Common Properties, Implicit Function, Rational Contractive Map and Related Common Fixed Point Theorems in Multiplicative Metric Space

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13 Number (017) pp Research India Publications Common Properties Implicit Function Rational Contractive Map and Related Common Fixed Point Theorems in Multiplicative Metric Space 1 Nisha Sharma Arti Mishra 1 Department of mathematics Manav Rachna International University Faridabad Haryana India. Abstract The purpose of this paper is to prove common fixed point theorems using rational contraction for weekly compatible mappings along with property (E.A) common limit range properties satisfying implicit functions in the setup of multiplicative metric space. Keyword: Rational contractive maps property (E.A) common limit range properties contractive map common fixed point multiplicative metric space. AMS Subject Classification: 46S40 47H10 54H5 1. INTRODUCTION AND PRELIMINARIES Common Fixed point theorems satisfying certain contraction conditions has many applications and has been at the Centre of various research activities also it is the key concepts to solve many problems of calculus. It is well known that set of all positive real numbers is not complete with respect to usual metric but it is observed that set of all real numbers is a complete multiplicative metric space with respect to the multiplicative absolute value function. This problem was solved in 008 by Bashirov [1] by introducing the new metric space named Multiplicative Metric Space. The theory of convergence in multiplicative metric space and related fixed point theorems in multiplicative metric space was introduced by Özavsar and Cevikel initiated [6]. Let us start with topological definitions of multiplicative metric space. The notations R and R + are used to represent the set of real numbers and the set of all positive real numbers respectively.

2 88 Nisha Sharma and Arti Mishra Definition 1.1. [6] Let X be a nonempty set. A multiplicative metric is a mapping d: X x X R + satisfying the following conditions: (1.1) d(xy) 1 xy Xand d(xy) = 1 if and only if x = y; (1.) d(xy) = d(yx) x y X; (1.3) d(xy) d(xz)+d(zy) x y X (multiplicative triangle inequality). Definition 1.. [6] Let (X d) be a multiplicative metric space. Then a sequence x n } in X is said to be (1.4) a multiplicative convergent to x if for every multiplicative open ball B ε (x)= y d( x y ) ε ε > 1 there exists a natural number N such that n N then x n B ε (x) i.e. d(x n x) 1 as n. (1.5) a multiplicative Cauchy sequence if ε > 1 for all m n >N that is d(x n x m ) 1 as n. (1.6) we call a multiplicative metric space complete if every multiplicative Cauchy sequence in it is multiplicative convergent to x X. The concept of multiplicative contraction was given by Özavsar and Cevikel [6] in 01 some fixed point theorem were also proved. Definition 1.3. [6] Let f be a mapping of a multiplicative metric space (X d) into itself. Then f is said to be a multiplicative contraction if there exist a real constant [01) such that d(fxfy) d (x y)for all xy X. Definition 1.4. [ ] Let f and gbe two mappings of a multiplicative metric space (X d) into itself then f and g are said to be (1.7) commutative mapping if fgx=gfx x X. (1.8) Weak commutative mapping if d(fgx gfx) d(fx gx) x X. (1.9) Weakly compatible if f and g commute at coincidence points that is ft=gt for somet X. Implies that fgt = gft. (1.10) E.A property if there exist a sequence x n } in X such that lim fx n = lim gx n = t for some t X. (1.11) CLRg property (common limit range of g property)if there exist a sequence x n } in X such that lim fx n = lim gx n = gt for some t X. (1.1) CLRf property (common limit range of f property)if there exist a sequence x n } in X such that lim fx n = lim gx n = ft for some t X.

3 Common Properties Implicit Function Rational Contractive Map and Related 89. MAIN RESULTS Implicit functions was used by Popa [9] it was a good contractive condition in multiplicative metric space. Many authors have been using the concept of multiplicative metric space e.g. [see ]. To prove the main result we define a suitable class of the implicit function involving three real non-negative arguments as follows: Let Ψ is family of functions such that ф: R + 3 R + is continuous and increasing in each coordinate variable and i. ф (t.t1 1 t) t.t1 ii. ф (1 t.t1 t1) t.t1 iii. ф (t 1 1) t iv. ф (1 t 1) t v. ф (t t 1) t for every t t1 R + (t t1 1). It is obvious that ф (1 1 1)= 1. There exist many functionsф Ψ. Now we prove the following theorems for weakly compatible mappings satisfying implicit function in a multiplicative metric space. Theorem.1. Let A B γ θ be self mappings of a multiplicative metric space (X d) such that (E1) γ X BX and θ X AX (E) d(γx θ y) d(axby)[d(θ yγx)+d(axθ y)] d(byax)+d(γxby) d(θ yγx)[d(byax)+d(γxby)] d(axγx)[d(byθ y)+d(byax)] d(θ yγx)+d(axθ y) d(axγx)+d(θ yγx) for all x y X where (0 1 ) and ф Ψ; (E3) let us suppose that the pairs (A γ) and (B θ) are weakly compatible; (E4) One of the subspaces AX or BX or γ X or θ X is complete Then A B and have a unique common fixed point. Proof. Let x 0 be any arbitrary point of metric space X. since γ X BX hence x 1 X such that γx 0 = Bx 1 = y 0 for this x 1 x X such that Ax = θ x 1 = y 1. Similarly an inductive sequence y n } can be defined in such a way that γx n = Bx n+1 = y n Ax n+ = θ x n+1 = y n+1. Next we prove that y n } is a multiplicative sequence in X. in fact n N we have }}

4 90 Nisha Sharma and Arti Mishra From (E) we haves d( y n y n+1 ) d(γx n θ x n+1 ) d(ax n Bx n+1 )[d(θ x n+1 γx n ) + d(ax n θ x n+1 )] d(bx n+1 Ax n ) + d(γx n Bx n+1 ) d(θ x n+1 γx n )[d(bx n+1 Ax n ) + d(γx n Bx n+1 )] d(θ x n+1 γx n ) + d(ax n θ x n+1 ) d(ax n γx n )[d(bx n+1 θ x n+1 ) + d(bx n+1 Ax n )] d(ax n γx n ) + d(θ x n+1 γx n ) } } d( y n 1 y n )[d( y n+1 y n ) + d( y n 1 y n+1 )] d( y n y n 1 ) + d( y n y n ) d( y n+1 y n )[d( y n y n 1 ) + d( y n y n )] d( y n+1 y n ) + d( y n 1 y n+1 ) d( y n 1 y n )[d( y n y n+1 ) + d( y n y n 1 )] d( y n 1 y n ) + d( y n+1 y n ) } } d( y n 1 y n )[d( y n+1 y n ) + d( y n 1 y n ). d( y n y n+1 )] d( y n y n 1 ) + 1 d( y n+1 y n )[d( y n y n 1 ) + 1] d( y n+1 y n ) + d( y n 1 y n ). d( y n y n+1 ) d( y n 1 y n )[d( y n y n+1 ) + d( y n y n 1 )] d( y n 1 y n ) + d( y n+1 y n ) } } d( y n 1 y n ). d( y n y n+1 ) 1 d( y n 1 y n )}} d( y n y n+1 ) d ( y n 1 y n ). d ( y n y n+1 ) This implies that d( y n y n+1 ) d 1 ( y n 1 y n ) On substituting h = (0 1 ) 1 d( y n y n+1 ) d h ( y n 1 y n ) In a similar way we have d( y n+1 y n+ ) = d(tx n+1 γx n+ ) = d(γx n+ Tx n+1 )

5 Common Properties Implicit Function Rational Contractive Map and Related 91 d(ax n+ Bx n+1 )[d(tx n+1 γx n+ ) + d(ax n+ Tx n+1 )] d(bx n+1 Ax n+ ) + d(γx n+ Bx n+1 ) d(tx n+1 γx n+ )[d(bx n+1 Ax n+ ) + d(γx n+ Bx n+1 )] d(tx n+1 γx n+ ) + d(ax n+ Tx n+1 ) d(ax n+ γx n+ )[d(bx n+1 Tx n+1 ) + d(bx n+1 Ax n+ )] d(ax n+ γx n+ ) + d(tx n+1 γx n+ ) } } d( y n+1 y n )[d( y n+1 y n+ ) + d( y n+1 y n+1 )] d( y n y n+1 ) + d( y n+ y n ) d( y n+1 y n+ )[d( y n y n+1 ) + d( y n+ y n )] d( y n+1 y n+ ) + d( y n+1 y n+1 ) d( y n+1 y n+ )[d( y n y n+1 ) + d( y n y n+1 )] d( y n+1 y n+ ) + d( y n+1 y n+ ) } } 1 d( y n+1 y n+ ). d( y n+1 y n ) d( y n y n+1 )}} d( y n+1 y n+ ) d ( y n y n+1 ). d ( y n+1 y n+ ) This implies that d( y n+1 y n+ ) d 1 ( y n y n+1 ) On substituting h = (0 1 ) 1 d( y n+1 y n+ ) d h ( y n y n+1 ) Hence d( y n y n+1 ) d h1 ( y n 1 y n ) d h ( y n y n 1 ) d hn ( y 0 y 1 ) For all n let mn N such that m n. Using the triangular multiplicative inequality we obtain d( y m y n ) d( y m y m 1 ). d( y m 1 y m ) d( y n+1 y n ) d hm 1 ( y 1 y 0 ). d hm ( y 1 y 0 ) d hn ( y 1 y 0 ) d1 h( hn y 1 y 0 ) This implies that d( y m y n ) 1 as n and m therefore y n } is a multiplicative Cauchy sequence in X. Now suppose that AX is complete there exist u AX such that y n+1 = θ x n+1 = A x n+ u (n ).

6 9 Nisha Sharma and Arti Mishra Consequently we can find v X such that Av=u. Further a multiplicative Cauchy sequence y n } has a convergent subsequence y n+1 } therefore the sequence y n } converges and hence a subsequence y n } also converges. Thus we have y n = γ x n = B x n+1 u(n ). We claim that γ v=u. if possible γ v u substituting x=v and y=x n+1 have d(γv θ x n+1 ) in (E) we d(av Bx n+1 )[d(θ x n+1 γv) + d(av θ x n+1 )] d(bx n+1 Av) + d(γv Bx n+1 ) d(θ x n+1 γv)[d(bx n+1 Av) + d(γv Bx n+1 )] d(θ x n+1 γv) + d(av θ x n+1 ) d(av γv)[d(bx n+1 θ x n+1 ) + d(bx n+1 Av)] d(av γv) + d(θ x n+1 γv) } } Taking n On the two sides of the above inequality d(γv u) d(u u)[d(u γv) + d(u u)] d(u γv)[d(u u) + d(γv u)] d(u u) + d(γv u) d(u γv) + d(u u) d(u γv)[d(u u) + d(u u)] d(u γv) + d(u γu) }} 1 d(u γv) 1}} d(u γv) d (u γv) a contradiction since (0 1 ) This implies that d(γ vu) = 1 implies γ v = u. Since u = γ v γx BX there exist w X such that u=bw. Claim that θ w=u if possible Tw u. substituting x=v and y=w in (E) we have d(u θ w ) = d(γ v θ w) d(av Bw)[d(θ w γv) + d(av θ w)] d(bw Av) + d(γv Bw) d(θ w γv)[d(bw Av) + d(γv Bw)] d(av γv)[d(bw θ w) + d(bw Av)] d(θ w γv) + d(av θ w) d(av γv) + d(θ w γv) }} As Av = u = γ v = Bw we have

7 Common Properties Implicit Function Rational Contractive Map and Related 93 d(u u)[d(θ w u) + d(u θ w)] d(u u) + d(u u) d(θ w u)[d(u u) + d(u u)] d(u u)[d(u θ w) + d(u u)] d(θ w u) + d(u θ w) d(u u) + d(θ w u) }} d(θ w u) 11}} d(u θ w) d (u Tw) A contradiction since (0 1 ) implies u = θ w. Hence we get u=av= γ v that is v is a coincidence point of A γ also u=bw= θ w that is w is coincidence point of B and θ. Therefore Av= γ v=bw= θ w=u. Since the pairs A γ and B are weakly compatible we have γ u= γ (Av)=A(γ v)=au=w1 (say) And θ u= θ (Bw)=B(θ w)=bu=w (say) From (E) we have d(w1 w) = d(γ uvu) d(au Bu)[d(θ u γu) + d(au θ u)] d(bu Au) + d(γu Bu) d(θ u γu)[d(bu Au) + d(γu Bu)] d(au γu)[d(bu θ u) + d(bu Au)] d(θ u γu) + d(au θ u) d(au γu) + d(θ u γu) }} Using symmetry and above conditions of w1 and w we have d(w 1 w )[d(w w 1 ) + d(w 1 w )] d(w w 1 ) + d(w 1 w ) d(w w 1 )[d(w w 1 ) + d(w 1 w )] d(w w 1 ) + d(w 1 w ) d(w 1 w ) d(w 1 w ) 1}} d(w 1 w ) d (w 1 w ) d(w 1 w 1 )[d(w w ) + d(w w 1 )] d(w 1 w 1 ) + d(w w 1 ) }}

8 94 Nisha Sharma and Arti Mishra on the other hand since (0 1 ) implies d(w 1 w ) = 1 which implies that w 1 = w and hence we have γ u=au= θ u=bu. Again using (E) and symmetry of multiplicative metric space we have d(γ v θ u) d(avbu)[d(θ uγv)+d(avθ u)] d(buav)+d(γvbu) d(θ uγv)[d(buav)+d(γvbu)] d(avγv)[d(buθ u)+d(buav)] d(θ uγv)+d(avθ u) d(avγv)+d(θ uγv) }} ф d(γv θ u)[d(γv γv) + d(θ u γv)] d(θ u θ u) + d(θ u γv) d(tu Sv)[d(Tu Sv) + d(sv Tu)] d(tu Sv) + d(sv Tu) d(γv θ u) d(γv θ u) 1}} d(γv θ u)[d(θ u γv) + d(γv θ u)] d(θ u γv) + d(γv θ u) d(γv γv)[d(θ u θ u) + d(θ u γv)] d(γv γv) + d(θ u γv) }} d(γv θ u) d (γv θ u) on the other hand since (0 1 ) implies d(γv θ u) = 1 i. e. γv = θ u(u= θ u) and hence we have u= γ u=au= θ u=bu. Therefore u is a common fixed point of A B γ and Similarly we can complete the proof for the different case in which BX or θ X or γ X is comsplete. UNIQUENESS Let p and q are two different common fixed points of A. B γ θ then using symmetry of multiplicative metric space and using equation (E) d(p q) = d(γp θ q) d(ap Bq)[d(θ q γp) + d(ap θ q)] d(bq Ap) + d(γp Bq) d(θ q γp)[d(bq Ap) + d(γp Bq)] d(ap γp)[d(bq θ q) + d(bq Ap)] d(θ q γp) + d(ap θ q) d(ap γp) + d(θ q γp) }} d(p q)[d(q p) + d(p q)] d(q p) + d(p q) d(q p)[d(q p) + d(p q)] d(p p)[d(q q) + d(q p)] d(q p) + d(p q) d(p p) + d(q p) }} d(p q) d(p q) 1 }} d(p q) d (p q)

9 Common Properties Implicit Function Rational Contractive Map and Related 95 on the other hand since (0 1 ) implies d(p q) = 1 i. e. p = q which proves the uniqueness. Corollary.. Let A B γ be mappings of a multiplicative metric space (X d) into itself satisfying (E5) γ X BX and γ X AX (E6) d(γx γy) d(axby)[d(γyγx)+d(axγy)] d(byax)+d(γxby) d(γyγx)[d(byax)+d(γxby)] d(axγx)[d(byγy)+d(byax)] d(γyγx)+d(axγy) d(axγx)+d(γyγx) For all x y X where (0 1 ) and ф Ψ; (E3) let us suppose that the pairs (A γ) and (B γ) are weakly compatible; (E4) One of the subspaces AX or BX or γ X is complete Then A B and γ have a unique common fixed point. In theorem.1. if we put θ = γ then we obtain the corollary. }} Corollary.3.Let γ and itself satisfying be mappings of a multiplicative metric space (X d) into (E7) d(γx θ y) d(xy)[d(θ yγx)+d(xθ y)] d(yx)+d(γxy) d(θ yγx)[d(yx)+d(γxy)] d(xγx)[d(yθ y)+d(yx)] d(θ yγx)+d(xθ y) d(xγx)+d(θ yγx) }} For all xy X where (0 1 ) and ф Ψ (E8) One of the subspaces γ X or θ X is complete Then γ and have a unique common fixed point. In theorem.1. if we put A=B=1 then we obtain the corollary.3. Theorem.4. Let A B γ θ be mappings of a multiplicative metric space (X d) into itself satisfying the conditions (E1) (E) (E3) and the following conditions: (E9) one of the subspaces AX or BX or γ X or θ X is closed subset of X (E10) the pairs (A γ) and (B θ) satisfy the E.A. property. Then A B γ θ have a unique common fixed point. Proof. Suppose that the pairs A γ satisfies the E.A property. Then a sequence x n } In X such that lim A x n = lim γ x n = z for some z X. Since SX BX a sequence y n } in X such that γ x n =B y n

10 96 Nisha Sharma and Arti Mishra Hence lim B y n = z Now suppose that BX is closed subset of X a point u X such that Bu=z. We will show that lim θ y n = z from inequality (E) we have d(γx n θ y n ) ф d(ax n By n )[d( θ y n γx n )+d(ax n θ y n )] d(by n Ax n )+d(γx n By n ) d( θ y n γx n )[d(by n Ax n )+d(γx n By n )] d( θ y n γx n )+d(ax n θ y n ) d(ax n γx n )[d(by n θ y n )+d(by n Ax n )] d(ax n γx n )+d( θ y n γx n ) } } Taking n approaches to infinity and using the symmetry property of multiplicative metric space we have d (z lim θ y n ) d ( lim θ y n z) 11}} d (z lim θ y n ) d (z lim θ y n ) [using (iii)] since (0 1 ) implies d (z lim θ y n ) = 1 i. e. z = lim θ y n. Thus we have lim A x n = lim B x n = lim γ x n = lim θ x n (say) for some u in X. substituting x= x n and y=u in (E) we have Taking n we have d(bu θ u) d(γx n θ u) ф = z = Bu d(ax n Bu)[d(θ uγx n )+d(ax n θ u)] d(buax n )+d(γx n Bu) d(θ uγx n )[d(buax n )+d(γx n Bu)] d(θ uγx n )+d(ax n θ u) d(ax n γx n )[d(buθ u)+d(buax n )] d(ax n γx n )+d(θ uγx n ) } } d(bu Bu)[d(θ u Bu) + d(bu θ u)] d(bu Bu) + d(bu Bu) d(θ u Bu)[d(Bu Bu) + d(bu Bu)] d(bu Bu)[d(Bu θ u) + d(bu Bu)] d(θ u Bu) + d(bu θ u) d(bu Bu) + d(θ u Bu) }} d(bu θ u) d(bu θ u) 11}}

11 Common Properties Implicit Function Rational Contractive Map and Related 97 d(bu θ u) d (Bu θ u) On the other hand since (0 1 ) it implies that d(bu θ u) = 1 i. e. Bu = θ u. It is given that B and θ are weakly compatible we have B θ u = θ Bu and then BBu = θ θ u = B θ u = θ But also θ X AX v X such that θ u=av. Next we claim that Av= γ v if possible Av γv putting x=v and y=u we have d(γv θ u) d(av Bu)[d(θ u γv) + d(av θ u)] d(bu Av) + d(γv Bu) d(θ u γv)[d(bu Av) + d(γv Bu)] d(θ u γv) + d(av θ u) d(av γv)[d(bu θ u) + d(bu Av)] d(av γv) + d(θ u γv) } } d(γv Av) ф d(avav)[d(avγv)+d(avav)] d(avav)+d(γvav) d(avγv)[d(avav)+d(γvav)] d(avγv)+d(avav) d(avγv)[d(avav)+d(avav)] d(avγv)+d(avγv) } } [since θ u=av and Bu=Av] d(γv Av) 1 d(av γv) 1}} d (γv Av) d (γv Av) Which is a contradiction because the range of this implies that d(γv Av) = 1 i. e. γv = Av. Hence we have Bu = θ u = Av = γ v. Since the pair A γ are weakly compatible we have A γ v= γ Av and then γ γ v = γ Av = A γ v = AAv. Next we claim that γ Av = Av if possible γ Av Av on substituting x=av and y=u we have d(γ AvAv) = d(γav θ u) d(aav Bu)[d(θ u γav) + d(aav θ u)] d(bu AAv) + d(γav Bu) d(θ u γav)[d(bu AAv) + d(γav Bu)] d(θ u γav) + d(aav θ u) d(aav γav)[d(bu θ u) + d(bu AAv)] d(aav γav) + d(θ u γav) } }

12 98 Nisha Sharma and Arti Mishra ф d(savav)[d(avsav)+d(savav)] d(avsav)+d(savav) d(avsav)[d(avsav)+d(savav)] d(avsav)+d(savav) d(savsav)[d(avav)+d(avsav)] d(savsav)+d(avsav) } } [since AAv = γav θ u = Av = Bu = Av] d(γav Av) d(av γav) 1}} d(γav Av) d (γav Av) This is a contradiction since (0 1 ) hence Av = γav hence γ Av = Av = AAv. Hence Av is a common fixed point of A and γ. Similarly BBu=Bu= θ Bu i.e. Bu is a common fixed point of B and θ as Av=Bu Av is a common fixed point of A B γ and θ. Similarly we can complete the proof for cases in which AX or θ X or γ X is closed subset of X. Uniqueness Let Av and Pu are two distinct common fixed points of A B γ and θ. Using symmetry of multiplicative metric space and (E) we have d(av Pu) = d(γav θ Pu) d(aav BPu)[d(θ Pu γav) + d(aav θpu)] d(bpu AAv) + d(γav BPu) d(θ Pu γav)[d(bpu AAv) + d(γav BPu)] d(θ Pu γav) + d(aav θ Pu) d(aav γav)[d(bpu θ Pu) + d(bpu AAv)] d(aav γav) + d(θ Pu γav) } } d(av Pu)[d(Pu Av) + d(av Pu)] d(pu Av) + d(av Pu) d(pu Av)[d(Pu Av) + d(av Pu)] d(pu Av) + d(av Pu) d(av Av)[d(Pu Pu) + d(pu Av)] d(av Av) + d(pu Av) } } d(av Pu) d(av Pu) 1}} d(av Pu) d (Av Pu) A contradiction since (0 1 ) hence d(av Pu) = 1 i. e. Av = Pu. Now we prove the following theorems for weakly compatible mappings with common limit range property satisfying the implicit function in a multiplicative metric space.

13 Common Properties Implicit Function Rational Contractive Map and Related 99 Lemma.5.Let A B γ θ be mappings of a multiplicative metric space (X d) satisfying the conditions (E1) and (E) and the following condition: (E11) the pairs (A γ) satisfies CLRA property or the pair (B ) satisfies CLRB property Then the pairs (A γ) and (B θ) share the common limit in the range of A property or B property. Proof. Let us consider that the pair (A γ) satisfies the common limit range of A property. Then a sequence x n } in X such that Lim A x n = lim γ x n = Az for some z X. Since γ X BX so for each x n there exists y n in X such thst γ x n lim B y n =Az. hence we have lim γ x n = lim A x n = lim B y n =Az. =B y n. Then Now we claim that lim θ y n =Az. On substituting x = x n and y = y n in (E) we have d(γx n θ y n ) d(ax n By n )[d( θ y n γx n ) + d(ax n θ y n )] d(by n Ax n ) + d(γx n By n ) d( θ y n γx n )[d(by n Ax n ) + d(γx n By n )] d( θ y n γx n ) + d(ax n θ y n ) d(ax n γx n )[d(by n θ y n ) + d(by n Ax n )] d(ax n γx n ) + d( θ y n γx n ) } } Taking n we have d (Az lim θ y n ) ф d(az Az) [d ( lim θ y n Az) + d (Az lim θ y n )] d(az Az) + d(az Az) d ( lim θ y n Az) [d(az Az) + d(az Az)] d ( lim θ y n Az) + d (Az lim θ y n ) d(az Az)[d (Az lim θ y n ) + d(az Az)] d(az Az) + d( lim θ y n Az) d (Az lim θ y n ) d ( lim θ y n ) 11}} d (Az lim θ y n ) d (Az lim θ y n ) } }

14 300 Nisha Sharma and Arti Mishra This is a contradiction since (0 1 ) hence d (Az lim θ y n ) = 1. Therefore Az = lim θ y n and hence lim θ y n = Az. i.e. lim θ y n = lim B y n = lim γ x n = lim A x n = Az Then the pairs (A γ) and (B θ) share the common limit range of A property. for the other pair (B θ) which shares common limit range of B property Since the pair (B θ) satisfies common limit range of B property. Then a sequence y n } in X such that lim B y n = lim θ y n = Bz for some z X. Since θ X AX so for each y n x n in X such that θ y n = A x n. Then lim A x n =Bz. hence we have lim A x n = lim B y n = lim θ y n = Bz. Now we claim that lim Bz On substituting x = x n and y = y n in (E) we have d(γx n θ y n ) γ x n =Bz. If possible lim γ x n d(ax n By n )[d(θ y n γx n ) + d(ax n θ y n )] d(by n Ax n ) + d(γx n By n ) d( θ y n γx n )[d(by n Ax n ) + d(γx n By n )] d( θ y n γx n ) + d(ax n θ y n ) d(ax n γx n )[d(by n θ y n ) + d(by n Ax n )] d(ax n γx n ) + d( θ y n γx n ) } } Taking n we have d ( lim γx n Bz) ф d(bz Bz) [d (Bz lim γx n ) + d(bz Bz)] d(bz Bz) + d ( lim γx n Bz) d (Bz lim γx n ) [d(bz Bz) + d ( lim γx n Bz)] d (Bz lim γx n ) + d(bz Bz) d (Bz lim γx n ) [d(bz Bz) + d(bz Bz)] d (Bz lim γx n ) + d(bz lim γx n ) d ( lim γx n Bz) 1 d(bz γx n ) 1}} d ( lim γx n Bz) d ( lim γx n Bz) } }

15 Common Properties Implicit Function Rational Contractive Map and Related 301 This is a contradiction since (0 1 ) hence Therefore lim γ x n =Bz. Then the pairs (A γ) and (B θ) share the common limit range of B property. Theorem.6.let A B γ θ be mappings of a multiplicative metric space (X d) satisfying the conditions (E1) and (E) and (E11) Then the pairs A γ and B have a coincidence point. Moreover assume that the pairs (A γ) and (B θ) are weakly compatible. Then A B γ and θ have a unique common fixed point. Proof. Using the lemma (.5) the pairs (A γ) and (B θ) share the common limit range of A property that is there exist two sequences x n } and y n } in X such that lim θ y n = lim B y n = lim γ x n = lim A y n =Av for some v X. First we claim that Av= γ v substituting x=v and y=y n in (E) we have d(γv θ y n ) d(av By n )[d( θ y n γv) + d(av θ y n )] d(by n Av) + d(γv By n ) d( θ y n γv)[d(by n Av) + d(γv By n )] d( θ y n γv) + d(av θ y n ) d(av γv)[d(by n θ y n ) + d(by n Av)] d(av γv) + d( θ y n γv) } } Taking n approaches to infinity we have d(γv Av) d(av Av)[d(Av γv) + d(av Av)] d(av Av) + d(γv Av) d(av γv)[d(av Av) + d(γv Av)] d(av γv) + d(av Av) d(av γv)[d(av Av) + d(av Av)] d(av γv) + d(av γv) } } d(γv Av) 1 d(av γv) 1}} d(γv Av) d (γv Av) On the other hand (0 1 ) implies that d(γ vav)=1 i.e. γ v=av. Since γ X BX w Xsuch that Bw= γ v. Now we claim that Bw= θ w if possible Bw θ w. Putting x=v and y=w in (E) we have d(bw θ w) = d(γv θ w)

16 30 Nisha Sharma and Arti Mishra d(av Bw)[d(θ w γv) + d(av θ w)] d(bw Av) + d(γv Bw) d(θ w γv)[d(bw Av) + d(γv Bw)] d(θ w γv) + d(av θ w) d(av γv)[d(bw θ w) + d(bw Av)] d(av γv) + d(θ w γv) } } d(bw Bw)[d(θ w Bw) + d(bw θ w)] d(bw Bw) + d(bw Bw) d(θ w Bw)[d(Bw Bw) + d(bw Bw)] d(θ w Bw) + d(bw θ w) d(bw Bw)[d(Bw θ w) + d(bw Bw)] d(bw Bw) + d(θ w Bw) } } [since Av=Bw γ v=bw] d(bw θ w) 1 d(θ w Bw) 1}} d(bw θ w ) = d (Bw θ w) Which is a contradiction hence since (0 1 ) Bw= θ w and hence θ w=av= γ v=bw. since (A γ) and (B θ) are weakly compatible and γ v=av and θ w=bw. Hence A γ v = γ Av = AAv = γ γ v θ Bw = B θ w = BBw = θ θ w. Finally we claim that γ Av=Av. Putting x=av and y=w in (E) we have d(γ AvAv) = d(γ Av θ w) d(aav Bw)[d(θ w γav) + d(aav θ w)] d(bw AAv) + d(γav Bw) d(θ w γav)[d(bw AAv) + d(γav Bw)] d(θ w γav) + d(aav θ w) d(aav γav)[d(bw θ w) + d(bw AAv)] d(aav γav) + d(θ w γav) } } [since AAv = γav θ w = Av Bw = Av] d(γ AvAv) ф d(aavav)[d(avγav)+d(γavav)] d(avγav)+d(γavav) d(avγav)[d(avγav)+d(γavav)] d(avγav)+d(γavav) d(γavγav)[d(avav)+d(avγav)] d(γavγav)+d(avγav) } } d(γav Av) d(γav A) d(γav Av) 1}} d(γ AvAv ) = d (γav Av)

17 Common Properties Implicit Function Rational Contractive Map and Related 303 which is a contradiction hence γ Av=Av and hence γ Av=Av=A γ v=aav= γ γ v which implies that Av is a common fixed point of A and γ. also one can easily prove that BBw= θ θ w= θ Bw=B θ w i.e. Bw is common fixed pointof B and θ. As Av=Bw Av is a common fixed point of A B γ and θ. The uniqueness follows easily from (E). This completes the proof. REFERENCES [1] A.E Bashirov E.M.kurpnaraA. Ozyapici Multiplicative calculus and its applications J. Math Anal. Appl. 337 (008) Doi: /j.jmaa [] Chahn yong Jung Prveen kumar sanjay Kumar Shin Ming Kang Common fixed point theorems for weakly compatible mapping satisfying implicit functions in multiplicative metric space. international J. Pure and Applies Math.10 No. 3 (015) Doi: /ijpam.v10i3.1 [3] F. Gu l.m. cui anf Y.H. Wn some fixed point theorems for new contractive type mappings J. Qiqihar Univ.19 (013)85-89 [4] G. Jungck Common fixed points for noncontineous nonself maps on nonmetric spaces Far east J. Math. Sci. 4 (1996) [5] M. Aamri D.El Moutawaskil Some new common fixed point theorems under strict contractive conditions J. Math. Anal. 70 (00) Doi: /S00-47X(0) [6] M. Özavsar A.C. Çevikel Fixed points of multiplicative contraction mappings on multiplicative metric space. arxiv: v1[math.gm]01 [7] S. Sharma B. Deshpande On compatible mapping satisfying an implicit relation in common fixed point consideration Tamkang J. Math.33 (00)45-5 [8] W. Sintunavarat P. Kumam Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric space J. Appl. Math.001 (011)Article ID pages doi: /011/ [9] V. Popa M. MocanuAltering distance and common fixed points under implicit relations Hacet.J. Math. Stat.38(009) [10] V. Popa A fixed point theorem for mappings in d-complete topological spaces Math. Moravica3 (1999) 43-48

18 304 Nisha Sharma and Arti Mishra