if < x y x 0 < F d , x g w a, b x. Moreover < g XnŽ x n . y g 9Ž x 0 .< F «r3 and < x n y x 0 < F d if}say}n G N. Hence, g 9Ž x0 . y
« 3
F g Xn Ž x n . F g Xn Ž x . F max g Xn Ž x 0 y d . , g Xn Ž x 0 q d . 4 , Ž B6 .
if n G N and < x y x 0 < F d . From Theorem B1 and its counterpart with concave functions it follows that g XnŽ x . ª g 9Ž x . as n ª q`, uniformly for x g w a, x 0 y d x j w x 0 q d , b x. Hence, there exists a number N1 G N such that < g XnŽ x . y g 9Ž x .< F «r3 for all x g w a, x 0 y d x j w x 0 q d , b x, provided that n G N1. In particular, max g XnŽ x 0 y d ., g XnŽ x 0 q d .4 F max g 9Ž x 0 y d ., g 9Ž x 0 q d .4 q «r3 F g 9Ž x 0 . q Ž2r3. « if n G N1. Then, by ŽB6. it follows that < g XnŽ x . y g 9Ž x 0 .< F Ž2r3. « if n G N1 and < x y x 0 < - d , whence < g Xn Ž x . y g 9 Ž x . < F < g Xn Ž x . y g 9 Ž x 0 . < q < g 9 Ž x 0 . y g 9 Ž x . < F « if n G N1 and x g w x 0 y d , x 0 q d x. Consequently, < g XnŽ x . y g 9Ž x .< F « for all x g w a, b x if n G N1. This proves that g XnŽ x . ª g 9Ž x . as n ª q` uniformly for x g w a, b x. Case 28. Suppose that g Ž x ., x g w a, b x is either strictly convex or strictly concave. By a variant of the argument in Case 18 it follows that}respectively}x n ª a as n ª q` or x n ª b as n ª q`. In the convex case, the proof in Case 18 holds if x 0 , x 0 y 2 h and x 0 y d are substituted by a. In the concave case the proof in Case 18 holds if x 0 ,
MONOTONE CROSS-RATIO
599
x 0 q 2 h and x 0 q d are substituted by b, and if the inequality ŽB5. is replaced by g Xn Ž b y h . y
1 h
g Xn Ž b y 2 h . y g Xn Ž b y h . Ž x n y b q h .
- g Xn Ž x n . - g Xn Ž b . . This completes the proof of the theorem. Remark. Theorem B2 remains valid if all the functions are weakly concave instead of weakly convex.
REFERENCES 1. K. M. Andersen and A. Sandqvist, On the cross-ratio of four solutions of a first order ordinary differential equation, J. Differential Equations 108 Ž1994., 89]99. 2. K. M. Andersen and A. Sandqvist, On the characteristic exponent function for a first order ordinary differential equation, Differential Integral Equations 8 Ž1995., 717]728. 3. P. S. Bullen, A criterion for n-convexity, Pacific J. Math. 36 Ž1971., 81]98. 4. E. Hille, ‘‘Lectures on Ordinary Differential Equations,’’ Addison]Wesley, Reading, MA, 1969.