Occasionally Weakly Compatible Mappings

Transcription

1 Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 3, Available online a hp://pubs.sciepub.com/jan/3/3/2 Science and Educaion Publishing DOI: /jan Occasionally Weakly Compaible Mappings Ami Kumar Govery *, Mama Singh 2 1 School of Sudies in Mahemaics, Vikram Universiy, Ujjain (M.P., India 2 Deparmen of Mahemaical Science and Compuer Applicaion, Bundelkhand Universiy, Jhansi (U.P., India *Corresponding auhor: amigovery@gmail.com Absrac In his paper, he concep of compaible maps of ype (A and occasionally weakly compaible maps in fuzzy meric space have been applied o prove common fixed poin heorem. A fixed poin heorem for six self maps has been esablished using he concep of compaible maps of ype (A and occasionally weakly compaible maps, which generalizes he resul of Cho [16]. Keywords: common fixed poins, fuzzy meric space, compaible maps, compaible maps of ype (A and occasionally weakly compaible maps Cie This Aricle: Ami Kumar Govery, and Mama Singh, Occasionally Weakly Compaible Mappings. Turkish Journal of Analysis and Number Theory, vol. 3, no. 3 (2015: doi: /jan Inroducion The concep of Fuzzy ses was iniially invesigaed by Zadeh [13] as a new way o represen vagueness in everyday life. Subsequenly, i was developed by many auhors and used in various fields. To use his concep in Topology and Analysis, several researchers have defined Fuzzy meric space in various ways. In his paper we deal wih he Fuzzy meric space defined by Kramosil and Michalek [11] and modified by George and Veeramani [20]. Recenly, Grebiec [1] has proved fixed poin resuls for Fuzzy meric space. In he sequel, Singh and Chauhan [12] inroduced he concep of compaible mappings of Fuzzy meric space and proved he common fixed poin heorem. Jungck e. al. [2] inroduced he concep of compaible maps of ype (A in meric space and proved fixed poin heorems. Using he concep of compaible maps of ype (A, Jain e. al. [18] proved a fixed poin heorem for six self maps in a fuzzy meric space. Singh e. al. [7,8] proved fixed poin heorems in a fuzzy meric space. Recenly in 2012, Jain e. al. [4,5] and Sharma e. al. [6] proved various fixed poin heorems using he conceps of semi-compaible mappings, propery (E.A. and absorbing mappings. The concep of occasionally weakly compaible mappings in meric spaces is inroduced by Al- Thagafi and Shahzad [14] which is mos general among all he commuaiviy conceps. Recenly, Khan and Sumira [15] exended he noion of occasionally weakly compaible maps o fuzzy meric space. In his paper, a fixed poin heorem for six self maps has been esablished using he concep of compaible maps of ype (A occasionally weakly compaible mappings, which generalizes he resul of Cho [16]. For he sake of compleeness, we recall some definiions and known resuls in Fuzzy meric space. 2. Definiions, lemmas, Remarks, Proposiions Definiion 2.1. [10] A binary operaion : [ 0,1] [ 0,1] [ 0,1] is called a -norm if ( [0, 1], is an abelian opological monoid wih uni 1 such ha a b c d. Whenever a c and b d for aa, bb, cc, dd. [0,1]. Examples of -norms are: aa bb = aaaa aaaaaa aa bb = mmmmmm {aa, bb}. Definiion 2.2. [10] The 3-uple (XX, MM, is said o be a Fuzzy meric space if X is an arbirary se, is a coninuous -norm and M is a Fuzzy se in x 2 [ 0, saisfying he following condiions: For all xx, yy, zz XX and ss, > 0 (FM-1 MM (xx, yy, 0 = 0, (FM-2 MM (xx, yy, = 1 ffffff aaaaaa > 0 iiiiii xx = yy, (FM-3 MM (xx, yy, = MM (yy, xx,, (FM-4 MM(xx, yy, MM(yy, zz, ss = MM(xx, zz, + ss, (FM-5MM (xx, yy, : [0, [0, 1] iiii llllllll cccccccccccccccccccc, (FM-6 lim nn MM(xx, yy, = 1 Noe ha MM (xx, yy, can be considered as he degree of nearness beween x and y wih respec o. We idenify xx = yy wih MM (xx, yy, = 1 for all > 0. The following example shows ha every meric space induces a Fuzzy meric space. Example 2.1. [10] Le (XX, dd be a meric space. Define aa bb = mmmmmm {aa, bb} and M( xy,, = + d( xy, for all xy, Xand > 0. Then (XX, MM, is a Fuzzy meric space. I is called he Fuzzy meric space induced by dd.

2 Turkish Journal of Analysis and Number Theory 79 Definiion 2.3. [10] A sequence { x n } in a Fuzzy meric space (X, M, is said o be a Cauchy sequence if and only if for each ε > 0, > 0, here exiss n 0 N such ha M (X n, X m, > 1 ε for all n, m n 0. The sequence {x n } is said o converge o a poin x in X if and only if for each ε > 0, > 0 here exiss n 0 N such ha M (x n, x, > 1 ε for all n, m n 0. A Fuzzy meric space (X, M, is said o be complee if every Cauchy sequence in i converges o a poin in i. Definiion 2.4. [12] Self mappings AA and SS of a Fuzzy meric space (XX, MM, are said o be compaible if and only if MM (AAAAxx nn, SSSSxx nn, 1 for all > 0, whenever {xx nn } is a sequence in XX such ha SSSSnn, AAAAnn pp for some pp in XX as nn. Definiion 2.5. [18] Self maps AA and SS of a Fuzzy meric space (XX, MM, are said o be compaible maps of ype (AA if MM(AAAAxx nn, SSSSxx nn, 1and MM(SSAAxx nn, AAAAxx nn, 1 for all > 0, whenever {xx nn } is a sequence in XX such ha SSxx nn, AAxx nn pp for some pp in XX as nn. Definiion 2.6. [15] Two maps AA and SS from a Fuzzy meric space (XX, MM, ino iself are said o be Occasionally weakly compaible (owc if and only if here is a poin xx XX, which is coincidence poin of AA and SS a which AA and SS commue. Remark 2.1. [18] The concep of compaible maps of ype (AA and occasionally weakly compaibiliy is more general han he concep of compaible maps in a Fuzzy meric space. Proposiion 2.1. [18] In a Fuzzy meric space (XX, MM, limi of a sequence is unique. Lemma 2.1. [1] Le (XX, MM, be a fuzzy meric space. Then for all xx, yy XX, MM(xx, yy,. is a non-decreasing funcion. Lemma 2.2. [16] Le (XX, MM, be a fuzzy meric space. If here exiss kk (0, 1 such ha for all xx, yy XX, MM(xx, yy, kkkk MM(xx, yy, > 0, heeee xx = yy. Lemma 2.3. [18] Le {xx nn } be a sequence in a fuzzy meric space (XX, MM,. If here exiss a number kk (0, 1 such ha MM(xx nn +2, xx nn +1, kkkk MM(xx nn+1, xx nn, > 0 and nn NN. Then {xx nn } is a Cauchy sequence in XX. Proposiion 2.2. [18] Le AA and SS be concep of compaible maps of ype (AA of a complee fuzzy meric space (XX, MM, wih coninuous -norm defined by aa bb = mmmmmm {aa, bb} for all aa, bb [0, 1] and SSSS = TTTT for some uu in XX. Then SSSSSS = TTTTTT = SSSSSS = TTTTTT. Lemma 2.4. [3] The only -norm saisfying rr rr rr for all rr [0, 1] is he minimum -norm, ha is aa bb = mmmmmm {aa, bb} for all aa, bb [0, 1]. 3. Main Resul Theorem 3.1. Le (XX, MM, be a complee fuzzy meric space and le AA, BB, SS, TT, PP and QQ be mappings from XX ino iself such ha he following condiions are saisfied: (a PP(XX SSSS(XX, QQ(XX AAAA(XX; (b AAAA = BBBB, SSSS = TTTT, PPPP = BBBB, QQQQ = TTTT; (c eiher PP or AAAA is coninuous; (d (PP, AAAA is compaible maps of ype (AA and (QQ, SSSS is occasionally weakly compaible ; (e here exiss qq (0, 1 such ha for every xx, yy XX and > 0 M ( Px, Qy, q M ( ABx, STy,* M ( Px, ABx, v * M ( Qy, STy, * M ( Px, STy,. Then AA, BB, SS, TT, PP and QQ have a unique common fixed poin in XX. Proof: Le xx 0 X. From (a here exis xx 1, xx 2 XX such ha PPxx 0 = SSSSxx 1 and xx 1 = AAAAxx 2. Inducively, we can consruc sequences {xx nn } and {yy nn } in XX such ha PPxx 2nn 2 = SSSSxx 2nn 1 = yy 2nn 1 and QQxx 2nn 1 = AAAAxx 2nn = yy 2nn ffffff nn = 2, 3,.... Sep 1. Pu xx = xx 2nn and yy = xx 2nn+1 in (e, we ge ( 2n, 2n+ ( 2n, 2n+ * ( 2n, 2n, ( 2n+ 1 2n+ 1 2n 2n+ 1 ( 2n, 2n+ * ( 2n+ 2n, M ( y y M ( y y M Px Qx q M ABx STx M Px ABx * M Qx, STx, * M ( Px, STx,. = M y y M y y * 2n+ 2, 2n+ * 2n+ 2n+ M( y, y,* M( y, y,. 2n 2n+ 1 2n+ 1 2n+ 2 From lemma 2.1 and 2.2, we have MM(yy 2nn+1, yy 2nn +2, qqqq MM(yy 2nn, yy 2nn +11,. Similarly, we have MM(yy 2nn+2, yy 2nn +3, qqqq MM(yy 2nn+1, yy 2nn +2,. Thus, we have MM(yy nn +1, yy nn +2, qqqq MM(yy nn, yy nn+1, ffffff nn = 2,... M( yn, yn+ M( yn, yn+ / q M( yn 2, yn / q M ( y, y, / q 1 asn, 1 2 and hence MM(yy nn, yy nn+1, 1 as nn for any > 0. For each εε > 0 and > 0, we can choose nn 0 NN such ha MM(yy nn, yy nn +1, > 1 εε for all nn > nn 0. For mm, nn NN, we suppose mm nn. Then we have M( yn, ym, M( yn, yn+ / m n * M( yn+ yn+ 2, / m n * * M( ym ym, / m n ( 1 ε *1 ( ε * *1 ( ε ( m n imes M( yn, ym, (1 ε and hence {yy nn } is a Cauchy sequence in XX. Since (XX, MM, is complee, {yy nn } converges o some poin zz XX. Also is subsequence s converges o he same poin i.e. zz XX i.e., {QQxx 2nn+1 } zz and {SSSSxx 2nn+1 } zz (1 {PPxx 2nn } zz and {AAAAxx 2nn } zz. (2 n

3 80 Turkish Journal of Analysis and Number Theory Case I. Suppose AAAA is coninuous. Since AAAA is coninuous, we have (AAAA 2 xx 2nn AAAAAA and AAAAAAxx 2nn AAAAAA. As (PP, AAAA is compaible pair of ype (AA, we have PPPPPPxx 2nn AAAAAA. Sep 2. Pu xx = AAAAxx 2nn and yy = xx 2nn+1 in (e, we ge (,, M PABx2n Qx2n+ 1 q M ( ABABx2n, STx2n+ * M ( PABx2n, ABABx2n, * M ( Qx2n+ STx2n+ * M ( PABx2n, STx2n+. Taking, we ge M ( ABz, z, q M ( ABz, z, * M ( ABz, ABz, * M ( z, z, * M ( ABz, z, M ( ABz, z,* M ( ABz, z, ii. ee. MM(AAAAAA, zz, qqqq MM(AAAAAA, zz,. Therefore, by using lemma 2.2, we ge AAAAAA = zz. (3 Sep 3. Pu xx = zz and yy = xx2nn + 1 in (e, we have M ( Pz, Qx2n+ q M ( ABz, STx2n+ * M ( Pz, ABz, * M ( Qx2n+ STx2n+ * M ( Pz, STx2n+. Taking nn and using equaion ( we ge MPzzq (,, Mzz (,, * MPzz (,, * M(, zz,* M( Pzz,, M ( Pz, z,* M ( Pz, z, i.e. MM(PPPP, zz, qqqq MM(PPPP, zz,. Therefore, by using lemma 2.2, we ge PPPP = zz. Therefore, AAAAAA = PPPP = zz. Sep 4. Puing xx = BBBB and yy = xx 2nn+1 in condiion (e, we ge M ( PBz, Qx2n+ q M ( ABBz, STx2n+ * M ( PBz, ABBz, * M ( Qx2n+ STx2n+ * M ( PBz, STx2n+. As BBBB = PPPP, AAAA = BBBB, so we have ( = ( = ( ( P Bz B Pz Bzand AB Bz = ( BA( Bz = B( ABz = Bz. Taking nn and using ( we ge M ( Bz, z, q M ( Bz, z,* M ( Bz, Bz,* M (, z z,* M ( Bz, z, M ( Bz, z,* M ( Bz, z, i.e. MM(BBBB, zz, qqqq MM(BBBB, zz,. Therefore, by using lemma 2.2, we ge BBBB = zz and also we have AAAAAA = zz AAAA = zz. Therefore, Az = Bz = Pz = z (4 Sep 5. As PP(XX SSSS(XX, here exiss uu XX such ha zz = PPPP = SSSSSS. Puing xx = xx 2nn and yy = uu in (e, we ge M ( Px2n, Qu, q M ( ABx2n, STu,* M ( Px2n, ABx2n, * M Qu, STu, * M ( Px2 n, STu,. ( Taking nn and using (1 and (2, we ge M(, zquq, M(,,* zz M(,, zz * MQuz (,,* M(, zz,* M(, zz, M ( Qu, z, i.e. MM(zz, QQQQ, qqqq MM(zz, QQQQ,. Therefore, by using lemma 2.2, we ge QQQQ = zz. Hence SSSSSS = zz = QQQQ. Since (QQ, SSSS is occasionally weakly compaible, herefore, by proposiion (2.2, we have QQQQQQQQ = SSSSSSSS. Thus QQQQ = SSSSSS. Sep 6. Puing xx = xx 2nn and yy = zz in (e, we ge ( 2n, Qz, q ( ( M Px M ABx2n, STz, * M ( Px2n, ABx2n, * M Qz, STz, * M ( Px2n, STz,. Taking nn and using (2 and sep 5, we ge M(, zqzq, M(, zqz,* M(,, zz * MQzQz (,,* M(, zqz, M(, zqz,* M(, zqz, ii. ee. MM(zz, QQQQ, qqqq MM(zz, QQQQ,. Therefore, by using lemma 2.2, we ge Qz = z. Sep 7. Puing xx = xx 2nn and yy = TTTT in (e, we ge M ( Px2n, QTz, q M ( ABx2n, STTz,* M ( Px2n, ABx2n, * M ( QTz, STTz, * M ( Px2n, STTz,. As QQQQ = TTTT and SSSS = TTTT, we have QTz = TQz = Tz and ST ( Tz = T ( STz = TQz = Tz. Taking nn, we ge M(, z Tz, q M(, z Tz,* M(,, z z * M ( Tz, Tz,* M (, z Tz, M (, z Tz,* M (, z Tz, i.e. MM(zz, TTTT, qqqq MM(zz, TTTT, Therefore, by using lemma 2.2, we ge TTTT = zz. Now SSSSSS = TTTT = zz implies SSSS = zz. Hence Combining (4 and (5, we ge Sz = Tz = Qz = z. (5 Az = Bz = Pz = Qz = Tz = Sz = z. Hence, zz is he common fixed poin of AA, BB, SS, TT, PP and QQ.

4 Turkish Journal of Analysis and Number Theory 81 Case II. Suppose PP is coninuous. As PP is coninuous, PP 2 xx 2nn PPPP and PP(AAAAxx 2nn PPPP. As (PP, AAAA is compaible pair of ype (AA, (AAAAPPxx 2nn PPPP. Sep 8. Puing xx = PPxx 2nn and yy = xx 2nn+1 in condiion (e, we have M ( PPx2n, Qx2n+ q M ( ABPx2n, STx2n+ * M ( PPx2n, ABPx2n, * M ( Qx2n+ STx2n+ * M ( PPx2n, STx2n+. Taking nn, we ge M ( Pz, z, q M ( Pz, z, * M ( Pz, Pz, * M( zz,, * M( Pzz,, M ( Pz, z,* M ( Pz, z, i.e. MM(PPPP, zz, qqqq MM(PPPP, zz,. Therefore by using lemma 2.2, we have PPPP = zz. Furher, using seps 5,6,7, we ge QQQQ = SSSSSS = SSSS = TTTT = zz. Sep 9. As QQ(XX AAAA(XX, here exiss ww XX such ha zz = QQQQ = AAAAAA Pu xx = ww and yy = xx 2nn+1 in (e, we ge M ( Pw, Qx2n+ q M ( ABw, STx2n+ * M ( Pw, ABw, * M ( Qx2n+ STx2n+ * M ( Pw, STx2n+. Taking nn, we ge M ( Pw, z, q M ( z, z, * M ( Pw, z, * M( zz,, * M( Pwz,, M ( Pw, z,* M ( Pw, z, i.e. MM(PPPP, zz, qqqq MM(PPPP, zz,. Therefore, by using lemma 2.2, we ge PPPP = zz. Therefore, AAAAAA = zz = PPPP. As (PP, AAAA is compaible pair of ype (AA, hen by proposiion (2.2, we have PPPP = AAAAAA. Also, from sep 4, we ge BBBB = zz. AAAA = BBBB = PPPP = zz. Furher, using seps 5, 6, 7, we ge QQQQ = SSSSSS = SSSS = TTTT = zz i.e. zz is he common fixed poin of he six maps AA, BB, SS, TT, PP and QQ in his case also. Uniqueness: Le uu be anoher common fixed poin of AA, BB, SS, TT, PP and QQ. Then AAAA = BBBB = PPPP = QQQQ = SSSS = TTTT = uu. Pu xx = zz and yy = uu in (e, we ge (,, (,, * (,, M Pz Qu q M ABz STu M Pz ABz Taking nn, we ge * M ( Qu, STu, * M ( Pz, STu,. M(, zuq, M(, zu,* M(,, zz * Muu (,, * M( zu,, M(, zu,* M(, zu, i.e. MM(zz, uu, qqqq MM(zz, uu,. Therefore by using lemma (2.2, we ge zz = uu. Therefore zz is he unique common fixed poin of self maps AA, BB, SS, TT, PP and QQ. Remark 3.1. If we ake BB = TT = II, he ideniy map on XX in heorem 3. hen condiion (bb is saisfied rivially and we ge Corollary 3.1. Le (XX, MM, be a complee fuzzy meric space and le AA, SS, PP and QQ be mappings from XX ino iself such ha he following condiions are saisfied: (a PP(XX SS(XX, QQ(XX AA(XX; (b eiher AA or PP is coninuous; (c (PP, AA is compaible maps of ype (AA and (QQ, SS is occasionally weakly compaible; (d here exiss qq (0, 1 such ha for every xx, yy XX and > 0 M ( Px, Qy, q M ( Ax, Sy,* M ( Px, Ax, * M ( Qy, Sy, * M ( Px, Sy,. Then AA, SS, PP and QQ have a unique common fixed poin in XX. Remark 3.2. In view of remark 3. corollary 3.1 is a generalizaion of he resul of Cho [16] in he sense ha condiion of compaibiliy of he pairs of self maps has been resriced o compaibiliy of ype (AA occasionally weakly compaible and only one map of he firs pair is needed o be coninuous. Acknowledgemen Auhors are hankful o he referee for his valuable commens. References [1] George and P. Veeramani, On some resuls in Fuzzy meric spaces, Fuzzy Ses and Sysems 64 (1994, [2] Jain and B. Singh, A fixed poin heorem for compaible mappings of ype (A in fuzzy meric space, Aca Ciencia Indica, Vol. XXXIII M, No. 2 (2007, [3] Jain, M. Sharma and B. Singh, Fixed poin heorem using compaibiliy of ype (β in Fuzzy meric space, Chh. J. Sci. & Tech., Vol. 3 & 4, ( , [4] Jain, V.H. Badshah, S.K. Prasad, Fixed Poin Theorem in Fuzzy Meric Space for Semi-Compaible Mappings, In. J. Res. Rev. Appl. Sci. 12 (2012, [5] Jain, V.H. Badshah, S.K. Prasad, The Propery (E.A. and The Fixed Poin Theorem in Fuzzy Meric, In. J. Res. Rev. Appl. Sci. 12 (2012, [6] Sharma, A. Jain, S. Chaudhary, A noe on absorbing mappings and fixed poin heorems in fuzzy meric space, In. J. Theoreical Appl. Sci. 4 (2012, [7] Singh, A. Jain, A.K. Govery, Compaibiliy of ype (ββ and fixed poin heorem in Fuzzy meric space, Appl. Mah. Sci. 5 ( [8] Singh, A. Jain, A.K. Govery, Compaibiliy of ype (A and fixed poin heorem in Fuzzy meric space,in. J. Conemp. Mah. Sci. 6 ( [9] Singh and M.S. Chouhan, Common fixed poins of compaible maps in Fuzzy meric spaces, Fuzzy ses and sysems, 115 (2000,

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