MA Xue-min,LI Bao-lin,LIN Chang-wei

(1.Department of Mathematics,Dingxi Advanced Teacher’s College,Dingxi 743000,China;2.College of Mathermatics and Statistics,Northwest Normal University,Lanzhou 730070,China;3.Department of Geography,Dingxi Advanced Teacher’s College,Dingxi 743000,China)

Continuous Dependence of Bounded Φ-variation Solutions on Parameter for a Class of Discontinuous Systems

MA Xue-min1,LI Bao-lin2,LIN Chang-wei3

(1.Department of Mathematics,Dingxi Advanced Teacher’s College,Dingxi 743000,China;2.College of Mathermatics and Statistics,Northwest Normal University,Lanzhou 730070,China;3.Department of Geography,Dingxi Advanced Teacher’s College,Dingxi 743000,China)

The functions of bounded Φ-variation are development and generalization of bounded variation functions in the usual sense.Henstock-Kurzweil integral is a very useful tool for some discontinuous systems.In this paper,by using Henstock-Kurzweil integral,we establish theorems of continuous dependence of bounded Φ-variation solutions on parameter for a class of discontinuous systems on the base of Φ-function.These results are essential generalizations of continuous dependence of bounded variation solutions on parameter for the systems.

discontinuous systems;Henstock-Kurzweil integral;bounded Φ-variation solution;continuous dependence

§1. Introduction

Consider discontinuous system

where x=(x1,x2,···,xn)T,f:G→Rnis Carath´eodory function,G=Bc×(a,b),Bc= {x∈Rn|‖x‖＜c},−∞＜a＜b＜+∞.This class of discontinuous systems have been found many practical applications such as in the development of discontinuous oscillatory theory and in the theory of modern control[14].The functions of bounded Φ-variation were introduced by Musielak and Orlicz[411].The Henstock-Kurzweil integral has since proved useful in the study of high oscillation function.It encompasses Newton,Riemann and Lebesgue integrals[13].Existence and uniqueness theorem of bounded Φ-variation solutions for this class of discontinuous systems were established in[4-5].Furthermore,theorems of continuous dependence of bounded Φ-variation solutions for this systems on parameters are discussed.

§2. Preliminaries

Let Φ(u)denote a continuous and increasing function defned for u≥ 0 with Φ(0)= 0,Φ(u)＞0 for u＞0 and satisfying the following conditions

(Δ2):There exist u0≥0 and a＞0 such that Φ(2u)≤aΦ(u)for u0≥u＞0;

(c):Φ(u)is a convex function.

We consider the function x:[a,b]→Rn,x(t)is of bounded Φ-variation over[a,b]if for any partition π:a=t0＜t1＜···＜tm=b,we have

VΦ(x;[a,b])is called Φ-variation of x(t)over[a,b].We always assume Φ(u)satisfying(Δ2)and (c).

We denote by BVΦthe class of all functions of bounded Φ-variation with x(a)=0.

Defnition 2.1[1-5]A function x(t)is called Henstock-Kurzweil integrable over[a,b]if there is an A∈Rnsuch that given ε＞0,there is a positive function δ(t):[a,b]→(0,+∞), such that

for every δ-fne partition D:

satisfying ξi−δ(ξi)＜ti−1≤ξi≤ti＜ξi+δ(ξi),i=1,2,···,k.

The element A∈Rnis called the Henstock-Kurzweil integral of x(t)over the interval[a,b] and will be denoted by

RemarkThe Henstock-Kurzweil integral has all the standard properties one normally expects for any kind of integrals and we shall not mention them here.

Defnition 2.2[4-5]A Carath´eodory function f:G→Rnbelongs to the class FΦ(G,h,ω), if

(1)There is a positive function δ:[a,b]→(0,+∞),satisfying τ∈[u,v]⊂(τ−δ(τ),τ+ δ(τ))⊂(a,b)and x∈Bcfor every[u,v],we have

(2)Satisfying τ∈[u,v]⊂(τ−δ(τ),τ+δ(τ))⊂(a,b)and x,y∈Bcfor every[u,v],we have

where h:[a,b]→ R is a nondecreasing function and continuous from the left on the [a,b], ω:[0,+∞)→R is a continuous and increasing function with ω(0)=0,ω(r)＞0 for(r＞0);

(3)For every step function Ψ(t)on[α,β]⊆ [a,b],the function f(Ψ(t),t)is Henstock-Kurzweil integrable on[α,β].

Theorem 2.1[7]IfΦ(u)satisfed (c),then the Φ-variation of a monotonic function x(t)in[a,b]is equal to Φ[|x(b)−x(a)|].

Corollary 2.1[5]Assume that f:G→Rnsatisfes the condition(2).If x:[α,β]→Rn,[α,β]⊂[a,b]is a solution of(1)then x is of bounded Φ-variation and continuous on [α,β]and

§3.Prime Results

Lemma 3.1Assume thatf:G→ Rnsatisfes the condition(2).Ifx:[α,β]→Rn,[α,β]⊂[a,b]is such that(x(t),t)∈G exists,then for every pair s1,s2∈[α,β],the inequality

holds.

ProofFor arbitrary s1,s2∈[α,β],s1＜s2,by Defnition 2.1 and (2),there is a positive function δ(t):[s1,s2]→ (0,+∞)such that for any division of[s1,s2]given by s1=t0＜t1＜···＜tn=s2,{τ1,τ2,···,τn},satisfying τi∈[ti−1,ti]⊂(τi−δ(τi),τi+ δ(τi)),i=1,2,···,n.We have

and since ε can be taken arbitrarily small we obtain the desired result

Lemma 3.2Assume thatfk:G→ Rnbelong to the class FΦ(G,h,ω)fork= 0,1,···and that

for(x,t)∈G.If x:[c,d]→Bcis a function of bounded Φ-variation and continuous on [c,d]⊂[a,b]then

ProofLet ε＞0 be given.Assume that η＞0 is such that

Since x:[c,d]→Bcis a function of bounded Φ-variation and continuous on [c,d],for η＞0,there is a fnite step function ψ:[c,d]→Bc,such that

for τ∈[c,d].Therefore

forτ∈[c,d],t1,t2∈[c,d]and k=0,1,···,becausefk∈FΦ(G,h,ω).The integralsandevidently exist.By Defnition 2.1 and (3),there is a positive function δ(τ):[c,d]→ (0,+∞)such that for any division of[c,d]given by c=t0＜t1＜···＜tm=d,{τ1,τ2,···,τm},satisfying τj∈[tj−1,tj]⊂(τj−δ(τj),τj+ δ(τj)),j=1,2,···,m.We have

By Theorem 2.1,we have

for every k=0,1,···.

Assume that ψ(τ)=cjfor τ∈(sj−1,sj),(sj−1,sj)⊂[c,d].By Defnition 2.1,we have

By(4)

Since ψ is a fnite step function from(6)by using the additive of the integral we obtain the relation

By(5),we have

by the choice of η.Taking k→∞ on both sides of this inequality,we have

and since ε can be taken arbitrarily small we obtain the desired result

Theorem 3.1Assume thatfk:G→ Rnbelong to the class FΦ(G,h,ω)fork= 0,1,···and that

for(x,t)∈G.Let xk:[α,β]→Rn,k=1,2,···be solutions of the discontinuous systems

n[α,β].Then x:[α,β]→Ris of bounded Φ-variation and continuous on[α,β]and it is a solution of the discontinuous system

on[α,β].

Proof By Lemma 3.1 and Theorem 2.1,

for every k=1,2,···and s1,s2∈[α,β].By Corollary 2.1,then

for every k=1,2,···and s1,s2∈[α,β].By Lemma 3.2,we have

Assume that ε＞0.By continuity of the function ω:[0,+∞)→R and ω(0)=0,there exists a δ＞0,such that

So whenever k≥p(t),we have

For s1,s2∈[α,β],by Theorem 2.1,for any division given by s1=t0＜t1＜···＜tm=s2and any{τ1,τ2,···,τm}satisfying τi∈[ti−1,ti]⊂(τi−δ(τi),τi+δ(τi)),i=1,2,···,m,since fk:G→Rnbelong to the class FΦ(G,h,ω)for k=0,1,···,we have

By(10)whenever k≥p(t),we have

Therefore,we have

By(9)and(11),we have

for every[s1,s2]⊂[α,β].Therefore

for every s1,s2∈[α,β],which means that x:[α,β]→Rnis a solution of(7)on the interval [α,β].

Theorem 3.2Assume thatfk:G→ Rnbelong to the class FΦ(G,hk,ω),where hk:[a,b]→R is a nondecreasing function and continuous for k=0,1,···.

Assume further that

for every a≤t1≤t2≤b.Suppose that

for (x,t)∈G.Let xk:[α,β]→ Rn,k=1,2,··· be a sequence of solutions of the discontinuous systems

on[α,β]⊂(a,b)such that

Then x:[α,β]→Rnis a continuous function of bounded Φ-variation on[α,β]and it is a solution of the discontinuous system

on the interval[α,β].

ProofSince xkis a solution of(13),by Lemma 3.1 and Theorem 2.1,we have

for k=1,2,···and s1,s2∈[α,β],s1＜s2.Then

By(14)and(12)we have also

Assume that ε＞0.By continuity of the function ω:[0,+∞)→R and ω(0)=0,there exists a δ＞0,such that

So whenever k≥p(t),we have

For [s1,s2]⊂[α,β],by the Defnition 2.1,for any division given by s1=t0＜t1＜···＜tm=s2and any{τ1,τ2,···,τm}satisfying τi∈[ti−1,ti]⊂(τi−δ(τi),τi+δ(τi)),i= 1,2,···,m,since fk:G→Rnbelong to the class FΦ(G,hk,ω)for k=0,1,···,we have

By(18)whenever k≥p(t),we have

Therefore,we have

By(17)and(19),we have

for every[s1,s2]⊂[α,β].Therefore

for every s1,s2∈[α,β],which means that x:[α,β]→ Rnis a solution of(15)on the interval[α,β].

RemarkNotice that if

by Theorem 1.15 of[7],we have

where BV[α,β]represents the class of all functions of bounded variation in usual sense on [α,β].

For example Φ(u)=up(1＜p＜∞),we have

Therefore,Theorem 3.1 and Theorem 3.2 are essential generalization of results in the paper[3].

[1]WU Cong-xin,LI Bao-lin,STANLEY LEE E.Discontinuous systems and the Henstock-Kurzweil integral[J]. J of Mathematical Analysis and Applications,1999,229(1):119-136.

[2]WU Cong-xin,LI Bao-lin.Bounded variation solutions for discontinuous systems[J].J of Mathematics Study,1998,31(4):417-427.

[3]LI Bao-lin,MA Xue-min.Continuous dependence on a parameter of bounded variation solutions for discontinuous systems[J].J of Mathematics Study,2007,40(2):159-163.

[4]LIANG Xue-feng,HE Wan-sheng,LI Bao-lin.Uniqueness of bounded Φ-variation solutions for a class of discontinuous system[J].Chinese Journal of Engineering Mathematics,2012,29(5):757-762.

[5]XIAO Yan-ping,LI Bao-lin.Bounded Φ-variation solutions to a kind of discontinuous systems[J].Chinese Journal of Engineering Mathematics,2008,25(3):489-494.

[6]LI Bao-lin,WU Cong-xin.Continuous dependence of bounded Φ-variation solutions on parameters for Kurzweil equations[J].J of Northeast Mathematics,2005,21(4):421-430.

[7]MUSIELAK J,ORLICZ W.On generalized variations(I)[J].Studia Mathematics,1959,18(1):11-41.

[8]LESNIEWZ R,ORLICZ W.On generalized variations(II)[J].Studia Mathematics,1973,45(1):71-109.

[9]MUSIELAK J,ORLICZ W.On space of functions of fnite generalized variations[J].Bulletin de I’Academie Polonaise des Science(III),1957,5(1):389-392.

[10]MATUSZEWSKSA W,ORLICZ W.On certain properties of Φ-function[J].Bulletin of the Polish Academy of Sciences-Mathematics,Astronomy and Physics,1961,9(1):445-450.

[11]MENG Cheng-hui,CUI Yun-an,WANG Ting-fu.Nearly uniformly convex in Orlicz sequence spaces equipped with Orlicz norm[J].Chinese Quarterly Journal of Mathematics,2000,15(1):1-8.

O175.14

A

1002–0462(2015)04–0610–10

Received date:2014-03-28

Foundation item:Supported by the National Natural Science Foundation of China(10771171);Supported by the 555 Innovation Talent Project of Gansu Province(GS-555-CXRC);Supported by the Technique Innovation Project of Northwest Normal University(NWNU-KJCXGC-212);Supported by the Youth Foundation of Dingxi Advanced Teachers College(1333)

Biographies:MA Xue-min(1981-),female,native of Longxi,Gansu,a lecturer of Dingxi Advanced Teacher’s College,M.S.D.,engages in generalized ordinary diferential equation;LI Bao-lin(1963-),male,native of Tianshui,Gansu,a professor of Northwest Normal University,Ph.D.,engages in generalized ordinary diferential equation;LIN Chang-wei(1983-),female,native of Zhuanghe,Liaoning,a lecturer of Dingxi Advanced Teacher’s College,M.S.D.,engages in remote sensing and landscape ecology.

2000 MR Subject Classifcation:34A20,34G10