In the present paper, we shall prove a fixed point theorem by using generalized weak C- contraction of integral type. Our result is generalization of  very known results. Key words: Metric space, fixed point, weak C- contraction. AMS Subject Classification: 54H2

Bhardwaj, Ramakant

Prajapati, Pravin B.

Rajput, S.S.

Tiwari, Ankur

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  180 A Fixed Point Theorem for  Weakly C - Contraction Mappings of Integral Type.  1Pravin B. Prajapati, 2Ramakant Bhardwaj ,3S.S.Rajput,2Ankur Tiwari 1S.P.B. Patel Engineering College, Linch 1The Research Scholar of CMJ University, Shillong (Meghalaya) 2Truba Institute of Engineering & Information Technology, Bhopal (M.P) 3Prof. & Head Deptt.of Mathematics Govt. P.G.College,Gadarwara E-mail:   1pravinprajapati86@gmail.com  , 2drrkbhardwaj100@gmail.com  ABSTRACT In the present paper, we shall prove a fixed point theorem by using generalized weak C- contraction of integral type. Our result is generalization of  very known results. Key words: Metric space, fixed point, weak C- contraction. AMS Subject Classification:  54H25  1    Introduction and Preliminaries Let (X, d) be a complete metric space and T:  →  a self-map of X. Suppose that   =  ∈ |  =  is the set of fixed points of f. The classical Banach’s fixed point theorem is one of the pivotal results of functional analysis. by using the following contractive definition: there exists k  [0, 1) such that     x, y ∈   , we have    d,  ≤kd (x,y)  .                                               (1.1)  If the  metric  space  (X,d)  is  complete  then  the  mapping  satisfying   (1.1)  has a  unique  fixed point .  Inequality (1.1) implies continuity of T. A  natural  question is that  whether  we can find  contractive  conditions  which  will  imply existence of  fixed point in a  complete  metric space  but  will not imply continuity . Kannan [10,11]  established  the  following  result  in which  the  above  question  has been answered  in the  affirmative. If  T :  →   where  (X,d)  is  complete  metric  space , satisfies  the  inequality  d,  ≤k[d (x,Tx) +d(y,Ty)]                                                 (1.2) where 0<  <   and  x, y ∈ , then T has a unique fixed point. The mapping T  need not  be  continuous .The mapping  satisfying (1.2)  are called Kannan type mappings. There  is a large literature  dealing  with Kannan type  mappings  and their  generalization some of  which  are  noted in [8] ,[17] and [19]. A similar  contractive  condition  has been introduced by  Chatterjee [6]. We call this contraction a C- contraction. Definition1.1 C-contraction Let T :  →   where  (X,d)  is  a metric space is called a C – contraction if there exists 0<  <    such that for all x, y ∈   the following  inequality holds:        d,  ≤k[d (x,Ty) +d(y,Tx)]                                                            (1.3) Theorem 1.1 A C- contraction defined on a complete  metric  space has a unique fixed point. In establishing  theorem 1.1 there is no requirement  of  continuity of the C-contraction. It has been established in [15] that  inequalities (1.1),(1.2) and (1.3) are independent of  one another. C- contraction  and  its  generalizations  have been discussed in a number of  works some of which are noted in [4] ,[8], [9] and [19]. Banach’s contraction mapping  theorem  has been  generalized in a number of recent papers. As for example, asymptotic  contraction  has been  introduced  by Kirk[12] and  generalized Banach contraction  conjecture  has been  proved in [1] and [14]. Particularly a weaker contraction  has been  introduced  in Hilbert spaces in[2].The following is the corresponding  definition in metric space. Definition1.2 Weakly contractive mapping A mapping  T :  →   where  (X,d)  is  complete  metric  space is said to be weakly contractive if   d,  ≤  d(x,y) −, ,                             (1.4) Where x, y ∈  ,  :[0,∞ → [0,∞ is  continuous  and non-decreasing,   (x) = 0  if and only if  x = 0  and lim→ ψ x = ∞. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  181 There are a number of works in which weakly contractive  mappings have been considered. Some of these works are noted in [3],[7],[13], and [16]. In the present work in the same spirit we introduce a generalization of C- contraction. Definition1.3 Weak C- contraction: A mapping T :  →  , where  (X,d)  is  a metric space is  said to be weakly C – contractive or a  weak C- contraction if for  all  x, y ∈  ,        d,  ≤   [d (x,Ty) +d(y,Tx)] −d (x,Ty), d(y,Tx) )                   (1.5)  where    : [0, ∞ → [0, ∞  is a  continuous  mapping such that   (x,y) = 0  if and only if  x = y= 0 . If we take   (x,y) = k(x+y)  where 0<  <   then (1.5) reduces to  (1.4), that is weak C – contractions are  generalizations  of  C – contractions.   In a recent paper of Branciari [20]  obtained a fixed point  result for a single mapping  satisfying  an  analogue of a Banach’s contraction principle for integral type inequality as  below: there exists  c  [0,1)  such that     x, y ∈   , we have    % &'' (),)*+ ≤k% &''  (,*+                                                    Where  & : ,- → ,-  is a Lebesgue – integrable  mapping which is summable, non-negative  and  such that  for  each  . >0, % &''0+ >0 . Our main result is extended and modified to  the weak C – contraction mapping in  integral type . MAIN RESULT Let   T :  →   where  (X,d)  is  complete  metric  space be a  weak C-contraction,   which is satisfying  the  following  property: % &'' (),)*+ ≤ 1 % &'' (,)*-2*,)+   + 3 % &''  456 (,),2*,)*+                             − % &''7  (6,89,(9,86,(,),2*,)*+                             (2.1)                                                                                                                        Then T has a unique fixed point. Where  1 , 3 ∈ [0,1)  with  21 +  3 ≤1   and  & : ,- → ,-  is a Lebesgue – integrable  mapping which is summable,non negative  and  such that  for  each  . >0, % &''0+ >0   and   : [0, ∞ → [0, ∞  is a  continuous  mapping such that   (x,y) = 0  if and only if  x = y= 0 . Proof   :   Let + ∈X and for all n≥ 1 , =-= T=  . If   =-= == T=. Then  =   is a   fixed point of T. So we assume,=- ≠ =.  Putting   x = =?  and   y = = in  (2.1)  we  have  for  all  n =  0,1,2, ……. % &'' (@ ,@AB+   =  % &'' ()@CB ,)@+                                ≤ 1 % &'' (@CB,   )@-2@,)@CB+                                +   3 % &''456  (@CB,)@CB,2@,)@+                                − % &''7  (@CB,8@,(,@8@CB,(@CB,)@CB,2@,)@+                                =   1 % &'' (@CB,   @AB-2@ ,@+                                 + 3 % &''456  (@CB,@,2@,@AB+                               − % &''7  (@CB,   @AB,(,@@,(@CB,@,2@,@AB+  Since T is Weakly C – contraction, this gives that ψ  d=?,   =-, 0, d=?, =, =, =-  = 0   and    % &'' (@ ,@AB+ ≤ 1 % &'' (@CB,   @AB+                                  + 3 % &''456  (@CB,@,2@,@AB+                             (2.2)                        Now here arise two cases: Case I: -   If we choose   max  d=?, =, = , =- }  =  d=?, = Then (2.2) can be written as   % &'' (@ ,@AB+ ≤ 1 % &'' (@CB,   @+    +1 % &'' (@,   @AB+                                     +3 % &''(@CB,@+   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  182 (1- 1) % &'' (@ ,@AB+ =  (1 + 3) % &''(@CB,@+              % &'' (@ ,@AB+  =  F-G?F% &''(@CB,@+             % &'' (@,@AB+ ≤K% &''(@CB,@+  where k =  F-G?F≤  1  HIJK L ∶ : -   If we choose   max  d=?, =, = , =- }  =  d=, =- Then (2.2) can be written as   % &'' (@ ,@AB+ ≤ 1 % &'' (@CB,   @+    +1 % &'' (@,   @AB+                                 +3 % &''(@,@AB+  [1 – (1 + 3)]% &'' (@,@AB+   =  1 % &'' (@CB,   @+  % &'' (@ ,@AB+  =   F – F-G  % &'' (@CB,   @+  % &'' (@ ,@AB+ ≤ k % &''(@CB,@+    ,where  k =  F – F-G  ≤  1        (2.3)                                    From above both cases:                 % &'' (@,@AB+ ≤  % &''(@CO,@CB+                                                    ≤ P % &''(@CQ ,@CO+                                                - - - - - -                                             ≤ = % &''(R ,B+  Taking limit as n → ∞ , we get limS → % &'' (@,@AB+  = 0 ,  as  k  [0,1)                                            (2.4)                                                                    Now we prove that  = is a Cauchysequence.  Suppose it is not.Then there exists an  T > 0 and sub sequence UVWXand U=WXsuch that M(p)  n(p)  m(p+1)  with    =W, VW)≥ T, =W?, VW )< T                                                                                                (2.5) Now    VW? , =W?)≤ VW? , VW)   +   VW, =W? )     < VW? , VW)   +  T                                                                    (2.6)                                                                                                        From   (2.4), (2.6), we get    limY→ % &''2Z[CB ,@[CB + ≤ % &''\+                                                   (2.7)                                                                                          Using (2.3), (2.5), and (2.7)    we  get ,                             % &''\+ ≤ % &''2@[,Z[+                                            k% &''2@[CB,Z[CB+                                            k% &''\+  Which is contradiction, since k  (0, 1). therefore  = is a Cauchy  sequence  Since  ( X,d) is  complete  metric space , therefore  have  call the  limit z . From (2.1), we get % &'' ()],@AB+ =  % &'' ()],)@+                               ≤ 1 % &'' (],)@-2@ ,)]+                                + 3 % &''456  (],)],2@,)@+                            − % &''7  (^,8@,(@ ,8^,(],)],2@,)@+  Taking limit as n → ∞ , we get % &'' ()],]+ ≤ 1 % &'' (],)]+  + 3 % &'' (],)]+                             =    (1 + 3 ) % &'' (],)]+  Which is Contradiction         Therefore   _ = z  That is z is a fixed point of T in X. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  183 Uniqueness  : Let w is  another  fixed point  of  T in X  such  that  z  w , then we have  From (2.1), we get % &'' (],`+   =  % &'' ()],)`+                          ≤ 1 % &'' (],)`-2`,)]+   + 3 % &''456  (],)],2`,)`+                         − % &''7  (^,8`,(`,8^,(],)],2`,)`+   % &'' (],`+ ≤ 21 % &'' (],`+  Which is contradiction  So  z = w  that is , z  is  unique  fixed  point  of  T  in  X . Acknowledgement: One of the author (Dr. R.K. B.) is thankful to MPCOST Bhopal for the project No 2556   REFERENCES [1`] Arvanitakis.A.D, A proof of the generalized Banach contraction conjecture, Proc.Amer.Math. Soc., 131(12) (2003), 3647-3656. [2] Alber .Ya.I and Guerre- Delabrieer. S, Principles of weakly contractive maps in Hilbert spaces, new results in operator theory, Advances and Appl. Birkhauser Verlag, Basel 98 (1997), 7-22.  [3] Beg. I and Abbas. M, Coincidence  point and  Invariant  Approximation  for  mappings  satisfying  generalized  weak contractive condition,  Fixed point  Th. And Appl. (2006), Article ID 74503, 1-7. [4] Berinde. V, Error estimates for approximating fixed points of quasi contractions, General Mathematics, 13, (2005), 23-34. [5] Berinde. V, Picard Iteration  converges faster than Mann  Iteration  for a class of  quasi contractive  operators,  Fixed Point  Theory  and  Applications (2004), 97-105. [6] Chatterjea. S.K, Fixed Point theorems, C.R. Acad. Bulgare Sci, 25, (1972), 727-730.  [7] Chidume. 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A Fixed Point Theorem for  Weakly C - Contraction Mappings of Integral Type

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A Fixed Point Theorem for Weakly C - Contraction Mappings of Integral Type

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