Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 351 https://internationalpubls.com Some Applications via Coupled Fixed Point Theorems for (𝛂, 𝛗)-H- Contraction Mappings in Partial b- Metric Spaces Kavvampalli Jyothirmayi Rani 1*, V .Naga Raju 2 1* Research scholar, Department of mathematics, Osmania University, Hyderabad, Telangana, India. Mail.id : jyothirmai.ran2013@gmail.com 2. Professor, Department of mathematics, Osmania University, Hyderabad, Telangana, India. Mail id : viswanag2007@gmail.com *Corresponding Author Article History: Received: 15-05-2024 Revised: 23-06-2024 Accepted: 10-07-2024 Abstract: This work establishes unique common coupled fixed point theorems for given mapping in complete partial b-metric spaces with the concept of (α, ϕ)-H-contraction in the context of partial b-metric spaces. (α, ϕ)-H-contraction Furthermore, we show how the results may be used and present applications to integral equations and Homotopy theory. Introduction In previous work, authors have discussed various fixed point theorems on partial b-metric spaces with (ψ, ϕ)-weakly contractive mappings, α−ψ-contractive type, Suzuki type contractions, rational contraction and H-weak contractions. In our work, with the help of (α, ϕ)-H-contraction, we investigated coupled fixed point theorems in partial b-metric spaces. Objectives: Finding the unique common fixed points for a given mapping in partial b-metric spaces via (α, ϕ)-H-contraction Methods with the help of α-admissible mapping, H-rational type, (α, ϕ)−H-contraction we have shown coupled fixed point findings in complete partial b-metric spaces Results: We obtained unique common coupled fixed point results via (α, ϕ)−H-contraction type for the given mapping in complete partial b-metric spaces. Conclusions: This present study uses contractive mappings of the H type in the reference of partial b-metric space to give some fixed point results, appropriate examples that illustrate the main findings, In addition, boundary value problems and homotopy applications are given. Keywords: Partial b-metric space, ω-compatible, H-type rational Contraction, Coupled fixed point. 2020 Mathematics Subject Classification. 54H25, 47H10, 54E50. 1. Introduction The principle of Banach contraction [1] holds significant importance in fixed point theory due to its widespread application across various mathematical and mathematical sciences fields. The concept of b-metric spaces was established by Czerwik ([2],[3]) in 1993, while Matthews [4] introduced the idea of partial metric spaces in the year 1994.In 2013 Shukla [5] combined the concepts of the idea of partial metric spaces and b-metric spaces . Mustafa [6] introduced a modified version of partial b-metric spaces that is dependent on b-metric spaces and demonstrated some common fixed point solutions for ( , )  -weakly contractive mappings.. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 352 https://internationalpubls.com Samet et al. [7] (2011) proved fixed point theorems for such mappings in the complete metric spaces and introduced α- ψ-contractive type mappings to obtain a very general structure that combines several existing fixed point theorems .They also developed the concepts of α- contractive and α-admissible mappings. Karapinar and Samet [8] enhanced the findings in [7] by creating the idea of generalized α-ψ-contractive type mappings. In particular the theory proved by Jaggi [9] in 1977 meets a contractive criterion of rational type .A rational type contraction is a novel contractive condition that was created by Dass and Gupta [10] in 1975. In 1987 Guo and Lakshmikantham [11]first introduced the idea of coupled fixed point Later employing a weak contractivity type assumption. Bhaskar and Lakshmikantham [12] created a novel fixed point theory for a mixed monotone mapping in a metric space driven by partial ordering. Refer to relevant references and study results in ([13]-[20]) for additional details on coupled fixed point outcomes. This work proves common coupled fixed point theorem for two mappings satisfying ( , )  - H- type contractive constraints in the partial b-metric space. We also examine at numerous boundary value problems and homotopy applications, with examples. In partial b-metric space, this work establishes a common coupled fixed point theorem for two mappings meeting ( , )  - H-type contractive constraints. Along with examples, we also look at number of boundary value issues and homotopy applications. Objectives Finding the unique common fixed points for a given mapping in partial b-metric spaces via (α, ϕ)-H- contraction Methods With the help of α-admissible mapping, H-rational type, (α, ϕ)−H-contraction we have shown coupled fixed point findings in complete partial b-metric spaces 2. Preliminaries: Definition 2.1. ([5]) Let 1v be a given real number and ℑ be a nonempty set. A partial b- metric is defined as a function : [0, )→ bς i f the following criteria are met for each 1 2 3, , æ æ æ . 1 2 1 1 1 2 2 2( 1)   ( , ) ( , ) ( , )if and only= = =æ æ æ æ æ æ æ æb b b bς ς ς ς 1 1 1 2( 2)   ( , ) ( , )æ æ æ æb b bς ς ς 1 2 2 1( 3)   ( , ) ( , )=æ æ æ æb b bς ς ς 1 2 1 3 3 2 3 3( 4) ( , ) ( ( , ) ( , ) ( , )). + −æ æ æ æ æ æ æ æb b b b bvς ς ς ς ς The pair ( , ) bς is called a partial b−metric space. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 353 https://internationalpubls.com Remark 2.2. The class of partial b-metric spaces ( ), bς is more extensive than the class of partial metric spaces because a partial metric space is a specific instance of a partial b-metric space ( ), bς when d = 1. In addition, the class of partial b- metric spaces, denoted as ( ), bς ,is larger than the class of b-metric spaces because a b-metric space is a specific case of a partial b- metric space ( ), bς where the self- distance ( )1 1P æ ,æ is equal to 0. The examples shown demonstrate that both a b-metric onand a partial b-metric on  do not necessarily have to satisfy the conditions stated in ([5]-[6]). Example 2.3. ([5])Assume that [0,1) = . 2 2 1 2 1 2 1 2( ; ) [max{ , }]  | |= + −b z z z z z zς , is the formula to create a function. bς . For every 1 2, . z z the pair ( ), bς is called a partial b-metric space when 2 1= v . However, bς is neither a b- metric nor a partial metric on  . Definition 2.4. ([6])Every partial b-metric bς defines a b-metric d bς , where 1 2 1 2 1 1 2 2 1 2( , ) 2 ( , ) ( , ) ( , ),      ,d for all= − − ς ς ς ς b b b bz z z z z z z z z z Definition 2.5. ([6])In a partial b-metric space ( , ) bς , a sequence }p{æ is defined as follows (i) The  lim  ( , ) ( , )pconvergent toward a target if then → − = æ æ æ æ æb b b p ς ς ς (ii) In bς if , lim ( , )q q→ b p p ς æ æ exists and is finite, then bς - Cauchy sequence (iii) A ( , ) bς partial b-metric space bς is said to be bς -complete if and only if, for each bς - Cauchy sequence }p{æ in  , converges to a point æ such that , lim ( , )   lim  ( , ) ( , )q q→ → = =b p b p b p p ς ς ςæ æ æ æ æ æ Lemma 2.6. ([6]) A sequence  næ is a bς -Cauchy sequence in a partial b-metric space ( ), , bς if and only if it is a bς -Cauchy sequence in the b-metric space ( , )d bς . Lemma 2.7. ([6]) If and only if the b-metric space ( ), d bς is bς -complete, a partial b-metric space ( , ) bς qualifies as bς -complete. Additionally, , lim ( , ) 0 q q d → = b p p ς æ æ     lim ( , ) lim  ( , ) ( , ).q q→ →  = =b p b b p ς ς ςæ æ æ æ æ æ Definition 2.8. ([12]) Let a nonempty set be  . If ( , ) ( )and= =S Sæ æ œ œ œ,æ , Then An element ( , )    æ œ is referred to as a coupled fixed point of the mapping :→S . Definition 2.9. ([13])Suppose 2  :     → S and :    → f are two mappings. A point ( , )æ œ is a connected coincident point of S and f if ( , ) , ( ) .= =S f S fæ œ æ œ,æ œ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 354 https://internationalpubls.com Definition 2.10. ([13])Suppose 2 :     → S and :    → f are two mappings. A point ( , )æ œ is a coupled common point of S and f if ( , ) , ( ) .= =æ œ æ=æ œ,æ œ=œS f S f Definition 2.11. ([13]) Let ( ,  ) bς denote a partial b-metric space. Weakly compatible pairs are those where ( ( , )) ( )=f S S f fæ œ æ , œ whenever for all , æ œ such that ( , ) , ( ) .= =S f S fæ œ æ œ,æ œ Definition 2.12. ([7]) Consider 2 :     → S and 2: R + → .If 1 2, y y , then S is  - admissible. 1 2 1 2 2 1( , ) 1 ( ( , ), ( , )) 1implies  y y S y y S y y Definition 2.13. ([7])Suppose 2:   , : →  →S f and 2:   R + → are mappings. If 1 2, y y , then S and f are  -admissible. 1 2 1 2 2 1( , ) 1 ( ( , ), ( , )) 1implies  fy fy S y y S y y Definition 2.14. ([19], [20]) A rational type contraction :  → S in the complete metric space ( , )d is referred to as H-rational type, If 0 2 1   + +  for every 1 2, y y then the following inequality holds 2 2 1 1 1 2 2 1 1 2 2 1 2 d( , )[1 d( , )]  ( , ) ( , ) ( ( , ) ( , )) 1 ( , ) d d d d d    +  + + + + y Sy y Sy Sy Sy y y y Sy y Sy y y Let ∆ be a family of functions :[0,  )    [0,  )  →  that meet the following requirements. a)  is non-decreasing; b) ( ) 0 and ( ) 0 iff 0s s s s s    = = We now prove our primary result. 3. Main Results & Discussion Definition3.1. Consider ( , ) bς as a partial b-metric space with coefficient v≥ 1 and 2: . R Let + → 2:   , : →  → S f be two mappings. If 0 2 1   + +  and ,  for ev e r y 1 2 1 2, , , ;æ æy y then  ( , ) - H- contraction if 1 1 1 1 1 2 1 2 2 2 ( , ),    ( , ) ( ( , ), ( , )) max     ( , )             æ æ æ æ æ b b b fy f fy f S y y S fy f ς ς ς Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 355 https://internationalpubls.com 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 ( , ( , ))[1 ( , ( , ))] , 1 ( , ) max ( , ( , ))[1 ( , ( , ))] 1 ( , ) f f   +     +  +    +     +   b b b b b b f S fy S y y f S fy S y y fy f ς ς ς ς ς ς æ æ æ æ æ æ æ æ 1 1 2 1 1 2 2 2 1 2 2 1 ( , ( , )), ( , ( , )),    max max          ( , ( , )) ( , ( , ))       + +          b b b b fy S y y f S fy S y y f S ς ς ς ς æ æ æ æ æ æ (3.1) Theorem 3.2. Consider ( , ) bς be a Partial b- metric space with the coefficient v ≥ 1 and 2:   :and →  →S f be two mappings satisfying ( , )  − H-contraction. Assume 2(3.2.1) ( ) ( ) ( )and is complete subspace of    S f f (3.2.2) and are admissible mappings −S f 0 0 0 0 0 0(3.2.3) , ( ( , ), ( , )) 1,   y S y S f fyæ æ æ (3.2.4) ( , )S f is weakly compatible pair. Then S and f have a UCCFP (unique common coupled fixed point) in  . Proof. Let 0 0,  æy be arbitrary points in  . From (3.2.1), there exist sequences        , , , , 0z z z z in suchthat for all z y æ œ B 1( , )z z z z+= =æ œS y fy 1( , )z z z z+= =æ æ BS y f Case (i): If for some 0z , we have 0 0 1 0 0 0 1     ( , )z z z z z+ + = = =œ œ æS y fy 0 0 1 0 0 0 1 ( , )z z z z z + + = = =æ æB B S f then 0 0 (   , )z zœ B is common coupled fixed point of S and f Case (ii): Suppose that 1   z z+œ œ and 1z z+B B for all 0z  . Since S and f are α-admissible, we have 0 1 0 0 1 1 1 2( , ) 1 ( ( , ), ( , )) ( , ) 1    = æ æfy fy S y S y fy fy Recursively, we find that 1( , )  1z z   + f f , for all 0,1,...z = From (3.1), (3.2.2) and (3.2.3), we have that 1 1 1( , )    ( ( , ), ( , ))z z z z z z+ + +=œ œ æ æb b S y S yς ς 1 1 1         ( , ) ( ( , ), ( , ))z z z z z z + + + æ æbfy fy S y S yς Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 356 https://internationalpubls.com 1 1 ( , ), max ( , ) z z z z  + +           æ æ b b fy fy f f ς ς 1 1 1 1 1 1 ( , ( , )), ( , ( , )),    max max        ( , ( , )) ( , ( , )) z z z z z z z z z z z z  + + + + + +      + +          æ æ æ æ æ æ b b b b fy S y fy S y f S y f S y ς ς ς ς 1 1 11 1 1 1 1 ( , )[1 ( , )] , 1 ( , )( , ),   max max (   ,   ) (   ,   )[1 (   , )] 1 (   ,   ) z z z z z zz z z z z z z z z z   + − −− − + − −   +      +       +      +        +    b b bb b b b b ς ς ςς ς ς ς ς œ œ œ œ œ œœ œ B B B B B B B B 1 1 1 1 ( , ), ( , ),    max    max    ( , ) ( , ) z z z z z z z z  − + − +        + +              b b b b ς ς ς ς œ œ œ œ B B B B Since ( )s s  for all 0s  , then we obtain 1 1 1 1 1 ( , ), ( , ), ( , )  max max ( , ) ( , ) z z z z z z z z z z   − + + − +      +        b b b b b ς ς ς ς ς œ œ œ œ œ œ B B B B 1 1 1 1 ( , ), ( , ), max max ( , ) ( , ) z z z z z z z z  − + − +      + +          b b b b ς ς ς ς œ œ œ œ B B B B (3.2) Similarly, we can prove that 1 1 1 1 1 ( , ), ( , ), ( , )  max max ( , ) ( , ) z z z z z z z z z z  − + + − +      +        œ œ œ œ λB B B B B B b b b b b ς ς ς ς ς 1 1 1 1 ( , ), ( , ), max max ( , ) ( , ) z z z z z z z z  − + − +      + +          œ œ œ œ B B B B b b b b ς ς ς ς (3.3) Combining (3.2) and (3.3), we get 1 1 1 1 1 1 ( , ), ( , ), ( , ), max max max ( , ) ( , ) ( , ) z z z z z z z z z z z z  + − + + − +        +            œ œ œ œ œ œ λ B B B B B B b b b b b b ς ς ς ς ς ς 1 1 1 1 1 1 1 1 ( , ( , ))[1 ( , ( , ))] , 1 ( , ) max ( , ( , ))[1 ( , ( , ))] 1 ( , ) z z z z z z b z z z z z z z z z z f    + + + + + + + +  +    +  +    +     +   æ æ æ æ æ æ æ æ b b b b b fy S y fy S y f f S y f S y f f ς ς ς ς ς Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 357 https://internationalpubls.com 1 1 1 1 ( , ), ( , ), max max ( , ) ( , ) z z z z z z z z  − + − +      + +          œ œ œ œ B B B B b b b b ς ς ς ς Implies that 1 1 1 1 ( , ), ( , ), max max ( , ) ( , )1 z z z z z z z z    + − + −    +     − −    œ œ œ œ λB B B B b b b b ς ς ς ς 2 2 1 2 1 ( , ), max ( , )1 b z z b z z    − − − −  +      − −    œ œ λ B B 0 1 0 1 ( , ), max 0 ( , )1 n as n     +   → →   − −   λ B B b b u uς ς It follows that 1 1lim ( , ) lim (   ,   ) 0.z z z z z z + + → → = =œ œ B Bb bς ς (3.4) From (3.4) and ( bς 2), we have that lim ( , ) lim (   ,   ) 0.z z z z z z→ → = =œ œ B Bb bς ς (3.5) From definition of d bς , (3.4) and (3.5), we have that 1 1lim ( , ) lim (   ,   ) 0.b z z b z z z z d d+ + → → = =œ œ B B (3.6) To show that  zœ and  zB are Cauchy sequences, we proceed by using triangle property. 1 1 1 1(   ,   ) ( (   ,   ) (   ,   )) (   ,   )z w z z z w z z+ + + + + −œ œ œ œ œ œ œ œb b b bvς ς ς ς 1 1(   ,   ) (   ,   )z z z w+ + +œ œ œ œb bv vς ς 1 1 2 2 2 2   (   ,   ) ( ( (   ,   ) (   , )) (   ,   ))z z z z z w z z+ + + + + + + + −œ œ œ œ œ œ œ œb b b bv v vς ς ς ς 2 2 1 1 2 2   (   ,   ) (   ,   ) (   ,   )z z z z z w+ + + + + +œ œ œ œ œ œb b bv v vς ς ς 2 3 1 1 2 2 3 1(   ,   ) (   ,   ) (   ,   ) ... (   ,   )w z z z z z z z w w − + + + + + − + + + +œ œ œ œ œ œ œ œb b b bv v v vς ς ς ς 1 0 1 0 12 0 1 0 1 2 0 13 0 1 (   ,   ), (   ,   ),         max max (   ,   ) (   ,   )1 1 (   ,   ),   max ... (   ,   )1 z z z          + +    + +     + +       − − − −        +  + +   − −    b b b b b b v v v œ œ œ œ λ λ œ œ λ ς ς ς ς ς ς B B B B B B 1 0 1 0 1 ( , ),      max (   ,   )1 w w z    − −  +  +    − −    œ œ λ B B b b v ς ς Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 358 https://internationalpubls.com 2 0 12 0 1 (   ,   ),         1 ... max (   ,   )1 1 1 z             + + +       + + +         − − − − − −         œ œ λ λ λ B B b b v v v ς ς 0 1 0 1    (   ,   ),1 max 0    (   ,   ) 1 1 z as z       +     − −  → →  +   −   − −  œ œλ λ B B b b v v ς ς Since    0 1 1    +   − −λ , that is , lim (   ,   )   0.  z w z w→ =œ œbς Thus, {   }zœ is a Cauchy sequence in ( , ). bς From definition of d bς , and Eq.(3.5), we have that . , , lim (   ,   ) 2   lim (   ,   ) 0z w z w z w z w d → → = =œ œ œ œ b bς ς Therefore, {   }zœ is a Cauchy sequence in ( , ). bς similarly, we can show that   zB is a Cauchy sequence in ( , ).d bς Suppose ( )f is complete subspace of  . Then {   }zœ and   zB converges to , ( ( ), ),in d b f ςö thus there exist , ( ) æy f such that lim  z z→ = =œ fyö and lim  z z→ = = fæB That is 1 1lim ( , ) 0, lim ( , ) 0z z z z d d+ + → → = =æ b b fy fς ςö for some ,= = æfy fö , we have that 1 , ( , ) lim ( , ) lim ( , ) lim ( , ) 0.z w z z z w z z + → → → = = = =b b b bfy fy fy fyς ς ς ςö ö ö ö (3.7) And 1 , ( , ) lim (   ,   ) lim (   , ) lim (   , ) 0.z w z z z w z z + → → → = = = =b b b bf f f fς ς ς ςæ æ æ æ (3.8) Assume that ( , )S f is  -admissible mapping. Therefore, there is a sub sequence {   }zkœ and {   } kzB of {   }zœ and   zB respectively such that 1( , ) 1z z + œ œ and 1( , ) 1z z + B B for all        z N then ( , )  1 kz fyœ and ( , )     1  .  kz fæB Now we claim that ( , ) and ( , )= =æ æS y S yö From (3.1), we have ( ( , ), ) ( ( , ), ( , ))z z z=æ œ æ æb bS y S y S yς ς      ( , ) ( ( , ), ( , ))z Z Z bfy fy S y S yς æ æ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 359 https://internationalpubls.com ( , ( , ))[1 ( , ( , ))] , 1 , )( , ),   max max ( , ) ( , ( , )[1 ( , ( , ))] 1 ( , ) Z Z Z zz z z z z z    +     +      +      +        +   æ æ æ æ æ æ æ æ æ æ b b bb b b b b fy S y fy S y (fy fyfy fy f f f S y f S y f f ς ς ςς ς ς ς ς ( , ( , ), ( , ( , )), max max ( , ( , )) ( , ( , )) Z Z Z Z z zf       + +          b b b b fy S y fy S y S y f S y ς ς ς ς æ) æ æ æ æ æ 1 1 ( , )[1 ( , ( , ))] 1 ( , )( , ),   max max ( , ) ( , )[1 ( , ( , ))] 1 ( , ) z z ZZ Z z z z   − −  +     +      +      +        +   œ œ æ λ æ æ æ B B b b bb b b b b S y fyfy f S y f ς ς ςς ς ς ς ς ö öö 1 1 ( , ( , ), ( , ),   max max ( , ( , )) ( , ) z z z z  − −      + +          b b b b S y S y ς ς ς ς æ œ œ æ B B ö Letting z → ∞ in the previous inequality, we get that ( , ( , )), ( ( , ), )       max    ( , ( , ))            æ æ æ b b b S y S y S y ς ς ς ö ö ( , ( , )), max ( , ( , )),         æ æ b b S y S y ς ς ö Similarly, we can prove ( , ( , )),     ( ( , ), ) max   ( , ( , ))         æ æ æ b b b S y S y S y ς ς ς ö Therefore, ( ( , ), ), ( , ( , )),   max max   ( ( , ), ) ( , ( , ))              b b b b S y S y S y S y ς ς ς ς æ æ æ æ ö ö ( , ) = =æS y fyö and ( , ) = =æ æS y f follow as  <1.Therefore a coupled coincidence point of S and f is ( , )æy . As a weakly compatible pair ( , )S f , we have 2 ( ( , )) ( , ) ( , )= = = =æ æf f y f S y S fy f Sö ö 2 (( , )) ( , ) ( , )= = = =æ æ æf f f y S f fy S ö We now establish that =f and =fö ö we can see from (3.1) that ( , ) ( ( , ), ( , ))z z z = b bf S S yς ςœ æ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 360 https://internationalpubls.com     ( , ) ( ( , ), ( , ))z Z Z  bf fy S S yς æ ( , ( , )[1 ( , ( , ))] , 1 ( , )( , ),   max max ( , ) ( , ( , )[1 ( , ( , ))] 1 ( , ) z z z zz z z z z z    +       +       +      +         +   b b bb b b b b fy S y f S f fyf fy f f f S y f S f f ς ς ςς ς ς ς ς æ λ æ æ æ æ ( , ( , )), ( , ( , )),    max max ( , ( , )) ( , ( , )) z z z z z z        + +          b b b b f S fy S y f S f S y ς ς ς ς æ æ æ 1 11 1 1 1 ( , )[1 ( , ( , ))] 1 ( , )( , ),   max max ( , ) ( , )[1 ( , ( , ))] 1 ( , ) z z zz z z z z   − −− − − −  +       +       +      +         +   b b bb b b b b f S ff f f S f ς ς ςς ς ς ς ς œ œ œœ λ B B B B 1 1 ( , ( , ), ( , ),   max max ( , ( , ) ( , ) b b z z b b z z      − −        + +          f S f S œ œ B B In the inequality above, letting ,z→ we have that ( , ), ( ,  )         max          ( , ) b b b                   f f f ( , ),       max          ( , ) b b            f f Similarly, we can prove that ( , ), ( ,  )        max          ( , ) b b b             f f f Therefore, ( , ), ( , ), max    max        ( , ) ( , ) b b b b                     f f f f As 1  . It follows that ( , ) =  = S f and ( , ) = =S f . Therefore ( , ) is CCFP ( common coupled fixed point) of S and f for uniqueness let us suppose * *( , ) be another CCFP of S and f such that *   , and * Now from (3.1), we have that ( , )       ( ( , ), ( , ))b b     =  S S      ( , ) ( ( , ), ( , ))b       f f S S Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 361 https://internationalpubls.com * * * ** * * * * * ( , ( , )[1 ( , ( , ))] 1 ( , )( , ),   max max ( , ) ( , ( , )[1 ( , ( , ))] 1 ( , ) ,b b bb b b b b            +   +     +  +  +                            f S f S f ff f f f f S f S f f * * * * * * ( , ( , )), (( , ( , )), max max ( , ( , )) (( , ( , )) b b b b             + +           f S f S f S f S * * * * * * ( , ), ( , ),   max max ( , ) ( , ) b b b b                  +                 * * * * ( , ), ( , ), max max ( , ) ( , ) b b b b             + +           * * * * ( , ), ( , ),   max max ( , ) ( , ) b b b b                  +                 * * * * ( , ), ( , ), max max ( , ) ( , ) b b b b              + +           Since ( )s s  for all 0s  , then we obtain * * * ( , ), ( , )     ( 2 )   ( , ) b b b max             + +     Since, 0 2 1   + +  , we have * * * ( , ),     ( , ) ( , ) b b b max             Therefore,    * * * * max   ( , ), ( , ) ( , ), ( , )b b b bmax         it is a contradiction. Therefore the U C C FP ( unique common coupled fixed point) of S and f is ( , ) . Corollary 3.3. Let ( , )b be a Partial b- metric space with the coefficient d 1 and 2:  → S be a mapping satisfying H-contraction, for all 1 2 1 2, , , y y æ æ with positive real numbers , ,   such that 0 2 1   + +  , Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 362 https://internationalpubls.com 1 1 2 1 1 2 1 11 1 1 2 1 2 2 2 2 2 1 2 2 1 2 2 ( , ( , )[1 ( , ( , ))] , 1 ( , )( , ), ( ( , ), ( , ) max max ( , ) ( , ( , )[1 ( , ( , ))] 1 ( , ) b b bb b b b b b           +   +     +    +     +  S y S y y yy S y y S y S y S y y y æ æ æ ææ æ æ æ æ æ æ æ 1 1 2 1 1 2 2 2 1 2 2 1 ( , ( , )), ( , ( , )),   max max ( , ( , )) ( , ( , )) b b b b           + +          y S y y S y S y y S æ æ æ æ æ æ In  there is a unique coupled fixed point for S . Example 3.4. Let {1, 2,3} = and 2 b :    [0, )   →  be defined as   2 max , ( ; ) 1 0 1 b if for for          − +   = =   = =  æ æ æ æ æ æ . Then ( , )b is a complete partial b-metric space with coefficient 4 1= v Define 2 :   (1,1) 1, (1,2) 1, (1,3) 2, (2,1) 1be as → = = = =S S S S S , (2, 2) 1, (2,3) 2, (3,1) 1, (3, 2) 1, (3,3) 1= = = = =S S S S S , and :→ f by 1 1, 2 3, 3 2= = =f f f . Also, define 2 :[0, ) [0, ) ( ) 7 t as t  →  = and 2 1 , {1,2,3} :   ( , ) 0 for R as for otherwise    +   → =   æ æ We show that S , f are α-admissible mappings. Let , æ , if ( , ) 1  f fæ then ,   f fæ and so ( , )   S f fæ implies that ( ( , ), ( , )) 1   S S f fæ æ .Therefore, the predication holds. Obviously, (1,1) 1 1= =S f implies that (1, 1) is a coupled coincidence point of S and f . Moreover ( )   {1, 2,3} =f and 2( )   {1, 2}. =S Hence, 2( ) ( )   S f and also   (1,1) ( 1, 1) (1,1) 1 1, = = = =S S f f fS f then (S , f ) is ω-compatible. Then, S and f with 1 1 1 , , 3 4 6   = = = , satisfy all the requirements of theorem 3.2. According to Theorem 3.2, S and f have a unique coupled fixed point which is (1, 1). 3.1 Application to BVP. In this section, we investigate the existence of a unique solution to a boundary value problem as an application of Corollary 3.3. Think about the boundary value problem. ( ) ( ) 2 2 ( , ( ), ( )) 0, [0,1], 0 1 0 d I d           + =  = = = (3.9) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 363 https://internationalpubls.com The associated Greens function   : I I I  → to Eq. 3.9, can be defined as follows (1 ) 0 1 ( , ) (1 ) 0 1 s t if s t s t t s if t s −     = −      We have the following properties of the Greens function  : a)   ( , ) 0 for all , [0,1];s t s t   b) 0 1 0 1  sup ( , ) 4 t t s d    = Let ( )C I = represents the set of continuous functions defined on I. Define the mapping b : [0, )d →  by b 2 2( , ) || ( ) || sup | ( ) ( ) |    , , .d s s s I = − = −   f g f g f g f g Then obviously, the pair ( , )bd is a complete with 2=v . The associated integral operator 2: →S to Eq. 3.9 is defined by 1 0 ( , )( )   ( , ) ( , ( ), ( ))s s d     = S f g f g It is noted that the operator S has a fixed point that solves Eq. 3.9. The condition under which the BVP has a solution is given by the following theorem. Theorem 3.5. Let the function :   ( )  ( )     IX C I XC I R → is continuous and satisfies the following condition: 2 2 2 2 2 ( ) ( , )( ) | ( , , ) ( , , ) | 16 | ( ) ( ) | | ( ) ( , )( ) |   ( ) ( , )( ) s s s s s s s s s s      −    −  − + − +   + −   g S g f f g h l f h h S h l l S l h for all , , , , ( )s I C I f g h l and , ,   (0,1)   with 2 1  + +  . Then the BVP Eq.3.9 has a solution. Proof. To accomplish this proof, Corollary 3.3 will be used. The operator 2:  → S defined above is continuous since the function  is continuous. We continue as follows to demonstrate that the mapping S forms a H− contraction 1 2 2 0  | ( , )( ) ( , )( ) |  | ( , )( ( , ( ), ( )) ( , ( ), ( ))) |s s s g d        − =  −S f g S h l f h l 2 2 21 2 2 0 ( ) ( ) ( ) ( , )( ) ( , ) 16 ( ) ( , )( ) ( ) ( , )( ) s s s s s d s s s s          − −    +    + − + −      f h g S g f h S h l l S l h Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 364 https://internationalpubls.com Since 1 2 0 1 (sup ( , ) )   16 s d   = for all ,s I thus, taking supremum on both sides of above inequality, we have ( , ( , ))    ( ( , ), ( , ))       ( , ) ( , ( , )) ( , ( , )) b b b b b d d d d d            + +       l S l h S f g S h l f h h S h l g S g f Now for any partial b-metric onb  , we can have a b-metric onbd  by ( , ) ( , ) 0b b if d if    =  = f g f g f g f g The last inequality can be written as: ( , ( , ))    ( ( , ), ( , ))       ( , ) ( , ( , )) ( , ( , )) b b b b b          + +   +  l S l h S f g S h l f h h S h l g S g f ( , ( , )[1 ( , ( , )] , 1 ( , )( , ), max max ( , ) ( , ( , )[1 ( , ( , )] 1 ( , ) b b bb b b b b          +   +     +    +     +  h S h l f S f g f hf h g l l S l h g S g f g l ( , ( , )), ( , ( , )),   max max ( , ( , )) ( , ( , )) b b b b           + +          f S f g h S h l g S g f l S l h Therefore the BVP (3.9) has a solution  in according to Corollary 3.3 4. APPLICATION TO HOMOTOPY In this section, we study the existence of a unique solution to homotopy theory. Theorem 4.1. Let ( , )b be complete partial b-metric space the coefficient 1v , U and U be an open and closed subset of  such that  U U . Suppose 2 : [0,1]p X →A U be an operator with following conditions are satisfying, 0 )   ( , , ), ( , , ),P P   A Aæ æ œ œ œ æ for each , Uæ œ and [0,1]  (here U is boundary of U in  ); 1) , , , , [0,1] 0 2 1for all and with such that       + + x UBæ œ ( , ),  ( ( , , ), ( , , ) )         max      ( , ) b b p p b              v A A x x æ æ œ œ B B Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 365 https://internationalpubls.com ( , ( , , ))[1 ( , ( , , ))] , 1 ( , ) max ( , ( , , ))[1 ( , ( , , ))] 1 ( , ) b p b p b b p b p b            +    +  +   +   +  A x A x A x A x æ æ œ æ œ œ æ œ B B B B ( , ( . , )), ( , ( , , )), max max ( , ( , , )) ( , ( , , )) b p b p b p b p               + +           A A x A x A x æ æ œ œ æ B B B B 2 ) 0 ( ( , , ), ( , , )) | |b p pM M         −A Aæ œ æ œ for every , Uæ œ and , [0,1].   Then (., 0)pA has a coupled fixed point (.,1)p A has a coupled fixed point. Proof. Consider the set { [0,1] : ( , , ), ( , , )for some , }.p p  =  = = A A Uæ æ œ œ œ æ æ œA We have that 2(0,0)    A since (., 0)pA has a coupled fixed point in U 2, proving that the set A is non- empty set. We will demonstrate that A is both open and closed in [0, 1]. Consequently, A = [0, 1] may be obtained by the connectedness. Consequently there is a coupled fixed point for (.,1)pA in U 2. We first demonstrate the closure of A in [0, 1]. Let 1{ }   z z  =  A where .z→ [0,1]z →  showing t h a t A is necessary. Considering that z A for 1 1  1, 2,3, ...,    ,      ( , , ), ( , , ).z z z p z z z z p z z zz and that  + +=   = =U A Aæ œ æ æ œ œ œ æ Think about 1( , )b z z +æ æ 1 1 1( ( , , ), ( , , ))b p z z z p z z z  − − −= A Aæ œ æ œ 1 1 1 1 1 1 1 ( ( , , ), ( , , )) ( ( , , ), ( , , )) ( ( , , ), ( , , )) b p z z z p z z z b p z z z p z z z b p z z z p z z z          − − − − − − −      +   −  A A v A A A A æ œ æ œ æ œ æ œ æ œ æ œ 1 1 1 1 1( ( , , ), ( , , )b p z z z p z z z z zM    − − − − − + −v A A væ œ æ œ Letting ,z→ we get 1 1 1 1 1lim ( , ) lim ( ( , , ), ( , , )) 0.b z z b p z z z p z z z z z    + − − − − → →  +v A Aæ æ æ œ æ œ From (τ1) we obtain 1 1 1 ( , ), lim ( , ) lim max ( , ) b z z b z z z z b z z     − + → → −        æ æ æ æ œ œ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 366 https://internationalpubls.com 1 1 1 1 1 1 1 1 1 1 ( , ( , , ))[1 ( , ( , , ))] , 1 ( , ) lim max ( , ( , , ))[1 ( , ( , , ))] 1 ( , ) b z p z z z b z p z z z b z z z b z p z z z b z p z z z b z z            − − − − − → − − − − − +    +  +   +   +  A A A A æ æ œ æ æ œ æ œ œ œ æ œ œ æ œ œ 1 1 1 1 1 1 1 1 (( , ( , , )), lim max ( , ( , , )) b z p z z z z b z p z z z      − − − − → − − − −   +     A A æ æ œ œ œ æ ( , ( , , )), lim max ( , ( , , )) b z p z z z z b z p z z z     →   +     A A æ æ œ œ œ æ 1 1 1 1 ( , ), ( , ), lim max lim max ( , ) ( , ) b z z b z z z z b z z b z z       − + → → − +      +        æ æ æ æ œ œ œ œ 1 1 1 1 ( , ), ( , ), lim max max ( , ) ( , ) b z z b z z z b z z b z z      − + → − +      + +          æ æ æ æ œ œ œ œ Similarly 1 1 1 1 1 ( , ), ( , ), lim ( , ) lim max lim max ( , ) ( , ) b z z b z z b z z z z z b z z b z z        − + − → → → − +      +        æ æ æ æ œ œ œ œ œ œ 1 1 1 1 ( , ), ( , ) lim max max ( , ) ( , ) b z z b z z z b z z b z z      − + → − +      + +          æ æ æ æ œ œ œ œ Therefore, we have 1 1 ( , ) lim max ( , ) b z z z b z z   + → +       æ æ œ œ 1 1 1 1 ( , ), ( , ), lim max lim max ( , ) ( , ) b z z b z z z z b z z b z z       − + → → − +      +        æ æ æ æ œ œ œ œ 1 1 1 1 ( , ), ( , ), lim max max ( , ) ( , ) b z z b z z z b z z b z z      − + → − +      + +          æ æ æ æ œ œ œ œ It follows that 1 1 1 1 ( , ) ( , ), lim max lim max ( , ) ( , )1 b z z b z z z z b z z b z z       + − → → + −    +     − −    æ æ æ æ œ œ œ œ 2 12 2 1 ( , ), lim( ) max ( , )1 b z z z b z z     − − → − −  +    − −   æ æ œ œ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 367 https://internationalpubls.com 0 1 0 1 ( , ), lim( ) max ( , )1 bz z b    →  +    − −   æ æ œ œ It follows that 1 1lim ( , ) 0 lim ( , ).b z z b z z z z  + + → → = =æ æ œ œ (4.1) From ( 2),b lim ( , ) 0 lim ( , ).b z z b z z z z   → → = =æ æ œ œ (4.2) We now demonstrate that the Cauchy sequences in ( , )b are { }zæ and { }zœ . Conversely, let’s say that neither { }zæ nor { }zœ is Cauchy. A monotonic rising series of natural numbers { }kw and { }kz with 0ò exists such that z ,k kw ( , ) ( , ) k k k kb w z b w z  æ æ œ œò ò (4.3) and 11( , ) ( , ) k k k kb w z b w z  −−  æ æ œ œò ò (4.4) From (4.3) and (4.4) we obtain    ( , ) k kb w z æ æò 1 1 1 1 ( ( , ) ( , ) ( , )) k k k k k kb w w b w z b w w   + + + +  + −v æ æ æ æ æ æ Using k → as the limit and working from (4.1) to (4.2), we obtain that 1 1 1, 1   lim ( , ) lim ( ( , , ), ( , ))  k k k k k k k kb w z b p w w w p z z z k k     + − − −→ →  =v v A Aæ æ æ œ æ œò 1 1, ( , ), lim max ( , ) k k k k b w z k b w z    − − →          æ æ œ œ 1 1 1, 1 1, 1, 1, 1 1 1 ( , ( , , ))[1 ( , ( , , ))] , 1 ( , ) lim max ( , ( , , ))[1 ( , ( , , ))] 1 ( , ) k k k k k k k k k k k k k k k k k k k k b z p z z z b w p w w w b w z z b z p z z z b w p w w w b w z            − − − − − − − − − − → +    +  +   +   +  A A A A æ æ œ æ æ œ æ œ œ œ æ œ œ æ œ œ ( , ( , , )), lim max ( , ( , , )) k k k k k k k k b w p w w w z b w p w w w     →    +      A A æ æ œ œ œ æ 1 1 1, 1 1, 1, 1 1 ( , ( , , )), lim max ( , ( , , )) k k k k k k k k b z p z z z z b z p z z z      − − − − − − − − →    +      æ æ œ œ œ æ A A Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 368 https://internationalpubls.com 1 1 1, 1, , ( , ), ( , ), lim max lim max ( , ) ( , ) k k k k k k k k b w z b z z k z b w z b z z       − − − − → →         +           æ æ æ æ œ œ œ œ 1 1 1, 1, , ( , ), ( , ), lim max max ( , ) ( , ) k k k k k k k k b w w b z z z b w w b z z      + − + − →          + +             æ æ æ æ œ œ œ œ 1 1 ( , ), lim max ( , ) k k k k b w z k b w z    − − →          æ æ œ œ 1 2 1 2, 2 ( , ), lim max ( , ) k k k k b w z k b w z    − − − − →          æ æ œ œ 1 0 1 0 ( , ), lim max ( , ) bk k b   →        æ æ œ œ 0 Hence, 1,  is in conflict with 0ò . Therefore in ( , )b , { }zæ is a Cauchy sequence and , lim ( , ) 0b z w z w  → =æ æ . Similarly, we can demonstrate that { }zœ is a Cauchy sequence in ( , )b and , lim ( , ) 0b z w z w  → =œ œ . Given the completeness of ( , )b , , x UB exist. 1 , ( , ) lim ( , ) lim ( , ) lim ( , ) 0b b z b z b z w z z z w    + → → → = = = =æ æ æ æB B B B 1 , ( , ) lim ( , ) lim ( , ) lim ( , ) 0b b z b z b z w z z z w    + → → → = = = =x x x xæ æ œ œ From Lemma 2.7, we get lim ( , ( , , )) ( , ( , , )).b z p b p z     → =A x A xæ B B B Now, 1  ( , ( , , ))     ( ( , , ), ( , , ))b z p b p z z z p    + =A x A A xæ æ œB B ( ( , , ), ( , , )) ( ( , , ), ( , , )) ( ( , , ), ( , , )) b p z z z p z z b p z z p b p z z p z z          +   − A A A A x v A A æ œ æ œ æ œ æ œ æ œ B ( ( , , ), ( , , )).z b p z z pM      − +v v A A xæ œ B Letting z → ∞, we obtain   ( , ( , , ))    lim ( ( , , ), ( , , ))b p b p z z p z      → A x v A A xæ œB B B ( , ),     lim max ( , ) b z z b z   →       x æ œ B Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 369 https://internationalpubls.com ( , ( , , ))[1 ( , ( , , ))], 1 ( , )     lim max ( , ( , , ))[1 ( , ( , , ))] 1 ( , ) b p b z p z z z b z z b p b z p z z z b z            → +    +  +   +   +  A x A x A x A x æ æ œ æ œ œ æ œ B B B B ( , ( , , )), ( , ( , , )),     lim max max ( , ( , , )) ( , ( , , )) b z p z z z b p z b z p z z z b p         →      + +           A A x A x A x æ æ œ œ œ æ B B B ( , ( , , )),    ( )max ( , ( , , )) b p b p          +     A x x A x B B B It follows that ( , ( , , )), (1 )max 0 ( , ( , , )) b p b p         − −     A x x A x B B B This suggests that both ( , ( , , )) 0b p  =A xB B and ( , ( , , )) 0b p  =x A x B . In order for ( , , )p  =A xB B and ( , , )p  =A x xB to be equal . Therefore .A Thus in  0,1 A is closed. Let A contain 0 . When 0 0 0 0( , , )p = Aæ æ œ t h e n 0 0, Uæ œ as well as 0 0 0 0( , , )p = Aœ œ æ . 0( , )bB r  Uæ and 0( , )bB r  Uœ Since U is open Choose 0 0(   ,   )   − +ò ò such that 0 1 | | zM  −   Then for 0 0 0 0( , ) { / ( , ) ( , )} b b bB r r   =   +æ æ æ æ æ æ æ and 0 0 0 0( , ) { / ( , ) ( , )} b b bB r r   =   +œ œ œ œ œ œ œ . Now we have 0 0 0 0  ( ( , , ), )    ( ( , , ), ( , , ))b p b p p    =A A Aæ œ æ æ œ æ œ 0 0 0 0 0 0 0 ( ( , , ), ( , , )) ( ( , , ), ( , , )) ( ( , , ), ( , , )) b p p b p p b p p          +     −  A A A A v A A æ œ æ œ æ œ æ œ æ œ æ œ 0 0 0 0 0( ( , , ), ( , , )).b p pM      − +v v A Aæ œ æ œ 0 0 0 01 1 ( ( , , ), ( , , )).b p pzM    −  +v v A Aæ œ æ œ Letting ,z→ we obtain 0 0 0 0 0  ( ( , , ), )     ( ( , , ), ( , , ))b p b p p    A v A Aæ œ æ æ œ æ œ 0 0 ( , ), max ( , ) b b           æ æ œ œ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 370 https://internationalpubls.com 0 0 0 0 0 0 0 0 0 0 ( ( , , ))[1 ( , ( , , ))] , 1 ( , ) max ( , ( , , )[1 ( , ( , , ))] 1 ( , ) b p b p b b p b p b            +    +  +   +   +  ,A A A A æ æ œ æ æ œ æ æ œ œ æ œ œ æ œ œ 0 0 0 0 0 0 0 0 ( , ( , , )), ( ( , , )), max max ( , ( , , )) ( , ( , , ) b p b p b p b p               + +           A ,A A A æ æ œ æ æ œ œ æ œ œ œ æ  0 0( 2 )max    ( , ), ( , )b b     + + æ æ œ œ  0 0max ( , ), ( , )b b  æ æ œ œ Similarly 0 0 0( ( , , ), ) max{ ( , ), ( , )}.b p b b   A œ æ œ æ æ œ œ Thus 0 0 0 0max{ ( ( , , ), ), ( ( , , ), )}    max{ ( , ), ( , )}b p b p b b     A Aæ œ æ œ æ œ æ æ œ œ 0 0 0 0max{ ( , ), ( , )}b br r  + +æ æ œ œ For every constant 0 0 0 0( , ), (., ) : ( , )  ( , )  b bpthis means that B r B r     − + →U æ æò ò and 0 0(., ) : ( , ) ( , ) b bp B r B r  →A œ œ . Since 1 ( ) also holds, Theorem 4.1 is satisfied in all of its conditions. From this, we infer that there is a coupled fixed point for (., )p A in 2 U . However, since 0( ) holds, this coupled fixed point has to reside in 2U . For any 0 0     (     , ).   − +ò ò A . Therefore 0 0(     , ) − +  Aò ò Hence, in [0, 1]. A is open. We employ the identical method for the opposite inference. Conclusion This paper presents several fixed point results and appropriate examples that demonstrate the major findings in the context of partial b-metric space using contractive mappings of the ( , )  –H type. Applications to homotopy and boundary value problems are also provided. References [1] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fundam. Math. 3 (1922), 133 181. [2] S. Czerwik, Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Osrav., 1(1993): 5-11. http://dml.cz/dmlcz/120469 [3] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat.Fis. Univ. Modena. 46 (1998): 263-276. [4] S. G. Matthews, Partal metric topology. Proc. 8th Summer Conference on General Topology and Applications, Ann. N.Y. Acad. Sci., 728 (1994): 183-197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 371 https://internationalpubls.com [5] S. Shukla, Partial b-metric spaces and fixed point theorems. Mediterranean Journal of Mathematics,(2013). https://link.springer.com/article/10.1007/s00009-013-0327-4 [6] Z. Mustafa, J. Rezaei Roshan, V. Parvaneh and Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, Journal of Inequalities and Applications, (2013),2013:562.. http://www.journalofinequalitiesandapplications.com/content/2013/1/562 [7] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-φ-contractive type mappings, Nonlinear Anal. 75 (2012), no. 4,21542165 https://doi.org/10.1016/j.na.2011.10.014 [8] E. Karapinar and B. Samet, Generalized α-φ contractive type mappings and related fixed point theorems with applications, Abstr. Appl. Anal. 2012 (2012), Article ID 793486. [9] D. S. Jaggi, Some unique fixed point theorems, Indian Journal of Pure and Applied Mathematics 8(2)(1977), 223- 230. [10] Dass, B.K.; Gupta, S. An extension of banach contraction principle through rational expression, Indian J. Pure Appl. Math. 1975, 6, 14551458. [11] D. J. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal., 11(1987), 623632. 1 https://doi.org/10.1016/0362-546X(87)90077-0 [12] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Non-linear Anal., 65 (2006), 13791393. 1.4, 1 doi:10.1016/J.NA.2005.10.017 [13] M. Abbas, M. Ali Khan, S. Radenovic, Common coupled fixed point theorems in cone metric spaces for ω- compatible mappings, Appl. Math. Comput.2010, 217(1),195-202. https://doi.org/10.1016/j.amc.2010.05.042 [14] N. Mangapathi, K.R.K.Rao, B.S.Rao, M.I.Pasha, On certain common coupled fixed points of rational contraction mappings in partial b-metric Spaces and its applications, Mathematical Statistician and Engineering Applications, Vol. 71 No. 4 (2022),735-752. https://doi.org/10.17762/msea.v71i4.552 [15] E. Karapinar, Coupled Fixed Point on Cone Metric Spaces, Gazi Univ. J. Sci., 1 (2011)5158. [16] P. Naresh, G. Upender Reddy, B. Srinuvasa Rao, Existence suzuki type fixed point results in Ab-metric spaces with application, Int. J. Anal. Appl. (2022), 20:67 https://etamaths.com/index.php/ijaa/article/view/2683 [17] K.P.R.Rao, G.N.V.Kishore, V.C.C.Raju, A coupled fixed point theorem for two pairs of ω-compatible maps using altering distance function in partial metric space, J.Adv. Res. Pure Math.2012, 4(4), 96-114. [18] W. Long, B. E. Rhoades, M. Rajovic, Coupled coincidence points for two mappings in metric spaces and cone metric spaces, Fixed Point Theory Appl., 2012 (2012), 9 pages. https://fixedpointtheoryandalgorithms.springeropen.com/ [19] Angel A Pereira, S. N. Leena Nelson, Unique fixed point theorem for H-contraction mapping in complete metric space, Mathematical Statistician and Engineering Applications, Vol 71 No.2 (2022),684 – 692. doi: 10.1186/1687-1812-2012-66 [20] A.Pereira, S.N.Leena Nelson, Unique fixed point theorem for H-contraction mapping in partially ordered metric space, Eur. Chem. Bull. 2023, 12 (Si6), 2687 2693. https://doi.org/10.17762/msea.v71i2.2435 https://link.springer.com/article/10.1007/s00009-013-0327-4 http://www.journalofinequalitiesandapplications.com/content/2013/1/562 https://doi.org/10.1016/j.na.2011.10.014 https://doi.org/10.1016/0362-546X(87)90077-0 https://doi.org/10.1016/J.NA.2005.10.017 https://doi.org/10.1016/j.amc.2010.05.042 https://etamaths.com/index.php/ijaa/article/view/2683 https://fixedpointtheoryandalgorithms.springeropen.com/ https://doi.org/10.1186/1687-1812-2012-66 https://doi.org/10.17762/msea.v71i2.2435