含参量Cauchy系统有界变差解的唯一性(英文)

AbstractBy using Henstock-Kurzweil integral， uniqueness of bounded variation solutions of cauchy systems involving parameter is discussed in the condition of generalization ofω.

Key wordsCauchy systems involving parameter； Henstock-Kurzweil integral； Bounded variation solutions； Uniqueness

CLC numberO175.14Document codeA

Explain the signs in this paper： Assume that I = （a，b），？∞< a < b < +∞，x ：[a，b]Rnis an Rn？valued function defined on [a，b]. Here∥·∥stands for the Euclidean norm in Rn. B（x0，c2） = {x∈Rn；∥x？x0∥< c2}，c2> 0.

Definition 1[1？5]A function x（t） is called Henstock-Kurzweil integrable over[a，b] if there is an A∈Rnsuch that givenε> 0， there is a positive functionδ（t） ： [a，b]（0，+∞）， such that

where h ： IR is a continuous from the left，nondecreasing function defined on I andω： [0，+∞）R is continuous， increasing function withω（0） = 0，ω（r） > 0 for （r > 0）.

（3） for every step functionΨ（t） on [α，β]？I， the function f（t，Ψ（t），μ） is Henstock-Kurzweil integrable on [α，β].

Lemma 1[6？7]Letψ： [a，b][0，+∞），h ： [a，b][0，+∞）， be given whereψis bounded and h is continuous from the left，nondecreasing function on the interval[a，b].

Suppose that the functionω： [0，+∞）R is continuous，increasing function withω（0） = 0，ω（r） > 0 for （r > 0）.

3Main Results

Therefore∥xμ（s）？yμ（s）∥= 0 for s∈（t0，t0+η] and the result is proved.

RemarkAssume thatω（r） = Lr，r≥0，L is a positive constant. Then the result of Theorem 1 is the special result in reference [3].

The local uniqueness for increasing values of t can be extended to the global uniqueness for increasing values t as following：

Theorem 2Assume that f∈F（G，h，ω） and that xμ（t） ： [α1，β1]Rn，yμ（t） ：[α2，β2]Rnare two bounded variation solutions of （1）， If the condition （3） is satisfied and if xμ（s） = yμ（s） for some s∈[α1，β1]∩[α2，β2]， then xμ（s） = yμ（s） for all t∈[α1，β1]∩[α2，β2]∩[s，b）.

ProofThe intersection s∈[α1，β1]∩[α2，β2]∩[s，b） is a closed interval of the form [s，c] where c < b. Denote

M = {t∈[s，c]；xμ（σ） = yμ（σ），σ∈[s，t]}.

If s = c then there is nothing to prove. Assume that s < c and putβ= supM， we evidently haveβ≤c.

Because the solutions xμ（t） and yμ（t） are continuous from the left in virtue of the assumption that the function h is continuous from the left， we have [s，β]？M， and we have to prove thatβ= c. If the inequalityβ< c holds， then Theorem 1 could be used to show that there is aη> 0 such that xμ（σ） = yμ（σ） forσ∈[β，β+η]， becauseβ∈M and xμ（β） = yμ（β）. This contradicts the definition ofβand consequently M = [s，c].

[2] Wu Congxin， Li Baolin. Bounded Variation Solutions for Discontinuous Systems[J]. J of Math Study， 1998， 31（4）： 417-427.

[3] Li Baolin， Xiao Yanping， Zang Zilong. Bounded Variation Solutions of Cauchy Systems Involving Parameter[J]. J of Northwest Normal University， 2005， 41（4）： 12-16.

[4] Li Baolin， Ma Xuemin. Continuous Dependence on a Parameter of Bounded Variation Solutions for Discontinuous Systems[J]. J of Math Study， 2007， 40（2）： 159-163.

[5] Jiang Xudong， Li Baolin. Uniqueness of BoundedΦ？Variation Solutions for a class of Impulsive Differential Systems at Fixed Time[J]. J of Gansu Sciences， 2010， 22（2）： 23-128.

[6] Li Baolin， Shang Dequan. Uniqueness of BoundedΦ？Variation Solutions for Kurzweil Equations[J]. J of Lanzhou University， 2007， 43（2）：107-111.

[7] Schwabic S. Generalized Ordinary Differential Equations[M]. Singapore， World Scientific， 1992.