Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 506 https://internationalpubls.com Common fixed point theorems for weakly compatible mappings in Multiplicative Metric Space Chiranjeevi Perala 1 ,Thirupathi Thota 2 , Srinivas Veladi 3 1 . Department of Mathematics, Osmania University, Hyderabad, Telangana ,India, Email: chiranjeeviperala@gmail.com 2 .Department of Mathematics, Sreenidhi Institute of Science and Technology, Hyderabad,Telangana, India Email: thotathirupathi1986@gmail.com 3 .Department of Mathematics, University College of Science, Osmania University, Hyderabad,Telangana, India Email:srinivasmaths4141@gmail.com Article History: Received: 04-06-2024 Revised: 03-07-2024 Accepted: 30-07-2024 Abstract The aim of this paper is to establish three common fixed point theorems in Multiplicative Metric Space (MMS). By utilizing the conditions concepts of non-continuous and non- compatible mappings.The first theoem is generated by applying the concepts of semi- compatible mapping, WCM and reciprocally continuous mappings.The second theorem is established by using the concepts of strongly semi compatible mappings,conditionally reciprocally continuous mappings and and OWC mappings. These conditions are weaker than the existing conditions like compatible mappings and cotinuous which generalizes the theorem of Afrah A. N. Abdou. Keywords: Multiplicative Metric Space (MMS), fixed point, semi-compatible mapping, reciprocally continuous mappings , weakly compatible mapping, strongly semi compatible mappings, conditionally reciprocally continuous mappings and and OWC mappings. 2020 Mathematics Subject Classification: 54H25, 47H10, 54E50. 1. Introduction Fixed point theory is one of the most exciting problems in current mathematics, and it may be an existing area of functional analysis.Furthermore, this topic has served as a platform for various researchers during the last several years. Within the subject of analysis,the theory of metric spaces has grown significantly.We know that the set of positive real numbers is not complete in metric space. Inorder to overcome this problem the concept known as MMS was introduced by Bashirove in 2008. Further Ozavsar and Cevikel in 2017 investigated and developed the multiplicative contraction principle and proved some common fixed point results. Thereafter, the theory of a multiplicative metric space has been developed by many authors[2,3,4,5,6,7,8,9].In this process Afrah A. N. Abdou [1] poved a theorem in MMS in 2016.The purpose of this paper is to prove two common fixed point theorems on MMS utilizing the concept like semi-compatible mapping,WCM,reciprocally continuous mappings,strongly semi compatible mappings,conditionally reciprocally continuous mappings and and OWC mappings. Additionally we provided three examples to validate our results. We first provide some helpful definitions and examples before presenting our findings. Preliminaries mailto:chiranjeeviperala@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 507 https://internationalpubls.com Definition 2.1 Let set and then is said to be MMS if it meets the requirements as below: (i) for all and (ii) for all (iii) for all (multiplicative triangle inequality) Definition 2.2 A sequence { } in a MMS is said to be (i) multiplicative convergent sequence to if for every multiplicative open ball । , thereexists a positive integer such that for all i.e ( ) as . (ii) multiplicative cauchy sequence if for all such that ( ) for all i.e ( ) as . Definition 2.3 The pair of mapping of a is said to be (i) Compatible if ( ) , whenever a sequence { } in like that for some . (ii) Weakly compatible if for some such that . (iii) Occasionally weakly compatible if for such that implies that (iii) Reciprocally continuous if ( ) ( ) for a sequence { } in such that for some (iv) Semi compatible if ( ) whenever { } is a sequence in such that for some (v) Conditionally semi compatible if a sequence { } is a sequence in such that is non empty for a sequence { } satisfying for some then ( ) and ( ) . (vi) Strongly semi-compatible mappings if they satisfy OWC and conditionally semi-compatible property. (vii) Conditionally reciprocally continuous if and only if ( ) ( ) whenever a sequence { } satisfying is non-empty there exists another sequence { } satisfying = as Now we present the example for strongly semi-compatible mappings and conditionally reciprocally continuous mappings. Example 2.1: Let [ ] and [ be defined as | | then is a MMS. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 508 https://internationalpubls.com Define the self-mappings and as { { Consider a sequence given by for ( ) [ ( ) ] and ( ) . Therefore is non-empty. Consider another sequence given by for ( ) 0 ( ) 1 and ( ) * + . Therefore (say) . This gives and . Further ( ) * + [ ( ) ] and ( * [ ( ) ] [ ] Therefore and . This shows that the couple is conditionally semi-compatible in MMS. Now and , showing that 1,2 are coincidence points. Then implies ( ) and implies ( ) . As a result, the two self maps and are OWC. Therefore the self maps and are strongly semi-compatible mappings in MMS. Example 2.2: Let [ ] and [ be defined as | | then is a MMS. Define the self-mappings and as 2 { Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 509 https://internationalpubls.com Consider a sequence sequences , - where . ( ) * ( )+ and ( ) *( ) + . Therefore (non-empty). Furether and . Which gives . Hence the couple is not compatible. Consider a sequence sequences , - where . ( ) [ ( ) ] and ( ) * + . Therefore (say). Furthermore ( ) and ( ) . Also we absorve that Thus but . Hence two self-maps G and I are conditionally reciprocally continuous in MMS. Afrah A.N.Abdou [1] established the following Theorem in MMS. Theorem 2.1 Assume that is a complete MMS and the mappings and are defined on such that (B1) (B2) 0 . 2 3/1 for all ,where . Where [ [ is a continuous and monotone increasing function such that for all (B3) one of , and is continuous (B4) both the pairs and are compatible. Then the four maps and share a unique common fixed point in . We now generalize Theorem 2.1 as below. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 510 https://internationalpubls.com Now we proceed to our main result. 3. Results and Discussion 3.1 Theorem: Suppose that in a complete MMS , the four self-mappings and meeting the requirements (C1) (C2) 0 . 2 3/1 for all ,where . Where [ [ is a continuous and monotone increasing function such that for all (C3) the pair is reciprocally continuous and semi-compatible (C4) the pair is weakly compatible mappings. Then there exists a unique common fixed point for the above mappings. Proof: Since By , we can consider a point there exists such that At this a point in and so on. Likewise, we are able to define ; for Now it is possible to establish that th e sequence { } ( ) ( ) 0 . 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3/1 [ ( { ( ) ( ) ( )})] [ ( { ( ) ( ) })] [ ( )] .[ ( )] which implies that ( ) [ ( )] [ ( )] Where . Similaly,we have ( ) ( ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 511 https://internationalpubls.com 0 . 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3/1 [ ( { ( ) ( ) ( )})] [ ( { ( ) ( ) })] [ ( )] .[ ( )] which implies that ( ) [ ( )] [ ( )] Thus it follows that, for all ( ) [ ( )] [ ( )] [ ] Therefore, with , by the multiplicative triangle inequality , we obtain ( ) ( ) ( ) [ ] [ ] [ ] [ ] . Which means that ( ) as . Hence { } is multiplicative cauchy sequence. By the completeness of as Accordingly, the sequences as By (C3) the couple is reciprocally continuous ( ) ( ) (1) Also the couple is semi-compatible, we have ( ) (2) From (1) and (2) we get (3) Since which gives then thereexists such that since as . Which implies (4) Now we prove that Putting and in (C2) we have ( ) 0 . 2 ( ) ( ) ( ) ( ) ( ) ( ) 3/1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 512 https://internationalpubls.com 0 . 2 3/1 [ ( )] [ ( )] Which gives . Therefore Since the couple is WCM and is coincidence point then we have . (5) Putting and in (C2) we have ( ) 0 . 2 ( ) ( ) ( ) ( ) ( ) ( ) 3/1 0 . 2 3/1 [ ( )] [ ] Which implies (6) From (5) and (6) we obtain . (7) In the inequality (C2) putting and 0 . 2 3/1 0 . 2 3/1 [ ( )] [ ] Which implies (8) From (3) and (8) it gives (9) From (7) and (9) we have . This demonstrates that the common fixed point for the maps above is . For Uniqueness: Assume that be another fixed point then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 513 https://internationalpubls.com Putting and in 0 . 2 3/1 0 . 2 3/1 [ ( )] [ ] , a contradiction which implies . This demonstrates is the unique common fixed point of four self-mappings. 3.1 Example: Suppose [ ] be defined in MMS with | | . We define self-mappings and as { { { and { Clearly * + ( ] ( ] * + ( + * + and * + ( + * + * + ( + * + so that condition is fulfilled. Suppose we have a sequence , - for . Now ( ) * ( )+ and ( ) 0 ( ) 1 . This gives Also ( ) [ ( ) ] and ( ) [ ( ) ( ) ] . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 514 https://internationalpubls.com This gives Therefore Also ( ) ( ) . This implies ( ) ( ( ) ( ) * * + * + and ( ) [ ( ) ] * + [ ( ) ( ) ] . This implies ( ) and ( ) . This proves that the couple is reciprocally continuous and semi-compatible in MMS. Further ( ) ( ) which imliies that is the coincidence point of and . Hence ( ) ( ) and ( ) ( ) which gives ( ( ) ( )* Hence the couple is weakly compatible mappings. But ( ) 0 ( ) 1 * + 0 ( ) ( ) 1 and ( ) 0 ( ) 1 * ( )+ [ ( ( )* ] . Similarly ( ) . ( ) / * + 0 ( ) 1 and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 515 https://internationalpubls.com ( * [ ( ) ] [ ] [ ( ) ] Therefore ( ) and ( ) . Demonstrating that the compatibility condition is not satisfied. Also ( ) ( ) ( ) ( ) . It has been observed that the unique common fixed point to four self-mapping and is . We now demonstrate another theorem on MMS. 3.2 Theorem: Suppose that in a complete MMS , the four self-mappings and meeting the requirements (D1) (D2) 0 . 2 3/1 for all ,where . Where [ [ is a continuous and monotone increasing function such that for all (D3) the pair is strongly semi-compatible and conditinally reciprocally continuous (D4) the pair is OWC. Then there exists a unique common fixed point for the above mappings. Proof: As in theorem (3.1), is cauchy sequence. By the completeness of as Accordingly, the sequences as By (D3) the couple is strongly semi-compatible whenever ( ) ( ) (non- empty) implies is a sequence in so that forsome then and as (1) Also the coulple is conditinally reciprocally continuous implies and as (2) From (1) and (2) it gives (3) Since implies such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 516 https://internationalpubls.com Therefore . (4) Now we show that . In the inequality putting and 0 . 2 3/1 0 . 2 3/1 [ ( )] [ ] Which implies (5) From (4) and (5) it gives (say). (6) But the couple is strongly semi-compatible mappings and Implies . Since the couple is OWC implies which gives implies . Now we show that . In the inequality (D2) putting and 0 . 2 3/1 as this implies 0 . 2 3/1 [ ( )] [ ] implies . Therefore (7) Now we show that In the inequality (D2) putting and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 517 https://internationalpubls.com 0 . 2 3/1 as this implies 0 . 2 3/1 [ ( )] [ ] implies . Therefore (8) From (7) and (8) we get . Thus is a fixed point that is common to all four self-maps and . For Uniqueness: Assume that be another fixed point then Putting and in 0 . 2 3/1 0 . 2 3/1 [ ( )] [ ] , a contradiction which implies . This demonstrates is the unique common fixed point of four self-mappings. 3.2 Example: Suppose [ ] be defined in MMS with | | . We define self-mappings and as { { Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 518 https://internationalpubls.com { and { Clearly * + , - * + * + * ) * + and * + * ) * + * + [ [ so that condition (D1) is fulfilled. Suppose we have a sequence , - for . Now ( ) 0 ( ) ( ) 1 and ( ) 0 ( ) 1 . This gives and . ( ) . ( ) / * + 0 ( ) ( ) 1 and ( * [ ( ) ( ) ] [ ( . ( )/ ) ] ( ) . Now take another sequence , - for . Now ( ) 0 ( ) 1 and ( ) 0 ( ) ( ) 1 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 519 https://internationalpubls.com This gives and . ( ) . ( ) ( ) / . ( ) ( ) / * + * + 0 ( ) ( ) 1 and ( ) 0 ( ) ( ) 1 [ ( . ( )/ ) ] Also ( ) and ( ) This shows that the self-maps and are conditionally reciprocally continuous in MMS. Similarly ( ) and ( ) This shows that the self-maps and are conditionally semi compatible in MMS. Moreover and are coincidence points ( ( ) ( )* implies ( ( ) ( )* ,but ( ) and ( ) . As a result, the couple satisfies OWC. Therefore the self-maps and T are strongly semi compatible mappings in MMS. But ( ) 0 ( ) 1 * + [ (( )* ] and ( * [ ( ) ] [ ] [ ( ) ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 520 https://internationalpubls.com Similarly ( ) . ( ) / * + * ( )+ and ( * [ ( *] [ ] [ .( )/ ] Therefore ( ) and ( ) . Demonstrating that the compatibility condition is not satisfied. Further ( ) ( ) which imliies that is the coincidence point of and . ( ) ( ) which imliies that is the coincidence point of and . Hence ( ) ( ) and ( ) ( ) which gives ( ( ) ( )* also ( ) ( ) and ( ) ( ) which gives ( ( ) ( )* . Hence the couple and are weakly compatible mappings but not compatible. Also ( ) ( ) ( ) ( ) . It has been observed that the unique common fixed point to four self-mapping and is . 4. Conclusion In this paper the concepts of semi-compatible mapping, WCM , reciprocally continuous , conditionally reciprocally continuous mappings and and OWC mappings are used for obtaining the generalization of existing common fixed point theorems proved in [1] which are weaker than those classes of compatible mappings and continuous mappings. Further, these results are also substantiated with appropriate examples. References: [1] Abdou, Afrah AN. "Common fixed point results for compatible-type mappings in multiplicative metric spaces." J. Nonlinear Sci. Appl 9 (2016): 2244-2257. [2] B Vijayabaskerreddy and V Srinivas Fixed point results on multiplicative semi-metric space. Journal of Scientific Research, 12(3):341–348, 2020. http://dx.doi.org/10.3329/jsr.v12i3.44754 [3] Kidane KOYAS, Alemayehu GEBRE, and Aynalem KASSAYE. A common fixed point theorem for generalized weakly contractive mappings in multiplicative metric spaces. Advances in the Theory of Nonlinear Analysis and its Application, 4(1):1–13, 2020. https://doi.org/10.31197/atnaa.573903 [4] B.Vijayabaskerreddy and V. Srinivas. Some results in multiplicative metric space using absorbing mappings. Indian Journal of Science and Technology, 13(39): 4161–4167, 2020. https://doi.org/10.17485/IJST/v13i39.1629 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 521 https://internationalpubls.com [5] V.Srinivas and K.Mallaiah. 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