Analytic Solutions of Special Functional Equations


©

,fgg 

,,)( Dxxxf 



D .f

g g

,f

g

,g

.f

F XDU 

X

X

)( H

.H

R, vu vu  [,] vu

R[,:] vug

,R)(lim)(lim 


wxgxg
vxux

[,] u [.,[ v

[,][,:] vuvuf 



[,])),(()( vuxxfgxg 

 [,] vu

;)(lim,)(lim uxfvxf
vxux






ff 
 1

[;,] vu

[;,] vu

,1},{N}),{\[,](  nnvuCg
n

 });{\[,]( vuCf
n



},{\[,] vu

[),,(]
2

vuCg  0)( g R


)(lim:
1

xf
x

})\{[,(])[,(]
21

 vuCvuCf  ;1)(  f

[),,(]
3

vuCg  0)( g R


)(lim:
1

xf
x

,)(lim:
2

Rxf
x


 

 })[\{,(][),(]
32

 vuCvuCf 

;
)(

)(

3

2
)(

2





g

g
f






g , f    ;1 f

,|:
[,] ul

gg  ;|:
[,] vr

gg




],])}()(;[,[sup{))(()(
000

1

0
 uxxgxgvxxggxf

lrlr





[,[)}()(;],]inf{))(()(
000

1

0
vxxgxguxxggxf

rlrl
 




.f

 ,\[,] vux    ., xgx

,g

g

      .,,, 1111 xxxgxxgx 



  .1xxf 

           .:,\[,],1   fvuxxfgxgxg

x  xfu ],,]  [,,[ v

v . x [,[ v

 xf ],,] u

 .u

[,0]  G

  ,0[,,0][,0:]  gg ],0] 

[,,[ 

    .limlim
0




xgxg
xx

,, Ghg 

hg ,

,
hg

ff 

hg
ff , ,g h

[.,0],,  Ghg hgghg  ,, 

G

;
hg

ff 

;,|:,|:,
[,[],0]

11
Ggggggghgh

rlllrr









[,0[[,0[:    ,00  

.gh 



X )( XIzom


XXT :


X

X


X

,X lD

};;{   xXxD
l

r
D

};;{   xXxD
r

XDg
ll
:

}.{\,))(()( 
ll

DxXIzomxg 




XDg
rr
:

}.{\,))(()( 
rr

DxXIzomxg 




)()( 
rl

gg  ),()(
rl

gRgR 

)( gR ,g )( xg g x

        .,

,:

rrll

rl

DxxgxgDxxgxg

XDDDg



 

DDF :

DxxFgxg  )),(()(

F D



1
F FF 

 1
;D



,)}()(;sup{))(()(
000

1

0 llrrlr
DxxgxgDxxggxF 



.)}()(;inf{))(()(
000

1

0 rrllrl
DxxgxgDxxggxF 



,
~
g g

.

      ,0~~,  wgzgwzH

 
 

  
 .lim

0

0

~

~

lim,0, zw
zwg

zg

w

H

zz











 

 ,\[,] vu

f

.

g
~

g

g
~

[.,] vu

f
~

,
~~~
fgg  f

~

f [.,] vu



f
~

.

     
 

 

 

 
    

       
 

 

 

 
       .0

!12

~

!2

~
~~

,0
!12

~

!2

~
~~

122
2

122
2

























































zfzf
k

g

k

g
zfgzfg

zz
k

g

k

g
zgzg

kk
k

kk
k

km 2

m g
~



 .g ,
~~~
fgg 

    .lim  fzfz 

   
 

      

 
         

,
~~

!2/
~

!2/
~~~

2/1

2

2
k

k

k

zfzfkg

zzkg

z

fzf
































 .  f 
~

k2

f
~

    .1
~

  ff

□

V 



   

   .0Im,Re;,0Im,Re;

,0Im,Re;,0Im,Re;









zzVzVzzVzV

zzVzVzzVzV

rr

ll





V 

 VfVf
~

:
~



                    .
~

,
~

,
~

,
~

,
~~ 


lrlrrlrl

VfVfVfVfVfVfVfVfidff 

f
~

f f
~

,0 x f

□

W [,] vu f
~

.W

  





zf
z

1

1 z   2 zzf

    
 

  








2

2

~
11










z
f

z
z

zf
z

.1 z



,1

1

2




n

nan

1, na n

,11 a 2, na n

    .2,   zzfzzf

  





zf
z

1

.

□

   zzf

    










1

2

1

.1,

n

n
n

n

n anzaz 

    ,2,1,0,1,

2

2
1

1

 








nmpapazazh nn

n

nm

nn

n

n
n 

nn mp ,



,

 
 

 
.

~

~

2
~

zg

zg
zzf






         
 

      .0
2

~
~~~

0
22

zzfzzf
zg

zzfzgzgzfg 




  0 zzf

  ,zzf 

□

g
~

U

,H  U ,

f
~

       

        

       ,~~2
~

,0Im
~

0Im

,
~~

,
~~~

1






UgUgUUf

zUfzU

UUffUgUfg



.U

□



        .0,0,0,expexp   axxfxfxx
aa

       .0/:,0,expexp  abxxbfxfbxx 

[,0][,0:] f

    ;0,0  ff

b/1 ;f

ff 
 1

[;,0] 

 xf

      

       [;,/1[,expexp];/1,0]inf

],/1,0],expexp[;,/1[sup

0000

0000





bxbxxbxxbxxf

bxbxxbxxbxxf

f f
~

[,,0]    ,
~~

,1/1
~

idffbf   f
~

,/1 b   .
2

12~

bz

bz

b
zzf






   bxxxg  exp 0  
11

/1


 bebg

 ,,0 1b  1bg 0
[.,[

1



b

g ,
~
g

           .,1,,exp1~,0~ 11 Cznnbznbzbzgbg nnn  

  ,0~ 1  bg

□



g

[,,] vu ,
1

b g

   .[,,] vgugvu 

    fbzbzzzg
~

,,0Re,exp
~ 1

  f

[,0] 

   HAAXX  ,

.H

      

 .0,;

,,;,; 111

HhhVhXVX

XUUVVUXVAXXAUUAHUX







X

    

     ,,],/1[;

,[/1,[;

1

1

rlr

l

DDDIbvbVXVD

IbbuVXVD

















 V .V

,: DDF 

       .,expexp DUUbFUFbUU 

Ib
1

;F

F FF 
 1

;D



F lD rD rD ;lD

F

      

       ;,expexp;inf

,,expexp;sup

0000

0000

rl

lr

DUbUUbUUDUUF

DUbUUbUUDUUF





,AU  ,
1

b

     .2)2(
11 

 bUIbUIbUUF

   bUUUg  exp ,, rl DD

g [,] vu

,, 21 lDUU 

.2,1,  jDU rj 21 , UU

 1,0t

     

   

   

   

 

   

     .1

1

11

21

21

21

221

21

1

21

21

2121

UtgUgt

ddxgtddxgt

ddtxxtgtUUtg

EE

UU

EE

UU

EE

UU



















rl DD ,

 ,\ 1 IbDU l




     0exp  IbUbUUg l

   .\,0 1 IbDUUg rr




rl gg ,



     ,:, 121 UFUgrUg ll  F f

,f

          .,/1 2112 rDUvbUfUFU  

        ,/1,, 1Ubuttfgtg rl 

,1U

 

 

 

 

         .21
1

1

1

1

1 rrrUE

U

rUE

U

ll gRUgUfgdtfgdtgUg  


   .lr grgr 

□

f

       ,0,11   xeeee xfxfxx

    .0,1ln   xexf x

    .0Re,1ln
~




zezf
z

    [,0],1   xeexg xx

.2ln

,f

         .22 xxfxxfxxfxxf eeeeeeee  



 
,2ln,0 


xee

xxf

□

f
~

      
2

2

~
,1,

~~~












z

z
zgzzfgzg

  .
1

~




z

z
zf