EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5730 ISSN 1307-5543 – ejpam.com Published by New York Business Global New Proofs of Fixed Point Theorems on Quasi-Metric Spaces Sehie Park1,2 1 The National Academy of Sciences, Republic of Korea, Seoul 06579 2 Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea Abstract. Based on our 2023 Metatheorem in Ordered Fixed Point Theory, we give very simple proofs of known fixed point theorems for various multimap classes on quasi-metric spaces. Such classes are represented by the Banach contractions, the Rus-Hicks-Rhoades maps, the Nadler multimaps, Covitz-Nadler multimaps, and others. Consequently, we obtain simple proofs of a large number of known theorems on extremal elements, fixed points, stationary points for several classes of maps or multimaps. Finally, we add some known theorems for which our Metatheorem does not work. 2020 Mathematics Subject Classifications: 06A75, 47H10, 54E35, 54H25, 58E30, 65K10 Key Words and Phrases: Fixed point, quasi-metric space, Banach contraction, Rus-Hicks- Rhoades map, Nadler multimap, Covitz-Nadler multimap, extremal point, stationary point 1. Introduction Let (X, d) be a metric space. A Banach contraction T : X → X is a map such that, for some r ∈ (0, 1), d(Tx, Ty) ≤ r d(x, y) ∀ x, y ∈ X. There have been appeared thousands of articles related to generalizations of the Banach contraction. Recently, we introduced the Rus-Hicks-Rhoades (RHR) map T : X → X for some r ∈ (0, 1) satisfying d(Tx, T 2x) ≤ r d(x, Tx) ∀ x ∈ X. We found that the class of the Rus-Hicks-Rhoades maps contains a large number of maps including Banach contractions. See our recent works [33], [35], [39]. Moreover, we found that the RHR maps characterize the metric completeness. See [39], [40]. In our previous work [40], we classified known fixed point theorems equivalent to quasi- metric completeness based on our 2023 Metatheorem in Ordered Fixed Point Theory [30], DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5730 Email addresses: park35@snu.ac.kr, sehiepark@gmail.com (S. Park) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 2 of 21 [31], [32]. In the present paper, we give very simple proofs of fixed point theorems for various multimap classes in [40]. Such classes are represented by the Banach contractions, the Rus-Hicks-Rhoades maps, the Nadler multimaps, Covitz-Nadler multimaps, and oth- ers. Consequently, we obtain another large number of new theorems on extremal elements, fixed points, stationary points for several classes of maps or multimaps. The present paper is organized as follows. For the preliminaries, we follow [16], [34], [40]. In Section 2, we introduce our Rus-Hicks-Rhoades Contraction Principle (Theorem P). Section 3 devotes to derive Theorem H which is equivalent formulations of the quasi- metric completeness as an application of our traditional Metatheorem and Theorem P. Let {0} denote the family of theorems which are equivalent to the metric completeness; see [40]. In Sections 4-7, we introduce major results in the subfamilies {α}-{ϵ} of the family {0} corresponding to each equivalent formulation in Theorem H. In Section 8, we give a few example of fixed point theorems which can not applicable our Theorem H. Finally, Section 9 is for epilogue. 2. The Rus-Hicks-Rhoades Contraction Principle For quasi-metric spaces, the convergence of a sequence, Cauchy sequences, complete- ness, orbits, and orbital continuity are routinely defined as in [16],[40]. The following Rus-Hicks-Rhoades (RHR) Contraction Principle was obtained in [33], [35-37], [39], [40]: Theorem P. Let (X, q) be a quasi-metric space and let T : X → X be an RHR map; that is, q(T (x), T 2(x)) ≤ α q(x, T (x)) ∀ x ∈ X, (p) where 0 < α < 1. (i) If X is T -orbitally complete, then, for each x ∈ X, there exists a point x0 ∈ X such that lim n→∞ Tn(x) = x0 and q(Tn(x), x0) ≤ αn 1− α q(x, T (x)), n = 1, 2, · · · , q(Tn(x), x0) ≤ α 1− α q(Tn−1(x), Tn(x)), n = 1, 2, · · · . (ii) x0 is a fixed point of T , and, equivalently, (iii) T : X → X is orbitally continuous at x0 ∈ X. Let f : X → X be a (Banach) contraction of a metric space (X, d), that is, there exists r ∈ (0, 1) such that d(Tx, Ty) ≤ r d(x, y) ∀ x, y ∈ X. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 3 of 21 Theorem P is a far-reaching generalization of the so-called Banach Contraction Prin- ciple originated from Banach [4] in 1922 as the following: Theorem. (Banach) If 10 U(X) be a continuous operator in E, the counter-domain of U(X) is contained in E1. 20 There exists a number 0 < M < 1 which implies, for every X ′ and X ′′, the inequality ||U(X ′)− U(X ′′)|| ≤ M.||X ′ −X ′′||. —- there exists an element X such that X = U(X). Here E and E1 is a normed space and its complete subset, resp. 3. A Form of Our New 2023 Metatheorem Let (X, q) be a quasi-metric space and Cl(X) denote the family of all nonempty closed subsets of X (not necessarily bounded). For A,B ∈ Cl(X), set H(A,B) = max{sup{q(a,B) : a ∈ A}, sup{q(b, A) : b ∈ B}}, where q(a,B) = inf{q(a, b) : b ∈ B}. Then H is called a generalized Hausdorff distance and it may have infinite values. Recently, as a basis of Ordered Fixed Point Theory [30], we obtained the 2023 Metathe- orem and Theorem H including Nadler’s fixed point theorem [28] in 1969 and its extended version by Covitz-Nadler [9] in 1970. Theorem H. ([36],[38],[39]) Let (X, q) be a quasi-metric space and 0 < r < 1. Then the following equivalent statements hold: (0) (X, q) is complete. (α) For a multimap T : X → Cl(X), there exists an element v ∈ X such that H(Tv, Tw) > r q(v, w) for any w ∈ X\{v}. (β) If F is a family of maps f : X → X such that, for any x ∈ X\{fx}, there exists a y ∈ X\{x} satisfying q(fx, fy) ≤ r q(x, y), then F has a common fixed element v ∈ X, that is, v = fv for all f ∈ F. (γ) If F is a family of maps f : X → X satisfying q(fx, f2x) ≤ r q(x, fx) for all x ∈ X\{fx}, then F has a common fixed element v ∈ A, that is, v = fv for all f ∈ F. (δ) Let F be a family of multimaps T : X → Cl(X) such that, for any x ∈ X\Tx, there exists y ∈ X\{x} satisfying H(Tx, Ty) ≤ r q(x, y). Then F has a common fixed element v ∈ X, that is, v ∈ Tv for all T ∈ F. (ϵ) If F is a family of multimaps T : X → Cl(X) satisfying H(Tx, Ty) ≤ r q(x, y) for all x ∈ X and any y ∈ Tx\{x}, then F has a common stationary element v ∈ X, that is, {v} = Tv for all T ∈ F. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 4 of 21 (η) If Y is a subset of X such that for each x ∈ X\Y there exists a z ∈ X\{x} satisfying H(Tx, Tz) ≤ r q(x, z) for a T : X → Cl(X), then there exists a v ∈ X ∩ Y = Y . Proof. The equivalency (α)− (η) follows from our 2023 Metatheorem in [30]-[32]. When F is a singleton, (β)-(ϵ) are denoted by (β1)-(ϵ1), respectively, They are also logically equivalent to (α)-(η). Note that (α) =⇒ (γ1) follows from Theorem P for quasi-metric spaces. The equivalency of (0) and (γ1) is given in [36], [39]. Then Theorem H holds. □ Remark 3.1. (1) (α) =⇒ (β1) implies the Banach Contraction Principle, which does not characterize the metric completeness. (2) Moreover, (α) =⇒ (δ1) and (α) =⇒ (ϵ1) extend the well-known fixed point theo- rems of Nadler [28] and Covitz-Nadler [9]. The following (α) =⇒ (γ1) of Theorem H also follows from Theorem P. It is the basis in the present article and will be called the RHR theorem, which characterizes the metric completeness. Theorem H(γ1). Let (X, δ) be a complete quasi-metric space and 0 < r < 1. (γ1) If a map f : X → X satisfies δ(fx, f2x) ≤ r δ(x, fx) for all x ∈ X\{fx}, then f has a fixed element v ∈ X, that is, v = fv. Consequently, this is a close relative of Theorems of Rus [48] and Hicks-Rhoades [14]. This is rather surprising and they are all close relatives of the Banach Contraction Prin- ciple; see [4]. 4. The subfamily {α} Based on our 2023 Metatheorem [30] and Theorem H, we obtained the following in [38], [39]: Theorem H(α). Let (X, q) be a quasi-metric space and 0 < r < 1. Then the following statements are equivalent: (0) (X, q) is complete. (α1) For a map f : X → X, there exists an element v ∈ X such that q(fv, fw) > r q(v, w) for any w ∈ X\{v}. (α) For a multimap T : X → Cl(X), there exists an element v ∈ X such that H(Tv, Tw) > r q(v, w) for any w ∈ X\{v}. If X is T -orbitally complete, then (α) holds; see [40]. 5. The subfamily {β} of the extended Banach type Theorem H(β). Let (X, q) be a quasi-metric space and 0 < r < 1. Then the following statements are equivalent: S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 5 of 21 (0) (X, q) is complete. (β) If F is a family of maps f : X → X such that, for any x ∈ X\{fx}, there exists a y ∈ X\{x} satisfying q(fx, fy) ≤ α q(x, y), then F has a common fixed element v ∈ X, that is, v = fv for all f ∈ F. In this section, we want to collect certain extensions of the Banach contraction satis- fying Theorem H(β). There are a huge number of extensions of the Banach Contraction Principle, however, we have known only one example of the family {β}. Recall that the Banach Contraction Principle follows from Theorem H(β). The origin of the subfamily {β} is Banach’s theorem [4] in 1922. 6. The subfamily {γ} of the Rus-Hicks-Rhoades type The following originates in [35],[36]: Theorem H(γ). Let (X, q) be a quasi-metric space and 0 < r < 1. Then the following statements are equivalent: (0) (X, q) is complete. (γ) If F is a family of maps f : X → X satisfying q(fx, f2x) ≤ α q(x, fx) for all x ∈ X\{fx}, then F has a common fixed element v ∈ X, that is, v = fv for all f ∈ F. Note that the f -orbital completeness of (X, q) for any RHR map f : X → X in F implies (γ). Such map is traditionally called as graphic contraction, iterative contraction, weakly contraction, Banach mapping, . . . . We prefer to call it a weak contraction. Many generalizations of the Banach principle are of the Rus-Hicks-Rhoades type; see [40]. Rhoades [46] in 1978 noted that the analogues of most of the conditions in his well- known list [45] could be extended to the RHR type of contractive definitions; see also [41]. In this section, we list some typical old and new examples of Theorem H(γ). Most of examples hold for quasi-metric spaces, but we state their original forms. Kannan [17] in 1969 Recall the following: Theorem 6.1. (Kannan) Let (X, d) be a complete metric spaces and T : X → X be a Kannan contraction mapping, i.e., d(Tx, Ty) ≤ λ[d(x, Tx) + d(y, Ty)] ∀ x, y ∈ X, where λ ∈ [0, 1/2). Then T has a unique fixed point. Comment: This is a simple consequence of Theorem P for quasi-metric spaces. In fact, for y = Tx, we have d(Tx, T 2x) ≤ λ 1− λ d(x, Tx) and 0 ≤ λ 1− λ < 1. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 6 of 21 Kannan’s example does not require the continuity of the map at every point, although maps satisfying his condition are continuous at fixed points; see Theorem P(ii). Reich [44] in 1971 The following theorem proved by Reich generalizes Banach’s fixed point theorem and Kannan’s fixed point theorem. Theorem 6.2. (Reich) Let f be a selfmap on a complete metric space (X, d). If there exist constants a, b, c ∈ [0, 1) with a+ b+ c < 1 such that d(fx, fy) ≤ ad(x, fx) + bd(y, fy)) + cd(x, y) ∀ x, y ∈ X, then f has a unique fixed point. Comment: Note that f is an RHR map and that Theorems P and H(γ1) are applicable to f . The uniqueness follows from the contractive condition. Rus, Reich, Ćirić in 1971-2001 The following theorem was proved by Reich, Rus and Ćirić independently to combine and improve both Banach and Kannan fixed point theorems. Theorem 6.3. (Rus-Reich-Ćirić) Let (X, d) be a complete metric spaces and T : X → X be a Rus-Reich-Ćirić contraction mapping, i.e., d(Tx, Ty) ≤ λ[d(x, y) + d(x, Tx) + d(y, Ty)] for all x, y ∈ X, where λ ∈ [0, 1/3). Then T has a unique fixed point. Comment: This is a simple consequence of the RHR theorem for quasi-metric spaces. In fact, for y = Tx, we have d(Tx, T 2x) ≤ 2λ 1− λ d(x, Tx) and 0 ≤ 2λ 1− λ < 1. Therefore, Theorems P and H(γ1) are applicable to T . Ćirić [6] in 1974 Theorem 6.4. (Ćirić) Let (M,d) be a quasi-metric space, and let T : M → M be a given mapping. Suppose that the following conditions are satisfied: (i) T is orbitally continuous on M; (ii) (M,d) is T-orbitally complete; (iii) There exists a constant q ∈ (0, 1) such that min{d(Tx, Ty), d(x, Tx), d(y, Ty)} −min{d(x, Ty), d(y, Tx)} ≤ q d(x, y), ∀ x, y ∈ M. Then, for every x ∈ M , the Picard sequence {Tnx} converges to a fixed point of T. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 7 of 21 Comment: For y = Tx, we can consider the following: min{d(Tx, T 2x), d(x, Tx), d(Tx, T 2x)} −min{d(x, T 2x), d(Tx, Tx)} ≤ q d(x, Tx) for all x, y = Tx ∈ M . Case 1: If d(Tx, T 2x) ≤ d(x, Tx), then d(Tx, T 2x) ≤ q d(x, Tx), possible. Case 2: If d(Tx, T 2x) ≥ d(x, Tx), then d(x, Tx) ≤ q d(x, Tx) implies x = Tx. In any case, Theorems P and H(γ1) are applicable to T . Dass and Gupta [10] in 1975 Theorem 6.5. (Dass-Gupta) Let (X, d) be a complete metric space and T : X → X be a mapping. If there exist k1, k2 ∈ [0, 1), with k1 + k2 < 1 such that d(Tx, Ty) ≤ k1 · d(y, Ty) 1 + d(x, Tx) 1 + d(x, y) + k2 · d(x, y), for all x, y ∈ X, then T has a unique fixed point u ∈ X and the sequence {Tnx} converges to the fixed point u for all x ∈ X. Comment: Note that T is an RHR map and can be applied Theorems P and H(γ1). The uniqueness follows from the contractive condition. Ran and Reurings [43] in 2004 In this paper, the following fixed point theorem in a partially ordered metric space is proved: Theorem 6.6. (Ran-Reurings) Let (M,≤) be an ordered set and d be a metric on M such that (M,d) is a complete metric space. Let U : M → M be a nondecreasing mapping, i.e. Ux ≤ Uy, for every x, y ∈ M with x ≤ y. Suppose that there exists x0 ∈ M with x0 ≤ Ux0 and L ∈ [0, 1) such that d(Ux,Uy) ≤ Ld(x, y) ∀ x, y ∈ M with x ≤ y. If U is continuous, then it has a fixed point in M. Comment: Note that U is an RHR map and can be applied Theorems P and H(γ1). Suzuki [49] in 2008 Suzuki generalized the Banach contraction principle as follows: Theorem 6.7. (Suzuki) Let (X, d) be a complete metric space and T be a mapping on X. Define a nonincreasing function θ from [0, 1) onto (1/2, 1] by θ(r) =  1 if 0 ≤ r ≤ ( √ 5− 1)/2, (1− r)r−2 if ( √ 5− 1)/2 ≤ r ≤ 2−1/2, (1 + r)−1 if 2−1/2 ≤ r < 1. (1) S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 8 of 21 Assume there exists r ∈ [0, 1) such that θ(r)d(x, Tx) ≤ d(x, y) =⇒ d(Tx, Ty) ≤ r d(x, y) for all x, y ∈ X. Then there exists a unique fixed point z of T. Moreover limn T nx = z for all x ∈ X. Comment: Note that T is an RHR map and Theorems P or H(γ1) can be applied. Now it suffices to show the uniqueness of the fixed point z. If w ∈ X is a fixed point, then θ(r)d(z, Tz) ≤ d(z, w) =⇒ d(Tz, Tw) ≤ r d(z, w). Hence d(z, w) = d(Tz, Tw) ≤ r d(z, w) and consequently d(z, w) = 0. This is a simple proof of Theorem 6.7. There have been appeared a large number of variants of Suzuki’s theorem. Many of them can be improved by easy proofs as shown as above; see [38]. In order to show uselessness of Theorems 2 and 3 in [49] and other works of Suzuki, let us consider any function θ′ : [0,+∞) → [0, 1]: Theorem 6.8. (Park) Replace the function θ in Theorem 6.7 by θ′. Then the conclusion of Theorem 6.7 holds. Proof. Note that, by putting y = Tx, T becomes an RHR map. Then by Theorems P or H(γ1), T has a fixed point and its uniqueness follows as in our proof of Theorem 6.7 in [36]. □ Altun and Erduran [1] in 2011 The authors present a fixed-point theorem for a single-valued map in a complete metric space using implicit relation, which is a generalization of several previously stated results including that of Suzuki [49]. The aim of [1] is to generalize the above results using the implicit relation technique in such a way that F (d(Tx, Ty), d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx) ≤ 0, for x, y ∈ X, where F : [0,∞)6 → R is a function as given in Section 2 in [1] with 5 examples. Theorem 6.9. (Altun-Erduran) Let (X, d) be a complete metric space, and let T be a mapping on X. Define a nonincreasing function θ : [0, 1) → (1/2, 1] as in Suzuki [49]. Assume that there exists an F as above, such that θ(r)d(x, Tx) ≤ d(x, y) implies F (d(Tx, Ty), d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx) ≤ 0, for all x, y ∈ X, then T has a unique fixed-point z and limn T nx = z holds for every x ∈ X. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 9 of 21 Comment: In the proof, the authors showed that d(Tx, T 2x) ≤ r d(x, Tx) for all x ∈ X, that is, T is an RHR map. Hence T has a fixed point by Theorems P and H(γ1) and its uniqueness follows from properties of F . Khojasteh, Abbas, Costache [25] in 2014 Theorem 6.10. (Khojasteh et al.) Let (X, d) be a complete metric space and let T be a mapping from X into itself. Suppose that T satisfies the following condition: d(Tx, Ty) ≤ d(x, Ty) + d(y, Tx) d(x, Tx) + d(y, Ty) + 1 d(x, y) for all x, y ∈ X. Then (a) T has at least one fixed point ẋ ∈ X, (b) {Tnx} converges to a fixed point, for all x ∈ X; (c) if ẋ and ẏ are distinct fixed points of T, then d(ẋ, ẏ) ≤ 1/2. Comment: Note that T is an RHR map and can be applicable Theorems P and H(γ1). The (c) follows from the contractive condition. Karapinar [18] in 2018 We start our results by the generalization of the definition of Kannan type contraction via interpolation notion, as follows: Definition 6.11. Let (X, d) be a metric space. We say that the self-mapping T : X → X is an interpolative Kannan type contraction, if there exist a constant λ ∈ [0, 1) and α ∈ (0, 1) such that d(Tx, Ty) ≤ λ[d(x, Tx)]α · [d(y, Ty)]1−α for all x, y ∈ X with x ̸= Tx. Theorem 6.12. (Karapinar) Let (X, d) be a complete metric space and T be an inter- polative Kannan type contraction. Then T has a unique fixed point in X. Comment: Later, the author withdraw the uniqueness of fixed point. For y = Tx, we have d(Tx, T 2x) ≤ λ[d(x, Tx)]α · [d(Tx, T 2x)]1−α =⇒ [d(Tx, T 2x)]α ≤ λ[d(x, Tx)]α =⇒ d(Tx, T 2x) ≤ λ1/αd(x, Tx) with 0 < λ1/α < 1. Therefore, T is an RHR map for which Theorems P and H(γ1) can be applicable. Hence Theorem 6.12 can be stated for T -orbitally complete quasi-metric spaces. Karapinar, Agarwal, and Aydi [23] in 2018 As a correction of Theorem 6.12 of the previous paper, the authors stated: S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 10 of 21 Theorem 6.13. (Karapinar et al.) Let (X, ρ) be a complete metric space. A self-mapping T : X → X possesses a fixed point in X, if there exist constants λ ∈ [0, 1) and α ∈ (0, 1) such that ρ(Tζ, Tη) ≤ λ[ρ(ζ, T ζ)]α · [ρ(η, Tη)]1−α for all ζ, η ∈ X\Fix(T ). Notice that several variations of Reich contractions (in Theorem 3 in [23]) can be stated. We may state the following: ρ(Tζ, Tη) ≤ aρ(ζ, η) + bρ(ζ, T ζ) + cρ(η, Tη), where a, b, c ∈ (0,∞) such that 0 ≤ a+ b+ c < 1. In [23], the authors investigate the validity of the interpolation approach for Reich contractions in the context of partial metric spaces that was introduced by Matthews. Comment: We have two examples of RHR maps in [23] as follows: For the Reich-Rus-Ćirić contraction, by putting η = Tζ, we have ρ(Tζ, T 2ζ) ≤ 2λ ρ(ζ, T ζ) + λ ρ(Tζ, T 2ζ) =⇒ ρ(Tζ, T 2ζ) ≤ 2λ 1− λ ρ(ζ, T ζ), λ ∈ [0, 1/2). For the variation of the Reich contraction, by putting η = Tζ, we have ρ(Tζ, T 2ζ) ≤ (a+b)ρ(ζ, T ζ)+c ρ(Tζ, T 2ζ) =⇒ ρ(Tζ, T 2ζ) ≤ a+ b 1− c ρ(ζ, T ζ), 0 < a+ b 1− c < 1. Therefore, Theorems P and H(γ1) can be applied to such two examples for T -orbitally complete quasi-metric spaces. Karapinar, Alqahtani, and Aydi [24] in 2018 By using an interpolative approach, the authors recognize the Hardy-Rogers fixed point theorem in the class of metric spaces. The obtained result is supported by some examples. They also give the partial metric case, according to their result. A generalization of the Banach Contraction Principle is due to Hardy-Rogers as follows; see [13]. Theorem 6.14. (Karapinar et al.) Let (X, d) be a complete metric space. Let T : X → X be a given mapping such that d(Tθ, Tϑ) ≤ αd(θ, ϑ) + βd(θ, Tθ) + γd(ϑ, Tϑ) + δ 2 [d(θ, Tϑ) + d(ϑ, Tθ)], for all θ, ϑ ∈ X, where α, β, γ, δ are non-negative reals such that α+ β + γ + δ < 1. Then T has a unique fixed point in X. In [24], the authors introduce the concept of interpolative Hardy-Rogers type contrac- tions, and provide some examples illustrating the obtained result. They also extend their obtained result to partial metric spaces. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 11 of 21 Definition 6.15. Let (X, d) be a metric space. We say that the self-mapping T : X → X is an interpolative Hardy-Rogers type contraction if there exists λ ∈ [0, 1) and α, β, γ ∈ (0, 1) with α+ β + γ < 1, such that d(Tθ, Tϑ) ≤ λ[d(θ, ϑ)]β · [d(θ, Tθ)]α · [d(ϑ, Tϑ)]γ · [1 2 (d(θ, Tϑ) + d(ϑ, Tθ))]1−α−β−γ for all θ, ϑ ∈ X\Fix(T ). Theorem 6.16. (Karapinar et al.) Let (X, d) be a complete metric space and T be an interpolative Hardy-Rogers type contraction. Then, T has a fixed point in X. Comment: The map in Theorem 6.16 is an RHR map by putting ϑ = Tθ. In fact, we have d(Tθ, T 2θ) ≤ (α+ β)d(θ, Tθ) + δ 2 [d(θ, Tθ) + d(Tθ, T 2θ)] =⇒ (1− γ − δ/2)d(Tθ, T 2θ) ≤ (α+ β + δ/2)d(θ, Tθ), where α+ β + δ/2 < 1− γ − δ/2. For an interpolative Hardy-Roger contraction, by putting ϑ = Tθ, we have d(Tθ, T 2θ) ≤ λ[d(θ, Tθ)]α+β · [d(Tθ, T 2θ)]γ · [max{d(θ, Tθ), d(Tθ, T 2θ)}]1−α−β−γ =⇒ d(Tθ, T 2θ)1−γ ≤ λ[d(θ, Tθ)]α+β+1−α−β−γ or [d(Tθ, T 2θ)]1−γ ≤ λ[d(θ, Tθ)]α+β · [d(Tθ, T 2θ)]1−α−β−γ =⇒ d(Tθ, T 2θ)1−γ ≤ λ[d(θ, Tθ)]1−γ or d(Tθ, T 2θ)α+β ≤ λ[d(θ, Tθ)]α+β =⇒ d(Tθ, T 2θ) ≤ λpd(θ, Tθ) or d(Tθ, T 2θ) ≤ λqd(θ, Tθ). Note that 0 < λp := λ 1 1−γ < 1 and 0 < λq := λ 1 α+β < 1. Consequently, Theorems P and H(γ1) can be applied to such two examples on T - orbitally complete quasi-metric spaces. Karapinar [19] in 2019 The author collect and combine several non-unique fixed point results in the context of several distinct abstract spaces. The main goal is to give a brief background on the topic as well as the to underline the importance of the non-unique fixed points. By using the auxiliary functions, some of the given results are reformulated in a more general form to cover the existing results on the topic in the literature. Given nonunique fixed point theorems are due to Ćirić (1974), Achari (1976), Pachpatte (1979), Ćirić-Jotić (1998), and Karapinar [19] in 2019. For example, Theorem 6.17. (Karapinar) Let T : X → X be an orbitally continuous self-map on the T -orbitally complete metric space (X, d). Suppose there exist real numbers a1, a2, a3, a4, a5 and a self mapping T : X → X which satisfies E(x, y) ≤ a4d(x, y) + a5d(x, T 2x), where E(x, y) := a1d(Tx, Ty) + a2[d(x, Tx) + d(y, Ty)] + a3[d(y, Tx) + d(x, Ty)], S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 12 of 21 for all x, y ∈ X. Then, T has at least one fixed point. Comment: Here {ai}5i=1 are chosen to hold the RHR condition d(Tx, T 2x) ≤ r d(x, Tx) with r ∈ [0, 1) for all x ∈ X. Note that Theorem 6.17 holds for quasi-metric spaces, and Theorems P and H(γ1) can be applied. For some related results, see [20], [21]. Miñana and Valero [27] in 2019 Many G-metric fixed point results can be retrieved from classical ones given in the (quasi-)metric framework. Indeed, many G-contractive conditions can be reduced to a quasi-metric counterpart assumed in the statement of celebrated fixed point results. In this paper, we show that the existence of fixed points for the most part in the aforesaid G-metric fixed point results is guaranteed by a very general celebrated result by Park, even when the G-contractive condition is reduced to a quasi-metric one which is not considered as a contractive condition in any celebrated fixed point result. Moreover, in all those cases in which a quasi-metric contractivity can be raised, we show that the uniqueness of the fixed point is also derived from it. · · · We are able to show that most fixed point results obtained in G-metric spaces can be deduced from a fixed point result stated in quasi-metric spaces obtained by Park in [29]. To this end, let us recall such a result. Theorem 6.18. (Miñana-Valero) Let (X, τ) be a topological space, let d : X×X → [0, !‘Ä[ be a continuous map, such that d(x, y) = 0 ⇐⇒ x = y, and let f : X → X be a map. Suppose that there exist x, x0 ∈ X, such that the following conditions hold: 1. limn→∞ d(fn(x0), f n+1(x0)) = 0; 2. (fn(x0))n∈N converges to x with respect to τ ; 3. f is orbitally continuous at x with respect to τ . Then x ∈ Fix(f) = {y ∈ X : f(y) = y}. It must be stressed that Park’s original version of the preceding result was stated for lower semicontinuous mappings d. However, we have focused our attention on continuous ones, because it is enough for our announced purpose. Corollary 6.19. (Miñana-Valero) Let (X,G) be a G-metric space and let f : X → X be a mapping. Suppose that there exist x, x0 ∈ X, such that the following conditions hold: 1. limn→∞ dG(f n(x0), f n+1(x0)) = 0; 2. (fn(x0))n∈N converges to x with respect to τdG; 3. f is orbitally continuous at x with respect to τdG. Then x = f(x). Moreover, the authors have shown that the existence of fixed points in many of the aforesaid G-metric fixed point results is a consequence of Park’s celebrated result, even when the G-contractive condition is reduced to a quasi-metric one that is not considered as a contractive condition in the statement of any known fixed point result. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 13 of 21 Aouine and Aliouche [3] in 2021 Abstract: We prove unique fixed point theorems for a self-mapping in complete metric spaces and that the fixed point problem is well-posed. Examples are provided to illustrate the validity of our results and we give some remarks about three papers. · · · Afterwards, we apply our result to study the possibility of optimally controlling the solution of an ordinary di¡×erential equation via dynamic programming. Definition 6.20. (Reich-Zaslavski) Let (X, d) be a metric space and T : X → X a mapping. The fixed point problem of T is said to be well-posed if i) T has a unique fixed point z in X, ii) for any sequence {yn} inX such that limn→∞ d(Tyn, yn) = 0, we have limn→∞ d(yn, z) = 0. Theorem 6.21. (Aouine-Aliouche) Let (X, d) be a complete metric space and T a mapping from X into itself satisfying the following condition d(Tx, Ty) ≤ d(x, Ty) + d(y, Tx) d(x, Tx) + d(y, Ty) + 1 max{d(x, Tx), d(y, Ty)} for all x, y ∈ X. Then a) T has a unique fixed point z ∈ X, b) The fixed point problem of T is well-posed, and c) T is continuous at z. Comment: For y = Tx, d(x, Ty) + d(y, Tx) d(x, Tx) + d(y, Ty) + 1 < 1 and hence max{d(x, Tx), d(y, Ty)} = d(x, Tx). Therefore T is an RHR map and applicable Theorems P and H(γ1). Aouine [2] in 2022 In [2], Aouine prove a fixed point theorem for p-contraction mappings in partially ordered metric spaces. As an application, Aouine investigate the possibility of optimally controlling the solution of the ordinary differential equations. Definition 6.22. Let (X, d) be a metric space. A mapping T : Y ⊂ X → X is said to be a metric p-contraction (or simply p-contraction) mapping if Y is T -invariant and it satisfies the following inequality: d(T (x), T 2(x)) ≤ p(x)d(x, T (x)) ∀ x ∈ Y, where p : Y → [0, 1] is a function such that p(x) < 1 for all x ∈ Y and supx∈Y p(Tx) = á < 1. Further, if ⋂∞ n=0 T n(Y ) is a singleton set, where Tn(Y ) = T (Tn−1(Y )) for each n ∈ N and T 0(Y ) = Y , then T is said to be a strong p-contraction. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 14 of 21 Comment: Note that T is an RHR map. Romaguera [47] in 2022 The author stated: The above theorem suggests the following natural question (see Section 2 for notation and concepts). Question. Let F be a self map of a bicomplete (or at least, Smyth complete) quasi-metric space (X, d) and let c ∈ (0, 1) be a constant, such that for every x, y ∈ X, the following contraction condition holds: d(x, Fx) ≤ 2d(x, y) =⇒ d(Fx, Fy) ≤ c d(x, y). Under the above assumptions, does F admit a fixed point? In Section 3, the author gives an example showing that this question has a negative answer in the general quasi-metric context. Comment: Note that F is an RHR map. Petrov [42] in 2023 Petrov defined generalized Kannan type maps and obtained: Proposition 6.23. (Petrov) Let (X, d) be a metric space and let T : X → X be a generalized Kannan type metric with some λ ∈ [0, 2/3). If x is an accumulation point of X and T is continuous at x, then the inequality d(Tx, Ty) ≤ λ[d(x, Tx) + d(y, Ty)/2] holds for all points y ∈ X. Theorem 6.24. (Petrov) Let (X, d), |X| > 3, be a complete metric space and let the mapping T : X → X satisfy the following two conditions: (i) T (T (x)) ̸= x for all x ∈ X such that Tx ̸= x. (ii) T is a generalized Kannan-type mapping on X. Then T has a fixed point. The number of fixed points is at most two. Comment: For any point x ∈ X, the inequality in Proposition 6.22 for y = Tx implies (1− λ 2 )d(Tx, T 2x) ≤ λ d(x, Tx) or d(Tx, T 2) ≤ αd(x, Tx) where α = λ 2− λ ∈ [0, 1). Hence, T is an RHR map and has a fixed point by Theorems P and H(γ1). Suppose that there exists at least three pairwise distinct fixed points x, y and z. Then Tx = x, Ty = y and Tz = z, which contradicts to the definition of generalized Kannan-type map. Anyway, the concept of generalized Kannan-type maps can be replaced by RHR maps. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 15 of 21 7. The subfamilies {δ} and {ϵ} of the Covitz-Nadler Type Theorem H((δ)-(ϵ) generalize the celebrated multi-valued versions of the Banach con- tractions due to Nadler and Covitz-Nadler. Moreover, Theorem H gives a unified elemen- tary proof of them. Nadler [28] in 1969 Some fixed point theorems for multi-valued contraction mappings (m.v.c.m.) are proved, as well as a theorem on the behaviour of fixed points as the mappings vary. Theorem 7.1. (Nadler) Let (X, d) be a complete metric space. If F : X → BC(X) is a m.v.c.m., then F has a fixed point. 5. Added in proof. In a forthcoming paper with Professor Covitz on multi-valued con- traction mappings in generalized metric spaces the author has extended Theorems 5 and 6 of this paper to mappings into Cl(X) = {C : C is a nonempty closed subset of X} with the generalized Hausdorff distance. These results give an affirmative answer to problems posed in this remark and show that even boundedness of point images is not necessary. Comment: Nadler’s and Covitz-Nadler’s fixed point theorems are consequences of The- orem H(δ) with simple elementary proofs. In fact, Theorems H(δ)–(ϵ) are new theorems. Fierro and Pizarro [12] in 2023 From Text: In this note, we prove a fixed point existence theorem for set-valued func- tions by extending the usual Banach orbital condition concept for single valued mappings. As we show, this result applies to various types of set-valued contractions existing in the literature. Given a multimap T : X → BC(X), x0 ∈ X, and k ∈ [0, 1), we say T satis- fies the multivalued Banach orbital (MBO) condition at x0 with constant k, whenever for all x ∈ O(x0, T ), infy∈Tx d(y, Ty) ≤ k d(x, Tx), and that, T satisfies the strong multivalued Banach orbital (SMBO) condition at x0 with constant k, whenever for all x ∈ O(x0, T ), supy∈Tx d(y, Ty) ≤ k d(x, Tx). Theorem 7.2. (Fierro-Pizarro) Let T : X → BC(X) be a set-valued mapping satisfying the MBO condition at x0 ∈ X with constant k. Then, there exist x∗ ∈ X and a sequence {xn}n∈N converging to x∗ such that, for all n ∈ N, xn+1 ∈ Txn, and the following two conditions hold: (i) d(xn, Txn) ≤ d(xn, xn+1) ≤ kn d(x0, Tx0) and (ii) d(x∗, Txn) ≤ {kn+1/(1− k)}d(x0, Tx0) for all n ∈ N. Moreover, the following conditions are equivalent: (iii) x∗ ∈ Tx∗, (iv) GT is (x0, T )-orbitally lower semicontinuous at x∗, and (v) the function h : X → R, defined by h(x) = d(x, Tx), is lower semicontinuous at x∗. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 16 of 21 Comment: The authors only claimed the equivalency of (iii)-(v). In the present article, we showed that (iii)-(v) actually hold when X is T -orbitally complete quasi-metric space. 8. Theorems not belonging any of the above families There are relatively small number of theorems which does not belong to any of the families {α} − {ϵ}. We give only a few example of such theorems. Ćirić [7] in 1974 Ćirić proved a celebrated fixed point theorem as follows: Theorem 8.1. (Ćirić) Let T be a self map of a complete metric space (X, d). If there is a constant α ∈ (0, 1) such that d(Tx, Ty) ≤ αmax{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)}, for all x, y ∈ X, then T has a unique point z ∈ X and d(z, Tnx0) → 0 as n → ∞, for all x0 ∈ X. This theorem is not in {β}. Bogin [5] in 1976 Theorem 8.2. (Bogin) Let (X, d) be a complete metric space and F : X → X a mapping satisfying for each x, y ∈ X: d(Fx, Fy) ≤ ad(x, y) + b[d(x, Fx) + d(y, Fy)] + c[d(x, Fy) + d(y, Fx)] where a, b, c > 0 and a+ 2b+ 2c = 1. Then F has a unique fixed point in X. Note that F is not an RHR map. Jaggi [15] in 1977 Consider the following: Theorem 8.3. (Jaggi) Let (X, d) be a complete metric space and T : X → X be a continuous mapping. If there exist k1, k2 ∈ [0, 1), with k1 + k2 < 1 such that d(Tx, Ty) ≤ k1 · d(x, Tx) d(y, Ty) d(x, y) + k2 · d(x, y) for all distinct x, y ∈ X, then T possesses a unique fixed point in X. Comment: For y = Tx, we have the following d(Tx, T 2x) ≤ k1 · d(x, Tx) d(Tx, T 2x) d(x, Tx) + k2 · d(x, Tx), S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 17 of 21 which implies d(Tx, T 2x) ≤ (k1+k2)d(x, Tx) whenever d(x, Tx) ≤ d(Tx, T 2x). This leads a contradiction. Hence we have to assume d(Tx, T 2x) < d(x, Tx) for all x ∈ X. Therefore T can not have a fixed point. Ćirić [8] in 1993 Theorem 8.4. (Ćirić) Let K be a closed convex subset of a complete convex metric space X and T : K → K a mapping satisfying d(Tx, Ty) ≤ a d(x, y) + (1− a)max{d(x, Tx), d(y, Ty), b [d(x, Ty) + d(y, Tx)]} for all x, y ∈ K, where 0 < a < 1 and b ≤ 1 2 − 1−a2 10+6a2 . Then T has a unique fixed point. Note that T is not an RHR map. Feng and Liu [11] in 2006 Let (X, d) be a complete metric space. Cl(X) denotes the collection of all nonempty closed subsets. Let T : X → Cl(X) be a multi-valued mapping. Define a function f : X → R as f(x) = d(x, T (x)). For a positive constant b ∈ (0, 1), define the set Ixb ⊂ X as Ixb = {y ∈ T (x) : b d(x, y) ≤ d(x, T (x))}. The following theorem is the main result: Theorem 8.5. (Feng-Liu) Let (X, d) be a complete metric space, T : X → Cl(X) be a multi-valued mapping. If there exists a constant c ∈ (0, 1) such that for any x ∈ X there is y ∈ Ixb satisfying d(y, T (y)) ≤ c d(x, y), then T has a fixed point in X provided c < b and f is lower semi-continuous. Corollary 8.6. (Feng-Liu) Let (X, d) be a complete metric space, T : X → Cl(X) be a multi-valued mapping. If there exists a constant c ∈ (0, 1) such that for any x ∈ X, y ∈ T (x), d(y, T (y)) ≤ c d(x, y), then T has a fixed point in X provided f is lower semi-continuous. This extends the Covitz-Nadler fixed point theorem [9] or Theorem H(δ1) for metric spaces. Kumam, Dung, Sitytithakerngkiet [26] in 2015 Kumam et al. obtained in [26] the following improvement of Ćirić’s theorem. Theorem 8.7. (Kumam et al.) Let T be a self map of a complete metric space (X, d). If there is a constant α ∈ (0, 1) such that d(Tx, Ty) ≤ αmax{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx), S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 18 of 21 d(x, T 2x), d(Tx, T 2x), d(x, T 2y), d(T 2x, Ty) for all x, y ∈ X, then T has a unique fixed point z ∈ X and d(z, Tnx0) → 0 as n → ∞, for all x0 ∈ X. Some corollaries and multi-valued versions of them are added. Comment: The contractive condition implies the Hegedüs condition d(Tx, Ty) ≤ q diam{OT (x) ∪OT (y)}. Hence the main theorem was already known; see Park [29]. The map T is not in β-class. 9. Conclusion In this paper, we introduced Theorem H based on our previous 2023 Metatheorem. Theorem H claims that the six statements (α)-(ϵ) and (η) are equivalent and that they characterize the metric completeness (0). We classified multi-valued selfmaps on quasi- metric spaces satisfying each of the statements (α)-(ϵ). Such classes of multimaps have extremal elements, fixed points, common fixed points, stationary points, common station- ary points by the Metatheorem. For example, the subfamily {γ} consisting of the Rus-Hicks -Rhoades theorem and other theorems can be easily obtained by the statement (γ1) in the Metatheorem. The numbers of theorems in other subfamilies are relatively small. We add some examples of multimaps not belonging to any of {α}-{ϵ} classes. Usually such type of theorems have relatively long and difficult proofs. Consequently, we can destroy possible theorems in the subfamily {γ} on thousands of artificial metric type spaces. This will save the energy of many researchers. References [1] I. Altun, A. Erduran, A Suzuki type fixed-point theorem, Inter. J. Math. Math. Sci. 2011, Article ID 736063, 9 pages. doi:10.1155/2011/736063 [2] A.C. Aouiney, A fixed point theorem for p-contraction mappings in partially ordered metric spaces and application to ordinary differential equations, Bul. Acad. Stiint E A, Rep. Moldova. Math. 3(100) (2022) 15–21. ISSN 1024-7696, E-ISSN 2587-4322 [3] A.C. Aouiney, A. Aliouche, Fixed point theorems of Kannan type with an application to control theory, Appl. Math. E-Notes 21 (2021) 238–249. ISSN 1607-2510 [4] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922) 133–181. [5] J. Bogin, A generalization of a fixed point theorem of Goebel, Kirk, and Shimi, Canad. Math. Bull. 19(1) (1976) 7–12. [6] Lj.B. Ćirić, On some maps with a non-unique fixed point, Publ. Inst. Math. 17 (1974) 52–58. S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 19 of 21 [7] Lj.B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45 (1974) 267–273. [8] Lj.B. Ćirić, On some discontinuous fixed point mappings in convex metric spaces, Czechoslovak Mathematical Journal 43(2) (1993) 319–326. [9] H. Covitz, S.B. Nadler,, Jr. Multi-valued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970) 5–11. [10] B.K. Dass, S. Gupta, An extension of Banach contraction principle through rational expressions, Indian J. Pure Appl. Math. 6 (1975) 1455–1458. [11] Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi- valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103–112. [12] R. Fierro, S. Pizarro, Fixed points of set-valued mappings satisfying a Banach orbital condition, CUBO, A Math. Jour. 25(1) (2023) 151–159. [13] G.E. Hardy, T.D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16 (1973) 201–206. [14] T.L. Hicks, B.E. Rhoades, A Banach type fixed point theorem, Math. Japon. 24 (1979) 327–330. [15] D.S. Jaggi, Some unique fixed point theorems, Indian J. Pure Appl. Math. 8 (1977) 223–230. [16] M. Jleli, B. Samet, Remarks on G-metric spaces and fixed point theorems, Fixed Point Theory Appl. 2012:210, 2012. [17] R. Kannan, Some results on fixed point II. Amer. Math. Monthly 76 (1969) 405–408. [18] E. Karapinar, Revisiting the Kannan type contractions via interpolation, Adv. Th. Nonlinear Anal. Appl. 2(2) (2018) 85–87. https://doi.org/10.31197/atnaa.431135 [19] E. Karapinar, Ćirić type nonunique fixed points results: A review, Appl. Comput. Math. 18(1) (2019) 3–21. [20] E. Karapinar, Interpolative Kannan-Meir-Keeler type contraction, Adv. Theory Non- linear Anal. Appl. 2021(4) (2021) 611–614. [21] E. Karapinar, Revisiting Ćirić-type nonunique fixed point theorem via interpolation, Applied General Topology 22(2) (2021) 483–496. [22] E. Karapinar, R. Agarwal, Interpolative Rus-Reich-Ćirić type ¢¥ contractions via simulation functions, An. Şt. Univ. Ovidius Constanţa 27(3) (2019) 137–152. DOI: 10.2478/auom-2019-0038 [23] E. Karapinar, R.P. Agarwal, H. Aydi, Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces, Mathematics 2018, 6(11), 256. [24] E. Karapinar, O. Alqahtani, H. Aydi, On interpolative Hardy-Rogers type contrac- tions, Symmetry 2018, 11, 8; doi:10.3390/sym11010008 [25] F. Khojasteh, M. Abbas, S. Costache, Two new types of fixed point theorems in complete metric spaces, Abstr. Appl. Anal., 2014, Art. ID 325840, 5pp. [26] P. Kumam, N.V. Dung, K. Sitytithakerngkiet, A generalization of Ćirić fixed point theorems, Filomat 29 (2015) 1549–1556. [27] J.-J. Miñana. O. Valero, Are fixed point theorems in G-metric spaces an authentic generalization of their classical counterparts? J. Fixed Point Theory Appl. (2019) 21:70 https://doi.org/10.1007/s11784-019-0705-z S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 20 of 21 [28] S.B. Nadler, Jr. Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475- 488. [29] S. Park, A unified approach to fixed points of contractive maps. J. Korean Math. Soc. 16 (1980) 95–105. [30] S. Park, Foundations of ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 61(2) (2022) 247–287. [31] S. Park, Remarks on the metatheorem in ordered fixed point theory, Advanced Math- ematical Analysis and Its Applications (Edited by P. Debnath, D. F. M. Torres, Y. J. Cho) CRC Press (2023) 11–27. DOI: 10.1201/9781003388678-2 [32] S. Park, History of the metatheorem in ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 62(2) (2023) 373–410. [33] S. Park, Relatives of a theorem of Rus-Hicks-Rhoades, Letters Nonlinear Anal. Appl. 1 (2023) 57–63. ISSN 2985-874X [34] S. Park, Some metric fixed point theorems hold for quasi- metric spaces, Results in Nonlinear Analysis 6(4) (2023) 116–127. https://doi.org/10.31838/rna/2023.06.04.012. A revised and corrected version appears in Research Gate. [35] S. Park, Almost all about Rus-Hicks-Rhoades maps in quasi-metric spaces, Adv. Th. Nonlinear Anal. Appl. 7(2) (2023) 455–471. DOI 0.31197/atnaa.1185449 [36] S. Park, Comments on the Suzuki type fixed point theorems, Adv. Theory Nonlinear Anal. Appl. 7(3) (2023) 67–78. [37] S. Park, The use of quasi-metric in the metric fixed point theory, J. Nonlinear Convex Anal. 25(7) (2024) 1553–1564. [38] S. Park, New applications of the metatheorem in ordered fixed point theory, J. Non- linear Analysis Optimization 15(2) (2024) [39] S. Park, The realm of the Rus-Hicks-Rhoades maps in the metric fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 63(1) (2024) 1–50. [40] S. Park, On the family of theorems on metric completeness, European Journal of Pure and Applied Mathematics, 17(4) (2024) 2370–2383. [41] S. Park, B.E. Rhoades, Some fixed point theorems, Sem. in Math. Lecture Note Series, Yonsei Univ. 2 (1981) 113–125. [42] E. Petrov, Fixed point theorem for generalized Kannan type mappings, arXiv:2308.05419v1 [math.GN] 10Aug2023. [43] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Am. Math Soc. 132 (2004) 1435–1443. [44] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1971) 121–124. [45] B.E. Rhoades, A comparison of various definitions of contrctive definitions, Trans. Amer. Math. Soc. 226 (1977) 257–290. [46] B.E. Rhoades, A collection of contractive definitions, Math. Sem. Notes 6 (1978) 229–235. [47] S. Romaguera, Basic contractions of Suzuki-type on quasi-metric spaces and fixed point results, Mathematics 2022, 10, 3931. https://doi.org/10.3390/math10213931 S. Park / Eur. J. Pure Appl. Math, 18 (1) (2025), 5730 21 of 21 [48] I.A. Rus, Teoria punctului fix, II, Univ. Babes-Bolyai, Cluj, 1973. [49] T. Suzuki, A generalized Banach contraction principle that characterizes metric com- pleteness. Proc. Amer. Math. Soc. 136 (2008) 1861–1869.