On common fixed point theorems for weakly compatible mappings in Menger space

Transcription

1 Available olie at Advaces i Applied Sciece Research, 2016, 7(5): ISSN: CODEN (USA): AASRFC O commo fixed poit theorems for weakly compatible mappigs i Meger space Arihat Jai 1 ad Basat Chaudhary 2 1 Departmet of Applied Mathematics, Shri Guru Sadipai Istitute of Techology ad Sciece, Ujjai (M.P.) , Idia 2 Research Scholar, Departmet of Mathematics, Mewar Uiversity, Chittorgarh (Raj.), Idia ABSTRACT I this paper, the cocept of weak compatibility i Meger space has bee applied to prove a commo fixed poit theorem for six self maps. Our result geeralizes ad exteds the result of Pathak ad Verma [8]. Keywords: Probabilistic metric space, Meger space, commo fixed poit, compatible maps, weak compatibility. AMS Subject Classificatio: Primary 47H10, Secodary 54H25. INTRODUCTION There have bee a umber of geeralizatios of metric space. Oe such geeralizatio is Meger space iitiated by Meger [6]. It is a probabilistic geeralizatio i which we assig to ay two poits x ad y, a distributio fuctio F x,y. Schweizer ad Sklar [9] studied this cocept ad gave some fudametal results o this space. Sehgal ad Bharucha-Reid [10] obtaied a geeralizatio of Baach Cotractio Priciple o a complete Meger space which is a milestoe i developig fixed-poit theory i Meger space. Recetly, Jugck ad Rhoades [5] termed a pair of self maps to be coicidetally commutig or equivaletly weakly compatible if they commute at their coicidece poits. Sessa [11] iitiated the traditio of improvig commutativity i fixed-poit theorems by itroducig the otio of weak commutig maps i metric spaces. Jugck [4] soo elarged this cocept to compatible maps. The otio of compatible mappig i a Meger space has bee itroduced by Mishra [7]. I the sequel, Pathak ad Verma [8] proved a commo fixed poit theorem i Meger space usig compatibility ad weak compatibility. Usig the cocept of compatible mappigs of type (A), Jai et. al. [1,2] proved some iterestig fixed poit theorems i Meger space. Afterwards, Jai et. al. [3] proved the fixed poit theorem usig the cocept of weak compatible maps i Meger space. I this paper a fixed poit theorem for six self maps has bee proved usig the cocept of semi-compatible maps ad occasioally weak compatibility which turs out be a material geeralizatio of the result of Pathak ad Verma [8]. 2. Prelimiaries. Defiitio 2.1. A mappig F : R R + is called a distributio if it is o-decreasig left cotiuous with if { F t R } = 0 ad sup { F t R} = 1. We shall deote by L the set of all distributio fuctios while H will always deote the specific distributio 0, t 0 fuctio defied by H =. 1, t > 0 Defiitio 2.2. [7] A mappig t : [0, 1] [0, 1] [0, 1] is called a t-orm if it satisfies the followig coditios : (t-1) t(a, 1) = a, t(0, 0) = 0 ; 46

2 Arihat Jai ad Basat Chaudhary Adv. Appl. Sci. Res., 2016, 7(5): (t-2) t(a, b) = t(b, a) ; (t-3) t(c, d) t(a, b) ; for c a, d b, (t-4) t(t(a, b), c) = t(a, t(b, c)) for all a, b, c, d [0, 1]. Defiitio 2.3. [7] A probabilistic metric space (PM-space) is a ordered pair (X, F) cosistig of a o empty set X ad a fuctio F : X X L, where L is the collectio of all distributio fuctios ad the value of F at (u, v) X X is represeted by F u, v. The fuctio F u,v assumed to satisfy the followig coditios: (PM-1 ) F u,v (x) = 1, for all x > 0, if ad oly if u = v; (PM-2) F u,v (0) = 0; (PM-3) F u,v = F v,u ; (PM-4) If F u,v (x) = 1 ad F v,w (y) = 1 the F u,w (x + y) = 1, for all u,v,w X ad x, y > 0. Defiitio 2.4. [7] A Meger space is a triplet (X, F, t) where (X, F) is a PM-space ad t is a t-orm such that the iequality (PM-5) F u,w (x + y) t {F u, v (x), F v, w (y) }, for all u, v, w X, x, y 0. Defiitio 2.5. [7] A sequece {x } i a Meger space (X, F, t) is said to be coverget ad coverges to a poit x i X if ad oly if for each ε > 0 ad λ > 0, there is a iteger M(ε, λ) such that F x, x (ε) > 1 - λ for all M(ε, λ). Further the sequece {x } is said to be Cauchy sequece if for ε > 0 ad λ > 0, there is a iteger M(ε, λ) such that F x, x m (ε) > 1- λ for all m, M(ε, λ). A Meger PM-space (X, F, t) is said to be complete if every Cauchy sequece i X coverges to a poit i X. A complete metric space ca be treated as a complete Meger space i the followig way: Propositio 2.1. [7] If (X, d) is a metric space the the metric d iduces mappigs F : X X L, defied by F p,q (x) = H(x - d(p, q)), p, q X, where H(k) = 0, for k 0 ad H(k) = 1, for k >0. Further if, t : [0,1] [0,1] [0,1] is defied by t(a, b) = mi {a, b}. The (X, F, t) is a Meger space. It is complete if (X, d) is complete. The space (X, F, t) so obtaied is called the iduced Meger space. Defiitio 2.6. [8] Self mappigs A ad S of a Meger space (X, F, t) are said to be weak compatible if they commute at their coicidece poits i.e. Ax = Sx for x X implies ASx = SAx. Defiitio 2.7. [8] Self mappigs A ad S of a Meger space (X, F, t) are said to be compatible if F ASx,SAx (x) 1 for all x>0, wheever {x } is a sequece i X such that Ax,Sx u for some u i X, as. Remark 2.1. [8] The cocept of weakly compatible mappigs is more geeral tha that of compatible mappigs. Lemma 2.1. [8] Let (X, F, *) be a Meger space with t-orm * such that the family {* (x)} N is equicotiuous at x = 1 ad let E deote the family of all fuctios φ : R + R + such that φ is o-decreasig with lim φ =+, t > 0. If {y } N the coditio F y, y +1 F y-1, y, is a sequece i X satisfyig for all t > 0 ad α [ 1, 0], the {y } N is a Cauchy sequece i X. 47

3 Arihat Jai ad Basat Chaudhary Adv. Appl. Sci. Res., 2016, 7(5): Propositio 2.2. Let {x } be a Cauchy sequece i a Meger space (X, F, t) with cotiuous t-orm t. If the subsequece {x 2 } coverges to x i X, the {x } also coverges to x. Proof. As {x 2 } coverges to x, we have ε ε F x,x ( ε) t F x,x,f 2 x 2,x. 2 2 The limf ( ε) t(1,1), which gives limf ( ε ) = 1, ε > 0 ad the result follows. x,x x,x RESULTS Theorem 3.1. Let A, B, S, T, P ad Q be self mappigs o a Meger space (X, F, *) with cotiuous t-orm * satisfyig : (3.1.1) P(X) ST(X), Q(X) AB(X); (3.1.2) AB = BA, ST = TS, PB = BP, QT = TQ; (3.1.3) Oe of ST(X), Q(X), AB(X) or P(X) is complete; (3.1.4) The pairs (P, AB) ad (Q, ST) are weak compatible; (3.1.5) [1 + αf ABx, STy ] * F Px, Qy α mi{f Px, ABx * F Qy, STy, F Px, STy (2t) * F Qy, ABx + F ABx, STy * F Px,ABx * F Qy, STy * F Px, STy * F Qy, ABx for all x, y X, t > 0 ad φ E. The A, B, S, T, P ad Q have a uique commo fixed poit i X. Proof. Suppose x 0 X. From coditio (3.1.1) x 1, x 2 X such that Px 0 = STx 1 ad Qx 1 = ABx 2. Iductively, we ca costruct sequeces {x } ad {y } i X such that y = Px = STx ad y = Qx = ABx for = 0, 1, 2,.... Step I. Let us show that F y+2, y +1 F y+1, y. For, puttig x for x ad x for y i (3.1.5) ad the o simplificatio, we have [1 + αf ] * F ABx2+2, STx Px2+2, Qx α mi{f * F, F (2t) Px2+2, ABx Qx2+1, STx Px2+2, STx F Qx2+1, ABx F * F * F ABx2+2, STx Px2+2, ABx Qx2+1, STx * F * F Px2+2, STx Qx2+1, ABx [1 + αf y2+1, y ] * F y2+2, y α mi{f * F, F (2t) *F y2+2, y y2+1, y y2+2, y y2+1, y F y2+1, y 2 * F * F * F * F y2+2, y y2+1, y y2+2, y y2+1, y F + αf * F y2+2, y y2+1, y y2+2, y

4 Arihat Jai ad Basat Chaudhary Adv. Appl. Sci. Res., 2016, 7(5): α mi{f y2+2, y 2 (2t), F y2+2, y 2 + F y2+1, y 2 * F y2+2, y 2+1 * F y2+2, y 2 * 1 F y2+2, y αf y2+1, y 2 * F y2+2, y 2+1 α F y2+2, y 2 (2t) + F y2+1, y 2 * F y2+2, y 2+1 F y2+2, y αf y2+2, y 2 (2t) α F y2+2, y 2 (2t) + F y2+1, y2 * F y 2+2, y 2+1 * F y2+1, y 2 F y2+2, y 2+1 F y2+1, y 2 * F y2+2, y 2+1 or, F y2+2, y 2+1 F y2+1, y 2+2 * F y2, y 2+1 or, F y2+2, y 2+1 mi{f y2+1, y 2+2, F y2, y 2+1 }. If F y2+1, y 2+2 is chose 'mi' the we obtai F y2+2, y 2+1 F y2+2, y 2+1, t > 0 a cotradictio as φ is o-decreasig fuctio. Thus, F y2+2, y 2+1 F y2+1, y 2, t > 0. Similarly, by puttig x 2+2 for x ad x 2+3 for y i (3.1.5), we have F y2+3, y 2+2 F y2+2, y 2+1, t > 0. Usig these two, we obtai F y+2, y +1 F y+1, y, = 0, 1, 2,..., t > 0. Therefore, by lemma 2.1, {y } is a Cauchy sequece i X. Case I. ST(X) is complete. I this case {y 2 } = {STx 2+1 } is a Cauchy sequece i ST(X), which is complete. Thus {y 2+1 } coverges to some z ST(X). By propositio 2.2, we have {Qx 2+1 } z ad {STx 2+1 } z, (3.1.6) {Px 2 } z ad {ABx 2 } z. (3.1.7) As z ST(X) there exists u X such that z = STu. Step I. Put x = x 2 ad y = u i (3.1.5), we get [1 + αf ABx2, STu ] * F Px 2, Qu α mi{f Px2, ABx 2 * F Qu, STu, F Px2, STu (2t) * F Qu, ABx 2 + F ABx2, STu * F Px 2, ABx 2 * F Qu, STu * F Px2, STu * F Qu, ABx2. Lettig ad usig (3.1.6), (3.1.7), we get [1 + αf z, z ] * F z, Qu α mi{f z, z * F Qu, z, F z, z (2t) * F Qu, z + F z, z * F z, z * F Qu, z * F z, z * F Qu, z 49

5 Arihat Jai ad Basat Chaudhary Adv. Appl. Sci. Res., 2016, 7(5): F z, Qu + αf z, Qu α mi{f Qu, z, F Qu, z + F Qu, z * F Qu, z F Qu, z + αf Qu, z α mi{f Qu, z, F Qu, z * F z, z } + F Qu, z * F Qu, z * F z, z F Qu, z + αf Qu, z α F Qu, z + F Qu, z F Qu, z F Qu, z which is a cotradictio by lemma (2.1) ad we get Qu = z ad so Qu = z = STu. Sice (Q, ST) is weakly compatible, we have STz = Qz. Step III. Put x = x 2 ad y = Tz i (3.1.5), we have [1 + αf ABx2, STTz ] * F Px 2, QTz α mi{f Px2, ABx 2 * F QTz, STTz, F Px2, STTz (2t) * F QTz, ABx 2 + F ABx2, STTz * F Px 2, ABx 2 * F QTz, STTz * F Px2, STTz * F QTz, ABx 2. As QT = TQ ad ST = TS, we have QTz = TQz = Tz ad ST(Tz) = T(STz) = Tz. Lettig, we get [1 + αf z, Tz ] * F z, Tz α mi{f z, z * F Tz, Tz, F z, Tz (2t) * F Tz, z + F z, Tz * F z, z * F Tz, Tz * F z, Tz * F Tz, z F z, Tz + α{f z, Tz * F z, Tz } α mi{1 * F Tz, z + F z, Tz * 1 * 1 * F Tz, z F Tz, z + αf Tz, z α F Tz, z (2t) + F Tz, z * F Tz, z F Tz, z + αf Tz, z α {F Tz, z * F z, z } + F Tz, z *F Tz, z *F z, z F Tz, z + αf Tz, z α F Tz, z + F Tz, z F Tz, z F Tz, z which is a cotradictio ad we get Tz = z. Now, STz = Tz = z implies Sz = z. Hece, Sz = Tz = Qz = z. Step IV. As Q(X) AB(X), there exists w X such that z = Qz = ABw. Put x = w ad y = x 2+1 i (3.1.5), we get [1 + αf ABw, STx2+1 ] * F Pw, Qx2+1 α mi{f Pw, ABw * F Qx2+1, STx 2+1, F Pw, STx2+1 (2t) * F Qx2+1, ABw + F ABw, STx 2+1 * F Pw, ABw * F Qx2+1, STx 2+1 * F Pw, STx2+1 * F Qx2+1, ABw. Lettig, we get 50

6 Arihat Jai ad Basat Chaudhary Adv. Appl. Sci. Res., 2016, 7(5): [1 + αf z, z ] * α mi{ * F z, z, (2t) * F z, z + F z, z * * F z, z * * F z, z + α α mi{, + * + α α mi{, * F z, z } + * F z, z + α α mi{, } + + α α } + which is a cotradictio ad hece, we get Pw = z. Hece, Pz = z = ABz. Step V. Put x = z ad y = x 2+1 i (3.1.5), we have [1 + αf ABz, STx2+1 ] * F Pz, Qx2+1 α mi{f Pz, ABz * F Qx2+1, STx 2+1, F Pz, STx2+1 (2t) * F Qx2+1, ABz + F ABz, STx2+1 * F Pz, ABz * F Qx2+1, STx 2+1 * F Pz, STx2+1 * F Qx2+1, ABz. Lettig, we get [1 + α ] * α mi{f Pz, Pz * F z, z, (2t) * F z, Pz + * F Pz, Pz * F z, z * * F z, Pz + α{ * } α mi{1 * 1, (2t) * + * 1 * 1 * * F z, Pz + α α mi{1, + * + α α (2t) + * + α α{ * F z, z } + * * F z, z + α α{ * 1} + * 1 + α α + which is a cotradictio ad hece, Pz = z ad so z = Pz = ABz. Step VI. Put x = Bz ad y = x 2+1 i (3.1.5), we get [1 + αf ABBz, STx2+1 ] * F PBz, Qx2+1 α mi{f PBz, ABBz * F Qx2+1, STx 2+1, F PBz, STx2+1 (2t) * F Qx2+1, ABBz + F ABBz, STx 2+1 * F PBz, ABBz * F Qx2+1, STx 2+1 * F PBz, STx2+1 * F Qx2+1, ABBz. 51

7 Arihat Jai ad Basat Chaudhary Adv. Appl. Sci. Res., 2016, 7(5): As BP = PB, AB = BA so we have P(Bz) = B(Pz) = Bz ad AB(Bz) = B(AB)z = Bz. Lettig ad usig (3.1.6), we get [1 + α ] * α mi{f Bz, Bz * F z, z, (2t) * F z, Bz + * F Bz, Bz * F z, z * * F z, Bz + α{ * } α mi{1 * 1, + * 1 * 1 * + α α (2t) + * + α α { * F z, z } + * * F z, z + α α { * 1} + * 1 + α α + which is a cotradictio ad we get Bz = z ad so z = ABz = Az. Therefore, Pz = Az = Bz = z. Combiig the results from differet steps, we get Az = Bz = Pz = Qz = Tz = Sz = z. Hece, the six self maps have a commo fixed poit i this case. Case whe P(X) is complete follows from above case as P(X) ST(X). Case II. AB(X) is complete. This case follows by symmetry. As Q(X) AB(X), therefore the result also holds whe Q(X) is complete. Uiqueess : Let z 1 be aother commo fixed poit of A, B, P, Q, S ad T. The Az 1 = Bz 1 = Pz 1 = Sz 1 = Tz 1 = Qz 1 = z 1, assumig z z 1. Put x = z ad y = z 1 i (3.1.5), we get [1 + αf ABz, STz1 ] * F Pz, Qz1 α mi{f Pz, ABz * F Qz1, STz 1, F Pz, STz1 (2t) * F Qz1, ABz + F ABz, STz1 * F Pz, ABz * F Qz1, STz 1 * F Pz, STz1 * F Qz1, ABz [1 + αf z, z1 ] * F z, z1 α mi{f z, z * F z1, z 1, F z, z1 (2t) * F z1, z + F z, z 1 * F z, z * F z1, z 1 * F z, z1 * F z1, z F z, z1 + α{f z, z1 * F z, z1 } α mi{1, F z, z1 + F z, z1 * F z, z1 F z, z1 + αf z, z1 αf z, z1 + F z, z1 * F z, z1 * F z, z F z1, z + αf z 1, z α{f z 1, z * F z, z } + F z 1, z * 1 F z1, z + αf z 1, z αf z 1, z + F z 1, z 52

8 Arihat Jai ad Basat Chaudhary Adv. Appl. Sci. Res., 2016, 7(5): F z1, z F z 1, z which is a cotradictio. Hece z = z 1 ad so z is the uique commo fixed poit of A, B, S, T, P ad Q. This completes the proof. Remark 3.1. If we take B = T = I, the idetity map o X i theorem 3.1, the coditio (3.1.2) is satisfied trivially ad we get Corollary 3.1. Let A, S, P ad Q be self mappigs o a Meger space (X, F, *) with cotiuous t-orm * satisfyig : (i) P(X) T(X), Q(X) A(X); (ii) Oe of S(X), Q(X), A(X) or P(X) is complete; (iii) The pairs (P, A) ad (Q, S) are weak compatible; (iv) [1 + αf Ax, Sy ] * F Px, Qy α mi{f Px, Ax * F Qy, Sy, F Px, Sy (2t) * + F Ax, Sy * F Px,Ax * F Qy, Sy * F Px, Sy * F Qy, Ax for all x, y X, t > 0 ad φ E. F Qy, Ax The A, S, P ad Q have a uique commo fixed poit i X. Remark 3.2. I view of remark 3.1, corollary 3.1 is a geeralizatio of the result of Pathak ad Verma [8] i the sese that coditio of compatibility of the first pair of self maps has bee restricted to weak compatibility ad we have dropped the coditio of cotiuity i a Meger space with cotiuous t-orm. REFERENCES [1]. Jai, Arihat ad Sigh, Bijedra, Commo fixed poit theorem i Meger space through compatible maps of type (A), Chh. J. Sci. Tech. 2 (2005), [2]. Jai, Arihat ad Sigh, Bijedra, A fixed poit theorem i Meger space through compatible maps of type (A), V.J.M.S. 5(2), (2005), [3]. Jai, Arihat ad Sigh, Bijedra, Commo fixed poit theorem i Meger Spaces, The Aligarh Bull. of Math. 25 (1), (2006), [4]. Jugck, G., Compatible mappigs ad commo fixed poits, Iterat. J. Math. ad Math. Sci. 9(4), (1986), [5]. Jugck, G. ad Rhoades, B.E., Fixed poits for set valued fuctios without cotiuity, Idia J. Pure Appl. Math. 29(1998), [6]. Meger, K., Statistical metrics, Proc. Nat. Acad. Sci. USA. 28(1942), [7]. Mishra, S.N., Commo fixed poits of compatible mappigs i PM-spaces, Math. Japo. 36(2), (1991), [8]. Pathak, H.K. ad Verma, R.K., Commo fixed poit theorems for weakly compatible mappigs i Meger space ad applicatio, It. Joural of Math. Aalysis, Vol. 3, 2009, No. 24, [9]. Schweizer, B. ad Sklar, A., Statistical metric spaces, Pacific J. Math. 10 (1960), [10]. Sehgal, V.M. ad Bharucha-Reid, A.T., Fixed poits of cotractio maps o probabilistic metric spaces, Math. System Theory 6(1972), [11]. Sessa, S., O a weak commutativity coditio of mappigs i fixed poit cosideratio, Publ. Ist. Math. Beograd 32(46), (1982),