Existence And Uniqueness Of Stationary Solution Of Nonlinear Stochastic Differential Equati

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第61卷 第5期吉林大学学报(理学版)V o l .61 N o .52023年9月J o u r n a l o f J i l i nU n i v e r s i t y (S c i e n c eE d i t i o n )S e p2023d o i :10.13413/j .c n k i .jd x b l x b .2023011一致分数阶时滞微分方程边值问题解的存在性与唯一性张 敏,周文学,黎文博(兰州交通大学数理学院,兰州730070)摘要:用L e r a y -S c h a u d e r 度理论和B a n a c h 压缩映射原理研究一致分数阶时滞微分方程边值问题D β0+u (t )=f (t ,u (t -τ)), t ɪ[0,1],u (t )=φ(t ), t ɪ[-τ,0],u (0)+u ᶄ(0)=0, u (1)+u ᶄ(1)=ìîíïïïï0解的存在性与唯一性.在非线性项满足增长性条件和L i p s c h i t z 条件下,分别得到了该边值问题解的存在性与唯一性结果,并举例说明所得结果的适用性.关键词:一致分数阶导数;时滞;边值问题;L e r a y -S c h a u d e r 度理论;B a n a c h 压缩映射原理中图分类号:O 175.8 文献标志码:A 文章编号:1671-5489(2023)05-1007-07E x i s t e n c e a n dU n i q u e n e s s o f S o l u t i o n s f o rB o u n d a r y Va l u eP r ob l e m s o fC o n f o r m a b l eF r ac t i o n a lD e l a y D i f f e r e n t i a l E qu a t i o n s Z H A N G M i n ,Z HO U W e n x u e ,L IW e n b o(S c h o o l o f M a t h e m a t i c s a n dP h y s i c s ,L a n z h o u J i a o t o n g U n i v e r s i t y ,L a n z h o u 730070,C h i n a )A b s t r a c t :B y u s i n g L e r a y -S c h a u d e rd e g r e et h e o r y a n d B a n a c h c o n t r a c t i o n m a p p i n g p r i n c i p l e ,w e s t u d i e dt h e e x i s t e n c e a n d u n i q u e n e s s o fs o l u t i o n sf o r b o u n d a r y va l u e p r ob l e m s o fc o n f o r m a b l e f r a c t i o n a lde l a y d if f e r e n t i a l e qu a t i o n s D β0+u (t )=f (t ,u (t -τ)), t ɪ[0,1],u (t )=φ(t ), t ɪ[-τ,0],u (0)+u ᶄ(0)=0, u (1)+u ᶄ(1)=0ìîíïïïï,w h e n t h en o n l i n e a r t e r ms a t i s f i e d t h e g r o w t hc o n d i t i o na n d t h eL i ps c h i t z c o n d i t i o n ,w eo b t a i n e d t h e r e s u l t s o f e x i s t e n c e a n du n i q u e n e s s o f s o l u t i o n f o r t h eb o u n d a r y v a l u e p r o b l e mr e s p e c t i v e l y ,a n d g a v e a ne x a m p l e t o i l l u s t r a t e t h e a p p l i c a b i l i t y of t h e o b t a i n e d r e s u l t s .K e y w o r d s :c o n f o r m a b l e f r a c t i o n a l d e r i v a t i v e ;d e l a y ;b o u n d a r y v a l u e p r o b l e m ;L e r a y -S c h a u d e r d e g r e e t h e o r y ;B a n a c hc o n t r a c t i o nm a p p i n gp r i n c i pl e 收稿日期:2023-01-04. 网络首发日期:2023-07-13.第一作者简介:张 敏(1998 ),女,汉族,硕士研究生,从事分数阶微分方程的研究,E -m a i l :m z h a n g 20222022@126.c o m.通信作者简介:周文学(1976 ),男,汉族,博士,教授,从事非线性分析问题的研究,E -m a i l :w x z h o u 2006@126.c o m.基金项目:国家自然科学基金(批准号:11961039;11801243)和兰州交通大学校青年科学基金(批准号:2017012).网络首发地址:h t t ps ://k n s .c n k i .n e t /k c m s 2/d e t a i l /22.1340.o .20230713.1056.001.h t m l .Copyright ©博看网. All Rights Reserved.0 引 言分数阶微分方程的边值问题是分数阶微分系统理论的重要课题.目前,对分数阶微分方程边值问题的研究已取得了丰富成果,其中最主要的是基于R i e m a n n -L i o u v i l l e 和C a p u t o 分数阶导数的定义[1-9].但这两种导数均不满足经典链式法则,并且这两种导数的某些性质使得分数阶导数的应用很困难.因此,K h a l i l 等[10]提出了一种新的分数阶导数和分数阶积分的定义,称为一致分数阶导数和积分.这种新的分数阶导数的定义可满足经典的分数阶导数不能满足的一些性质,如乘积法则㊁商法则㊁链式法则㊁罗尔定理和中值定理等,并且其在生物物理学㊁电容理论㊁控制理论和实验数据拟合等领域应用广泛[11-13].但对带有时滞的分数阶微分方程边值问题的研究目前报道较少[14-16].Y a n g 等[17]利用S c h a e f e r 不动点定理和K r a s n o s e l s k i i s 不动点定理研究了一类非线性分数阶微分方程边值问题cD α0+u (t )=f (t ,u (t ),u ᶄ(t )),u (0)+u ᶄ(0)=0, u (1)+u ᶄ(1)={正解的存在性,其中0<t <1,1<αɤ2,f :[0,1]ˑ[0,+ɕ)ˑℝң[0,+ɕ)是连续函数,c D α0+是α阶C a p u t o 分数阶导数.X u [18]利用B a n a c h 压缩映射原理㊁L e r a y -S c h a u d e r 度理论和K r a s n o s e l s k i i s 不动点定理研究了一类分数阶微分方程边值问题cD q x (t )=f (t ,x (t )), t ɪ[0,1],x (1)=μʏ1x (s )d s , x ᶄ(0)+x ᶄ(1)={解的存在唯一性,其中1<q <2,f :[0,1]ˑX ңX 是连续函数,c D q 是q 阶C a p u t o 分数阶导数.基于上述研究,本文利用L e r a y -S c h a u d e r 度理论和B a n a c h 压缩映射原理考虑如下一类一致分数阶时滞微分方程边值问题:D β0+u (t )=f (t ,u (t -τ)), t ɪ[0,1],u (t )=φ(t ), t ɪ[-τ,0],u (0)+u ᶄ(0)=0, u (1)+u ᶄ(1)=ìîíïïïï0(1)解的存在性与唯一性,其中1<βɤ2,τ>0,f :[0,1]ˑℝңℝ是连续函数,D β0+是阶数为β的一致分数阶导数.1 预备知识定义1[10] 假设函数f :[0,ɕ)ңℝ,则f 的βɪ(n ,n +1]阶一致分数阶导数定义为D βf (t )=l i m εң0f (β⌉-1)(t +εt β⌉-β)-f (β⌉-1)(t )ε, t >0,(2)其中β是大于等于β的最小整数.式(2)右端极限存在,此时称函数f 是β阶可微的.特别地,当βɪ(1,2]时,D βf (t )=l i m εң0f ᶄ(t +εt 2-β)-f ᶄ(t )ε, t >0.(3) 注1 如果函数f 在(0,b )(b >0)上是β阶可微的,并且l i m t ң0+D βf (t )存在,则D βf (0)=l i m t ң0+D βf (t).注2 由一致分数阶导数定义可知,当β=1时,一致分数阶导数定义即为传统的一阶导数定义.引理1[10] 当βɪ(n ,n +1]并且f 在t >0处n +1阶可微时,有D βf (t )=t β⌉-βf(β⌉)(t ).(4) 证明:令k =εt β⌉-β,则ε=t β-β⌉k ,因此由定义1可得D βf (t )=l i m εң0f (β⌉-1)(t +εt β⌉-β)-f (β⌉-1)(t )ε=l i m k ң0t β⌉-βf (β⌉-1)(t +k )-f (β⌉-1)(t )k=t β⌉-βf (β⌉)(t ). 定义2[19]假设函数f :[0,ɕ)ңℝ,则f 的βɪ(n ,n +1]阶一致分数阶积分定义为8001 吉林大学学报(理学版) 第61卷Copyright ©博看网. All Rights Reserved.I βf (t )=1n!ʏt 0(t -s )n s β-n -1f (s )d s .(5)特别地,当βɪ(1,2]时,I βf (t )=ʏt 0(t -s )s β-2f (s )d s .引理2[19] 假设函数f :[0,ɕ)ңℝ连续,并且βɪ(n ,n +1],则有D βI βf (t )=f (t ).(6) 引理3[19]假设f :[0,ɕ)ңℝ是β阶可微函数,并且βɪ(n ,n +1],则有I βD βf (t )=f (t )+a 0+a 1t + +a nt n ,(7)其中a i ɪℝ,i =0,1,2, ,n .引理4 设函数f :[0,1]ˑℝңℝ是连续的,u (t )是边值问题(1)的解,则u (t )=ʏ10G (t ,s )f (s ,u (s -τ))d s ,t ɪ[0,1],φ(t ),t ɪ[-τ,0{],(8)其中格林函数G (t ,s)为G (t ,s )=(1-s )(2-t )sβ-2,0ɤs ɤt ɤ1,(1-t )(2-s )sβ-2,0ɤt ɤs ɤ1{.(9) 证明:由引理3知,有u (t )=I β0+f (t ,u (t -τ))-a 0-a 1t =ʏt 0(t -s )s β-2f (s ,u (s -τ))d s -a 0-a 1t ,(10)从而u ᶄ(t )=ʏts β-2f (s ,u (s -τ))d s -a 1.根据u (0)+u ᶄ(0)=0,有a 0+a 1=0;(11)根据u (1)+u ᶄ(1)=0,有a 0+2a 1-ʏ10(2-s )s β-2f (s ,u (s -τ))d s =0.(12)结合式(11),(12)可得a 0=-ʏ10(2-s )s β-2f (s ,u (s -τ))d s , a 1=ʏ10(2-s )s β-2f (s ,u (s -τ))d s .(13)将式(13)代入式(10)可得u (t )=ʏt 0(t -s )s β-2f (s ,u (s -τ))d s +ʏ10(2-s )s β-2f (s ,u (s -τ))d s -t ʏ1(2-s )s β-2f (s ,u (s -τ))d s =ʏt 0(1-s )(2-t )s β-2f (s ,u (s -τ))d s +ʏ1t(1-t )(2-s )sβ-2f (s ,u (s -τ))d s =ʏ10G (t ,s )f (s ,u (s -τ))d s . 引理5(A r z e l a -A s c o l i 定理)[20] 集合P ⊂C ([a ,b ])列紧的充分必要条件为:1)集合P 有界,即存在常数ψ,使得对∀u ɪP ,有u (t )ɤψ(∀t ɪ[a ,b ]);2)集合P 等度连续,即对∀ε>0,始终存在σ=σ(ε)>0,使得对于∀t 1,t 2ɪ[a ,b ],只要t 1-t 2<σ,即有u (t 1)-u (t 2)<ε(∀u ɪP ).2 主要结果设A 为C ([-τ,1],ℝ)按范数 u =m a x t ɪ[-τ,1]u (t )构成的B a n a c h 空间,在A 上定义一个算子Q ,Q u (t )=ʏ10G (t ,s )f (s ,u (s -τ))d s ,t ɪ[0,1],φ(t ),t ɪ[-τ,0]{. 假设条件:(H 1)函数f ɪC ([0,1]ˑℝ,ℝ),并且φɪC ([-τ,0],ℝ);9001 第5期张 敏,等:一致分数阶时滞微分方程边值问题解的存在性与唯一性 Copyright ©博看网. All Rights Reserved.(H 2)存在常数α,B >0,使得∀(t ,u )ɪ[0,1]ˑℝ,有f (t ,u )ɤαu +B ;(H 3)存在函数η(t )ɪL 1/2([0,1],ℝ+),使得∀t ɪ[0,1],当任取u ,v ɪℝ时,有f (t ,u )-f (t ,v )ɤη(t )u -v ,其中 η =ʏ10η2(s )d ()s 1/2.为方便,引入记号:Λ1=β+2β(β-1),Λ2=1(β-1)(2β-1)(2β-3),Λ3=2β2-β+1(β-1)(2β-1)(2β-3),32<βɤ2.定理1 如果条件(H 1)和(H 2)成立,并且αɪ(0,Λ-11),则边值问题(1)至少存在一个解.证明:由函数G (t ,s ),f (s ,u (s -τ))的连续性可知算子Q 是连续的,并且易证Q (A )⊂A .设P 是A 中的一个有界集,则存在常数M >0,使得对任意的u ɪP ,有 u ɤM .下面利用L e r a y -S c h a u d e r 度理论证明边值问题(1)正解的存在性,分以下3个步骤.1)证明算子Q (P )是一致有界的.对任意的u ɪP ,有Q u (t)=ʏ10G (t ,s )f (s ,u (s -τ))d s ɤʏ10G (t ,s )㊃f (s ,u (s -τ))d s ɤ(αu +B )ʏ10G (t ,s )d s ɤ(αM +B )ʏ10(2-s )(1-t )s β-2d s +ʏt(t -s )s β-2d []s =(αM +B )β+1β(β-1)(1-t )+1β(β-1)㊃t éëêêùûúúβɤ(αM +B )β+1β(β-1)+1β(β-1éëêêùûúú)=(αM +B )Λ1,因此,算子Q (P )是一致有界的.2)证明算子Q (P )是等度连续的.对任意的u ɪP ,t 1,t 2ɪ[-τ,1]且t 1<t 2:①当0ɤt 1<t 2ɤ1时,有Q u (t 2)-Q u (t 1)=ʏ10G (t 2,s )f (s ,u (s -τ))d s -ʏ1G (t 1,s )f (s ,u (s -τ))d s ɤʏ10G (t 2,s )-G (t 1,s )㊃f (s ,u (s -τ))d s ɤ(αu +B )ʏ10G (t 2,s )-G (t 1,s )d s ɤ (αM +B )ʏt 10G (t 2,s )-G (t 1,s )d s +ʏt 2t 1G (t 2,s )-G (t 1,s )d s +ʏ1t 2G (t 2,s )-G (t 1,s )d []s = (αM +B )ʏt 10{[(2-s )(1-t 2)s β-2-(2-s )(1-t 1)s β-2]+[(t 2-s )s β-2-(t 1-s )s β-2]}d s + (αM +B )ʏt 2t 1{[(2-s )(1-t 2)s β-2-(2-s )(1-t 1)s β-2]+(t 2-s )s β-2}d s + (αM +B )ʏ1t 2[(2-s )(1-t 2)s β-2-(2-s )(1-t 1)s β-2]d s =(αM +B )ʏt 10(t 1-t 2)(2-s )s β-2d s +ʏt 10(t 2-t 1)s β-2d s +ʏt 2t 1(t 1-t 2)(2-s )s β-2d [s + ʏt 2t 1(t 2-s )s β-2d s +ʏ1t 2(t 1-t 2)(2-s )s β-2d ]s ɤ(αM +B )(t β2-t β1)-(β+1)(t 2-t 1)β(β-1); ②当-τɤt 1<t 2ɤ0时,有Q u (t 2)-Q u (t 1)ɤφ(t 2)-φ(t 1);③当-τɤt 1<0<t 2ɤ1时,有Q u (t 2)-Q u (t 1)ɤQ u (t 2)-Q u (0)+Q u (0)-Q u (t 1)ɤʏ10G (t 2,s )-G (0,s )㊃f (s ,u (s -τ))d s +φ(0)-φ(t 1)ɤ(αM +B )ʏ10G (t 2,s )d s +φ(0)-φ(t 1)ɤ0101 吉林大学学报(理学版) 第61卷Copyright ©博看网. All Rights Reserved.(αM +B )t β2β(β-1)+φ(0)-φ(t 1)ɤ(αM +B )t β2-t β1β(β-1)+φ(0)-φ(t 1). 在上面3种情形中,当t 1ңt 2时,总有Q u (t 2)-Q u (t 1)ң0,表明Q (P )是等度连续的.故由引理5可知,Q (P )是列紧的,从而算子Q :A ңA 是全连续的.3)利用L e r a y -S c h a u d e r 度理论证明问题(1)正解的存在性.定义范数 φ [-τ,0]=m a x t ɪ[-τ,0]φ(s ).假设当γɪ[0,1],u ɪA 时,u =γQ u ,则u (t )=γQ u (t )ɤQ u (t)ɤʏ10G (t ,s )㊃f (s ,u (s -τ))d s ,t ɪ[0,1],φ(t ),t ɪ[-τ,0{],ɤʏ10G (t ,s )(αu +B )d s ,t ɪ[0,1],φ(t ),t ɪ[-τ,0{],ɤ(αu +B )ʏ10(2-s )(1-t )s β-2d s +ʏt 0(t -s )s β-2d []s ,t ɪ[0,1],φ(t ),t ɪ[-τ,0{],ɤ(α u +B )Λ1,t ɪ[0,1], φ [-τ,0],t ɪ[-τ,0{],从而 u ɤB Λ11-αΛ1 φìîíïïïɤT .令ω=T +1,B ω={u ɪA : u <ω},则u ʂγQ u ,对任意的u ɪ∂B ω,γɪ[0,1].定义一个映射:F γ(u )=u -γQ u ,则F γ(u )=u -γQ u ʂ0,对任意的u ɪ∂B ω,γɪ[0,1].因此,由L e r a y -S c h a u d e r 度的同伦不变性,有d e g (F γ,B ω,θ)=d e g (I -γQ ,B ω,θ)=d e g (F 1,B ω,θ)=d e g (F 0,B ω,θ)=d e g (I ,B ω,θ)=1ʂθ.从而根据L e r a y -S c h a u d e r 度的可解性可知,方程F 1(u )=u -Q u =0在B ω上至少存在一个解,进而边值问题(1)至少有一个正解.证毕.定理2 如果条件(H 1)和(H 3)成立,并且 η (Λ2+Λ3)<1,则边值问题(1)存在唯一解.证明:假设s u p t ɪ[0,1]f (t ,0)=ζ<ɕ.定义B δ={u ɪA : u ɤδ}为A 中的有界闭球,并选择δȡζΛ11- η (Λ2+Λ3).下面利用B a n a c h 压缩映射原理证明边值问题(1)解的存在唯一性,分以下两个步骤.1)证明Q (B δ)⊂B δ.对任意的u ɪB δ,有Q u (t)ɤʏt 0(t -s )s β-2f (s ,u (s -τ))d s +ʏ10(1-t )(2-s )s β-2f (s ,u (s -τ))d s ɤʏt 0(t -s )s β-2[f (s ,u (s -τ))-f (s ,0)+f (s ,0)]d s +ʏ10(1-t )(2-s )s β-2[f (s ,u (s -τ))-f (s ,0)+f (s ,0)]d s ɤ u ʏt(t -s )s β-2η(s )d s +ζʏt(t -s )s β-2d s +u (1-t )ʏ10(2-s )s β-2η(s )d s +ζʏ10(1-t )(2-s )s β-2d s ɤ u ʏt(t s β-2-s β-1)2d ()s 1/2ʏtη2(s )d ()s 1/2+ζβ(β-1)t β+ u (1-t )ʏ10(2s β-2-s β-1)2d []s 1/2ʏ10η2(s )d ()s 1/2+(β+1)ζβ(β-1)(1-t )ɤ1101 第5期张 敏,等:一致分数阶时滞微分方程边值问题解的存在性与唯一性 Copyright ©博看网. All Rights Reserved.1(β-1)(2β-1)(2β-3) u η t β-1/2+ζβ(β-1)t β+2β2-β+1(β-1)(2β-1)(2β-3) u η (1-t )+(β+1)ζβ(β-1)(1-t )ɤδ η (Λ2+Λ3)+ζΛ1,则 Q u ɤδ.表明算子Q 将B δ中的有界子集映为B δ中的有界子集,即Q (B δ)⊂B δ.2)证明算子Q 为压缩映射.对任意的u ,v ɪA :①当t ɪ[0,1]时,有Q u (t )-Qv (t )ɤʏt 0(t -s )s β-2f (s ,u (s -τ))-f (s ,v (s -τ))d s +ʏ10(1-t )(2-s )s β-2f (s ,u (s -τ))-f (s ,v (s -τ))d s ɤ u -v ʏt(t -s )s β-2η(s )d s + u -v (1-t )ʏ10(2-s )s β-2η(s )d s ɤu -v ʏt(t s β-2-s β-1)2d ()s 1/2ʏtη2(s )d ()s 1/2+u -v (1-t )ʏ10(2s β-2-s β-1)2d ()s 1/2ʏ10η2(s )d ()s 1/2ɤ1(β-1)(2β-1)(2β-3) u -v ㊃ ηt β-1/2+2β2-β+1(β-1)(2β-1)(2β-3) u -v ㊃ η (1-t )ɤ η (Λ2+Λ3) u -v ; ②当t ɪ[-τ,0]时,有Q u (t )-Q v (t )=φ(t )-φ(t )=0.由①,②可得Q u -Q v [-τ,1]ɤ η (Λ2+Λ3) u -v [-τ,1]. 因为 η (Λ2+Λ3)<1,所以算子Q 为压缩映射.即由B a n a c h 压缩映射原理可知算子Q 存在唯一的不动点,故边值问题(1)存在唯一解.3 应用实例考虑下列一致分数阶时滞微分方程边值问题:D 7/40+u (t )=e -3t s i n 1/2t 5(2+t )2㊃u (t -τ)1+u (t -τ), t ɪ[0,1],u (t )=φ(t ), t ɪ[-τ,0],u (0)+u ᶄ(0)=0,u (1)+u ᶄ(1)=ìîíïïïïïï0(14)解的存在性与唯一性.证明:在边值问题(14)中,β=74,函数f (t ,u (t ))=e -3t s i n 1/2t 5(2+t)2㊃u 1+u 是连续的,满足条件(H 1);对任意的u ,v ɪℝ,t ɪ[0,1],有f (t ,u (t -τ))-f (t ,v (t -τ))ɤe -3t s i n 1/2t 5(2+t )2u -v ɤe -3t s i n 1/2t ㊃u -v .所以存在η(t )=e -3t s i n 1/2t ɪL 1/2([0,1],ℝ+),满足条件(H 3),且 η =0.1667.又因为Λ2=1(β-1)(2β-1)(2β-3)ʈ1.0328, Λ3=2β2-β+1(β-1)(2β-1)(2β-3)ʈ2.3944.所以 η (Λ2+Λ3)ʈ0.5713<1.因此根据定理2可知,边值问题(14)存在唯一解.2101 吉林大学学报(理学版)第61卷Copyright ©博看网. 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第27卷第1期吉首大学学报(自然科学版)Vol.27No.1 2006年1月J ournal of J ishou University(Natural Science Edi ti on)Jan.2006Article ID:1007-2985(2006)01-0023-04A Sufficient Condition for General Equilibrium inIndivisib le Markets XLI Chun-lin1,LI Xiu-jun2,YAO Ran-ting2(1..Maths and Stat.School,Hebei University of Economics and Business,Shijiazhuang050061,China;2.Department of M aths,Hebei University of Technology,T ianjin300130,China)Abstract:This paper generalizes the general equilibrium theorem to the case of indivisible markets where preferences is strictly convex,con-ti nuous and strongly monotonous.Key words:fixed point theorem;indivi sible markets;general equilibriu mCLC number:O177Document code:A1IntroductionThe application of fixed point theorem in ec onomics is far more widely used than it is in mathematics.In microeco-nomics,fixed point theore m is a general solution to the existence of equilibrium,particularly from Brouwer.s and Kaku-tani.s theorems.Brouwer.s theorem states that/a continuous func tion from a non-empty compact conve x set into itself has a fixed point0.But in economics,a correspondence is much more general than a function,so Kakutani.s theorem, which states that/a compac t and conve x-valued correspondence from a compact conve x set into itself has a fixed point, if it is upper he m-i continuous0,i.e.if its image does not explode rapidly,is more widely used.It is also well kno wn that/a compact and convex-valued correspondence from a compact convex set into itself has a fixed point,if it.s lower hem-i continuous0[1-2],i.e.if its image does not shrink rapidly.Unfortunately,those theorems,respectively,carry over to the situation where goods and markets are indivisible.For example,in the real world,most goods can only be ident-i fied by integer.We can buy one apple,but what.s the meaning of0.01apples?If we consider goods in integer spaces, the original fixed point theorems fail to hold.This paper attempts to close this gap.Many economists contribute a lot to the e xistence and uniqueness of general equilibrium through the centuries.For e xample,in the middle of20th century,Debreu gave the first proof of existence of equilibrium by calculus foundations[3]. Then based upon n goods and n prices,Smale found another solution in terms of Global Ne wton Method[4].After that,De-breu wrote of the radical change of ma thematical tools from calc ulus to conve xity and algebraic topology[3].Bettzuge has pointed out that the uniqueness can be derived via the Mitjushin-Polterovich theorem in incomplete market[5].Almost at the sa me time,Nash brought about Kakutani.s theorem in the e xistence of equilibrium in game theory,which has flour-ished the application of fixed point theory in general equilibrium theory.Danilov et al have given definition of/discrete c onvexity0in their o wn way,respectively.[6-8]This paper gives a sufficient condition on the indivisible goods markets and price vectors,which allo ws genera-l izing the Kakutani.s theorem to the case where the price might not be spanned in the compac t convex set.X Received date:2005-10-17Biography:LI Chun-lin(1963-),male,was born in Xingtai City,Hebei Province,professor,mater adviser of Hebei University of Economics and Business;research area are functional analysis and mathematical modeli ng,quantity economics and stati stics analysis.2 Edgeworth EconomyFirst,let .s c onsider a pure exchange economy by Edge worth box.Because goods are indivisible,we define a con -sumption set by Z 2+,and preferences <i defined on this set.The endo wment vector X i =(X 1i ,X 2i ),X i I Z 2+.The to -tal endowment of each l is X l =X l 1+X l 2,X li I Z 2+.The allocation in this ec onomy is an assignment of a nonnegative consumption vector to each consumer:x =(x 1,x 2)=((x 11,x 21),(x 12,x 22)) x I Z 4+.We say an allocation is feasible for the economy ifx l 1+x l 2[X l for l =1,2.(1)That is,if the total consumption of each commodity is no more than the ec onomy .s a ggre gate endowment of it (Note that in this notion of feasibility,we are implicitly assuming that there is free disposal of commodities).Fig .1 An Edgeworth BoxThe feasible allocation for which equality holds in(1)c ould be called nonwasteful.Nonwasteful feasible a-llocations can be depicted by means of an Edge worth box,shown in fig. 1.In the Edgeworth box,consumer 1.s quantities aremeasured in the usual way,with the southwest corner asthe origin.In contrast,consumer 2.s quantities are mea -sured using the northeast corner as the origin.For bothconsumers,the vertical dimension measures quantities ofgood 2,and the horizontal dimension measures quantitiesof good 1.The length of the box is X 1,the economy .s to -tal endowment of good 1.It .s height is X 2,the economy .stotal endowment of good 2.Any point in the box representsa nonwasteful division of the economy .s total endowment between consumer 1and consumer 2.For example,fig.1de -picts the endo wment vector X =((X 11,X 21),(X 12,X 22))of the two c onsumers.As is characteristic of general equilibrium theory,the wealth of a consumer is not given exogenously.Rather,for any prices p =(p 1,p 2),consumer i .s wealth equals the market value of his endowments of c om modities,p #X i =p 1X 1i =p 2X 2i .Wealth levels are therefore deter mined by the values of prices.Hence,given the consumer .s endow -ment vector X i ,his budget set can be viewed solely as a func tion of prices:B i (p )={x i I Z 2+:p #x i [p #X i }.C ontrast to continuous commodity spaces,although we can also draw a budget line,through the endowment point X with slope -(p 1/p 2),as shown in fig.2,only those integral points make sense.Observe that only allocations on the budget line are affordable to both consumers simultaneously at prices (p 1,p 2).Fig.3illustrates how the consumption vector demanded by consumer 1can be determined for any price vec tor p .Given p ,the consumer de mands his most preferred point in B 1(p ),which can be expressed using his de mand function as x 1(p ,p #X 1).In discrete market,the optimal allocation might not be unique,the allocation will be determined by the market,if it .s not feasible,the price vector p varies,the budget line pivots around the endowment point X ,and the demanded consumptions trace out a curve,denoted by OC 1,as it shows in fig.4,that is called the offer curve of consumer 1.Note that this c urve passes through the endo wment point.Because at every p the endowment vec tor w 1=(X 11,X 21)is affordable to consumer 1,it follows that this consumer must find every point on his offer curve at least as good as his endowment point.At a mar -ket equilibrium where consumers take prices as given,markets should be clear.That is,the c onsumers should fulfill 24吉首大学学报(自然科学版)第27卷their desired purchases and sales of c om modities at the going market prices.That is,demand should equal supply.This gives as the notion of equilibrium presented in fig.5.In indivisible spaces,if such feasible nonwasteful allocation ex -ists,the allocation is an equilibrium allocation;the price is an equilibrium price.Since we can .t do calculus in integral set,we first suppose the commodity spaces are divisible and find the optimal allocation in that case,then transfer to in -tegral space,there must be some connection between the twospaces.Fig .2 C onsumer .s Budget Set Fig .3 Consumption Vector Dem anded by C onsumer1Fig .4 Offer Curve of Consumer 1 Fig .5 Notion of Equilibrium Presented3 General Equilibrium Theorem in Indivisible GoodsDefinition 1 A Walrasian (or competitive)equilibrium[9-10]for an Edge worth box economy is a price vector p *and an allocation x *=(x *1,x *2)in the Edge worth box such that for i =1,2,x *i <i x i c x i c I B i (p *).Proposition 1 x *is a Pareto improve ment of initial endowment X if there is a Walrasian (or competitive)equ-i librium for an Edge worth box economy and an allocation x *=(x *1,x *2)in the Edge worth box such that x *X X .Proof P i ,x *i I arg max x i I B i (p )u i (x i ),X i I B i (p )]u i (x *i )\u i (X i ),P i ]x *is a Pareto improvementof initial endo wment X .Proposition 2 There must be an equilibrium allocation.Proof We can al w ays find an optimal allocation in finite points.Thus we can find a discrete equilibrium allocation from continuous equilibrium allocation.If E =(x 1,x 2)=((x 11,x 21),(x 12,x 22))is a continuous equilibrium alloca tion,define the subspace of Z 4as 25第1期 李春林,等:不可分市场中一般均衡的一个充分条件26吉首大学学报(自然科学版)第27卷C(E)={x I Z4:+x-E+[1}z I Z4,+z+=sup|z i|.i=1,+,4Proposition3There is an equilibrium allocation x*I C(E)in Edge worth box economy when the two consum-ers.preferences are strictly monotonous.Proof If E I C(E)is an equilibrium allocation,the c onclusion is obvious.If E|C(E),suppose P x I C(E)is not an equilibrium allocation,that is y|C(E)is a feasible allocation and Pareto optimal than x.For v i,s.t,u i(y i)>u i(x i),y i I B i(p).Since the preferences are continuous,there is a linear allocation of E and y indifferent with some points in C(E).Also based on the convexity of preferences,for i, u i(y i)[u i(x i)contradict to the assumption.4ConclusionIn this paper,we have studied the existence of equilibrium in indivisible markets where preferences<i is strictly convex,continuous and strongly monotonous.References:[1]B ORDER K.Fixed Poi nt Theorems with Applications to Economics and Game Theory[M].London:Cambridge University Press,1985.[2]BROOME J.Approxi mate Equilibrium in Economics wi th Indivisible Commodities[J].Journal of Economic Theory,1972,5:224-249.[3]DEBREU G.Theory of Value[M].New York:Wiley,1959.[4]SMALE S.Dynamics in General Equilibrium Theory[J].American Economics Review,1976,66:288-294.[5]MARC OLIVER BE TTZUGE.An Extension of a Theorem by Mitjushin and Polterovich to Incomplete Markets[J].Journal of M ath-ematical Economics,1998,30:285-300.[6]DANILOV V,KOSHEVOY G,MUROTA K.Discrete Convexity and Eq uilibrium in Economics with Indivisible Goods and Money[J].Mathematical Social Sciences,2001,41:251-273.[7]MUROTA K.Discrete Convex Analysis[J].Mathematical Programming,1998,83:313-371.[8]TAKUYA IIMURA.A Discrete Fixed Point Theorm and Its Applications[J].Journal of Mathematical Economics,2003,39:725-742.[9]SMALE S BLOCK,HURWICZ L.The Stability of the Competi tive Equili bri um I[J].Econometrica,1958,26:522-552.[10]SMALE S B LOCK,HURWICZ L.The Stability of Competitive Equilibrium II[J].Econometrica,1959,27:82-109.不可分市场中一般均衡的一个充分条件李春林1,李秀军2,姚冉婷2(1.河北经贸大学数学与统计学学院,河北石家庄050061;2.河北工业大学理学院,天津300130)摘要:给出了不可分市场中一般均衡存在的一个充分条件:消费者偏好函数是强凸、连续和严格单调的.关键词:不动点定理;不可分市场;一般均衡中图分类号:O177文献标识码:A(责任编辑向阳洁)。

社会福利函数的存在性与唯一性——兼其在收入分配中的应用作者：欧阳葵1，王国成22013-2-26 8:49:21 来源:数量经济技术经济研究（1.西北大学经济管理学院；2.中国社会科学院数量经济与技术经济研究所）【摘要】本文提出了最低正义准则，并证明了唯一满足不相关选择的独立性、序数可比性和最低正义准则的社会福利函数是罗尔斯主义，唯一满足不相关选择的独立性、匿名性、比率不可比性和最低正义准则的社会福利函数是纳什社会福利函数。

在常相对风险规避个体效用函数和纳什社会福利函数假设下，如果收入分配总是最优的，那么国民收入增长意味着每个人的收入都在增长，社会福利是国民收入的严格增函数，对称性最优所得税制意味着个体税后收入和社会福利都与起征点无关。

关键词：功利主义，罗尔斯主义，纳什社会福利函数，最低正义，收入分配中图分类号：F061.4 文献标识码：AExistence and Uniqueness of Social Welfare FunctionAbstract: This paper proposes the principle of minimal justice and shows that Rawlsianism is the only social welfare function satisfying independence of irrelevant alternatives, ordinal comparability, and minimal justice, and Nash social welfare function is the only social welfare function satisfying independence of irrelevant alternatives, anonymity, ratio-scale non-comparability, and minimal justice. Furthermore, under the assumption of constant-relative-risk-aversion utility function and Nash social welfare function, if the income distribution is always optimal, then the increase of national income necessarily implies the increase of everyone’s income, social welfare is stri ctly increasing in national income, and under optimal symmetrical taxation, tax-free threshold does not affect after-tax income and social welfare.Keywords: Utilitarian, Rawlsianism；Nash social welfare function, Minimal Justice, Income distribution引言简单地说，改革可视为从一系列可行配置中选择一个社会最优配置的过程。

寻找地球以外的生命有什么意义英语作文Exploring the Significance of Searching for Extraterrestrial LifeThe search for extraterrestrial life has captivated the human imagination for centuries. As we gaze into the vast expanse of the cosmos, the possibility of discovering intelligent life beyond our planet has become an enduring fascination. This endeavor, often referred to as the search for extraterrestrial intelligence (SETI), holds immense significance that extends far beyond mere scientific curiosity.One of the primary motivations behind the search for extraterrestrial life is the profound implications it could have for our understanding of the universe and our place within it. The discovery of even the most primitive forms of life beyond Earth would revolutionize our perception of the cosmos, challenging the long-held belief that Earth is the sole bastion of life in the universe. Such a revelation would shatter the anthropocentric view that has dominated much of human thought and history, forcing us to reconsider our own significance and the uniqueness of our existence.Moreover, the search for extraterrestrial life holds the potential touncover valuable insights into the origins and evolution of life itself. By studying the conditions and mechanisms that give rise to life in other planetary systems, scientists could gain a deeper understanding of the fundamental processes that underlie the emergence and development of living organisms. This knowledge could have far-reaching implications for fields such as biology, astrobiology, and evolutionary science, potentially leading to breakthroughs in our comprehension of the origins of life on Earth.The search for extraterrestrial life also carries immense philosophical and existential significance. The discovery of intelligent life beyond our planet would force us to confront profound questions about the nature of consciousness, the universality of intelligence, and the possibility of shared values and experiences across vastly different civilizations. Such an encounter could challenge our most deeply held beliefs and assumptions about the human condition, our place in the cosmic order, and our relationship with the rest of the universe.Furthermore, the search for extraterrestrial life holds the potential to inspire and unite humanity in unprecedented ways. The pursuit of this grand scientific endeavor has the power to transcend political, cultural, and ideological boundaries, fostering a shared sense of wonder and a collective effort to explore the unknown. The discovery of extraterrestrial life, or even the mere possibility of its existence, could serve as a catalyst for global cooperation, as nations andindividuals come together to unravel the mysteries of the universe.Beyond the scientific and philosophical implications, the search for extraterrestrial life also holds profound implications for our own future as a species. The discovery of advanced civilizations beyond Earth could provide invaluable insights into the challenges and opportunities that may lie ahead for humanity. By studying the technological, social, and environmental trajectories of other intelligent species, we may gain crucial perspectives on the sustainable development of our own civilization, potentially guiding us towards a more harmonious and resilient future.Moreover, the search for extraterrestrial life could have significant practical applications, leading to advancements in fields such as communication technology, space exploration, and the development of new materials and energy sources. The technological innovations and scientific breakthroughs that arise from this endeavor could have far-reaching benefits for humanity, improving our quality of life and expanding the boundaries of human knowledge and capabilities.In conclusion, the search for extraterrestrial life holds immense significance that transcends the realm of scientific curiosity. It has the potential to revolutionize our understanding of the universe, challenge our most fundamental beliefs, inspire global cooperation, and guide us towards a more sustainable and resilient future. As wecontinue to explore the vast expanse of the cosmos, the search for extraterrestrial life remains a pursuit of profound importance, one that holds the power to transform our very conception of ourselves and our place in the grand tapestry of the universe.。

聚丙烯(PP)晶体结构规整,具备易加工、抗冲击强度、抗挠曲性以及电绝缘性好等优点,它的应用十分广泛,特别是在纤维和长丝、薄膜挤压、注塑加工等方面,是合成树脂中消费增速最快、用途最广的品种。

随着催化剂技术的进步、设备制造能力的提高和市场对新产品需要的不断增加,聚丙烯生产工艺也在不断的改进和完善。

1聚丙烯生产工艺发展根据反应介质及反应器构型的不同,聚丙烯生产工艺主要有:淤浆法、本体法(包括本体-气相法组合)和气相法。

1.1浆液法世界上最早用于聚丙烯生产,直到20世纪80年代,它还占主要地位。

特点是将丙烯溶于惰性烃类稀释剂中进行聚合,主要有意大利的Montecatini 工艺、美国Hercules 工艺、日本三井东压化学工艺、美国Amoco 工艺、日本三井油化工艺以及索维尔工艺等。

该工艺流程长,成本高,操作与投资费用较高。

除生产少量高性能的塑料合金外,自20世纪80年代以后,新、改建的大型聚丙烯装置基本不再采用。

1.2本体法(本体-气相法组合)该工艺特点是反应体系中不加任何其它溶剂,将催化剂直接分散在液相丙烯中进行聚合反应。

20世纪70年代后期的装置大都基于此法。

本体法工艺有过多种工艺路线。

根据聚合反应器的不同,可分为釜式聚合工艺和管式聚合工艺,经过多年的发展和竞争,目前应用较多的主要有Basell 公司的Spheripol 工艺、日本三井化学公司的Hypol 工艺和Borealis 公司的的Borstar 工艺等。

Spheripol 工艺自1982年首次工业化以来,是迄今为止最成功、应用最为广泛的聚丙烯生产工艺。

它是一种液相预聚合同液相均聚和气相共聚相结合的聚合工艺,采用一个或者多个环管反应器和一个或多个串联的气相流化床反应器,在环管反应器中进行均聚和无规共聚,在气相流化床中生产抗冲共聚物。

虽然流程相比之下较长,但设备简单,投资不高,操作稳定可靠,产品性能好。

Hypol 工艺于20世纪80年代初期开发成功,采用HY-HS-II 催化剂(TK-II),是一种多级聚合工艺。