Quadruple Coincidence Point Results in Partially Ordered Metric Spaces

Transcription

1 BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN p ISSN o wwwimviblorg / JOURNALS / BULLETIN OF IMVI Vol doi: /BIMVI10105G Former BULLETIN OF SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN o ISSN X p Quadruple Coincidence Point Results in Partially Ordered Metric Spaces Vishal Gupta and Raman Deep Abstract In this paper we prove quadruple coincidence point theorems for mixed g-monotone mappings satisfying the compatibility property in partially ordered metric space 1 Introduction The fixed point theorems in metric spaces are playing a major role to construct methods in mathematics and to solve problems in applied mathematics and sciences The existence of a fixed point in partially ordered metric and G-metric spaces has been considered in [1]-[] and [5]-[10] The notion of coupled fixed points have been introduced by Guo and Laksmikantham [3] in connection with monotone operators which is further generalized by Choudhury [1] Bessem Samet [2] and many more Berinde and Borcut [7] introduced the concept of triple fixed point and proved some related theorems The concept of quadruple fixed point is considered by Erdal Karapinar [] Mustafa [10] Here our aim is to prove a unique quadruple coincidence point theorem for g-monotone mappings satisfying the compatibility property in partially ordered metric space 2 Preliminaries Definition 21 [10] Let X be partially ordered set and F : X X We say that F has the mixed g-monotone property if for any x y z w X x 1 x 2 X gx 1 gx 2 F x 1 y z w F x 2 y z w y 1 y 2 X gy 1 gy 2 F x y 2 z w F x y 1 z w z 1 z 2 X gz 1 gz 2 F x y z 1 w F x y z 2 w 2010 Mathematics Subject Classification 5H25 7H10 Key words and phrases Quadruple coincidence point partially ordered sets metric spaces g-monotone property 5

2 6 GUPTA AND DEEP w 1 w 2 X gw 1 gw 2 F x y z w 2 F x y z w 1 Definition 22 [10] An element x y z w X is called a quadruple coincidence point of F : X X and g : X X if the following conditions are satisfied F x y z w = g x F y z w x = g y F z w x y = g z F w x y z = g w Definition 23 [5] Let X d be a metric space and {x n } X The mappings f g : X X are said to be compatible if lim d fgx n gfx n = 0 n whenever {x n } is a sequence in X such that for some x X such that lim fx n = lim gx n = x n n Now we define a mapping d : X X X on X d by: d x y z w u v h l = d x u y v z h w l which will be denoted for convenience by d Also let ψ denotes all functions ϕ : [0 [0 which satisfy: 1 ϕ is non-decreasing 2 ϕ t < t for all t > 0 3 lim r t +ϕ r < t for all t > 0 3 Main Result Theorem 31 Let X be a partially ordered set and X d be a complete metric space Let F : X X be a mapping having the mixed g-monotone property on X such that there exist four elements x 0 y 0 z 0 w 0 X with 31 gx 0 F x 0 y 0 z 0 w 0 gy 0 F y 0 z 0 w 0 x 0 gz 0 F z 0 w 0 x 0 y 0 and gw 0 F w 0 x 0 y 0 z 0 Suppose there exist ϕ ψ M 0 such that 32 d F x y z w F u v h l d gx gu gy gv gz gh gw gl ϕ x y z u v h l X with gx gu gy gv gz gh and gw gl Also let F X g X and F g being continuous monotone increasing and compatible mappings Then F and g have quadruple coincidence point in X Proof Suppose x 0 y 0 z 0 w 0 X be given by 31 As F X g X therefore we can choose x 1 y 1 z 1 w 1 X such that gx 1 = F x 0 y 0 z 0 w 0 gy 1 = F y 0 z 0 w 0 x 0 gz 1 = F z 0 w 0 x 0 y 0 gw 1 = F w 0 x 0 y 0 z 0 Then we have gx 0 gx 1 gy 0 gy 1 gz 0 gz 1 and gw 0 gw 1 In the same way we have gx 2 = F x 1 y 1 z 1 w 1 gy 2 = F y 1 z 1 w 1 x 1 gz 2 = F z 1 w 1 x 1 y 1 gw 2 = F w 1 x 1 y 1 z 1

3 QUADRUPLE COINCIDENCE POINT 7 Since F has mixed g-monotone property therefore we have gx 0 gx 1 gx 2 gy 2 gy 1 gy 0 gz 0 gz 1 gz 2 and gw 2 gw 1 gw 0 Continuing this process we can construct four sequences {gx n } {gy n } {gz n } and {gw n } such that gx n = F x n 1 y n 1 z n 1 w n 1 gx n+1 = F x n y n z n w n gy n+1 = F y n z n w n x n gy n = F y n 1 z n 1 w n 1 x n 1 gz n = F z n 1 w n 1 x n 1 y n 1 gz n+1 = F z n w n x n y n gw n+1 = F w n x n y n z n gw n = F w n 1 x n 1 y n 1 z n 1 Now for any n N we have d gx n+1 gx n = d F x n y n z n w n F x n 1 y n 1 z n 1 w n 1 d gxn gx n 1 gy n gy n 1 gz n gz n 1 gw n gw n 1 33 ϕ d gy n gy n+1 = d F y n 1 z n 1 w n 1 x n 1 F y n z n w n x n d gyn 1 gy n gz n 1 gz n gw n 1 gw n gx n 1 gx n 3 ϕ d gz n+1 gz n = d F z n w n x n y n F z n 1 w n 1 x n 1 y n 1 d gzn gz n 1 gw n gw n 1 gx n gx n 1 gy n gy n 1 35 ϕ d gw n gw n+1 = d F w n 1 x n 1 y n 1 z n 1 F w n x n y n z n d gwn 1 gw n gx n 1 gx n gy n 1 gy n gz n 1 gz n 36 ϕ Due to equations we obtain 37 d gx n+1 gx n gy n gy n+1 gz n+1 gz n gw n gw n+1 d gxn gx n+1 gy n gy n 1 gz n gz n 1 gw n gw n 1 ϕ Let d n = d gx n gx n+1 gy n gy n+1 + d gz n gz n+1 gw n gw n+1 Then equation 37 implies d n ϕ dn 1 d n < d n 1 Thus d n is decreasing sequence Therefore there is some d > 0 such that 3 lim n d n = d Now we claim that d = 0 If not then taking n of both sides of equation 36 we get d lim n ϕ d n < d which is a contradiction Hence d = 0 that is 39 lim n [d gx n gx n+1 gy n gy n+1 gz n gz n+1 gw n gw n+1 ] = 0 Now we will prove that {gx n } {gy n } {gz n } and {gw n } are Cauchy sequences Suppose to contrary that at least one of these sequences is not a Cauchy sequence

4 GUPTA AND DEEP Then there exist an ϵ > 0 for which we can find subsequences of integers m k and n k with n k > m k > k such that [ d gxnk gx mk gynk gy mk d gz nk gz mk gwnk gw mk ] ϵ Further corresponding to m k we can choose n k in such a way that it is the smallest integer with n k > m k and satisfying equation 310 then [ d gxnk 1 gx mk gynk 1 gy mk d gz nk 1 gz mk gwnk 1 gw mk ] < ϵ From equation and applying triangle inequality we have ϵ r k = d gx nk gx mk gynk gy mk gznk gz mk gw nk gw mk d gx nk gx nk 1 gynk gy nk 1 gz nk gz nk 1 gwnk gw nk 1 Letting k in above inequality with keeping in mind equation 3 we conclude that 312 lim k r k = ϵ Again employing triangle inequality we obtain r k = d 313 gx nk gx mk gynk gy mk gznk gz mk gw nk gw mk 31 d nk mk gx nk+1 gx mk+1 gynk+1 gy mk+1 gz nk+1 gz mk+1 gwnk+1 gw mk+1 As n k > m k we have gx nk gx mk gy nk gy mk gz nk gz mk and gw nk gw mk Using equation 32 we obtain 315 d rk gx nk+1 gx mk+1 ϕ Similarly 316 d gy mk+1 gy nk+1 = ϕ rk 317 d gz nk+1 gz mk+1 = ϕ rk 31 d gw mk+1 gw nk+1 = ϕ rk

5 QUADRUPLE COINCIDENCE POINT 9 Due to equation and keeping in view the property of function ϕ we get 319 d gx nk+1 gx mk+1 gymk+1 gy nk+1 + d gz nk+1 gz mk+1 gwmk+1 gw nk+1 < rk Hence from equation 31 and 31 we get r k < d nk mk + r k Taking k and using equation 39 we conclude r k < r k It is a contradiction Thus {gx n } {gy n } {gz n } and {gw n } are Cauchy sequences in X and since X is a complete metric space therefore there exist x y z w X such that 320 lim n F x n y n z n w n = lim n gx n = x 321 lim n F y n z n w n x n = lim n gy n = y 322 lim n F z n w n x n y n = lim n gz n = z 323 lim n F w n x n y n z n = lim n gw n = w Now as F and g are compatible mappings we have 32 lim n d g F x n y n z n w n F gx n gy n gz n gw n = lim n d g F y n z n w n x n F gy n gz n gw n gx n = lim n d g F z n w n x n y n F gz n gw n gx n gy n = lim n d g F w n x n y n z n F gw n gx n gy n gz n = 0 Since F is continuous for all n 0 we get d gx F gx n gy n gz n gw n d gx g F x n y n z n w n g F x n y n z n w n F gx n gy n gz n gw n On applying n and combining equation 31 and 322 we obtain F x y z w = gx F y z w x = gy F z w x y = gz and F w x y z = gw Hence we conclude that F and g have a quadruple coincidence point in X Theorem 32 In addition to the hypothesis of Theorem 31 suppose that for every x y z w x 1 y 1 z 1 w 1 in X there exists u v h l that is comparable to x y z w and x 1 y 1 z 1 w 1 then F and g have a unique quadruple coincidence point

6 50 GUPTA AND DEEP Proof From Theorem 31 the set of quadruple fixed points of F and g is nonempty Suppose x y z w and x 1 y 1 z 1 w 1 are quadruple coincidence points of F and g that is And gx = F x y z w gy = F y z w x gz = F z w x y gw = F w x y z and gx 1 = F x 1 y 1 z 1 w 1 gy 1 = F y 1 z 1 w 1 x 1 gz 1 = F z 1 w 1 x 1 y 1 gw 1 = F w 1 x 1 y 1 z 1 We shall show that gx = gx 1 gy = gy 1 gz = gz 1 and gw = gw 1 By assumption there exist u v h l X that is comparable to x y z w and x 1 y 1 z 1 w 1 Now we define sequences {gu n } {gv n } {gh n } and {gl n }as follows: u 0 = u v 0 = v h 0 = h l 0 = l gu n+1 = F u n v n h n l n gv n+1 = F v n h n l n u n gh n+1 = F h n l n u n v n and gl n+1 = F l n u n v n h n for all n N Since u v h l being comparable with x y z w we may assume that x y z w u v h l = u 0 v 0 h 0 l 0 Applying mathematical induction it is easy to prove that Due to equation 32 we obtain x y z w u n v n h n l n for all n N 32d gx gu n+1 = d F x y z w F u n v n h n l n d gx gun gy gv n gz gh n gw gl n ϕ Analogously d gvn gy gh n gz gl n gw gu n gx 329 d gv n+1 gy ϕ d gz ghn gw gl n gx gu n gy gv n 330 d gz gh n+1 ϕ d gw gln gx gu n gy gv n gz gh n 331 d gw gl n+1 ϕ On adding equation and using the property of function ϕ we have d gx gu n+1 gy gv n+1 gz gh n+1 gw gl n+1 d gx gun+1 gy gv n+1 gz gh n+1 gw gl n+1 ϕ or d gx gu n+1 gy gv n+1 gz gh n+1 gw gl n+1 < d gx gu n+1 gy gv n+1 gz gh n+1 gw gl n+1

7 QUADRUPLE COINCIDENCE POINT 51 Thus the sequence {d gx gu n gy gv n gz gh n gw gl n } is decreasing therefore there exist δ 0 such that 33 lim n [d gx gu n gy gv n gz gh n gw gl n ] = δ Suppose that δ > 0 taking limit as n in equation 330 we have ϕ δ 335 δ It is a contradiction Hence δ = 0 that is By this we obtain lim [d gx gu n gy gv n gz gh n gw gl n ] = 0 n 336 lim n d gx gu n = lim n d gy gv n = lim n d gz gh n = lim n d gw gl n In the same way it is easy to show that 337 lim d gx 1 gu n = lim d gy 1 gv n = lim d gz 1 gh n = lim d gw 1 gl n n n n n On account of equation 335 and 336 we have gx = gx 1 gy = gy 1 gz = gz 1 and gw = gw 1 Hence the result Example 31 Let R d be a complete metric space with the usual metric defined on R Consider g : X X and F : X X be defined as g x = 7 9 x y + z w x and F x y z w = Also suppose ϕ : [0 [0 be given by ϕ t = 6 7 t Now for all x y z u v h l X satisfying gx gu gv gy gz gh and gl gw the LHS of the condition of equation 31 is x y + z w d F x y z w F u v h l = d u v + h l = x y + z w Now the RHS of equation 32 becomes ϕ dgxgu+dgygv+dgzgh+dgwgl = u v + h l x u + y v + z h + w l we find that the hypothesis of equation 32 are satisfied Also is the unique quadruple fixed point of F and g

8 52 GUPTA AND DEEP References [1] B S Choudhury A Kundu A coupled coincidence point result in partially ordered metric spaces for compatible mappings Nonlinear Analysis: Theory Methods and Applications [2] B Samet C Vetro Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces Nonlinear Analysis: Theory Methods and Applications [3] D Guo V Lakshmikantham Coupled fixed points of nonlinear operators with applications Nonlinear Analysis: Theory Methods and Applications [] E Karapinar V Berinde Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces Banach J Math Anal [5] G Jungck Compatible mappings and common fixed points Internat J Math Math Sci [6] R P Agarwal M A El-Gebeily D O Regan Generalized contractions in partially ordered metric spaces Appl Anal [7] V Berinde M Borcut Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces Nonlinear Analysis: Theory Methods and Applications [] V Gupta R Deep Coupled Coincidence Point Result in Partially Ordered G-Metric Space for Compatible Mapping In: Advancements in the era of Multi - Disciplinary Systems Elsevier [9] V Lakshmikantham L Ciric Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces Nonlinear Analysis: Theory Methods and Applications [10] Z Mustafa H Aydi E Karapinar Mixed g-monotone property and quadruple fixed point theorems in partially ordered metric spaces Fixed Point Theory and Applications :71 pp1-19 Doi: 10116/ Received by editors ; revised version ; available online Department of Mathematics Maharishi Markandeshwar University Mullana Ambala Haryana India address: vishalgmn@gmailcom Department of Mathematics Maharishi Markandeshwar University Mullana Ambala Haryana India address: ramandeepvirk02@gmailcom