A Common Fixed Point Theorem for Six Mappings in G-Banach Space with Weak-Compatibility

Transcription

1 Mathematical Theory and Modelin Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Enineerin Applications 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.) 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

2 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) 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

3 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 = 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 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 y αmax y βmax (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 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

4 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 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

5 Mathematical Theory and Modelin 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

6 Mathematical Theory and Modelin 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) 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 Ciric Lj.B and J.S Ume some fixed point theorems for weakly compatible mappin s J.Math.Anal.Appl.314(2) (2006) 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 Sinh,S.L. and Sinh,s.p. A fixed point theorm Indian journals pure and Applied. Math.,vol no.11(1980), 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)

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