º¯ÊýÏÊýÒ»ÖÂÊÕÁ²ÐÔ¼°ÆäÓ¦Óà ÏÂÔر¾ÎÄ

2 º¯ÊýÏÊýµÄÏà¹Ø¸ÅÄî½éÉÜ

2.1 º¯ÊýÁм°ÆäÒ»ÖÂÊÕÁ²ÐÔ

¶¨Òå1 Éè

f1,f2,?£¬fn,?

ÊÇÒ»Áж¨ÒåÔÚͬһÊý¼¯EÉϵĺ¯Êý£¬³ÆΪ¶¨ÒåÔÚEÉϵĺ¯ÊýÁУ¬Ò²¿É¼òµ¥µÄд×÷£º

{fn}»òfn£¬n?1,2,?.

Éèx0?E£¬ÒÔx0´úÈë{fn}¿ÉµÃÊýÁÐ

f1(x0),f2(x0),?£¬fn(x0),?

ÈôÊýÁÐ{fn(x0)}ÊÕÁ²£¬Ôò³Æº¯ÊýÁÐ{fn}ÔÚµãx0ÊÕÁ²£¬x0³ÆΪº¯ÊýÁÐ{fn}µÄÊÕÁ²µã.ÈôÊýÁÐ

{fn(x0)}·¢É¢£¬Ôò³Æº¯ÊýÁÐ{fn}ÔÚµãx0·¢É¢.Èôº¯ÊýÁÐ{fn}ÔÚÊý¼¯D?EÉÏÿһµã¶¼ÊÕÁ²£¬Ôò

³Æ{fn}ÔÚÊý¼¯DÉÏÊÕÁ².ÕâʱDÉÏÿһµãx£¬¶¼ÓÐÊýÁÐ{fn(x)}µÄÒ»¸ö¼«ÏÞÖµÓëÖ®Ïà¶ÔÓ¦£¬ÓÉÕâ¸ö¶ÔÓ¦·¨ÔòËùÈ·¶¨µÄDÉϵĺ¯Êý£¬³ÆΪº¯ÊýÁÐ{fn}µÄ¼«ÏÞº¯Êý.Èô¼«ÏÞº¯Êý¼Ç×÷f£¬ÔòÓÐ limfn(x)?f(x)£¬x?D

n??»ò fn(x)?f(x) (n??)£¬x?D. ʹº¯ÊýÁÐ{fn}ÊÕÁ²µÄÈ«ÌåÊÕÁ²µã¼¯ºÏ£¬³ÆΪº¯ÊýÁÐ{fn}µÄÊÕÁ²Óò.

¶¨Òå2 É躯ÊýÁÐ{fn}Ó뺯Êýf¶¨ÒåÔÚͬһÊý¼¯DÉÏ£¬Èô¶ÔÈθøµÄÕýÊý?£¬×Ü´æÔÚijһÕýÕûÊýN£¬Ê¹µÃµ±n?Nʱ£¬¶ÔÒ»ÇÐx?D£¬¶¼ÓÐ

fn(x)?f(x)??£¬ Ôò³Æº¯ÊýÁÐ{fn}ÔÚDÉÏÒ»ÖÂÊÕÁ²ÓÚf£¬¼Ç×÷

fn(x)?f(x) (n??), x?D. ×¢£º±¾ÎÄÓá°?¡±±íʾһÖÂÊÕÁ².

Óɶ¨Òå¿´µ½£¬Èç¹ûº¯ÊýÁÐ{fn}ÔÚDÉÏÒ»ÖÂÊÕÁ²£¬ÄÇô¶ÔÓÚËù¸øµÄ?£¬²»¹ÜDÉÏÄÄÒ»µãx£¬×Ü´æÔÚ¹«¹²µÄN(?)£¨¼´NµÄÑ¡È¡½öÓë?Óйأ¬ÓëxµÄÈ¡ÖµÎ޹أ©£¬Ö»Òªn?N£¬¶¼ÓÐ

fn(x)?f(x)??.

2

ÓÉ´Ë¿ÉÒÔ¿´µ½º¯ÊýÁÐ{fn}ÔÚDÉÏÒ»ÖÂÊÕÁ²£¬±ØÔÚDÉÏÿһµã¶¼ÊÕÁ².·´Ö®£¬ÔÚDÉÏÿһµã¶¼ÊÕÁ²µÄº¯ÊýÁÐ{fn}£¬ÔÚDÉϲ»Ò»¶¨Ò»ÖÂÊÕÁ².

2.2 º¯ÊýÏÊý¼°ÆäÒ»ÖÂÊÕÁ²ÐÔ

¶¨Òå3 Éè{un(x)}ÊǶ¨ÒåÔÚÊý¼¯EÉϵÄÒ»¸öº¯ÊýÁУ¬ ±í´ïʽ

u1(x)+u2(x)+¡­+un(x)+¡­£¬x?E £¨1£©

³ÆΪ¶¨ÒåÔÚEÉϵĺ¯ÊýÏÊý£¬¼ò¼ÇΪ

?un?1?n(x) »ò?un(x)¡£³Æ

Sn(x)??uk(x)£¬x?E£¬n?1,2,?

k?1nΪº¯ÊýÏÊýµÄ²¿·ÖºÍº¯ÊýÁС£

Èôx0?E£¬ÊýÏÊý

u1(x0)?u2(x0)???un(x0)?? £¨2£© ÊÕÁ²£¬¼´²¿·ÖºÍSn(x0)??uk?1nk(x0)µ±n??ʱ¼«ÏÞ´æÔÚ£¬Ôò³Æ¼¶Êý£¨1£©ÔÚµãx0ÊÕÁ²£¬x0³ÆΪ

¼¶Êý£¨1£©µÄÊÕÁ²µã£®Èô¼¶Êý£¨2£©·¢É¢£¬Ôò³Æ¼¶Êý£¨1£©ÔÚµãx0·¢É¢.Èô¼¶Êý£¨1£©ÔÚEµÄij¸ö×Ó¼¯

DÉÏÿµã¶¼ÊÕÁ²£¬Ôò³Æ¼¶Êý£¨1£©ÔÚDÉÏÊÕÁ²£®ÈôDΪ¼¶Êý£¨1£©È«ÌåÊÕÁ²µãµÄ¼¯ºÏ£¬ÕâʱÔò³ÆD

Ϊ¼¶Êý£¨1£©µÄÊÕÁ²Óò£®¼¶Êý£¨1£©ÔÚDÉÏÿһµãxÓëÆäËù¶ÔÓ¦µÄÊýÏÊý£¨2£©µÄºÍS(x)¹¹³ÉÒ»¸ö¶¨ÒåÔÚDÉϵĺ¯Êý£¬³ÆΪ¼¶Êý£¨1£©µÄºÍº¯Êý£¬²¢Ð´×÷

u1(x)?u2(x)???un(x)???S(x)£¬x?D, ¼´

limSn(x)?S(x)£¬x?D£®

n?? Ò²¾ÍÊÇ˵£¬º¯ÊýÏÊý£¨1£©µÄÊÕÁ²ÐÔ¾ÍÊÇÖ¸ËüµÄ²¿·ÖºÍº¯ÊýÁеÄÊÕÁ²ÐÔ£®

¶¨Òå4 Éè{Sn(x)}ÊǺ¯ÊýÏÊýÁ²ÓÚº¯ÊýS(x),Ôò³Æº¯ÊýÏÊý

?un(x)µÄ²¿·ÖºÍº¯ÊýÁУ®Èô{Sn(x)}ÔÚÊý¼¯DÉÏÒ»ÖÂÊÕ

?un(x)ÔÚDÉÏÒ»ÖÂÊÕÁ²ÓÚº¯ÊýS(x)£¬»ò³Æ?un(x)ÔÚDÉÏÒ»

ÖÂÊÕÁ²(»ª¶«Ê¦·¶´óѧÊýѧϵ£¬2001).

[2]

3

2.3 Ò»ÖÂÊÕÁ²º¯ÊýÏÊýµÄÐÔÖÊ

¶¨Àí1 £¨Á¬ÐøÐÔ£©Èôº¯ÊýÏÊýºÍº¯ÊýÔÚ?a,b?ÉÏÒ²Á¬Ðø.

ËüÖ¸³ö£º£¨ÎÞÏÞÏÇóºÍÔËËãÓëÇó¼«ÏÞÔËËã¿ÉÒÔ½»»»Ë³Ðò£¬¼´

?un(x)ÔÚÇø¼ä?a,b?ÉÏÒ»ÖÂÊÕÁ²£¬ÇÒÿһÏÁ¬Ðø£¬ÔòÆä

?(limux?x0n(x))?lim(?un(x)).

x?x0¶¨Àí2 (ÖðÏîÇó»ý)Èôº¯ÊýÏÊý

?un(x)ÔÚ?a,b?ÉÏÒ»ÖÂÊÕÁ²£¬ÇÒÿһÏîun(x)¶¼Á¬Ðø£¬Ôò

ba??baun(x)dx???un(x)dx.

´Ë¶¨ÀíÖ¸³ö£¬º¯ÊýÏÊýÔÚÒ»ÖÂÊÕÁ²µÄÇé¿öÏ£¬ÇóºÍÔËËãÓëÇó»ý·ÖÔËËã¿ÉÒÔ½»»»Ë³Ðò. ¶¨Àí3 (ÖðÏîÇóµ¼)Èôº¯ÊýÏÊý

?un(x)ÔÚ?a,b?ÉÏÿһÏÓÐÁ¬ÐøµÄµ¼º¯Êý£¬x?[a,b]Ϊ

?uµÄÊÕÁ²µã£¬ÇÒu(x)?n(x)ÔÚ?a,b?ÉÏÒ»ÖÂÊÕÁ²£¬Ôò ?n

?(dxudn(x))?d(?un(x)). dx´Ë¶¨ÀíÖ¸³ö£¬º¯ÊýÏÊýÔÚÒ»ÖÂÊÕÁ²µÄÇé¿öÏ£¬ÇóºÍÔËËãÓë΢·ÖÔËËã¿ÉÒÔ½»»»Ë³Ðò(ÌÕ¹ðÐ㣬2005).

[3]

4

3 º¯ÊýÏÊýµÄÒ»ÖÂÊÕÁ²ÐÔÅбð·¨

3.1 Ò»°ã·½·¨

Åбðº¯ÊýÏÊýÒ»ÖÂÊÕÁ²¼ÈÊÇÊýѧ·ÖÎöÖеÄÒ»¸öÖصã,ÓÖÊÇÒ»¸öÄѵã.Ò»°ãµÄÇé¿öÏÂ,Ö¤Ã÷Ò»ÖÂÊÕÁ²»áÀûÓÃÒ»ÖÂÊÕÁ²µÄ¶¨Ò壬¼´¶¨Òå4À´Ö¤Ã÷.

¶¨Òå4µÄÌõ¼þÌ«Ç¿£¬º¯ÊýÏÊý¹Ì¶¨Ò»µãx?D,ÀûÓÃÊýÁеÄÐÔÖʵõ½ÒÔ϶¨Àí£º

¶¨Àí4 £¨Ò»ÖÂÊÕÁ²µÄ¿ÂÎ÷×¼Ôò£©º¯ÊýÏÊý

?unn(x)ʵ¼ÊÉÏÊÇÒ»¸öÌØÊâÊýÁÐ.ÊÜ´ËÆô·¢,

?u(x)ÔÚÊý¼¯DÉÏÒ»ÖÂÊÕÁ²µÄ³äÒªÌõ¼þΪ£º

¶ÔÈθøµÄÕýÊý?£¬×Ü´æÔÚijÕýÕûÊýN£¬Ê¹µÃµ±n?Nʱ£¬¶ÔÒ»ÇÐx?DºÍÒ»ÇÐÕýÕûÊýp£¬¶¼ÓÐ

Sn?p(x)?Sn(x)??

»ò un?1(x)?un?2(x)???un?p(x)??.

´Ë¶¨ÀíÖе±p?1ʱ£¬µÃµ½º¯ÊýÏÊýÒ»ÖÂÊÕÁ²µÄ±ØÒªÌõ¼þ. ÍÆÂÛ º¯ÊýÏÊýÊÕÁ²ÓÚÁã.

É躯ÊýÏÊýÔÚDÉϵĺͺ¯ÊýΪS(x)£¬³Æ

rn(x)?S(x)?Sn(x) Ϊº¯ÊýÏÊý

?un(x)ÔÚÊý¼¯DÉÏÒ»ÖÂÊÕÁ²µÄ±ØÒªÌõ¼þΪ£ºº¯ÊýÁÐ?un(x)?ÔÚDÉÏÒ»ÖÂ

?un(x)µÄÓàÏî.

¶¨Àí5 º¯ÊýÏÊý

?un(x)ÔÚÊý¼¯DÉÏÒ»ÖÂÊÕÁ²ÓÚS(x)µÄ³äÒªÌõ¼þÊÇ£º

n??x?D limsuprn(x)?limsupS(x)?Sn(x)?0.

n??x?DÖ¤Ã÷ ±ØÒªÐÔ ÒòΪ

?un(x)ÔÚÇø¼äDÉÏÒ»ÖÂÊÕÁ²,ËùÒÔ???0£¬?N?0£¬Ê¹µÃµ±n?Nx?Dʱ£¬¶ÔÒ»ÇÐx?D,¶¼ÓÐSn(x)?S(x)??£¬¼´rn(x)??£¬ËùÒÔsuprn(x)??£¬ËùÒÔ

limsuprn(x)?0.

n??x?D³ä·ÖÐÔ Éè

?un(x)ÔÚDÉϲ»Ò»ÖÂÊÕÁ²,¼´??0?0£¬?N?0,?n0?N,?x0?D£¬Ê¹µÃ

x?Dn??x?DSn0(x0)?S(x0)??0£¬¼´suprn0(x)??0£¬ËùÒÔlimsuprn(x)?0.ÓëÒÑ֪ì¶Ü(Àîá°£¬2003)[4].

Àý1 Èôfn(x)ÔÚ?a,b?ÉÏ¿É»ý£¬n?1,2,?ÇÒf(x)Óëg(x)ÔÚ?a,b?É϶¼¿É»ý£¬

5