韋麗蘭, 黃力人

(華南師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院,廣東廣州 510631)

變分不等式理論是當(dāng)今非線性分析的重要組成部分,它在力學(xué)、微分方程、控制論、數(shù)理邏輯、對策論和非線性規(guī)劃等理論和應(yīng)用學(xué)科都有廣泛的應(yīng)用.

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研究變分不等式問題和非線性互補(bǔ)問題的一個基本問題是討論解的存在性.研究變分不等式問題和非線性互補(bǔ)問題解的存在性有很多方法,例如Browder不動點定理、KKM定理、拓?fù)涠壤碚摵蛷?qiáng)制性條件等.

SMITH在文獻(xiàn)[2]中引入“例外序列”的概念來研究非線性互補(bǔ)問題解的存在性問題,但只限于K=的情形.ISAC注意到例外序列和拓?fù)涠鹊穆?lián)系,在文獻(xiàn)[3]中提出例外簇的概念并借助拓?fù)涠鹊睦碚搧硌芯恳话惴蔷€性互補(bǔ)問題解的存在性.

在例外簇概念提出以后,很多學(xué)者用例外簇的概念研究變分不等式問題和非線性互補(bǔ)問題解的存在性,使例外簇概念得到了豐富和發(fā)展,可參見文獻(xiàn)[1]、[4]-[14].ISAC和ZHAO[4]首次把例外簇的概念從n空間推廣到更一般的無限維Hilbert空間.2006年,BIANCHI[1]等在賦范空間中引入一個強(qiáng)變分不等式SVIP的例外簇的概念,推廣了文獻(xiàn)[12]和文獻(xiàn)[13]例外簇的概念.

本文的第2部分給出一些定義和基本概念.第3部分在賦范空間中提出一個強(qiáng)變分不等式SVIP的α-例外簇的概念,推廣文獻(xiàn)[1]所給出的例外簇的概念,并給出相應(yīng)的解的存在性定理,得到擇一型“強(qiáng)變分不等式或者有解,否則存在一個α-例外簇”.

1 預(yù)備知識

正規(guī)對偶映射J:X→2X*定義如下:

注1[1]若X是Hilbert空間H,記H=H*,則J(x)=x.

即

于是

因此,〈x*,x〉≥‖x*‖‖x‖.另一方面,〈x*,x〉≤‖x*‖‖x‖.所以

〈x*,x〉=‖x*‖‖x‖.

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〈x*,y〉≤‖x*‖‖y‖≤‖x*‖‖x‖=〈x*,x〉.

2 主要結(jié)果

下面給出文獻(xiàn)[1]中例外簇的定義.

注2 BIANCHI在文獻(xiàn)[1]中研究0-例外簇.

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由條件(d4)和tr>0,有

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參考文獻(xiàn)：

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