Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 3(5), 83-85, May (2014) Res.J.Recent Sci. International Science Congress Association        
83
 
The Existence of General Solution of Non-linear partial differential equation in General case in complex spaceN. Taghizadeh and S. Norozpour* Department of Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O Box 1914, Rasht, IRAN Available online at: 
www.isca.in
, 
www.isca.me
Received 23rd July 2013, revised 6th October 2013, accepted 20th November 2013Abstract  In this paper, we discuss on the existence of general solution of non-linear partial dierential equation 
 in complex space by using Fixed point theorem and contraction function. Keywords: Partial differential equation, complex space, fixed point theorem, Holomorphic function, contraction function IntroductionSupposed D  C and S  , weakly and strongly singular operators   
 and  
  are defined as follows: 
1()()
Tfzdd
z
x
zh
px
-


, 
1()()()
fzdd
z
x
zh
px
-
P=

,                                               so that  =  + i  , z = x + i y and  
()
()
Tfz
fz
z
¶
 and  
()
()
Tfz
fz
z
¶
=P and if  
f
 is a bounded in  
D
 then   
()
D
Tfz
is bounded and Holder Continuous. An interest to the developing of this area is connected rst of all with dierent types applications of generalized analytic functions. The most known constructions are those generalized analytic functions of Vekua type dened as a solution to elliptic system of dierential equations in complex domains containing so called (F, G)-derivatives. Dierent methods are developed for the study of the corresponding differential equations and the corresponding boundary value problems.  Now, we represent the most results in this direction. In relation to this subject, mentioned topics have been studied as follows: i. Mamourian A, Esralian E, Taghizadeh.N. On the existence of general solution of rst order elliptic systems by Fixed-point theorem. ii. N.Taghizadeh. On the uniqueness of solution of rst order non-linear complex elliptic systems of partial differential equations in Sobolev space. iii. N.Taghizadeh, M.Akbari, the existence and uniqueness of solution of non-linear partial differential equations by Fixed-point theorem in sobolev space and their results. Existence of general solution in complex space. Firstly, we assume that 
(
)
(
)
  0    1
wCD
Î
 is an arbitrary solution of   
(,,,)0
wwFzw
zz
¶¶
=
¶¶. We dene two functions as follows: 
,,,,,,
wwwww
GzwTFzw
zzzzz
¶¶¶¶¶

=-

¶¶¶¶¶

  (2) ()()
,,,.
D
ww
zwzTGzw
zz
¶¶

=-

¶¶

                        (3) By differentiating  
  partially with respect to z and  
 we obtain that: 
,,,0
wwwGzwzzzz¶¶¶¶

=-=

¶¶¶¶

                             (4) 
,,,.
D
ww
Gzw
zz
zz¶¶¶

P

¶¶

                  
(5)                  
Since 
()()
,,, 0    1
wwGzwCDzz¶¶
Î
¶¶the following estimates hold: 
,,
,
,,,DD
D
wwTGzwzzaa¶¶


¶¶

£+          (6) 
Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 3(5), 83-85, May (2014) Res. J. Recent Sci.   International Science Congress Association 
84
 
,,,,,
.
DD
D
wwGzwzzaa¶¶¶¶£P¶¶  (7) It follows from the equation (4) and Wely’s lemma that 
j
  is  holomorphic in  
D
 and it belongs to the complex space according to (6), so, we deduce that, if  
w
is a solution of (1)  then it is in the form:  ()()
,,,
ww
wzzTGzw
zz
¶¶

=+

¶¶

                      (8) where  
j
 is holomorphic in  
D
. Theorem 1.2:  supposed 
Í


 and 
¥
. A function  
(
)
(
)
  0    1
wCD
Î
 is a solution of the partial differential equation (1) if and only if for a function 
(
)
 
CD
 and holomorphic in 
D
, 
(,)
wh
  satisfied the system below: ()()
()
,,,(),,
w
wzzTGzwh
z
w
hzzGzwh
z
jj¶

=+

¶

¶

+P


¶

                         (9) Proof: Firstly, supposed  
w
is a solution of (1), as it is proved, 
w
is in form (8) where  
j
 is holomorphic in
(
)
 
CD
 and 
G
is in form (2). By differentiating  (8)  partially with respect to 
z
we obtain that: 
,,,zz
ww
Gzw
zz
¶¶

P

¶¶
¶¶

we denote: 
w
h
z
¶
=
then ()
,,(),
w
zGzwhzh
z
j¶

+P

¶

so 
(,)
wh
 is a solution of the system (9). Now, we suppose 
(,)
wh
 is a solution of the following system. By differentiating the first equation in this system partially with the respect to 
z
 and 
z
 we obtain: 
,,,
,,,()
()wwGzwhzz
Gzwhhz
=+¶¶¶¶P=             (10) on substitution  
w
h
z
¶
=
 in (10) we obtain the following result. Remark: We denote the set of all pairs 
(,)
wh
 for which both  
w
 and  
h
 belong to the space 
(
)
 
CD
. The norm in this space shall be defined as follows: 
,,,
(,)(,)
DDD
whMaxwhaaathus making 
(
)
 
CD
 a Banach space. Theorem 2.2:  If  i. The domain 
D
has a finite area. ii. As a function of the variables zD
ÎÍ


 ,  
,,,
ww
Gzw
zz
¶¶


¶¶

  is  a continuous of its variables. iii. The function  
G
 satisfy in the Lipschitz condition of the form: 
12,,,,,,wwGzwh
LwwLhh
Gzwhzz¶¶¶¶
£-+-


almost everywhere in  
D
,whereas constant 
2
L
   is strictly less than 1 and 
1
L
 is arbitrary positive number. There exist 
(
)
{
}
,()(,)|, whDhCww
D
ÎW=Î  such that  
,,,
ww
Gzw
zz
¶¶


¶¶

Î
()
p
D
. Then differential equation (1) has unique solution in the complex space. Proof: We denote 
(
)
{
}
()(,)|, DwhwC
D
W=Îand we define the norm as follows 
,,,
(,)(,)
DDD
whMaxwhaaathen the set 
()
p
D
 is a  Banach space. For  a pair  
(,)()
p
whD
ÎW we define an operator L as follows: 
:()()
(,)(,)
pp
LDD
LwhWH
W®W
=  
Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 3(5), 83-85, May (2014) Res. J. Recent Sci.   International Science Congress Association 
85
where         (12) where 
j
  is  holomorphic  in  
D
 and it belongs to 
(
)
 
CD
. We will prove  L satisfy in Lipschitz condition. Supposed  
(,),(,)()
p
whwhD
ÎW  so: 
(,)(,)
LwhWH
=
where  and where          .                                                                              (13) Then  
(,)(,)(,)(,)
LwhLwhWHWH
-=-

 
(),()
WWHH
=--

 
(,)
MaxWWHH
=--

 .  According to the systems (12) and (13): 
WW
-

=
,,,,,,DD
ww
TGzwhTGzwh
zz
¶¶

-

¶¶


 =
,,,,)
(,,wwTGzwhGzwhzz¶¶

-

¶¶


 
12
()()
ADLwwLhh
£-+-

  
12
()()((,)(,))
ADLLwhwh
£+-

and similarly: 
12
()()((,)(,)).
HHBDLLwhwh
-£+-

This mean that: 
12
(,)(,)()max((),())(,)(,).
WHWHLLADBDwhwh
-£+-

We denote 12
()max((),())
KLLADBD
=+then 
(,)(,)(,)(,).
WHWHKwhwh
-£-

Now, if  
01
K
££
 then the operator  is contraction function in 
()
p
D
 and such as, there exists a unique fixed element 
(,)
wh
 of the operator  which is also a solution of (9). The corresponding  
w
 is then by theorem (1.2) a general solution of differential equation (1).  ConclusionIn this paper, we discuss on the existence of general solution of Non-linear partial dierential equation in general case in complex space. We deduce that the proposed method can be extended in other spaces. References1.Vekua I.N., Elliptic rst order system of partial dierential equations boundary value problems, Matem, Sbornik, 31, 217-314 (1952) 2.Vekua I.N., Generalized analytic functions, 2nd ed,Nauka, Moscow, (in Rus-sian) (1988)3.Mamourian A., Esralian E. and Taghizadeh N., On the existence of general solution of rst order elliptic systems by Fixed-point theorem, Non-linear Analysis theory and Application, 30(1), 5351-5356 (1997)4.Taghizadeh N., On the uniqueness of solution of rst order non-linear complex elliptic systems of partial differential equations in Sobolev space, Honam Mathematical, 27(2),205-209 (2005)5.Taghizadeh N. and Akbari M., the existence and uniqueness of solution of non-linear partial differential equations by Fixed-point theorem in sobolev space and their results, International journal of Applied Mathematics, 22(6), 879-885 (2009)()()
()
,,,(),,WzzTGzwh
z
w
HzzGzwh
z
jj¶

=+

¶

¶

+P

¶

	=	
(,)(,)
LwhWH
=

()()
()
,,,(),,WzzTGzwh
z
w
HzzGzwh
z
jj¶

=+

¶

¶

+P

¶

()()
()
,,,,,,()HzWzzTGzwh
z
w
zGzwh
z
jj=¶

=+

¶

¶

+P

¶	





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