STUDY OF GENERALIZED α- ADMISSIBLE MODIFIED ALMOST z- CONTRACTIONS VIA SIMULATION FUNCTIONS SURENDRA KUMAR TIWATI , ANAND MOHAN DUBEY, A V SENTHIL KUMAR 1. Introduction and preliminaries Consider N⊬ = N ∪ {0}, where N denotes the set of positive integers. As usual R indicates the set of real numbers.Furthermore, we set R0 + = [0, ∞]. Many problems in several branches of mathematics are well known to be transformed into invariant point problems in the form T x = x for self mapping T . It is worth noting that based on the work of Banach S. [7] in1922, known as the Banach contraction principle (BCP ), the metric fixed point theory took off. Alot of authrs studied generalizations of this principle. In addition, Berinde [9, 10] introduced almost contractions which exhibits new features with respect to the ones of the particular results in coprated as follows: Definition 1.1. Let (X, d) be a metric space. A self mapping Γ on X is called an almost contraction if there are constants δ ∈ [0, 1) and ∃L ≥ 0 such that d(Γx,Γy) ≤ δd(x, y) + Ld(y,Γx),∀x, y ∈ X. (1) Berinde [9, 10] investigated that every almost contraction mapping defined in a complete metric space has at least one fixed point. Subsequently, Babu et al.[6] defined the class of mapping satisfying condition (B) as follows: Definition 1.2. Let (X, d) be a metric space. A self mapping Γ on X issaid to be satisfy condition (B) if there are constants δ ∈ [0, 1) and ∃L ≥ 0 such that d(Γx,Γy) ≤ δd(x, y) + LQd(y,Γx),∀x, y ∈ X. (2) where Q(x, y) = min{d(x,Γx), d(y,Γy), d(x,Γy), d(y,Γx)} They proved a fixed point theorem for such mappings in complete metric spaces. They also discussed quasi-contraction, almost contraction and the class of mappingd that satisfy condition (B) in detail. Iseki et al. [20] presented definition of almost z- contraction as follows: Date: ∗Corresponding author: Surendra Kumar Tiwari. 2020 Mathematics Subject Classification. 54H25,47H10,55M20, , .... Key words and phrases. Almost z−- contraction, Modified Almost z-contraction, Simulation Function, α- Admissible Mapping. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2217 Abstract. In this paper, we introduce generalized α -admissible modified almost z -contraction with the help of simulation function and obtain fixed point results in the setting og metric space and verified with an example. The presented results extend, generalize and unify several related fixed point finding in the existing literature. Article History: Received: 10-01-2025 Revised: 15-02-2025 Accepted: 08-03-2025 Definition 1.3. Let (X, d) be a metric space and ζ. A self mapping Γ : X → X is called an almost z-contraction if there are constants θ ≥ o such that ζ(d(Γx,Γy), d(x, y) + LQ(x, y),∀x, y ∈ X, (3) where Q(x, y) is defined as in Definition 1.2. Also they investigated the existence and uniqueness of a fixed point of an almost z- contraction in metric space with simulation functions. Authos [25, 26, 27, 28] demostrated that almost contractions type mappings have a unique fixede point in deferent metric spaces. In sequel, P. Bunpatcharacharoen et al. [11] modified almost type z- contraction mapping in metric space and ob- tained fixed point. Khojsteh et al. [29] Originated the notion of z- contractions by usung a specific family of functions called simulation functions and proved a version of BCP. Subsequently, many researchers generalized this idea in many ways (See [30, 31, 32, 33, 34, 35, 36, 37, 38] ) and proved various interesting results in the arena of fixed point theory by using simulation functions. Definition 1.4. [29] A function ζ; [o,∞)2 → R is called a simulation function if ζ satisfies the following conditions: (ζ1) ζ(o, o) = o; (ζ2) (t, s) < s− t for all t, s > o; (ζ3) if {tn}, {sn} are sequence in (o,∞) such that lim n→∞ tn = lim n→∞ sn > 0, then lim supn→∞ζ(tn, sn) < 0. The following function ζ : [o,∞) × [o,∞) → R belongs to z. Definition 1.5. [29] A funtion Γ : X → X is called a z- contraction with respect to a simulation function ζ ∈ Z on metric space(X,d), if the following condition is satisfied ζ(d(Γx.Γy), d(x, y)) ≤ ∀x, y ∈ X. (4) Remark 1.6. [29] It is clear from the defintion of simulation function thar ζ(t, s) < o for all t ≥ s > o. Therefore, if Γ is a Z-contraction with respect to simulation functiomn ζ, then d(Γx,Γy) < d(x, y) for all x, y ∈ X. (5) Theorem 1.7. [29] Let (X, d) be a complete metric space and Γ : X → X be a z-contraction with respect to ζ. Then Γ has a unique fixed point u ∈ X and for every x0 ∈ X, the Picard sequence {xn} where xn = Γxn−1 for all n ∈ N converges to this fixed point of Γ. It is worth mentioning that the Banch contraction is a perfect example of z- contractions by taking (ζ, s) = λs − 1, where λ ∈ [0, 1), as the corresponding simulation function. Argoubi et al. [4] shown that the condition (ζ1) to be reduntant one ine above definition 1.4 of simulation functionand so redefined it as : Definition 1.8. [4] Asimulation function is a mapping ζ : [o,∞)2 → Rsatisfies the foolowing conditions: (i) ζ(t, s) < s− t for all t, s > 0; (ii) if {tn} and {sn} are sequences in (0,∞) such that lim n→∞ tn = lim n→∞ sn > 0, and tn < sn,then lim supn→∞ ζ(tn, sn) < 0. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2218 n→∞ tn = lim n→∞ Roldan-Lopez-de Hierrow et al.[36] modified the notion of simulation function replacing (ζ3) by(ζ ′ 3) of definition 1.4 as : (ζ ′ 3) if {tn} and {sn} are sequences in (0, ∞) such that lim sn > 0, and tn < sn, then lim supn→∞ ζ(tn, sn) < 0. It is clear that, if the function ζ satisfies the conditions (ζ1) − (ζ3), we say that ζ is a simulation function according to the sense of Khojasteh et al [29]. If it satisfies (ζ2) − (ζ3),it is a simulation function according to the sense of Argoubi et al.[4]. and if it satis fies (ζ),(ζ2) ,and (ζ ′ 3), then it is a simulation function according to the sense of Roldan- Lopez- de- Hierro et al. [36]. Samet et al.[39] introduced a new category of contractive type mappings known as α−ψ contractive type mapping. The results obtained by Samet et al.[39] extended and generalized the existing fixed point results in the literature, in particular the Banach contraction principle. Further, Karapinar E. and Samet[16] generalized the α − ψ comntractive type mappings and obtained various fixed point theo- rems for this generalized class of contractive mappings. In 2013, Hussain et al. [17] introduced α-admissible mappings and proved fixed point theorems in metric space. Subsequently, Abdeljawad[1] introduced a pair of α-admissible mappings satisfying new sufficient contractive conditions, whixh are different from those in[14, 16] and obtained fixed point and common fixed point theorems. After- ward, some authors have obtained fixed point theorems for some kinds of α- admissible mappins (see [1], 2, 3, 5, 6,7, 8,12,13,14, 15,18, 20, 21,22, 24, 38,40,41 and 42 ). Definition 1.9. [39] Let Γ : X → X and α : X × X → R+ be the functions. Then Γ is called α -admissible if α(x, y) ≥ 1 ⇒ α(Γx,Γy) ≥ 1, Karapinar E. [23] introduced the notion of α- admissible z- contraction and generalized the results of Samet et al.[39]and Khojasteh et al. [29]. Very recently, Dipti et al. [43] presented some fixed point results in complete metric spaces using generalized α admissible mappings embedded in the simulation functions. Definition 1.10. [15] LetΓ : X → X and α : X×X → R+ be a functions. Then we say that Γ is an α-orbital admissible if α(x,Γx) ≥ 1 implies α(Γx,Γ2x) ≥ 1. Moreover, Γ is called a triangular α- orbital admissible if Γ-orbital admissible and α(x, y) ≥ 1 and α(y,Γy) ≥ 1 implies α(x,Γy) ≥ 1, forall x, y ∈ X. Definition 1.11. [23] Let Γ : X → X be a self map defined on a metric space(X, d). If there exist ζ ∈ Z and α : X ×X → R+ such that ζ(α(x, y)d(Γx,Γy)d(x, y)) ≥ 0,∀x, y ∈ X (6) Then Γ is called an α-admiossible z- contraction with respect to ζ. Theorem 1.12. [23] Let (X, d) be a complete metric space and let Γ : X → X be an α-admissible z- contraction with respect to ζ. Suppose that (i) Γ is triangular α- orbital admissble; (ii) there exist x0 ∈ X such that α(X0,Γx0) ≥ 1; (iii) Γ is continuous. Then there exists u ∈ X such that Γu = u. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2219 Very recently, Dipti et al. [43] presented some fixed point results in complete metric spaces using generalized α admissible mappings embedded in the simula- tion functions. Definition 1.13. [43] Let(X, d) be a metric space, Γ : X → X be a self mapping, there exists ζ ∈ Z and α : X × X → X[0,∞). Then continuous mapping Γ is called generalized α -admissible almost z- contraction with respect to ζ and β ∈ G and L ≥ 0 such that for all x, y ∈ X, ζ(α(x,Γx)α(y,Γy)d(Γx,Γy),K(x, y) + LQ(x, y) ≥ 0 (7) for all distinct x, y ∈ X, where zeta is a simulation function in the sense of Defi- nition 1. Also K(x, y) = β(E(x, y))E(x, y) + LN(x, y) (8) , where E(x, y) = d(x, y) + |d(x,Γx) − d(y,Γy)| (9) and N(x, y) = min{d(x,Γx), d(y,Γy), d(x,Γy), d(y,Γx)}. (10) Definition 1.14 ([11]). Let (X, d) be a metric space and ζ. We say that Γ : X → X is a modified almost type z-contraction if there are constants L ≥ o such that ζ(d(Γx,Γy),K(x, y) + LQ(x, y),∀x, y ∈ X, (11) where, K(x, y) = max { d(x, y), [1 + d(x,Γx)d(y,Γy)] 1 + d(x, y) } and Q(x, y) = min{d(x,Γx), d(y,Γy), d(x,Γx), d(y,Γy)} Remark 1.15. If Γ is a modified almost type z-contraction with respect to ζ ∈ Z, then d(Γx,Γy) < K(x, y) + LQ(x, y) ∀x, y ∈ X, (9) Inspired and motivated by the combining the ideas in[11],[20], [25], [27] and [43], we introduce a new clas of mappings, and define generalized α -admissible modified almost z contraction with respect to ζ in the setting of metric space and obtain the existence and uniqueness of fixed point of such map. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2220 2. Main results Finaly, we give the following definition which will be used in our main results. Definition 2.1. Let(X, d) be a metric space, Γ : X → X be a self mapping, there exists ζ ∈ Z and α : X × X → X[0,∞). Then continuous mapping Γ is called generalized α-admissible modified almost z- contraction with respect to ζ and L ≥ 0 such that for all x, y ∈ X, ζ(α(x,Γx)α(y,Γy)d(Γx,Γy),K(x, y) + LQ(x, y)) ≥ 0 (10) where ζ is a simulation function in the sense of Definition 1.4. Also K(x, y) = max { d(x, y), [1 + d(x,Γx)]d(y,Γy) 1 + d(x, y) , [1 + d(x,Γy)]d(y,Γx) 1 + d(x, y) } (11) and Q(x, y) = min { d(x,Γx), d(y,Γy), d(x,Γy), d(y,Γx), d(x,Γy)d(y,Γx) 1 + d(x, y) , d(x,Γx)d(y,Γy) 1 + d(x, y) } . (12) We can now state the main finding of this paper. Theorem 2.2. Let (X, d) be a complete metric space and Γ : X → X is a gener- alized α-admissible modified almost z- contraction with respect to ζ. Furthermore, we suppose for all x, y ∈ X such that: (i) Γ is triangular α- orbital admissible; (ii) there exists x0 ∈ Xsuch that α(x0, γx0) ≥ 1; (iii) Γ is continuous. (iv) α(x,Γx) ≥ 1 Then Γ has a unique fixed point x∗ ∈ X. Proof. By(ii), there exists x0 ∈ X such that α(x0.Γx0) ≥ 1,and let {xn} be the iterative sequence Xdefined by xn+1 = Γxn, forall n ∈ N (13) If there exists some nonnegative integer n such that xn = xn+1 = Γxn, then xn is a fixed point of Γ. Therfore, to continue our proof, we assume that xn ̸= xn+1 for all n ∈ N. Since Γ is an α- admissible mapping, we have α(x0, x1) = α(x0,Γx0) ⇒ α(Γx0,Γx1) = α(x1, x2) ≥ 1. (14) By induction, we get α(xn, xn+1) ≥ 1, forall n ∈ N ∪ {0}. (15) Applying the codition (10), putting x = xn−1 and y = xn and by using (15), we have 0 ≤ ζ(α(xn−1,Γxn−1), α(xn,Γxn), d(Γxn−1,Γxn),K(xn−1, xn) + LQ(xn−1, xn)) = ζ(α(xn−1, xn), α(xn, xn+1), d(xn, xn+1)K(xn−1, xn) + LQ(xn−1, xn) < K(xn−1, xn) + LQ(xn−1, xn) − α(xn−1, xn), α(xn, xn+1), d(xn, xn+1) (16) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2221 Also, where K(xn−1, xn) =max { d(xn−1, xn), [1 + d(xn−1,Γxn−1)]d(xn,Γxn) 1 + d(x, y) , [1 + d(xn−1,Γxn)]d(xn,Γxn−1) 1 + d(x, y) } ≤ max { d(xn−1, xn), [1 + d(xn−1, xn)]d(xn, xn+1) 1 + d(xn−1, xn) , [1 + d(xn−1, xn+1)]d(xn, xn) 1 + d(xn−1, xn) } = max{d(xn−1, xn), d(xn, xn+1)} (17) and Q(xn−1, xn) = min{d(xn−1,Γxn−1), d(xn,Γxn), d(xn−1,Γxn), d(xn,Γxn−1), d(xn−1,Γxn)d(xn,Γxn−1) 1 + d(xn−1, xn) , d(xn−1,Γxn−1)d(xn,Γxn) 1 + d(xn−1, xn) } ≤ min{d(xn−1, xn), d(xn, xn+1), d(xn−1, xn+1), d(xn, xn), d(xn−1, xn+1)d(xn, xn) 1 + d(xn−1, xn) , d(xn−1, xn)d(xn, xn+1) 1 + d(xn−1, xn) } ≤ min { d(xn−1, xn), d(xn, xn+1), d(xn−1, xn+1), 0, 0, d(xn−1, xn)d(xn, xn+1) 1 + d(xn−1, xn) } =0. (18) By (16), and taking in account (15),(17) ,and(18), we derive that 0 < max{d(xn−1, xn), d(xn, xn+1)} − α(xn+1, xn), α(xn, xn+1)d(xn, xn+1) (19) which implies that d(xn, xn+1 ≤ α(xn−1, xn), α(xn, xn+1)d(xn, xn+1) < max{d(xn−1, xn), d(xn, xn+1)} ∀n ≥ 1. (20) If max {d(xn−1, xn), d(xn, xn+1)} = d(xn, xn+1) for some n ≥ 1, then from (20), we get d(xn, xn+1) ≤ α(xn−1, xn), α(xn, xn+1)d(xn, xn+1) < d(xn, xn+1). (21) which is contradiction, therefore, max{d(xn−1, xn), d(xn, xn+1)} = d(xn−1, xn) (22) Hence d(xn, xn+1) ≤ α(xn−1, xn), α(xn, xn+1), d(xn, xn+1) < d(xn−1, xn). (23) Consequently, we deduce that {d(xn−1, xn)} is a monotonically decreasing se- quence for nonnegative reals and bounded below by zero. so, there esists r ≥ 0 such that lim n→∞ d(xn−1, xn) = r (24) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2222 we claim that lim n→∞ d(xn−1, xn) = 0. (25) (26) On the contrary, assume that r > 0 and using equation(23) we have the following lim n→∞ α(xn−1, xn), α(xn, xn+1), d(xn, xn+1) = r (27) Now, we take tn = {α(xn−1, xn), α(xn, xn+1), d(xn, xn+1) and sn = {d(xn−1, xn)}. then lim n→∞ tn = lim n→∞ sn = r (28) Since T is a generalized α- admissible modified almost z- contraction with respect to ζ ∈ Z. Therfore, by(ζ3) and equation (21) and taking limit as n → ∞, we have lim supn→∞ ζ(tn, sn) < 0 i.e. 0 ≤ lim supn→∞ ζ(α(xn−1, xn), α(xn, xn+1), d(xn, xn+1), d(xn−1, xn) < 0 (29) This is a contradiction. Then we deduce that r = 0, that is, we have following lim n→∞ d(xn−1, xn) = 0 (30) Now, we will show that sequence {xn} is aCauchy sequence in X.Assume that {xn} is not a Cauchy sequence, then there exists ϵ > 0 and two sequences {xnk }, {xmk }: mk > nk > k such that d(xmk , xnk ) ≤ ϵ. (31) and d(xmk , xnk−1) ≤ ϵ, for all m , n, k ∈ N (32) By applying the triangal inequality and using equations (30) and (31), we get the following ϵ < d(xmk , xnk ) ≤ d(xmk , xnk−1) + d(xnk−1, xnK ) ≤ d(xnk−1, xnK ) + ϵ (33) Taking k → ∞ in equation (33) and using equation(30), we get lim n→∞ d(xmk , xnk ) = ϵ. (34) Again, using the triangal inequality, we have d(xmk , xnk ) ≤ d(xmk , xnk−1) + d(xnk−1, xnk ) ≤ d(xmk , xnk−1) + d(xmk−1, xnk−1) + d(xnk−1, xnk ) (35) Again, we have d(xmk−1, xnk−1) ≤ d(xmk−1, xmk ) + d(xmk , xnk−1) + d(xmk−1, xmk ) (36) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2223 By taking the limit as k → ∞ in equation (35) ,(36) ,and using (30) we deduce that lim n→∞ d(xmk−1, xnk−1) < ϵ. (37) By the same reasoning as above, we get that lim n→∞ d(xmk , xnk−1) = lim n→∞ d(xmk−1, xnk ) = ϵ. (38) Since T is triangular α-orbital admisssible, we have αd(xmk−1, xnk−1) ≤ 1. (39) Moreover, since T is a generalized α -admissible modified almost z-contraction with respect to ζ, 0 ≤ ζ(α(xmk−1,Γxmk−1), α(xnk−1,Γxnk−1), d(Γxmk−1,Γxnk−1)K(xmk−1, xnk−1) + LQ(xmk−1, xnk−1) = ζ(α(xmk−1, xmK ), α(xnk−1, xnk ), d(xmk , xnk ),K(xmk−1, xnk−1) + LQ(xmk−1, xnk−1) It follows from condition(ζ2), we get 0 < K(xmk−1, xnk−1) + LQ(xmk−1, xnk−1) − ζ(α(xmk−1, xmK ), α(xnk−1, xnk ), d(xmk , xnk ) (40) Hence, 0 < d(xmk , xnk ) ≤ α(xmk−1, xmk ), α(xnk−1, xnk )d(xmk , xnk ) < K(xmk−1, xnk−1),+LQ(xmk−1, xnk−1) (41) Also, where K(xmk−1, xnk−1) = max{d(xmk−1, xnk−1), [1 + d(xmk−1,Γxmk−1)]d(xnk−1,Γxnk−1) 1 + d(xmk−1, xnk−1) [1 + d(xmk−1,Γxnk−1)]d(xnk−1,Γxmk−1) 1 + d(xmk−1, xnk−1) } = max{d(xmk−1, xnk−1), [1 + d(xmk−1, xmk )]d(xnk−1, xnk ) 1 + d(xmk−1, xnk−1) [1 + d(xmk−1, xnk )]d(xnk−1, xmk ) 1 + d(xmk−1, xnk−1) }. (42) and, Q(xmk−1, xnk−1) = min{d(xmk−1,Γxmk−1), d(xnk−1,Γxnk−1), d(xmk−1,Γxnk−1), d(xnk−1,Γxmk−1), d(xmk−1,Γxnk−1)d(xmk−1,Γxmk−1) 1 + d(xmk−1, xnk−1) , d(xmk−1,Γxmk−1)d(xnk−1,Γxnk−1) 1 + d(xmk−1, xnk−1) } = min{d(xmk−1, xmk ), d(xnk−1, xnk ), d(xmk−1, xnk ), d(xnk−1, xmk ), d(xmk−1, xnk )d(xmk−1, xmk ) 1 + d(xmk−1, xnk−1) , d(xmk−1, xmk )d(xnk−1, xnk ) 1 + d(xmk−1, xnk−1) } (43) Taking limit ask → ∞ in (42),(43) using (30), (34), (37) and (38), we get lim k→∞ K(xmk−1, xnk−1) = ϵ. (44) and, lim k→∞ Q(xmk−1, xnk−1) = 0. (45) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2224 From (34), (37) ,(40), and (44) (45), also condition (ζ3), we get 0 ≤ lim supk→∞ζ(α(xmk−1, xmk ), α(xnk−1, xnk ), d(xmk , xnk )K(xmk−1, xnk−1) + LQ(xmk−1, xnk−1) ≤ lim supk→∞ζ(α(xmk−1, xmk ), α(xnk−1, xnk ), d(xmk , xnk )K(xmk−1, xnk−1) < 0. This is a contradiction. Hence {xn} is a Cauchy sequence. Since (X, d) is com- plete metric space, there exists x∗ ∈ X such that lim n→∞ d(xn, x ∗) = 0. (46) Now, we shall show that Γx∗ = x∗. Since Γ is continuous, we obtain that Γx∗ = Γ( lim n→∞ xn) = lim n→∞ Γ(xn) = lim n→∞ xn+1 = x∗. Thus, x∗ is a fixed point of Γ. To prove the uniqueness of the fixed point.Assume that, x∗ andy∗ be two fixed point of Γ and hence x∗, y∗ ∈ Fix(Γ) ,which is a generalized α-admissible modified self mapping of metric spaceX, d). Then d(x∗, y∗) > 0. By (10), we have that 0 ≤ ζ(α(x∗,Γx∗), α(y∗,Γy∗)d(Γx∗,Γy∗),K(x∗, y∗) + LQ(x∗, y∗) (47) where K(x∗, y∗) = d(x∗, y∗) and Q(x∗, y∗) = 0. Then, by (47), we get 0 ≤ ζ(α(x∗, x∗), α(y∗, y∗), d(x∗, y∗)K(x∗, y∗) + LQ(x∗, y∗) < K(x∗, y∗) + LQ(x∗, y∗) − α(x∗, x∗), α(y∗, y∗), d(x∗, y∗) 0 < d(x∗, y∗) − d(x∗, y∗) = 0. (48) which is a contradiction.Thus we have x∗ = y∗. Hence Γ has a unique fixed point. □ Theorem 2.3. Let (X, d) be a complete metric space and Γ : X → X is a generalized α-admissible modified almost z- contraction with respect to ζsatisfying the following conditions: (i) Γ is triangular α- orbital admissible; (ii) there exists x0 ∈ Xsuch that α(x0,Γx0) ≥ 1; (iii) if {xn} is a sequence in X such that α(xn, xn+1) ≤ 1 for all n ∈ N ∪ {0} and xn → x ∈ X as n→ ∞, then there exists a subsequence {xn(k) } of {xn} such that α(xn(k) , x∗) ≤ 1; (iv) α(x, y) ≥ 1, for all x, y ∈ Fix(Γ), where Fix(Γ) denotes the set of fixed point of Γ. Then, Γ has a unique fixed point x∗ ∈ X. Proof. By(ii), suppose x0 ∈ X such that α(x0,Γx0) ≥ 1. there exists xn ∈ X such that xn+1 = Γxn, for all n ∈ N. We have by Theorem 2.2, {xn} is a Cauchy sequence such that lim n→∞ d(xn, xn+1) = 0. Since (X, d) is complete, there exists x∗ ∈ X such that xn → x∗.By (15) and the condition (iii) , there exists a subsequence {xn(k) } of {xn} such that α(xn(k) , x∗) ≤ 1 for all k ∈ N. Using (10), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2225 we have 0 ≤ ζ(α(xnk ,Γxnk ), α(x∗,Γx∗), d(Γxnk ,Γx∗)K(xnk , x∗) + LQ(xnk , x∗) = ζ(α(xnk , xnk+1), α(x∗,Γx∗)d(xnk+1,Γx ∗),K(xnk , x∗) + LQ(xnk , x∗) < K(xnk , x∗) + LQ(xnk , x∗) − α(xnk , xnk+1), α(x∗,Γx∗), d(xnk+1,Γx ∗). (49) Hence, d(xnk+1,Γx ∗) ≤ α(xnk , xnk+1), α(x∗,Γx∗)d(xnK+1,Γx ∗) < K(xnk , x∗) + LQ(xnk , x∗). (50) Also, where K(xnk , x∗) = max{d(xnk , x∗), [1 + d(xnk ,Γxnk )]d(x∗,Γx∗) 1 + d(xnk , x∗) [1 + d(xnk ,Γx∗)]d(x∗,Γxnk ) 1 + d(xnk , x∗) } = max{d(xnk , x∗), [1 + d(xnk , xnk+1)]d(x∗,Γx∗) 1 + d(xnk , x∗) , [1 + d(xnk ,Γx∗)]d(x∗, xnk+1) 1 + d(xnk , x∗) }. (51) and Q(xnk , x∗) = min{d(xnk ,Γxnk ), d(x∗,Γx∗), d(xnk ,Γx∗), d(x∗,Γxnk ), d(xnk ,Γx∗)d(x∗,Γxnk ) 1 + d(xnk , x∗) , d(xnk ,Γx∗)d(x∗,Γx∗) 1 + d(xnk , x∗) } = min{d(xnk , xnk+1), d(x∗,Γx∗), d(xnk ,Γx∗), d(x∗, xnk+1), d(xnk ,Γx∗)d(x∗, xnk+1) 1 + d(xnk , x∗) , d(xnk ,Γx∗)d(x∗,Γx∗) 1 + d(xnk , x∗) }. (52) Taking k → 0 in the equation(50) and(51) we derive that K(xnK , x ∗) = d(x∗,Γx∗) and Q(xnk , x∗) = 0. (53) From (50) ,by using (53), we get d(xnk+1,Γx ∗) < d(x∗,Γx∗)for all k ∈ N (54) By (49), (54), and the condition (ζ3), we have 0 ≤ lim supn→∞ζ(α(xnk ,Γxnk ), α(x∗,Γx∗), d(Γxnk ,Γx∗)K(xnk , x∗) + LQ(xnk , x∗) < 0. This is contradiction.Hence therefore, x∗ is a fixed point og Γ. Now ,assume that, there exists x∗, y∗ ∈ X such that x∗ = Γx∗ and y∗ = Γy∗ with x∗ ̸= y∗. Since Γ is Generalized α-admissible modified almost z contraction selfmapping of a metric space (X, d). So by assumption (iv) from Theorem 2.3, we hav α(x⋆, y⋆) ≥ 1. (55) Therefore, from (10) and |zeta2 that, 0 ≤ ζ(α(x⋆,Γx⋆), α(y⋆,Γy⋆)d(Γx⋆,Γy⋆),K(x⋆, y⋆) + LQ(x⋆, y⋆) = ζ(α(x⋆, x⋆), α(y⋆, y⋆), d(x⋆, y⋆)K(x⋆, y⋆) + LQ(x⋆, y⋆) < K(x⋆, y⋆) + LQ(x⋆, y⋆) − α(x⋆, x⋆), α(y⋆, y⋆), d(x⋆, y⋆) (56) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2226 Also , where K(x⋆, y⋆) = max { d(x∗, y∗) [1 + d(x∗, x∗)]d(y∗, y∗) 1 + d(x∗, y∗) , [1 + d(x∗, y∗)]d(y∗, x∗) 1 + d(x∗, y∗) } = max{d(x∗, y∗), d(Y ∗, x∗)} = d(x∗, y∗) (57) and Q(x⋆, y⋆) = min{d(x∗, x∗), d(y∗, y∗), d(x∗, y∗), d(y∗, x∗), d(x∗, y∗)d(y∗, x∗) 1 + d(x∗, y∗) , d(x∗, x∗)d(y∗, y∗) 1 + d(x∗, y∗) } = min { 0, 0, d(x∗, y∗), d(y∗, x∗), d(x∗, y∗)d(y∗, x∗) 1 + d(x∗, y∗) , 0, } = 0. (58) From (56) together with (57) and (58), we deduce that 0 < d(x∗, y∗) ≤ α(x∗, x∗), α(y∗, y∗)d(x∗, y∗) < d(x∗, y∗). this is contradiction. Thus , we have x⋆ = y⋆. Hence, Γ has a unique fixed point. □ Corollary 2.4. Let (X, d) be a metric space, Γ : X → X is a generalized α- admissible modified almost z- contraction self mapping. There exists ζ ∈ Z and α : X × X → [0,∞) be a function with α(x,Γx) = 1 ,and α(y,Γy) = 1, for all x, y ∈ X such that ζ(d(Γx,Γy)K(x, y) + LQ(x, y) ≥ 0, forall x, y ∈ X, andL ≥ 0, Also where K(x, y) = max { d(x, y), [1 + d(x,Γx)]d(y,Γy) 1 + d(x, y) , [1 + d(x,Γy)]d(y,Γx) 1 + d(x, y) } and Q(x, y) = min { d(x,Γx), d(y,Γy), d(x,Γy), d(y,Γx), d(x,Γy)d(y,Γx) 1 + d(x, y) , d(x,Γx)d(y,Γy) 1 + d(x, y) } . Then Γ has a unique fixed point x⋆ ∈ X. Corollary 2.5. Let (X, d) be a metric space, Γ : X → X is a generalized α- admissible modified almost z- contraction self mapping. There exists ζ ∈ Z and α : X ×X → [0,∞) be a function with α(x,Γx) = 1 , α(y,Γy) = 1,and Q = 0. for all x, y ∈ X such that ζ(d(Γx,Γy)K(x, y) ≥ 0, forall x, y ∈ X, andL ≥ 0, Also where K(x, y) = max { d(x, y), [1 + d(x,Γx)]d(y,Γy) 1 + d(x, y) , [1 + d(x,Γy)]d(y,Γx) 1 + d(x, y) } Then Γ has a unique fixed point x⋆ ∈ X. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2227 Example 2.6. Let [0, 4] be endowed with metric space d(x, y) = |x − y| for all x, y ∈ X and Γ : X → X be defined by Γx = 4 − x. Consider ζ(t, s) = αs − t, where α ∈ [0, 1), for all t ≥ 0 and L ≥ 0 and α : X ×X → [0,∞) be defined by α(x, y) = { 1 if , x, y ∈ [0, 1] 0, otherwise } Note that Γ is triangular α-orbital admissible if α(x,Γx) ≥ 1 ⇒ α(Γx,Γ2x) ≥ 1, and α(x, y) ≥ 1 and α(y,Γy) ⇒ α(x,Γy) ≥ 1. Since α(x, y) > 1 and x, y ∈ [0, 1]. Then ,we have α(x,Γx) = (x, 4) = 1 and α(y,Γy) = 1 for all x, y ∈ X. In fact, for all x ̸= y, then we have ζ(d(Γx,Γy)d(x, y)) = α |x− y| − |4 − x− (4 − y)| = α |x− y| − |x− y| < |x− y| − |x− y| = 0 Now we show that Γ is a generalized α admissible modified almost z -contractionwith respect to ζ ∈ Z .Now ζ(d(Γx,Γy)K(x, y)) + LQ(x, y) = K(x, y) + LQ(x, y) − d(Γx,Γy) = α [|K(x, y) + LQ(x, y)|] − |4 − x− (4 − y)| , = α [|K(x, y) + LQ(x, y)|] − |x− y| , wher K(x, y) = max { |x− y| , [1 + |x− (4 − x)|] |y − (4 − y)| 1 + |x− y| , [1 + |x− (4 − y)|] |y − (4 − x)| 1 + |x− y| } = max { |x− y| , [1 + |2x− 4|] |2y − 4| 1 + |x− y| , [1 + |x− 4 + y|] |y − 4 − x| 1 + |x− y| } and, Q(x, y) = min{|x− (4 − x)| , |y − (4 − y)| , |x− (4 − y)| , |y − (4 − x)| , |x− (4 − y)| . |y − (4 − x)| 1 + |x− y| |x− (4 − y)| . |y − (4 − x)| 1 + |x− y| } = min { |2x− 4| , |2y − 4| , |x+ y − 4| , |x+ y − 4| 1 + |x− y| , |2x− 4| . |2y − 4| 1 + |x− y| } . We deduce that ζ(d(Γx,Γy)K(x, y)) + LQ(x, y) = α[max { |x− y| , [1 + |2x− 4|] |2y − 4| 1 + |x− y| , [1 + |x+ y − 4|] |x+ y − 4| 1 + |x− y| } + Lmin { |2x− 4| , |2y − 4| , |x+ y − 4| , |x+ y − 4| 1 + |x− y| , |2x− 4| . |2y − 4| 1 + |x− y| } ] − |x− y| . Hence, we get two cases: Case(i): If x = y, then, ζ(d(Γx,Γy)K(x, y)) + LQ(x, y) = α[{[1 + |2x− 4|] |2x− 4| , [1 + |2x− 4|] |2x− 4|} + L {|2x− 4| , |2y − 4| , |2x− 4| , |2x− 4| , |2x− 4| . |2y − 4|}]; = α {[1 + |2x− 4|] |2x− 4| + L |2x− 4|} ≥ 0. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2228 Case(ii): Without loss of generality, suppose thatx > y, then ζ(d(Γx,Γy)K(x, y)) + LQ(x, y) = α[ { [1 + |2x− 4|] |2y − 4| 1 + |x− y| , [1 + |2x− 4|] |2y − 4| 1 + |x− y| } + L { |2y − 4| , |2y − 4| , |2y − 4| , |2y − 4| 1 + |x− y| , |2y − 4| . |2y − 4| 1 + |x− y| } ] − |x− y| . = α [1 + |2x− 4|] |2y − 4| 1 + |x− y| + αL |2y − 4| − |x− y| . 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Raman University, Bilaspur, chhattisgarh- India. Email address: sk10tiwari@gmail.com 2 Teacher,School Education Department , Takhatpur, Bilaspur, Chhattisgarh, -India. Email address: dubeyanandmohan767@gmail.com 2 TDepartment of MCA ,Hidustan College of Arts and Science, Coimbatore Tamilnadu, -India. Email address: avsenthilkumar@yahoo.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2231