GENERALIZED α-ADMISSIBLE ALMOST z-CONTRACTIONS INVOLVING SIMULATION FUNCTIONS IN A METRIC SPACE DIPTI, ANIL KUMAR DUBEY, URMILA MISHRA Abstract. In this paper, we present some fixed point results in com- plete metric spaces using generalized α-admissible mappings embedded in the simulation function. These results serve to generalize and unify several related fixed point results found in the existing literature. To validate our findings, we provide a specific example illustrating the ap- plication of these results. 1. Introduction Let N0 = N ∪ {0}, where N represents the set of positive integers. As usual R indicates the set of all real numbers. Furthermore, we set R+ 0 := [0,∞). Khojasteh et al.[14] introduced the notion of z-contraction by using a new class of auxiliary function called simulation function. They [14] proved several fixed point theorems and showed that many results in the literature are simple consequences of their obtained results. Definition 1. [14] A function ζ : [0,∞)× [0,∞) → R is called a simulation function if ζ satisfies the following conditions: (ζ1) ζ(0, 0) = 0. (ζ2) ζ(t, s) < s− t, for all t, s > 0. (ζ3) If {tn}, {sn} are sequences in (0,∞) such that limn→∞ tn = limn→∞sn = l ∈ (0,∞), then limn→∞ sup ζ(tn, sn) < 0. In [14], the following unique fixed point theorem is established. Theorem 2. [14] Let (X, d) be a metric space and T : X → X be a z- contraction with respect to a simulation function ζ, that is ζ(d(Tx, Ty), d(x, y)) ≥ 0 for all x, y ∈ X. Then T has a unique fixed point. It is worth mentioning that the Banach contraction is an example of z- contractions by defining ζ : [0, ∞) × [0, ∞) → R via ζ(t, s) = λs − t , for all s, t ∈ [0, ∞), where λ ∈ [0, 1). Argoubi et al. [2] modified Definition (1 ) as follows. Definition 3 . [2] A simulation function is a function ζ : [0, ∞)×[0, ∞) → R that satisfies t he f ollowing conditions: 2000 Mathematics Subject Classification. 54H25, 47H10, 55M20. Key words and phrases. Almost z-contraction, Simulation Function, α-admissible Map- ping. Corresponding Author-Dr. Urmila Mishra. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 354 Received: 15-08-2024, Revised: 01-10-2024, Accepted: 20-10-2024 DIPTI, ANIL KUMAR DUBEY, URMILA MISHRA, (i) ζ(t, s) < s− t, for all s, t > 0; (ii) If {tn} and {sn} are sequences in (0,∞) such that limn→∞ tn = limn→∞sn = l ∈ (0,∞), then limn→∞ sup ζ(tn, sn) < 0. It is clear that any simulation function in the sense of Khojasteh et al. [14] (Definition 1) is also a simulation function in the sense of Argoubi et al. [2] (Definition 3). The converse is not true. Very recently many fixed point results by using simulation functions have been provided. We have used some important article related to this paper [1, 4, 9, 10, 12, 15, 16, 17, 19]. In 2012, Samet et al.[20] introduced the concept of α-contraction and α-admissible and established various fixed point results for such class of mapping defined on complete metric space. There after the existence of fixed point of α-admissible contraction type mappings in different metric spaces have been studied by several authors (see [8, 9, 10, 16, 18]) and references cited there in. Definition 4. [20] Consider two mappings f : X → X and α : X ×X → [0,∞). Then f is called α-admissible mapping if for all x, y ∈ X with α(x, y) ≥ 1 implies α(fx, fy) ≥ 1. In this paper, we introduce the concept of generalized α-admissible almost z-contraction with respect toζ. We also establish the existence of fixed point for this class of mappings in complete metric spaces. The presented theorems extends, generalizes and improve many existing results in the literature, in particular the results [3, 7, 11, 13, 17]. 2. Main Results Here we put forward the notion of Geraghty functions and Geraghty con- tractions were discussed by Geraghty [11]. Definition 5. [11] A function β : [0,∞) → (0, 1) is called Geraghty function if {rn} ⊂ [0,∞) and limn→∞ β(rn) = 1− implies rn → 0+ as n → ∞. Definition 6. [11] A mapping T : X → X is called Geraghty contraction if there exists a Geraghty function β such that d(Tx, Ty) ≤ β(d(x, y))d(x, y), for all x, y ∈ X. The concept of Geraghty contraction mapping has been used in many works for example (see[3, 8, 18]). Berinde [5, 6] extended the class of con- tractive mappings, introducing the notion of almost contractions as follows. Definition 7. Let (X, d) be a metric space. A self mapping T on X is called an almost contraction if there are constants λ ∈ (0, 1) and θ ≥ 0 such that d(Tx, Ty) ≤ λd(x, y) + θd(y, Tx) for all x, y ∈ X. Berinde [5, 6] proved that every almost contraction mapping defined in a complete metric space has at least one fixed point. Subsequently, many authors [7, 9, 13] demonstrated that almost contractions type mappings have a unique fixed point in different metric spaces. By using the concept of Geraghty function (β) and almost contractions, we introduce the following: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 355 GENERALIZED α-ADMISSIBLE ALMOST ...... FUNCTIONS IN A METRIC SPACE Definition 8. Let (X, d) be a metric space, f : X → X be a self mapping, there exists ζ ∈ Z and α : X × X → [0,∞). Then continuous mapping f 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, fx)α(y, fy)d(fx, fy),K(x, y)) ≥ 0 (2.1) for all distinct x, y ∈ X, where ζ is a simulation function in the sense of Definition 1. Also K(x, y) = β(E(x, y))E(x, y) + LN(x, y), (2.2) where E(x, y) = d(x, y) + |d(x, fx)− d(y, fy)| and N(x, y) = min{d(x, fx), d(y, fy), d(x, fy), d(y, fx)}. Now we prove our main result. Theorem 9. Let (X, d) be a complete metric space, f is a generalized α- admissible almost z-contraction with respect to ζ furthermore, we suppose for all x, y ∈ X such that: (i) f is α-admissible; (ii) there exists x0 ∈ X such that α(x0, fx0) ≥ 1; (iii) for every sequence {xn} ∈ X such that α(xn, fxn) ≥ 1 for all n ∈ N ∪ {0} and {xn} converges to x, then α(x, fx) ≥ 1; (iv) α(x, fx) ≥ 1 for all x ∈ Fix(f). Then f has a unique fixed point x∗ in X. Proof. On account of (ii), there is a point x0 ∈ X such that α(x0, fx0) ≥ 1. There exists xn ∈ X such that xn = fxn−1 for all n ∈ N. Since f is α- admissible, we obtain α(fx0, fx1) = α(x1, x2) ≥ 1 implies α(fx1, fx2) = α(x2, x3) ≥ 1. By induction, we get α(xn, xn+1) ≥ 1, for all n ∈ N ∪ {0}. (2.3) If xn = xn+1 for some n ∈ N ∪ {0}, then xn = xn+1 = fxn and hence xn is a fixed point of f . Therefore, we can assume that xn ≠ xn+1 for all n ∈ N. Then we get d(xn, xn+1) > 0, so by (2.1), we have 0 ≤ ζ(α(xn, fxn)α(xn−1, fxn−1)d(fxn, fxn−1),K(xn, xn−1)) = ζ(α(xn, xn+1)α(xn−1, xn)d(xn+1, xn),K(xn, xn−1)) < K(xn, xn−1)− α(xn, xn+1)α(xn−1, xn)d(xn+1, xn), (2.4) where K(xn, xn−1) = β(E(xn, xn−1))E(xn, xn−1) + LN(xn, xn−1). Also, N(xn, xn−1) = min{d(xn, fxn), d(xn−1, fxn−1), d(xn, fxn−1), d(xn−1, fxn)} = min{d(xn, xn+1), d(xn−1, xn), d(xn, xn), d(xn−1, xn+1)} = 0, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 356 DIPTI, ANIL KUMAR DUBEY, URMILA MISHRA, and E(xn, xn−1) = d(xn, xn−1) + |d(xn, fxn)− d(xn−1, fxn−1)| = d(xn, xn−1) + |d(xn, xn+1)− d(xn−1, xn)| = d(xn, xn+1). Therefore, K(xn, xn−1) = β(d(xn, xn+1))d(xn, xn+1), from (2.4), we get 0 ≤ β(d(xn, xn+1))d(xn, xn+1)− α(xn, xn+1)α(xn−1, xn)d(xn+1, xn) which implies that d(xn, xn+1) ≤ β(d(xn, xn+1))d(xn, xn+1) < d(xn, xn+1), (2.5) a contradiction. Consequently, we deduce that d(xn, xn+1) < d(xn−1, xn) for each n ∈ N. Thus, we conclude that the sequence {d(xn−1, xn)} is a monotonically de- creasing sequence of non-negative reals and bounded from below by zero. So, there is some r ≥ 0 such that limn→∞ d(xn−1, xn) = r. It is ev- ident that limn→∞E(xn−1, xn) = r. As a next step, we will show that limn→∞ d(xn, xn−1) = 0. We assert that r = 0. Suppose, in contrast that r ̸= 0, then since f is generalized α-admissible almost z-contraction with respect to ζ ∈ Z therefore by (ζ3) and equation (2.5), and taking limit as n → ∞, we have 0 ≤ lim n→∞ sup ζ(α(xn, xn+1)α(xn−1, xn)d(xn+1, xn),K(xn, xn−1)) < 0. Therefore lim n→∞ β(E(xn−1, xn)) = 1 ⇒ lim n→∞ E(xn−1, xn) = 0. Attendantly, r = 0 and also r = lim n→∞ d(xn, xn−1) = 0. (2.6) Now, we will show that sequence {xn} is a Cauchy sequence. Assume that {xn} is not a Cauchy sequence, then there exists ϵ > 0 and sequences {xnk }, {xmk } : mk > nk > k such that d(xmk , xnk ) > ϵ and d(xmk−1, xnk ) ≤ ϵ for all m,n, k ∈ N. Therefore, by the triangle inequality, we have that ϵ < d(xmk , xnk ) ≤ d(xmk , xmk−1) + d(xmk−1, xnk ) ≤ d(xmk , xmk−1) + ϵ. (2.7) Letting k → ∞, using (2.6) and (2.7), we get lim n→∞ d(xmk , xnk ) = ϵ. (2.8) Since f is a generalized α-admissible almost z-contraction with respect to ζ, 0 ≤ ζ(α(xmk−1, xmk )α(xnk−1, xnk )d(xmk , xnk ),K(xmk−1, xnk−1)) It follows from condition (ζ2), we get 0 < K(xmk−1, xnk−1)− α(xmk−1, xmk )α(xnk−1, xnk )d(xmk , xnk ) d(xmk , xnk ) = d(fxmk−1, fxnk−1) < K(xmk−1, xnk−1). (2.9) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 357 GENERALIZED α-ADMISSIBLE ALMOST ...... FUNCTIONS IN A METRIC SPACE Also, K(xmk−1, xnk−1) = β(E(xmk−1, xnk−1))E(xmk−1, xnk−1) +LN(xmk−1, xnk−1), (2.10) where N(xmk−1, xnk−1) = min{d(xmk−1, xmk ), d(xnk−1, xnk ), d(xmk−1, xnk ), d(xnk−1, xmk )} and E(xmk−1, xnk−1) = d(xmk−1, xnk−1) + |d(xmk−1, xmk )− d(xnk−1, xnk )|. Letting k → ∞, using (2.6) and (2.8), we get lim k→∞ K(xmk−1, xnk−1) = ϵ. (2.11) By (2.8), (2.9), (2.11) and the condition (ζ3), we get 0 ≤ lim n→∞ sup 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. Thus limm,n→∞ d(xn, xm) exists and is equal to zero. Since (X, d) is complete, there exists x∗ ∈ X such that lim n→∞ d(xn, x ∗) = 0. (2.12) Now we shall show that fx∗ = x∗. Since f is continuous, we drive the desired results obviously, that is fx∗ = f( lim n→∞ xn) = lim n→∞ f(xn) = lim n→∞ xn+1 = x∗. Suppose we have (iii), 0 = lim m,n→∞ d(xm, xn) = lim n→∞ d(xn, x ∗) = d(x∗, x∗) and α(x∗, fx∗) ≥ 1. Moreover, 0 ≤ ζ(α(xn, fxn)α(x ∗, fx∗)d(fxn, fx ∗),K(xn, x ∗)) = ζ(α(xn, xn+1)α(x ∗, fx∗)d(xn+1, fx ∗),K(xn, x ∗)) < K(xn, x ∗)− α(xn, xn+1)α(x ∗, fx∗)d(xn+1, fx ∗), (2.13) where K(xn, x ∗) = β(E(xn, x ∗))E(xn, x ∗) + LN(xn, x ∗). Also, N(xn, x ∗) = min{d(xn, fxn), d(x∗, fx∗), d(xn, fx∗), d(x∗, fxn)} = min{d(xn, xn+1), d(x ∗, fx∗), d(xn, fx ∗), d(x∗, xn+1)} = 0. (2.14) And E(xn, x ∗) = d(xn, x ∗) + |d(xn, fxn)− d(x∗, fx∗)| = 0 + |0− d(x∗, fx∗)| = d(x∗, fx∗), as n → ∞. (2.15) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 358 DIPTI, ANIL KUMAR DUBEY, URMILA MISHRA, By (2.13), (2.14) and (2.15), we get d(xn+1, fx ∗) = d(fxn, fx ∗) ≤ α(xn, xn+1)α(x ∗, fx∗)d(fxn, fx ∗) < d(x∗, fx∗). (2.16) By letting n → ∞ in (2.13), together with the observation above, we have 0 ≤ lim n→∞ supζ(α(xn, fxn)α(x ∗, fx∗)d(fxn, fx ∗),K(xn, x ∗)) < 0. This is a contradiction. Hence, therefore x∗ is a fixed point of f i.e. fx∗ = x∗. Suppose that x∗ and u∗ be two fixed points of f and hence x∗, u∗ ∈ Fix(f) which is a generalized α-admissible almost z-contraction self mappings of a metric space (X, d). By (2.1), we have that 0 ≤ ζ(α(x∗, fx∗)α(u∗, fu∗)d(fx∗, fu∗),K(x∗, u∗)), (2.17) where K(x∗, u∗) = β(E(x∗, u∗))E(x∗, u∗) + LN(x∗, u∗). (2.18) Also E(x∗, u∗) = d(x∗, u∗) + |d(x∗, fx∗)− d(u∗, fu∗)| = d(x∗, u∗) (2.19) and N(x∗, u∗) = min{d(x∗, fx∗), d(u∗, fu∗), d(x∗, fu∗), d(u∗, fx∗)} = 0. (2.20) Therefore, from (2.17), (2.18), (2.19) and (2.20) we get that 0 ≤ ζ(α(x∗, fx∗)α(u∗, fu∗)d(fx∗, fu∗), d(x∗, u∗)) = ζ(α(x∗, x∗)α(u∗, u∗)d(x∗, u∗), d(x∗, u∗)). This is a contradiction. Thus, we have x∗ = u∗. Hence f is a unique fixed point. □ Theorem 10. Let (X, d) be a complete metric space, f is a generalized α-admissible almost z-contraction with respect to ζ. Assume that (i) f is a α-admissible, (ii) there exists x0 ∈ X such that α(x0, fx0) ≥ 1, (iii) X is a regular and for every sequence {xn} in X such that α(xn, xn+1) ≥ 1 for all n ∈ N ∪ {0} and we have α(xm, xn) ≥ 1 for all m,n ∈ N with m < n, (iv) α(x, y) ≥ 1, for all x, y ∈ Fix(f). Then f has a unique fixed point x∗ in X. Proof. By (ii), let x0 ∈ X such that α(x0, fx0) ≥ 1. There exist xn ∈ X such that xn = fxn−1 for all n ∈ N. We have by Theorem 9, {xn} is a Cauchy sequence such that limn→∞ d(xnxn+1) = 0. Thus limm,n→∞ d(xn, xm) exists and is equal to 0. Since (X, d) is complete, there exists x∗ ∈ X such that lim n→∞ d(xn, x ∗) = 0, (2.21) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 359 GENERALIZED α-ADMISSIBLE ALMOST ...... FUNCTIONS IN A METRIC SPACE then lim m,n→∞ d(xm, xn) = lim n→∞ d(xn, x ∗) = d(x∗, x∗) = 0. Since X is regular, therefore there exists a subsequence {xnk } of {xn} such that α(xnk , x∗) ≥ 1 for all k ∈ N. Therefore 0 ≤ ζ(α(xnk , fxnk )α(x∗, fx∗)d(fxnk , fx∗),K(xnk , x∗)) = ζ(α(xnk , xnk+1 )α(x∗, fx∗)d(xnk+1 , fx∗),K(xnk , x∗)) < K(xnk , x∗)− α(xnk , xnk+1 )α(x∗, fx∗)d(xnk+1 , fx∗), (2.22) where K(xnk , x∗) = β(E(xnk , x∗))E(xnk , x∗) + LN(xnk , x∗). (2.23) Also E(xnk , x∗) = d(xnk , x∗) + |d(xnk , fxnk )− d(x∗, fx∗)| = d(xnk , x∗) + |d(xnk , xnk+1 )− d(x∗, fx∗)| = d(x∗, fx∗) for large k, (2.24) and N(xnk , x∗) = min{d(xnk , fxnk ), d(x∗, fx∗), d(xnk , fx∗), d(x∗, fxnk )} = min{d(xnk , xnk+1 ), d(x∗, fx∗), d(xnk , fx∗), d(x∗, xnk+1 )} = 0. (2.25) Therefore K(xnk , x∗) = d(x∗, fx∗). Consequently, we have d(xnk+1, fx ∗) = d(fxnk , fx∗) ≤ α(xnk , fxnk )α(x∗, fx∗)d(fxnk , fx∗) < d(x∗, fx∗) for all k ∈ N. (2.26) By (2.22), (2.26) and the condition (ζ3), we get 0 ≤ limn→∞supζ(α(xn, fxn)α(x ∗, fx∗)d(fxn, fx ∗),K(xn, x ∗)) < 0. This is a contradiction. Hence, therefore x∗ is a fixed point of f . Suppose that x∗ and u∗ be two fixed points of f and hence x∗, u∗ ∈ Fix(f) which is a generalized α-admissible almost z-contraction self -mappings of a metric space (X, d). By (2.1), we have that 0 ≤ ζ(α(x∗, fx∗)α(u∗, fu∗)d(fx∗, fu∗),K(x∗, u∗)), (2.27) where K(x∗, u∗) = d(x∗, u∗), by using (2.18), (2.19) and (2.20). This to- gether with (2.27) shows that 0 ≤ ζ(α(x∗, fx∗)α(u∗, fu∗)d(fx∗, fu∗),K(x∗, u∗)) = ζ(α(x∗, x∗)α(u∗, u∗)d(x∗, u∗), d(x∗, u∗)). This is a contradiction. Thus, we have x∗ = u∗. Hence f has a unique fixed point. □ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 360 DIPTI, ANIL KUMAR DUBEY, URMILA MISHRA, Corollary 11. Let (X, d) be a complete metric space, f : X → X be a self-mapping, there exists ζ ∈ Z and α : X ×X → [0,∞) be a function with α(x, y) = 1 for all x, y ∈ X such that ζ(d(fx, fy),K(x, y)) ≥ 0 for all distinct x, y ∈ X, where K(x, y) = β(E(x, y))E(x, y) + LN(x, y) and E(x, y) = d(x, y) + |d(x, fx)− d(y, fy)| and N(x, y) = min{d(x, fx), d(y, fy), d(x, fy), d(y, fx)}. Then f has a unique fixed point x∗ in X. Example 12. Let X = [0, 1] endowed with metric d(x, y) = |x − y| for all x, y ∈ X. Let ζ(t, s) = s−t and considering β : [0,∞) → [0, 1) as β(t) = 1 1+t for all t ≥ 0 and L ≥ 0. Let f : X → X be defined by f(x) = x 3 for all x ∈ [0, 1] and α : X ×X → [0,∞) be defined by α(x, y) = { 1, if, x, y ∈ [0, 1]; 0, otherwise. Note that f is an α-admissible if α(x, fx) ≥ 1 implies α(fx, f2x) ≥ 1. Now by definition of α and x, y ∈ [0, 1], we have α(x, fx) = α(x, x3 ) = 1. Simi- larly α(y, fy) = 1 for all x, y ∈ X. From above, it is clear that f is a generalized α-admissible mapping. Now ζ(d(fx, fy),K(x, y)) = K(x, y)− d(fx, fy) = β(E(x, y))E(x, y) + LN(x, y)− 1 3 |x− y| = E(x, y) 1 + E(x, y) + LN(x, y)− 1 3 |x− y| ≤ 5 3d(x, y) 1 + 5 3d(x, y) + LN(x, y)− 1 3 |x− y| = 5 3 |x− y| 1 + 5 3 |x− y| + LN(x, y)− 1 3 |x− y| ≥ 0. Therefore, f is generalized α-admissible almost z-contraction with respect to ζ ∈ Z. Hence all the assumptions of Theorem 9 and Corollary 11 are satisfied and hence f has a unique fixed point. References [1] A. S. Alharbi, H. Alsulami and E. Karapinar: On the power of simulation and admissible functions in metric fixed point theory, Jour. Func. Spaces, Volume 2017, Article ID 2068163, 7 pages. https://doi.org/10.1155/2017/2068163. [2] H. Argoubi, B. Samet, C. Vetro: Nonlinear contractions involving simulation func- tions in a metric space with a partial order,Jour. Nonlinear Sci. Appl., 8 (2015), 1082-1094. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 361 GENERALIZED α-ADMISSIBLE ALMOST ...... FUNCTIONS IN A METRIC SPACE [3] H. Aydi, A. Felhi, H. Afshari: New Geraghty type contractions on metric-like spaces, Jour. Nonlinear Sci. Appl., 10 (2017), 780-788. [4] H. Aydi, A. Felhi, E. Karapinar and F. A. Alojail: Fixed points on quasi-metric spaces via simulation functions and consequences, Jour. Math. Anal., 9(2018) (2), 10-24. [5] V. Berinde: Approximating fixed point of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9 (1)(2004), 43-53. [6] V. Berinde: General constructive fixed point theorems for Ciric-type almost contrac- tions in metric spaces, Carpathian Jour. Math, 24 (2)(2018), 10-19. [7] P. Bunpatcharacharoen, S. Saelee, and P. Saipara: Modified almost type z- contraction, Tahi Jour. Math., 18(1) (2020), 252-260. [8] S. Chandok: Some fixed point theorems for (α, β)-admissible Geraghty type contrac- tive mappings and related results, Mathematical Sciences, 9 (2015), 127-135. [9] A. Dewangan, A. K. Dubey, U. Mishra and R. P. Dubey: Fixed point results for (α, β)-admissible almost z-contractions in metric-like space via simulation function, Facta Universitatis (NIS), SER. MATH. INFORM, 37(3) (2022) , 529-540. [10] A. Felehi, H. Aydi, D. Zhang: Fixed point for α-admissible contractive mappings via simulation functions, Jour. Nonlinear Sci. Appl., 9(10) (2016), 5544-5560. [11] M. Geraghty: On contractive mappings, Proc. Amer. Math. Soc., 40(2) (1973), 604-608. [12] E. Karapinar: Fixed point results via simulation functions, Filomat, 30(8) (2016), 2343-2350. [13] E. Karapinar and V. M. L. Hima Bindu: Discussion on the almost z-contraction, Open Mathematics, 18 (2020), 448-457. [14] F. Khojasteh, S. Shukla and S. Radenovic: A new approach to the study of fixed point theorems via simulation functions, Filomat, 29 (6)(2015), 1189-1194. [15] S. Mishra, A. K. Dubey, U. Mishra and H. G. Hyun: Some fixed point theorems for rational (α, β, z)-contraction mappings under simulation functions and cyclic (α, β)- admissibility, Nonlinear Func. Anal. Appl., 27 (4)(2022), 751-771. [16] S. Mishra, A. K. Dubey, U. Mishra and R. P. Dubey: On some fixed point results for cyclic (α, β)-admissible almost z-contraction in metric-like space with simulation function, Communication in Math. Appl. 13 (1)(2022), 223-233. [17] A. Padcharoen and P. Sukprasert: On admissible mapping via simulation functions, Aust. Jour. Math. Anal. Appl., 13 (1)(2021), 1-10. [18] O. Popescu: Some new fixed point theorems for α-Geraghty contractive type maps in metric spaces, Fixed Point Theory Appl., 2014 (2014), Art. ID 190. [19] A. F. Roldan-Lopez-de Hierro, E. Karapinar, C. Roldan-Lopez-de-Hierro, J. Martinez-Moreno: Coincidence point theorems on metric spaces via simulation func- tions, Jour. Comp. Appl. Math., 275 (2015), 345-355. [20] B. Samet, C. Vetro and P. Vetro: Fixed point theorems for (α−ψ) contractive type mappings, Jour. Nonlinear Anal., 75 (4)(2012), 2154-2165. (Dipti) Department of Mathematics, Dr. C. V. Raman University, Kargi Road Kota, Bilaspur (C.G.) Email address: diptisharma2704@gmail.com (Anil Kumar Dubey) Department of Applied Mathematics, Bhilai Institute of Technology, Bhilai House, Durg (Chhattisgarh), India Email address: anilkumardby70@gmail.com (Urmila Mishra) Department of Mathematics, Vishwavidyalaya Engg. Col- lege, Ambikapur (Chhattisgarh) (A Constituent College of CSVTU, Bhilai), India Email address: mishra.urmila22@csvtu.ac.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 362 diptisharma2704@gmail.com mailto:anilkumardby70@gmail.com mailto:mishra.urmila22@csvtu.ac.in 1. Introduction 2. Main Results References