International Journal of Analysis and Applications Volume 19, Number 6 (2021), 904-914 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-904 Received August 15th, 2021; accepted October 4th, 2021; published November 3rd, 2021. 2010 Mathematics Subject Classification. 54H25, 47H10. Key words and phrases. self-mappings; occasionally weakly compatible mappings; probabilistic 2-metric space; CLR’S-property. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 904 SOME RESULTS BY USING CLR’s-PROPERTY IN PROBABILISTIC 2-METRIC SPACE V. SRINIVAS1, K. SATYANNA2,* 1Department of Mathematics, University College of Science, Osmania University, Hyderabad, Telangana State, India 2Department of Mathematics, M.A.L.D. Government Degree College, Gadwal, Palamoor University, Mahaboobnagar, Telangana State, India *Corresponding author: satgjls@gmail.com ABSTRACT. The aim of this paper is to generate two fixed point theorems in probabilistic 2-metric space by applying CLR’S-property and occasionally weakly compatible mappings (OWC), these two results generalize the theorem proved by V. K. Gupta, Arihant Jain and Rajesh Kumar. Further these results are justified with suitable examples. 1. INTRODUCTION Menger [1] pioneered the statistical metric(SM) space theory. One of the major achievements was the translation of probabilistic concepts into geometry. Menger used the notation of new distance distribution function from p to q by a Fpq. B. Schweizer, and A. Sklar [2] introduced a new notion of a probabilistic-norm. This norm naturally generates topology, convergence ,continuity and completeness in SM-space. Mishra [3] used compatible mappings and generated some fixed points in Menger space. Altumn Turkoglu [4] proved some more results of SM-space by utilizing the implicit relation in multivalued mappings. Zhang, Xiaohong, Huacan He, and Yang Xu [5] employed the Schweizer-Sklar t-norm established fuzzy https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-904 Int. J. Anal. Appl. 19 (6) (2021) 905 logic system to contribute in development of SM-space. Sehgal, V. M., and A. T. Bharucha-Reid [6] used classical Banach contraction to establish the first result of Menger space for coincidence points. Weakly compatible mappings were generalized by Al-Thagafi and Shahzad [7], by introducing occasionally weakly compatible mappings. Futher Chauhan, Sunny, Wutiphol Sintunavarat, and Poom Kumam[9] proved some more theorems by using CLR’S-property in fuzzy metric space. Further some more results can be witnessed by using the concepts of sub sequentially continuous and semi compatible mappings in Menger space [10]. 2. PRELIMINARIES Definition 2.1 [8] A continuously t-norm is mapping t: [0, 1] × [0, 1] → [0, 1] and it satisfies the following properties (𝑡1) 𝑡 𝑖𝑠 𝑎𝑏𝑒𝑙𝑖𝑎𝑛 & 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑣𝑒 (𝑡2) 𝑡(𝛾, 1) = 𝛾, ∀ 𝛾 ∈ [0, 1] (𝑡3) 𝑡 (𝛾, 𝜔) ≤ 𝑡 (𝛼, 𝜗) 𝑓𝑜𝑟 𝛾 ≤ 𝛼 𝑎𝑛𝑑 𝜔 ≤ 𝜗 ∀ 𝛾, 𝛼, 𝜗, 𝜔 ∈ [0, 1]. Definition 2.2 [8] The pair (𝑋, 𝐹) named as Probabilistic 2-metric space (2-PM space) where 𝑋 ≠ ∅ and 𝐹 ∶ 𝑋 × 𝑋 × 𝑋 → 𝐿 here L is the set of all distribution functions and the F value at (𝑢, 𝑣, 𝑤) ∈ 𝑋 × 𝑋 × 𝑋 is represented by 𝐹𝑢,𝑣,𝑤 and obeys properties as under a) 𝐹𝑢,𝑣,𝑤 (0) = 0, b) For all distinct u, v in X ∃ w ∈ X with 𝐹𝑢,𝑣,𝑤 (t) < 1 for some t > 0, c) 𝐹𝑢,𝑣,𝑤 (t) = 1 ∀ t > 0, If any two of the three points have to be the same, d) 𝐹𝑢,𝑣,𝑤 ( t) = 𝐹𝑣,𝑤,𝑢 (t) = 𝐹𝑤,𝑢,𝑣 ( t), e) 𝐹𝑢,𝑣,𝑤 ( 𝑡𝑎) = 𝐹𝑣,𝑤,𝑢 (𝑡𝑏) = 𝐹𝑤,𝑢,𝑣 ( 𝑡𝑐) = 1 then 𝐹𝑢,𝑣,𝑤 (𝑡𝑎 + 𝑡𝑏 + 𝑡𝑐 ) = 1. Definition 2.3 [8] A sequence 〈𝑥𝑛 〉 in 2-Menger space ( 𝑋, 𝐹, 𝑡) is (i) Converges to 𝛽 if for each 𝜖 ∗ > 0, 𝑡 > 0, ∃ N(𝜖 ∗) ∈ N implies 𝐹𝑥𝑛,𝛽,𝑎 (ϵ *) > 1 –t, ∀ 𝑎𝜖 𝑋 𝑎𝑛𝑑 𝑛 ≥ 𝑁 (𝜖 ∗). (ii) Cauchy if for each 𝜖 > 0, 𝑡 > 0, ∃ N(𝜖) ∈ N implies 𝐹𝑥𝑛,𝑥𝑚,𝑎 (𝜖) > 1 – t, ∀ 𝑎 ∈ 𝑋 𝑎𝑛𝑑 𝑛, 𝑚 ≥ 𝑁(𝜖). (iii) If the cauchy sequence converges in X then it is referred as a complete 2-Menger space. Int. J. Anal. Appl. 19 (6) (2021) 906 Definition 2.4 [8] Self-mappings P, S in 2-Menger space ( 𝑋, 𝐹, 𝑡) are called (i) Compatible If 𝐹𝑃𝑆𝑥𝑛,𝑆𝑃𝑥𝑛, 𝑎(δ) → 1, ∀ a ∈ X and δ > 0 whenever a sequence 〈𝑥𝑛〉 in X ∋ 𝑃𝑥𝑛,𝑆𝑥𝑛 → 𝑧 where z is an element of X as n → ∞. (ii) Weakly compatible if the mappings commute at their coincidence points. (iii) Occasionally weakly compatible (OWC) if ∃ x in X ∋ Px = Sx ⇒ PSx = SPx. Remark 2.5 Two weakly compatible mappings are obviously OWC mappings, but the converse does not have to be the case. Example 2.6. By treating 𝑋 = [0, 1] and d be the usual metric on X and for all 𝑡1 ∈ [0, 1], define 𝐹𝑢, 𝑣, 𝑎(𝑡1) = { 𝑡1 𝑡1+⎸𝛼−𝛽⎹ , 𝑡1 > 0 0 , 𝑡1 = 0 ∀ 𝛼, 𝛽 in X and fixed a, 𝑡1 > 0. Define mappings 𝑃, 𝑆 ∶ 𝑋 → 𝑋 𝑎𝑠 𝑃(𝑥) = 𝑥2 2 , x ∈ [0, 1] and 𝑆(𝑥) = 𝑥 3 , x ∈ [0, 1]. We notice that the pair (P, S) has two coincidence points 0, 2 3 . If x = 2 3 then P ( 2 3 ) = S ( 2 3 ) = 2 9 (2.6.1) 𝑃𝑆 ( 2 3 ) = 𝑃( 2 9 ) = 2 81 , (2.6.2) 𝑆𝑃( 2 3 ) = S( 2 9 ) = 2 27 . (2.6.3) From (2.6.2) and (2.6.3) PS( 2 3 ) ≠ SP( 2 3 ). At x = 0, P ( 0 ) = S (0 ) and 𝑃𝑆( 0 ) = 𝑆𝑃( 0 ). This shows the mappings P, S are OWC but not weakly compatible. Definition 2.7 [9] “Self maps P and S of a 2-Menger space (X, F, t) are said to satisfy CLR’S – property (common limit range property) if there exists a sequence 〈𝑥𝑛 〉 ∈ 𝑋 ∋ 𝑃𝑥𝑛,𝑆𝑥𝑛 → 𝑆𝑧, for some element z ∈ 𝑋 as n → ∞. This example shows that mappings P, S satisfy CLR’S- property but they do not have closed ranges. Example 2.8. Take X = (0, 1] and 𝑡 ∈ [0, 1], define Int. J. Anal. Appl. 19 (6) (2021) 907 𝐹𝑢,𝑣(t) = { 𝑡 𝑡+⎸𝛼−𝛽⎹ , 𝑡 > 0 0 , 𝑡 = 0 ∀ 𝛼, 𝛽 in X and 𝑡 > 0. 𝐷𝑒𝑓𝑖𝑛𝑒 𝑃, 𝑆 ∶ 𝑋 → 𝑋 𝑎𝑠 𝑃(𝑥 ) = { 1 − 𝑥, 𝑥 ∈ (0, 𝟐 𝟑 ) 𝑥, 𝑥 ∈ [ 𝟐 𝟑 , 1] (2.8.1) and 𝑆(𝑥) = { 2𝑥, 𝑥 ∈ (0, 2 3 ] 1, 𝑥 ∈ ( 2 3 , 1] . (2.8.2) Consider a sequence 𝑥𝑛 = 1 3 − 1 3𝑛 for n = 1, 2, 3… then (2.8.3) P𝑥𝑛 = 1 − ( 𝟏 𝟑 − 1 3𝑛 ) = 𝟐 𝟑 + 1 3𝑛 → 𝟐 𝟑 (2.8.4) S𝑥𝑛 = 2( 𝟏 𝟑 − 1 3𝑛 ) = 𝟐 𝟑 − 2 9𝑛 → 𝟐 𝟑 as n → ∞. (2.8.5) Thus 𝑃𝑥𝑛,𝑆𝑥𝑛 → 𝑆( 𝟏 𝟑 ) = 𝟐 𝟑 as n → ∞. (2.8.6) Where P(X) = ( 1 3 , 1], S(X) = (0, 4 3 ] U{1} this shows that P, S are satisfy CLR’S- property but they do not have closed ranges. Now we give the statement of Theorem (A). It is proved by V. K. Gupta et al. Theorem (A) [8] “ Let A, B, S and T be self -mappings on a complete probabilistic 2-metric space (X̃, F, t) satisfying: (A1) A(X̃) ⊆ T(X̃), B(X̃) ⊆ S( X̃) (𝐴2) one of A(X̃), B(X̃), T(X̃) or S(X̃) is complete, (𝐴3) pairs (A, S) and (B, T) are weakly compatible, (𝐴4) 𝐹𝐴𝑥. 𝐵𝑦, 𝑎 (t) ≥ r 𝐹𝑆𝑥. 𝑇𝑦, 𝑎 (t) for all x, y and t > 0, where r: [0, 1] → [0, 1] is some continuous function such that r(t) > t for each o < t < 1, then A,B,S and T have unique common fixed point in X̃ ’’. We now generalize Theorem(A) as under. 3. MAIN RESULT Theorem 3.1 Let A, B, S and T be self -mappings on a complete probabilistic 2-metric space (X̃, F, 𝑡 ∗) satisfying : (3.1.1) A(X̃) ⊆ T(X̃) , B(X̃) ⊆ S(X̃), (3.1.2) the pairs (A, S), (B, T) share the CLR’S property with OWC, Int. J. Anal. Appl. 19 (6) (2021) 908 (3.1.3) 𝐹𝐴𝑥. 𝐵𝑦, 𝑎 (𝑡 ∗ ) ≥ r 𝐹𝑆𝑥. 𝑇𝑦, 𝑎 (𝑡 ∗ ) for all x, y and 𝑡 ∗ > 0, where r: [0, 1] → [0, 1] is some continuous function such that r(𝑡 ∗ ) > 𝑡 ∗ for each o < 𝑡 ∗ < 1 then A, B, S and T have unique common fixed point in X̃. Proof: Iteratively the sequences 〈𝑦𝑛 〉 and 〈𝑥𝑛 〉 can be constructed as x0 ∈ X̃ ⇒ Ax0 ∈ A(X̃) ⊆ T( X̃), ∃ x1 ∈ X̃ in such a way that Ax0 = Tx1, Bx1 ∈ B( X̃) ⊆ S( X̃) then we have x2 ∈ X̃ with Bx1 = Sx2 〈𝑦2𝑛 〉 = A𝑥2𝑛= T𝑥2𝑛+1and 〈𝑦2𝑛+1〉 = B𝑥2𝑛+1 = S𝑥2𝑛+ 2. (3.1.4) Now our claim is to show 〈𝑦𝑛 〉 is cauchy sequence. For this take 𝑥 = 𝑥2𝑛, 𝑦 = 𝑥2𝑛+1 in (3.1.3) we get 𝐹𝐴𝑥2𝑛, 𝐵 𝑥2𝑛+1, 𝑎 (𝑡 ∗) ≥ 𝑟𝐹𝑆 𝑥2𝑛 . 𝑇 𝑥2𝑛+1 (𝑡 ∗), (3.1.5) ⇒𝐹𝑦2𝑛, 𝑦2𝑛+1, 𝑎 (𝑡 ∗) ≥ 𝑟𝐹 𝑦2𝑛−1 . 𝑦2𝑛, 𝑎 (𝑡 ∗) > 𝐹 𝑦2𝑛−1 . 𝑦2𝑛, 𝑎 (𝑡 ∗). (3.1.6) Similarly 𝐹𝑦2𝑛+1, 𝑦2𝑛+2, 𝑎 (𝑡 ∗) > 𝐹 𝑦2𝑛 . 𝑦2𝑛+1, 𝑎 (𝑡 ∗). (3.1.7) In general we have 𝐹𝑦𝑛+1. 𝑦𝑛, 𝑎 (𝑡 ∗) > 𝐹𝑦𝑛, 𝑦𝑛−1, 𝑎 (𝑡 ∗) for all values of n. Then < 𝐹𝑦𝑛+1. 𝑦𝑛, 𝑎 (𝑡 ∗) > is an increasing sequence bounded above by 1 therefore it must converge to L, where L ≤ 1. If L < 1 then 𝐹𝑦𝑛+1. 𝑦𝑛, 𝑎 (𝑡 ∗) = L > r(1) > 1 as a result of the contradiction, L= 1. Hence 𝐹𝑦𝑛+1. 𝑦𝑛, 𝑎 (t) = 1 for all n and p. As a result, because Cauchy sequence exists in complete space X, it has a limit z in X̃ and consequently each sub sequence has the same limit z. That is Ax2n , Sx2n → z and Bx2n+1, Tx2n+1 → z as n → ∞. (3.1.8) On using CLRS-Property of (A, S) , (B, T) implies there are sequences (an ) as well as (bn ) in order for Aan , San , B bn , T bn → Sµ as n → ∞ for some µ in X̃. (3.1.9) To prove z = Sµ put x = a2n , y = x2n+5 in (3.1.3) we get 𝐹A𝑎2𝑛. 𝐵x2𝑛+5, 𝑎 (𝑡 ∗) ≥ r (𝐹S𝑎2𝑛. 𝑇x2𝑛+5, 𝑎 (𝑡 ∗) ) as n → ∞ (3.1.10) ⇒ 𝐹Sµ. z, 𝑎 (𝑡 ∗) ≥ r (𝐹Sµ. z, 𝑎 (𝑡 ∗)).> 𝐹Sµ. z, 𝑎 (𝑡 ∗). (3.1.11) Resulting 𝐹Sµ. z, 𝑎 (𝑡 ∗). > 𝐹Sµ. z, 𝑎 (𝑡 ∗) (3.1.12) which is a contradiction. Hence Sµ =z. (3.1.13) Int. J. Anal. Appl. 19 (6) (2021) 909 Claim A µ = S µ. Put x = µ, y = x2n+3 in (3.1.3) we get 𝐹Aµ. 𝐵x2𝑛+3, 𝑎 (𝑡 ∗) ≥ r (𝐹Sµ. 𝑇x2𝑛+3, 𝑎 (𝑡 ∗) ) as n → ∞ (3.1.14) ⇒ 𝐹Aµ. z, 𝑎 (𝑡 ∗) ≥ r (𝐹Sµ. z, 𝑎 (𝑡 ∗)) using (3.1.13) (3.1.15) ⇒ 𝐹Aµ. z, 𝑎 (𝑡 ∗) ≥ r (𝐹z. z, 𝑎 (𝑡 ∗)) = r(1) = 1. (3.1.16) This results A µ = S µ = z. (3.1.17) Since the pair (A, S) obeys OWC resulting A µ = S µ ⇒ SA µ = AS µ. That is Az = Sz. (3.1.18) Claim Az = z. Substitute y = x2n+3, x = z in (3.1.3) we have 𝐹Az. 𝐵x2𝑛+3, 𝑎 (𝑡 ∗) ≥ r 𝐹Sz. 𝑇x2𝑛+3, 𝑎 (𝑡 ∗ ) letting n → ∞. (3.1.19) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗ ) ≥ 𝑟𝐹𝑆𝑧 . 𝑧, 𝑎 ( 𝑡 ∗) using (3.1.18) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝐴𝑧 . 𝑧, 𝑎 ( 𝑡 ∗) > 𝐹𝐴𝑧 . 𝑧, 𝑎 ( 𝑡 ∗), (3.1.20) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗ ) > 𝐹𝐴𝑧 . 𝑧, 𝑎 ( 𝑡 ∗). (3.1.21) This is a contradiction. Thus z = Az. Resulting Az = Sz = z. (3.1.22) Since Az ∈ A(X̃) ⊆ T(X̃) then ∃ ρ ∈ X̃ such that Az = T 𝜌. (3.1.23) Claim z = B𝜌. By employing x = x4n, y = 𝜌 of (3.1.3) we obtain 𝐹𝐴𝑥2𝑛, 𝐵𝜌 (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑥 2𝑛 . 𝑇𝜌 , 𝑎 ( 𝑡 ∗) as n → ∞ . (3.1.24) From (3.1.22) & (3.1.23) ⇒ 𝐹𝑧, 𝐵𝜌 (𝑡 ∗) ≥ 𝑟𝐹𝑧 . 𝑇𝜌 , 𝑎 ( 𝑡 ∗ ) = r(1) = 1. (3.1.25) Thus z = B𝜌 = T𝜌. Since the pair of mappings (B, T) obeys OWC, this results B𝜌 = T𝜌 ⇒ BT𝜌 = TB𝜌. That is Bz = Tz. (3.1.26) Claim z = Bz. By substituting y = z, x = z in (3.1.3) results 𝐹𝐴𝑧, 𝐵𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑧 . 𝑇𝑧, 𝑎 ( 𝑡 ∗) using (3.1.22) &(3.1.26) (3.1.27) 𝐹𝑧, B𝑧, a (𝑡 ∗ ) ≥ 𝑟𝐹𝑧 . 𝐵𝑧, 𝑎 ( 𝑡 ∗) > 𝐹𝑧 . 𝐵𝑧, 𝑎 ( 𝑡 ∗). (3.1.28) Resulting 𝐹𝑧, B𝑧, a (𝑡 ∗) > 𝐹𝑧 . 𝐵𝑧, 𝑎 ( 𝑡 ∗). It is impossible. Therefore Bz =z. Combining all we get Az = Bz = z = Sz = Tz. Int. J. Anal. Appl. 19 (6) (2021) 910 Thus 𝑧 is the required common fixed point for these mappings A, B, S and T. Uniqueness: Assume z1 is second common fixed point. Now assume z ≠ z1. By considering y = z1 , x = z in (3.1.4) we obtain 𝐹𝐴𝑧, 𝐵𝑧1 , a (𝑡∗) ≥ 𝑟𝐹𝑆𝑧 . 𝑇𝑧1, 𝑎 (𝑡 ∗) 𝐹𝑧, 𝑧1, a (𝑡 ∗) ≥ 𝑟𝐹𝑧 . 𝑧1, 𝑎 (𝑡 ∗) > 𝐹𝑧 . 𝑧1, 𝑎 (𝑡 ∗) 𝐹𝑧, 𝑧1, a (𝑡 ∗) > 𝐹𝑧 . 𝑧1, 𝑎 (𝑡 ∗) which is absurd. Hence z = z1. As a result, four self- mappings A, B, S, and T have the only one common fixed point. Now we justify our theorem as under. 3.2 Example Let us take X = [0, 𝜋] and each 𝑡 ∈ [0, 1], define 𝐹𝑢,𝑣( t) = { 𝑡 𝑡+⎸𝛼−𝛽⎹ , 𝑡 > 0 0 , 𝑡 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝛼, 𝛽 𝑖𝑛 𝑋 , 𝑡 > 0. Define mappings 𝑃, 𝑆, 𝑇 & 𝑄 ∶ 𝑋 → 𝑋 𝑎𝑠 𝐴 (𝑥) = 𝐵(𝑥) = { 2𝑒 −𝜋𝑥 , 𝑥 ∈ [0, 𝜋 2 ) 𝜋 − 𝑥, 𝑥 ∈ [ 𝜋 2 , 𝜋] (3.2.1) 𝑎𝑛𝑑 𝑆(𝑥) = 𝑇(𝑥) = { 2𝑒 −𝜋𝑥 2 , 𝑥 ∈ [0, 𝜋 2 ) 𝑥, 𝑥 ∈ [ 𝜋 2 , 𝜋] (3.2.2) Now A(X) = B(X) = [0, 2] and S(X) = T(X) = [0, 𝜋] implies 𝐴(𝑋 ) ⊆ 𝑇(𝑋) 𝑎𝑛𝑑 𝐵(𝑋 ) ⊆ 𝑆(𝑋). Clearly 𝜋 2 and 1 are coincidence points for the mappings B, T. At x = 𝜋 2 , B( 𝜋 2 ) = T ( 𝜋 2 ) and 𝐵𝑇 ( 𝜋 2 ) = 𝐵 ( 𝜋 2 ) = 𝜋 2 , 𝑇𝐵 ( 𝜋 2 ) = T( 𝜋 2 ) = 𝜋 2 implies 𝐵𝑇 ( 𝜋 2 ) = 𝑇𝐵 ( 𝜋 2 ). At x = 1, B(1) = T(1) and BT(1) ≠ TB(1). Thus the pairs (A, S), (B, T) satisfy OWC but are not weakly compatible. If 𝑥𝑛 = 𝜋 2 − 1 𝑛 for all n ≥ 1. Then (3.2.3) S𝑥𝑛= T𝑥𝑛= S ( 𝜋 2 − 1 𝑛 ) = 𝜋 2 − 1 𝑛 → 𝜋 2 . (3.2.4) A𝑥𝑛= B𝑥𝑛= A ( 𝜋 2 − 1 𝑛 ) = 𝜋 – ( 𝜋 2 − 1 𝑛 ) = 𝜋 2 + 1 𝑛 → 𝜋 2 as n → ∞. (3.2.5) Int. J. Anal. Appl. 19 (6) (2021) 911 ⇒ A𝑥𝑛, S𝑥𝑛, T𝑥𝑛, B𝑥𝑛 → S( π 2 ) as n → ∞. (3.2.6) This gives the pairs of maps (A, S), (B, T) sharing the CLR’S property with OWC. Thus A, B, S and T satisfy all the norms of Theorem and having the unique commonly fixed point at 𝜋 2 as A ( 𝜋 2 )= S( 𝜋 2 ) = B( 𝜋 2 ) = T( 𝜋 2 ) = 𝜋 2 . Now we present another generalization of Theorem (A) as under. Theorem 3.3 Let A, B, S and T be self -mappings on a complete probabilistic 2-metric space (X̃, F, 𝑡∗) satisfying : (3.3.1) A(X̃) ⊆ T(X̃), B(X̃) ⊆ S( X̃) (3.3.2) the pair (A, S) satisfies CLR’S property with OWC and (B, T) satisfies OWC. (3.3.3) Further 𝐹𝐴𝑥. 𝐵𝑦, 𝑎 (𝑡 ∗) ≥ r 𝐹𝑆𝑥. 𝑇𝑦, 𝑎 (𝑡 ∗) for all elements x, y in X̃ and 𝑡 ∗> 0 r is continuous self-map on [0, 1] such that r(𝑡 ∗) > 𝑡∗ for each o < 𝑡∗< 1. Then A, B, S and T have unique common fixed point in X̃. Proof: Take the constructed sequences < xn> , < yn> in Theorem (3.1) as 〈𝑦2𝑛 〉 = A𝑥2𝑛= T𝑥2𝑛+1and 〈𝑦2𝑛+1〉 = B𝑥2𝑛+1 = S𝑥2𝑛+ 2 . (3.3.4) It is already shown that 〈𝑦𝑛 〉 as cauchy sequence. As a result each sub sequence has the same limit point z in complete space X̃. That is Ax2n , Sx2n → z and Bx2n+1, Tx2n+1 → z. The pair (A, S) obeys CLRS-property this implies there is a sequence 〈𝑧𝑛 〉 such that 𝐴𝑧𝑛 , 𝑆𝑧𝑛 → 𝑆𝑣 for some v in X̃. Claim z = Sv. By putting y = x2n+1 , x = zn, in (3.3.3), that results 𝐹𝐴𝑧𝑛, 𝐵𝑥2𝑛+1, a (𝑡∗) ≥ 𝑟𝐹𝑆𝑧𝑛 . 𝑇𝑥2𝑛+1 , 𝑎 (𝑡 ∗) as n → ∞ (3.3.5) ⇒𝐹𝑆𝑣, 𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑣 . 𝑧, 𝑎 (𝑡 ∗) > 𝐹𝑆𝑣 . 𝑧, 𝑎 (𝑡 ∗), (3.3.6) ⇒𝐹𝑆𝑣, 𝑧, a (𝑡 ∗) > 𝐹𝑆𝑣 . 𝑧, 𝑎 (𝑡 ∗) . (3.3.7) This is absurd. As a result Sv =z. (3.3.8) Claim Av = Sv. (3.3.9) By inserting x = v, y = x2n+3 in (3.3.3), that results 𝐹𝐴𝑣, 𝐵𝑥 2𝑛+3, 𝑎 (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑣 . 𝑇𝑥 2𝑛+3 , 𝑎 (𝑡 ∗) letting as n → ∞ (3.3.10) ⇒ 𝐹𝐴𝑣, 𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑣 . 𝑧, 𝑎 (𝑡 ∗) using (3.3.8) (3.3.11) ⇒ 𝐹𝐴𝑣, 𝑆𝑣, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑣 . 𝑆𝑣, 𝑎 (𝑡 ∗) = r(1) = 1. (3.3.12) Int. J. Anal. Appl. 19 (6) (2021) 912 ⇒ Av = Sv = z. (3.3.13) Since the pair (A, S) satisfies OWC property, that results Av = Sv ⇒ SAv = ASv. This gives Az = Sz. (3.3.14) Claim Az = z. By replacing y = x2n+1, x = z in (3.3.3), as a result 𝐹𝐴𝑧, 𝐵𝑥2𝑛+1, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑧 . 𝑇𝑥2𝑛+1 , 𝑎 (𝑡 ∗). As n → ∞ (3.3.15) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑧 . 𝑧, 𝑎 (𝑡 ∗), using ( 3.3.14) (3.3.16) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝐴𝑧 . 𝑧, 𝑎 (𝑡 ∗) > 𝐹𝐴𝑧 . 𝑧, 𝑎 (𝑡 ∗) , (3.3.17) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗) > 𝐹𝐴𝑧 . 𝑧, 𝑎 (𝑡 ∗). This is a contradiction. Consequently Az = z. (3.3.18) By combining (3.3.14) and (3.3.18) gives z = Sz = Az. (3.3.19) Since Az ∈ A( X̃) ⊆ T( X̃) then ∃ 𝜔 ∈ X̃ such that Az = T 𝜔. (3.3.20) Claim z = B 𝜔. By using x = x4n, y = 𝜔 of (3.3.4), we obtain 𝐹𝐴𝑥 2𝑛,, 𝐵𝜔 (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑥 2𝑛, . 𝑇𝜔 , 𝑎 (𝑡 ∗). (3.3.21) Taking limit as n → ∞ and from (3.3.19) and (3.3.20) we get ⇒ 𝐹𝑧, 𝐵𝜔 (𝑡 ∗) ≥ 𝑟𝐹𝑧 . 𝑧, 𝑎 (𝑡 ∗) = r(1) = 1. (3.3.22) Thus z = B 𝜔 = T 𝜔. (3.3.23) Since the pair (B, T) obeys OWC property gives B 𝜔 = T 𝜔 ⇒ BT 𝜔 = TB 𝜔 implying Bz = Tz. (3.3.24) Claim z = Bz. Applying x = y = z in (3.3.3), this resulting 𝐹𝐴𝑧, 𝐵𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑧 . 𝑇𝑧, 𝑎 (𝑡 ∗), (3.3.25) using (3.3.19) and (3.3.24) ⇒ 𝐹𝑧, B𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑧 . 𝐵𝑧, 𝑎 (𝑡 ∗) > 𝐹𝑧 . 𝐵𝑧, 𝑎 (𝑡 ∗), (3.3.26) ⇒ 𝐹𝑧, B𝑧, a (𝑡 ∗) > 𝐹𝑧 . 𝐵𝑧, 𝑎 (𝑡 ∗). (3.3.27) Contradicting the fact implies Bz =z. As a result Az = Bz = Sz = Tz = z. As a consequence four self-mappings A, B, S, and T, there is a fixed point commonly. Uniqueness can be easily proved as in the Theorem (3.1). Int. J. Anal. Appl. 19 (6) (2021) 913 Now the Theorem (3.3) can be supported by discussing with suitable example. 3.4 Example We choose X = [0, 1], d be usual metric on X and each 𝑡 ∈ [0, 1], define 𝐹𝑢,𝑣 ( t) = { 𝑡 𝑡+⎸𝛼−𝛽⎹ , 𝑡 > 0 0 , 𝑡 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝛼, 𝛽 𝑖𝑛 𝑋 , 𝑡 > 0. Choose mappings 𝑃, 𝑆, 𝑇 & 𝑄 ∶ 𝑋 → 𝑋 𝑎𝑠 𝑃 (𝑥) = 𝑄(𝑥) = { 1 − 2𝑥, 𝑥 ∈ [0, 0.2] 𝑥2, 𝑥 ∈ (0.2, 1] (3.4.1) 𝑎𝑛𝑑 𝑆(𝑥) = 𝑇(𝑥) = { 3𝑥, 𝑥 ∈ [0, 0.2] 𝑥3, 𝑥 ∈ (0.2, 1] . (3.4.2) Now P(X) = Q(X) = (0.04, 1] and S(X) = T(X) = [0, 1] so that 𝑃(𝑋 ) ⊆ 𝑇(𝑋) 𝑎𝑛𝑑 𝑄(𝑋 ) ⊆ 𝑆(𝑋). Clearly 0.2 and 1 are coincedence points of the graphs Q, T. At x = 0.2, Q (0.2) = T(0.2) = 0.6 but QT(0.2) = Q(0.6) = 0.36, TQ(0.2) = T(0.6) = 0.216 At x = 1, Q (1) = T(1) and 𝑄𝑇(1) = 𝑄(1) = 1 = 𝑇(1) = 𝑇𝑄(1). This demonstrates that the pairs (P, S), (Q, T) are OWC mappings, although they are not weakly compatible. If we choose 𝑥𝑛 = 1 − 4 3𝑛 for all n ≥ 1. Then (3.4.3) P𝑥𝑛= P ( 1 − 4 3𝑛 ) = ( 1 − 4 3𝑛 )2 → 1 (3.4.4) S𝑥𝑛= S ( 1 − 4 3𝑛 ) = ( 1 − 4 3𝑛 )3 → 1 as n → ∞. 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