COMMON FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN COMPLEX VALUED b-metric SPACES

I S S N 3 4 7-9 J o u r a l o f A d v a c e s i M a t h e m a t i c s COMMON FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN COMPLEX VALUED b-metric SPACES Ail Kumar Dube, Madhubala Kasar, Ravi Prakash Dube 3 Departmet of Mathematics, Bhilai Istitute of Techolog, Bhilai House, Durg, Chhattisgarh 4900, Idia ailkumardb70@gmail.com Govermet Higher Secodar School, Titurdih, Durg, Chhattisgarh 4900, Idia madhubala.kasar@gmail.com 3 Departmet of Mathematics, Dr. C.V. Rama Uiversit, Kota, Bilaspur, Chhattisgarh 4953, Idia ravidube_963@ahoo.co.i KeWords: Complex valued b-metric space, weakl compatible maps, (E.A. propert, (CLRg propert. 00 MSC : 47H0, 54H5. Abstract I this paper, we prove a commo fixed poit theorem for weakl compatible mappigs i complex valued b-metric space ad also improve the coditio of cotractio of the results of M. Kumar et al.[7]. Further, we prove commo fixed poit theorems for weakl compatible mappigs with (E.A. propert ad (CLRg propert.. Itroductio ad Prelimiaries I 0, Azam et al. [] itroduced the otio of complex valued metric spaces ad established sufficiet coditios for the existece of commo fixed poits of a pair of mappigs satisfig a cotractive coditio. Complex valued metric space is a geeralizatio of classical metric space. After the establishmet of complex valued metric space, ma authors have cotributed with their works i this space. Some of these results are described i [7,8,,3,4]. I 03, Rao et al.[0] itroduced the cocept of complex valued b-metric space which is a geeralizatio of complex valued metric space. The, some other authors geeralized this cocept ad proved several commo fixed poit ad fixed poit theorems i complex valued b-metric spaces (see [3,4,5,9] ad the refereces cotaied therei. Aamri ad Moutawakil[] itroduced the otio of (E.A. propert. Recetl, Verma ad Pathak [] proved commo fixed poit theorem for two pairs of weakl compatible mappigs with (E.A. propert, ad a commo fixed poit theorem usig (CLRg propert which was itroduced b Situavarat ad Kumam []. I this paper, we prove a commo fixed poit theorem for weakl compatible mappigs satisfig a cotractive coditio of ratioal tpe i the frame work of complex valued b-metric spaces. Further applicatio of perpert (E.A. ad commo limit rage (CLRg propert are emploed. Let us start b defiig some importat defiitios, basic otatios ad ecessar results from existig literature. Let C be the set of complex umbers ad z z. Defie a partial order o C as follows:, C if ad ol if Re( z Re( z, Im( z Im( z. Thus z z if oe of the followig holds: I particular, we write z satisfied. z Æ if z z ad oe of (C,(C ad (C3 is satisfied ad we write z z if ol (C3 is Remark.. We ote that the followig statemets hold: a, br ad a b az bz z C. 0 z Æz z < z. z z ad z z3 z z 3. Defiitio.[0] Let X be a oempt set ad let s be a give real umber. A fuctio d : X X C is called a complex valued b-metric o X if, z X the followig coditios are satisfied: 0 ad d ( = 0 if ad ol if x =, d ( =,, 704 P a g e

I S S N 3 4 7-9 J o u r a l o f A d v a c e s i M a t h e m a t i c s s[ z z, ]. pair ( X, d is called a complex valued b-metric space. Example.3[0] Let X = [0,]. Defie a mappig d : X X C b = x i x, all X. The ( X, d is complex valued b-metric space with s =. Defiitio.4[0] Let (, d (i A poit X be a complex valued b-metric space. for x X is called iterior poit of a set A X wheever there exists r C r = X : r A (ii A poit (iii A subset (iv A subset. 0 such that x X is called a limit poit of a set A wheever for ever 0 r C, r ( Ax. A X is called a ope set wheever each elemet of A is a iterior poit of A. A X is called closed set wheever each limit poit of A belogs to A. (v A sub-basis for a Hausdorff topolog o X is a famil F = { r : x X ad0 r}. Defiitio.5[0] Let (, d (i If for ever C, X be a complex valued b-metric space, c with 0 c there is N N such that > N, ( x, c, coverget, x coverges to x ad x is the limit poit of xas. x x be a sequece i X ad x X. d the x is said to be x = or x. We deote this b lim x (ii If for ever c C, with 0 c there is N N such that > N, x, x c, x is said to be a Cauch sequece. (iii If ever Cauch sequece i X is coverget, the (, d Lemma.6[0] Let (, d m where m N, the X is said to be a complete complex valued b-metric space. X be a complex valued b-metric space ad let x be a sequece i X. The coverges to x if ad ol if x, 0 as. Lemma.7[0] Let (, d X be a complex valued b-metric space ad let x be a sequece i X. The a Cauch sequece if ad ol if (, 0 d x x m as, where N m. I 996, Jugck [6] itroduced the cocept of weakl compatible maps as follows: Defiitio.8. Two self maps I 00, Aamri et al. [] itroduced the otio of E.A. propert as follows: f ad g are said to be weakl compatible if the commute at coicidece poits. Defiitio.9. Two self mappigs f ad g of a metric space (, d a sequece { x } i X such that lim fx = lim gx = t for some t i X. I 0, Situavarat et al. [] itroduced the otio of ( CLR g propert as follows: Defiitio.0. Two self mappigs x x is X are said to satisf E.A. propert if there exists ( g f ad g of a metric space ( X, d are said to satisf there exists a sequece { x } i X such that lim fx = lim gx = gx for some x i X. CLR propert if. Mai Result I this sectio, we shall prove our results relaxig the coditio of complex valued b-metric space beig complete. Theorem.. Let ( X, d be a complex valued b-metric space with the coefficiet s ad let f g : X X If there exist a mappigs A, B, C, D, E : X (0, such that X : (i A( A( ; ; ; D( D( ad E( E(,,. 705 P a g e

I S S N 3 4 7-9 J o u r a l o f A d v a c e s i M a t h e m a t i c s (ii fx gx, (iii ( sa B C D E <, (iv f, I gx is complete subspace of X, the f ad g have a coicidece poit. Moreover, if f ad g are weakl compatible the f ad g have a uique commo fixed poit. Proof. Let x, From (ii, we ca costruct sequeces x } ad } i X b 0 X = gx = fx, = 0,,,. From (, we have, = fx, gx gx, gx = A ( fx gx Thus, we have Sice,, fx gx, > (,, 0 0 where = <. Now (.,., d we have A( D( 0.. d,,, So for m >,, s, s, m m s s Therefore, m,, 0 0 Hece, as. is a Cauch sequece i gx. But gx is a complete subspace of X, so there is a u i gx such that u as. Let v g u. The gv = u. Nowe shall porve that fv = u. x = v ad = x Lettig, we have i (, we get, that is d ( fv 0 implies that fv = u. Thus, fv = u = gv ad hece v is the coicidece poit of f ad g. No sice f ad g are weakl compatible, so u = fv = gv implies that fu = fgv = gfv = gu. Now we claim that gu = u. Let, if possible, gu u. From (, we have that is Sice gu > gu, we have which implies that ( A u, a cotradictio. Hece gu = u = fu. Therefore u is the commo fixed poit of f ad g. { { 706 P a g e

I S S N 3 4 7-9 J o u r a l o f A d v a c e s i M a t h e m a t i c s For the uiqueess, let w be aother commo fixed poit of f ad g such that w u. From (, we have that is Sice u > u, we have u ( A w u implies that ( A w, a cotradictio. Hece f ad g have a uique commo fixed poit. Corollar.. Let ( X, d be a complex valued b-metric space with the coefficiet s ad let f, g : X X. If there exist a mappigs A, B, C, D: X (0, such that X : (i A( A( ; ; ; D( D(, (ii fx gx, (iii ( sa B C D <, (iv f, I gx is complete subspace of X, the f ad g have a coicidece poit. Moreover, if f ad g are weakl compatible the f ad g have a uique commo fixed poit. Proof. B puttig E = 0 i Theorem., we get the result of Corollar.. Corollar.3. Let ( X, d be a complex valued b-metric space with the coefficiet s ad let f, g : X X. If there exists mappig A : X (0, such that X : (i A( fg A(, (ii fx gx, (iii ( sa <, (iv d (. (3 I gx is complete subspace of X, the f ad g have a coicidece poit. Moreover, if f ad g are weakl compatible the f ad g have a uique commo fixed poit. Proof. B puttig = C = D = E = 0 B i Theorem., we get the result of Corollar.3. Theorem.4. Let ( X, d be a complex valued b-metric space with the coefficiet s ad let f g : X X If there exist a mappigs A, B, C, D, E : X (0, such that X : (i A( A( ; ; ; D( D( ad E( E(, (ii fx gx, (iii ( sa B C D E <, (iv f,,. If f ad g satisf (CLRg propert ad f ad g are weakl compatible the f ad g have a uique commo fixed poit. Proof. Sice f ad g satisf the (CLRg propert, there exists a sequece x } i X such that for some x i X. { 707 P a g e

From (4, we have fx Lettig, we have which implies that ( 0 No let = fx = gx. fx A( gx. gx d that is fx = gx. I S S N 3 4 7-9 J o u r a l o f A d v a c e s i M a t h e m a t i c s u Sice f ad g are weakl compatible mappigs, therefore fgx = gfx which implies that fu = fgx = gfx = gu. No we claim that gu = u. Let, if possible, gu u. From (4, we have that is Sice gu > gu, the we have which implies that ( u, A a cotradictio. Hece gu u = fu Therefore, u is the commo fixed poit of f ad g. =. the uiqueess, let w be aother commo fixed poit of f ad g such that w u. From (4, we have d ( u = f fu is u u u A( w u w u. Sice u > u, the we have implies that ( w, Corollar.5. Let A a cotradictio. Hece f ad g have a uique commo fixed poit. of Theorem.4 ad the followig: (6 f ad g be self mappigs of a complex valued b-metric space (, d i X, where ( sa C <. The ad g Corollar.6. Let f have a uique commo fixed poit. B puttig B = D = E = 0 i Theorem.4, we get the Corollar.5. of Theorem.4 ad the followig: (7 f ad g be self mappigs of a complex valued b-metric space (, d i X, where ( sc <. The ad g Corollar.7. Let f have a uique commo fixed poit. B puttig A = 0 i Corollar.5, we get the Corollar.6. Theorem.4 ad the followig: (8 f ad g be self mappigs of a complex valued b-metric space (, d d ( X satisfig all coditios X satisfig all coditios X satisfig all coditio of 708 P a g e

i X, where ( sa <. The ad g I S S N 3 4 7-9 J o u r a l o f A d v a c e s i M a t h e m a t i c s f have a uique commo fixed poit. B puttig C = 0 i Corollar.5, we get the Corollar.7. Theorem.8. Let ( X, d be a complex valued b-metric space with the coefficiet s ad let f ad g be self mappigs satisfig all coditios of Theorem.4 ad the followig: (a f ad g satisf E.A. Propert ad f ad g are weakl compatible, (b gx is a closed subset of X. The f ad g have a uique commo fixed. Sice f ad g satisf the.a. (9 for some x i X. E propert, there exists a sequece x No gx is closed subset of X, therefore lim gx = ga, lim fx = ga. fa =. We claim that ga i X such that for some a X. So from (9, we have From (4, we have fx Lettig, we have which implies that ( 0 ga, A( ga, ga. gx d, that is fa = ga. No we show that fa is the commo fixed poit of f ad g. Let, if possible fa ffa. Sice f ad g are weakl compatible, From (4, we have gfa = fga implies that ffa = fga = gfa = gga. f g g g ga, D( g E( g f = A( f f f ga, D( f E( f 709 P a g e

= A( f f that is, A( f f. Sice >, we have ( A f, Hece = fa = gfa. I S S N 3 4 7-9 J o u r a l o f A d v a c e s i M a t h e m a t i c s a cotradictio, sice ( A C. ffa Thus fa is commo fixed poit of f ad g. Fiall, we show that the commo fixed poit is uique. For this, let u ad v be two commo fixed poit of f ad g such that u v. that is Sice v > v, we have v ( A u v, a cotradictio, sice ( A C. Hece f ad g have a uique commo fixed poit. Corollar.9. Let ( X, d be a complex valued b-metric space with the coefficiet s ad let f ad g be self mappigs satisfig the followig: (i fx gx, (ii f ad g satisf E.A. propert ad f ad g are weakl compatible, (iii ( sa C <, (iv, (0 i X. The f ad g have a uique commo fixed poit. Proof. B puttig = D = E = 0 B i Theorem.8, we get the Corollar.9. Corollar.0. Let f ad g be self mappigs of a complex valued b-metric space (, d coditios of Corollar.9 ad the followig: ( i X, where ( sc <. The f ad g have a uique commo fixed poit. Proof. B puttig A = 0 i Corollar.9, we get the Corollar.0. Corollar.. Let f ad g be self mappigs of a complex valued b-metric space (, d coditios of Corollar.9 ad the followig: d ( ( i X, where ( sa <. X satisfig all X satisfig all 700 P a g e

I S S N 3 4 7-9 J o u r a l o f A d v a c e s i M a t h e m a t i c s The f ad g have a uique commo fixed poit. Proof. B puttig C = 0 i Corollar.9, we get the Corollar.. Refereces. M. Aamri ad D. El. MoutawaKil, Some ew commo fixed poit theorems uder strict cotractive coditios, J. Math. Aal. Appl. 7(, 8-88(00.. A.Azam, B.Fisher ad M. Kha, Commo fixed poit theorems i complex valued metric spaces, Numer. Fuct. Aal. Optim. 3(3,43-53(0. 3. A.K. Dube, Commo fixed poit results for cotractive mappigs i complex valued b-metric spaces, Noliear Fuct. Aal. ad Appl., Vol.0, No., 57-68(05. 4. A.K. Dube, Rita Shukla, R.P. Dube, Some Fixed Poit Theorems i Complex Valued b-metric Spaces, Joural of Complex Sstems, Vol. 05, Article ID 83467, 7 Pages, http.//dx.doi.org/0.55/05/83467. 5. A.K. Dube, Rita Shukla, R.P. Dube, Some commo fixed poit theorems for cotractive mappigs i complex valued b-metric spaces, Asia Joural of Mathematics ad Applicatios, Vol. 05, Article ID ama066, 3 pages. 6. G. Jugck, Commo fixed poits for o-cotiuous o-self mappigs o o-metric spaces, Far East J. Math. Sci. 4(, 99-(996. 7. M. Kumar, P. Kumar, S. Kumar, Commo fixed poit theorems i Complex valued metric spaces, Joural of Aalsis & Number Theor, (, 03-09(04. 8. S. Kumar, M. Kumar, P. Kumar, S.M. Kag, Commo fixed poit theorems for weakl compatible mappigs i complex valued metric spaces, It. J. of Pure ad Applied Mathematics, 9(3, 403-49(04. 9. A.A. Mukheimer, Some commo fixed poit theorems i complex valued b-metric spaces, The Scietific World Joural, Vol.04(04, Article ID 58785, 6 pages. 0. K.P.R Rao, P.R Swam, J.R., Prasad, A commo fixed poit theorem i complex valued b-metric spaces, Bulleti of Mathematics ad Statistics Research, Vol., Issue,-8(03.. R.K. Verma, H.K. Pathak, Commo fixed poit theorems usig propert (E.A. i complex valued metrid spaces, Thai J. of Mathematics, (, 347-355(03.. W. Situavarat, P. Kumam, Commo fixed poit theorems for a pair of weakl compatible mappigs i fuzz metric spaces, Joural of Applied Mathematics, 0, Article ID 637958, 4 pages. 3. W. Situavarat, P. Kumam, Geeralized commo fixed poit theorems i complex valued metric spaces ad applicatios, Joural of Iequalities ad Applicatios, (0, 0:84 4. W. Situavarat, Y.J. Cho, P. Kumam, Ursoh itegral equatios approach b commo fixed poits i complex valued metric spaces, Adva ces i Differece Equatios, 03(03:49. 70 P a g e