Theorem for Expansion Mapping Without Continuity in Cone Metric Space

Theorem for Expansion Mapping Without Continuity in Cone Metric Space

Anushri A. Aserkar

Rajiv Gandhi College of Engineering and Research, Nagpur, India

Abstract

In the present paper we prove a unique common fixed point theorem for expansion mapping without continuity for four self mappings. We use the condition of weakly compatibility to prove the fixed point. In this result the cone is not necessarily a normal cone. The result is an extension and generalisations of many results available in the literature

Keywords: Cone metric space, Expansion mapping, weakly compatible mappings.

1. Introduction

   In 1922, Banach proved a common fixed point theorem which ensures, under appropriate conditions, the existence and uniqueness of a fixed point. This result of Banach is known as Banachs fixed point theorem or Banach contraction principle. Many authors have extended, generalized and improved Banachs fixed point theorem in different ways.

   In 1984, Wang et al. [10] presented some interesting work on expansion mappings in metric spaces which correspond to some contractive mappings in [7].Further, Khan et al. [4] generalized the result of

   [10] by using functions. Also, Rhoades [8] and Taniguchi [9] generalized the results of Wang [10] for a pair of mappings. Kang [3] generalized the result of Khan et al. [4], Rhoades [8] and Taniguchi [9] for expansion mappings. Daffer and Kaneko [1] defined an expanding condition for a pair of mappings and proved some common fixed point theorems for two mappings in complete metric spaces.

   Manjusha P. Gandhi

   Yeshwantrao Chavan College of Engineering, Nagpur, India

   Recently, Huang and Zhang [2] introduced the concept of a cone metric space as a generalization

   of a metric space. They proved the properties of sequences in cone metric spaces and obtained various fixed point theorems for contractive mappings.

   We have proved fixed point theorem for expansion mapping for four mapping in cone metric space. The theorem is an extensions and generalizations of Yan Han and Shaoyuan Xu [13], Wasfi Shatanawi and Fadi Awawdeh [11], Xianjiu Huang, Chuanxi Zhu and Xi Wen[12]

2. Preliminary

   We need to use the following fundamental concepts throughout this paper.

   1. Cone

      Let E be a real Banach space and P E .Then the set P is called a cone if and only if

      1. P is closed, non empty and P;

      2. a, b R, a, b 0, x, y Pax + byP

      3. P (-P) = .

   2. Partial ordered cone

      For given cone P E ,we define a partial ordering

      with respect to P by x y if and only if y – xP. We shall write x y for y – xP0, where P0 stands for interior of P. Also we will use x < y to indicate that x y and x y .

   3. Cone metric space

      Let X be a non empty set. Suppose that the mappings d : X X E satisfies:

      1. 0 d(x, y) for all x, yX and

         d(x, y) =0 if and only if x = y;

      2. d(x, y) = d( y, x) for all x, yX;

      3. d(x, y) d(x, z) +d(z, y) for all x, y, zX .Then d is called a cone metric on X and (X ,d) is called a cone

      metric space.

   4. Example 1

      Let E R 2 , P {(x, y)E : x, y 0} R 2 , X R and d : X X E such that

      d(x, y) ( x y , x y )), here 0 is a constant. Then (X, d) is a cone metric space

   5. Expansion mapping

      Let (X, d) be a complete cone metric space. If f is a mapping of X into itself and if there exists a constant q

      > 1 such that d (f(x), f(y)) q d(x , y) for each x, yX, then f is called as the expansion mapping in

      X.

      .

   6. Convergent Sequence

      Let (X, d) be a cone metric space. The sequence {xn} in X is said to be a convergent sequence if for every c

      Let f and g be two self-maps defined on a set X . Then f and g are said to be weakly compatible if they commute at coincidence points. That is, if fu =gu for

      some uX, then fgu=gfu.

      2.10 Coincidence Point

      Let f and g be self-maps on a set X . If w = fx = gx, for some x in X, then w is called coincidence point of f and g.

      Our theorem is an extension and generalization of Yan Han and Shaoyuan Xu [13], Wasfi Shatanawi and Fadi Awawdeh [11], Xianjiu Huang, Chuanxi Zhu and Xi Wen[12]

3. Main theorem

   Let (X, d) be a complete cone metric space. Suppose A, B, P, Q are self mappings on X itself and each of it are surjective.

   (i)A(X) Q(X) , B(X) P(X).

   (ii)(A,P) and (B,Q) are weakly compatible. (iii)Suppose for ,, , , such that ,, , , [0,1) and + + > 1

   d(Px, Qy) d(Ax, Px) + d(By, Qy) +

   ……(1)

   d(Ax, By) + d(Ax, Qy) + d(Px, By)

   for all x.y X. Either 1+ > or 1+ > Then A, B, P, Q has a unique common fixed point in X.

   E with 0 < c, there is n0 N such that for all n n0, d (xn, x) < c for some x X. We denote this by

   Proof: Let x0

   is an arbitrary point in X.

   n

   n

   lim x = x .

   n

   1. Cauchy Sequence

      Let (X, d) be a cone metric space. The sequence {xn} in X is said to be a Cauchy sequence if for all c E with 0 c, there is no N such that d (xm , xn) c, for all m, n n0.

   2. Complete cone metric space

      A cone metric space (X, d) is said to be complete if every Cauchy sequence in X is convergent in X.

   3. Weakly Compatible

   A, B, P, Q are surjective.

   There exists {x2n} ,{y2n}X such that

   A x2n = Q x2n+1= y2n and B x2n+1 =P x2n+2 = y2n+1 for all n.

   Case-1:

   Putting x= x2n, y= x2n+1 in (1) , we get

   d(Px2n , Qx2n+1) d(Ax2n , Px2n ) +

   d(Bx2n+1, Qx2n+1) + d(Ax2n , Bx2n+1) + (Ax2n , Qx2n+1) + d(Px2n , Bx2n+1)…….(2)

   d(y2n-1, y2n ) d(y2n , y2n-1) + d(y2n+1, y2n )

   + d(y2n , y2n+1)+ d(y2n , y2n )

   + d(y2n-1, y2n+1)

   d(y2n-1, y2n ) d(y2n , y2n-1) + d(y2n+1, y2n )

   + d(y2n , y2n+1) +

   (1 – + ) d(y2n-1, y 2n-2)

   ( + + ) d(y2n , y 2n-1)

   d(y2n+1,y2n ) – d(y2n , y2n-1)

   d(y , y ) (1- + )

   2n 2n-1

   2n 2n-1

   ( + + )

   d(y2n-1, y 2n-2)

   (1 – + ) d(y2n , y 2n-1)

   ( + + ) d(y2n+1, y 2n)

   (1-+)

   here (1 – + ) > 0 (1 + )> ( + + )> (1 – + ) + + > 1

   d(y2n , y 2n-1) k d(y2n-1, y 2n-2)

   ..(6)

   d(y2n , y 2n+1) (++)

   d(y2n , y 2n-1)

   where k = (1-+)

   (++)

   ……..(7)

   and k < 1.

   d(y2n , y 2n+1)

   (1-+)

   (++)

   d(y2n , y 2n-1)

   From case – I or case – II, weget

   here (1 – + ) > 0 1 + > and ( + + ) >(1 – + ) + + > 1

   d(y2n , y 2n+1) h d(y2n , y 2n-1)

   …..(3)

   d(y2n1, y2n ) hd(y2n , y2n-1)

   h kd(y2n-1, y2n-2 )

   .

   .

   Case-II:

   where h =

   (1-+)

   (++)

   …………(4)

   and h < 1

   d(y2n1, y2n )(kh)n d(y1, y0 )

   and

   d(y2n , y2n-1) kd(y2n-1, y2n-2 )

   Putting x = x2n ,y = x2n-1 in (1) , we get

   d(Px2n , Qx2n-1) d(Ax2n , Px2n ) +

   d(Bx2n-1, Qx2n-1) + d(Ax2n , Bx2n-1) .

   + d(Ax2n , Qx2n-1) + d(Px2n , Bx2n-1) .

   k hd(y2n-2 , y2n-3 )

   k(hk)d(y2n-3, y2n-4 )

   d(y2n-1, y2n-2 ) = d(Px, Qx2n-1) d(y2n , y2n-1)

   + d(y2n-1, y2n-2 ) + d(y2n , y2n-1)

   + d(y2n , y2n-2 ) + d(y2n-1, y2n-1)…..(5)

   d(y2n , y2n-1) d(y2n , y2n-1 +

   d(y2n , y2n1) k(kh)n d(y1, y0 )

   k < 1andh < 1 kh < 1 i.e. kh [0,1)

   Hence for n > m

   d(y2n1, y2m-1) d(y2n1, y2n ) d(y2n , y2n-1)

   … d(y2m, y2m-1)

   d(y2n-1, y2n-2 ) + d(y2n , y2n-1)+ d(y2n , y2n-2 ) + d(y2n-1, y2n-1)

   d(y2n , y2n-2 ) d(y2n , y2n-1)-d(y2n-1, y2n-2 )

   d(y2n , y2n-1) d(y2n , y2n-1) +

   d(y2n-1, y2n-2 ) + d(y2n , y2n-1) + d(y2n ,y2n-1) – d(y2n-1, y2n-2 )

   m

   i=n

   (kh)i d(y1, y0 )

   i=m

   i=n

   k (kh)i d(y1, y0 )

   (kh)m 1-(kh)n-m+1

   1-kh

   d(y1, y0 ) +

   Bq = y Bq =Qq

   As (B, Q ) are weakly compatible

   (kh)m 1-(kh)n-m+1 d(y1,y0 ) k 1-kh

   (1+k)(kh)m 1-(kh)n-m+1d(y1,y0 )

   BQ q = Q Bq

   By = Qy

   = 1-kh

   0

   as n, m

   Again putting x = x 2n in (1) we get

   For c 0 , we can find some > 0 such that

   c – x int P, where x < i.e. x c.

   d(Px2n , Qy) d(Ax2n , Px2n ) + d(By, Qy) +

   d(Ax2n , By) + d(Ax2n , Qy) + d(Px2n , By)

   For this, we can find a natural number N such that

   (1+k)(kh)m 1-(kh)n-m+1d(y ,y )

   as n we get

   d(y, By) d(y, y) + d(By, By) + (y, By) +

   1-kh

   1 0 for n, m > N.

   d(y, By) + d(y, By)

   Thus we get d(y2n1, y2m-1) c for n > m > N Thus{yn}isacauchysequence.

   As X is complete, there exists some yX such that

   yn y X .It is equivalent to say that y2n y X

   and y2n+1 y X .

   A x2n = Q x2n+1 = y2n y X. and

   B x2n+1 = P x2n+2 = y2n+1 y X.

   d(y, By) 0 + 0 + (y, By) + d(y, By) + d(y, By)

   (1- ( + + )) d(y,By)=0 By = y

   By = Qy = y

   Now, putting x=p in (1) we get

   d(Ps, Qy) d(As, Ps) + d(By, Qy) + d(As, By)

   + d(As, Qy) + d(Ps, By)

   d(y, y) d(Ap, y) + d(y, y) + d(Ap, y) +

   A,B,P,Q are onto mappings thus there exists p, q, r, sX such that Ap = Qq = y and Br = Ps = y

   Now we will show that p = q = r = s = y

   ( + + ) d(Ap,y) 0

   Ap = y

   d(Ap, y) + d(y, y)

   Putting x= x 2n and y =q in (1) we get,

   d(Px2n , Qq) d(Ax2n , Px2n ) + d(Bq, Qq) +

   d(Ax2n , Bq) + d(Ax2n , Qq) + d(Px2n , Bq)

   As n

   d(y, y) d(y, y) + d(Bq, y) + d(y, Bq) + d(y, y)

   + d(y, Bq)

   0( d(y, Bq)

   d(y,Bq) = 0

   Bq = y

   (A, P) are weakly compatible

   APp=PAp Ay = Py

   Now put putting x = y in (1) we get

   d(Py, Qy) d(Ay, Py) + d(By, Qy) + d(Ay, By)

   + d(Ay, Qy) + d(Py, By) d(Ay, y) d(Ay, Ay) + d(y, y) + d(Ay, y) +

   d(Ay, y) + d(Ay, y)

   d(Ay,y)=0

   Ay=y

   Ay=By=Py=Qy=y

   Now to prove the uniqueness of the fixed point Let if possible there are two fixed points say y and y*

   Ay = By = Py = Qy =y and A y* = B y*= P y* = Q y* = y*

   Putting x = y and y = y* in (1),we get

   d(Py, Qy*) d(Ay, Py) + d(By*, Q y*) +

   d(Ay, B y*) + d(Ay, Q y*) + d(Py, B y*) d(y, y*) d(y, y) + d(y*, y*) + d(y, y*) +

   d(y, y*) + d(y, y*)

   0 (( + + ) -1)d(y, y*)

   which is a contradiction.Thus y = y*

   i.e. fixed point is unique.

   1. Corollary-1.

      Let (X, d) be a complete cone metric space. Suppose A, P are self mappings on X itself and each of it are surjective.

      (i)A(X) P(X),

      (ii)(A,P) are weakly compatible. (iii)Suppose for ,, , , such that , , , , [0,1) and + + > 1

      d(Px, Py) d(Ax, Px) + d(Ay, Py) + d(Ax, Ay)

      + d(Ax, Py) + d(Px, Ay) for all x.y X. Either 1+ > or 1+ > Then A, P have a unique common fixed point in X.

      Proof: In Theorem -1, if we put P=Q and A=B, we get the proof.

   2. Corollary-2.

   Let (X, d) be a complete cone metric space. Suppose P is self mapping on X itself and it are surjective.

   Suppose for , , , , such that

   , , , , [0,1) and + + > 1

   d(Px, Py) d(x, Px) + d(y, Py) + d(x, y) +

   d(x, Py) + d(Px, y)

   for all x. y X. . Either 1+ > or 1+ > Then A, P have a unique common fixed point in X.

   Proof: Putting A=I, identity mapping we get the proof.

   3..3 Corollary-3.

   Let (X, d) be a complete cone metric space. Suppose A, P are self mappings on X itself and each of it are surjective.

   (i)A(X) P(X)

   (ii)(A, P) are weakly compatible.

   (iii)Suppose for , , such that , , [0,1) and

   + + > 1

   d(Px, Py) d(Ax, Px) + d(Ay, Py) + d(Ax, Ay) for all x. y X .Then A, P have a unique common fixed point in X.

   Proof: In the corollary-1 if we put = = 0 we get the proof.

   1. Corollary-4.

      Let (X, d) be a complete cone metric space. Suppose P is self mapping on X itself and it is surjective. Suppose for , , such that , , [0,1) and + + > 1

      d(Px, Py) (Px, x) + d(Py, y) + d(x, y) for all x.y X . Then P has a unique common fixed point in X.

      Proof: In the corollary-3, if we put A=I, identity mapping, we get the proof.

   2. Corollary-5.

      Let (X, d) be a complete cone metric space. Suppose P is self mapping on X itself and it is surjective.

      For > 1.

      d(Px, Py) d(x, y) for all x.y X . Then P has a unique common fixed point in X.

      Proof: In the corollary-4, if we put = = 0 , we get the proof.

   3. Corollary-6.

   Let (X, d) be a complete cone metric space. Suppose A, P are self mappings on X itself and each of it are surjective.

   (i)A(X) P(X)

   (ii)(A, P) are weakly compatible.

   (iii)Suppose for such that > 1

   d(Px, Py) d(Ax, Ay) for all x. y X .Then A, P have a unique common fixed point in X.

   Proof: In the corollary-1, if we put = = = , we get the proof.

4. Remark

   1. Corrollary-2 is the main result of Yan Han and Shaoyuan Xu [13]

   2. Corrollary-3 is the main theorem of Wasfi Shatanawi and Fadi Awawdeh [11]

   3. Corrollary-4 is the theorem 2.1 from Xianjiu Huang, Chuanxi Zhu and Xi Wen[12]

   4. Corrollary-6 is theorem 2.3 from Xianjiu Huang, Chuanxi Zhu and Xi Wen[12].

5. Acknowledgment

   The authors are thankful to the affiliated college authorities for financial support given by them.

6. References

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8. Rhoades, BE: Some Fixed Point Theorems For Pairs Of Mappings. Jnanabha 15(1985), 151-156

9. Taniguchi, T: Common Fixed Point Theorems On Expansion Type Mappings On Complete Metric Spaces. Math. Jpn. 34(1989),139-142

10. Wang, SZ, Li, BY, Gao, ZM, Iseki, K: Some Fixed Point Theorems On Expansion Mappings. Math. Jpn. 29(1984), 631-636

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    Coincidence Point Theorems ForExpansive Maps In Cone Metric Spaces, Fixed Point Theory and Applications 2012,2012:19

    http://www.fixedpointtheoryandapplications.com/content/01 2/1/19

12. Xianjiu Huang, Chuanxi Zhu and Xi Wen, Fixed Point Theorems For ExpandingMappings In Cone Metric Spaces, math. reports 14(64), 2 (2012), 141148

13. Yan Han and Shaoyuan Xu, Some New Theorems Of Expanding Mappings Without Continuity In Cone Metric Spaces Fixed Point Theory and Applications 2013, 2013:3 http://www.fixedpointtheoryandapplications.com/content/201 3/1/3