付科程
(重慶師范大學(xué)數(shù)學(xué)學(xué)院,重慶 401331)
帶有q距離的向量Ekeland變分原理
付科程
(重慶師范大學(xué)數(shù)學(xué)學(xué)院,重慶 401331)
基于變分原理的形式和空間的多樣性,研究了帶有q距離的向量Ekeland變分原理在分離序列完備一致空間中的一些重要應(yīng)用.
向量Ekeland變分原理;q距離;分離序列完備一致空間
眾所周知,Ekeland變分原理在數(shù)學(xué)非線性分析理論中有著非常重要的地位,它在非線性分析、全局控制優(yōu)化理論、向量均衡問題、臨界點(diǎn)理論與博弈論等諸多領(lǐng)域中有著十分重要的意義和作用.許多學(xué)者對Ekeland變分原理從空間和表達(dá)形式上進(jìn)行了推廣,得出了各種類型的Ekeland變分原理.基于此,研究了在分離序列完備一致空間中,帶有q距離的向量Ekeland變分原理及其一些應(yīng)用問題.
定義1[5]設(shè)X是一個分離一致空間,有一個實(shí)值映射P:X×X→0,+∞[)滿足以下條件:
2)對任意序列{yn}?X,若滿足p(yn,ym)→0(m>n→∞),則稱{yn}為柯西列,p(yn,y)→0等價于yn→y;
3)p(z,x)=0且p(z,y)=0,則x=y(tǒng),對每個x,y,z∈X,稱p為p距離.如果條件2)變?yōu)楦醯臈l件2*):對任意序列{yn}?X,若滿足p(yn,ym)→0(m>n→∞)是柯西列,且在X內(nèi),p(yn,y)→0包含著yn→y,則稱p為q距離.
引理1[2]Y是拓?fù)湎蛄靠臻g,K?Y是一個閉凸錐,ko∈K-K,定義非線性標(biāo)量化函數(shù)ξko:Y→R∪{+∞}
則函數(shù)ξko(y)有以下幾種性質(zhì):
(i)ξko(y)為真;
(ii)ξko(y)下半連續(xù);
(iii)ξko(y)次線性;
(iv)ξko(y)為K-單調(diào)(即y1≤ky2,ξko(y1)≤ξko(y2));
定義2Y是拓?fù)湎蛄靠臻g,K?Y是一個閉凸錐,則關(guān)于K的預(yù)序“≤K”有如下定義:?y1,y2∈Y,有y1≤Ky2當(dāng)且僅當(dāng)y2-y1∈K.
定義3函數(shù)f:X→Y∪{∞}被稱為關(guān)于序列p下單調(diào)的,如果序列{xn}?X滿足xn→且f(xn+1)≤Kf(xn),?n∈N,則有f(ˉx)≤Kf(xn),?n∈N.
定義4設(shè)(X,U)是一致空間,p是q距離,f:X→Y∪{∞}關(guān)于≤K的下單調(diào)特征函數(shù)(如果dom f={x∈X,f(x)∈Y}≠?)(X,U)關(guān)于(p,f)序完備,如果在X中有一個序列{xn}滿足p(xn,xm)→0(m>n→∞),f(xn+1)≤Kf(xn),?n∈N,則?ˉx∈x,使得
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Vector Ekeland’s Variation Principle With q Distance
FU Ke-cheng
(School of Mathematics,Chongqing Normal University,Chongqing 401331,China)
Based on the diversity of variation principle in form and space,this paper researches some important application of vector Ekeland’s variation principle with q distance in complete uniform space of separation sequence.
Vector Ekeland variation principle;q distance;complete uniform space of separation sequence
0176
A
1672-058X(2015)04-0032-04
10.16055/j.issn.1672-058X.2015.0004.009
2014-08-06;
2014-09-25.
付科程(1989-),男,重慶潼南人,碩士研究生,從事向量優(yōu)化理論及應(yīng)用研究.