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A Common Fixed Point Theorem for Six Mappings in G-Banach Space with Weak-Compatibility Ranjeeta Jain Infinity Management and Engineering college,Pathariya jat Road Sagar (M.P.)470003 ABSTRACT The aim of this paper is to introduce the concept of G-Banach Space and prove a common fixed point theorem for six mappings in G-Banach spaces with weak–compatibility. Keywords: Fixed point , common fixed point , G-Banach Space , Continuous mappings , weak compatible mappings. Introduction : This is well know that the fundamental contraction principle for proving fixed points results is the Banach Contraction principle. There have been a number of generalization of metric space and Banach space. One such generalization is G-Banach space.The concept of G-Banach space is introduce by Shrivastava R,Animesh,Yadav R.N.[4 ] which is a probable modification of the ordinary Banach Space. Recently in 2012,R.K. Bharadwaj [2 ] introduced fixed point theorems in G-Banach Space through weak compatibility and gave the following fixed point theorems for four mappingsTheorems[A]: Let X be a G-Banach Space , such that ∇ Satisfy property with α - α ≤ 1. If A , B , S and T be mapping from X into itself satisfying the following condition: I. A(X) ⊆ T(X) , B(X) ⊆ S(X) and T(X) or S(X) is a closed subset of X. II. The Pair (A,S) and (B,T) are weakly compatible, III. For all x,y ∈ X

 || Sx - Ax ||g || Sx - By ||g || Tx - Ax ||g || Ty - By ||g   ≤ k1  ∇   || Sx Ty || || Sx Ty || g g    || Sx - Ax ||g || Ty - By ||g || Sx - By ||g || Ty - Ax || g   + k 2 max  ∇   || Sx Ty || || Sx Ty || g g   + k 3 (|| Sx - Ax || g ∇ || Ty - By || g ∇ || Sx - By || g ∇ || Ty - Ax || g ∇ || Sx - Ty || g )

|| Ax - By || g

Where k1 , k2 , k3 > 0 and 0 < k1 + k2 + k3 < 1. Then A , B , S and T have a unique common fixed point in X. In 1980, Singh and Singh[5] gave the following theorem on metric space for self maps which is use to our main resultTheorems[B]: Let P, Q and T be self maps of a metric space (X, d) such that (i) PT = TP and QT = TQ, (ii) P(X) ∪ Q(X) ⊆ T(X), (iii) T is continuous, (iv) d(Px , Qy ) ≤ c λ (x, y), where

λ (x, y) = max{d(Tx, Ty), d(Px, Tx), d(Qy, Ty),

1 [d(Px, Ty)+d(Qy, Ty)]} 2

for all x, y ∈ X and 0 ≤ < 1. Further if (v) X is complete then P, Q, T have a unique common fixed point in X. Just we recall the some definition of G-Banach space for the sake of completeness which as followsN be the set of natural numbers and R+ be the set of all positive numbers let binary operation ∇ : R+ × R+ → R+ satisfies the following conditions: i. ∇ is associative and commutative, ii. ∇ is continuous. Five typical example are as follows: i. a ∇ b = max (a,b) ii. a ∇ b = a+b iii. a ∇ b =a.b iv. a ∇ b = a.b+a+b

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v.

a∇ b =

ab max(a, b,1)

Definition 1: The binary operation ∇ is said to satisfy α-property if there exists a positive real number α , such that a ∇ b ≤ α max (a,b) for every a,b Є R+ Example:If we define a ∇ b = a + b for each a,b Є R+ then for α ≥ 2, we have a ∇ b ≤ α max (a,b) if we define a ∇ b =

ab for each a,b Є R+ then for α ≥ 1, we have max(a, b,1)

a ∇ b ≤ α max (a,b) Definition 2: Let x be a nonempty set ,A Generalized Normed Space on X is a function ║ ║g : x × x → R+ that satisfies the following conditions for each x,y,z, Є X 1. ║x-y║g > 0 2. ║x-y║g = 0 if and only if x = y 3. ║x-y║g = ║y-x║g 4. ║ α x║g = │ α │║x║g for any scalar α 5. ║x-y║g ≤ ║x-z║g ∇ ║z-x║g The pair (x, ║ ║g) is called generalized Normed Space or simply G-Normed Space. Definition 3: A Sequence in X is said converges to x if ║xn-x║g→0 , as n → ∞. That is for each є > 0 there exists n0 Є N such that for every n ≥ n0 implies that ║xn-x║g < є Definition 4: A sequence {xn} is said to be Cauchy sequence if for every є > 0 there exists n0 є N such that ║xm-xn║g < є for each m,n ≥ n0. G-Normed Space is said to be G-Banach Space if for every Cauchy sequence is converges in it. Definition 5: Let (x, ║ ║g) be a G-Normed Space for r>0 we define Bg (x,r) = {y є X : ║x-y║g < r } Let X be a G-normed Space and A be a subset of X, then for every x є A, there exists r > 0 such that Bg (x,r) ⊆ A , then the subset A is called open subset of X. A subset A of X is said to be closed if the complement of A is open in X. Definition 6: Let A and S be mappings from a G-Banach space X into itselt. Then the mappings are said to be weakly compatible if they are commute at there coincidence point that is Ax= Sx implies that ASx = SAx Here we generalized and extend the results of R.K.Bhardwaj[2] (theorem A) for six mappings opposed to four mappings in G-Banach space using the concept of weak-compatibility. Main Result : THEOREM(1) : Let X be a G-Banach Space , such that ∇ Satisfy property with α - α ≤ 1. If P , Q , A , B , S and T be mapping from X into itself satisfying the following condition: I. A(X) ⊆ Q(X) ∪ T(X) , B(X) ⊆ P(X) ∪ S(X) and T(X) or S(X) is a closed subset of X. II. The Pair (A,S) and (P,S) , (B,T) and (Q ,T) are weakly compatible, III. For all x,y ∈ X || Ax - By || g ≤

|| Px - Ax ||g || Qy - By ||g || Sx - Ty ||g || Qy - By ||g    || Sx - By ||g || Ty - Ax ||g ∇ ∇ ∇ || Sx - Py ||g || Qy - Ax ||g + || Px - Ax ||g   ≤ α max    || Px - Qx ||g + || Bx - Ty ||g    || Sx - By ||g    || Px - By ||g     || Sx - Ty || g ∇ || Px - Sx || g ∇ || Qy - By || g ∇ || Px - Ty || g ∇ || Qy - Sx || g ∇    || Sx - Ty || g || Qy - Sx || g + β max  || Px - Ax || g || Qy - By || g  ∇   || Px - Qy || g || Px - Qy || g   Where α , β > 0 and 0 < α + β < 1. Then P , Q , A , B , S and T have a unique common fixed point in X. Proof: Let x0 be an arbitrary point in X then by (i) we choose a point x1 in X such that y0 = Ax0 = Tx1 = Qx1 and y1 = Bx1 = Sx2 = Px2 . 207

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In general there exists a sequence {yn} such that y2n = Ax2n = Tx2n+1 = Qx2n+1 and y2n+1 = Bx2n+1 = Sx2n+2 = Px2n+2 , for n = 1 , 2 , 3 , -------------we claim that the sequence {yn} is a Cauchy sequence. By (iii) we have || y 2n - y 2n +1 ||g = || Ax 2n - Bx 2n +1 ||g

|| Px - Ax2n ||g || Qx2n+1 - Bx2n+1 ||g  ||Sx2n - Bx2n+1 ||g || Tx2n+1 - Ax2n ||g ∇ 2n ∇ ||Sx2n - Px2n+1 ||g  ≤ α max   ||Sx2n - Tx2n+1 ||g || Qx2n+1 - Bx2n+1 ||g || Px2n - Qx2n ||g + || Bx2n - Tx2n+1 ||g  || Qx - Ax || + || Px - Ax || ∇ || Px2n - Bx2n+1 ||g 2n+1 2n g 2n 2n g  

      ||Sx2n - Bx2n+1 ||g    ||Sx2n - Tx2n+1 ||g ∇|| Px2n - Sx2n ||g ∇|| Qx2n+1 - Bx2n+1 ||g ∇|| Px2n - Tx2n+1 ||g ∇||Qx2n+1 - Sx2n ||g ∇   + β max|| Px2n - Ax2n ||g ||Qx2n+1 - Bx2n+1 ||g ||Sx2n - Tx2n+1 ||g ||Qx2n+1 - Sx2n ||g  ∇   || Px2n - Qx2n+1 ||g || Px2n - Qx2n+1 ||g   || y - y || || y - y ||   || y 2n-1 - y 2n−1 ||g || y 2n - y 2n ||g ∇ 2n-1 2n g 2n 2n+1 g ∇  || y 2n-1 - y 2n ||g   ≤ α max    || y 2n-1 - y 2n ||g || y 2n - y 2n+1 ||g || y 2n-1 - y 2n-1 ||g + || y 2n - y 2n+1 ||g   ∇  || y 2n-1 - y 2n ||g   || y - y || + || y || y 2n-1 - y 2n ||g   2n-1 - y 2n ||g  2n 2n g 

|| y 2n-1 - y 2n ||g ∇ || y 2n-1 - y 2n-1 ||g ∇ || y2n - y 2n+1 ||g ∇ || y2n-1 - y 2n ||g ∇ || y 2n - y2n-1 ||g ∇   + β max || y 2n-1 - y 2n ||g || y 2n - y2n+1 ||g || y 2n-1 - y2n ||g || y2n - y 2n-1 ||g  ∇   || y 2n-1 - y 2n ||g || y2n-1 - y2n ||g   || y 2n - y 2n +1 ||g ≤ ( α + β ) || y 2n -1 - y 2n || g That is by induction we can show that || y 2n - y 2n +1 ||g ≤ ( α + β )n || y 0 - y1 || g || y 2n - y 2n +1 ||g As n → ∞ || y 2n - y 2n +1 ||g → 0 , for any integer m ≥ n It follows that the sequence {yn} is a Cauchy sequence which converges to y ∈ X . This implies that lim n → ∞ yn = lim n → ∞ Ax2n = lim n → ∞ Tx2n+1 = lim n → ∞ Qx2n+1 = lim n → ∞ Bx2n+1 = lim n → ∞ Sx2n+2 = lim n → ∞ Px2n+2 = y Now let us assume that T(X) is closed subset of X, then there exists v ∈ X such that Tv = Qv = y We now prove that Bv = y , By (iii) we get

|| Ax 2n - Bv ||g

|| Px2n - Ax2n ||g || Qv - Bv ||g  || Sx2n - Bv ||g || Tv - Ax2n ||g ∇ ∇ || Sx2n - Pv ||g  ≤ α max  || Px2n - Qx2n ||g + || Bx2n - Tv ||g  || Sx2n - Tv ||g || Qv - Bv ||g ∇  || Qv - Ax || + || Px - Ax ||  || Px2n - Bv ||g  2n g 2n 2n g 

|| y - Bv || g ≤

       || Sx2n - Bv ||g    || Sx 2n - Tv ||g ∇ || Px 2n - Sx 2n ||g ∇ || Qv - Bv ||g ∇ || Px 2n - Tv ||g ∇ || Qv - Sx 2n ||g ∇    + β max || Px 2n - Ax2n ||g || Qv - Bv ||g || Sx 2n - Tv ||g || Qv - Sx 2n ||g  ∇   || Px 2n - Qv ||g || Px 2n - Qv ||g  

+ β ) || y - Bv || g

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Which is contradiction , it follows that Bv = y = Tv = Qv . Since (B,T) and (Q,T) are weakly compatible mappings , then we have BTv = TBv and QTv = TQv By = Ty and Qy = Ty Which implies By = Qy. Now we prove that By = y for this by using (iii) we get

|| Ax 2n - By || g || Px2n - Ax2n ||g || Qy- By||g  || Sx2n - By||g || Ty - Ax2n ||g ∇ ∇ || Sx2n - Py ||g  ≤ α max  || Px2n - Qx2n ||g + || Bx2n - Ty ||g  || Sx2n - Ty ||g || Qy- By||g ∇  || Qy- Ax || + || Px - Ax ||  || Px2n - By||g  2n g 2n 2n g 

|| Ax 2n

       || Sx2n - By||g    || Sx2n - Ty ||g ∇ || Px2n - Sx2n ||g ∇ || Qy - By ||g ∇ || Px2n - Ty ||g ∇ || Qy - Sx2n ||g ∇   + β max || Px2n - Ax2n ||g || Qy - By ||g || Sx2n - Ty ||g || Qy - Sx2n ||g  ∇   || Px2n - Qy ||g || Px2n - Qy ||g   - By || g ≤ ( α + β ) || y - By || g

Which is contradiction. Thus By = y = Ty = Qy --------------------(A) Since B(X) ⊆ P(X) ∪ S(X) , there exists w ∈ X. such that sw = y = Pw . we show that Aw = y From (iii) we have

|| Aw - By || g || Pw - Aw || g || Qy - By || g   || Sw - By || g || Ty - Aw || g ∇ ∇ || Sw - Py || g  ≤ α max  || Pw - Qw || g + || Bw - Ty || g  || Sw - Ty || g || Qy - By || g  || Qy - Aw || + || Pw - Aw || ∇  || Pw - By || g  g g 

       || Sw - By || g     || Sw - Ty || g ∇ || Pw - Sw || g ∇ || Qy - By || g ∇ || Pw - Ty || g ∇ || Qy - Sw || g ∇    + β max || Pw - Aw || g || Qy - By || g || Sw - Ty || g || Qy - Sw || g  ∇   || Pw - Qy || g || Pw - Qy || g   || Aw - y || g ≤ ( α + β ) || Aw - y ||g Which is contradiction , so that Aw = y = Sw = Pw . Since (A , S) and (P , S) are weakly compatible , then ASw = Saw and PSw = SPw Ay = Sy Py = Sy Therefore Ay = Py Now we Show that Ay = y, From (iii) we have

|| Py - Ay || g || Qy - By || g   || Sy - By || g || Ty - Ay || g ∇ ∇ || Sy - Py || g  || Ay - By || g ≤ α max  || Py - Qy || g + || By - Ty || g  || Sy - Ty || g || Qy - By || g  || Qy - Ay || + || Py - Ay || ∇  || Py - By || g  g g 

209

       || Sy - By || g   

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 || Sy - Ty || g ∇ || Py - Sy || g ∇ || Qy - By || g ∇ || Py - Ty || g ∇ || Qy - Sy || g ∇    + β max || Py - Ay || g || Qy - By || g || Sy - Ty || g || Qy - Sy || g  ∇   || Py - Qy || g || Py - Qy || g   || Ay - y || g ≤ ( α + β ) || Ay - y || g Which is contradiction thus Ay = y and therefore Ay = Sy = Py = y ---------------------------------------------(B) Now from equation (A) and (B) we get Ay = By = Py = Qy = Sy = Ty = y . Hence y is a common fixed point of A ,B , S , T , P and Q . Uniqueness: Let us assume that x is another fixed point of A ,B , S , T , P and Q different from y in X. Then from (iii) we have || Ax - By || g ≤

|| Px - Ax ||g || Qy- By ||g || Sx - Ty ||g || Qy - By ||g   || Sx - By ||g || Ty - Ax ||g ∇ ∇ ∇ || Sx - Py ||g || Qy - Ax ||g + || Px - Ax ||g   ≤ α max    || Px - Qx ||g + || Bx - Ty ||g    || Sx - By ||g   || Px - By ||g      || Sx - Ty || g ∇ || Px - Sx || g ∇ || Qy - By || g ∇ || Px - Ty || g ∇ || Qy - Sx || g ∇    || Sx - Ty || g || Qy - Sx || g + β max  || Px - Ax || g || Qy - By || g  ∇   || Px - Qy || g || Px - Qy || g  

|| x - y || g ≤ ( α + β ) || x - y || g Which is contradiction . Thus x = y . This completes the proof of the theorem. COROLLARY: Let X be a G-Banach Space such that satisfying the following condition:

∇ Satisfy α- property with α ≤ 1. If T be a mapping from X into itself,

|| T r x - T s y ||g

  || x - T r x ||g || y - T s y || g || x - y || g || y - T s y ||g  || x - T s y || g || y - T r x ||g ∇  ∇ ∇  || x - y || g || y - T r x ||g + || x - T r x ||g   ≤ α max   || x - T r x || g + || y - T s y || g   r  || x - T x || g    s || x - T y || g     r s s r  || x - y || g ∇ || x - T x || g ∇ || y - T y || g ∇ || x - T y || g ∇ || y - T x || g ∇    r s r r   + β max || x - T x || g || y - T y || g || x - T x || g || y - T x || g   ∇   || x y || || x y || g g   For non-negative α , β suvh that 0 < has a unique common fixed point in X.

α +β

210

< 1 and r , s ∈ N(set of natural number). Then T

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REFERENCES : 1. Ahmad A. and Shakil M. “Some fixed point theorems in Banach spaces ”, Nonlinear func. Anal. Appl.11(2006) 343-349. 2. Bhardwaj R.K. “ Some contraction on G-Banach Space”,Global journal of Science frontier research Mathematics and Decision Sciences vol.12 issue 1 version 1.o, jaanuary (2o12).pp.81-89 3. Ciric Lj.B and J.S Ume “some fixed point theorems for weakly compatible mapping s”J.Math.Anal.Appl.314(2) (2006) 488-499. 4. Shrivastava R ,Animesh,yadav R.N. “some Mapping on G-Banach Space” international journal of Mathematical Science and Engineering Application, Vol.5, No. VI , (2011),pp.245-260. 5. Singh,S.L. and Singh,s.p. “A fixed point theorm” Indian journals pure and Applied. Math.,vol no.11(1980), 1584-1586. 6. Saini R.K. and Chauhan S.S “Fixed point theorem for eight mappings on metric space”journal of mathematics and system science,vol.1 no.2 (2005). 7. V.Popa,” A general fixed point theorems for weakly compatible mapping satisfying an Implict relation”,Filomat,19(2005) 45-51.

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