Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 275 https://internationalpubls.com Fixed Point Results in G Metric Space via Α-Series K. Varalakshmi 1, *, G.Upender Reddy2 1 Research Scholar, Department of Mathematics, Osmania University, Telangana, India. E-mail: varak2121@gmail.com. 2 Associate Professor of Mathematics, Nizam College (A), Osmania University, Telangana, India. E-mail: yuviganga@gmail.com. Article History: Received: 19-10-2024 Revised: 03-12-2024 Accepted: 11-12-2024 Abstract: Introduction: Mustafa and sims [1] introduced the concept of G-metric space in 2005. Afterwards, Mustafa et al and many authors [3]-[19] obtained some common fixed-point theorems, coupled and tripled fixed point results for mappings satisfying different contractive conditions in G metric space. In this study we prove fixed point results in G metric space via α-series by using some conditions that are a sequence of a mappings and a self-mapping. Objectives: To show tripled fixed-point theorems and common fixed point theorems by using sequence of mappings and self a self-mapping via α-series and shown an example which supports the main result. Methods: In recent study the authors worked on fixed point results by using different contractions such as Suzuki type contraction, Rational type contraction, cyclic contraction, F-contraction, Mier keeler contraction, (ψ, ϕ)-weakly contractive mappings and Integral type contractions etc. in G-metric spaces. In this study we proved fixed point results in G metric space via α-series by using sequence of mappings. Results: Obtained Unique common fixed point and tripled fixed point results in G metric spcace via α-series. Conclusion: In this study we present unique tripled fixed-point and common fixed-point results for a sequence of mappings and a self-mapping in G metric space via α-series and discussed corollary with supporting example. Keywords: G-metric space, tripled fixed point, α-series, compatible mapping, weakly reciprocally continuous mappings. 1.Introduction: Mustafa and sims [1] introduced the concept of G-metric space in 2005. Afterwards, Mustafa et al and many authors [3]-[19] obtained some common fixed-point theorems, coupled and tripled fixed point results for mappings satisfying different contractive conditions in G metric space. In 2014, Sihag et al [21] proposed an α-series to find a common fixed point by utilizing the sequence of mappings and self- mappings. Chang and Ma [22] presented coupled fixed point in 1991.Later this concept has attracted numerous researchers [23]-[28] across various fields. The notation of tripled fixed point was initiated by Berinde and Borcut [32],[33] in partially ordered metric spaces and also presented the concept of tripled coincidence point and obtained tripled coincidence point outcomes. Later many authors obtained common tripled fixed-point results by using different contraction conditions in different Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 276 https://internationalpubls.com metric spaces. In this paper we prove tripled fixed-point results in G metric space by utilizing the sequence of mappings via α- series and we provided illustrations to support our results. 2.Objectives: To show tripled fixed-point theorems and common fixed point theorems by using sequence of mappings and self a self-mapping via α-series and shown an example which supports the main result 3.Methodology: Defnition 3.1 [1] Consider Ɱ be a non-void set and G: Ɱ 3 → [0,∞) is a mapping such that (G1) G (ꝕ, ꝙ, 𝟉) = 0 if ꝕ = ꝙ = 𝟉 for all ꝕ, ꝙ, 𝟉 𝜖 Ɱ (G2) G (ꝕ, ꝕ, ꝙ) > 0 for ꝕ, ꝙ 𝜖 Ɱ with ꝕ≠ꝙ (G3) G (ꝕ, ꝕ, ꝙ) ≤ G (ꝕ, ꝙ, 𝟉) for all ꝕ, ꝙ, 𝟉 𝜖 Ɱ with ꝙ ≠ 𝟉 (G4) G (ꝕ, ꝙ, 𝟉) = G (ꝙ, 𝟉, ꝕ) = G (𝟉, ꝙ, ꝕ) = _ _ _ _ _ (symmetry) (G5) G (ꝕ, ꝙ, 𝟉) ≤ G (ꝕ, 𝝏, 𝝏) + G (𝝏, ꝙ, 𝟉) for all 𝝏, ꝕ, ꝙ, 𝟉 ϵ Ɱ. Then the function G is termed as G –metric on Ɱ and (Ɱ, G) is a G-metric space. Definiton 3.2 [22]: Assume Ɱ ≠∅ and a mapping E: Ɱ 2 → Ɱ. An element (λ, μ) ϵ Ɱ 2 is a coupled fixed point of E if Ɱ (λ, μ) = λ & Ɱ (μ, λ)= μ. Definition 3.3[32]: An element (ꝕ, ꝙ, ν) ϵ Ɱ3 be a tripled fixed point of mapping 𝛄: Ɱ3→Ɱ if 𝛄 (ꝕ, ꝙ, ν) = ꝕ, 𝛄 (ꝙ, ꝕ, ꝙ) = ꝙ and 𝛄 (ν, ꝙ, ꝕ) = ν. Definition 3.4 [33]: A trio (𝞎, ꝙ, Ꝛ) ϵ ℱ3 is a tripled coincidence point of the mappings 𝕲: ℱ3 → ℱ and 𝒻: 𝓕→𝓕 if 𝒻(𝞎)=𝓕 (𝞎, ꝙ, Ꝛ), 𝒻(ꝙ)=𝓕 (ꝙ, 𝞎, ꝙ) and 𝒻(Ꝛ)=𝓕 (Ꝛ, ꝙ, 𝞎). Definition 3.5[34]: Let (K, ≤) be a poset and 𝕻: K3→K. If 𝕻 (λ, μ, ν) is monotone-non increasing in μ and monotone non-decreasing in λ & ν then 𝕻 has mixed monotone property. i.e, for any λ, μ, ν ϵ K. λ1, λ2, ϵ K, λ1 ≤ λ2 ⟹ 𝕻 (λ1, μ, ν) ≤ 𝕻 (λ2, μ, ν) μ1, μ2 ϵ K, μ1 ≤ μ2 ⟹ 𝕻 (λ, μ1, ν) ≥ 𝕻 (λ, μ2, ν) ν1, ν2 ϵ K, ν1 ≤ ν2 ⟹ 𝕻 (λ, μ, ν1) ≤ 𝕻(λ, μ, ν2) Definition 3.6[34]: Let (E, ≤) be a partially ordered set and ϔ: E3→E and g: E→E be two maps. If ϔ (λ, μ, ν) is monotone non increasing in μ and ϔ (λ, μ, ν) is monotone non decreasing in λ and ν then ϔ has g-mixed monotone property. i.e, for any λ, μ, ν ϵ E. λ1, λ2, ϵ E, g(λ1) ≤ g (λ2) ⟹ ϔ (λ1, μ, ν) ≤ ϔ (λ2, μ, ν) μ1, μ2 ϵ E, g(μ1) ≤ g(μ2) ⟹ ϔ (λ, μ1, ν) ≥ ϔ (λ, μ2, ν) ν1, ν2 ϵ E, g(ν1) ≤ g(ν2) ⟹ ϔ(λ, μ, ν1) ≤ ϔ(λ, μ, ν2) Definition 3.7[24]: Let (U, d) be a metric space and mappings E and g where E: U3→U and g: U→U are compatible if lim 𝑛→∞ 𝑑 (𝑔(𝐸(𝜆𝑛, 𝜇𝑛, 𝜈𝑛)), 𝐸(𝑔𝜆𝑛, 𝑔𝜇𝑛, 𝑔𝜈𝑛)) = 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 277 https://internationalpubls.com lim 𝑛→∞ 𝑑 (𝑔(𝐸(𝜇𝑛, 𝜆𝑛, 𝜇𝑛)), 𝐸(𝑔𝜇𝑛, 𝑔𝜆𝑛, 𝑔𝜇𝑛)) = 0 lim 𝑛→∞ 𝑑(𝑔(𝐸(𝜈𝑛, 𝜇𝑛, 𝜆𝑛), 𝐸(𝑔𝜈𝑛, 𝑔𝜇𝑛, 𝑔𝜈𝑛)) = 0 Whenever {𝜆𝑛}, {𝜇𝑛}, { 𝜈𝑛}𝑎𝑟𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 𝑖𝑛 𝑋, 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 lim 𝑛→∞ 𝐸(𝜆𝑛, 𝜇𝑛,𝑣𝑛) = lim 𝑛→∞ 𝑔(𝜆𝑛) = 𝜆, lim 𝑛→∞ E (𝜇𝑛, 𝜆n, μn) = lim 𝑛→∞ 𝑔(𝜇n) = μ lim 𝑛→∞ 𝐸(𝑣𝑛, 𝜇𝑛,𝜆𝑛) = lim 𝑛→∞ 𝑔(𝑣𝑛) = 𝑣 For all 𝜆, 𝜇, 𝑣, 𝜖𝑈 Definition 3.8[24]: The mapping E: Ɱ3→Ɱ and 𝕻: Ɱ→Ɱ are (i) Reciprocally continuous if lim 𝑛→∞ 𝔓 (𝐸(𝜆𝑛,𝜇𝑛,𝑣𝑛)) = 𝔓(𝜆) 𝑎𝑛𝑑 lim 𝑛→∞ 𝐸(𝔓(𝜆𝑛), 𝔓(𝜇𝑛), 𝔓(𝑣𝑛)) = 𝐸(𝜆, 𝜇, 𝑣) lim 𝑛→∞ 𝔓 (𝐸(𝜇𝑛,𝜆𝑛,𝜇𝑛)) = 𝔓(𝜇) 𝑎𝑛𝑑 lim 𝑛→∞ 𝐸(𝔓( 𝜇𝑛), 𝔓(𝜆𝑛), 𝔓(𝜇𝑛)) = E (μ, λ, μ) and lim 𝑛→∞ 𝔓 (𝐸(𝑣𝑛 , 𝜇𝑛,𝜆𝑛,)) = 𝔓(𝜈) 𝑎𝑛𝑑 lim 𝑛→∞ 𝐸(𝔓(𝑣𝑛), 𝔓(𝜇𝑛),𝔓(𝜆𝑛,)) = E(ν,μ,λ) whenever {𝜆𝑛,},{𝜇𝑛},{𝑣𝑛}are sequences in Ɱ such that lim 𝑛→∞ 𝐸(𝜆𝑛,𝜇𝑛,𝑣𝑛)= lim 𝑛→∞ 𝔓(𝜆𝑛) = 𝜆 lim 𝑛→∞ 𝐸(𝜇𝑛,𝜆𝑛,𝜇𝑛,)= lim 𝑛→∞ 𝔓(𝜇𝑛) = 𝜇 lim 𝑛→∞ 𝐸 (𝐸(𝑣𝑛, 𝜇𝑛,𝜆𝑛,))= lim 𝑛→∞ 𝔓(𝑣𝑛) =ν (ii) Weakly reciprocally continuous if lim 𝑛→∞ 𝔓(𝐸( 𝜆𝑛, 𝜇𝑛, 𝜈𝑛) = 𝔓(𝜆𝑛) or lim 𝑛→∞ 𝐸(𝔓( 𝜆𝑛), 𝔓( 𝜇𝑛), 𝔓( 𝜈𝑛)) =E(𝜆, μ, ν) lim 𝑛→∞ 𝔓(𝐸(𝜇𝑛 , 𝜆𝑛, 𝜇𝑛) = 𝕻(μ) or lim 𝑛→∞ 𝐸(𝔓( 𝜇𝑛), 𝔓(𝜆𝑛), 𝔓(𝜈𝑛)) = E (μ, λ, μ) lim 𝑛→∞ 𝔓(𝐸(𝜈𝑛 , 𝜇𝑛, 𝜆𝑛) = 𝔓(ν) or lim 𝑛→∞ 𝐸(𝔓(𝜈𝑛), 𝔓( 𝜇𝑛), 𝔓(𝜆𝑛)) = E (ν, μ, λ) Whenever {λn}, {μn}, {νn} are sequences in Ɱ such that lim 𝑛→∞ 𝐸( 𝜆𝑛, 𝜇𝑛, 𝜈𝑛) = lim 𝑛→∞ 𝔓( 𝜆𝑛) = λ lim 𝑛→∞ 𝐸( 𝜇𝑛, 𝜆𝑛, 𝜇𝑛) = lim 𝑛→∞ 𝔓( 𝜇𝑛) = μ lim 𝑛→∞ 𝐸( 𝜈𝑛, 𝜇𝑛, 𝜆𝑛) = lim 𝑛→∞ 𝔓( 𝜈𝑛) = ν for some λ, μ, ν ∈ Ɱ. Definition 3.9[21]: Assume 〈𝑎𝑛〉 be a sequence of non-negative real numbers. If there exist 0 < α < 1 and nα ∈ N such that ∑ 𝑎𝑖 𝑘 𝑖=1 ≤ αk for each k ≥ nα then the series ∑ 𝑎𝑛 ∞ 𝑛=1 is called an α-series. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 278 https://internationalpubls.com Lemma 3.10 [34] Let (K, ≤) be a poset and g: K→K and {Tn}nϵN: K3 → K be a sequence of mappings such that Tn(K 3) ⊆ g(K) then Tn has g-mixed monotone property if g(λ) ≤ g(ꝕ), g(ꝙ) ≤ g(μ) and g(ν) ≤ g(𝝏) for any λ,μ,ν,ꝕ,ꝙ,𝝏 ∈ K imply Tn(λ, μ, ν) ≤ Tn+1(ꝕ, ꝙ, 𝝏) Tn+1(ꝙ, ꝕ, ꝙ) ≤ Tn(μ, λ, μ) (1) Tn(ν, μ, λ) ≤ Tn+1(𝝏, ꝙ, ꝕ) In our main proof we construct the sequence as follows Assume λ0, µ0, ν0 ∈ K such that g(λ0) ≤ T0(λ0, μ0, ν0), g(μ0) ≥T0(μ0, λ0, μ0), g(ν0) ≤ T0(ν0, μ0, λ0) (2) Since T0(K 3) ⊆ g(K) we choose λ1, μ1, ν1∈ K such that g(λ1) = T0(λ0, μ0, ν0), g(μ1) =T0(μ0, λ0, ν0), g(ν1) =T0(ν0, μ0, λ0) again, we choose λ2, μ2, ν2 ∈ K such that g(λ2) = T1(λ1, μ1, ν1), g(μ2) = T1(μ1, λ1, μ1), g(ν2) = T1(ν1, μ1, λ1) continuing this process, we construct three sequences {λm}, {μm}, {νm} such that g(λm+1) =Tm(λm, μm, νm), g(μm+1) = Tm(μm, λm, μm), g(νm+1) = Tm (νm, μm, λm) for all m ≥ 0 (3) By mathematical induction, we prove that g(λm) ≤ g(λm+1), g(μm) ≥ g(μm+1) and g(νm ) ≤ g(νm+1) for all m ≥ 0 . (4) since condition 2 holds, in view of g(λ1) =T0(λ0, μ0, ν0), g (μ 1) =T0(μ0, λ0, μ 0), g(ν1) =T0(ν0, μ0, λ0) we obtain g(λ0) ≤ g(λ1), g(μ0) ≥ g(μ1), g(ν0 ) ≤ g(ν1). That is condition … 4 is true for m=0. Now we consider condition 4 is true for some m>0 From (3) and (4) we deduce g(λm+1) =Tm (λm, μm, νm) ≤ Tm+1(λm+1, μm+1, νm+1) = g(λm+2) g(μm+2) = Tm+1(μm+1, λm+1, μm+1) ≤ Tm (μm, λm, μm) = g(μm+1) g(νm+1 ) =Tm(μm , λm , νm) ≤ Tm+1(νm+1, μm+1, λm+1 ) = g(νm+2) we conclude that 4 holds for all m ≥ 0 by the mathematical induction ∴ we have g(λ0) ≤ g(λ1) ≤ g(λ2) ≤. . .. g(λm+1) ≤ . . . . . .. g(μ0) ≥ g(μ1 ) ≥ g(μ2) ≥ . . . . . . g(μm+1) ≥ . . . . . . g(ν0 ) ≤ g(ν1) ≤ g(ν2 ) ≤ . .. . .. . g(νm+1) ≤ . . . . . . definitions 2.7 and 2.8 are revised in main results as follows by using above considerations. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 279 https://internationalpubls.com 4.Results and Discussion Definition 4.1: Consider (𝓜, G) be a generalized metric space, {Tm}m∈ N: 𝓜3 →𝓜 be a sequence of mappings and a self-mapping g: 𝓜→𝓜 are compatible if lim 𝑚→∞ 𝐺(g(Tm (λm, μm, νm)), Tm (gλm, gμm, gνm), Tm (gλm, gμm, gνm)) = 0 lim 𝑚→∞ 𝐺(g(Tm (μm, λm, μm)), Tm (gμm, gλm, gμm), Tm (gμm, gλm, gμm)) = 0 lim 𝑚→∞ 𝐺( g(Tm (νm, μm, λm)), Tm(gνm, gμm, gλm), Tm (gνm, gμm, gλm)) = 0 Whenever {λm}, {μm, {νm} are sequences in 𝓜 such that lim m→∞ Tm (λm, μm, νm) = lim m→∞ g(λm+1) =λ lim m→∞ Tm (μm, λm, μm) = lim m→∞ g(μm+1) = μ lim m→∞ Tm (νm, μm, λm) = lim m→∞ g(νm+1) =ν for some λ, μ, ν ∈ 𝓜. Definition 4.2: Assume (𝓠, G) be a G- metric space, the mappings {Tm}m∈ N: 𝓠3 →𝓠 and g: 𝓠→𝓠 are weakly reciprocally continuous if lim 𝑚→∞ g(𝑇𝑚(𝜉𝑚, 𝜂𝑚, 𝜃𝑚)) = g(𝜉) lim 𝑚→∞ g(𝑇𝑚(𝜂𝑚, 𝜉𝑚, 𝜂𝑚)) = g(𝜂) lim 𝑚→∞ g(𝑇𝑚(𝜃𝑚, 𝜂𝑚, 𝜉𝑚)) = g(𝜃) whenever {𝜉𝑚}, {𝜂𝑚}, {𝜃𝑚} are sequences in 𝓠 such that lim 𝑚→∞ 𝑇𝑚(𝜉𝑚, 𝜂𝑚, 𝜃𝑚)) = lim 𝑚→∞ g(𝜉𝑚+1) = 𝜉 lim 𝑚→∞ 𝑇𝑚 (𝜂𝑚, 𝜉𝑚, 𝜂𝑚) = lim 𝑚→∞ g(𝜂𝑚+1) = 𝜂 lim 𝑚→∞ 𝑇𝑚( 𝜃𝑚, 𝜂𝑚, 𝜉𝑚) = lim 𝑚→∞ g(𝜃𝑚+1) = 𝜃 for some 𝜉, 𝜂, 𝜃 ϵ 𝓠. Theorem 4.3 : Assume (℧,G) be a partially ordered complete G-metric space . Let a sequence of mappings Tn :℧ 3→℧ and a self-mapping g:℧ →℧ such that Tn(℧ 3) ⊆ g(℧) and g(℧) is closed. {Tn}nϵN & g has g-mixed monotone property and both are continuous, compatible, weakly reciprocally continuous and satisfying the below conditions. 1. There exists 𝜆0, 𝜇0, 𝜈0 ϵ X such that g(λ0) ≤ T0(λ0, μ0, ν0), g(μ0) ≥T0(μ0, λ0, μ0), g(ν0) ≤ T0(ν0, μ0, λ0) holds. 2. {Tn}nϵN and g satisfies the condition G(Tn(λ, μ, ν),Tj(𝜻,𝜼,𝝒),Tj(𝝆,𝝈,𝝉)) ≤ γn,j[G(gλ,Tn(λ, μ,ν), Tn(λ, μ,ν)) +G(g𝜻,Tj(𝜻,𝜼,𝝒),Tj(𝝆,𝝈,𝝉)] + δn,j G(gλ, g𝜻, g𝝆) (5) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 280 https://internationalpubls.com for λ, μ, ν, 𝜻, 𝜼, 𝝒, 𝝆, 𝝈, 𝝉 ϵ ℧ and 0 ≤ γ n, j, δ n, j < 1, n≠j=1,2,….. i.e n, j ϵN if ∑ ( 𝛾𝑛,𝑛+1+ 𝛿𝑛,𝑛+1 1−𝛾𝑛,𝑛+1 ∞ 𝑛=1 ) is an α-series then g and {Tn}nϵN have a tripled coincidence point. 3. If{Tn}nϵN and g have tripled coincidence point comparable with respect to g then {Tn}nϵN and g have a unique tripled fixed point. Proof: For any λ0, μ0, ν0 ϵ ℧. We consider three sequences {𝜆𝑚},{𝜇𝑚}, {𝜈𝑚} constructed above by taking gλm+1=Tm (λm, μm, νm), gμm+1=Tm (μm, λm, 𝝁m), gνm+1=Tm (νm, μm, λm) By condition (5) G (gλ1, gλ2, gλ2) = G (T0(λ0, μ0, ν0), T1(λ1, μ1, ν1), T1(λ1, μ1, ν1)) ≤ γ0,1[G (gλ0, T0(λ0, μ0, ν0), T0(λ0, μ0, ν0)) + G ((gλ1, T1(λ1, μ1, ν1), T1(λ1, μ1, ν1))] + δ0,1G (gλ0, gλ1, gλ1) ⟹G (gλ1, gλ2, gλ2) ≤ γ0,1[G (gλ0, gλ1, gλ1) + G (gλ1, gλ2, gλ2)] +δ0,1 G (gλ0, gλ1, gλ1) ≤ (γ 0,1+ δ0,1) (1−γ 0,1) G (gλ0, gλ1, gλ1) similarly G(gλ2, gλ3, gλ3) ≤ (γ 1,2+ δ1,2) (1−γ 1,2) G(gλ1,gλ2, gλ2) ≤ ( γ 1,2+δ1,2 1−γ 1,2 )( γ 0,1+δ0,1 1−γ 0,1 ) G (gλ0, gλ1, gλ1) Repeating the above, we obtain G (gλm, gλm+1, gλm+1) ≤ ∏ ( 𝜸𝒏,𝒏+𝟏+𝜹𝒏,𝒏+𝟏 𝟏−𝜸𝒏,𝒏+𝟏 m−1 n=0 ) G (gλ0, gλ1, gλ1) ---------(6) Using similar procedure, we can also prove that G (gμm, gμm+1, gμm+1) ≤ ∏ ( 𝜸𝒏,𝒏+𝟏+𝜹𝒏,𝒏+𝟏 𝟏−𝜸𝒏,𝒏+𝟏 m−1 n=0 )G (gμ0, gμ1, gμ1) ------(7) and G (νm, νm+1, νm+1) ≤ ∏ ( 𝜸𝒏,𝒏+𝟏+𝜹𝒏,𝒏+𝟏 𝟏−𝜸𝒏,𝒏+𝟏 )m−1 n=0 G (gν0, gν1, gν1) -------(8) adding (6), (7), (8) we get G (gλm, gλm+1, gλm+1) + G (gμm, gμm+1, gμm+1) + G (νm, νm+1, νm+1) ≤ ∏ ( γ n,n+1 +δn,n+1 1−γ n,n+1 𝒎−1 n=0 )[G (gλ0, gλ1, gλ1) + G (gμ0, gμ1, gμ1) + G (gν0, gν1, gν1)] Let βm ≤ ∏ ( γ n,n+1 +δn,n+1 1−γ n,n+1 )m−1 n=0 β0 where βm = G (gλm, gλm+1, gλm+1) + G (gμm, gμm+1, gμm+1) + G(νm, νm+1,νm+1). For p > 0 and by using (G5), we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 281 https://internationalpubls.com G(gλm,gλm+p, gλm+p)+ G(gμm,gμm+p,gμm+p)+ G(νm,νm+p,νm+p) ≤ G(gλm,gλm+1, gλm+1)+ G(gμm,gμm+1,gμm+1)+ G(νm,νm+1,νm+1)+ G(gλm+1,gλm+2, gλm+2)+ G(gμm+1,gμm+2,gμm+2)+ G(νm+1,νm+2,νm+2)+----+ G(gλm+p-1, gλm+p, gλm+p)+ G(gμm+p-1,gμm+p,gμm+p)+ G(νm+p-1,νm+p,νm+p) ≤∏ ( γ n,n+1 +δn,n+1 1−γ n,n+1 )β 0 𝒎−1 n=0 + ∏ ( γ n,n+1 +δn,n+1 1−γ n,n+1 )β 0 𝒎 n=0 + ---- +∏ ( γ n,n+1 +δn,n+1 1−γ n,n+1 )𝒎+𝒎−2 n=0 β 0 ≤∑ ∏ ( γ n,n+1 +δn,n+1 1−γ n,n+1 )𝑚+𝑚−1 𝑚=0 𝑚−1 𝑚=0 β0 ≤∑ ∏ ( γ n,n+1 +δn,n+1 1−γ n,n+1 )𝑚−1 𝑚=0 𝑚+𝑚−1 𝑚=𝑚 β0 G(gλm,gλm+p, gλm+p)+ G(gμm,gμm+p,gμm+p)+ G(νm,νm+p,νm+p) ≤∑ ∏ ( γn,n+1+δn,n+1 1−γn,n+1 )𝑘−1 𝑛=0 𝑚+𝑝−1 𝑘=𝑚 β0 ……(9) By the fact that the arithmetic mean is greater than or equal to geometric mean for non-negative real numbers and α, nα are like in definition 2.9 for m≥ nα follows as below ∑ ∏ ( γ n,n+1 +δn,n+1 1−γ n,n+1 )𝑘−1 𝑛=0 𝑚+𝑝−1 𝑘=𝑚 ≤ ∑ [ 1 𝑘 ∑ ( γ n,n+1 +δn,n+1 1−γ n,n+1 )𝑘−1 𝑛=0 𝑚+𝑝−1 𝑘=𝑚 ]K ………………………… (10) From (9) and (10) ∴ G(gλm,gλm+p, gλm+p)+ G(gμm,gμm+p,gμm+p)+ G(νm,νm+p,νm+p) ≤ ∑ [ 1 𝑘 ∑ ( γ n,n+1 +δn,n+1 1−γ n,n+1 )𝑘−1 𝑛=0 𝑚+𝑝−1 𝑘=𝑚 ]K β0 ≤ (∑ αkm+p−1 k=m )β Where α = 1 𝑘 ∑ ( γ n,n+1 +δn,n+1 1−γ n,n+1 )𝑘−1 𝑛=0 ≤ (αm + αm+1 +αm+2+-----+αm+p−1) β0 ≤ αm(1+α1+α2+------+αp−1) β0 ≤ ( αm 1−α )β0 G (gλm, gλm+p, gλm+p) + G (gμm, gμm+p, gμm+p) + G (gνm, gνm+p, gνm+p) + G (gνm, gνm+p, gνm+p) ≤ ( αm 1−α )β0 Now letting the limit as m, 𝑝 → ∞, we obtain lim 𝑚→∞ G (gλm, gλm+p, gλm+p) + G ((gμm, gμm+p, gμm+p) + G (gνm, gνm+p, gνm+p) = 0 since 0 < α <1 Further which implies that lim 𝑚→∞ G(gλm, gλm+p, gλm+p) = 0 lim 𝑚→∞ G( g𝜇𝑚, g𝜇𝑚+𝑝, g𝜇𝑚+𝑝) = 0 (11) lim 𝑚→∞ 𝐺(gν𝑚, gν𝑚+𝑝, gν 𝑚+𝑝) ) = 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 282 https://internationalpubls.com Thus, {g𝜆𝑚}, { gμm}, { gνm} are Cauchey sequences in ℧. Since g(℧) is closed and by the completeness of ℧ there exists (r, s, t) ϵ ℧3 with lim m→∞ {gλm}=g(r)=λ, lim 𝑚→∞ {g𝜇𝑚}=g(s)=μ,lim n→∞ {gνm}=g(t)=ν By construction lim m→∞ g(λm+1) = lim m→∞ Tm(𝜆𝑚, 𝜇𝑚, 𝜈𝑚)=λ lim m→∞ g(μ m+1 ) = lim m→∞ Tm(𝜇𝑚, 𝜆𝑚, 𝜇𝑚)=μ and lim m→∞ g(νm+1) = lim m→∞ Tm(𝜈𝑚, 𝜇𝑚, 𝜆𝑚)=ν. Now by compatibility and weakly reciprocally continuous of g and {Tn}nϵN we have lim m→∞ Tm(g𝜆𝑚, g𝜇𝑚, g𝜈𝑚) = g(λ) lim m→∞ Tm(g𝜇𝑚, g𝜆𝑚, g𝜇𝑚) = g(μ) (12) lim m→∞ Tm(g𝜈𝑚, g𝜇𝑚, g𝜆𝑚) = g(ν). Suppose {Tn}nϵN is continuous & using G(5), we have G (Tn (λ, μ, ν), Tm (gλm, gμm, gνm), Tm (gλm, gμm, gνm)) ≤ G (Tn (λ, μ, ν), g (Tm (λm, μm, νm)), g (Tm (λm, μm, νm))) + G (g (Tm (λm, μm, νm)), Tm (gλm, gμm, gνm), Tm (gλm, gμm, gνm)) Similarly G (Tn (μ, λ, μ), Tm (gμm, gλm, gμm), Tm (gμm, gλm, gμm)) ≤ G (Tn (μ, λ, μ), g (Tm (μm, λm, μm)), g (Tm (μm, λm, μm))) + G (g (Tm (μm, λm, μm)), Tm (gμm, gλm, gμm), Tm(gμm, gλm, gμm)) And G (Tn (ν, μ, λ), Tm (gνm, gμm, gλm), Tm (gνm, gμm, gλm)) ≤ G (Tn (ν, μ, λ), g (Tm (νm, μm, λm)), g (Tm (νm, μm, λm))) + G (g (Tm (νm, μm, λm)), Tm (gνm, gμm, gλm), Tm (gνm, gμm, gλm)) Taking limit as m→∞, and weakly reciprocally continues we get G (Tn (λ, μ, ν), gλ, gλ) ≤ G (Tn (λ, μ, ν), gλ, gλ) + G (gλ, gλ, gλ) G (Tn (λ, μ, ν), gλ, gλ)) = 0. Similarly, G (Tn (μ, λ, μ), gμ, gμ) = 0 and G (Tn (ν, μ, λ), gν, gν) = 0 i.e., Tn (λ, μ, ν) = gλ Tn (μ, λ, μ) = gμ Tn (ν, μ, λ) = gν Thus (λ, μ, ν) is tripled coincidence point of {Tn} and g. Now we prove it is unique. Assume (ꝕ, ꝙ, r) is another tripled coincidence points. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 283 https://internationalpubls.com i.e, g(ꝕ)=Tn(ꝕ, ꝙ,r), g(ꝙ)=Tn(ꝙ,ꝕ,ꝙ) and g(r)=Tn(r,ꝙ,ꝕ) then we prove that g(λ)= g(ꝕ), g(μ)=g(ꝙ) and g(v)=g(r) since the set of tripled coincidence points are comparable, applying condition (5), we obtain G(gλ, gꝕ ,gꝕ) = G(Tn(λ,μ,ν),Tm(ꝕ ,ꝙ,r) ,Tm(ꝕ,ꝙ,r)) ≤ γn,m[G(gλ,Tn(λ,μ,ν),Tn(λ,μ,ν))+G(gp,Tm(p,q,r),Tm(p,q,r))]+δn,mG(gλ,gp,gp) G(gλ, gp ,gp) ≤ γn,m[G(gλ, gλ ,gλ)+G(gp,gp,gp)]+δn,m(gλ,gp,gp) G(gλ, gp, gp) = 0 as δn,m < 1 ∴ gλ =gp In the same way we can prove g(μ)=g(q) and g(v) =g(r) Hence g and {Tn}nϵN have a unique tripled point of coincidence. i.e, (gλ, gμ, gν). {Tn}nϵN and g are weakly compatible since they are compatible, i.e, they commute at their coincidence points, i.e, λ=gλ =Tn(λ, μ, ν), μ=gμ =Tn(μ ,λ, μ), ν=gν =Tn(ν, μ, λ). Thus, {Tn}nϵN and g have unique tripled common fixed point whenever they are weakly compatible. COROLLARY 4.4: Assume (ℬ, G, ≤) is a complete partially ordered G – metric space. Let {Tn}nϵN: ℬ3 →ℬ be a sequence of mappings and g is an identity mapping for 𝜌, 𝝈, 𝝉,𝝒,𝜼,𝜻 ∈ ℬ , {Tn}nϵN with 𝝆 ≤ 𝝒, 𝜼 ≤ 𝝈, 𝝉 ≤ 𝜻 or 𝝒 ≤ 𝝆,𝝈 ≤ 𝜼, 𝜻 ≤ 𝝉, satifies the following conditions. (i) Tm(𝝆,𝝈,𝝉) ≤ Tm+1(𝝒,𝜼, 𝜻) (ii) G(Tn(𝝆,𝝈,𝝉), Tm(𝝒,𝜼, 𝜻),Tm(𝞁,𝝰,𝝏)) ≤ γn,m [(ρ, Tn(𝝆,𝝈,𝝉),Tn(𝝆,𝝈,𝝉))+G(𝝒,Tm(𝝒,𝜼, 𝜻) ,Tm(𝞁,𝝰,𝝏))]+δn,m G(𝝆,𝝒,𝞁) with 0 ≤ γn,m ,δn,m < 1 and n, m ∈N If ∑ ( 𝛾𝑛,𝑛+1+ 𝛿𝑛,𝑛+1 1−𝛾𝑛,𝑛+1 ∞ 𝑛=1 ) is an α- series and ℬ is regular ,then there exists a tripled fixed point of {Tn}n∈N. i.e, ∃ (𝝆,𝝈,𝝉) ∈ℬ3 such that 𝝆=Tn(𝝆,𝝈,𝝉), 𝝈= Tn(𝝈,𝝆,𝝈),𝝉=Tn(𝝉,𝝈,𝝆) for n∈N. Theorem 4.5: Consider (ℬ, G, ≤) be a partially ordered complete G-metric space and is regular.Let g and {Tm}m∈N is same as in theorem 3.3 and lim 𝑛⟶∞ sup 𝛽𝑚,𝑛 < 1, 0 ≤ 𝛽𝑖,𝑗 , 𝛾𝑖,𝑗 < 1𝑎𝑛𝑑 𝑙𝑒𝑡 𝑔 𝑎𝑛𝑑 {𝑇𝑚} 𝑠𝑎𝑡𝑖𝑓𝑖𝑒𝑠 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 (2), (3) 𝑎𝑛𝑑 (5) 𝑡ℎ𝑒𝑛 𝑔 𝑎𝑛𝑑 {Tm}m∈N have tripled coincidence point. Proof: from the theorem ,sequences {𝑔𝜆𝑚}, {𝑔𝜇𝑚} 𝑎𝑛𝑑 {𝑔𝜐𝑚} 𝑎𝑟𝑒 𝑐𝑎𝑢𝑐ℎ𝑦 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 𝑖𝑛 𝐺(ℬ).Since {𝑔𝜆𝑚} 𝑎𝑛𝑑 {𝑔𝜐𝑚} 𝑎𝑟𝑒 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 𝑎𝑛𝑑 {𝑔𝜇𝑚} 𝑖𝑠 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦: 𝑏𝑦 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑖𝑡𝑦 𝑜𝑓 (ℬ, G, ≤), we have 𝑔𝛼𝑚 ≤ 𝞪, ρ ≤ 𝑔𝜌𝑚, 𝑔𝜎𝑚 ≤ 𝝈 for all m ≥ 0 then by (5), we obtain G(𝑇𝑚(𝑔𝛼𝑚, 𝑔𝜌𝑚 , 𝑔𝜎𝑚), 𝑇𝑛(𝛼, 𝜌, 𝜎), 𝑇𝑛(𝛼, 𝜌, 𝜎)) ≤ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 284 https://internationalpubls.com 𝛽𝑚,𝑛[𝐺(𝑔(𝑔𝛼𝑚), 𝑇𝑚(𝑔𝛼𝑚, 𝑔𝜌𝑚, 𝑔𝜎𝑚), 𝑇𝑚(𝑔𝛼𝑚, 𝑔𝜌𝑚, 𝑔𝜎𝑚)) + 𝐺(𝑔𝛼, 𝑇𝑛(𝛼, 𝜌, 𝜎), 𝑇𝑛(𝛼, 𝜌, 𝜎)) + 𝛾𝑚,𝑛(𝐺(𝑔(𝑔𝛼𝑚), 𝑔𝛼, 𝑔𝛼) Taking as m→∞, we obtain 𝑇𝑚(𝛼, 𝜌, 𝜎) = 𝑔𝛼 𝑎𝑠 𝛽𝑚,𝑛 < 1, Similarly we can prove 𝑇𝑚(𝜌, 𝛼, 𝜌) = 𝑔𝝆 And 𝑇𝑚(𝜎, 𝜌, 𝛼) = 𝑔𝝈 Thus (𝛼, 𝜌, 𝜎) 𝑖𝑠 𝑡𝑟𝑖𝑝𝑙𝑒𝑑 𝑐𝑜𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑐𝑒 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 {𝑇𝑚}𝑚∈𝑁 𝑎𝑛𝑑 𝑔 Example 4.6: Consider E = [0,1] and G(𝛼, 𝜌, 𝜎) = max{|𝛼 − 𝜌|, |𝜌 − 𝜎|, |𝜎 − 𝛼|}. Clearly (E, G) is complete G metric space Define 𝛾𝑖,𝑗 = 1 42𝑖+1 , 𝛿𝑖,𝑗 = 1 4𝑖 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖, 𝑗 = 1,2, … … … .. Consider the mapping Tn: E 3→E and g: E→E by Ti (𝝰, 𝝆, 𝝈) = α+ρ+σ 3i , g(𝝰)=2𝝰 for all 𝝰, 𝝆, 𝝈 ϵ E, n=1,2,3------ Assume i 𝔪 ≥ 𝒻, ρ < 𝔫 ≤ ℊ, σ > 𝔬 ≥ 𝒽 G(Ti(𝝰, 𝝆,𝝈),Tj(𝔪,𝔫,𝔬),Tj(𝒻, 𝓰, 𝓱))= | 𝛼+𝜌+𝜎 3𝑖 − 𝒻+ℊ+𝒽 3𝑗 | and G(gα, Ti(α, ρ, σ) , Ti(α, ρ, σ)) + G(g𝔪, Tj(𝔪, 𝔫, 𝔬), Tj(𝒻, ℊ, 𝒽)=|2𝛼 − α+ρ+σ 3i |+|2𝔪 − 𝒻+ℊ+𝑐 3𝑗 | 𝐺(𝑔𝞪, 𝒈𝖒, 𝒈𝓯) = |𝟔𝞪 − 𝟔𝓯| Since 𝛾𝑖,𝑗, 𝛿𝑖,𝑗 < 1, condition (5) satisfied for all α, ρ, σ, 𝔪, 𝔫, 𝔬, 𝒻, ℊ, 𝒽 ϵ E with α > 𝔪 ≥ 𝒻, ρ < 𝔫 ≤ ℊ, σ > 𝔬 ≥ 𝒽 Moreover, the series ∑ ( γ i,i+1 +δi,i+1 1−γ i,i+1 )∞ 𝑖=1 =∑ 1 42𝑖+1+ 1 4𝑖 1−4 1 2𝑖+1 ∞ 𝑖=1 =∑ 4𝑖+1+1 42𝑖+1−1 ∞ 𝑖=1 is an α-series with α = 1 4 . Unique common tripled fixed point for g and Tn is (0,0,0) Theorem 4.7: Let (ℋ, G) be complete G metric space and {Tn}: ℋ → ℋ be a sequence of self-mappings such that G(Tj(ꝕ),Tk(ꝙ),Tk(υ)) ≤ βj,k [G(ꝕ,Tj(ꝕ),Tj(ꝕ))+G(ꝙ,Tk(ꝙ),Tk(ꝙ))+G(υ,Tk(υ),Tk(υ))] +γj k G(ꝕ,ꝙ,υ) (A) for ꝕ, ꝙ, υ ∈ ℋ with ꝕ ≠ ꝙ, 0 ≤ βj,k,, γj, k < 1 2 , j,k=1,2,---- If ∑ 𝛽𝑗,𝑗+1+𝛾𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 ∞ 𝑗=1 is an α-series then {Tn} has unique common fixed point in ℋ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 285 https://internationalpubls.com Proof: For any ꝕ0∈ℋ, we can consider the sequence ꝕn= Tn(ꝕn-1), n=1,2---- From (A) we have G(ꝕ1,ꝕ2,ꝕ2)= G(T1ꝕ0, T2ꝕ1, T2ꝕ1) ≤ β1,2[G(ꝕ0, T1ꝕ0, T1ꝕ0)+ G(ꝕ1, T2ꝕ1, T1ꝕ1)+G(ꝕ1,T2ꝕ1, T2ꝕ1)]+ γ1,2G(ꝕ0, ꝕ1, ꝕ1) ≤ β1,2[G(ꝕ0, ꝕ1, ꝕ1)+G(ꝕ1, ꝕ2, ꝕ2)+G(ꝕ1, ꝕ2, ꝕ2)]+ γ1,2G(ꝕ0, ꝕ1, ꝕ1) ≤ β1,2[G(ꝕ0, ꝕ1, ꝕ1)+2G(ꝕ1, ꝕ2, ꝕ2)]+ γ1,2G(ꝕ0, ꝕ1, ꝕ1) G(ꝕ1,ꝕ2, ꝕ2) ≤ ( 𝛽1,2+𝛾1,2 1−2𝛽1,2 ) G(ꝕ0, ꝕ1, ꝕ1). Also we get G(ꝕ2, ꝕ3, ꝕ3)] = G(T2ꝕ1, T3ꝕ2, T3ꝕ2) ≤ ( 𝛽2,3+𝛾2,3 1−2𝛽2,3 )G(ꝕ1, ꝕ2, ꝕ2) ≤ ( 𝛽2,3+𝛾2,3 1−2𝛽2,3 )( 𝛽1,2+𝛾1,2 1−2𝛽1,2 ) G(ꝕ0, ꝕ1, ꝕ1). Repeating the above reasoning we obtain G(ꝕn, ꝕn+1,ꝕn+1) ≤ ∏ ( 𝛽𝑗,𝑗+1+𝛾𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 )𝑛 𝑗=1 G(ꝕ0, ꝕ1, ꝕ1). For p>0 and by G5 G(ꝕn, ꝕn+p, ꝕn+p) ≤ G(ꝕn, ꝕn+1,ꝕn+1)+ G(ꝕn+1, ꝕn+2,ꝕn+2)+ G(ꝕn+2, ꝕn+3,ꝕn+3)+--- + G(ꝕn+p-1, ꝕn+p, ꝕn+p) ≤ ∏ ( 𝛽𝑗,𝑗+1+𝛾𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 )𝑛 𝑗=1 +∏ ( 𝛽𝑗,𝑗+1+𝛾𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 )𝑛+1 𝑗=1 +---- +∏ ( 𝛽𝑗,𝑗+1+𝛾𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 ) 𝑛+𝑝−1 𝑗=1 ] G(ꝕ0, ꝕ1, ꝕ1)). ≤ ∑ ∏ ( 𝛽𝑗,𝑗+1+𝛾𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 )𝑛+𝑘 𝑗=1 𝑝−1 𝑘=0 G(ꝕ0, ꝕ1, ꝕ1). ≤ ∑ ∏ ( 𝛽𝑗,𝑗+1+𝛾𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 )𝑘 𝑗=1 𝑛+𝑝−1 𝑘=𝑛 G(ꝕ0, ꝕ1, ꝕ1) By using the fact that the Geometric mean is less than or equal to arithmetic mean for non-negative real numbers and let α and nα for m≥ nα as in definition 2.9 follows as below ≤ ∑ [ 1 𝑘 ∑ ( 𝛽𝑗,𝑗+1+𝛾𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 )𝑘 𝑗=1 ]𝑛+𝑝−1 𝑘=𝑛 𝑘 G(ꝕ0, ꝕ1, ꝕ1). ∴ G(ꝕn, ꝕn+p, ꝕn+p)≤ ∑ [ 1 𝑘 ∑ ( 𝛽𝑗,𝑗+1+𝛾𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 )𝑘 𝑗=1 ]𝑛+𝑝−1 𝑘=𝑛 𝑘 G(ꝕ0, ꝕ1, ꝕ1). ≤ (∑ 𝛼𝑘𝑛+𝑝−1 𝑘=𝑛 )G(ꝕ0, ꝕ1, ꝕ1). ≤ (𝛼𝑛 + 𝛼𝑛+1 + 𝛼𝑛+2 +---+𝛼𝑛+𝑝−1)G(ꝕ0, ꝕ1, ꝕ1). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 286 https://internationalpubls.com ≤ 𝛼𝑛(1+α+𝛼2+𝛼3+------+𝛼𝑝−1) G(ꝕ0, ꝕ1, ꝕ1). ≤ 𝛼𝑛 1−𝛼 G(ꝕ0, ꝕ1, ꝕ1). Let n, p →∞ , G(ꝕn, ꝕn+p, ꝕn+p)→0 since 0 < α < 1. Thus, {ꝕn} is a Cauchy sequence in ℋ .Due to completeness of ℋ it converges to ς in ℋ . For m>0, we have G(ꝕn,Tm(ς),Tm(ς)) = G(Tn(ꝕn-1),Tm(ς), Tm(ς)) ≤ 𝛽𝑛,𝑚[G(ꝕn-1, ꝕn, ꝕn)+ G(ς, Tmς, Tmς )+ G(ς, Tmς, Tmς)]+ γn,m (ꝕn-1, ς, ς) Letting n→∞, as ꝕn→ς we obtain G(ς, Tmς, Tmς) ≤ 𝛽𝑛,𝑚[G(ς,ς,ς)+2G(ς, Tmς, Tmς)+γn,mG(ς, ς, ς) G(ς, Tmς, Tmς) ≤2𝛽𝑛,𝑚 G(ς, Tmς, Tmς) Since 𝛽𝑛,𝑚< 1 2 , G(ς, Tmς, Tmς) = 0 ∴ Tmς = ς ς is fixed point of {Tm}. Assume ϑ is another fixed point of {Tm}, i.e., Tm(ϑ)=ϑ. Such that ς≠ϑ. G(ς, ϑ, ϑ) = G(Tmς, Tmϑ, Tmϑ) ≤ 𝛽𝑛,𝑚[G(ς, Tmς, Tmς)+ G(ϑ, Tmϑ, Tmϑ)+ G(ϑ, Tmϑ, Tmϑ)]+ γn,mG(ς, ϑ, ϑ) G(ς, ϑ, ϑ) ≤ 𝛽𝑛,𝑚[G(ς,ς,ς)+G(ϑ, ϑ, ϑ)+ G(ϑ, ϑ, ϑ)]+ γn,mG(ς, ϑ, ϑ). G(u,ϑ,ϑ) ≤ γn,mG(u, ϑ, ϑ). , which is a contradiction since γn,m< 1 2 ∴ ς = ϑ ∴ ς is a unique fixed point of {Tm}. Corollary 4.8: Consider a sequence of self mappings be {Tn} and (ℋ, G) be a complete G- metric space satisfies the following contraction such that G(Tjꝕ, Tkꝙ, Tkυ) ≤ 𝛽𝑗,𝑘[G(ꝕ, Tjꝕ,Tjꝕ)+G(ꝙ,Tkꝙ,Tkꝙ)+ G(υ,Tkυ,Tkυ) (B) for ꝕ, ꝙ,υ ∈ ℋ with ꝕ ≠ ꝙ, 0≤𝛽𝑗,𝑘< 1 2 j,k=1,2,3---- then {Tn} has a unique fixed point in ℋ if ∑ ( 𝛽𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 )∞ 𝑗=1 is an α-series Example 4.9. Let ℋ= [0,1] and G(ꝕ,ꝙ,υ) = 𝑚𝑎𝑥{|ꝕ− ꝙ|, |ꝙ− 𝜐|, |𝜐 −ꝕ|}. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 287 https://internationalpubls.com Clearly (ℋ, G) is a complete Generalized metric space. Define βj,k= 1 2+3𝑘 for all j,k = 1,2,3,…. and Tj(ꝕ) = ꝕ 3𝑗 for all ꝕϵ ℋ and j = 1,2,…. Assume j < k and ꝕ> ꝙ ≥ υ so we have G(Tjꝕ, Tkꝙ, Tkυ) = 𝑚𝑎𝑥{| ꝕ 3𝑗 − ꝙ 3𝑘 |,| ꝙ 3𝑘 − 𝜐 3𝑘 |,| 𝜐 3𝑘 − ꝕ 3𝑗 |} = | ꝕ 3𝑗 − ꝙ 3𝑘 | G(ꝕ, Tjꝕ,Tjꝕ)+G(ꝙ,Tkꝙ,Tkꝙ)+ G(υ,Tkυ,Tkυ) = |ꝕ − ꝕ 3𝑗 | +|ꝙ− ꝙ 3𝑘 | +|𝜐 − 𝜐 3𝑘 | Clearly G(Tjꝕ, Tkꝙ, Tkυ) ≤ G(ꝕ, Tjꝕ,Tjꝕ)+G(ꝙ,Tkꝙ,Tkꝙ)+ G(υ,Tkυ,Tkυ). ∴ condition (B) satisfied for all ꝕ, ꝙ, υ ∈ ℋ with ꝕ≠ꝙ ∑ ( 𝛽𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 )∞ 𝑗=1 = ∑ 1 2+3𝑗 1−2 1 2+3𝑗 ∞ 𝑗=1 = ∑ 1 2+3𝑗 2+3𝑗−2 2+3𝑗 ∞ 𝑗=1 = ∑ 1 3𝑗 ∞ 𝑗=1 is an α series with α = 1 3 . By corollary 3.8, {Tn} has a unique fixed point 0 ∈ ℋ. Theorem 4.10: Assume (ℋ,G) be a complete G metric space.{Tn} be a sequence of self mappings on ℋ such that G(𝑇𝑗 𝑝(ꝕ), 𝑇𝑘 𝑝(ꝙ), 𝑇𝑘 𝑝(𝜐)) ≤ 𝛽𝑗,𝑘[G(ꝕ, 𝑇𝑗 𝑝(ꝕ), 𝑇𝑗 𝑝(ꝕ))+G(ꝙ, 𝑇𝑘 𝑝(ꝙ), 𝑇𝑘 𝑝(ꝙ))+ G(υ, 𝑇𝑘 𝑝(𝜐), 𝑇𝑘 𝑝(𝜐))+ γj,kG(ꝕ,ꝙ,υ). For ꝕ,ꝙ,υ ∈ ℋ, ꝕ≠ꝙ, 0≤𝛽𝑗,𝑘, γj,k< 1 2 j,k=1,2,3----- If ∑ ( 𝛽𝑗,𝑗+1+𝛾𝑗,𝑗+1 1−2𝛽𝑗,𝑗+1 )∞ 𝑗=1 is an α-series, then {Tn} has unique common fixed point in ℋ. 5.CONCLUSION In this study we present unique tripled fixed-point and common fixed-point results for a sequence of mappings and a self-mapping in G metric space via α-series with supporting examples. References [1] Z. Mustafa; A new structure for generalized metric spaces with applications to fixed point theory, PH. D thesis, The university of Newcastle, Callaghan, Australia (2005). [2] Z.Mastafa,B.Sims, A new approach to generalized metric spaces, J.Nonlinear Convex Anal; 7 (2) ,289 – 297 (2006). [3] Mustafa, Z, Obiedat, H, Awawdeh, F: Some fixed-point theorem for mapping on complete G-metric spaces. Fixed Point Theory Appl 2008, 12 (2008). ID 189870. 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