Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 41, 2021, pp. 28-44 28 A Result on b-metric space using − compatible mappings Thirupathi Thota* Srinivas Veladi† Abstract The objective of this paper is to generate a common fixed point theorem in b-metric space using  -compatible and -continuous mappings. This result generalizes the Theorem proved by J.R. Roshan and others. Further our findings are supported by discussing some valid examples. Keywords: b-metric space; fixed point;  -fixed point;  -compatible;  -continuous mappings. 2010 AMS subject classification: 54H25, 47H10. *Mathematics Department, Sreenidhi Institute of Science & Technology, Ghatkesar, Hyderabad, Telangana, India-501301. E-mail: thotathirupathi1986@gmail.com. †Mathematics Departments, University College of Science, Saifabad, Osmania University, Hyderabad, India. srinivasmaths4141@gmail.com. *Received on September 26, 2021. Accepted on December 5, 2021. Published on December 31, 2021. doi: 10.23755/rm.v41i0.666. ISSN: 1592-7415. eISSN: 2282-8214. A Result on b-metric space using − compatible mappings 29 1. Introduction Fixed point theory plays an important role in mathematics and it is fast growing in the fields of analysis because of its applications in mathematics and allied subjects. Several authors [1, 2, 3, 4] established many results in fixed point theory using various weaker conditions. In the recent past, b-metric space was emerged as one of the generalizations of metric space. During this period Czerwik [5] introduced the concept of −b metric space. In recent years, number of well-known fixed point theorems have been established in b-metric space such as [6, 7, 8, 9]. The concept of alpha compatible and alpha continuous mappings were introduced in metric space [10] and some results were established in the recent past under certain weaker conditions. J.R. Roshan, N. Shobkolaei, S.Sedghi and M.Abbas [11] proved a common fixed point theorem using compatible and continuous mappings in b-metric space. In this paper, we use the concept of  -compatible and  -continuous maps and generate a fixed point theorem in b-metric space. 2.2. Preliminaries Definition 2.1. A function + → RXXd : where X is a nonempty set and 1m is a b-metric space if and only if for each X ,, (i)  == 0),(d (ii) ),(),(  dd = (iii)   .),(),(),(  ddmd + Definition 2.2. Two self maps M, N of a b-metric space X is said to be compatible if d ( ) 0, =KK NMMN  whenever sequence in X such that  == KK NM for some as Definition 2.3. A point X is said to be an  - fixed point of map XXM →: if ( )  =M . Remark 2.4. A fixed point is not necessarily  - fixed point and  - fixed point is not necessarily a fixed point. If  =1, the identity map then they coincide. Example2.5. Let M,  : RR → be defined by M(u) = 1 2 −u and .1)( 3 += uu   k  X .→k Thirupathi Thota and Srinivas Veladi 30 Then 0)1()0)(( =−= M and 1)0()1)(( == M . Therefore 0 and 1 are - fixed points but not fixed points. Example 2.6. Let M,  : RR → be defined by M(u) = 3 2 u and .)( 3 uu = Here 0)0()0)(( == M and M(0) = 0. Therefore 0 is - fixed point of M which is also fixed point of M. Definition 2.7. A pair of self maps M and N of a b-metric space X is called − commuting if ( ) ( ) )()()()( MNNM = for all X . The preceding example show the relation between commuting and − commuting mappings. Example 2.8. Let M, N,  : RR → be defined by 4 )( uuM = , uuN =)( and uu 3)( = for all Ru  . 24 )()()( uuuMuMNThen === and 24 )()( uuNuNM == . Therefore )()( uNMuMN = . Hence M and N are commuting mappings. `Also for Ru  , 44 3)())(( uuuM ==  , .3)())(( uuuN == 2524 3)3()3)(())(()( uuuMuNM ===  .33)3()3)(())(()( 244 uuuNuMNand ===  Therefore ).)(()())(()( uNMuMN   Hence M and N are not − commuting mappings. Example 2.9. Suppose    00:,, −→− RRNM  given by 4 )( uuM = , 5 )( uuN = and u u 1 )( = for all .Ru  2 05 )()( uuMuMNThen == and 2 04 )()( uuNuNM == . Therefore )()( uNMuMN = . Hence M and N are commuting mappings. Also for  0− Ru , 4 4 1 )())(( u uuM == , 5 5 1 )())(( u uuN == 2 0 2 055 111 )())(()( u uu T u MuNM =      =            =      =  . A Result on b-metric space using − compatible mappings 31 2 0 2 044 111 )())(()( u uu S u NuMNand =      =            =      =  . Therefore ))(()())(()( uMNuNM  = Hence M and N are − commuting mappings. Example 2.10. Let M, N,  : RR → be defined by 4 )( uuM = , uuN 4)( = and 4 )( u u = for all Ru  . 444 4)4()4()( uuuMuMN === , 44 4)()( uuNuNM == . Therefore )()( uNMuMN  . Hence M and N are not commuting mappings. Also for Ru  , 4 )())(( 4 4 u uuM == , uuuN == )4())((  4 )())(())(()( 4 4 u uuMuNM ===  4 )( 4 )())(()( 4 4 4 u u u NuMN ==        =  . Therefore, ).)(()())(()( uMNuNM  = Hence M and N are − commuting mappings. Example 2.11. Suppose    00:,, −→− RRNM  given by 3 )( uuM = , 2 3)( uuN = and 2 1 )( u u = for all Ru  . 632 3)3()( uuMuMN == , 63 3)()( uuNuNM == . Therefore )()( uNMuMN  . Hence M and N are not commuting mappings. Also for  0− Ru , 6 3 1 )())(( u uuM == , 42 2 3 1 )3())(( u uuN == 2 46 1 2344 9 9 1 9 1 9 1 )())(()( u uu M u MuNM =      =            =      =  . 9 1 3 11 )())(()( 2 4 1 266 u uu S u NuMN =            =            =      =  . Therefore, ).)(()())(()( uMNuNM   Thirupathi Thota and Srinivas Veladi 32 Hence, M and N are not − commuting mappings. Definition 2.12. A pair of self maps of a b-metric space (X, d) is called weakly − commuting mappings if )( M and )( N are weakly commuting maps. i.e ( ) ( )( ) ( )))((),)(()()()(),()()( uNuMduMNuNMd   for all .Xu  Definition 2.13. The self maps M,Nofa b-metric space X are called  - compatible maps if ( )M and ( )N are compatible if whenever  nu is a Sequence in X such that ( )( ) thenXuNuMd nn ,)(),)(( →  ( ) ( )( ) .0)()()(),()()( →→ nasuMNuNMd nn  Definition 2.14. Two self maps M and N are called weakly − compatible if )( M and )( N are weakly compatible,i.e )( M and )( N commute at their coincidence points. Remark 2.15. It may be observed that − commuting maps are weakly − commuting maps,weakly − commuting maps are − compatible maps and − compatible mappings are weakly − compatible maps.But converse is not true in each case. These facts are presented in the following example. Example2.16. Let M, N,  :    00 −→− RR given by 5)( uuM = , 4 )( uuN = and u u 1 )( = for all Ru  . Here 5 5 1 )())(( u uuM == , 4 4 1 )())(( u uuN == 20 204 11 )())(()( u uu MuNM =      =      =  . 20 205 11 )())(()( u uu NuMNand =      =      =  . Therefore, ))(()())(()( uMNuNM  = Hence, N and M are − commuting mappings. Also for  0− Ru ( ) ( )( ) 0)()()(),()()( 2 2020 =−= uuuMNuNMd  ( ) 2 5 2 45 111 )(,)( u u uu uNuMd − =−= A Result on b-metric space using − compatible mappings 33 ( ) ( )( ) 0)()()(),()()( =uMNuNMd  ( ).)(,)( 1 2 5 uNuMd u u = −  Therefore M and N are weakly − commuting maps. Now ( ) 5 1 )( n n u uM = , ( ) 4 1 )( n n u uN = ( ) .10 11 ))((),)(( 2 45 →→−= n nn nn uas uu uNuMd  Hence M and N are − compatible mappings. Definition 2.16. The self mapping M of a b-metric space (X,d) is said to be − continuous if ( )M is continuous .In other words for every ,0 for all 0 such that ( ) ( ) .)(,)(,   vMuMdyxd The following theorem was proved in [11]. Theorem 2.17[10]. Let f, g, S and T be four self mappings defined on a complete b-metric space(X, d) with the following conditions: (C1) ( ) ( )XTXf  and ( ) ( )XSXg  (C2) ( ) ( ) ( ) ( ) ( ) ( )( )       + TvfudgvSudTvgvdSDufudTvSud k q gvfud ,, 2 1 ,,,,,,max, 4 holds for every Xvu , with 10  q . (C3) The self mappings T and S are both continuous (C4) two pairs ( )Sf , and ( )Tg , are compatible. Then the above four maps will be having a unique fixed point which is common. Now we prove the generalization of Theorem (2.17) in the preceding Theorem under some modified conditions. To do so, we'll need to recall the following lemmas. Thirupathi Thota and Srinivas Veladi 34 Lemma 2.18[10]. Let ( )dX , be a −b metric space with 1k and two sequences }{ nu and }{ nv are b-convergent to u and v respectively. Then we have ( ) ).,(),(suplim,inflim),( 1 2 2 vudkvudvudvud k nn n nn n  →→ Lemma 2.19[9]. Let M and N be − compatible mappings from a b-metric .,)(lim)(limsuchthat itselfintod)(X,space XsomeforuNuM n j n j == →→  Then ( )  == → )()()(lim MuNM n j , if M is − continuous. 3. Main Result Theorem 3.1. Let M, N, P and Q be four self maps and  as defined on a b- metric space (X,d) which is complete with the given conditions: (b1) ( )( ) ( )( )XQXM   and ( )( ) ( )( )XPXN   (b2) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) , , , 2 1 ,, ,, ,, max, 4                         +  vQuMd vNuPd vQvNd uPuMd vQuPd k q vNuMd       holds for every Xvu , with ( ).1,0q (b3) The mappings Q and P are − continuous (b4) the pair of maps ( )PM , and ( )QN , are − compatible. Then the above four maps will be having a unique fixed point which is common. Proof: Using the condition (b1) for the point Xu 0  Xu 1 such that ( ) ( ) . 10 uQuM  = For this point 1u we can select a point Xu 2 such that ( ) ( ) 21 uPuN  = and so on. Continuing this process it is possible to construct a sequence }{ jv such that ( ) ( ) 1222 +== jjj uQuMv  and ( ) ( ) 221212 +++ == jjj uPuNv   .0j We now demonstrate that }{ j v is a cauchy sequence. Take ( ) ( ) ( )( ) 122122 ,, ++ = jjjj uNuMdvvd  A Result on b-metric space using − compatible mappings 35 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ))},(),(( 2 1 ),,(),,(),,(max{ 122122 1212221224 ++ +++ +  jjjj jjjjjj uQuMduNuPd uQuNduPuMduQuPd k q   ))}.,(),(( 2 1 ),,(),,(),,(max{ 221212 2121222124 jjjj jjjjjj vvdvvd vvdvvdvvd k q + = +− +−− } 2 ),( ),,(),,(max{ 1212 1222124 +− +− = jj jjjj vvd vvdvvd k q ))}.,(),(( 2 ),,(),,(max{ 1222121222124 +−+− + jjjjjjjj vvdvvd k vvdvvd k q If ( ) ( ) jjjj vvdvvd 212122 ,, −+  for some j, then the above inequality gives ( ) ( ) 1223122 ,, ++  jjjj vvd k q vvd a contradiction. Hence ( ) ( ) jjjj vvdvvd 212122 ,, −+  for all Nj  . Now the above inequality gives ( ) ( ).,, 2123122 jjjj vvd k q vvd −+  -------------------(1) Similarly ( ) ( ).,, 12223212 −−−  jjjj vvd k q vvd ------------(2) From (1) and (2) we have ( ) ( ),,, 211 −−−  jjjj vvdvvd  where 1 3 = k q  and 2j . Hence for all 2j ,we obtain ( ) ( ) ),(..........,, 01 1 211 vvdvvdvvd j jjjj − −−−   .----------(3) So for all lj  , we have ( ) ( ) ),(..........),(,, 1 1 21 2 1 jj lj lllllj yydkyydkyykdyyd − −− +++ +++ . Now from (3),we have ( ) ),()..............(, 01 1112 vvdkkkvvd jljll lj −−−+ +++  ),(....)..........1( 01 22 vvdkkk l +++  ).,( 1 01 vvd k k l   −  Taking limits as →jl, ,we have 0),( → lj vvd as k is less than one. Thirupathi Thota and Srinivas Veladi 36 Therefore }{ j v is a cauchy sequence in X and by completeness of X, it converges to some point  in X such that ( ) ( ) ( ) ( )  ==== + → + → + →→ 2212122 limlimlimlim j j j j j j j j uPuNuQuM . Since P is − continuous, therefore ( ) ( ) ( ) PuPP j j = + → 22 lim and ( ) ( ) ( ) .lim 2  PuMP j j = → By (b4) we have (M,P) is − compatible, ( ) ( ) ( ) ( )( ) 0,lim 22 = → jj j uMPuPMd  so by Lemma (2.19), we have ( ) ( ) ( ) PuPM j j = → 2 lim . Now putting j uPu 2 )(= and 12 + = j uv in (b2), we get take sup limit as →j on both the sides and by Lemma (2.18),we get ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) , ,)( ,)( 2 1 ,, ,)(,)( ,,)( max,)( 122 122 1212 22 122 4122                           +  + + ++ + + jj jj jj jj jj jj uQuPMd uNuPPd uQuNd uPPuPMd uQuPPd k q uNuPMd       ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) , ,)(suplim ,)(suplim 2 1 ,,suplim ,)(,)(suplim ,,)(suplim max ,)(lim ,)( 122 122 1212 22 122 4 1222                                   +   + → + → ++ → → + → + → jj j jj j jj j jj j jj j jj j uQuPMd uNuPPd uQuNd uPPuPMd uQuPPd k q uNuPMd k Pd        ( ) ( ) ( ) ( ) ( )( )           +    ,)(,)( 2 1 ,,,)(,)(,,)( max 2 4 PdPd dPPdPdk k q ( ) ( )          ,)( ,0,0,,)( max 2 4 Pd Pdk k q ( ) ( )          ,)( ,0,0,,)( max 2 4 Pd Pdk k q = ( ) ,)(2 4 Pdk k q A Result on b-metric space using − compatible mappings 37 = ( ).,)( 2 Pd k q Therefore ( ) ( ) ,)(,)( PqdPd  . As ,10  q so .)(  =P Since Q is − continuous, therefore ( ) ( ) ( ) TuTT j j = + → 22 lim and ( ) ( ) ( ) .lim 2  QuNQ j j = → Since the pair (N,Q) is − compatible, we have ( ) ( ) ( ) ( )( ) .0,lim 22 = → jj j uNQuQNd  So by Lemma (2.18) we have ( ) ( ) ( ) QuQN j j = → 2 lim . Now putting j uu 2 = and ( ) 12 + = j uQv  in (b2), we get ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )                           +  + + ++ + + 122 122 1212 2 122 4122 , , 2 1 ,, ,, ,, max, jj jj jj j jj jj uQQuMd uQNuPd uQQuQNd uPuMd uQQuPd k q uQNuMd       take sup limit as →j on both the sides and by Lemma (2.18),we get ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )                +  +     Qd Qdk Qdk k q uQNuMdQd jj , , 2 ,0,0,, max,, 2 2 4122 ( )( ) ( ) )(,, 2 Qd k q Qd  which implies that .)(  Q= Therefore .)()(  == QP ---------(4) Again putting =u and 12 + = j uv in (b2) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) , , , 2 1 ,, ,, ,, max, 12 12 1212 12 412                           +  + + ++ + + j j jj j j uQMd uNPd uQuNd PMd uQPd k q uNMd       take sup limit as →j on both the sides and by Lemma (2.18) we get Thirupathi Thota and Srinivas Veladi 38 ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) , , , 2 ,, ,,,, max, 2 2 22 4                         +       Md Pdk dk PMdkPdk k q Md = ( )( )., 2 Md k q This implies that ( )( ) .0, =Md That gives ( )  =M as .10  q Again putting =u and =v in (b2), we get ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) , , , 2 1 ,, ,, ,, max, 4                         +        QMd NPd QNd PMd QPd k q NMd ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) , ,, 2 1 ,,,,,, max, 4           +     dNd Nddd k q Nd ( )( ) ( )( ) ( )( )( ) , 0, 2 1 ,,,0,0 max, 4           +     Nd Nd k q Nd ( )( ) ( )( ) ( )( )( ) , 0, 2 1 ,,,0,0 max, 4           +     Nd Nd k q Nd ( )( ) , 2 Nd k q = ( )( ) ,Nqd which implies that ( )( ) 0, = Nd ( ) N= . Therefore ( ) ( ) . == NM ----------(5) Hence from (4) and (5) we obtain  ==== )()()()( NMQP . Therefore  is a common − fixed point of M, N, P and Q. A Result on b-metric space using − compatible mappings 39 Uniqueness: Assume that )(   is another common fixed point of the four mappings M, N, P, and Q. Put =u and =v in (b2) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )                         +        QMd NPd QNd PMd QPd k q NMd , , 2 1 ,, ,, ,, max, 4 ( ) ( ) ( ) ( ) ( )( )           +    ,, 2 ,,,,,, max 2 222 4 dd k dkdkdk k q ( ) ),(,0,0,,max 22 4  dkdk k q = ),( ),( 2 42   dk k q k d = ).,(),(  qdd  10  qAs ,so . = Hence the four maps M, N, P and Q will be having a unique common fixed point. Now we give an illustration to support our result. Example 3.2: Suppose  1,0=X is a b-metric space ( ) 2 , vuvud −= where Xvu , . Define the four self maps M, N, P , Q and  as follows ( )           =  + === 1 2 1 , 6 1 2 1 , 2 1 2 1 0, 6 2 )(;)( u u u u uNuMuu ; Thirupathi Thota and Srinivas Veladi 40 ( )      −  + == 1 2 1 ,1 2 1 0, 6 34 )( uu u uPuQ  ; ( ) (        == 2 1 33.0,16.0)( XNXM , ( ) ( )       == 2 1 ,0XQXP , clearly the condition (b1) is satisfied. Take a sequence as j u j 1 2 1 −= for . Now 6 5 6 1 6 5 6 2 1 2 1 lim 1 2 1 limlim =−=       +− =      −= →→→ n j j MMu jj j j and 6 5 6 3 1 2 1 4 lim 1 2 1 limlim =         +      − =      −= →→→ j j QPu jj j j that is a sequence }{ ju in X such that 6 5 limlim == →→ j j j j PuMu . Similarly . 6 5 limlim == →→ j j j j QuNu Also andkas j M j M j MPMPu j →=      −=               +      − =      −= 2 1 3 2 6 5 6 3 1 2 1 4 1 2 1 . 6 1 6 1 6 5 1 6 1 6 5 6 2 1 2 1 1 2 1 →=+−=      −=               +      − =      −= jas nn P j P j PMPMu j 0j  A Result on b-metric space using − compatible mappings 41 ( ) .0 36 4 6 5 2 1 6 5 , 2 1 ,lim 2 =−=      = → dPMuMPudthatSo jj j ( ) .0,lim  → jj j QNuNQudSimilarly Showing that the pairs ( )PM , and ( )QN, are not compatible mappings. ( ) ( ) →=      −=               +      − =      −= jas j j j MuMAgain j 6 5 6 1 6 5 6 2 1 2 1 1 2 1 )(  ( ) ( ) . 6 5 3 2 6 5 6 3 1 2 1 4 1 2 1 )( →=      −=               +      − =      −= jas j j j PuPalsoand j  ( ) ( ) ( ) andjas j M j MuPM j →=      =            −=      −= 6 1 6 1 3 2 6 5 3 2 6 5  ( ) ( ) ( ) . 6 1 6 1 6 1 6 1 6 5 1 6 1 6 5 →=      +=       +−=      −= jas nnj QuMP j  ( ) ( ) ( ) ( )( ) .0 6 1 6 1 6 1 , 6 1 ,lim 2 =−=      = → duMPuPMdthatSo jj j  ( ) ( ) ( ) ( )( ) .0,lim = → jj j uQNuNQdSimilarly  Showing that the pairs ( )PM  , and ( )QN  , are  -compatible mappings. Now we fulfill the requirement that the mappings M, N, P and Q satisfy the condition (b2). Thirupathi Thota and Srinivas Veladi 42 We have ( ) 0)( =uM , ( ) 16 )( v vN = ( ) . 2561616 ,0)(,)( 22 vvv dvNuMd ==      = Also ( ) uuP =)( and ( ) 4 )( v vQ = , ( ) , 44 ,)(,)( 2 v u v udvQuPd −=      = ( ) ( ) ,,0)(,)( 2uuduPuMd == ( ) , 256 9 4164 , 16 )(,)( 22 vvvvv dvTvgd =−=      = ( ) , 1616 ,)(,)( 2 v u v udvguSd −=      = ( ) . 164 ,0)(,)( 2 vv dvQuMd =      = Substituting all these in the inequality (b2), we obtain                 +      −      − 16162 , 256 9 ,, 4 max 256 2222 22 2 2 4 2 vv u kv kuk v uk k qv If we choose 5.0=u , 9.0=v and k = 2 we obtain  4948.0,112.0,1,3024.0max 16 00316.0 2 k q  )1( 16 00316.0 q  ).1,0(05.0)1( 16 00316.0 = q q Hence the condition (b2) is satisfied. 4. Conclusion This work is focused to generate the existence of common fixed point theorem proved by J.R.Roshan and others mentioned in Theorem (2.17) by employing A Result on b-metric space using − compatible mappings 43 some weaker conditions − compatible and − continuous mappings instead of compatible and continuous mappings. At the end of the theorem our result is justified with a suitable example. References [1] V.Srinivas and K.Mallaiah. A Resulton Multiplicative Metric Space. Journal of Mathematical and computational science, 10(5),1384-1394, 2020. [2] V.Nagaraju,Bathini Raju and P.Thirupathi. Common fixed point theorem for four self maps satisfying common limit point property. Journal of Mathematical and Computational Science, 10(4), 1228-1238, 2020. [3] B.Vijayabaskar Reddy and V.Srinivas. Fixed point results on multiplicative semi metric space. Journal of Scientific Research, 12(3), 341-348, 2020. [4] V.Srinivas,T.Thirupathi and K.Mallaiah. 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