Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 433 https://internationalpubls.com Tripled Fixed Point Results in 𝐆𝐛-Metric Spaces V. Rajitha1, G. Upender Reddy2 1Research Scholar,Department of Mathematics, Osmania University, Telangana, India. E-mail address: rajivan07@gmail.com 2Associate Professor of Mathematics, Nizam College (A), Osmania University, Telangana, India. E-mail address: yuviganga@gmail.com Article History: Received: 07-05-2024 Revised: 25-06-2024 Accepted: 09-07-2024 Abstract In recent work authors were discussed fixed point results with various contractions like (ψ,Ο•)-weakly contractive mappings, cyclic contraction, E.A property, Suzuki-type contraction etc. in complete Gb-metric spaces, With the help of completeness property and continuous function we obtained unique tripled fixed point in Gb-metric spaces. Objectives: To show tripled fixed point theorems in Gb-metric spaces via new type of contraction and shown illustrate an example which supports the main result. Methods: In recent work authors were discussed fixed point results with various contractions like (ψ, Ο•)-weakly contractive mappings, cyclic contraction, E.A property, Suzuki-type contraction etc. in complete Gb-metric spaces, here we have showed tripled fixed point results by using new type of contraction. Results: Unique tripled fixed points with new type of contraction in Gb-metric spaces. Conclusions: In this work we have obtained TFP results by using a new type of contraction and discussed corollary also an example which supports the main result. Keywords: tripled coincident point (TCIP); tripled fixed point (TFP); Gb-metric space (Gb-MS); Cauchy sequence (CS); continuous function; completeness property. 1. Introduction Fixed point theory is a key tool in nonlinear functional analysis, with applications in computer science, chemistry, biology, and engineering. The Banach contraction principle, which asserts that each contraction has a unique fixed point in complete metric spaces, is a cornerstone of this study. Many authors are interested in fixed point theory, particularly the Banach contraction principle, due to its possible applications in the above domains. Investigating the presence and uniqueness of a fixed point for many contraction type mappings in diverse metric spaces is especially natural and intriguing. In 2008, Mustafa and Sims introduced G-metric space as new generalizations of metric spaces. In 2014, Aghajani et al. generalised metric spaces. They created a Gb-metric space by combining G- metric and b-metric definitions. They also noted that Gb-MS are effectively larger than G-MS. The G-MS is a subset of the Gb-MS when s = 1. They also proved that every Gb-MS topologically equals a b-MS. Many articles have published on this MS (See [1]-[15]). In this current work we have discussed with new type of contraction on 𝐺𝑏-MS, unique TFP results and some corollary also shown an example of our work. mailto:rajivan07@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 434 https://internationalpubls.com Before the main work we will discuss some basic definitions. 2. Methodology Definition 2.1 Let Ὣ is a nonempty set, ₒ𝑏: Ὣ3β†’ ℝ+ is a function satisfying the following properties: (ₒ𝑏1) ₒ𝑏 (α΄‚, α΄”, 𝔣) = 0, if α΄‚=α΄” =𝔣; (ₒ𝑏2) 0<ₒ𝑏 (α΄‚, α΄‚, 𝔣) = 0, for α΄‚, 𝔣 Ο΅ Ὣ and α΄‚ β‰  𝔣; (ₒ𝑏3) ₒ𝑏 (α΄‚, α΄‚, 𝔣) ≀ ₒ𝑏 (α΄‚, 𝔣, α΄”)for all α΄‚, α΄‚, 𝔣 ∈ Ὣ with 𝔣 β‰  α΄”; (ₒ𝑏4) ₒ𝑏 (α΄‚, α΄”, 𝔣) = ₒ𝑏 (α΄”, 𝔣, α΄‚) = ₒ𝑏 (𝔣, α΄‚, α΄”) = …….. (Symmetry of three variables); (ₒ𝑏5) ₒ𝑏 (α΄‚, α΄”, 𝔣) ≀ s[ₒ𝑏 (α΄‚, π“Š, π“Š) + ₒ𝑏 (π“Š, α΄”, 𝔣)] for all α΄‚, α΄”, 𝔣, π“Š ∈ Ὣ (Rectangular inequality). The function ₒ𝑏 is called metric on Ὣ and the pair (Ὣ,ₒ𝑏) is called ₒ𝑏-MS. Example 2.2 Let (Ὣ, d) is a MS. The function ₒ𝑏: Ὣ3β†’ ℝ+ defined by ₒ𝑏 (α΄‚, α΄”, 𝔣) = max{d(α΄‚, α΄”),d(α΄”, 𝔣),d(𝔣,α΄‚)} and ₒ𝑏 (α΄‚, α΄”, 𝔣) = d(α΄‚, α΄”)+d(α΄”, 𝔣)+d(𝔣,α΄‚) for all α΄‚, α΄”, 𝔣 ∈ Ὣ. Then (Ὣ, ₒ𝑏) is a ₒ𝑏-MS. Definition 2.3 Let (Ὣ,ₒ𝑏) be ₒ𝑏-MS, for any Ξ΅>0, a sequence {ℐ𝑛} is a ₒ𝑏-CS, if ₒ𝑏(ℐ𝑛, β„π‘š, ℐ𝑙) < Ξ΅ for all n, m, l β‰₯ N. Definition 2.4 Let (Ὣ,ₒ𝑏) is ₒ𝑏-MS. A sequence {ℐ𝑛} is called a ₒ𝑏-convergent to𝔖 ∈ Ὣ, if for anyΞ΅ > 0, there is 𝑁 ∈ β„• such that ₒ𝑏(𝔖, ℐ𝑛, ℐ𝑛) < Ξ΅ for all n β‰₯ 𝑁. Definition 2.5 A ₒ𝑏-MS (Ὣ,ₒ𝑏) is called ₒ𝑏-complete. If every ₒ𝑏-CS in Ὣ is ₒ𝑏-convergent in Ὣ. Definition 2.6 Let (Ὣ,ₒ𝑏) is ₒ𝑏-MS. A mapping β„‹: Ὣ3 β†’ Ὣ is said to be continuous if for any {ℐ𝑛}, {ὡ𝑛}, {ϱ𝑛}β‚’-convergent sequences to Ο°, β„“, β„˜ then {β„‹(ℐ𝑛, ὡ𝑛, ϱ𝑛)} is ₒ𝑏- convergent to β„‹(Ο°, β„“, β„˜). Definition 2.7 Let β„‹: Ὣ3 β†’ Ὣ and β„±: Ὣ β†’ Ὣ be any two mappings, then the mappings are said to be commute if β„±(β„‹(Ο°, β„“, β„˜)) = β„‹(β„±(Ο°), β„±(β„“), β„±(β„˜)) for allΟ°, β„“, β„˜ ∈ Ὣ. 3. Results and Discussion Definition 3.1 Let β„‹: Ὣ3 β†’ Ὣ be a mapping. An element (α΄‚, α΄”, 𝔣) is called TFP of the mappingβ„‹. If β„‹(α΄‚, α΄”, 𝔣) = α΄‚, β„‹( α΄”, 𝔣, α΄‚) = α΄” and β„‹(𝔣, α΄‚, α΄”) = 𝔣 Definition 3.2 Let β„‹: Ὣ3 β†’ Ὣ be a mapping and β„±: Ὣ β†’ Ὣ be two mappings, then an element (α΄‚, α΄”, 𝔣) is TCIP of the mappings β„± and β„‹. If β„‹(α΄‚, α΄”, 𝔣) = β„±α΄‚, β„‹( α΄”, 𝔣, α΄‚) = β„±α΄” and β„‹(𝔣, α΄‚, α΄”) = ℱ𝔣 Definition 3.3 Let β„‹: Ὣ3 β†’ Ὣ be a mapping and β„±: Ὣ β†’ Ὣ be two mappings, then an element (α΄‚, α΄”, 𝔣) is common TFP of the mappings β„± and β„‹. If β„‹(α΄‚, α΄”, 𝔣) = β„±α΄‚ = α΄‚, β„‹( α΄”, 𝔣, α΄‚) = β„±α΄” = α΄” and β„‹(𝔣, α΄‚, α΄”) = ℱ𝔣 = 𝔣 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 435 https://internationalpubls.com Theorem 3.4Let (Ὣ,ₒ𝑏) is ₒ𝑏-MS. Let β„‹: Ὣ3 β†’ Ὣ and β„±: Ὣ β†’ Ὣ be two mappings such that [ₒ𝑏(β„‹(α΄‚, α΄”, 𝔣), β„‹(Ο°, β„“, β„˜), β„‹(Ο°, β„“, β„˜)) + ₒ𝑏(β„‹( α΄”, 𝔣, α΄‚), β„‹(β„“, β„˜, Ο°), β„‹(β„“, β„˜, Ο°)) + ₒ𝑏(β„‹(𝔣, α΄‚, α΄”), β„‹(β„˜, Ο°, β„“), β„‹(β„˜, Ο°, β„“))] ≀ πœƒ[ₒ𝑏(β„±α΄‚, β„±Ο°, β„±Ο°) + ₒ𝑏(β„±α΄”, β„±β„“, β„±β„“) + ₒ𝑏(ℱ𝔣, β„±β„˜, β„±β„˜)]------------------- (3.4.1) for all α΄‚, α΄”, 𝔣, Ο°, β„“, β„˜ ∈ Ὣ. If β„‹and β„± satisfies the following conditions i) β„‹(Ὣ3) βŠ‚ β„±(Ὣ); ii) β„±(Ὣ) is ₒ𝑏-complete; iii) β„± is ₒ𝑏-continuous and commutes with β„‹; If πœƒ ∈ [0,1) then there exists a unique common TFP for β„‹ and β„± inὫ. Proof.Letℐ0, ὡ0, Ο±0be arbitrary in Ὣ then from (i) we can construct sequences {ℐ𝑛}, {ὡ𝑛} and {ϱ𝑛} inὫ such that ℱℐ𝑝+1 = β„‹(ℐ𝑝, ὡ𝑝, ϱ𝑝), ℱὡ𝑝+1 = β„‹(ὡ𝑝, ϱ𝑝, ℐ𝑝) and ℱϱ𝑝+1 = β„‹(ϱ𝑝, ℐ𝑝, ὡ𝑝)-------- (3.4.2) let ℑ𝑝 = ₒ𝑏(ℱℐ𝑝, ℱℐ𝑝+1, ℱℐ𝑝+1) + ₒ𝑏(ℱὡ𝑝, ℱὡ𝑝+1, ℱὡ𝑝+1) + ₒ𝑏(ℱϱ𝑝, ℱϱ𝑝+1, ℱϱ𝑝+1) for all 𝑝 ∈ β„•. Now by using equation (3.4.1), we have ℑ𝑝 = ₒ𝑏(ℱℐ𝑝, ℱℐ𝑝+1, ℱℐ𝑝+1) + ₒ𝑏(ℱὡ𝑝, ℱὡ𝑝+1, ℱὡ𝑝+1) + ₒ𝑏(ℱϱ𝑝, ℱϱ𝑝+1, ℱϱ𝑝+1) =ₒ𝑏 (β„‹(β„π‘βˆ’1, α½΅π‘βˆ’1, Ο±π‘βˆ’1), β„‹(ℐ𝑝, ὡ𝑝, ϱ𝑝), β„‹(ℐ𝑝, ὡ𝑝, ϱ𝑝)) + ₒ𝑏 (β„‹(α½΅π‘βˆ’1, Ο±π‘βˆ’1, β„π‘βˆ’1), β„‹(ὡ𝑝, ϱ𝑝, ℐ𝑝), β„‹(ὡ𝑝, ϱ𝑝, ℐ𝑝)) + ₒ𝑏 (β„‹(Ο±π‘βˆ’1, β„π‘βˆ’1, α½΅π‘βˆ’1), β„‹(ϱ𝑝, ℐ𝑝, ὡ𝑝), β„‹(ϱ𝑝, ℐ𝑝 , ὡ𝑝)) ≀ πœƒ[ₒ𝑏(β„±β„π‘βˆ’1, ℱℐ𝑝, ℱℐ𝑝) + ₒ𝑏(β„±α½΅π‘βˆ’1, ℱὡ𝑝, ℱὡ𝑝) + ₒ𝑏(β„±Ο±π‘βˆ’1, ℱϱ𝑝, ℱϱ𝑝)] = πœƒβ„‘π‘βˆ’1. Which yields that ℑ𝑝 ≀ πœƒπ‘β„‘0, βˆ€π‘ ∈ β„•. Now for all π‘š, 𝑛 ∈ β„• with π‘š > 𝑛 and by using (ₒ𝑏5), we get ₒ𝑏(ℱℐ𝑛, β„±β„π‘š , β„±β„π‘š) + ₒ𝑏(ℱὡ𝑛, β„±α½΅π‘š, β„±α½΅π‘š) + ₒ𝑏(ℱϱ𝑛, β„±Ο±π‘š, β„±Ο±π‘š) ≀ 𝑠[ₒ𝑏(ℱℐ𝑛, ℱℐ𝑛+1, ℱℐ𝑛+1) + ₒ𝑏(ℱℐ𝑛+1, β„±β„π‘š, β„±β„π‘š)] + 𝑠[ₒ𝑏(ℱὡ𝑛 , ℱὡ𝑛+1, ℱὡ𝑛+1) + ₒ𝑏(ℱὡ𝑛+1, β„±α½΅π‘š, β„±α½΅π‘š)] + 𝑠[ₒ𝑏(ℱϱ𝑛, ℱϱ𝑛+1, ℱϱ𝑛+1) + ₒ𝑏(ℱϱ𝑛+1, β„±Ο±π‘š, β„±Ο±π‘š)] ≀ [𝑠ₒ𝑏(ℱℐ𝑛, ℱℐ𝑛+1, ℱℐ𝑛+1) + 𝑠2ₒ𝑏(ℱℐ𝑛+1, ℱℐ𝑛+2, ℱℐ𝑛+2) + 𝑠2ₒ𝑏(ℱℐ𝑛+2, β„±β„π‘š, β„±β„π‘š)] + [𝑠ₒ𝑏(ℱὡ𝑛 , ℱὡ𝑛+1, ℱὡ𝑛+1) + 𝑠2ₒ𝑏(ℱὡ𝑛+1, ℱὡ𝑛+2, ℱὡ𝑛+2) + 𝑠2ₒ𝑏(ℱὡ𝑛+2, β„±α½΅π‘š, β„±α½΅π‘š)] + [𝑠ₒ𝑏(ℱϱ𝑛, ℱϱ𝑛+1, ℱϱ𝑛+1) + 𝑠2ₒ𝑏(ℱϱ𝑛+1, ℱϱ𝑛+2, ℱϱ𝑛+2) + 𝑠2ₒ𝑏(ℱϱ𝑛+2, β„±Ο±π‘š, β„±Ο±π‘š)] ……………………………………………………………………………………………. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 436 https://internationalpubls.com ≀ [sₒ𝑏(ℱℐ𝑛, ℱℐ𝑛+1, ℱℐ𝑛+1) + 𝑠2ₒ𝑏(ℱℐ𝑛+1, ℱℐ𝑛+2, ℱℐ𝑛+2) + 𝑠3ₒ𝑏(ℱℐ𝑛+2, ℱℐ𝑛+3, ℱℐ𝑛+3) + β‹― + π‘ π‘šβˆ’1ₒ𝑏(β„±β„π‘šβˆ’2, β„±β„π‘šβˆ’1, β„±β„π‘šβˆ’1) + π‘ π‘šβ‚’π‘(β„±β„π‘šβˆ’1, β„±β„π‘š, β„±β„π‘š)] + [sₒ𝑏(ℱὡ𝑛 , ℱὡ𝑛+1, ℱὡ𝑛+1) + 𝑠2ₒ𝑏(ℱὡ𝑛+1, ℱὡ𝑛+2, ℱὡ𝑛+2) + 𝑠3ₒ𝑏(ℱὡ𝑛+2, ℱὡ𝑛+3, ℱὡ𝑛+3) + β‹― + π‘ π‘šβˆ’1ₒ𝑏(β„±α½΅π‘šβˆ’2, β„±α½΅π‘šβˆ’2, β„±α½΅π‘šβˆ’1) + π‘ π‘šβ‚’π‘(β„±α½΅π‘šβˆ’1, β„±α½΅π‘š, β„±α½΅π‘š)] + [𝑠ₒ𝑏(ℱϱ𝑛, ℱϱ𝑛+1, ℱϱ𝑛+1) + 𝑠2ₒ𝑏(ℱϱ𝑛+1, ℱϱ𝑛+2, ℱϱ𝑛+2) + 𝑠3ₒ𝑏(ℱϱ𝑛+2, ℱϱ𝑛+3, ℱϱ𝑛+3) + β‹― + π‘ π‘šβˆ’1ₒ𝑏(β„±Ο±π‘šβˆ’2, β„±Ο±π‘šβˆ’1, β„±Ο±π‘šβˆ’1) + π‘ π‘šβ‚’π‘(β„±Ο±π‘šβˆ’1, β„±Ο±π‘š, β„±Ο±π‘š)] ≀ 𝑠[ₒ𝑏(ℱℐ𝑛, ℱℐ𝑛+1, ℱℐ𝑛+1) + ₒ𝑏(ℱὡ𝑛, ℱὡ𝑛+1, ℱὡ𝑛+1) + ₒ𝑏(ℱϱ𝑛, ℱϱ𝑛+1, ℱϱ𝑛+1)] + 𝑠2[ₒ𝑏(ℱℐ𝑛+1, ℱℐ𝑛+2, ℱℐ𝑛+2) + ₒ𝑏(ℱὡ𝑛+1, ℱὡ𝑛+2, ℱὡ𝑛+2) + ₒ𝑏(ℱϱ𝑛+1, ℱϱ𝑛+2, ℱϱ𝑛+2)] + β‹― + π‘ π‘šβˆ’1[ₒ𝑏(β„±β„π‘šβˆ’2, β„±β„π‘šβˆ’1, β„±β„π‘šβˆ’1) + ₒ𝑏(β„±α½΅π‘šβˆ’2, β„±α½΅π‘šβˆ’2, β„±α½΅π‘šβˆ’1) + ₒ𝑏(β„±Ο±π‘šβˆ’2, β„±Ο±π‘šβˆ’1, β„±Ο±π‘šβˆ’1)] + π‘ π‘š[ₒ𝑏(β„±β„π‘šβˆ’1, β„±β„π‘š, β„±β„π‘š) + ₒ𝑏(β„±α½΅π‘šβˆ’1, β„±α½΅π‘š, β„±α½΅π‘š) + ₒ𝑏(β„±Ο±π‘šβˆ’1, β„±Ο±π‘š, β„±Ο±π‘š)] ≀ 𝑠ℑ𝑛 + 𝑠2ℑ𝑛+1 + β‹― … … … … … … + π‘ π‘šβˆ’1β„‘π‘šβˆ’2 + π‘ π‘šβ„‘π‘šβˆ’1 ≀ π‘ πœƒπ‘›β„‘0 + 𝑠2πœƒπ‘›+1β„‘0 + β‹― … … … … … … + π‘ π‘šβˆ’1πœƒπ‘šβˆ’2β„‘0 + π‘ π‘šπœƒπ‘šβˆ’1β„‘0 ≀ [π‘ πœƒπ‘› + 𝑠2πœƒπ‘›+1 + β‹― … … … … … … + π‘ π‘šβˆ’1πœƒπ‘šβˆ’2 + π‘ π‘šπœƒπ‘šβˆ’1]β„‘0 ≀ [π‘ πœƒπ‘› + 𝑠2πœƒπ‘›+1 + β‹― … … … … … … ]β„‘0 ≀ π‘ πœƒπ‘›[1 + π‘ πœƒ + π‘ πœƒ2 + β‹― … … … … … ]β„‘0 ≀ π‘ πœƒπ‘› 1βˆ’π‘ πœƒ β„‘0 β†’ 0 as 𝑛 β†’ ∞. Thereforeβ‚’(ℱℐ𝑛, β„±β„π‘š, β„±β„π‘š) + β‚’(ℱὡ𝑛, β„±α½΅π‘š, β„±α½΅π‘š) + β‚’(ℱϱ𝑛, β„±Ο±π‘š, β„±Ο±π‘š) β†’ 0 as 𝑛, π‘š β†’ ∞. So, we can conclude that the sequences {ℱℐ𝑛}, {ℱὡ𝑛}, {ℱϱ𝑛} are CS in Ὣ. But, β„±(Ὣ) is ₒ𝑏-complete, there exists 𝜌, 𝜎, 𝔑 ∈ β„±(Ὣ) such that lim π‘›β†’βˆž ℱℐ𝑛 β†’ 𝜌, lim π‘›β†’βˆž ℱὡ𝑛 β†’ 𝜎 and lim π‘›β†’βˆž ℱϱ𝑛 β†’ 𝔑 𝑖. 𝑒. , lim π‘›β†’βˆž ₒ𝑏(ℱℐ𝑛, 𝜌, 𝜌) = 0, lim π‘›β†’βˆž ₒ𝑏(ℱὡ𝑛, 𝜎, 𝜎) = 0 π‘Žπ‘›π‘‘ lim π‘›β†’βˆž ₒ𝑏(ℱϱ𝑛, 𝔑, 𝔑) = 0. βˆ’ βˆ’ (3.4.3) Since β„± is continuous, lim π‘›β†’βˆž ₒ𝑏(ℱℱℐ𝑛, β„±πœŒ, β„±πœŒ) = 0, lim π‘›β†’βˆž ₒ𝑏(ℱℱὡ𝑛, β„±πœŽ, β„±πœŽ) = 0 π‘Žπ‘›π‘‘ lim π‘›β†’βˆž ₒ𝑏(ℱℱϱ𝑛, ℱ𝔑, ℱ𝔑) = 0. ---- (3.4.4) Since β„± commutes with β„‹ and by equation (3.4.1) ₒ𝑏(ℱℱℐ𝑛+1, β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜌, 𝜎, 𝔑)) + ₒ𝑏(ℱℱὡ𝑛, β„‹(𝜎, 𝔑, 𝜌), β„‹(𝜎, 𝔑, 𝜌)) + ₒ𝑏(ℱℱϱ𝑛 , β„‹(𝔑, 𝜌, 𝜎), β„‹(𝔑, 𝜌, 𝜎)) = ₒ𝑏 (β„±(β„‹(ℐ𝑛, ὡ𝑛, ϱ𝑛)), β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜌, 𝜎, 𝔑)) + ₒ𝑏 (β„±(β„‹(ὡ𝑛, ϱ𝑛, ℐ𝑛)), β„‹(𝜎, 𝔑, 𝜌), β„‹(𝜎, 𝔑, 𝜌)) + ₒ𝑏 (β„±(β„‹(ϱ𝑛, ℐ𝑛, ὡ𝑛)), β„‹(𝔑, 𝜌, 𝜎), β„‹(𝔑, 𝜌, 𝜎)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 437 https://internationalpubls.com = ₒ𝑏 (β„‹((ℱℐ𝑛, ℱὡ𝑛, ℱϱ𝑛)), β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜌, 𝜎, 𝔑)) + ₒ𝑏 (β„‹((ℱὡ𝑛, ℱϱ𝑛, ℱℐ𝑛)), β„‹(𝜎, 𝔑, 𝜌), β„‹(𝜎, 𝔑, 𝜌)) + ₒ𝑏 (β„‹((ℱϱ𝑛, ℱℐ𝑛, ℱὡ𝑛)), β„‹(𝔑, 𝜌, 𝜎), β„‹(𝔑, 𝜌, 𝜎)) ≀ πœƒ[ₒ𝑏(ℱℱℐ𝑛, β„±πœŒ, β„±πœŒ) + ₒ𝑏(ℱℱὡ𝑛, β„±πœŽ, β„±πœŽ) + ₒ𝑏(ℱℱϱ𝑛, ℱ𝔑, ℱ𝔑)]. From (3.4.4)we get ₒ𝑏(ℱℱℐ𝑛+1, β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜌, 𝜎, 𝔑)) + ₒ𝑏(ℱℱὡ𝑛, β„‹(𝜎, 𝔑, 𝜌), β„‹(𝜎, 𝔑, 𝜌)) + ₒ𝑏(ℱℱϱ𝑛 , β„‹(𝔑, 𝜌, 𝜎), β„‹(𝔑, 𝜌, 𝜎)) β†’ 0 π‘Žπ‘  𝑛 β†’ ∞. On the other hand, β„± is continuous, ₒ𝑏(ℱℱℐ𝑛+1, β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜌, 𝜎, 𝔑)) + ₒ𝑏(ℱℱὡ𝑛, β„‹(𝜎, 𝔑, 𝜌), β„‹(𝜎, 𝔑, 𝜌)) + ₒ𝑏(ℱℱϱ𝑛, β„‹(𝔑, 𝜌, 𝜎), β„‹(𝔑, 𝜌, 𝜎)) β†’ ₒ𝑏(β„±πœŒ, β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜌, 𝜎, 𝔑)) + ₒ𝑏(β„±πœŽ, β„‹(𝜎, 𝔑, 𝜌), β„‹(𝜎, 𝔑, 𝜌)) + ₒ𝑏(ℱ𝔑, β„‹(𝔑, 𝜌, 𝜎), β„‹(𝔑, 𝜌, 𝜎)) 𝑛 β†’ ∞. It gives ₒ𝑏(β„±πœŒ, β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜌, 𝜎, 𝔑)) + ₒ𝑏(β„±πœŽ, β„‹(𝜎, 𝔑, 𝜌), β„‹(𝜎, 𝔑, 𝜌)) + ₒ𝑏(ℱ𝔑, β„‹(𝔑, 𝜌, 𝜎), β„‹(𝔑, 𝜌, 𝜎)) = 0 ₒ𝑏(β„±πœŒ, β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜌, 𝜎, 𝔑)) = ₒ𝑏(β„±πœŽ, β„‹(𝜎, 𝔑, 𝜌), β„‹(𝜎, 𝔑, 𝜌)) = ₒ𝑏(ℱ𝔑, β„‹(𝔑, 𝜌, 𝜎), β„‹(𝔑, 𝜌, 𝜎)) = 0 β‡’ β„±πœŒ = β„‹(𝜌, 𝜎, 𝔑), β„±πœŽ = β„‹(𝜎, 𝔑, 𝜌) and ℱ𝔑 = β„‹(𝔑, 𝜌, 𝜎)------------(3.4.5) ∴ (𝜌, 𝜎, 𝔑) is the TCIP of β„± and β„‹. Now, ₒ𝑏(β„±πœŒ, β„±πœŽ, β„±πœŽ) + ₒ𝑏(β„±πœŽ, ℱ𝔑, ℱ𝔑) + ₒ𝑏(, ℱ𝔑, β„±πœŒ, β„±πœŒ) = ₒ𝑏(β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜎, 𝔑, 𝜌), β„‹(𝜎, 𝔑, 𝜌))+ₒ𝑏(β„‹(𝜎, 𝔑, 𝜌), β„‹(𝔑, 𝜌, 𝜎), β„‹(𝔑, 𝜌, 𝜎)) +ₒ𝑏(β„‹(𝔑, 𝜌, 𝜎), β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜌, 𝜎, 𝔑)) Then from equation (3.4.1) ≀ πœƒ[ₒ𝑏(β„±πœŒ, β„±πœŽ, β„±πœŽ) + ₒ𝑏(β„±πœŽ, ℱ𝔑, ℱ𝔑) + ₒ𝑏(, ℱ𝔑, β„±πœŒ, β„±πœŒ)] which gives ₒ𝑏(β„±πœŒ, β„±πœŽ, β„±πœŽ) + ₒ𝑏(β„±πœŽ, ℱ𝔑, ℱ𝔑) + ₒ𝑏(ℱ𝔑, β„±πœŒ, β„±πœŒ) = 0 β‡’ β„±πœŒ = β„±πœŽ, β„±πœŽ = ℱ𝔑 and ℱ𝔑 = β„±πœŒ.--------------------------------------(3.4.6) By using (ₒ𝑏5) and equation (3.4.1) ₒ𝑏(𝜌, β„±πœŒ, β„±πœŒ) + ₒ𝑏(𝜎, β„±πœŽ, β„±πœŽ) + ₒ𝑏(𝔑, ℱ𝔑, ℱ𝔑) ≀ 𝑠[ₒ𝑏(𝜌, ℱℐ𝑝+1, ℱℐ𝑝+1) + ₒ𝑏(ℱℐ𝑝+1, β„±πœŒ, β„±πœŒ)]+ s[ₒ𝑏(𝜎, ℱὡ𝑝+1, ℱὡ𝑝+1) + ₒ𝑏(ℱὡ𝑝+1, β„±πœŽ, β„±πœŽ)]+𝑠[ₒ𝑏(𝔑, ℱϱ𝑝+1, ℱϱ𝑝+1) + ₒ𝑏(ℱϱ𝑝+1, ℱ𝔑, ℱ𝔑)] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 438 https://internationalpubls.com ≀ 𝑠[ₒ𝑏(𝜌, ℱℐ𝑝+1, ℱℐ𝑝+1) + ₒ𝑏(𝜎, ℱὡ𝑝+1, ℱὡ𝑝+1)+ₒ𝑏(𝔑, ℱϱ𝑝+1, ℱϱ𝑝+1)] +𝑠[ₒ𝑏(ℱℐ𝑝+1, β„±πœŒ, β„±πœŒ) + ₒ𝑏(ℱὡ𝑝+1, β„±πœŽ, β„±πœŽ)+ₒ𝑏(ℱϱ𝑝+1, ℱ𝔑, ℱ𝔑)] =𝑠[ₒ𝑏(𝜌, ℱℐ𝑝+1, ℱℐ𝑝+1) + ₒ𝑏(𝜎, ℱὡ𝑝+1, ℱὡ𝑝+1)+ₒ𝑏(𝔑, ℱϱ𝑝+1, ℱϱ𝑝+1)] +𝑠[ₒ𝑏 (β„‹(ℐ𝑝 , ὡ𝑝, ϱ𝑝), β„‹(𝜌, 𝜎, 𝔑), β„‹(𝜌, 𝜎, 𝔑))+ₒ𝑏 (β„‹(ὡ𝑝, ϱ𝑝, ℐ𝑝), β„‹(𝜎, 𝔑, 𝜌), β„‹(𝜎, 𝔑, 𝜌))+ ₒ𝑏 (β„‹(ϱ𝑝, ℐ𝑝 , ὡ𝑝), β„‹(𝔑, 𝜌, 𝜎), β„‹(𝔑, 𝜌, 𝜎))] ≀ 𝑠[ₒ𝑏(𝜌, ℱℐ𝑝+1, ℱℐ𝑝+1) + ₒ𝑏(𝜎, ℱὡ𝑝+1, ℱὡ𝑝+1)+ₒ𝑏(𝔑, ℱϱ𝑝+1, ℱϱ𝑝+1)] +π‘ πœƒ[ₒ𝑏(ℱℐ𝑝, β„±πœŒ, β„±πœŒ) + ₒ𝑏(ℱὡ𝑝, β„±πœŽ, β„±πœŽ) + ₒ𝑏(ℱϱ𝑝, ℱ𝔑, ℱ𝔑)] π‘Žπ‘  𝑝 β†’ ∞, we get ₒ𝑏(𝜌, β„±πœŒ, β„±πœŒ) + ₒ𝑏(𝜎, β„±πœŽ, β„±πœŽ) + ₒ𝑏(𝔑, ℱ𝔑, ℱ𝔑) ≀ [ₒ𝑏(𝜌, 𝜌, 𝜌) + ₒ𝑏(𝜎, 𝜎, 𝜎)+ₒ𝑏(𝔑, 𝔑, 𝔑)]+ +π‘ πœƒ[ₒ𝑏(𝜌, β„±πœŒ, β„±πœŒ) + ₒ𝑏(𝜎, β„±πœŽ, β„±πœŽ)+ₒ𝑏(𝔑, ℱ𝔑, ℱ𝔑)] ≀ π‘ πœƒ[ₒ𝑏(𝜌, β„±πœŒ, β„±πœŒ) + ₒ𝑏(𝜎, β„±πœŽ, β„±πœŽ)+ₒ𝑏(𝔑, ℱ𝔑, ℱ𝔑)]. Which gives 𝜌 = β„±πœŒ, 𝜎 = β„±πœŽ and 𝔑 = ℱ𝔑.-------------------------------------- (3.4.7) From equations (3.4.5), (3.4.6) and (3.4.7), we conclude that (𝜌, 𝜌, 𝜌) is common TFP of β„± and β„‹. Now we will prove that uniqueness property, if possible let us assume that 𝔏 = ℱ𝔏 = β„‹(𝔏, 𝔏, 𝔏)is another TFP of β„± and β„‹. Now consider, 3ₒ𝑏(𝜌, 𝔏, 𝔏) = ₒ𝑏 (β„‹((𝜌, 𝜌, 𝜌)), β„‹(𝔏, 𝔏, 𝔏), β„‹(𝔏, 𝔏, 𝔏)) ≀ 3π‘ πœƒ[β‚’(β„±πœŒ, ℱ𝔏, ℱ𝔏)] ≀ 3π‘ πœƒβ‚’(𝜌, 𝔏, 𝔏) It is contradiction, therefore (𝜌, 𝜌, 𝜌) is unique common TFP of β„± and β„‹. Corollary 3.5Let (Ὣ, ₒ𝑏) is complete β‚’b-MS. Let β„‹: Ὣ3 β†’ Ὣ be a mapping such that ₒ𝑏(β„‹(α΄‚, α΄”, 𝔣), β„‹(Ο°, β„“, β„˜), β„‹(Ο°, β„“, β„˜)) + ₒ𝑏(β„‹( α΄”, 𝔣, α΄‚), β„‹(β„“, β„˜, Ο°), β„‹(β„“, β„˜, Ο°))+ₒ𝑏(β„‹(𝔣, α΄‚, α΄”), β„‹(β„˜, Ο°, β„“), β„‹(β„˜, Ο°, β„“)) ≀ πœƒ[ₒ𝑏(α΄‚, Ο°, Ο°) + ₒ𝑏(α΄”, β„“, β„“) + ₒ𝑏(𝔣, β„˜, β„˜)] for all all α΄‚, α΄”, 𝔣, Ο°, β„“, β„˜ ∈ Ὣ and if πœƒ ∈ [0,1) then there exists a unique TFP for β„‹ and inὫ. Example 3.6 LetὫ = [0,1], define a mappingₒ𝑏:Ὣ3 β†’ Ὣ as ₒ𝑏(α΄‚, α΄”, 𝔣) = 1 9 {|α΄‚ βˆ’ α΄”| + |α΄” βˆ’ 𝔣| + |α΄‚ βˆ’ 𝔣|}2 for allα΄‚, α΄”, 𝔣 ∈ Ὣ. Then it is clear that (Ὣ, ₒ𝑏) is ₒ𝑏-MS. Now define β„‹: Ὣ3 β†’ Ὣ byβ„‹(α΄‚, α΄”, 𝔣)=1 βˆ’ α΄‚2 32 βˆ’ 3 α΄”2 32 βˆ’ 5 𝔣2 32 and β„±: Ὣ β†’ Ὣ by β„±(α΄‚) = α΄‚ 4 for all α΄‚, α΄”, 𝔣, Ο°, β„“, β„˜ ∈ Ὣ, we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 439 https://internationalpubls.com ₒ𝑏(β„‹(α΄‚, α΄”, 𝔣), β„‹(Ο°, β„“, β„˜), β„‹(Ο°, β„“, β„˜)) + ₒ𝑏(β„‹( α΄”, 𝔣, α΄‚), β„‹(β„“, β„˜, Ο°), β„‹(β„“, β„˜, Ο°))+ₒ𝑏(β„‹(𝔣, α΄‚, α΄”), β„‹(β„˜, Ο°, β„“), β„‹(β„˜, Ο°, β„“)) = ₒ𝑏 (1 βˆ’ α΄‚2 32 βˆ’ 3 α΄”2 32 βˆ’ 5 𝔣2 32 , 1 βˆ’ Ο°2 32 βˆ’ 3 β„“2 32 βˆ’ 5 β„˜2 32 , 1 βˆ’ Ο°2 32 βˆ’ 3 β„“2 32 βˆ’ 5 β„˜2 32 ) +ₒ𝑏 (1 βˆ’ α΄”2 32 βˆ’ 3 𝔣2 32 βˆ’ 5 α΄‚2 32 , 1 βˆ’ β„“2 32 βˆ’ 3 β„˜2 32 βˆ’ 5 Ο°2 32 , 1 βˆ’ β„“2 32 βˆ’ 3 β„˜2 32 βˆ’ 5 Ο°2 32 ) +ₒ𝑏 (1 βˆ’ 𝔣2 32 βˆ’ 3 α΄‚2 32 βˆ’ 5 α΄”2 32 , 1 βˆ’ β„˜2 32 βˆ’ 3 Ο°2 32 βˆ’ 5 β„“2 32 , 1 βˆ’ β„˜2 32 βˆ’ 3 Ο°2 32 βˆ’ 5 β„“2 32 ) ≀ 1 8 {|α΄‚2 βˆ’ Ο°2|2+|α΄”2 βˆ’ β„“2|2 + |𝔣2 βˆ’ β„˜2|2} ≀ 1 8 {|α΄‚ βˆ’ Ο°| + |α΄” βˆ’ β„“| + |𝔣 βˆ’ β„˜|}2. And if we consider, ₒ𝑏(α΄‚, Ο°, Ο°) + ₒ𝑏(α΄”, β„“, β„“) + ₒ𝑏(𝔣, β„˜, β„˜) = 2 9 {|α΄‚ βˆ’ Ο°|2 + |α΄” βˆ’ β„“|2 + |βˆ’β„˜|2} = 2 9 [|α΄‚ βˆ’ Ο°| + |α΄” βˆ’ β„“| + |𝔣 βˆ’ β„˜|]2. From we conclude that ₒ𝑏(β„‹(α΄‚, α΄”, 𝔣), β„‹(Ο°, β„“, β„˜), β„‹(Ο°, β„“, β„˜)) + ₒ𝑏(β„‹( α΄”, 𝔣, α΄‚), β„‹(β„“, β„˜, Ο°), β„‹(β„“, β„˜, Ο°))+ₒ𝑏(β„‹(𝔣, α΄‚, α΄”), β„‹(β„˜, Ο°, β„“), β„‹(β„˜, Ο°, β„“)) ≀ πœƒ[ₒ𝑏(α΄‚, Ο°, Ο°) + ₒ𝑏(α΄”, β„“, β„“) + ₒ𝑏(𝔣, β„˜, β„˜)]. Then from corollary, we can conclude (0, 0, 0) unique TFP of β„‹ . 4. Conclusion In this work we have obtained TFP results by using a new type of contraction and discussed some corollary also an example which supports the main result. Refrences [1] D. Srilatha, V. Kiran. (2023). A Study on Tripled Fixed Point Results in GJS - Metric Space. 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