Common Random Fixed Point Theorems under Contraction of rational Type in Multiplicative Metric Space
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- Λυσιστράτη Λιάπης
- 5 χρόνια πριν
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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13 Number ) pp Research India Publications Common Random Fixed Point Theorems under Contraction of rational Type in Multiplicative Metric Space 1 Nisha Sharma 2* Arti Mishra 3 Pooja Khurana 4 Vandana Kumari 5 Vishnu Narayan Mishra 1 23 Department of Mathematics Manav Rachna International University Faridabad Haryana India. 4 Department of Mathematics Govt. College Tigaon Haryana India. 5 Sardar Vallabhbhai National Institute of Technology Surat Gujarat India. * corresponding author: Abstract This is the paper inspired and yeasted by [1] we obtain common random fixed point theorems satisfying contractive mapping in the setup of multiplicative metric space and additionally we are going to prove compatible and weak compatible mappings again under rational contractive maps to prove the common random fixed point theorems in multiplicative metric space. Keywords. Random fixed point compatible mapping weak compatible mapping implicit function common fixed point multiplicative metric space. AMS Subject Classification: 46S40 47H10 54H25 I. INTRODUCTION The initiation and research in the direction of random approximation and random fixed point theorems was introduced by the Prague School of probabilistic in 1950s. since then a lot of efforts have been devoted to random fixed point theory and applications see5-14] [18]).Whereas In 1976 some sufficient conditions have been given by Bharucha Reid[14] for a stochastic analogue of Schauder s fixed point theorem which was defined in a separable metric space. One of the powerful tools in current mathematical applications that we have are fixed point theorems. Wide class of problems of mechanics physics etc. have been solved by the extended and
2 3704 Nisha Sharma et al generalized fixed point theorems. Many fixed point theorems have been proved in [references]. In 2008 it was found that the set of all positive real numbers is not complete with respect to usual metric by Bashirov et al. [4] which was overcome by the introduction of the concept of multiplicative metric space. Many authors Abbas [2] kang et al. [15] Sarvar and Badshah-e [17] have been proved many fixed point theorems also they tried to give more results on multiplicative metric space. Beg et al. [5 8 10] studied the interrelation of common random fixed points and random coincidence point of compatible random operator and proved the random fixed points for random operators. II. PRELIMINARIES Definition 1.1. [3] ) Let X be a nonempty set. A multiplicative metric is a mapping Ω: X x X R + satisfying the following conditions: 1.1) Ω xy) 1 xy X and Ω xy)=1 if and only if x=y; 1.2) Ω xy)= Ω yx) xy X; 1.3) Ω xy) Ω xz). Ω zy) xy X multiplicative triangle inequality). Then the mapping Ω together with X i.e. X Ω) is a multiplicative metric space. Definition 1.2. [15] ) Let Xd) be a multiplicative metric space and F:R x X X be a function where X is a non empty set. Then function g:r X is said to be a random common fixed point of the function F is Ftgt))=gt) for all t R. Definition 1.3. [15] ) Let X Ω) 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 Ω x y ) ε ε > 1} there exists a natural number N such that n N then x n B ε x) i.e. Ω x n x) 1 as n. 1.5) a multiplicative Cauchy sequence if ε > 1 for all m n >N that is Ω 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. 1.7) Let Xd) be a multiplicative metric space. Two mappings αγ: R x X X are said to be
3 Common Random Fixed Point Theorems under Contraction of rational Type 3705 a. Compatible for each t R if lim n Ωαt γt θ n t)))) γt αt θ n t)))=1 Whenever lim n α t γt θ n t))) lim n γ t αt θ n t))) exist in X and lim α t γt θ n t)))=lim γ t αt θ n t))) t R where θ n } is a n n sequence of mappings. b. Weakly compatible for each t R if γt θt)) = αt θt)) for some mapping θ implies γ t αt θt))) = α t γt θt))) for every t R. III. MAIN RESULTS Now we prove the following theorems for compatible and weakly compatible mappings satisfying implicit function in a multiplicative metric space. Theorem 2.1. Let us consider complete multiplicative metric space X Ω) and a function Φ: R x X X be a function satisfying the following conditions E1) ΩΦt x) Φt y)) For all x y X where t R and [01). Then Φ have a unique random fixed point. ΩyΦtx)). ΩΦtx)Φty)) [ΩyΦty))+ΩxΦty)) 1+Ωxy) [ Ωxy)ΩyΦtx))+ΩΦty)x)ΩyΦtx)) ΩyΦtx))+ΩΦtx)Φty)) [ Ωxy)+ΩΦtx)Φty)) ] ΩxΦtx))+ΩyΦty)) [ΩxΦty))+ΩxΦtx)) 1+ΩΦtx)Φty)) ] } 2 Proof. A sequence θ n }can be defined as θ n t) = Φt θ n 1 t)) where θ n is a arbitrary function defined as θ n : R X for t R and n=0123 If we define θ n+1 t) = θ n t) for some n and t R then θ n t) is a random fixed point Let θ n+1 t) θ n t) for each n and t R then for all t R substituting x = θ n 1 t) and y = θ n t) in E1) we have Ω θ n t) θ n+1 t))= Ω Φt θ n 1 t)) Φt θ n t)))
4 3706 Nisha Sharma et al Ω θ n t)φt θ n 1 t))) ΩΦt θ n 1 t))φt θ n t))). 2 [ Ω θ n t)φt θ n t)))+ω θ n 1 t)φt θ n t))) 1+Ω θ n 1 t) θ n t)) [ Ω θ n 1 t) θ n t)).ω θ n t)φt θ n 1 t)))+ωφt θ n t)) θ n 1 t)).ω θ n t)φt θ n 1 t))) Ω θ n t)φt θ n 1 t)))+ωφt θ n 1 t))φt θ n t))) [ Ω θ n 1 t) θ n t))+ωφt θ n 1 t))φt θ n t))) Ω θ n 1 t)φt θ n 1 t)))+ω θ n t)φt θ n t))) [ Ω θ n 1 t)φt θ n t)))+ω θ n 1 t)φt θ n 1 t))) ] 1+ΩΦt θ n 1 t))φt θ n t))) Ω θ n t) θ n t)) Ω θ n t) θ n+1 t)). [ Ω θ n t) θ n+1 t))+ω θ n 1 t) θ n+1 t)). ] 1+Ω θ n 1 t) θ n t)) [ Ω θ n 1 t) θ n t)).ω θ n t) θ n t))+ω θ n 1 t) θ n+1 t)).ω θ n t) θ n t)) Ω θ n t) θ n t))+ω θ n+1 t) θ n t)) [ Ω θ n 1 t) θ n t))+ω θ n t) θ n+1 t)) Ω θ n 1 t) θ n t))+ω θ n t) θ n+1 t)) [ Ω θ n 1 t) θ n+1 t))+ω θ n 1 t) θ n t)) ] 1+Ω θ n t) θ n+1 t)) 1 Ω θ n t) θ n+1 t)). [ Ω θ n t) θ n+1 t))+ω θ n 1 t) θ n t))ω θ n t) θ n+1 t)) ] 1+Ω θ n 1 t) θ n t)) [ Ω θ n 1 t) θ n t))+ω θ n 1 t) θ n t))ω θ n t) θ n+1 t)) 1+Ω θ n+1 t) θ n t)) [ Ω θ n 1 t) θ n t))+ω θ n t) θ n+1 t)) Ω θ n 1 t) θ n t))+ω θ n t) θ n+1 t)) [ Ω θ n 1 t) θ n t)).ω θ n t) θ n+1 t))+ω θ n 1 t) θ n t)) ] 1+Ω θ n t) θ n+1 t)) } Ω 2 θ n 1 t) θ 2 n t))} } 2 2 } hence we have Ω θ n t) θ n+1 t)) Ω θ n 1 t) θ n t)) Ω 2 θ n 2 t) θ n 1 t)) Ω 3 θ n 3 t) θ n 2 t)) Ω n θ 0 t) θ 1 t))
5 Common Random Fixed Point Theorems under Contraction of rational Type 3707 For n m N having m<n we have Ω θ m t) θ n t)) Ω θ m t) θ m+1 t)). Ω θ m+1 t) θ m+2 t)) Ω θ n 1 t) θ n t)) Ω m θ 0 t) θ 1 t)). Ω m+1 θ 0 t) θ 1 t)) Ω n 1 θ 0 t) θ 1 t)) Ω m 1 θ 0 t) θ 1 t)) Taking n approaches to infinity Ω θ m t) θ n t)) 1. This implies that θ m t)} is a multiplicative Cauchy sequence. Since the space is complete it implies that the convergent point say θt)) belongs to X. now it is to be claim that Φt θt)) = θt) substituting x= θt) and y= θ n t) in E1) ΩΦt θt) ) θ n+1 t)) ΩΦt θt) ) Φt θ n t))) Ω θ n t)φtθt) )) ΩΦtθt) )Φt θ n t))). 2 [ Ω θ n t)φt θ n t)))+ωθt) Φt θ n t))) 1+Ωθt) θ n t)) [ Ωθt) θ n t))ω θ n t)φtθt) ))+ΩΦt θ n t))θt) )Ω θ n t)φtθt) )) Ω θ n t)φtθt) ))+ΩΦtθt) )Φt θ n t))) [ Ωθt) θ n t))+ωφtθt) )Φt θ n t))) Ωθt) Φtθt) ))+Ω θ n t)φt θ n t))) [ Ωθt) Φt θ n t)))+ωθt) Φtθt) )) 1+ΩΦtθt) )Φt θ n t))) ] } Ω θ n t)φtθt) )) ΩΦtθt) ) θ n+1 t)). [ Ω θ n t) θ n+1 t))+ωθt) θ n+1 t)) 1+Ωθt) θ n t)) [ Ωθt) θ n t))ω θ n t)φtθt) ))+Ω θ n+1 t)θt) )Ω θ n t)φtθt) )) Ω θ n t)φtθt) ))+ΩΦtθt) ) θ n+1 t)) [ Ωθt) θ n t))+ωφtθt) ) θ n+1 t)) Ωθt) Φtθt) ))+Ω θ n t) θ n+1 t)) [ Ωθt) θ n+1 t))+ωθt) Φtθt) )) 1+ΩΦtθt) ) θ n+1 t)) ] } 2
6 3708 Nisha Sharma et al Taking n approaches to infinity we have ΩΦt θt) ) θt)) Hence we have [ [ Ωθt)Φtθt) )) ΩΦtθt) )θt)). Ωθt)θt))+Ωθt) θt)) 1+Ωθt) θt)) Ωθt) θt))ωθt)φtθt) ))+Ωθt)θt) )Ωθt)Φtθt) )) Ωθt)Φtθt) ))+ΩΦtθt) )θt)) Ωθt) θt))+ωφtθt) )θt)) [ Ωθt) Φtθt) ))+Ωθt)θt)) ] [ } 1.1 Ωθt) θt))+ωθt) Φtθt) )) 1+ΩΦtθt) )θt)) ] } 2 Implies ΩΦt θt) ) θt)) =1 Φt θt) ) = θt) Uniqueness Let t) be another common random fixed point of Φ. Then using the inequality E1) Ωθt) θt))=ω Φt θt)) Φt θt))) Ωθt)Φtθt))). ΩΦtθt))Φtθt))) [Ωθt)Φtθt)))+Ωθt)Φtθt))) 1+Ωθt)θt)) [ Ωθt)θt))Ωθt)Φtθt)))+ΩΦtθt))θt))Ωθt)Φtθt))) Ωθt)Φtθt)))+ΩΦtθt))Φtθt))) [ Ωθt)θt))+ΩΦtθt))Φtθt))) Ωθt)Φtθt)))+Ωθt)Φtθt))) 2 [ Ωθt)Φtθt)))+Ωθt)Φtθt))) ] 1+ΩΦtθt))Φtθt))) }
7 Common Random Fixed Point Theorems under Contraction of rational Type 3709 Ωθt)θt)). Ωθt)θt)) [Ωθt)θt))+Ωθt)θt)) 1+Ωθt)θt)) [ Ωθt)θt))Ωθt)θt))+Ωθt)θt))Ωθt)θt)) Ωθt)θt))+Ωθt)θt)) [ Ωθt)θt))+Ωθt)θt)) ] Ωθt)θt))+Ωθt)θt)) [Ωθt)θt))+Ωθt)θt)) 1+Ωθt)θt)) ] } 2 [since θt) and θt) are random fixed points of Φ ] 1.1. = [Ωθt) θt))]. } [Ωθt) θt))]. 1 Ωθt) θt)) Ω 2 2 θt) θt))} 2 Ωθt) θt)) Ω θt) θt)) which is a contradiction since [01). Hence we have θt) = θt). This completes the proof. Theorem 2.2. Let us consider complete multiplicative metric space X Ω) and functions Φ α γ and β: R x X X be four functions satisfying the following conditions: E2) αtx) β tx) and ΦtX) γtx) E3) Ω αt xt)) Φt yt)))
8 3710 Nisha Sharma et al Ωβtyt))Φtyt)))[Ωαtxt))γtxt))+Ωβtyt))γtxt))] ΩΦtyt))βtyt)))+ΩΦtyt))αtxt))) Ωαtxt))Φtyt)))[Ωβtyt))Φtyt))+Ωαtxt))γtxt))] ΩΦtyt))αtxt)))+Ωγtxt))βtyt))) Ωβtyt))γtxt)))[ΩΦtyt))βtyt)))+Ωγ txt))αtxt))] Ωβtyt)) γtxt)))+ωαtxt))φtyt))) Ω γtxt))βtyt)))[ωφtyt))βtyt)) +ΩΦtyt))αtxt))] Ωαtxt))γtxt))+Ωβtyt)) γtxt))) )} For all x y X where t R and [01). Then Φ α γ and β have a unique random fixed point is and only if one of the following conditions are satisfied: E4) either α or γ is continuous the pair α γ) is compatible and the pair β Φ) is weakly compatible. E5) either Φ or β is continuous the pair Φ β)is compatible and the pair α γ ) is weakly compatible. Proof: Let the function θ 0 : R X be an arbitrary function on X by E2) there exists a function θ 1 : R X such that for t R can be defined as t θ 0 t))= β t θ 1 t)) and for this function ϑ 1 : R X we can choose a function θ 2 : R X such that for t R can be defined as γ t θ 2 t))= Φ t θ 1 t)) and so on. By using the principle of mathematical induction we can define a sequence y n t)} t R of functions as follows: y 2n+1 t)= βt θ 2n+1 t))= αt θ 2n t)) y 2n t)= γt θ 2n t))= Φt θ 2n 1 t)) 2.1) Using E3) and 2.1) we have Ωy 2n+1 t) y 2n t)) = Ω αt θ 2n t)) Φt θ 2n 1 t)))
9 Common Random Fixed Point Theorems under Contraction of rational Type 3711 Ωβt θ 2n 1 t))φt θ 2n 1 t)))[ωαt θ 2n t))γt θ 2n t))+ωβt θ 2n 1 t))γt θ 2n t))] ΩΦt θ 2n 1 t))βt θ 2n 1 t)))+ωφt θ 2n 1 t))αt θ 2n t))) Ωαt θ 2n t))φt θ 2n 1 t)))[ωβt θ 2n 1 t))φt θ 2n 1 t))+ωαt θ 2n t))γt θ 2n t))] ΩΦt θ 2n 1 t))αt θ 2n t)))+ωγt θ 2n t))βt θ 2n 1 t))) Ωβt θ 2n 1 t))γt θ 2n t)))[ωφt θ 2n 1 t)βt θ 2n 1 t)))+ωγt θ 2n t))αt θ 2n t))] Ωβt θ 2n 1 t)) γt θ 2n t)))+ω αt θ 2n t))φt θ 2n 1 t))) Ω γt θ 2n t))βt θ 2n 1 t)))[ωφt θ 2n 1 t))βt θ 2n 1 t)) +ΩΦt θ 2n 1 t))αt θ 2n t))] Ωαt θ 2n t))γt θ 2n t))+ωβt θ 2n 1 t)) γt θ 2n t))) )} Ωy 2n 1 t)y 2n t))[ωy 2n+1 t)y 2n t)+ωy 2n 1 t)y 2n t))] Ωy 2n t)y 2n 1 t))+ωy 2n t)y 2n+1 t)) Ωy 2n+1 t)y 2n t))[ωy 2n 1 t)y 2n t))+ωy 2n+1 t)y 2n t))] Ωy 2n t)y 2n+1 t))+ωy 2n t)y 2n 1 t)) Ωy 2n 1 t)y 2n t))[ωy 2n t)y 2n 1 t))+ωy 2n t)y 2n+1 t))] Ωy 2n 1 t) y 2n t))+ω y 2n+1 t)y 2n t)) Ωy 2n 1 t)y 2n t))[ωy 2n t)y 2n 1 t)) +Ωy 2n t)y 2n+1 t))] Ωy 2n+1 t)y 2n t))+ωy 2n 1 t)y 2n t)) )} Ωy 2n 1 t) y 2n t)) Ωy 2n+1 t) y 2n t)) Ωy 2n 1 t) y 2n t)) Ωy 2n 1 t) y 2n t)))} if Ωy 2n+1 t) y 2n t)) Ω y 2n+1 t) y 2n t)) a contradiction since [01). Implies Ωy 2n+1 t) y 2n t)) Ω y 2n 1 t) y 2n t)) 2.2) Ωy 2n+2 t) y 2n+1 t)) = Ω Φt θ 2n+1 t)) αt θ 2n t))) = Ω αt θ 2n t)) Φt θ 2n+1 t))) Ωβt θ 2n+1 t))φt θ 2n+1 t)))[ωαt θ 2n t))γt θ 2n t))+ωβt θ 2n+1 t))γt θ 2n t))] ΩΦt θ 2n+1 t))βt θ 2n+1 t)))+ωφt θ 2n+1 t))αt θ 2n t))) Ωαt θ 2n t))φt θ 2n+1 t)))[ωβt θ 2n+1 t))φt θ 2n+1 t))+ωαt θ 2n t))γt θ 2n t))] ΩΦt θ 2n+1 t))αt θ 2n t)))+ωγt θ 2n t))βt θ 2n+1 t))) Ωβt θ 2n+1 t))γt θ 2n t)))[ωφt θ 2n+1 t))βt θ 2n+1 t)))+ωγt θ 2n t))αt θ 2n t))] Ωβt θ 2n+1 t)) γt θ 2n t)))+ω αt θ 2n t))φt θ 2n+1 t))) Ω γt θ 2n t))βt θ 2n+1 t)))[ωφt θ 2n+1 t))βt θ 2n+1 t)) +Ω Φt θ 2n+1 t))αt θ 2n t))] Ωαt θ 2n t))γt θ 2n t))+ωβt θ 2n+1 t)) γt θ 2n t))) )}
10 3712 Nisha Sharma et al Ωy 2n+1 t)y 2n+2 t))[ωy 2n+1 t)y 2n t))+ωy 2n+1 t)y 2n t))] Ωy 2n+2 t)y 2n+1 t))+ωy 2n+2 t)y 2n+1 t)) Ωy 2n+1 t)y 2n+2 t))[ωy 2n+1 t)y 2n+2 t))+ωy 2n+1 t)y 2n t))] Ωy 2n+2 t)y 2n+1 t))+ωy 2n t)y 2n+1 t)) Ωy 2n+1 t)y 2n t))[ωy 2n+2 t)y 2n+1 t)+ωy 2n t)y 2n+1 t))] Ωy 2n+1 t) y 2n t))+ω y 2n+1 t)y 2n+2 t)) Ωy 2n t)y 2n+1 t))[ωy 2n+2 t)y 2n+1 t)) +Ωy 2n+2 t)y 2n+1 t)] Ωy 2n+1 t)y 2n t))+ωy 2n+1 t)y 2n t)) )} Ωy 2n+1 t) y 2n t)) Ωy 2n+1 t) y 2n+2 t)) Ωy 2n+1 t) y 2n t)) Ωy 2n+2 t) y 2n+1 t)))} if Ωy 2n+2 t) y 2n+1 t)) Ω y 2n+2 t) y 2n+1 t)) a contradiction since [01). Implies Ωy 2n+2 t) y 2n+1 t)) Ω y 2n+1 t) y 2n t)) 2.3) Using 2.2) and 2.3) we have hence 2.4) Ωy n+1 t) y n t)) Ω y n t) y n 1 t)) for all n Ωy n+1 t) y n t)) Ω y n t) y n 1 t)) Ω 2 y n 1 t) y n 2 t))... Ω n y 1 t) y 0 t)) Let m n N with m >n. then we have Ω y m t) y n t)) Ω y m t) y m 1 t)). Ω y m 1 t) y m 2 t)) Ω y n+1 t) y n t)) Ω m 1 y 1 t) y 0 t)). Ω m 2 y 1 t) y 0 t)) Ω n y 1 t) y 0 t))
11 Common Random Fixed Point Theorems under Contraction of rational Type 3713 Ω n 1 y 1 t) y 0 t)) Taking n Ω y m t) y n t)) 0 hence yn t)} is a multiplicative Cauchy sequence. Since X is complete so ynt)} is a multiplicative convergent to a point yt) X as n approaches to infinity and also subsequences of ynt)} also are multiplicative convergent to a point yt) X and t R βt θ 2n+1 t)) yt) Φt θ 2n+1 t)) yt) 2.5) αt θ 2n t)) yt) γt θ 2n t)) yt) Now since the pair α γ is compatible then by 2.5) we obtain lim Ωα t γt θ 2n t))) γ t αt θ 2n t))) = ) n Suppose that γ is continuous it follows that γ t γt θ 2n t))) γt yt)) γ t αt θ 2n t))) γt yt)) 2.7) Then by definition of compatibility we have α t γt θ 2n t))) γ t yt)) t R. 2.8) Using condition E3) we have Ω α t γt θ 2n t))) Φt θ 2n+1 t))) Ωβt θ 2n+1 t))φt θ 2n+1 t)))[ωαtγt θ 2n t)))γtγt θ 2n t)))+ωβt θ 2n+1 t))γtγt θ 2n t)))] ΩΦt θ 2n+1 t))βt θ 2n+1 t)))+ωφt θ 2n+1 t))αtγt θ 2n t)))) Ωαtγt θ 2n t)))φt θ 2n+1 t)))[ωβt θ 2n+1 t))φt θ 2n+1 t))+ωαtγt θ 2n t)))γtγt θ 2n t)))] ΩΦt θ 2n+1 t))αtγt θ 2n t))))+ωγtγt θ 2n t)))βt θ 2n+1 t))) Ωβt θ 2n+1 t))γtγt θ 2n t))))[ωφt θ 2n+1 t))βt θ 2n+1 t)))+ωγ tγt θ 2n t)))αtγt θ 2n t)))] Ωβt θ 2n+1 t)) γtγt θ 2n t))))+ωαtγt θ 2n t)))φt θ 2n+1 t))) Ω γtγt θ 2n t)))βt θ 2n+1 t)))[ωφt θ 2n+1 t))βt θ 2n+1 t)) +ΩΦt θ 2n+1 t))αtγt θ 2n t)))] Ωαtγt θ 2n t)))γtγt θ 2n t)))+ωβt θ 2n+1 t)) γtγt θ 2n t)))) )}
12 3714 Nisha Sharma et al Letting n approaches to infinity we have Ω γt yt)) yt)) Ωyt)yt))[Ωγtyt))γtyt))+Ωyt)γtyt))] Ωyt)yt))+Ωyt)γtyt))) Ωγtyt))yt))[Ωyt)yt))+Ωγtyt))γtyt)))] Ωyt)γtyt)))+Ωγtyt)))yt)) Ωyt)γtyt))))[Ωyt)yt))+Ωγtyt))γtyt))] Ωyt) γtyt)))+ωγtyt))yt)) Ω γtyt))yt))[ωyt)yt)) +Ωyt)γtyt))] Ωγtyt))γtyt)))+Ωyt)γtyt))) )} Ω γt yt)) yt)) ) } Ω γt yt)) yt)) Ω γt yt)) yt)) a contradiction since [01). Which implies that γ tyt))=yt) 2.9) Now we prove that αt yt)) = yt). if possible αt yt)) yt) Using condition E3) Ω αt yt)) Φt θ 2n 1 t))) Ωβt θ 2n+1 t))φt θ 2n+1 t)))[ωαtyt))γtyt))+ωβt θ 2n+1 t))γtyt))] ΩΦt θ 2n+1 t))βt θ 2n+1 t)))+ωφt θ 2n+1 t))αtyt))) Ωαtyt))Φt θ 2n+1 t)))[ωβt θ 2n+1 t))φt θ 2n+1 t))+ωαtyt))γtyt))] ΩΦt θ 2n+1 t))αtyt)))+ωγtyt))βt θ 2n+1 t))) Ωβt θ 2n+1 t))γtyt)))[ωφt θ 2n+1 t))βt θ 2n+1 t)))+ωγ tyt))αtyt))] Ωβt θ 2n+1 t)) γtyt)))+ωαtyt))φt θ 2n+1 t))) Ω γtyt))βt θ 2n+1 t)))[ωφt θ 2n+1 t))βt θ 2n+1 t)) +ΩΦt θ 2n+1 t))αtyt))] Ωαtyt))γtyt))+Ωβt θ 2n+1 t)) γtyt))) )}
13 Common Random Fixed Point Theorems under Contraction of rational Type 3715 Taking n we have Ω αt yt)) yt)) Ωyt)yt))[Ωαtyt))yt)+Ωyt)yt)] Ωyt)yt))+Ωyt)αtyt))) Ωαtyt))yt))[Ωyt)yt))+Ωαtyt))yt)] Ωyt)αtyt)))+Ωyt)yt)) Ωyt)yt))[Ωyt)yt))+Ωyt)αtyt))] Ωyt) yt))+ωαtyt))yt)) Ω yt)yt))[ωyt)yt)) +Ωyt)αtyt))] Ωαtyt))yt))+Ωyt) yt)) )} [using 2.5)2.9)] Implies 1 Ω αt yt)) yt)) 1 1 )} Ω αt yt)) yt)) Ω αt yt)) yt)) a contradiction since [01). And thus we have αt yt)) = yt). Hence we have αt yt)) = γt yt)) = yt). 2.10) Now since yt) = αt yt)) αt X)) β tx) ht) X such that yt) = β tht)) for t R. Now we prove that Φt ht))=yt). if possible Φt ht)) yt) Using condition E3) we obtain Ω yt) Φt ht))) = Ω αt yt)) Φt ht)))
14 3716 Nisha Sharma et al Ωβtht))Φtht)))[Ωαtyt))γtyt))+Ωβtht))γtyt))] ΩΦtht))βtht)))+ΩΦtht))αtyt))) Ωαtyt))Φtht)))[Ωβtht))Φtht))+Ωαtyt))γtyt))] ΩΦtht))αtyt)))+Ωγtyt))βtht))) Ωβtht))γtyt)))[ΩΦtht))βtht)))+Ωγ tyt))αtyt))] Ωβtht)) γtyt)))+ωαtyt))φtht))) Ω γtyt))βtht)))[ωφtht))βtht)) +ΩΦtht))αtyt))] Ωαtyt))γtyt))+Ωβtht)) γtyt))) )} Hence we obtain Ωyt)Φtht)))[Ωyt)yt)+Ωyt)yt)] ΩΦtht))yt))+ΩΦtht))yt)) Ω yt) Φt ht))) h h Ωyt)Φtht)))[Ωyt)Φtht))+Ωyt)yt)] ΩΦtht))yt))+Ωyt)yt)) Ωyt)yt))[ΩΦtht))yt))+Ωyt)yt)] Ωyt) yt))+ωyt)φtht))) Ω yt)yt))[ωφtht))yt)) +Ωβtht))yt)] Ωyt)yt))+Ωyt) yt)) )} 1 Ω yt) Φt ht))) 1 Ω yt) Φt ht))) )} Ω yt) Φt ht))) Ω yt) Φt ht))) a contradiction since [01). And thus we have Φt ht)) = yt) for t R. hence we have Φt ht)) = βt ht)) = yt) for t R. 2.11) Since the pair Φ β) is weakly compatible we have Φ t βt ht))) = β t Φt ht))) that is Φt yt)) = βt yt)). 2.12)
15 Common Random Fixed Point Theorems under Contraction of rational Type 3717 From condition E3) and 2.10) we have Ω yt) Φt yt))) = Ω αt yt)) Φt yt))) Ωβtyt))Φtyt)))[Ωαtyt))γtyt))+Ωβtyt))γtyt))] ΩΦtyt))βtyt)))+ΩΦtyt))αtyt))) Ωαtyt))Φtyt)))[Ωβtyt))Φtyt))+Ωαtyt))γtyt))] ΩΦtyt))αtyt)))+Ωγtyt))βtyt))) Ωβtyt))γtyt)))[ΩΦtyt))βtyt)))+Ωγ tyt))αtyt))] Ωβtyt))γtyt)))+Ωαtyt))Φtyt))) Ω γtyt))βtyt)))[ωφtyt))βtyt)) +ΩΦtyt))αtyt))] Ωαtyt))γtyt))+Ωβtyt))γtyt))) )} Ωβtyt))Φtyt)))[Ωyt)yt))+ΩΦtyt))yt))] ΩΦtyt))Φtyt)))+ΩΦtyt))yt)) Ωyt)Φtyt)))[ΩΦtyt))Φtyt))+Ωyt)yt)] ΩΦtyt))yt))+Ωyt)Φtyt))) ΩΦtyt))yt))[ΩΦtyt))Φtyt)))+Ωyt)yt))] ΩΦtyt)) yt))+ωyt)φtyt))) Ωyt)Φtyt)))[ΩΦtyt))Φtyt)) +ΩΦtyt))yt)] Ωyt)yt)+ΩΦtyt)) yt)) )} [using 2.10) and 2.12)] Ω yt) Φt yt))) ) } Ω yt) Φt yt))) Ω yt) Φt yt))) a contradiction since [01). We get yt) = Φt yt)) = βt yt)) for t R 2.13) Using 2.10) and 2.13) we have yt)= Φt yt)) = βt yt))= αt yt)) = γt yt)) t R that is yt) is a common random fixed point of Φ β α and γ.
16 3718 Nisha Sharma et al For uniqueness let zt) X be another common random fixed point of Φ β α and γ using condition E3) Ωyt) zt)) = Ω αt yt)) Φt zt))) Ωβtzt))Φtzt)))[Ωαtyt))γtyt))+Ωβtzt))γtyt))] ΩΦtzt))βtzt)))+ΩΦtzt))αtyt))) Ωαtyt))Φtzt)))[Ωβtzt))Φtzt))+Ωαtyt))γtyt))] ΩΦtzt))αtyt)))+Ωγtyt))βtzt))) Ωβtzt))γtyt)))[ΩΦtzt))βtzt)))+Ωγ tyt))αtyt))] Ωβtzt)) γtyt)))+ωαtyt))φtzt))) Ω γtyt))βtzt)))[ωφtzt))βtzt)) +ΩΦtzt))αtyt))] Ωαtyt))γtyt))+Ωβtzt)) γtyt))) )} Ωzt)zt))[Ωyt)yt)+Ωzt)yt)] Ωzt)zt))+Ωzt)yt)) Ωyt)zt))[Ωzt)zt))+Ωyt)yt)] Ωzt))yt))+Ωyt)zt)) Ωzt)yt))[Ωzt)zt))+Ωyt)yt)] Ωzt) yt))+ωyt)zt)) Ω yt)zt))[ωzt)zt)) +Ωzt)yt)] Ωyt)yt)+Ωzt) yt)) )} Ωyt) zt)) Ω yt) zt)) which is a contradiction since [01). Which gives yt) = zt) for t R. Hence Φ β α and γ have unique common fixed point. Now suppose that α is continuous then we have α t αt θ 2n t))) αt yt))α t γt θ 2n t))) αt yt)) 2.14) Since the pairs α γ is compatible 2.6) implies that γ t αt θ 2n t))) αt yt)) 2.15) Using E3) we have Ω α t αt θ 2n t))) Φt θ 2n+1 t)))
17 Common Random Fixed Point Theorems under Contraction of rational Type 3719 Ω βt θ 2n+1 t)) Φt θ 2n+1 t))). [Ωαtαt θ 2n t)))γtαt θ 2n t)))+ωβt θ 2n+1 t))γtαt θ 2n t)))] ΩΦt θ 2n+1 t))βt θ 2n+1 t)))+ωφt θ 2n+1 t))αtαt θ 2n t)))) Ω α t αt θ 2n t))) Φt θ 2n+1 t))). [Ωβt θ 2n+1 t))φt θ 2n+1 t))+ωαtαt θ 2n t)))γtαt θ 2n t)))] ΩΦt θ 2n+1 t))αtαt θ 2n t))))+ωγtαt θ 2n t)))βt θ 2n+1 t))) Ω βt θ 2n+1 t)) γ t αt θ 2n t)))). [ΩΦt θ 2n+1 t))βt θ 2n+1 t)))+ωγ tαt θ 2n t)))αtαt θ 2n t)))] Ωβt θ 2n+1 t))γtαt θ 2n t))))+ωαtαt θ 2n t)))φt θ 2n+1 t))) Ω γ t αt θ 2n t))) βt θ 2n+1 t))). [ΩΦt θ 2n+1 t))βt θ 2n+1 t)) +ΩΦt θ 2n+1 t))αtαt θ 2n t)))] Ωαtαt θ 2n t)))γtαt θ 2n t)))+ωβt θ 2n+1 t))γtαt θ 2n t)))) )} Letting n approaches to infinity we have Ω αt yt)) yt)) Ωyt)yt))[Ωαtyt))αtyt))+Ωyt)αtyt)))] Ωyt)yt))+Ωyt)αtyt))) Ωαtyt))yt))[Ωyt)yt))+Ωαtyt))αtyt)))] Ωyt)αtyt)))+Ωαtyt))yt)) Ωyt)αtyt)))[Ωyt)yt))+Ωαtyt))αtyt))] Ωyt) αtyt)))+ωαtyt))yt)) Ω αtyt))yt))[ωyt)yt)) +Ωyt)αtyt))] Ωαtyt))αtyt)))+Ωyt)αtyt))) )} [using 2.14) and and 2.15) ] Ω αt yt)) yt)) Ω αt yt)) yt)) which is a contradiction since [01). Which gives αt yt)) = yt) for t R. 2.16) Since yt)= αt yt)) αt X) βt X) there exists ht) X such that yt) = βt ht)) for t R.
18 3720 Nisha Sharma et al Using condition E3) we obtain Ω α t αt θ 2n t))) Φt ht))) Ωβtht))Φtht)))[Ωαtαt θ 2n t)))γtαt θ 2n t)))+ωβtht))γtαt θ 2n t)))] ΩΦtht))βtht)))+ΩΦtht))αtαt θ 2n t)))) Ωαtαt θ 2n t)))φtht)))[ωβtht))φtht))+ωαtαt θ 2n t)))γtαt θ 2n t)))] ΩΦtht))αtαt θ 2n t))))+ωγtαt θ 2n t)))βtht))) Ωβtht))γtαt θ 2n t))))[ωφtht))βtht)))+ωγ tαt θ 2n t)))αtαt θ 2n t)))] Ωβtht)) γtαt θ 2n t))))+ωαtαt θ 2n t)))φtht))) Ω γtαt θ 2n t)))βtht)))[ωφtyt))βtht)) +ΩΦtht))αtαt θ 2n t)))] Ωαtαt θ 2n t)))γtαt θ 2n t)))+ωβtht)) γtαt θ 2n t)))) )} letting n approaches to infinity Ωyt)Φtht)))[Ωyt)yt)+Ωyt)yt))] ΩΦtht))yt))+ΩΦtht))yt))) Ω yt) Φt ht))) Ωyt)Φtht)))[Ωyt)Φtht))+Ωyt)yt))] ΩΦtht))yt))+Ωyt))yt)) Ωyt)yt))[ΩΦtht))yt))+Ωyt)yt))] Ωyt) yt)))+ωyt)φtht))) Ω yt)yt))[ωφtyt))yt)) +ΩΦtht))yt)] Ωyt)yt))+Ωyt) yt)) )} 1 Ω yt) Φt ht))) 1 ΩΦt ht)) ht) ) } [using 2.14) and 2.15)] Ω yt) Φt ht))) Ω yt) Φt ht))) which is a contradiction since [01). Φt ht)) = yt). Hence Φt ht)) = βt ht)) = yt )for t R. Since the pair Φ β) is weakly compatible the we have Φ t βt ht))) = β t Φt ht))) i. e. Φt yt)) = βt yt)) 2.17)
19 Common Random Fixed Point Theorems under Contraction of rational Type 3721 from condition E3) and 2.17) we have Ω αt θ 2n t)) Φt yt))) Ωβtyt))Φtyt)))[Ωαt θ 2n t))γt θ 2n t))+ωβtyt))γt θ 2n t))] ΩΦtyt))βtyt)))+ΩΦtyt))αt θ 2n t))) Ωαt θ 2n t))φtyt)))[ωβtyt))φtyt))+ωαt θ 2n t))γt θ 2n t))] ΩΦtyt))αt θ 2n t)))+ωγt θ 2n t))βtyt))) Ωβtyt))γt θ 2n t)))[ωφtyt))βtyt)))+ωγ t θ 2n t))αt θ 2n t))] Ωβtyt)) γt θ 2n t)))+ωαt θ 2n t))φtyt))) Ω γt θ 2n t))βtyt)))[ωφtyt))βtyt)) +ΩΦtyt))αt θ 2n t))] Ωαt θ 2n t))γt θ 2n t))+ωβtyt)) γt θ 2n t))) )} Taking n approaches to infinity we have ΩΦtyt))Φtyt)))[Ωyt)yt))+ΩΦtyt))yt))] ΩΦtyt))Φtyt)))+ΩΦtyt))yt)) Ωyt)Φtyt)))[ΩΦtyt))Φtyt))+Ωyt)yt))] ΩΦtyt))yt))+Ωyt)Φtyt))) ΩΦtyt))yt))[ΩΦtyt))Φtyt)))+Ωyt)yt))] ΩΦtyt)) yt))+ωyt)φtyt))) Ω yt)φtyt)))[ωφtyt))φtyt)) +ΩΦtyt))yt))] Ωyt)yt))+ΩΦtyt)) yt)) )} [using 2.17] Ω yt) Φt yt))) Ω yt) Φt yt))) which is a contradiction since [01). Hence we have yt) = Φt yt)) for t R. Using 2.17) we have Φt yt)) = βt yt)) = yt). 2.18) Since yt) = Φt yt)) Φt X) γt X) there exists gt) X such that γt gt)) = yt) for t R 2.19) Again using E3) we have Ω αt gt)) yt))= Ω αt gt)) Φt yt))
20 3722 Nisha Sharma et al Ωβtyt))Φtyt)))[Ωαtgt))γtgt))+Ωβtyt))γtgt))] ΩΦtyt))βtyt)))+ΩΦtyt))αtgt))) Ωαtgt))Φtyt)))[Ωβtyt))Φtyt))+Ωαtgt))γtgt))] ΩΦtyt))αtgt)))+Ωγtgt))βtyt))) Ωβtyt))γtgt)))[ΩΦtyt))βtyt)))+Ωγ tgt))αtgt))] Ωβtyt)) γtgt)))+ωαtgt))φtyt))) Ω γtgt))βtyt)))[ωφtyt))βtyt)) +ΩΦtyt))αtgt))] Ωαtgt))γtgt))+Ωβtyt)) γtgt))) )} Ωyt)yt))[Ωαtgt))yt))+Ωyt)yt)] Ωyt)yt))+Ωyt)αtgt))) Ωαtgt))yt))[Ωyt)yt))+Ωαtgt))yt))] Ωyt)αtgt)))+Ωyt)yt)) Ωyt)yt))[Ωyt)yt))+Ωyt)αtgt))] Ωyt) yt))+ωαtgt))yt)) Ω yt)yt))[ωyt)yt))+ωyt)αtgt))] Ωαtgt))yt))+Ωyt) yt)) )} [using 2.18) and 2.19)] Ω αt gt)) yt)) Ω αt gt)) yt)) which is a contradiction since [01). Hence αt Φt)) = yt). Using 2.19) we have αt gt)) = γt gt))=yt) for t R 2.20) Since the pair α γ) is compatible then α t γt gt))) = γ t αt gt))) i.e. t yt)) = γt yt)). Hence using 2.18) and 2.20) we obtain αt yt)) = γt yt)) = Φt yt)) = βt yt)) = yt) for t R that is yt) is the common random fixed point of α γ Φ and β. Uniqueness Let zt) be another fixed point Ωyt) zt)) = Ω αt yt)) Φt zt)))
21 Common Random Fixed Point Theorems under Contraction of rational Type 3723 Ωβtzt))Φtzt)))[Ωαtyt))γtyt))+Ωβtzt))γtyt))] ΩΦtzt))βtzt)))+ΩΦtzt))αtyt))) Ωαtyt))Φtzt)))[Ωβtzt))Φtzt))+Ωαtyt))γtyt))] ΩΦtzt))αtyt)))+Ωγtyt))βtzt))) Ωβtzt))γtyt)))[ΩΦtzt))βtzt)))+Ωγ tyt))αtyt))] Ωβtzt)) γtyt)))+ωαtyt))φtzt))) Ω γtyt))βtzt)))[ωφtzt))βtzt)) +ΩΦtzt))αtyt))] Ωαtyt))γtyt))+Ωβtzt)) γtyt))) )} Ωzt)zt))[Ωyt)yt))+Ωzt)yt)] Ωzt)zt))+Ωzt)yt)) Ωyt)zt))[Ωzt)zt))+Ωyt)yt))] Ωzt)yt))+Ωyt)zt)) Ωzt)yt))[Ωzt)zt))+Ωyt)yt)] Ωzt) yt))+ωyt)zt)) Ω yt)zt))[ωzt)zt)) +Ωzt)yt)] Ωyt)yt))+Ωzt)) yt)) )} Ωyt) zt)) Ω β yt) zt)) which is a contradiction since [01). yt) = zt). This completes the proof. Similarly we can prove the result using condition E5). In theorem if we put β=γ = 1 the identity mapping) we obtain the corollary Corollary 2.3. Let us consider complete multiplicative metric space X Ω) and functions Φ α: R x X X be two functions satisfying the following conditions E6) Ω αt xt)) Φt yt))) Ωyt)Φtyt)))[Ωαtxt))xt))+Ωyt)xt)] ΩΦtyt))yt))+ΩΦtyt))αtxt))) Ωαtxt))Φtyt)))[Ωyt)Φtyt))+Ωαtxt))xt))] ΩΦtyt))αtxt)))+Ωxt)yt)) Ωyt)xt))[ΩΦtyt))yt))+Ωxt)αtxt))] Ωyt) xt))+ωαtxt))φtyt))) Ω xt)yt))[ωφtyt))yt)) +ΩΦtyt))αtxt))] Ωαtxt))xt))+Ωyt) xt)) )}
22 3724 Nisha Sharma et al For all x y X where t R and [01). Then α and Φ have a unique random fixed point The proof of corollary 2.4 is immediate by assuming R to be a singleton set in Theorem 2.2. Corollary 2.4. Let us consider complete multiplicative metric space X Ω) and a function Φ α γ and β: X X be four function satisfying the following conditions E7) αx) β X) ΦX) γx) E8) Ωαx) Φy)) Ωβy)Φy))[Ωαx)γx))+Ωβy)γx))] ΩΦy)βy))+ΩΦy)αx)) Ωαx)Φy))[Ωβy)Φy))+Ωαx)γx))] ΩΦy)αx))+Ωγx)βy)) Ωβy)γx))[ΩΦy)βy))+Ωγx)αx)] Ωβy) γx)+ωαx)φy)) Ω γx)βy))[ωφy)βy)) +ΩΦy)αx)] Ωαx)γx))+Ωβy) γx)) )} For all x y X where and [01). Then Φ α γ and β have a unique random fixed point is and only if one of the following conditions are satisfied: E9) either α or γ is continuous the pair α γ is compatible and the pair βφ) is weakly compatible. E10) either Φ or β is continuous the pair Φ β is compatible and the pair α γ ) is weakly compatible. REFERENCES [1] S.M Kang P. Kumar S. Kumar P. nagpal and S.K. garg Common fixed points for compatible mappings and its variants in multiplicative metric spaces Int. j. Pure apll. Math ) [2] M. Abbas M. De la sen and T. Nazir Common fixed points of generalized rational type cocyclic mappings in multiplicative metric spaces discrete Dyn. Nat. Soc ) 1-10 [3] S. Banach Sur les operations abstraits et leur application aux equations integrals Fund. Maths ) [4] A.E. Bashirov E.M. Kurplanara and A. Ozyapici multiplicative calculus and its applications J. Math. Anal. App ) [5] I. Beg Random fixed points of random operators satisfying semi contractivity conditions Math. Japan )
23 Common Random Fixed Point Theorems under Contraction of rational Type 3725 [6] I. Beg Minimal displacement of Random variables under lipschitz random maps Topol. Methods Nonlinear Anal ) [7] I. Beg approximation of random fixed points in normed spaces Normed Linear Anal: Theory Methods and Analysis ) [8] I. Beg and N. Shahzad random Fixed Point theorems on product spaces J. app. Math. Stoch. Anal ) Nonlinear Analysis: Theory Methods and Applications ) [9] I. Beg and N. Shahzad Random Fixed Point of Random Multivalued operators on polish spaces Nonlinear Analysis: Theory Methods and Applications ) [10] I. Beg and N. Shahzad Random Fixed Point Theorems for non expansive and contractive-type random operators on Banach Space. J. Appl. Math. Stoch. Anal ) [11] I. Beg and N. Shahzad Random approximations and Random fixed point theorems J. Appl. Math. Stoch. Anal ) [12] I. Beg and N. Shahzad common Random Fixed Points of Random multivalued Operators on metric spaces Boll. Un. Mat. Ital. 9-A 1995) [13] I. Beg and N. Shahzad common Random Fixed Points of weakly inward operators in conical shells. J. Anal. Math. Stoch. Anal ) [14] A.T Bharucha-reid Random Integral Equations Academic Press new York [15] S.M Kang P. Kumar S. Kumar Chahn Yong Jung Common fixed points theorems in multiplicative metric space Int. j. Math. Anal ) [16] M. Ozavsar and A.C. Cevikal fixed point of Multiplicative contraction mappings on multiplicative metric space ArXiv: v1 [math.g.m] 2012) 14. [17] M. Sarwar and Badshah-e-Rome Some fixed point Theorems in multiplicative metric spaces ArXiv: v2 [math.gm] 2014) 1-19 [18] H.K. Xu Some Random fixed point Theorems for condensing and nonexpansive operators Proc. Amer. Math. Soc ) [19] Kamal Kumar Nisha Sharma Rajeev Jha Arti Mishra and Manoj Kumar Weak Compatibility and Related Fixed Point Theorem for Six Maps in Multiplicative Metric Space. Turkish Journal of Analysis and Number Theory vol. 4 no ): Doi: /tjant [20] Nisha Sharma Kamal Kumar Sheetal Sharma Rajeev Jha Rational Contractive Condition in Multiplicative Metric Space and Common Fixed
24 3726 Nisha Sharma et al Point Theorem ISSNOnline): ISSN Print): International Journal of Innovative Research in Science Engineering and Technology Vol. 5 Issue 6 June 2016 Copyright to IJIRSET DOI: /IJIRSET
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