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Discontinuous Systems and the Henstock–Kurzweil Integral
Journal of Mathematical Analysis and Applications 229, 119]136 Ž1999. Article ID jmaa.1998.6149, available online at http:rrwww.idealibrary.com on
Discontinuous Systems and the Henstock]Kurzweil Integral* Wu Congxin and Li Baolin Department of Mathematics, Harbin Institute of Technology, 150001 China
and E. Stanley Lee† Department of Industrial Engineering, Kansas State Uni¨ ersity, Manhattan, Kansas 66506-5101 Submitted by William F. Ames Received March 26, 1997
By using the Henstock]Kurzweil integral and the inequalities of this integral, the existence and uniqueness theorems for the solution of the discontinuous Caratheodory system are established. The results are generalizations of earlier investigations wJ. He and Po Chen, Ad¨ . in Math. 16 Ž1987., 17]32 Žin Chinese.; A. F. Filippov, Math. USSR-Sb. 51 Ž1960. Žin Russian.; O. Hajek, J. Differential Equations 32 Ž1979., 149]185x of this discontinuous system. Q 1999 Academic Press
1. INTRODUCTION To introduce the problem which is considered in this paper, let us first consider the general discontinuous system defined by the following ordinary differential equation, x9 s f Ž t , x . ,
Ž 1.1.
where x s Ž x 1 , x 2 , . . . , x n .T , x9 s dxrdt, f : G ª R n is a function with some discontinuity, G is an open region in R nq 1. * This work was supported by the Nature Science Foundation of China. † E-mail address: [email protected]. 119 0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
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CONGXIN, BAOLIN, AND LEE
Because the first work on this general discontinuous system which was carried out by Caratheodory in 1918 w12x, many investigators have studied this system. Furthermore, this discontinuous system has found many applications such as in the development of discontinuous oscillatory theory and in the theory of modern control. He and Chen w1x summarized some of the developments and their applications. In the literature w1]3x, the existence, uniqueness, and stability for the solution of the discontinuous Caratheodory and Filippov systems were obtained by using the Lebesgue integral and the solutions obtained are absolute continuous functions. However, there are discontinuous systems in which the right-hand side functions f Ž t, x . are not Lebesgue integrable on certain intervals and their solutions are not absolute continuous functions. To illustrate, consider the following example: EXAMPLE 1. Consider the following discontinuous system, x9 s t 2 x q h Ž t . , Ž 1.2. 2 y2 where < t < F 1, < x < F 1, and hŽ t . s Ž drdt .Ž t sin t .. If t / 0 and hŽ0. s 0, then f Ž t, x . s t 2 x Ž t . q hŽ t . is a highly oscillating function and is not Lebesgue integrable on < t < F 1. However, with x Ž0. s 0, the system Ž1.2. has the following solution, x Ž t . s et
3
r3
t yŽ s 3 r3.
H0 e
h Ž s . ds.
Ž 1.3.
The preceding integral is neither the Riemann integral nor the Lebesgue integral. It is the Henstock]Kurzweil integral and the solution x Ž t . of system Ž1.2. is not an absolutely continuous function on w0, 1x. In order to investigate this problem, we must use the Henstock]Kurzweil integral which encompasses the Newton, Riemann, and Lebesgue integrals w4]8x. This integral was introduced by Henstock and Kurzweil independently during 1957]1958 and was proven to be useful in the study of ordinary differential equations w4x. In this paper, the existence and uniqueness theorems for solutions of a generalized discontinuous Caratheodory system are established by using the Henstock]Kurzweil integral and the inequalities of this integral.
2. HENSTOCK]KURZWEIL INTEGRAL AND THE CONVERGENCE THEOREM DEFINITION 2.1 w4]8x. Let f : w a, b x ª R n be a function. f is said to be Henstock]Kurzweil integrable to A on w a, b x if for every e ) 0, there exists a positive function d Ž j . such that whenever a division D given by a s t 0 - t 1 - ??? - t n s b and
j 1 , j 2 , . . . , jn 4
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121
satisfies j i y d Ž j i . - t iy1 F j i F t i - j q d Ž j i . for i s 1, 2, . . . , k, we have k
Ý f Ž j i . Ž t i y t iy1 . y A
- e,
Ž 2.1.
is1
where Hab f Ž t . dt s A. The Henstock]Kurzweil integral has all the standard properties one normally expects of any integral w4]8x. Here we only mention the relation between the Henstock]Kurzweil integral and the Lebesgue integral. THEOREM 2.2 w4, 8x. If f is Lebesgue integrable on w a, b x, then f is Henstock]Kurzweil integrable. THEOREM 2.3 w4, 8x. If f is Henstock]Kurzweil integrable on w a, b x and nonnegati¨ e, then f is Lebesgue integrable. DEFINITION 2.4 w8x. A function F: w a, b x ª R n is said to be absolutely continuous in the restricted sense on set X or, in short, AC#Ž X ., if for every e ) 0 there exists h ) 0 such that for every finite or infinite sequence of nonoverlapping intervals w a i , bi x4 with a i , bi g X and satisfying Ý i < bi y a i < - h we have Ý i v Ž F; w a i , bi x. - e , where v denotes the oscillation of F over w a i , bi x. A function F is said to be generalized absolutely continuous in the restricted sense on w a, b x or, ACG#, if w a, b x is the union of a sequence of closed sets X i such that on each X i , the function F is AC#Ž X i .. Note that if F is AC#Žw a, b x., then F is absolutely continuous on w a, b x. It is known that if F is differentiable everywhere on w a, b x then F is ACG# on w a, b x, see w14, Chap. VII, Theorem 10.5x. Hence, if there is a solution of x9 s f Ž t, x ., for every t on w a, b x, then the solution is ACG#. EXAMPLE 2. Let F Ž t . s t 2 sin ty2 if t / 0 and F Ž0. s 0, then the solution of x9 s F9Ž x . x is x Ž t . s e F Ž t ., which is ACG# on w0, 1x. Note that F Ž t . is not a bounded variation on w0, 1x; thus, F Ž t . and e F Ž t . are not absolutely continuous on w0, 1x. THEOREM 2.5 w8x. A function f : w a, b x ª R n is Henstock]Kurzweil integrable on w a, b x if and only if there exists a continuous function F which is ACG# on w a, b x such that F9Ž t . s f Ž t . almost e¨ erywhere. THEOREM 2.6 w7, 8x ŽControlled Convergence Theorem.. If a sequence of Henstock]Kurzweil integrable function f n4 satisfies the following conditions: Ži. f nŽ x . ª f Ž x . almost e¨ erywhere in w a, b x as n ª `; Žii. the primiti¨ es FnŽ x . s Hab f nŽ s . ds of f n are ACG# uniformly in n; Žiii. the primiti¨ es Fn are equicontinuous on w a, b x, then f is Henstock]Kurzweil integrable on w a, b x and we ha¨ e Hab f nŽ x . dx ª Hab f Ž x . dx as n ª `.
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CONGXIN, BAOLIN, AND LEE
If conditions Žii. and Žiii. are replaced by the condition: Živ. g Ž x . F f nŽ x . F hŽ x . almost e¨ erywhere on w a, b x, where g and h are Henstock]Kurzweil integrable, then the result of Theorem 2.6 also holds.
3. GENERALIZED CARATHEODORY SYSTEMS In this section, a generalized Caratheodory system of the form Ž1.1. is defined by using the Henstock]Kurzweil integral. The main result of this section is an existence theorem for the solution to the generalized Caratheodory system. DEFINITION 3.1 w2x. Let right-hand side function f Ž t, x . of the system Ž1.1. be a Caratheodory function defined on an open region G; i.e., f is continuous in x for almost all t and measurable in t for each fixed x. Suppose there exists a Lebesgue integrable function mŽ t . on every bounded closed subregion G 0 ; G such that f Ž t , x . F m Ž t . , for Ž t , x . g G 0 ,
Ž 3.1.
then f Ž t, x . is said to satisfy the Caratheodory condition, and the system Ž1.1. is said to be a Caratheodory system. DEFINITION 3.2 w2x. A function x Ž t .: I ª R n Ž I represents an interval in R1 . is said to be a solution of the Caratheodory system Ž1.1. or, in short, C-solution, if x Ž t . satisfies the following conditions: Ži. x Ž t . is absolutely continuous on each compact subinterval of interval I; Žii. Ž t, x . g G for t g I; Žiii. x9Ž t . s f Ž t, x Ž t .. for almost everywhere t g I. Now we can generalize the Caratheodory system to the generalized Caratheodory system using the Henstock]Kurzweil integral. DEFINITION 3.3. Let the right-hand side function of the system Ž1.1. be a Caratheodory function defined on an open region G. Suppose that there exist two Henstock]Kurzweil integrable functions g Ž t . and hŽ t . for every bounded closed subregion G 0 ; G such that g Ž t . F f Ž t , x . F h Ž t . , for all x and almost all t with Ž t , x . g G 0 ,
Ž 3.2. then the f Ž t, x . satisfies the generalized Caratheodory condition on G, and the system Ž1.1. is a generalized Caratheodory system.
DISCONTINUOUS SYSTEMS
123
DEFINITION 3.4. A function x Ž t .: I ª R n Ž I represents an interval in R is said to be a solution of the generalized Caratheodory system Ž1.1. or GC-solution if x Ž t . satisfies the following conditions: 1.
Ži. x Ž t . is ACG# on each compact subinterval of I; Žii. Ž t, x . g G for t g I; Žiii. x9Ž t . s f Ž t, x Ž t .. for almost everywhere t g I. We remark that if g and h are Lebesgue integrable functions, then the previous generalized Caratheodory system reduces to the Caratheodory system, and the GC-solution also reduces to the C-solution. Now we give the existence theorem for the GC-solution for the preceding generalized Caratheodory system. THEOREM 3.5. Suppose that f Ž t, x . satisfies the conditions of Definitions 3.3, then there exists a GC-solution F of the generalized Caratheodory system Ž1.1. on some inter¨ al < t y t < F b , and satisfies F Žt . s j . Proof. Let Žt , j . g G be fixed and let G 0 be a bounded closed subregion of G, G0 :
< t y t < F a,
5 x y j 5 F b,
Ž 3.3.
then there exist two Henstock]Kurzweil integrable functions g Ž t . and hŽ t . such that for all x and for almost all t with Ž t, x . g G 0 we have 0 F f Ž t, x . y g Ž t . F hŽ t .. y g Ž t .. By Theorem 2.3, h y g is Lebesgue integrable. Let FŽ t, x. s f t, x q
ž
t
Ht g Ž s . ds
/
y gŽ t. ,
Ž 3.4.
then F is a Caratheodory function, 0 F F Ž t, x . F hŽ t . y g Ž t . for all Ž t, x . g GX0 , where GX0 is a bounded closed subregion of G 0 , such that xq
t
Ht g Ž s . ds y j
F b, for all Ž t , x . g GX0 ,
then x9 s F Ž t, x . is a Caratheodory system. By Caratheodory existence theorem, there is a function C on some interval < t y t < F b such that C9Ž t . s F Ž t, C Ž t .. almost everywhere in this interval and C Žt . s j . Let FŽ t. s CŽ t. q
t
Ht g Ž s . ds.
Ž 3.5.
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CONGXIN, BAOLIN, AND LEE
Then, for almost all t, F9 Ž t . s C9 Ž t . q g Ž t . s FŽ t, CŽ t. . q g Ž t. s f t, CŽ t. q
ž
t
Ht g Ž s . ds
y gŽ t. q gŽ t.
/
s f Ž t, FŽ t. . , and FŽt . s CŽt . q
t
Ht g Ž s . ds s j .
The proof is complete. EXAMPLE 3. Consider x9 s f Ž t , x . s g Ž t , x . q h Ž t . ,
Ž 3.6.
where 5 g Ž t, x .5 F g 1Ž t . for all < t < F 1, 5 x 5 F 1, g 1Ž t . is Lebesgue integrable on < t < F 1, and hŽ t . s Ž drdt .Ž t 2 sin ty2 . if t / 0 and hŽ0. s 0. Here t s j s 0. h is Henstock]Kurzweil integrable but not Lebesgue integrable and h Ž t . y g 1 Ž t . F f Ž t , x . F h Ž t . q g 1 Ž t . , for < t < F 1, 5 x 5 F 1. Thus, by Theorem 3.5, there exists a GC-solution F of x9 s f Ž t, x . with F Ž0. s 0. For instance, if g Ž t, x . s t 2 x, then this example is Example 1, FŽ t. s e t
3
r3
t ys 3 r3
H0 e
h Ž s . ds.
THEOREM 3.6. Let the right-hand side function f Ž t, x . of the system Ž1.1. be a Caratheodory function on the open region G, Let G 0 : < t y t < F a, 5 x y j 5 F b, be a fixed bounded closed subregion of G, and let f Ž t, w Ž t .. be Henstock]Kurzweil integrable on < t y t < F a for any step function w Ž t . defined on < t y t < F a with ¨ alues in 5 x y j 5 F b. Denote Fw Ž t . s Htt f Ž s, w Ž s .. ds. If Fw : w is a step function4 is ACG# uniformly in w and equicontinuous on < t y t < F a, then the system Ž1.1. is a generalized Caratheodory system. Proof. Notice that f Ž t, x . is a Caratheodory function. Thus, there exist two measurable functions uŽ t . and ¨ Ž t . defined on < t y t < F a with values in 5 x y j 5 F b such that f Ž t , uŽ t . . F f Ž t , x . F f Ž t , ¨ Ž t . . ,
Ž 3.7.
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DISCONTINUOUS SYSTEMS
for all Ž t, x . g G 0 wsee 14, Lemma 17.2x. Next, we should show that f Ž t, uŽ t .. and f Ž t, ¨ Ž t .. are Henstock]Kurzweil integrable by using the controlled convergence Theorem 2.6. First, there is a sequence k nŽ t .4 of step functions defined on < t y t < F a with values in 5 x y j 5 F b such that k nŽ t . ª uŽ t . almost everywhere as n ª `. Thus, f Ž t, k nŽ t .. ª f Ž t, uŽ t .. almost everywhere as h ª `. Let Fn Ž t . s
t
Ht f Ž s, k Ž s . . ds. n
Then FnŽ t .4 is ACG# uniformly in n and equicontinuous. By Theorem 2.6, f Ž t, uŽ t .. is Henstock]Kurzweil integrable. Similarly, f Ž t, ¨ Ž t .. is Henstock]Kurzweil integrable. Let g Ž t . s f Ž t, uŽ t .., hŽ t . s f Ž t, ¨ Ž t .., then by Ž3.7. we have g Ž t . F f Ž t , x . F hŽ t . . The proof is complete.
4. UNIQUENESS THEOREM FOR THE GENERALIZED CARATHEODORY SYSTEM In this section, the uniqueness theorems for the solutions of the generalized Caratheodory system is given by using some inequalities from the Henstock]Kurzweil integral. LEMMA 4.1 w8x. If f Ž t .: w a, b x ª R n is a Henstock]Kurzweil integrable function with respect to the function g Ž t .: w a, b x ª R n, then, for e¨ ery c g w a, b x, we ha¨ e s
lim
Ha f Ž t . dg Ž t . y f Ž c . Ž g Ž s . y g Ž c . .
lim
Hu f Ž t . dg Ž t . q f Ž c . Ž g Ž u . y g Ž c . .
sªc
uªc
b
c
s
Ha f Ž t . dg Ž t . ,
s
Hc f Ž t . dg Ž t . .
b
Ž 4.1. Ž 4.2.
LEMMA 4.2. Let f, g: w a, b x ª R n be functions for which the Henstock]Kurzweil-integral Hab f Ž t . dg Ž t . exists. If u, ¨ : w a, b x ª R1 are functions such that integral Hab uŽ t . d¨ Ž t . exists and if there is a positi¨ e function d : w a, b x ª Ž0, q`. such that < t y j < 5 f Ž j . Ž g Ž t . y g Ž j . . 5 F Ž t y j . u Ž j . Ž ¨ Ž t . y ¨ Ž j . . , Ž 4.3.
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CONGXIN, BAOLIN, AND LEE
for e¨ ery t g w j y d Ž j ., j q d Ž j .x, j g w a, b x. Then b
Ha f Ž t . dg Ž t .
F
b
Ha u Ž t . d¨ Ž t . .
Ž 4.4.
Proof. Assume that e ) 0 is given. Because the integrals Hab f Ž t . dg Ž t ., uŽ t . d¨ Ž t . exist, there is a positive function d 1 on w a, b x with d 1Ž s . F d Ž s . for s g w a, b x such that for every d 1-fine partition D of w a, b x,
Hab
j1, j2 , . . . , jk4 ,
a s t 0 - t 1 - ??? - t k s b and
with j i y d 1Ž j i . - t iy1 F j i F t i - j i q d 1Ž j i ., i s 1, 2, . . . , k, we have k
b
Ý f Ž j i . Ž g Ž t i . y g Ž t iy1 . . y H f Ž t . dg Ž t . a
is1 k
b
Ý u Ž j i . Ž ¨ Ž t i . y ¨ Ž t iy1 . . y H u Ž t . d¨ Ž t . a
is1
-e,
Ž 4.5.
- e.
Ž 4.6.
By Ž4.3., for j s 1, 2, . . . , k, we have f Ž j j . Ž g Ž t j . y g Ž t jy1 . . F f Ž j j . Ž g Ž t j . y g Ž j j . . q f Ž j j . Ž g Ž j j . y g Ž t jy1 . . F u Ž j j . Ž ¨ Ž t j . y ¨ Ž t jy1 . . . Hence, b
Ha
k
f Ž t . dg Ž t . F
b
Ý f Ž j i . Ž g Ž t i . y g Ž t iy1 . . y H f Ž t . dg Ž t . a
is1
k
q
Ý f Ž j i . Ž g Ž t i . y g Ž t iy1 . . is1 k
-eq
Ý u Ž j i . Ž ¨ Ž t i . y ¨ Ž t iy1 . . is1
- 2e q
b
Ha u Ž t . d¨ Ž t . .
Because e ) 0 is arbitrary, the inequality Ž4.4. is satisfied. COROLLARY 4.3. If f : w a, b x ª R1 , f Ž t .< F c for t g w a, b x where c is a constant, g: w a, b x ª R1 is of bounded ¨ ariation on w a, b x, and the integral
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127
Hab f Ž t . dg Ž t . exists then b
Ha f Ž t . dg Ž t .
F c Var ab g ,
Ž 4.7.
where Var ab g is the total ¨ ariation of g on w a, b x. LEMMA 4.4. Let h: w a, b x ª R1 be a nonnegati¨ e nondecreasing function which is continuous from the left on Ž a, b x. Assume that u: w0, q`. ª w0, q`. is a continuous nondecreasing function with primiti¨ e U: w0, q`. ª R1, then the integral Hab uŽ hŽ t .. dhŽ t . exists and b
Ha uŽ h Ž t . . dhŽ t . F U Ž h Ž b . . y U Ž h Ž a. . .
Ž 4.8.
Proof. The composition of functions u and h given by uŽ hŽ t .. for t g w a, b x is nondecreasing on w a, b x. Therefore, the integral Hab uŽ hŽ t .. dhŽ t . exists. Assume that e ) 0 is given. By the definition of the primitive U to u, for every g w0, q`. there exists u Ž s . ) 0 such that for every h ) 0 with 0 F
Ž 4.9.
Because lim t ª jq hŽ t . s hŽ jq. for j g w a, b ., there exists a dqŽ j . ) 0, dqŽ b . s 1 such that for t g Ž j , j q dqŽ j .x l w a, b x we have 0 F h Ž t . y h Ž jq . F u Ž h Ž jq . . . Let s s hŽ jq. and h s hŽ t . y hŽ jq. we obtain u Ž h Ž jq . . Ž h Ž t . yh Ž jq . . FU Ž h Ž t . . yU Ž h Ž jq . . q e Ž h Ž t . yh Ž jq . . . Further, we have u Ž h Ž j . . Ž h Ž jq . yh Ž j . . y U Ž h Ž jq . . yU Ž h Ž j . . s
q.
HhhŽ jj. Ž
u Ž h Ž j . . y u Ž s . ds
G 0, because uŽ hŽ j .. F uŽ s . for s g w hŽ j ., hŽ jq.x . Therefore, u Ž h Ž j . . Ž h Ž t . yh Ž j . . s u Ž h Ž j . . Ž h Ž t . y h Ž jq . . q u Ž h Ž j . . Ž h Ž jq . y h Ž j . . F UŽ hŽ t . . y UŽ hŽ j . . q e Ž hŽ t . y hŽ j . . ,
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CONGXIN, BAOLIN, AND LEE
for t g Ž j , j q dqŽ j .. l w a, b x. From the inequality Ž4.9. and from the left continuity of the function h at the point j g Ž a, b x, there is a dyŽ j . ) 0, dyŽ a. s 1 such that for t g w j y dyŽ j ., j x l w a, b x the inequality, uŽ hŽ j . . Ž hŽ t . y hŽ j . . F UŽ hŽ j . . y UŽ hŽ t . . q e Ž hŽ j . y hŽ t . . is satisfied. Let d Ž j . s min dyŽ j ., dqŽ j .4 for j g w a, b x, then for j g w a, b x and t g w j y d Ž j ., j q d Ž j .x l w a, b x we obtain by the preceding inequalities the relation, < t y j < < uŽ hŽ j . . Ž hŽ t . y hŽ j . . < F Ž t y j . Ž UŽ hŽ t . . q e hŽ t . y UŽ hŽ j . . y e hŽ j . . . By Lemma 4.2, this inequality implies b
b
Ha u Ž h Ž t . . dhŽ t . F Ha d U Ž h Ž t . . q e h Ž t . s U Ž h Ž b . . y U Ž h Ž a. . q e Ž h Ž b . y h Ž a. . . Because e ) 0 can be chosen arbitrarily small, the proof is complete. LEMMA 4.5. Let c : w a, b x ª w0, q`., h: w a, b x ª w a, q`. be gi¨ en where c is bounded and h is nondecreasing and continuous from the left on the inter¨ al w a, b x. Suppose that the function v : w0, q`. ª R1 is continuous, nondecreasing, v Ž0. s 0, v Ž r . ) 0 for r ) 0. For u ) 0, let V Ž u. s
u
Hu
0
1
vŽ r.
dr ,
Ž 4.10.
with some u 0 ) 0. The function V: Ž0, q`. ª R 1 is increasing, V Ž u 0 . s 0 lim uª 0y V Ž u. s a G y`, lim uªq` V Ž u. s b F q`. Assume that for j g w a, b x the inequality,
cŽj . Fkq
j
Ha v Ž c Ž t . . dhŽ t .
and
Ž 4.11.
holds, where k ) 0 is a constant. If V Ž k . q hŽ b . y hŽ a. - b then for j g w a, b x we ha¨ e
c Ž j . F Vy1 Ž V Ž k . q h Ž j . y h Ž a . . ,
Ž 4.12.
where Vy1 : Ž a , b . ª R1 is the in¨ erse function of the function V in Ž4.10..
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DISCONTINUOUS SYSTEMS
Proof. If we have V Ž l . q hŽ b . y hŽ a. - b for some l ) 0 then for all t g w a, b x we have
a - V Ž l . q h Ž t . y h Ž a. - b . Therefore the value of V Ž l . q hŽ t . y hŽ a. belongs to the domain of Vy1 provided t g w a, b x, and for t we can define Hl Ž t . s Vy1 Ž V Ž l . q h Ž t . y h Ž a . . . Define further
w Ž s . s v Ž Vy1 Ž V Ž l . q s . . ,
Ž 4.13.
for s g w0, b y V Ž l .x. At Vy1 Ž V Ž l . q s . there exists a derivative V9 of the function V and V9 Ž Vy1 Ž V Ž l . q s . . s
1 y1
vŽV
Ž V Ž l . q s. .
/ 0.
The well-known formula for the derivative of the inverse function leads to d ds
Vy1 Ž V Ž l . q s . s
1 y1
V9 Ž V
Ž V Ž l . q s. .
s v Ž Vy1 Ž V Ž l . q s . . s w Ž s. ,
Ž 4.14.
for s g w0, b y V Ž l .x. If now j g w a, b x is given, then using the definition of the function w from Ž4.13. we obtain j
j
y1
Ha v Ž H Ž t . . dhŽ t . s Ha v Ž V l
s
Ž V Ž l . q h Ž t . y h Ž a. . . dh Ž t .
j
Ha w Ž h Ž t . y h Ž a. . d Ž h Ž t . y h Ž a. . .
This together with Ž4.14. and Lemma 4.4 imply j
y1
Ha v Ž H Ž t . . dhŽ t . F V l
Ž V Ž l . q h Ž j . y h Ž a. . y Vy1 Ž V Ž l . .
s Hl Ž j . y l,
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CONGXIN, BAOLIN, AND LEE
and consequently for j g w a, b x we have the inequality, lq
j
Ha v Ž H Ž t . . dhŽ t . F H Ž j . . l
l
Assume that e 0 ) 0 is such that V Ž k q e 0 . q hŽ b . y hŽ a. - b . Let us take an arbitrary e g Ž0, e 0 . and set l s k q e . For this case the last inequality reads kqeq
j
Ha v Ž H
kq e
Ž t . . dh Ž t . F Hkq e Ž j . ,
and taking into account the relation Ž4.11. for every j g w a, b x we get
c Ž j . y Hkq e Ž j . F k q
j
Ha v Ž c Ž t . . dhŽ t . y k y e j
Ha v Ž H
y
s ye q
j
Ha
kq e
Ž t . dh Ž t .
v Ž c Ž t . . y v Ž Hkq e Ž t . . dh Ž t . . Ž 4.15.
Hence c Ž a. y Hkq e Ž j . F ye and also v Ž c Ž a.. y v Ž Hkq e Ž a.. F 0 because the function v is assumed to be nondecreasing. The functions c and Hkq e are bounded and therefore there is a constant K ) 0 such that
v Ž c Ž t . . y v Ž Hkq e Ž t . . - K , for t g w a, b x. Using Lemma 4.1 and Corollary 4.3 we obtain from the last two displayed inequalities,
c Ž j . y Hkq e Ž j . F ye q v Ž c Ž a . . y v Ž Hkq e Ž t . . Ž h Ž aq . y h Ž a . . q limq dª0
j
Haqd
v Ž c Ž t . . y v Ž Hkq e Ž t . . dh Ž t .
F ye q K limq h Ž j . y h Ž a q d . dª0
s ye q K h Ž j . y h Ž aq . . Because lim j ª aq hŽ j . s hŽ aq. , an h ) 0 can be found such that for j g Ž a, a q h . the inequality hŽ j . y hŽ aq. - erŽ2 K q 1. holds and
131
DISCONTINUOUS SYSTEMS
therefore also
c Ž j . y Hkq e Ž j . - ye q
Ke 2Kq 1
-y
e 2
- 0,
for j g Ž a, a q h .. Let us set T s sup t g w a, b x ; c Ž j . y Hkq e Ž j . - 0, for j g w a, t x 4 . As has been shown previously, we have T ) a and for j g w a, T . the inequality c Ž j . y Hkq e Ž j . - 0 and therefore v Ž c Ž j .. y v Ž Hkq e Ž j .. F 0 holds. The last conclusion is a consequence of the assumption that v is nondecreasing. By Ž4.15. and Lemma 4.1 we have
c Ž T . y Hkq e Ž T . F ye q limq dª0
Ty d
Ha
v Ž c Ž t . . y v Ž Hkq e Ž t . . dh Ž t .
q v Ž c Ž T . . y v Ž Hkq e Ž T . .
Ž h Ž T . y h Ž Ty . .
F ye - 0, because h Ž T . y h Ž Ty . s h Ž T . y limy h Ž t . s 0, tªT
and lim
Ty d
H
dª0 q a
v Ž c Ž t . . y v Ž Hkq e Ž t . . dh Ž t . F 0.
If we assume that T - b then we can repeat this procedure for j ) T by virtue of the inequality,
c Ž j . y Hkq e Ž j . F ye q
j
HT
v Ž c Ž t . . y v Ž Hkq e Ž t . . dh Ž t . ,
thus obtaining c Ž j . y Hkq e Ž j . - 0 for j g w T, T q h x for some h ) 0. Hence T s b and
c Ž j . - Hkq e Ž j . s Vy1 Ž V Ž k q e . q h Ž j . y h Ž a . . , for j g w a, b x. Because the function V is continuous and the last inequality holds for every sufficiently small e ) 0, we obtain the inequality Ž4.12.. The proof is complete.
132
CONGXIN, BAOLIN, AND LEE
COROLLARY 4.6. If c , h, and k satisfy the assumptions in lemma 4.5 and if for j g w a, b x the inequality,
cŽj . FkqL
j
Ha c Ž t . dhŽ t .
holds with a constant L ) 0 instead of Ž4.11., then for e¨ ery j g w a, b x the inequality,
c Ž j . F ke LŽ hŽ j .yhŽ a.. is satisfied. Lemma 4.5 represents a Bellman type inequality for the Henstock] Kurzweil integral. Results of this type are especially useful for deriving uniqueness results for the generalized Caratheodory system. DEFINITION 4.7. A solution x: wt , t q h x ª R n of the generalized Caratheodory system Ž1.1. is called locally unique for increasing values of t if for any solution y: wt , t q s x ª R n of the generalized Caratheodory system Ž1.1. with y Žt . s x Žt . there exists h1 ) 0 such that x Ž t . s y Ž t . for t g wt , t q h x l wt , t q s x l wt , t q h1 x. THEOREM 4.8. Assume that the system Ž1.1. is a generalized Caratheodory system and the right-hand side function f Ž t, x . of system Ž1.1. satisfies the following condition on G 0 : < t y t < F a, 5 x y j 5 F b, f Ž u , x . y f Ž u , y . Ž ¨ y u . F v Ž 5 x y y 5 . Ž h Ž ¨ . y h Ž u . . , Ž 4.16. for each inter¨ al w u, ¨ x with u g w u, ¨ x ; wt y a, t q ax and all x, y belonging to 5 x y j 5 F b, where h: wt y a, t q ax ª R1 is nondecreasing and continuous from the left. v : w0, q`. ª R1 is continuous, nondecreasing, v Ž r . ) 0 for r ) 0, v Ž0. s 0 and u
limq
¨ ª0
H ¨
1
vŽ r.
dr ,
Ž 4.17.
for e¨ ery u ) 0. Then e¨ ery solution x of the generalized Caratheodory system Ž1.1. such that x Žt . s j is locally unique for increasing ¨ alues of t.
DISCONTINUOUS SYSTEMS
133
Proof. Let x, y: wt , t q h x ª R n be solutions of the generalized Caratheodory system Ž1.1. such that x Žt . s y Žt . s j . Assume that e ) 0 is given. Because the Henstock]Kurzweil integrals Hrt w f Ž s, x Ž s .. y f Ž s, y Ž s ..x ds, Htt v Ž5 x Ž s . y y Ž s .5. dhŽ s . exist there is a positive function d 1 on wt , t x such that for every d 1-fine partition D of wt , t x,
t s t 0 - t 1 - ??? - t k s t and
u1 , u2 , . . . , uk4 ,
with u i y d 1Ž u i . - t iy1 F u i F t i - u i q d 1Ž u i ., i s 1, 2, . . . , k, we have t
f Ž s, x Ž s . . y f Ž s, y Ž s . . ds
Ht
F
t
f Ž s, x Ž s . . y f Ž s, y Ž s . . ds
Ht
k
y Ý f Ž u i , x Ž u i . . y f Ž u i , y Ž u i . . Ž t i y t iy1 . is1 k
q
f Ž u i , x Ž u i . . y f Ž u i , y Ž u i . . Ž t i y t iy1 .
Ý is1
-
e 2
k
Ý v Ž 5 x Ž u i . y y Ž u i . 5.Ž h Ž t i . y h Ž t iy1 . . ,
q
is1
and k
Ý v Ž 5 x Ž u i . y y Ž u i . 5.Ž h Ž t i . y h Ž t iy1 . . is1 k
F
Ý v Ž 5 x Ž u i . y y Ž u i . 5.Ž h Ž t i . y h Ž t iy1 . . is1 t
Ht v Ž 5 x Ž s . y y Ž s . 5. dhŽ s .
y t
Ht v Ž 5 x Ž s . y y Ž s . 5. dhŽ s .
q -
e 2
q
t
Ht v Ž 5 x Ž s . y y Ž s . 5. dhŽ s . ,
134
CONGXIN, BAOLIN, AND LEE
then 5 xŽ t. y yŽ t. 5 s
t
f Ž s, x Ž s . . y f Ž s, y Ž s . . ds
Ht
-eq
t
Ht v Ž 5 x Ž s . y y Ž s . 5. dhŽ s . .
Because e ) 0 is arbitrary, then 5 xŽ t. y yŽ t. 5 F s
t
Ht v Ž 5 x Ž s . y y Ž s . 5. dhŽ s . tq d
Ht
v Ž 5 x Ž s . y y Ž s . 5 . dh Ž s .
t
Htqdv Ž 5 x Ž s . y y Ž s . 5. dhŽ s . ,
q
where 0 - d - t y t . By Lemma 4.1, we have tq d
Ht
v Ž 5 x Ž s . y y Ž s . 5 . dh Ž s .
s v Ž 5 x Ž t . y y Ž t . 5 . h Ž tq . y h Ž t . q limq t1ª t
F
tq d
Ht
sup sg wt , t q d x
v Ž 5 x Ž s . y y Ž s . 5 . dh Ž s .
1
v Ž 5 x Ž s . y y Ž s . 5 . h Ž t q d . y h Ž tq .
s AŽ d . ,
Ž 4.18.
because v Ž5 x Žt . y y Žt .5 s v Ž0. s 0. Because the limit hŽtq. exists we have also lim d ª 0q AŽ d . s 0, therefore, x Ž t . y y Ž t . F AŽ d . q
t
Htqdv Ž 5 x Ž s . y y Ž s . 5. dhŽ s . ,
for t g wt q d , t q h x. Let u 0 ) 0 and set V Ž u. s
u
Hu
0
1
vŽ r.
dr.
DISCONTINUOUS SYSTEMS
135
Using Lemma 4.5 we obtain x Ž t . y y Ž t . F Vy1 Ž V Ž A Ž d . . q h Ž t . y h Ž t q d . . ,
Ž 4.19.
for t g wt q d , t q h x provided V Ž AŽ d .. q hŽt q h . y hŽt q d . - b , where b s lim uªq` V Ž u. F q`. Evidently, we have V Ž A Ž d . . q h Ž t q h . y h Ž t q d . F V Ž A Ž d . . q h Ž t q h . y h Ž tq . , and because lim d ª 0q AŽ d . s 0 and lim uª 0q V Ž u. s y`, we have lim V Ž A Ž d . . q h Ž t q h . y h Ž tq . s y`.
dª0 q
Hence there is a d 0 ) 0 such that for d g Ž0, d 0 . the inequality V Ž AŽ d .. q hŽt q h . y hŽtq. - b holds. Applying now the map V to both sides of Ž4.19., we have V Ž 5 x Ž t . y y Ž t . 5. F V Ž AŽ d . . q h Ž t . y h Ž t q h . , and this yields V Ž 5 x Ž t . y y Ž t . 5 . y V Ž A Ž d . . F h Ž t . y h Ž t q d . F h Ž t . y h Ž tq . . From the definition of V we therefore have 5 Ž .
HAŽxd .t yy t
1
Ž .5
vŽ r.
dr F h Ž t q h . y h Ž tq . .
Assume now that 5 x Ž t*. y y Ž t9.5 s k ) 0 for some t* g Žt , t q h x, then 1
k
HAŽ d . v Ž r .
dr F h Ž t q h . y h Ž tq . - q`,
for every d g Ž0, d 0 . such that d - t* y t . Now it is possible to use d ª 0q to obtain the inequality, limq
dª0
k
1
HAŽ d . v Ž r .
dr F h Ž t q h . y h Ž tq . - q`,
which contradicts the assumption on the function v . Therefore 5 x Ž t . y y Ž t .5 s 0 for t g Žt , t q h x. The proof is complete. COROLLARY 4.9. If the function v Ž r . s Lr, r ) 0, L ) 0 in Theorem 4.8, then the result in Theorem 4.8 also holds.
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CONGXIN, BAOLIN, AND LEE
The local uniqueness for increasing values of t can be extended to the global uniqueness for increasing values of t in the same manner as is done for the case of classical ordinary differential equations, we elaborate on this.
REFERENCES 1. J. He and P. Chen, Some aspects of the theory and applications of discontinuous differential equations, Ad¨ . in Math. 16 Ž1987., 17]32 Žin Chinese.. 2. A. F. Filippov, Differential equations with discontinuous right-hand side, Math. USSR-Sb. 51 Ž1960. Žin Russian.. 3. O. Hajek, Discontinuous differential equation I, II, J. Differential Equations 32 Ž1979., 149]185. 4. J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslo¨ ak Math. J. 7 Ž1957., 418]446. 5. R. Henstock, Definitions of Riemann type of the variational integrals, Proc. London Math. Soc. 11 Ž1961., 402]418. 6. R. Henstock, ‘‘Lectures on the Theory of Integration,’’ World Scientific, Singapore, 1988. 7. P. Y. Lee and T. S. Chew, A better convergence theorem for Henstock integrals, Bull. London Math. Soc. 17 Ž1985., 557]574. 8. P. Y. Lee, ‘‘Lanzhou Lectures on Henstock Integration,’’ World Scientific, Singapore, 1989. 9. W. F. Pfeffer, Integration by parts for the generalized Riemann]Stieltjes integral, J. Austral. Math. Soc. Ser. A 34 Ž1983., 229]231. 10. K. Ostaszewski and J. Sochacki, Gronwall’s inequality and the Henstock integral, J. Math. Anal. Appl. 127 Ž1987., 370]374. 11. B. Li, The topological structure on the space of ŽH. integrable functions, J. Math. 14 Ž1994., 61]68 Žin Chinese.. 12. C. Caratheodory, ‘‘Vorlesungen ¨ uber Relle Funktionen,’’ Teubner, Leipzig, 1918. 13. E. A. Coddington and N. Levinson, ‘‘Theory of Ordinary Differential Equations,’’ McGraw-Hill, New York, 1955. 14. S. Saks, ‘‘Theory of the Integral,’’ 2nd revised ed., Dover, New York, 1964. 15. P. S. Bullen and D. N. Sarkhel, On the solution of Ž d¨ rdx .a p s f Ž x, y ., J. Math. Anal. Appl. 127 Ž1987., 365]376.
Discontinuous Systems and the Henstock]Kurzweil Integral* Wu Congxin and Li Baolin Department of Mathematics, Harbin Institute of Technology, 150001 China
and E. Stanley Lee† Department of Industrial Engineering, Kansas State Uni¨ ersity, Manhattan, Kansas 66506-5101 Submitted by William F. Ames Received March 26, 1997
By using the Henstock]Kurzweil integral and the inequalities of this integral, the existence and uniqueness theorems for the solution of the discontinuous Caratheodory system are established. The results are generalizations of earlier investigations wJ. He and Po Chen, Ad¨ . in Math. 16 Ž1987., 17]32 Žin Chinese.; A. F. Filippov, Math. USSR-Sb. 51 Ž1960. Žin Russian.; O. Hajek, J. Differential Equations 32 Ž1979., 149]185x of this discontinuous system. Q 1999 Academic Press
1. INTRODUCTION To introduce the problem which is considered in this paper, let us first consider the general discontinuous system defined by the following ordinary differential equation, x9 s f Ž t , x . ,
Ž 1.1.
where x s Ž x 1 , x 2 , . . . , x n .T , x9 s dxrdt, f : G ª R n is a function with some discontinuity, G is an open region in R nq 1. * This work was supported by the Nature Science Foundation of China. † E-mail address: [email protected]. 119 0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
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CONGXIN, BAOLIN, AND LEE
Because the first work on this general discontinuous system which was carried out by Caratheodory in 1918 w12x, many investigators have studied this system. Furthermore, this discontinuous system has found many applications such as in the development of discontinuous oscillatory theory and in the theory of modern control. He and Chen w1x summarized some of the developments and their applications. In the literature w1]3x, the existence, uniqueness, and stability for the solution of the discontinuous Caratheodory and Filippov systems were obtained by using the Lebesgue integral and the solutions obtained are absolute continuous functions. However, there are discontinuous systems in which the right-hand side functions f Ž t, x . are not Lebesgue integrable on certain intervals and their solutions are not absolute continuous functions. To illustrate, consider the following example: EXAMPLE 1. Consider the following discontinuous system, x9 s t 2 x q h Ž t . , Ž 1.2. 2 y2 where < t < F 1, < x < F 1, and hŽ t . s Ž drdt .Ž t sin t .. If t / 0 and hŽ0. s 0, then f Ž t, x . s t 2 x Ž t . q hŽ t . is a highly oscillating function and is not Lebesgue integrable on < t < F 1. However, with x Ž0. s 0, the system Ž1.2. has the following solution, x Ž t . s et
3
r3
t yŽ s 3 r3.
H0 e
h Ž s . ds.
Ž 1.3.
The preceding integral is neither the Riemann integral nor the Lebesgue integral. It is the Henstock]Kurzweil integral and the solution x Ž t . of system Ž1.2. is not an absolutely continuous function on w0, 1x. In order to investigate this problem, we must use the Henstock]Kurzweil integral which encompasses the Newton, Riemann, and Lebesgue integrals w4]8x. This integral was introduced by Henstock and Kurzweil independently during 1957]1958 and was proven to be useful in the study of ordinary differential equations w4x. In this paper, the existence and uniqueness theorems for solutions of a generalized discontinuous Caratheodory system are established by using the Henstock]Kurzweil integral and the inequalities of this integral.
2. HENSTOCK]KURZWEIL INTEGRAL AND THE CONVERGENCE THEOREM DEFINITION 2.1 w4]8x. Let f : w a, b x ª R n be a function. f is said to be Henstock]Kurzweil integrable to A on w a, b x if for every e ) 0, there exists a positive function d Ž j . such that whenever a division D given by a s t 0 - t 1 - ??? - t n s b and
j 1 , j 2 , . . . , jn 4
DISCONTINUOUS SYSTEMS
121
satisfies j i y d Ž j i . - t iy1 F j i F t i - j q d Ž j i . for i s 1, 2, . . . , k, we have k
Ý f Ž j i . Ž t i y t iy1 . y A
- e,
Ž 2.1.
is1
where Hab f Ž t . dt s A. The Henstock]Kurzweil integral has all the standard properties one normally expects of any integral w4]8x. Here we only mention the relation between the Henstock]Kurzweil integral and the Lebesgue integral. THEOREM 2.2 w4, 8x. If f is Lebesgue integrable on w a, b x, then f is Henstock]Kurzweil integrable. THEOREM 2.3 w4, 8x. If f is Henstock]Kurzweil integrable on w a, b x and nonnegati¨ e, then f is Lebesgue integrable. DEFINITION 2.4 w8x. A function F: w a, b x ª R n is said to be absolutely continuous in the restricted sense on set X or, in short, AC#Ž X ., if for every e ) 0 there exists h ) 0 such that for every finite or infinite sequence of nonoverlapping intervals w a i , bi x4 with a i , bi g X and satisfying Ý i < bi y a i < - h we have Ý i v Ž F; w a i , bi x. - e , where v denotes the oscillation of F over w a i , bi x. A function F is said to be generalized absolutely continuous in the restricted sense on w a, b x or, ACG#, if w a, b x is the union of a sequence of closed sets X i such that on each X i , the function F is AC#Ž X i .. Note that if F is AC#Žw a, b x., then F is absolutely continuous on w a, b x. It is known that if F is differentiable everywhere on w a, b x then F is ACG# on w a, b x, see w14, Chap. VII, Theorem 10.5x. Hence, if there is a solution of x9 s f Ž t, x ., for every t on w a, b x, then the solution is ACG#. EXAMPLE 2. Let F Ž t . s t 2 sin ty2 if t / 0 and F Ž0. s 0, then the solution of x9 s F9Ž x . x is x Ž t . s e F Ž t ., which is ACG# on w0, 1x. Note that F Ž t . is not a bounded variation on w0, 1x; thus, F Ž t . and e F Ž t . are not absolutely continuous on w0, 1x. THEOREM 2.5 w8x. A function f : w a, b x ª R n is Henstock]Kurzweil integrable on w a, b x if and only if there exists a continuous function F which is ACG# on w a, b x such that F9Ž t . s f Ž t . almost e¨ erywhere. THEOREM 2.6 w7, 8x ŽControlled Convergence Theorem.. If a sequence of Henstock]Kurzweil integrable function f n4 satisfies the following conditions: Ži. f nŽ x . ª f Ž x . almost e¨ erywhere in w a, b x as n ª `; Žii. the primiti¨ es FnŽ x . s Hab f nŽ s . ds of f n are ACG# uniformly in n; Žiii. the primiti¨ es Fn are equicontinuous on w a, b x, then f is Henstock]Kurzweil integrable on w a, b x and we ha¨ e Hab f nŽ x . dx ª Hab f Ž x . dx as n ª `.
122
CONGXIN, BAOLIN, AND LEE
If conditions Žii. and Žiii. are replaced by the condition: Živ. g Ž x . F f nŽ x . F hŽ x . almost e¨ erywhere on w a, b x, where g and h are Henstock]Kurzweil integrable, then the result of Theorem 2.6 also holds.
3. GENERALIZED CARATHEODORY SYSTEMS In this section, a generalized Caratheodory system of the form Ž1.1. is defined by using the Henstock]Kurzweil integral. The main result of this section is an existence theorem for the solution to the generalized Caratheodory system. DEFINITION 3.1 w2x. Let right-hand side function f Ž t, x . of the system Ž1.1. be a Caratheodory function defined on an open region G; i.e., f is continuous in x for almost all t and measurable in t for each fixed x. Suppose there exists a Lebesgue integrable function mŽ t . on every bounded closed subregion G 0 ; G such that f Ž t , x . F m Ž t . , for Ž t , x . g G 0 ,
Ž 3.1.
then f Ž t, x . is said to satisfy the Caratheodory condition, and the system Ž1.1. is said to be a Caratheodory system. DEFINITION 3.2 w2x. A function x Ž t .: I ª R n Ž I represents an interval in R1 . is said to be a solution of the Caratheodory system Ž1.1. or, in short, C-solution, if x Ž t . satisfies the following conditions: Ži. x Ž t . is absolutely continuous on each compact subinterval of interval I; Žii. Ž t, x . g G for t g I; Žiii. x9Ž t . s f Ž t, x Ž t .. for almost everywhere t g I. Now we can generalize the Caratheodory system to the generalized Caratheodory system using the Henstock]Kurzweil integral. DEFINITION 3.3. Let the right-hand side function of the system Ž1.1. be a Caratheodory function defined on an open region G. Suppose that there exist two Henstock]Kurzweil integrable functions g Ž t . and hŽ t . for every bounded closed subregion G 0 ; G such that g Ž t . F f Ž t , x . F h Ž t . , for all x and almost all t with Ž t , x . g G 0 ,
Ž 3.2. then the f Ž t, x . satisfies the generalized Caratheodory condition on G, and the system Ž1.1. is a generalized Caratheodory system.
DISCONTINUOUS SYSTEMS
123
DEFINITION 3.4. A function x Ž t .: I ª R n Ž I represents an interval in R is said to be a solution of the generalized Caratheodory system Ž1.1. or GC-solution if x Ž t . satisfies the following conditions: 1.
Ži. x Ž t . is ACG# on each compact subinterval of I; Žii. Ž t, x . g G for t g I; Žiii. x9Ž t . s f Ž t, x Ž t .. for almost everywhere t g I. We remark that if g and h are Lebesgue integrable functions, then the previous generalized Caratheodory system reduces to the Caratheodory system, and the GC-solution also reduces to the C-solution. Now we give the existence theorem for the GC-solution for the preceding generalized Caratheodory system. THEOREM 3.5. Suppose that f Ž t, x . satisfies the conditions of Definitions 3.3, then there exists a GC-solution F of the generalized Caratheodory system Ž1.1. on some inter¨ al < t y t < F b , and satisfies F Žt . s j . Proof. Let Žt , j . g G be fixed and let G 0 be a bounded closed subregion of G, G0 :
< t y t < F a,
5 x y j 5 F b,
Ž 3.3.
then there exist two Henstock]Kurzweil integrable functions g Ž t . and hŽ t . such that for all x and for almost all t with Ž t, x . g G 0 we have 0 F f Ž t, x . y g Ž t . F hŽ t .. y g Ž t .. By Theorem 2.3, h y g is Lebesgue integrable. Let FŽ t, x. s f t, x q
ž
t
Ht g Ž s . ds
/
y gŽ t. ,
Ž 3.4.
then F is a Caratheodory function, 0 F F Ž t, x . F hŽ t . y g Ž t . for all Ž t, x . g GX0 , where GX0 is a bounded closed subregion of G 0 , such that xq
t
Ht g Ž s . ds y j
F b, for all Ž t , x . g GX0 ,
then x9 s F Ž t, x . is a Caratheodory system. By Caratheodory existence theorem, there is a function C on some interval < t y t < F b such that C9Ž t . s F Ž t, C Ž t .. almost everywhere in this interval and C Žt . s j . Let FŽ t. s CŽ t. q
t
Ht g Ž s . ds.
Ž 3.5.
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CONGXIN, BAOLIN, AND LEE
Then, for almost all t, F9 Ž t . s C9 Ž t . q g Ž t . s FŽ t, CŽ t. . q g Ž t. s f t, CŽ t. q
ž
t
Ht g Ž s . ds
y gŽ t. q gŽ t.
/
s f Ž t, FŽ t. . , and FŽt . s CŽt . q
t
Ht g Ž s . ds s j .
The proof is complete. EXAMPLE 3. Consider x9 s f Ž t , x . s g Ž t , x . q h Ž t . ,
Ž 3.6.
where 5 g Ž t, x .5 F g 1Ž t . for all < t < F 1, 5 x 5 F 1, g 1Ž t . is Lebesgue integrable on < t < F 1, and hŽ t . s Ž drdt .Ž t 2 sin ty2 . if t / 0 and hŽ0. s 0. Here t s j s 0. h is Henstock]Kurzweil integrable but not Lebesgue integrable and h Ž t . y g 1 Ž t . F f Ž t , x . F h Ž t . q g 1 Ž t . , for < t < F 1, 5 x 5 F 1. Thus, by Theorem 3.5, there exists a GC-solution F of x9 s f Ž t, x . with F Ž0. s 0. For instance, if g Ž t, x . s t 2 x, then this example is Example 1, FŽ t. s e t
3
r3
t ys 3 r3
H0 e
h Ž s . ds.
THEOREM 3.6. Let the right-hand side function f Ž t, x . of the system Ž1.1. be a Caratheodory function on the open region G, Let G 0 : < t y t < F a, 5 x y j 5 F b, be a fixed bounded closed subregion of G, and let f Ž t, w Ž t .. be Henstock]Kurzweil integrable on < t y t < F a for any step function w Ž t . defined on < t y t < F a with ¨ alues in 5 x y j 5 F b. Denote Fw Ž t . s Htt f Ž s, w Ž s .. ds. If Fw : w is a step function4 is ACG# uniformly in w and equicontinuous on < t y t < F a, then the system Ž1.1. is a generalized Caratheodory system. Proof. Notice that f Ž t, x . is a Caratheodory function. Thus, there exist two measurable functions uŽ t . and ¨ Ž t . defined on < t y t < F a with values in 5 x y j 5 F b such that f Ž t , uŽ t . . F f Ž t , x . F f Ž t , ¨ Ž t . . ,
Ž 3.7.
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DISCONTINUOUS SYSTEMS
for all Ž t, x . g G 0 wsee 14, Lemma 17.2x. Next, we should show that f Ž t, uŽ t .. and f Ž t, ¨ Ž t .. are Henstock]Kurzweil integrable by using the controlled convergence Theorem 2.6. First, there is a sequence k nŽ t .4 of step functions defined on < t y t < F a with values in 5 x y j 5 F b such that k nŽ t . ª uŽ t . almost everywhere as n ª `. Thus, f Ž t, k nŽ t .. ª f Ž t, uŽ t .. almost everywhere as h ª `. Let Fn Ž t . s
t
Ht f Ž s, k Ž s . . ds. n
Then FnŽ t .4 is ACG# uniformly in n and equicontinuous. By Theorem 2.6, f Ž t, uŽ t .. is Henstock]Kurzweil integrable. Similarly, f Ž t, ¨ Ž t .. is Henstock]Kurzweil integrable. Let g Ž t . s f Ž t, uŽ t .., hŽ t . s f Ž t, ¨ Ž t .., then by Ž3.7. we have g Ž t . F f Ž t , x . F hŽ t . . The proof is complete.
4. UNIQUENESS THEOREM FOR THE GENERALIZED CARATHEODORY SYSTEM In this section, the uniqueness theorems for the solutions of the generalized Caratheodory system is given by using some inequalities from the Henstock]Kurzweil integral. LEMMA 4.1 w8x. If f Ž t .: w a, b x ª R n is a Henstock]Kurzweil integrable function with respect to the function g Ž t .: w a, b x ª R n, then, for e¨ ery c g w a, b x, we ha¨ e s
lim
Ha f Ž t . dg Ž t . y f Ž c . Ž g Ž s . y g Ž c . .
lim
Hu f Ž t . dg Ž t . q f Ž c . Ž g Ž u . y g Ž c . .
sªc
uªc
b
c
s
Ha f Ž t . dg Ž t . ,
s
Hc f Ž t . dg Ž t . .
b
Ž 4.1. Ž 4.2.
LEMMA 4.2. Let f, g: w a, b x ª R n be functions for which the Henstock]Kurzweil-integral Hab f Ž t . dg Ž t . exists. If u, ¨ : w a, b x ª R1 are functions such that integral Hab uŽ t . d¨ Ž t . exists and if there is a positi¨ e function d : w a, b x ª Ž0, q`. such that < t y j < 5 f Ž j . Ž g Ž t . y g Ž j . . 5 F Ž t y j . u Ž j . Ž ¨ Ž t . y ¨ Ž j . . , Ž 4.3.
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CONGXIN, BAOLIN, AND LEE
for e¨ ery t g w j y d Ž j ., j q d Ž j .x, j g w a, b x. Then b
Ha f Ž t . dg Ž t .
F
b
Ha u Ž t . d¨ Ž t . .
Ž 4.4.
Proof. Assume that e ) 0 is given. Because the integrals Hab f Ž t . dg Ž t ., uŽ t . d¨ Ž t . exist, there is a positive function d 1 on w a, b x with d 1Ž s . F d Ž s . for s g w a, b x such that for every d 1-fine partition D of w a, b x,
Hab
j1, j2 , . . . , jk4 ,
a s t 0 - t 1 - ??? - t k s b and
with j i y d 1Ž j i . - t iy1 F j i F t i - j i q d 1Ž j i ., i s 1, 2, . . . , k, we have k
b
Ý f Ž j i . Ž g Ž t i . y g Ž t iy1 . . y H f Ž t . dg Ž t . a
is1 k
b
Ý u Ž j i . Ž ¨ Ž t i . y ¨ Ž t iy1 . . y H u Ž t . d¨ Ž t . a
is1
-e,
Ž 4.5.
- e.
Ž 4.6.
By Ž4.3., for j s 1, 2, . . . , k, we have f Ž j j . Ž g Ž t j . y g Ž t jy1 . . F f Ž j j . Ž g Ž t j . y g Ž j j . . q f Ž j j . Ž g Ž j j . y g Ž t jy1 . . F u Ž j j . Ž ¨ Ž t j . y ¨ Ž t jy1 . . . Hence, b
Ha
k
f Ž t . dg Ž t . F
b
Ý f Ž j i . Ž g Ž t i . y g Ž t iy1 . . y H f Ž t . dg Ž t . a
is1
k
q
Ý f Ž j i . Ž g Ž t i . y g Ž t iy1 . . is1 k
-eq
Ý u Ž j i . Ž ¨ Ž t i . y ¨ Ž t iy1 . . is1
- 2e q
b
Ha u Ž t . d¨ Ž t . .
Because e ) 0 is arbitrary, the inequality Ž4.4. is satisfied. COROLLARY 4.3. If f : w a, b x ª R1 , f Ž t .< F c for t g w a, b x where c is a constant, g: w a, b x ª R1 is of bounded ¨ ariation on w a, b x, and the integral
DISCONTINUOUS SYSTEMS
127
Hab f Ž t . dg Ž t . exists then b
Ha f Ž t . dg Ž t .
F c Var ab g ,
Ž 4.7.
where Var ab g is the total ¨ ariation of g on w a, b x. LEMMA 4.4. Let h: w a, b x ª R1 be a nonnegati¨ e nondecreasing function which is continuous from the left on Ž a, b x. Assume that u: w0, q`. ª w0, q`. is a continuous nondecreasing function with primiti¨ e U: w0, q`. ª R1, then the integral Hab uŽ hŽ t .. dhŽ t . exists and b
Ha uŽ h Ž t . . dhŽ t . F U Ž h Ž b . . y U Ž h Ž a. . .
Ž 4.8.
Proof. The composition of functions u and h given by uŽ hŽ t .. for t g w a, b x is nondecreasing on w a, b x. Therefore, the integral Hab uŽ hŽ t .. dhŽ t . exists. Assume that e ) 0 is given. By the definition of the primitive U to u, for every g w0, q`. there exists u Ž s . ) 0 such that for every h ) 0 with 0 F
Ž 4.9.
Because lim t ª jq hŽ t . s hŽ jq. for j g w a, b ., there exists a dqŽ j . ) 0, dqŽ b . s 1 such that for t g Ž j , j q dqŽ j .x l w a, b x we have 0 F h Ž t . y h Ž jq . F u Ž h Ž jq . . . Let s s hŽ jq. and h s hŽ t . y hŽ jq. we obtain u Ž h Ž jq . . Ž h Ž t . yh Ž jq . . FU Ž h Ž t . . yU Ž h Ž jq . . q e Ž h Ž t . yh Ž jq . . . Further, we have u Ž h Ž j . . Ž h Ž jq . yh Ž j . . y U Ž h Ž jq . . yU Ž h Ž j . . s
q.
HhhŽ jj. Ž
u Ž h Ž j . . y u Ž s . ds
G 0, because uŽ hŽ j .. F uŽ s . for s g w hŽ j ., hŽ jq.x . Therefore, u Ž h Ž j . . Ž h Ž t . yh Ž j . . s u Ž h Ž j . . Ž h Ž t . y h Ž jq . . q u Ž h Ž j . . Ž h Ž jq . y h Ž j . . F UŽ hŽ t . . y UŽ hŽ j . . q e Ž hŽ t . y hŽ j . . ,
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CONGXIN, BAOLIN, AND LEE
for t g Ž j , j q dqŽ j .. l w a, b x. From the inequality Ž4.9. and from the left continuity of the function h at the point j g Ž a, b x, there is a dyŽ j . ) 0, dyŽ a. s 1 such that for t g w j y dyŽ j ., j x l w a, b x the inequality, uŽ hŽ j . . Ž hŽ t . y hŽ j . . F UŽ hŽ j . . y UŽ hŽ t . . q e Ž hŽ j . y hŽ t . . is satisfied. Let d Ž j . s min dyŽ j ., dqŽ j .4 for j g w a, b x, then for j g w a, b x and t g w j y d Ž j ., j q d Ž j .x l w a, b x we obtain by the preceding inequalities the relation, < t y j < < uŽ hŽ j . . Ž hŽ t . y hŽ j . . < F Ž t y j . Ž UŽ hŽ t . . q e hŽ t . y UŽ hŽ j . . y e hŽ j . . . By Lemma 4.2, this inequality implies b
b
Ha u Ž h Ž t . . dhŽ t . F Ha d U Ž h Ž t . . q e h Ž t . s U Ž h Ž b . . y U Ž h Ž a. . q e Ž h Ž b . y h Ž a. . . Because e ) 0 can be chosen arbitrarily small, the proof is complete. LEMMA 4.5. Let c : w a, b x ª w0, q`., h: w a, b x ª w a, q`. be gi¨ en where c is bounded and h is nondecreasing and continuous from the left on the inter¨ al w a, b x. Suppose that the function v : w0, q`. ª R1 is continuous, nondecreasing, v Ž0. s 0, v Ž r . ) 0 for r ) 0. For u ) 0, let V Ž u. s
u
Hu
0
1
vŽ r.
dr ,
Ž 4.10.
with some u 0 ) 0. The function V: Ž0, q`. ª R 1 is increasing, V Ž u 0 . s 0 lim uª 0y V Ž u. s a G y`, lim uªq` V Ž u. s b F q`. Assume that for j g w a, b x the inequality,
cŽj . Fkq
j
Ha v Ž c Ž t . . dhŽ t .
and
Ž 4.11.
holds, where k ) 0 is a constant. If V Ž k . q hŽ b . y hŽ a. - b then for j g w a, b x we ha¨ e
c Ž j . F Vy1 Ž V Ž k . q h Ž j . y h Ž a . . ,
Ž 4.12.
where Vy1 : Ž a , b . ª R1 is the in¨ erse function of the function V in Ž4.10..
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DISCONTINUOUS SYSTEMS
Proof. If we have V Ž l . q hŽ b . y hŽ a. - b for some l ) 0 then for all t g w a, b x we have
a - V Ž l . q h Ž t . y h Ž a. - b . Therefore the value of V Ž l . q hŽ t . y hŽ a. belongs to the domain of Vy1 provided t g w a, b x, and for t we can define Hl Ž t . s Vy1 Ž V Ž l . q h Ž t . y h Ž a . . . Define further
w Ž s . s v Ž Vy1 Ž V Ž l . q s . . ,
Ž 4.13.
for s g w0, b y V Ž l .x. At Vy1 Ž V Ž l . q s . there exists a derivative V9 of the function V and V9 Ž Vy1 Ž V Ž l . q s . . s
1 y1
vŽV
Ž V Ž l . q s. .
/ 0.
The well-known formula for the derivative of the inverse function leads to d ds
Vy1 Ž V Ž l . q s . s
1 y1
V9 Ž V
Ž V Ž l . q s. .
s v Ž Vy1 Ž V Ž l . q s . . s w Ž s. ,
Ž 4.14.
for s g w0, b y V Ž l .x. If now j g w a, b x is given, then using the definition of the function w from Ž4.13. we obtain j
j
y1
Ha v Ž H Ž t . . dhŽ t . s Ha v Ž V l
s
Ž V Ž l . q h Ž t . y h Ž a. . . dh Ž t .
j
Ha w Ž h Ž t . y h Ž a. . d Ž h Ž t . y h Ž a. . .
This together with Ž4.14. and Lemma 4.4 imply j
y1
Ha v Ž H Ž t . . dhŽ t . F V l
Ž V Ž l . q h Ž j . y h Ž a. . y Vy1 Ž V Ž l . .
s Hl Ž j . y l,
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CONGXIN, BAOLIN, AND LEE
and consequently for j g w a, b x we have the inequality, lq
j
Ha v Ž H Ž t . . dhŽ t . F H Ž j . . l
l
Assume that e 0 ) 0 is such that V Ž k q e 0 . q hŽ b . y hŽ a. - b . Let us take an arbitrary e g Ž0, e 0 . and set l s k q e . For this case the last inequality reads kqeq
j
Ha v Ž H
kq e
Ž t . . dh Ž t . F Hkq e Ž j . ,
and taking into account the relation Ž4.11. for every j g w a, b x we get
c Ž j . y Hkq e Ž j . F k q
j
Ha v Ž c Ž t . . dhŽ t . y k y e j
Ha v Ž H
y
s ye q
j
Ha
kq e
Ž t . dh Ž t .
v Ž c Ž t . . y v Ž Hkq e Ž t . . dh Ž t . . Ž 4.15.
Hence c Ž a. y Hkq e Ž j . F ye and also v Ž c Ž a.. y v Ž Hkq e Ž a.. F 0 because the function v is assumed to be nondecreasing. The functions c and Hkq e are bounded and therefore there is a constant K ) 0 such that
v Ž c Ž t . . y v Ž Hkq e Ž t . . - K , for t g w a, b x. Using Lemma 4.1 and Corollary 4.3 we obtain from the last two displayed inequalities,
c Ž j . y Hkq e Ž j . F ye q v Ž c Ž a . . y v Ž Hkq e Ž t . . Ž h Ž aq . y h Ž a . . q limq dª0
j
Haqd
v Ž c Ž t . . y v Ž Hkq e Ž t . . dh Ž t .
F ye q K limq h Ž j . y h Ž a q d . dª0
s ye q K h Ž j . y h Ž aq . . Because lim j ª aq hŽ j . s hŽ aq. , an h ) 0 can be found such that for j g Ž a, a q h . the inequality hŽ j . y hŽ aq. - erŽ2 K q 1. holds and
131
DISCONTINUOUS SYSTEMS
therefore also
c Ž j . y Hkq e Ž j . - ye q
Ke 2Kq 1
-y
e 2
- 0,
for j g Ž a, a q h .. Let us set T s sup t g w a, b x ; c Ž j . y Hkq e Ž j . - 0, for j g w a, t x 4 . As has been shown previously, we have T ) a and for j g w a, T . the inequality c Ž j . y Hkq e Ž j . - 0 and therefore v Ž c Ž j .. y v Ž Hkq e Ž j .. F 0 holds. The last conclusion is a consequence of the assumption that v is nondecreasing. By Ž4.15. and Lemma 4.1 we have
c Ž T . y Hkq e Ž T . F ye q limq dª0
Ty d
Ha
v Ž c Ž t . . y v Ž Hkq e Ž t . . dh Ž t .
q v Ž c Ž T . . y v Ž Hkq e Ž T . .
Ž h Ž T . y h Ž Ty . .
F ye - 0, because h Ž T . y h Ž Ty . s h Ž T . y limy h Ž t . s 0, tªT
and lim
Ty d
H
dª0 q a
v Ž c Ž t . . y v Ž Hkq e Ž t . . dh Ž t . F 0.
If we assume that T - b then we can repeat this procedure for j ) T by virtue of the inequality,
c Ž j . y Hkq e Ž j . F ye q
j
HT
v Ž c Ž t . . y v Ž Hkq e Ž t . . dh Ž t . ,
thus obtaining c Ž j . y Hkq e Ž j . - 0 for j g w T, T q h x for some h ) 0. Hence T s b and
c Ž j . - Hkq e Ž j . s Vy1 Ž V Ž k q e . q h Ž j . y h Ž a . . , for j g w a, b x. Because the function V is continuous and the last inequality holds for every sufficiently small e ) 0, we obtain the inequality Ž4.12.. The proof is complete.
132
CONGXIN, BAOLIN, AND LEE
COROLLARY 4.6. If c , h, and k satisfy the assumptions in lemma 4.5 and if for j g w a, b x the inequality,
cŽj . FkqL
j
Ha c Ž t . dhŽ t .
holds with a constant L ) 0 instead of Ž4.11., then for e¨ ery j g w a, b x the inequality,
c Ž j . F ke LŽ hŽ j .yhŽ a.. is satisfied. Lemma 4.5 represents a Bellman type inequality for the Henstock] Kurzweil integral. Results of this type are especially useful for deriving uniqueness results for the generalized Caratheodory system. DEFINITION 4.7. A solution x: wt , t q h x ª R n of the generalized Caratheodory system Ž1.1. is called locally unique for increasing values of t if for any solution y: wt , t q s x ª R n of the generalized Caratheodory system Ž1.1. with y Žt . s x Žt . there exists h1 ) 0 such that x Ž t . s y Ž t . for t g wt , t q h x l wt , t q s x l wt , t q h1 x. THEOREM 4.8. Assume that the system Ž1.1. is a generalized Caratheodory system and the right-hand side function f Ž t, x . of system Ž1.1. satisfies the following condition on G 0 : < t y t < F a, 5 x y j 5 F b, f Ž u , x . y f Ž u , y . Ž ¨ y u . F v Ž 5 x y y 5 . Ž h Ž ¨ . y h Ž u . . , Ž 4.16. for each inter¨ al w u, ¨ x with u g w u, ¨ x ; wt y a, t q ax and all x, y belonging to 5 x y j 5 F b, where h: wt y a, t q ax ª R1 is nondecreasing and continuous from the left. v : w0, q`. ª R1 is continuous, nondecreasing, v Ž r . ) 0 for r ) 0, v Ž0. s 0 and u
limq
¨ ª0
H ¨
1
vŽ r.
dr ,
Ž 4.17.
for e¨ ery u ) 0. Then e¨ ery solution x of the generalized Caratheodory system Ž1.1. such that x Žt . s j is locally unique for increasing ¨ alues of t.
DISCONTINUOUS SYSTEMS
133
Proof. Let x, y: wt , t q h x ª R n be solutions of the generalized Caratheodory system Ž1.1. such that x Žt . s y Žt . s j . Assume that e ) 0 is given. Because the Henstock]Kurzweil integrals Hrt w f Ž s, x Ž s .. y f Ž s, y Ž s ..x ds, Htt v Ž5 x Ž s . y y Ž s .5. dhŽ s . exist there is a positive function d 1 on wt , t x such that for every d 1-fine partition D of wt , t x,
t s t 0 - t 1 - ??? - t k s t and
u1 , u2 , . . . , uk4 ,
with u i y d 1Ž u i . - t iy1 F u i F t i - u i q d 1Ž u i ., i s 1, 2, . . . , k, we have t
f Ž s, x Ž s . . y f Ž s, y Ž s . . ds
Ht
F
t
f Ž s, x Ž s . . y f Ž s, y Ž s . . ds
Ht
k
y Ý f Ž u i , x Ž u i . . y f Ž u i , y Ž u i . . Ž t i y t iy1 . is1 k
q
f Ž u i , x Ž u i . . y f Ž u i , y Ž u i . . Ž t i y t iy1 .
Ý is1
-
e 2
k
Ý v Ž 5 x Ž u i . y y Ž u i . 5.Ž h Ž t i . y h Ž t iy1 . . ,
q
is1
and k
Ý v Ž 5 x Ž u i . y y Ž u i . 5.Ž h Ž t i . y h Ž t iy1 . . is1 k
F
Ý v Ž 5 x Ž u i . y y Ž u i . 5.Ž h Ž t i . y h Ž t iy1 . . is1 t
Ht v Ž 5 x Ž s . y y Ž s . 5. dhŽ s .
y t
Ht v Ž 5 x Ž s . y y Ž s . 5. dhŽ s .
q -
e 2
q
t
Ht v Ž 5 x Ž s . y y Ž s . 5. dhŽ s . ,
134
CONGXIN, BAOLIN, AND LEE
then 5 xŽ t. y yŽ t. 5 s
t
f Ž s, x Ž s . . y f Ž s, y Ž s . . ds
Ht
-eq
t
Ht v Ž 5 x Ž s . y y Ž s . 5. dhŽ s . .
Because e ) 0 is arbitrary, then 5 xŽ t. y yŽ t. 5 F s
t
Ht v Ž 5 x Ž s . y y Ž s . 5. dhŽ s . tq d
Ht
v Ž 5 x Ž s . y y Ž s . 5 . dh Ž s .
t
Htqdv Ž 5 x Ž s . y y Ž s . 5. dhŽ s . ,
q
where 0 - d - t y t . By Lemma 4.1, we have tq d
Ht
v Ž 5 x Ž s . y y Ž s . 5 . dh Ž s .
s v Ž 5 x Ž t . y y Ž t . 5 . h Ž tq . y h Ž t . q limq t1ª t
F
tq d
Ht
sup sg wt , t q d x
v Ž 5 x Ž s . y y Ž s . 5 . dh Ž s .
1
v Ž 5 x Ž s . y y Ž s . 5 . h Ž t q d . y h Ž tq .
s AŽ d . ,
Ž 4.18.
because v Ž5 x Žt . y y Žt .5 s v Ž0. s 0. Because the limit hŽtq. exists we have also lim d ª 0q AŽ d . s 0, therefore, x Ž t . y y Ž t . F AŽ d . q
t
Htqdv Ž 5 x Ž s . y y Ž s . 5. dhŽ s . ,
for t g wt q d , t q h x. Let u 0 ) 0 and set V Ž u. s
u
Hu
0
1
vŽ r.
dr.
DISCONTINUOUS SYSTEMS
135
Using Lemma 4.5 we obtain x Ž t . y y Ž t . F Vy1 Ž V Ž A Ž d . . q h Ž t . y h Ž t q d . . ,
Ž 4.19.
for t g wt q d , t q h x provided V Ž AŽ d .. q hŽt q h . y hŽt q d . - b , where b s lim uªq` V Ž u. F q`. Evidently, we have V Ž A Ž d . . q h Ž t q h . y h Ž t q d . F V Ž A Ž d . . q h Ž t q h . y h Ž tq . , and because lim d ª 0q AŽ d . s 0 and lim uª 0q V Ž u. s y`, we have lim V Ž A Ž d . . q h Ž t q h . y h Ž tq . s y`.
dª0 q
Hence there is a d 0 ) 0 such that for d g Ž0, d 0 . the inequality V Ž AŽ d .. q hŽt q h . y hŽtq. - b holds. Applying now the map V to both sides of Ž4.19., we have V Ž 5 x Ž t . y y Ž t . 5. F V Ž AŽ d . . q h Ž t . y h Ž t q h . , and this yields V Ž 5 x Ž t . y y Ž t . 5 . y V Ž A Ž d . . F h Ž t . y h Ž t q d . F h Ž t . y h Ž tq . . From the definition of V we therefore have 5 Ž .
HAŽxd .t yy t
1
Ž .5
vŽ r.
dr F h Ž t q h . y h Ž tq . .
Assume now that 5 x Ž t*. y y Ž t9.5 s k ) 0 for some t* g Žt , t q h x, then 1
k
HAŽ d . v Ž r .
dr F h Ž t q h . y h Ž tq . - q`,
for every d g Ž0, d 0 . such that d - t* y t . Now it is possible to use d ª 0q to obtain the inequality, limq
dª0
k
1
HAŽ d . v Ž r .
dr F h Ž t q h . y h Ž tq . - q`,
which contradicts the assumption on the function v . Therefore 5 x Ž t . y y Ž t .5 s 0 for t g Žt , t q h x. The proof is complete. COROLLARY 4.9. If the function v Ž r . s Lr, r ) 0, L ) 0 in Theorem 4.8, then the result in Theorem 4.8 also holds.
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CONGXIN, BAOLIN, AND LEE
The local uniqueness for increasing values of t can be extended to the global uniqueness for increasing values of t in the same manner as is done for the case of classical ordinary differential equations, we elaborate on this.
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