萬軒,趙克全

(重慶師范大學數(shù)學學院,重慶 400047)

廣義Ekeland變分原理的應用

萬軒,趙克全

(重慶師范大學數(shù)學學院,重慶 400047)

研究了廣義Ekeland變分原理在擬度量空間中的一些重要應用.利用廣義Ekeland變分原理證明了函數(shù)f滿足關(guān)于α的Takahashiε-條件當且僅當f滿足關(guān)于相同α的Hamelε-條件.此外,利用關(guān)于α的Takahashiε-條件得到了一些重要結(jié)論.

廣義Ekeland變分原理;擬度量空間;Takahashiε-條件;Hemalε-條件

1 引言

眾所周知,自文獻[1-2]在1972年給出關(guān)于帶擾動的下半連續(xù)函數(shù)取嚴格極小值的變分原理以來,由于其在最優(yōu)化、控制論和非線性分析等領(lǐng)域有著廣泛的應用,出現(xiàn)了許多Ekeland變分原理的推廣形式和等價形式的研究[3-10].特別地,文獻[3]在概率度量空間中證明了Ekeland變分原理與Caristi不動點定理等價.文獻[4]在完備度量空間中研究了一般Ekelandε-變分原理以及它的Borwein-Preiss光滑ε-變分原理的一些應用.文獻[5]在完備度量空間中利用Ekeland變分原理證明了Takahashiε-條件的等價定理,以及在弱尖極小及不動點等中的應用.文獻[7]在完備度量空間中證明了廣義Ekeland變分原理,并利用其推導出廣義Caristi不動點定理,廣義Takahashi非凸最小化定理,非凸極小極大定理以及非凸平衡定理等相關(guān)給論.文獻[9]在擬度量空間上建立了具有Q-函數(shù)的Ekeland變分原理,并證明了多值映射下的Caristi-K irk型不動點定理、Takahashi定理與其等價和一些相關(guān)給論.

本文在完備擬度量空間下建立Takahashiε-條件及Hemalε-條件,并利用Ekeland變分原理證明其等價性及一些相關(guān)結(jié)論.

2 預備知識

為了得到本文的主要結(jié)果,首先引進下面的定義和引理.

定義2.1[9]設X為非空集合.映射d:X×X→?+使得對任意的x,y,z∈X滿足: (M1)d(x,y)≥0;(M2)d(x,y)=0當且僅當x=y;(M3)d(x,y)≤d(x,z)+d(z,y),則稱映射d是X上的擬度量,(X,d)稱為擬度量空間.

定義2.2[9]映射q:X×X→?+滿足:

(Q1)?x,y,z∈X,q(x,y)≤q(x,z)+q(z,y);

(Q2)如果x∈X和序列{yn}n∈??X,存在M=M(x)＞0使得q(x,yn)≤M和limn→∞yn=y(關(guān)于擬度量),則q(x,y)≤M;

(Q3)對任意的ε＞0,存在δ＞0使得q(x,y)≤δ和q(x,z)≤δ蘊含d(y,z)≤ε,則稱映射q為Q-函數(shù).

定義2.3[11]稱函數(shù)f:X→?∪{+∞}為從上面下半連續(xù)(簡記:lsca),如果對任意序列{xn}n∈??X收斂于某一點x∈X和對任意n∈?滿足f(xn+1)≤f(xn),有

根據(jù)文獻[5],下面給出在擬度量空間上的關(guān)于α的Takahashiε-條件和關(guān)于α的Hamel ε-條件的定義.設

定義2.4設(X,d)為擬度量空間,q:X×X→?+為Q-函數(shù),φ:(-∞,∞]→(0,∞)為非減函數(shù)和f:X→?∪{+∞}為有下界的真lsca.

函數(shù)f滿足關(guān)于α的Takahashiε-條件,如果存在α＞0,0＜ε≤+∞和對任意滿足infXf＜f(x)＜infXf+ε的x∈X,則存在y∈X,y/=x,使得

函數(shù)f滿足關(guān)于α的Hamelε-條件,如果存在α＞0,0＜ε≤+∞和對任意滿足infXf＜f(x)＜infXf+ε的x∈X,則存在z∈Z,使得

特別地,當ε=+∞時,則關(guān)于α的Takahashiε-條件和Hamelε-條件分別叫做關(guān)于α的Takahashi件和Hamel條件.

引理2.1設(X,d)為擬度量空間,q:X×X→?+為Q-函數(shù),φ:(-∞,∞]→(0,∞)為非減函數(shù)和f:X→?∪{+∞}為有下界真lsca.則對任意的0＜ε1＜ε2,關(guān)于α的Takahashi ε2-條件蘊含關(guān)于相同α的Takahashiε1-條件和關(guān)于α的Hamelε2-條件蘊含關(guān)于相同α的Hamelε1-條件.

證明對α＞0,0＜ε1＜ε2≤+∞和對任意滿足infXf＜f(x)＜infXf+ε1的x∈X,則對這些任意的x∈X也滿足infXf＜f(x)＜infXf+ε2,再利用f滿足關(guān)于α的Takahashi ε2-條件,則存在y∈X,y/=x,使得

即f滿足關(guān)于相同α的Takahashiε1-條件.

同理可得,關(guān)于α的Hamelε2-條件蘊含關(guān)于相同α的Hamelε1-條件.

3 主要結(jié)果

[1]Ekeland I．Sur lesprobemesvariationnels[J]．Com ptesRendusde l′Acadˊem iedes Sciences-Series I,1972,275: 1057-1059.

[2]Ekeland I.On the variational princip le[J].Journal of Mathematical Analysis and App lication,1974,47:324-353.

[3]Zhang Shisheng,Chen Yuqing,Guo Jinli.Ekeland′s variational princip le and Caristi′s fixed point theorem in p robabilistic m etric space[J].Acta.Mathem eticae App licatae Sinica,1991,7(3):217-228.

[4]Li Yongxin,Shi Shuzhong.A generalization of Ekeland′sε-variational p rincip le and its Borwein-Preiss sm ooth variant[J].Journal of Mathem atical Analysis and App lications,2000,246:308-319.

[5]Wu Zili.Equivalent formu lations of Ekeland′s variational princip le[J].Nonlinear Analysis,2003,55:609-615. [6]Monica Bianchi,Gˊabor Kassay,Rita Pini.Existence of equilibria via Ekeland′s princip le[J].Journal of Mathem atical Analysis and App lications,2005,305:502-512.

[7]Lin Laijiu,Du W S.Ekeland′s variational princip le,m inimax theorem sand existence of nonconvex equilibria in com p letem etric spaces[J].Journal of Mathem atical Analysis and App lications,2006,323:360-370.

[8]Lin Laijiu,Du W S.Some equivalent formu lations of the generalized Ekeland′s variational p rincip le and their app lications[J].Non linear Analysis,2007,67:187-199.

[9]A l-Hom idan S,Ansari Q H,Yao J C.Some generalizations of Ekeland-type variational princip le with app lications to equilibrium problem s and fixed point theory[J].Nonlinear Analysis,2008,69:126-139.

[10]Xiang Shuwen,Yin Wenshuang,Wang Changchun.Some new resu lts on stability of Ekeland′sε-variational p rincip le[J].Journal of Mathem atical Analysis and App lications,2008,339:802-810.

[11]Chen Yuqing,Cho Y J,Yang Li.Note on the resu lts with lower sem i-continuity[J].Bu lletin of the Korean Mathem atical Society,2002,39(4):535-541.

Applications of generalized Ekeland′s variational principle

Wan Xuan,Zhao Kequan
(Department of Mathematics,Chongqing Normal University,Chongqing 400047,China)

In this paper,some im portant app lications about generalized Ekeland′s variational p rincip le in quasi-m etric space are investigated.We proved that function f satisfies theε-condition of Takahashi with αiff f satisfies theε-condition of Hamel with the sameαby the generalized Ekeland′s variational princip le. Furtherm ore,som e im portant conclusions are obtained by theε-condition of Takahashiwithα.

generalized Ekeland′s variational princip le,quasi-m etric space,theε-condition of Takahashi, theε-condition of Hamel

O176;O177.91

A

1008-5513(2012)03-0363-07

2011-06-02.

國家自然科學基金(10831009);重慶市科委運籌學與系統(tǒng)工程重點實驗室專項經(jīng)費(CSTC,2011KLORSE02);重慶自然科學基金(2011BA 0030);重慶市教委科技項目(KJ110625).

萬軒(1987-),碩士生,研究方向：變分分析及應用研究.

2010 MSC:65K 10