SOME EASY TO REMEMBER ABSTRACT FORMS OF EKELANDS VARIATIONAL PRINCIPLE AND CARISTIS FIXED POINT THEOREM
Titel:
SOME EASY TO REMEMBER ABSTRACT FORMS OF EKELANDS VARIATIONAL PRINCIPLE AND CARISTIS FIXED POINT THEOREM
Auteur:
Árpád Száz
Verschenen in:
Applicable analysis and discrete mathematics
Paginering:
Jaargang 1 (2007) nr. 2 pagina's 335-339
Jaar:
2007
Inhoud:
If $X$ is a set, then an extended real-valued function $Phi$of $X^{2}$ is called an 'ecart on $X$. Moreover, if $d$and $Phi$ are 'ecarts on $X$, then for any $x, ,yin X$we write $xle y$ if and only if $d(x, ,y)lePhi(x, ,y)$.Thus, we call the pair $(d, ,Phi)$ admissible at a point$ain X$ if $X$ has a maximal element $b$ with $ale b$.Here, an important particular case is when $d$ is a certain metric on $X$and $Phi$ is of the form $Phi(x, ,y)=varphi(x)-psi(y)$.These definitions allow us to easily state and prove some easily rememberedabstract forms of {sc Ekeland}'s variational principle and {sc Caristi}'s fixed pointtheorem. For instance, we show that if ,$F$ ,is a relation on ,$X$ ,and $ain X$ such that there exists a pair ,$(,d,, ,Phi,)$ ,of 'ecarts on,$X$ ,which is admissible at ,$a$ ,and satisfies $d,(,x,, ,y,)le ,Phi,(,x,, ,y,)$ for all $xin X$ and $yin F(x)$,, ,then there exists $bin X$, with $d,(,a,, ,b,)lePhi,(,a,, ,b,)$,, ,suchthat $F,(b)subset{,b,}$,.
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