Mathematical Theory and Modelin Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Enineerin Applications www.iiste.or A Common Fixed Point Theorem for Six Mappins in G-Banach Space with Weak-Compatibility Ranjeeta Jain Infinity Manaement and Enineerin collee,pathariya jat Road Saar (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 mappins in G-Banach spaces with weak compatibility. Keywords: Fixed point, common fixed point, G-Banach Space, Continuous mappins, weak compatible mappins. Introduction : This is well know that the fundamental contraction principle for provin fixed points results is the Banach Contraction principle. There have been a number of eneralization of metric space and Banach space. One such eneralization 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 throuh weak compatibility and ave the followin fixed point theorems for four mappins- Theorems[A]: Let X be a G-Banach Space, such that Satisfy property with α - α 1. If A, B, S and T be mappin from X into itself satisfyin the followin 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 Ax -By Sx -Ax Sx -By Tx -Ax Ty -By k 1 Sx -Ty Sx -Ty Sx -Ax Ty -By Sx -By Ty -Ax + k 2 max Sx -Ty Sx -Ty +k 3 Sx -Ax Ty -By Sx -By Ty -Ax Sx -Ty ( ) Where k 1, k 2, k 3 > 0 and 0 < k 1 + k 2 + k 3 < 1. Then A, B, S and T have a unique common fixed point in X. In 1980, Sinh and Sinh[5] ave the followin theorem on metric space for self maps which is use to our main result- Theorems[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 ) cλ (x, y), where λ (x, y) = max{d(tx, Ty), d(px, Tx), d(qy, Ty), 2 1 [d(px, Ty)+d(Qy, Ty)]} 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 follows- N be the set of natural numbers and R + be the set of all positive numbers let binary operation : R + R + R + satisfies the followin 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 206
Mathematical Theory and Modelin Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Enineerin Applications v. a b = ab max( a, b,1) www.iiste.or 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) ab if we define a b = 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 : x x R + that satisfies the followin conditions for each x,y,z, Є X 1. x-y > 0 2. x-y = 0 if and only if x = y 3. x-y = y-x 4. α x = α x for any scalar α 5. x-y x-z z-x The pair (x, ) is called eneralized Normed Space or simply G-Normed Space. Definition 3: A Sequence in X is said converes to x if x n -x 0, as n. That is for each є > 0 there exists n 0 Є N such that for every n n 0 implies that x n -x < є Definition 4: A sequence {x n } is said to be Cauchy sequence if for every є > 0 there exists n 0 є N such that x m -x n < є for each m,n n 0. G-Normed Space is said to be G-Banach Space if for every Cauchy sequence is converes in it. Definition 5: Let (x, ) be a G-Normed Space for r>0 we define B (x,r) = {y є X : x-y < 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 B (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 mappins from a G-Banach space X into itselt. Then the mappins are said to be weakly compatible if they are commute at there coincidence point that is Ax= Sx implies that ASx = SAx Here we eneralized and extend the results of R.K.Bhardwaj[2] (theorem A) for six mappins opposed to four mappins in G-Banach space usin 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 mappin from X into itself satisfyin the followin 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 Px -Ax Qy -By Sx -Ty Qy -By Sx -By Ty -Ax Sx -Py Qy -Ax + Px -Ax αmax Px -Qx + Bx -Ty Sx -By Px -By Sx -Ty Px -Sx Qy -By Px-Ty Qy -Sx + β max Px -Ax Qy -By Sx -Ty Qy -Sx Px -Qy Px -Qy 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 x 0 be an arbitrary point in X then by (i) we choose a point x 1 in X such that y 0 = Ax 0 = Tx 1 = Qx 1 and y 1 = Bx 1 = Sx 2 = Px 2. 207
Mathematical Theory and Modelin Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Enineerin Applications In eneral there exists a sequence {y n } such that y = Ax = Tx +1 = Qx +1 and y +1 = Bx +1 = Sx +2 = Px +2, for n = 1, 2, 3, -------------- we claim that the sequence {y n } is a Cauchy sequence. By (iii) we have - + 1 = Ax -Bx+ 1 Qx+ 1-Bx+ 1 Sx -Bx+ 1 Tx+ 1 Sx- + 1 αmax Sx + -Tx+ 1 Qx+ 1-Bx+ 1 -Qx Bx-Tx + 1 Sx-Bx+ 1 Qx + 1 + -Bx + 1 Sx-Tx+ 1 -Sx Qx+ 1-Bx+ 1 -Tx+ 1 Qx+ 1-Sx + βmax Qx+ 1-Bx+ 1 Sx-Tx + 1 Qx+ 1-Sx -Qx + 1 -Qx + 1-1 + 1 y -1 1-1 αmax y + -1 + 1-1 -1 + 1-1 + -1-1 -1-1 -1 + 1-1 -1 + βmax -1 + 1-1 -1-1 -1 + 1 (α + β ) -1 That is by induction we can show that + 1 (α + β ) n y0 -y1 + 1 As n y -y + 0, for any inteer m n 1 www.iiste.or It follows that the sequence {y n } is a Cauchy sequence which converes to y X. This implies that lim n y n = lim n Ax = lim n Tx +1 = lim n Qx +1 = lim n Bx +1 = lim n Sx +2 = lim n Px +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 et Ax -Bv Qv- Bv Sx -Bv Tv- Ax Sx -Pv αmax Sx -Tv Qv- Bv -Qx + Bx -Tv Qv + -Bv Sx -Tv -Sx Qv- Bv -Tv Qv-Sx + β max Qv- Bv Sx -Tv Qv-Sx -Qv -Qv + ) y -Bv y -Bv (α β Sx -Bv 208
Mathematical Theory and Modelin Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Enineerin Applications Which is contradiction, it follows that Bv = y = Tv = Qv. Since (B,T) and (Q,T) are weakly compatible mappins, 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 usin (iii) we et Ax -By www.iiste.or Qy -By Sx -By Ty Sx-Py α max Sx + -Ty Qy- By -Qx Bx-Ty Sx-By Qy- Ax + -By Sx -Ty -Sx Qy -By -Ty Qy-Sx + β max Qy -By Sx -Ty Qy -Sx -Qy -Qy Ax -By (α β y -By + ) 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 Pw-Aw Qy -By Sw-By Ty -Aw Sw-Py α max Sw-Ty + Qy-By Pw-Qw Bw-Ty Sw-By Qy-Aw + Pw-Aw Pw-By Sw-Ty Pw-Sw Qy -By Pw-Ty Qy-Sw + β max Pw-Aw Qy -By Sw-Ty Qy -Sw Pw-Qy Pw-Qy Aw-y (α + β ) Aw-y 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 Ay -By Py -Ay Qy -By Sy -By Ty -Ay Sy -Py α max Sy Qy-By Py -Qy + By Qy-Ay + Py -Ay Py -By Sy -By 209
Mathematical Theory and Modelin www.iiste.or Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Enineerin Applications Sy Py -Sy Qy -By Py-Ty + β max Py-Ay Qy -By Sy Qy -Sy Py -Qy Py-Qy Ay -y (α + β ) Ay -y Qy-Sy Which is contradiction thus Ay = y and therefore Ay = Sy = Py = y ---------------------------------------------(B) Now from equation (A) and (B) we et 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 Px -Ax Qy -By Sx -Ty Qy -By Sx -By Ty -Ax Sx -Py Qy -Ax + Px -Ax αmax Px -Qx + Bx -Ty Sx -By Px -By Sx -Ty Px -Sx Qy -By Px-Ty Qy -Sx + β max Px -Ax Qy -By Sx -Ty Qy -Sx Px -Qy Px -Qy x -y (α β + ) x -y Which is contradiction. Thus x = y. This completes the proof of the theorem. COROLLARY: Let X be a G-Banach Space such that Satisfy α- property with α 1. If T be a mappin from X into itself, satisfyin the followin condition: r s T x -T y r s s x -T x y x -y y s r x -T y x r r x -y x + x -T x αmax r s x -T x + y r s x -T x x -T y r s s r x -y x -T x y x-t y x + r s r r β max x -T x y x -T x x x -y x -y For non-neative α, β suvh that 0 < α +β < 1 and r, s N(set of natural number). Then T has a unique common fixed point in X. 210
Mathematical Theory and Modelin www.iiste.or Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Enineerin Applications 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 mappin s J.Math.Anal.Appl.314(2) (2006) 488-499. 4. Shrivastava R,Animesh,yadav R.N. some Mappin on G-Banach Space international journal of Mathematical Science and Enineerin Application, Vol.5, No. VI, (2011),pp.245-260. 5. Sinh,S.L. and Sinh,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 eiht mappins on metric space journal of mathematics and system science,vol.1 no.2 (2005). 7. V.Popa, A eneral fixed point theorems for weakly compatible mappin satisfyin an Implict relation,filomat,19(2005) 45-51. 211
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