Coupled coincidence point results for probabilistic φ-contractions

Transcription

1 PURE MATHEMATICS RESEARCH ARTICLE Coupled coicidece poit results for probabilistic φ-cotractios Biayak S Choudhury 1, Pradyut Das 1 * P Saha 1 Received: 18 February 2016 Accepted: 29 May 2016 First Published: 01 July 2016 *Correspodig author: Pradyut Das, Departmet of Mathematics, Idia Istitute of Egieerig Sciece Techology, Shibpur, Howrah , West Begal, Idia pradyutdas747@gmailcom Reviewig editor: Hari M Srivastava, Uiversity of Victoria, Caada Additioal iformatio is available at the ed of the article Abstract: I this paper, we establish a ew coupled coicidece poit results i partially ordered probabilistic metric spaces by utilizig the Gauge fuctio We use the compatibility coditio betwee two mappigs We use mootoe mixed mootoe properties of fuctios with respect to the orderig Our mai result has several corollaries The mai result is supported with a example which shows that the corollaries are actually cotaied i our mai theorem The methodology is a combiatio of aalytic order theoretic approaches Subjects: Egieerig & Techology; Mathematics & Statistics; Physical Scieces; Scieces; Techology Keywords: Meger PM-space; Hadz ić type t-orm; φ-fuctio; partially ordered set; mixed mootoe property; Cauchy sequece; compatible mappigs; coupled coicidece poit AMS subject classificatios: 47H10; 54H25 1 Itroductio Probabilistic fixed poit theory has its origi i the work of Sehgal Bharucha-Reid i (1972 where they established a probabilistic versio of the famous Baach s cotractio mappig priciple The probabilistic cotractio mappig defied i the above-metioed work was exteded i may ways by the use of cotrol fuctios (Choudhury & Das, 2009; Ćirić, 2010; Fag, 2009; Fag, 2015; O Rega & Saadati, 2008; Xiao, Zhu, & Cao, 2011 Recetly such a extesio was doe by Fag i (2015 i which he itroduced a probabilistic φ- cotractio where φ is assumed to satisfy certai coditios makig it more geeral tha the class of Gauge fuctios which were used by some previous authors ABOUT THE AUTHORS Biayak S Choudhury is a professor of Mathematics at Idia Istitute of Egieerig Sciece Techology, Shibpur, West Begal, Idia His research iterests are i areas of pure mathematics, applied mathematics, theoretical physics i which he has published about 200 research articles He has also acted as Pricipal Ivestigators of several projects from Govermet Agecies of Idia Pradyut Das is a seior research studet pursuig his PhD work i the Departmet of Mathematics, Idia Istitute of Egieerig Sciece Techology, Shibpur, West Begal, Idia He has 20 research publicatios to his credit P Saha is a professor of Mathematics at Idia Istitute of Egieerig Sciece Techology, Shibpur, West Begal, Idia Her research areas are fuzzy fuctioal aalysis fuzzy logic PUBLIC INTEREST STATEMENT Probabilistic aalysis is a brach of mathematics which purports to systematically deal with ucertai situatios arisig i sciece, egieerig, ecoomics, fiace, may other areas of huma activities The preset article cotais ew results o a emergig area of research i probabilistic aalysis which has developed durig last five years The work is a part of both theoretical applied mathematics i which results are obtaied by utilizig mathematical cotrol fuctios The authors feel that there are large scopes of applicatios of these types of results deduced here 2016 The Author(s This ope access article is distributed uder a Creative Commos Attributio (CC-BY 40 licese Page 1 of 12

2 Meawhile coupled fixed poit results occupied a large place i metric fixed poit theory Although the cocept of coupled fixed poit was itroduced by Guo Lakshmikatham (1987, coupled fixed poit problems attracted large attetio oly after 2006 whe a coupled cotractio mappig theorem was proved by Bhaskar Lakshmikatham (2006 Several istaces of works o this topic are i (Bhaskar & Lakshmikatham, 2006; Choudhury & Kudu, 2010; Lakshmikatham & Ćirić, 2009; Luog & Thua, 2011; Mohiuddie & Alotaibi, 2012; Mursaleem, Mohiuddie, & Agarwal, 2012; Samet, 2010 A probabilistic coupled fixed poit result was first successfully itroduced by Hu & Ma, (2011 After that some works have followed i which coupled fixed coicidece poit results have bee established i probabilistic metric spaces (Choudhury & Das, 2014; Ćirić, Agarwal, & Samet, 2011; Doric, 2013 Our edeavor here is to establish a ew coupled coicidece poit results i probabilistic metric spaces by utilizig the Gauge fuctio used by Fag (2015 A compatibility coditio betwee mappigs has bee used Our mai result has several corollaries The mai result is supported with a example which shows that the corollaries the works which have bee exteded are actually cotaied i our mai theorem 2 Mathematical prelimiaries I this sectio, we discuss certai defiitios lemmas which will be ecessary for establishig the results of the ext sectio Throughout this paper (X, sts for a partially ordered set with partial order By x y we shall mea y x by x y we shall mea x y with x y Defiitio 21 (Schweizer & Sklar, 1983 A mappig F: R R + is called a distributio fuctio if it is o-decreasig left cotiuous with if F(t =0 sup F(t =1, where R is the set of real t R t R umbers R + deotes the set of all o-egative real umbers Defiitio 22 (HadŽĭć & Pap, 2001; Schweizer & Sklar, 1983 A biary operatio Δ:[0, 1] 2 [0, 1] is called a t-orm if the followig properties are satisfied: (i Δ is associative commutative, (ii Δ(a,1=a for all a [0, 1], (iii Δ(a, b Δ(c, d wheever a c b d, for all a, b, c, d [0, 1] Geeric examples of t-orm are Δ M (a, b =mi{a, b}, Δ P (a, b =ab etc Defiitio 23 (Schweizer & Sklar, 1983 A Meger space is a triplet (X, F, Δ, where X is a o empty set, F is a fuctio defied o X X to the set of distributio fuctios Δ is a t-orm, such that the followig are satisfied: (i F x,y (0 =0 for all x, y X, (ii F x,y (s =1 for all s < 0 if oly if x = y, (iii F x,y (s =F y,x (s for all s < 0, x, y X, (iv F x,y (u + v Δ(F x,z (u, F z,y (v for all u, v 0 x, y, z X Defiitio 24 (Schweizer & Sklar, 1983 Let (X, F, Δ be a Meger space Page 2 of 12

3 (i A sequece {x } X is said to be coverge to a poit x X if give ε>0, λ>0 we ca fid a positive iteger N ε,λ such that for all > N ε,λ F x,x(ε 1 λ (ii A sequece {x } is said to be a Cauchy sequece i X if give ε >0, λ>0 there exists a positive iteger N ε,λ such that F x,x m (ε 1 λ for all m, < N ε,λ (iii A Meger space (X, F, Δ is said to be complete if every Cauchy sequece is coverget i X (i (ii ca be equivaletly writte by replacig with > More ofte tha ot, they are writte i that way We have give them i the preset form for our coveiece i the proof of our results Defiitio 25 (Hadžić & Pap, 2001 A t-orm Δ be said to be a Hadz ić type t-orm if the family {Δ m } m>0 of its iterates defied for each t [0, 1] by Δ 0 (t =t, Δ 1 (t = Δ(t, t, i geeral, for all m > 1, Δ m (t =Δ(t, Δ m 1 (t is equi-cotiuous at t = 1, that is, give λ >0 there exists η(λ (0, 1 such that 1 t <η(λ Δ (m (t 1 λ for all m < 0 Lemma 26 (Chag, Cho, & Kag, 1994, p 24 Let (X, F, Δ be a Meger space with a cotiuous t- orm The, for every t > 0, x x, y y, imply lim if F x,y (t =F x,y (t I the followig lemma, we ote a property of a cotiuous fuctio Lemma 27 (Choudhury & Das, 2014 If f : R R is a cotiuous fuctio {a i,j } i=1, j = 1, 2,, are umber of sequeces such that lim if i a ik = a k for all k l for some l {a ii } i=1 is bouded The lim if f (a, a,, a =f(a, a i i1 i2 i 1 2, lim if a,, a i il Let (X, be a partially ordered set g: X X be a mappig The mappig g is said to be o-decreasig if, for all x 1, x 2 X, x 1 x 2 implies g(x 1 g(x 2 o-icreasig if, for all x 1, x 2 X, x 1 x 2 implies g(x 1 g(x 2 (Bhaskar & Lakshmikatham, 2006 Defiitio 28 (Bhaskar & Lakshmikatham, 2006 Let (X, be a partially ordered set G: X X X be a mappig The mappig G is said to have the mixed mootoe property if G is o-decreasig i its first argumet is o-icreasig i its secod argumet, that is, if, for all x 1, x 2 X, x 1 x 2 implies G(x 1, y G(x 2, y for fixed y X if, for all y 1, y 2 X, y 1 y 2 implies G(x, y 1 G(x, y 2, for fixed x X Defiitio 29 (Lakshmikatham & Ćirić, 2009 Let (X, be a partially ordered set G: X X X g: X X be two mappigs The mappig G is said to have the mixed g-mootoe property if G is mootoe g-o-decreasig i its first argumet is mootoe g-o-icreasig i its secod argumet, that is, if, for all x 1, x 2 X, gx 1 gx 2 implies G(x 1, y G(x 2, y for fixed y X if, for all y 1, y 2 X, gy 1 gy 2 implies G(x, y 1 G(x, y 2, for fixed x X Defiitio 210 (Bhaskar & Lakshmikatham, 2006 Let X be a oempty set A elemet (x, y X X is called a coupled fixed poit of the mappig G: X X X if Page 3 of 12

4 G(x, y =x, G(y, x =y Further Lakshmikatham Ćirić (2009 had itroduced the cocept of coupled coicidece poit Defiitio 211 (Lakshmikatham & Ćirić, 2009 Let X be a oempty set A elemet (x, y X X is called a coupled coicidece poit of the mappigs g: X X G : X X X if G(x, y =gx, G(y, x =gy Defiitio 212 (Lakshmikatham & Ćirić, 2009 Let X be a oempty set The mappigs g: X X G: X X X are commutig if, for all x, y X, gg(x, y =G(gx, gy A compatible pair {g, G} i a metric space (X, d, where g: X X G : X X X, was defied by Choudhury Kudu i (2010 Defiitio 213 (Choudhury & Kudu, 2010 Let (X, d be a metric space The mappigs g G where g: X X G: X X X, are said to be compatible if lim d(g(g(x, y, G(gx, gy = 0 lim d(g(g(y, x, G(gy, gx = 0, wheever {x } {y } are sequeces i X such that lim G(x, y = lim gx = x lim G(y, x = lim gy = y The followig is the defiitio of compatible pairs i Meger spaces Defiitio 214 (Doric, 2013 Let (X, F, Δ be a Meger space The mappigs g G where g: X X G: X X X, are said to be compatible if, for all t > 0, lim F g(g(x,y,g(gx,gy (t =1 lim F g(g(y,x,g(gy,gx (t =1, wheever {x } {y } are sequeces i X such that lim G(x, y = lim gx = x lim G(y, x = lim gy = y Out of several types of gauge fuctios utilized i extedig probabilistic cotractio mappig priciple, we cosider the followig two classes Defiitio 215 (Jachymski, 2010 the followig coditio: Let Φ deote the class of all fuctios φ : R + R + satisfyig lim φ (t =0 for all t < 0 Defiitio 216 (Fag, 2015 followig coditio: Let Φ w deote the class of all fuctios φ: R + R + satisfyig the Page 4 of 12

5 for each t > 0 there exists r t such that lim φ (r =0 Here Φ is a proper subclass of Φ w [see Fag (2015] Lemma 217 (Fag, 2015 Let φ Φ w, the for each t > 0 there exists r t such that φ(r < t 3 Mai results Theorem 31 Let (X, F, Δ be a complete Meger space where Δ is a cotiuous Hadz ić type t-orm o which a partial orderig is defied Let g: X X G: X X X be two mappigs such that G has the mixed g-mootoe property Let there exist φ Φ w such that F G(x,y,G(u,v (φ(t [F gx,gu (tf gy,gv (t] 1 2, (31 for all t > 0, x, y, u, v X with gx gu gy gv Let g be cotiuous, mootoic icreasig, G(X X g(x such that {g, G} is a compatible pair Also suppose either (a G is cotiuous or (b X has the followig properties: (i if a o-decreasig sequece {x } x, thex x for all 0, (32 (ii if a o-icreasig sequece {y } y, they y for all 0 (33 If there exist x 0 X such that gx 0 G(x 0 gy 0 G(y 0, the g G have a coupled coicidece poit i X, that is, there exist x, y X such that gx = G(x, y gy = G(y, x Proof By a coditio of the theorem, there exist x 0 X such that gx 0 G(x 0 gy 0 G(y 0 Sice G(X X g(x, it is possible to defie the sequeces {x } {y } i X as follows: gx 1 = G(x 0 gy 1 = G(y 0 gx 2 = G(x 1, y 1 gy 2 = G(y 1, x 1, i geeral, for all 0, gx +1 = G(x, y gy +1 = G(y, x (34 Next, for all 0, we prove that gx gx +1 (35 gy gy +1 (36 Sice gx 0 G(x 0 gy 0 G(y 0, i view of the facts that gx 1 = G(x 0 gy 1 = G(y 0, we have gx 0 gx 1 gy 0 gy 1 Therefore, (35 (36 hold for = 0 Page 5 of 12

6 Let (35 (36 hold for some = m, that is, gx m gx m+1 gy m gy m+1 As G has the mixed g-mootoe property, from (34, we get gx m+1 = G(x m, y m G(x m+1, y m G(y m+1, x m G(y m, x m =gy m+1 (37 Also, for the same reaso, we have gx m+2 = G(x m+1, y m+1 G(x m+1, y m G(y m+1, x m G(y m+1, x m+1 =gy m+2 (38 Now, (37 (38, imply gx m+1 gx m+2 gy m+1 gy m+2 The, by iductio, it follows that (35 (36 hold for all 0 Now, for all t > 0, 1, we have F gx,gx +1 (φ(t = F G(x 1,y 1,G(x,y (φ(t (by(34 [F gx 1,gx (tf gy,gy 1 (t] 1 2 (by (31, (35 (36 (39 Similarly, we have for all t > 0 F gy,gy (φ(t = F +1 G(y 1,x 1,G(y,x (φ(t ( by (34 [F gy 1,gy (tf gx,gx 1 (t] 1 2 [ by (39 (310] (310 Let P (t =[F gx 1,gx (tf gy,gy 1 (t] 1 2 (311 The P (tp (t F gx,gx +1 (φ(tf gy,gy +1 (φ(t, which implies that [P (t] 2 [P +1 (φ(t] 2, that is, [P +1 (φ(t] [P (t] By repeated applicatio of the above iequality, usig (39 (310, respectively, for all t > 1, > 1, we have that, F gx,gx +1 (φ (t P (φ 1 (t P 1 (t =[F gx0,gx 1 (tf gy0,gy 1 (t] 1 2, F gy,gy +1 (φ (t P (φ 1 (t P 1 (t =[F gx0,gx 1 (tf gy0,gy 1 (t] 1 2 Now we prove that lim F gx,gx (t =1 for all t < 0 limf +1 gy,gy (t =1, for all t < 0 +1 Sice F gx0,gx 1 (t 1, F gy0,gy 1 (t 1 as t, for ay ε (0, 1] there exists t 1 > 0 such that F gx0,gx 1 (t 1 > 1 ε F gy0,gy 1 (t 1 > 1 ε Sice φ Φ w, there exists t 0 t 1 such that lim φ (t 0 =1 Thus for each t > 0, there exists 0 1 such that φ (t 0 < t for all 0 The from the above, for all 0, F gx,gx (t F +1 gx,gx (φ (t +1 0 [F gx0,gx (t 1 0 F gy0,gy (t 1 0 ] > [(1 ε(1 ε] 2 > 1 ε Hece lim F gx,gx +1 (t =1 for all t > 0 Similarly, lim F gy,gy +1 (t =1, for all t > 0 Therefore from (311, Page 6 of 12

7 P (t 1 as (312 Sice φ Φ w, by Lemma 217, for ay t > 0 there exists r t such that φ(r < t Let 1 be give We ext show by iductio that for ay k 1, F gx,gx +k (t Δ k 1 (P (t φ(r F gy,gy +k (t Δ k 1 (P (t φ(r (313 (314 Sice Δ 0 (s =s, this true for k = 1 Let (313 (314 hold for some k F gx,gx (t =F +k+1 gx,gx (t φ(r+φ(r +k+1 Δ(F gx,gx (t φ(r, F +1 gx+1,gx (φ(r +k+1 Δ(F gx,gx (t φ(r, [F +1 gx,gx (rf +k gy,gy (r] 1 2 +k ( by (31, (35 (36 Δ(F gx,gx (t φ(r, [F +1 gx,gx (tf +k gy,gy (t] 1 2 (sice r t +k Δ(P (t φ(r, [Δ k 1 (P (t φ(r, Δ k 1 (P (t φ(r] 1 2 Δ(P (t φ(r, Δ k 1 (P (t φ(r Δ k (P (t φ(r Similarly, we have F gy,gy (t Δ k (P +k+1 (t φ(r Therefore, by iductio, (313 (314 hold for all k 1 t > 0 Now, we prove {gx } {gy } are Cauchy sequeces Sice, the t-orm Δ is of H-type, the family of iterates {Δ (p } is equi-cotiuous at the poit s = 1, that is, there exists δ (0, 1 such that Δ (p (s > 1 δ, (315 wheever 1 s > 1 ε p 1 By (311, we have a positive iteger 0 such that for all 0 (P (t φ(r > 1 δ It follows from (315, (313 (314 that F gx,gx +k (t Δ k 1 (P (t φ(r < 1 ε for all 0, k 1 F gy,gy (t Δ k 1 (P +k (t φ(r < 1 ε for all 0, k 1 This shows that {gx } {gy } are Cauchy sequeces Sice X is complete, there exist x, y X such that lim gx = x lim gy = y, that is, lim G(x, y =limgx +1 = x lim G(y, x =limgy = y +1 (316 Sice {g, G} is a compatible pair, usig cotiuity of g, we have gx = lim g(gx +1 gy = lim g(gy +1 = lim g(g(x, y = lim G(gx, gy = lim g(g(y, x = lim G(gy, gx (317 (318 Page 7 of 12

8 Now, we show that gx = G(x, y gy = G(y, x First let us assume that (a holds The, by cotiuity of G, lim G(gx, gy =G(lim gx, lim gy =G(x, y lim G(gy, gx =G(lim gy, lim gx =G(y, x The, from (317 (318, we coclude that gx = G(x, y gy = G(y, x Next we assume that (b holds By (35, (36, (317 (318, we have {gx } is a o-decreasig sequece with gx x {gy } is a o-icreasig sequece with gy y as The, by (311 (312, for all 0, we have gx x gy y (319 Sice g is mootoic icreasig, we have that g(gx gx g(gy gy (320 Now for all t > 0 usig the lemma 217, we have F gx,g(x,y (t Δ{F gx,g(gx(+1 (t φ(r, F g(gx (+1,G(x,y (φ(r} Takig limit i the above iequality, for all t > 0, we have F gx,g(x,y (t lim if Δ{F (t φ(r, F (φ(r} gx,g(gx (+1 g(gx (+1,G(x,y = Δ{limF gx,g(gx(+1 (t φ(r, lim if ( by lemma 27 cotiuity of Δ Now, usig (31 (320, we have F g(gx (+1,G(x,y (φ(r} = Δ{1, lim if F G(gx,gy,G(x,y(φ(r} (by lemma 27 (317 = lim if F (φ(r G(gx,gy,G(x,y F G(gx,gy,G(x,y (φ(r} [F g(gx,gx (rf g(gy,gy (r] 1 2 (321 Takig limif o both sides of the above iequality, we have lim if F G(gx,gy,G(x,y(φ(r} lim if [F (rf (r] 1 2 g(gx,gx g(gy,gy = lim [F g(gx,gx (rf g(gy,gy (r] 1 2 = 1 (by (317 (318 (322 From (321 (322 we coclude that F gx,g(x,y (t =1 for all t > 0, that is, gx = G(x, y Similarly we ca prove that gy = G(y, x, that is, g G have a coupled coicidece poit i X Page 8 of 12

9 Corollary 32 Let (X, F, Δ be a complete Meger space where Δ is a cotiuous Hadz ić type t-orm o which a partial orderig is defied Let g: X X G: X X X be two mappigs such that G has the mixed g-mootoe property Let there exist φ Φ w such that F G(x,y,G(u,v (φ(t [F gx,gu (tf gy,gv (t] 1 2, for all t > 0, x, y, u, v X with gx gu gy gv Let g be cotiuous, mootoic icreasig, G(X X g(x such that {g, G} is a commutig pair Also suppose either (a G is cotiuous or (b X has the followig properties: (i if a o-decreasig sequece {x } x, the x x for all 0, (ii if a o-icreasig sequece {y } y, the y y for all 0 If there are x 0 X such that gx 0 G(x 0 gy 0 G(y 0, the g G have a coupled coicidece poit i X, that is, there exist x, y X such that gx = G(x, y gy = G(y, x Proof Sice compatibility implies commutig coditio, Theorem 31 cotais Corollary 32 Later, with the help of a example we show that the above corollary is properly cotaied i the above Theorem 31 Corollary 33 Let (X, F, Δ be a complete Meger space where Δ is a cotiuous Hadz ić type t-orm o which a partial orderig is defied Let g: X X G: X X X be two mappigs such that G has the mixed g-mootoe property Let there exist φ Φ such that F G(x,y,G(u,v (φ(t [F gx,gu (tf gy,gv (t] 1 2, for all t > 0, x, y, u, v X with gx gu gy gv Let g be cotiuous, mootoic icreasig, G(X X g(x such that {g, G} is a compatible pair Also suppose either (a G is cotiuous or (b X has the followig properties: (i if a o-decreasig sequece {x } x, the x x for all 0, (ii if a o-icreasig sequece {y } y, the y y for all 0 If there are x 0 X such that gx 0 G(x 0 gy 0 G(y 0, the g G have a coupled coicidece poit i X, that is, there exist x, y X such that gx = G(x, y gy = G(y, x Proof Sice Φ is a proper subclass of Φ w, Theorem 31 cotais Corollary 33 Example 35 discussed i the followig shows that the above corollary is properly cotaied i Theorem 31 Combiig the above two corollaries we have partly a result obtaied i (Xiao et al, 2011 which we describe i the followig corollary Corollary 34 (Xiao et al, 2011 Let (X, F, Δ be a complete Meger space where Δ is a cotiuous Hadz ić type t-orm o which a partial orderig is defied Let g: X X G: X X X be two mappigs such that G has the mixed g-mootoe property Let there exist φ Φ such that F G(x,y,G(u,v (φ(t [F gx,gu (tf gy,gv (t] 1 2, Page 9 of 12

10 for all t > 0, x, y, u, v X with gx gu gy gv Let g be cotiuous, mootoic icreasig, G(X X g(x such that {g, G} is a commutig pair Also suppose either (a G is cotiuous or (b X has the followig properties: (i if a o-decreasig sequece {x } x, the x x for all 0, item (ii if a o-icreasig sequece {y } y, the y y for all 0 If there are x 0 X such that gx 0 G(x 0 gy 0 G(y 0, the g G have a coupled coicidece poit i X, that is, there exist x, y X such that gx = G(x, y gy = G(y, x Proof Sice compatibility implies commutig, Corollary 33 cotais Corollary 34 Example 35 Let X =[0, 1] Let for all t > 0, x, y X, F x,y (t =e x y t Let Δ(a, b =mi{a, b} for all a, b [0, 1] The (X, F, Δ is a complete Meger space Let φ: R + R + be defied by φ(t = t, 2 if t [0, 1, t + 4, 3 3 if t [1, 2], t + 4, 3 otherwise It is obvious φ Φ w but φ Φ From the defiitio of φ, we have φ(t t for all t 0 2 Let the mappig g: X X be defied as g(x = 5 6 x2 for all x X the mappig G: X X X be defied as G(x, y = x2 y 2 9 The G(X X g(x G satisfies the mixed g-mootoe property Let {x } {y } be two sequeces i X such that lim G(x, y =a, lim g(x =a, lim G(y, x =b lim g(y =b Now, for all 0, g(x = 5 6 x2, g(y =5 6 y2, G(x, y = x2 y2 4 G(y, x = y2 x2 4 Page 10 of 12

11 The ecessarily a = 0 b = 0 It the follows from lemma 26 that, for all t > 0, lim F g(g(x,y,g(g(x,g(y (t =1 lim F g(g(y,x,g(gy,g(x (t =1 Therefore, the mappigs G g are compatible i X Now we show that the coditio (31 holds G(x, y G(u, v 1 [ g(x g(u + g(y g(v ], x u, y v 4 Therefore, G(x,y G(u,v φ(t that is, G(x,y G(u,v φ(t Now F G(x,y,G(u,v (φ(t = e G(x,y G(u,v φ(t Hece (31 holds 1 4 [ g(x g(u + g(y g(v t g(x g(u + g(y g(v 2t e [ g(x g(u + g(y g(v ] 2t e g(x g(u 2t e g(y g(v 2t e g(x g(u t e g(y g(v t =(F g(x,g(u (tf g(y,g(v (t ] (sice φ(t t for all t 0, 2 Thus all the coditios of Theorem 31 are satisfied The, by a applicatio of the Theorem 31, we coclude that g F have a coupled coicidece poit Here, (0, 0 is a coupled coicidece poit of g F i X Remark 36 (i {g, G} are ot commutig but compatible, so Corollary 32 is properly cotaied i Theorem 31 (ii φ Φ, so the Corollary 34 is properly cotaied i Corollary 33 (iii For the reaso metioed i (ii, the Corollary 34 i properly cotaied i Theorem 31 Ope problems The gauge fuctio used i this paper is of a very geeral type It appears possible to use this fuctio to exted probabilistic cotractios as i (Fag, 2015 to defie a ew probabilistic cotractios as i the preset work The study of such cotractios, especially relatig to the existece of their fixed poit, is a class of problems which will be worthy of ivestigatio Ackowledgemet The suggestios of the leared referees have bee ackowledged Fudig The first author gratefully ackowledges the support from DST, West Begal, Idia [grat-i-aid umber 624 (sac/ ST/P/S]; [grat-i-aid umber T/ Misc-5/2012] Author details Biayak S Choudhury 1 biayak12@yahoocoi Pradyut Das 1 pradyutdas747@gmailcom P Saha 1 parbati_saha@yahoocoi 1 Departmet of Mathematics, Idia Istitute of Egieerig Sciece Techology Shibpur, Howrah , West Begal, Idia Citatio iformatio Cite this article as: Coupled coicidece poit results for probabilistic φ-cotractios, Biayak S Choudhury, Pradyut Das & P Saha, Coget Mathematics (2016, 3: Page 11 of 12

12 Refereces Bhaskar, T G, & Lakshmikatham, V (2006 Fixed poit theorems i partially ordered metric spaces applicatios Noliear Aalysis, 65, Choudhury, B S, & Das, K P (2009 A coicidece poit result i Meger spaces usig a cotrol fuctio Chaos, Solitos Fractals, 42, Choudhury, B S, & Das, P (2014 Coupled coicidece poit results for compatible mappigs i partially ordered probabilistic metric spaces Asia-Europea Joural of Mathematics, 7, Choudhury, B S, & Kudu, A (2010 A coupled coicidece poit result i partially ordered metric spaces for compatible mappigs Noliear Aalysis, 73, Chag, S S, Cho, Y J, & Kag, S M (1994 Probablistic metric spaces oliear operator theory Chegdu, Chia: Sichua Uiversity Press Ćirić, L (2010 Solvig the Baach fixed poit priciple for oliear cotractios i probabilistic metric spaces Noliear Aalysis, 72, Ćirić, L, Agarwal, R P, & Samet, B (2011 Mixed mootoe geeralized cotractios i partially ordered probabilistic metric spaces Fixed Poit Theory Applicatios, 2011, 56 Doric, D (2013 Noliear coupled coicidece coupled fixed poit theorems for ot ecessary commutative cotractive mappigs i partially ordered probabilistic metric spaces Applied Mathematics Computatio, 219, Fag, J X (2009 Commo fixed poit theorems of compatible weak compatible maps i Meger spaces Noliear Aalysis, 71, Fag, J X (2015 O φ- cotractios i probabilistic fuzzy metric spaces Fuzzy Sets Systems, 267, doi:101016/jfss Guo, D, & Lakshmikatham, V (1987 Coupled fixed poits of oliear operators with applicatios Noliear Aalysis, 11, HadžiĆ, O, & Pap, E (2001 Fixed poit theory i probabilistic metric spaces Dordrecht: Kluwer Academic Publishers Hu, X Q, & Ma, X Y (2011 Coupled coicidece poit theorems uder cotractive coditios i partially ordered probabilistic metric spaces Noliear Aalysis, 74, Jachymski, J (2010 O probabilistic φ- cotractios o Meger spaces Noliear Aalysis, 73, Lakshmikatham, V, & Ćirić, L (2009 Coupled fixed poit theorems for oliear cotractios i partially ordered metric spaces Noliear Aalysis, 70, Luog, N V, & Thua, N X (2011 Coupled fixed poits i partially ordered metric spaces applicatio Noliear Aalysis, 74, Mursaleem, M, Mohiuddie, S A, & Agarwal, R P (2012 Coupled fixed poit theorems for α ψ cotractive type mappigs i partially ordered metric spaces Fixed Poit Theory Applicatios, 2012, 228 Mohiuddie, S A, & Alotaibi, A (2012 O coupled fixed poit theorems for oliear cotractios i partially ordered G- metric spaces Abstract Applied Aalysis, 2012, 15 Art ID O Rega, D, & Saadati, R (2008 Noliear cotractio theorems i probabilistic spaces Applied Mathematics Computatio, 195, Samet, B (2010 Coupled fixed poit theorems for a geeralized Meir-Keeler cotractio i partially ordered metric spaces Noliear Aalysis, 72, Schweizer, B, & Sklar, A (1983 Probabilistic metric spaces North-Holl: Elsevier Sehgal, V M, & Bharucha-Reid, A T (1972 Fixed poit of cotractio mappigs o PM space Theory of Computig Systems, 6, Xiao, J Z, Zhu, X H, & Cao, Y F (2011 Commo coupled fixed poit results for probabilistic φ-cotractios i Meger spaces Noliear Aalysis, 74, The Author(s This ope access article is distributed uder a Creative Commos Attributio (CC-BY 40 licese You are free to: Share copy redistribute the material i ay medium or format Adapt remix, trasform, build upo the material for ay purpose, eve commercially The licesor caot revoke these freedoms as log as you follow the licese terms Uder the followig terms: Attributio You must give appropriate credit, provide a lik to the licese, idicate if chages were made You may do so i ay reasoable maer, but ot i ay way that suggests the licesor edorses you or your use No additioal restrictios You may ot apply legal terms or techological measures that legally restrict others from doig aythig the licese permits Coget Mathematics (ISSN: is published by Coget OA, part of Taylor & Fracis Group Publishig with Coget OA esures: Immediate, uiversal access to your article o publicatio High visibility discoverability via the Coget OA website as well as Taylor & Fracis Olie Dowload citatio statistics for your article Rapid olie publicatio Iput from, dialog with, expert editors editorial boards Retetio of full copyright of your article Guarateed legacy preservatio of your article Discouts waivers for authors i developig regios Submit your mauscript to a Coget OA joural at wwwcogetoacom Page 12 of 12