In this paper, we have generalized the result of Ferhan Sola Erduran [15] and Anupama Gupta [3] using sub compatible map of type (α) and subsequent continuity. We established a common fixed-point theorem for six maps

Dave, Neetu

Gupta, Anupama

Gupta, Arvind

English

AGPE The Royal Gondwana Research Journal

   Vol-4,  Issue-01,  January 2023   ISSN (E): 2583-1348 AGPE The Royal Gondwana Research Journal of History, Science, Economic, Political and Social science        24 Common fixed- point theorem in non-archimedean intuitionistic fuzzy metric …                           Abstract:    In this paper, we have generalized the result of Ferhan Sola Erduran [15] and Anupama Gupta [3] using sub compatible map of type (α) and subsequent continuity. We established a common fixed-point theorem for six maps.  Keywords: Weak non- Archimedean intuitionistic fuzzy metric space, subcompatible maps of type (α) & (β), common fixed point.  Introduction    The fuzzy set was present and explains by Zadeh [19] & fuzzy metric space introduced by Michalak and Kramosil [11], the notion of fuzzy metric space was modified by George and Veeramani [8] in distinct ways. Vasuki verified fixed point theorem for R-weakly commuting maps. Pant found the new conception of common fixed-point theorems. Intuitionistic fuzzy metric space defined by Alaca et.al.[1] with continuous t-norm and continuous t-conorm. Compatible maps, compatible maps of type (α) & (β) introduced by Turkoglu et.al.[18] in intuitionistic fuzzy metric space. Lately, Erduran [17] established the concept of weak non- Archimedean intuitionistic fuzzy metric space.        Main Result  Theorem: Let A, B, S, T, P and Q be self-maps of a weak non-Archimedean intuitionistic fuzzy metric space (X, M, N, ∗, ◊) and let the pair (P, AB) and (Q, ST) are subcompatible maps of type CORRESPONDING AUTHOR: RESEARCH  ARTICLE  Neetu Dave Research Scholar, Department of Mathematics,  Govt. MVM college Bhopal, Madhya Pradesh, India. Email: trivedi223@gmail.com COMMON FIXED- POINT THEOREM IN NON-ARCHIMEDEAN INTUITIONISTIC FUZZY METRIC SPACE BY USING SUBCOMPATIBLE MAPS OF  TYPE (α) WITH SIX MAPS    1Dr. Arvind Gupta, 2Dr. Anupama Gupta, 3Neetu Dave  1. Professor, Department of Mathematics, Govt. MVM college Bhopal (M.P) India. 2. Professor, Department of Mathematics, Govt. MVM college Bhopal (M.P) India.  3. Research Scholar, Department of Mathematics, Govt. MVM college Bhopal (M.P) India.    AGPE THE ROYAL GONDWANA RESEARCH JOURNAL  OF HISTORY, SCIENCE, ECONOMIC, POLITICAL AND SOCIAL SCIENCE ISSN (E): 2583-1348  A Peer reviewed  Open Accsess & Indexed   Volume 04  Issue 01   January 2023  www.agpegondwanajournal.co.in  Page No. 24-27       Vol-4,  Issue-01,  January 2023   ISSN (E): 2583-1348 AGPE The Royal Gondwana Research Journal of History, Science, Economic, Political and Social science        25 Common fixed- point theorem in non-archimedean intuitionistic fuzzy metric … (α) and subsequentially continuous. If   (       ) ∗ , (        )   (        )-    ,    * (        )  (         )+ {   ( (        )   (        )+]                      (1)          (       ) ∗ , (        )   (        )-     ,    * (        )  (         )+ {   ( (        )   (        )+]                         (2) For all x, y   X, t > 0, where     : [0, 1] → [0, 1] are continuous functions such that  ( )         ( )    for each s   (0, 1). Then A, B, S, T, P and Q have a unique fixed point in X.  Proof.  Since the pairs (P, AB) and (Q, ST) are subcompatible maps of type (α) and subsequentially continuous, then there exist two sequences {xn} and {yn} in X such that                              and satisfy         (              )   (        )           (              )   (        )              (            )   (        )                       (            )   (        )                                      and satisfy        (              )   (        )           (              )   (        )              (            )   (        )                    (            )   (        )     Therefore, Pz = ABz and Qw = STw, that is z is a coincidence point of P and AB; w is a coincidence point of Q and ST. Now, we prove that    . By using (1) and (2) for       and      , we get  (         ) ∗ , (          )   (          )-   ,    * (          )  (           )+ *   ( (          )  (          )+-  (         ) ∗ , (          )   (          )-    ,    * (          )  (           )+ *   ( (          )  (          )+- Taking the limit as n → ∞, we have  (     ) ∗ , (     )   (     )-    ,    * (     )  (     )+ *   ( (     )   (     )+-  (      ) ∗ , (     )   (     )-      ,    * (     )  (     )+ *   ( (     )   (     )+- That is  (      )    (     )  (      )     (     ) which yield z = w. Again using (1) and (2) for     and      , we obtain    Vol-4,  Issue-01,  January 2023   ISSN (E): 2583-1348 AGPE The Royal Gondwana Research Journal of History, Science, Economic, Political and Social science        26 Common fixed- point theorem in non-archimedean intuitionistic fuzzy metric …  (        ) ∗ , (        )   (          )-   ,    * (        )  (          )+ *   ( (         )  (         )+-  (        ) ∗ , (        )   (          )-    ,    * (        )  (          )+ *   ( (         )   (         )+- Taking the limit as n → ∞, we have  (      ) ∗ , (        )   (     )-   ,    * (        )  (       )+ *   ( (       )   (      )+-  (      ) ∗ , (        )   (     )-    ,    * (        )  (       )+ *   ( (       )   (      )+- That is                                              (      )   (      )                                               (      )    (      ) Which yield      =   Similarly, if we Again using (1) and (2) for      and     , we obtain                                              (      )    (      )                                               (      )    (      ) Which yield       . Therefore     is common fixed point of A, B, S, T, P and Q.  Uniqueness: -    Suppose that there exists another fixed point   of A, B, S, T, P and Q. then from (1) and (2), we have  (       ) ∗ , (        )   (        )-   ,    * (        )  (         )+ *   ( (        )   (        )+-          (       )    ,    *   (       )+ *   ( (       )   (       )+-         (       )   (       ) And   (       ) ∗ , (        )   (        )-    ,    * (        )  (         )+ *   ( (        )   (        )+-           (       )     ,    *   (       )+ *   ( (       )   (       )+-         (       )     (       ) Which yield     therefore uniqueness follows.   References 1) C. Alaca, D. Turkoglu, C. Yildiz, Fixed points in intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 29 (2006), 1073–1078. 1 2) K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Sys., 20 (1986), 87–96. 1    Vol-4,  Issue-01,  January 2023   ISSN (E): 2583-1348 AGPE The Royal Gondwana Research Journal of History, Science, Economic, Political and Social science        27 Common fixed- point theorem in non-archimedean intuitionistic fuzzy metric … A. Gupta, Fixed point in intuitionistic fuzzy metric space, mathematical theory and modeling, vol-.5, no.8,2015 3) H. Bouhadjera, C. Godet-Thobie, Common fixed point theorems for pairs of subcompatible maps, arXiv, 2011 (2011), 16 pages. 1, 1.14, 1.15 4) Z. Deng, Fuzzy pseudo-metric spaces, J. Math. Anal. Appl., 86 (1982), 74–95. 1 5) B. Dinda, T. K. Samanta, I. H. Jebril, Intuitionistic fuzzy Ψ-Φ-contractive mappings and fixed point theorems in non-Archimedean intuitionistic fuzzy metric spaces, Electron. J. Math. Anal. Appl., 1 (2013), 161–168. 1 6) M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl., 69 (1979), 205–230. 1 A. George, P. 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Schweizer, A. Sklar, Statistical metric space, Pac. J. Math., 10 (1960), 314–334. 1.1, 1.2 15) F. Sola Erduran, C. Yildiz, S. Kutukcu, A common fixed point theorem in weak non-Archimedean intuitionistic fuzzy metric spaces, Int. J. Open Problems Compt. Math., 7 (2014). 1, 1.7 16) D. Turkoglu, C. Alaca, C. Yildiz, Compatible maps and compatible maps of types (α) and (β) in intuitionistic fuzzy metric spaces, Demonsratio Math., 39 (2006), 671–684. 1, 1.9, 1.10, 1.11 17) L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353. 1            

COMMON FIXED- POINT THEOREM IN NON-ARCHIMEDEAN INTUITIONISTIC FUZZY METRIC SPACE BY USING SUBCOMPATIBLE MAPS OF  TYPE (α) WITH SIX MAPS

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COMMON FIXED- POINT THEOREM IN NON-ARCHIMEDEAN INTUITIONISTIC FUZZY METRIC SPACE BY USING SUBCOMPATIBLE MAPS OF TYPE (α) WITH SIX MAPS

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