EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6086 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Fixed Point Theorems for Θ− ϕ-Multivalued Contraction Mappings in Rectangular b-Metric Spaces Hafida Massit1, Mohamed Rossafi2, Zoran D. Mitrović3,∗, Ahmad Aloqaily4, Nabil Mlaiki4 1 Laboratory Analysis, Geometry and Applications, University of Ibn Tofail, Kenitra, Morocco 2 Laboratory Analysis, Geometry and Applications, Higher School of Education and Training, University of Ibn Tofail, Kenitra, Morocco 3 Faculty of Electrical Engineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina 4 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia Abstract. In this paper, we give some fixed point theorems for θ−ϕ−multivalued contractions in α−complete rectangular b−metric spaces. We establish some fixed point theorems including the α−admissible θ−ϕ−multivalued Kannan type and Reich type. Our results improve and generalize some results from the literature. 2020 Mathematics Subject Classifications: 41A58, 42C15, 46L05 Key Words and Phrases: Admissible mapping, fixed point, αcomplete spaces, Θ−ϕ−multivalued contraction, rectangular b−metric spaces 1. Introduction and preliminaries Many generalizations of the concept of metric spaces have been defined and some fixed theorems were proven in these spaces [1–13]. Particularly, b−metric spaces were intro- duced by Bakhtin [4] and Branciari [5] introduced generalized metric spaces. Recently, George et al [7] announced the concept of rectangular b−metric spaces. In 2017, Zheng et al [14] established some fixed point results for θ − ϕ−contractions in complete metric spaces. Nadler [15] extented the contraction principle to multivalued mappings. In this work, we introduce a notion of θ − ϕ−multivalued contraction mappings in rect- angular b−metric spaces. We obtain some fixed point theorems for θ − ϕ−multivalued ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6086 Email addresses: massithafida@yahoo.fr (H. Massit), mohamed.rossafi1@uit.ac.ma ( M. Rossafi), zoran.mitrovic@etf.unibl.org (Z. D. Mitrović), maloqaily@psu.edu.sa (A. Aloqaily), nmlaiki@psu.edu.sa (N. Mlaiki) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 2 of 19 contractions in α−complete rectangular b−metric spaces. We establish some fixed point theorems including the α−admissible θ−ϕ−multivalued Kannan type [16] and Reich type [17] in rectangular b−metric spaces. Our results improve and generalize some results from the literature. We believe that our paper may be interesting to researchers in fixed point theory, because using the methods presented in this paper, at the end we give several open problems. Definition 1. [13] Let U be a non-empty set and b ≥ 1. Suppose that the mapping d : U × U → [0,+∞) satisfies: (i) d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x) for all x, y ∈ U , (iii) d(x, y) ≤ b[d(x, u) + d(u, v) + d(v, y)] for all x, y ∈ U and for all distinct points u, v ∈ U \ {x, y}. Then (U , d) is called a rectangular b−metric space with coefficient b. Lemma 1. [9] Let (U , d) be a rectangular b−metric space and {xn} be a sequence in U such that lim n→+∞ d(xn, xn+1) = lim n→+∞ d(xn, xn+2) = 0. If {xn} is not a Cauchy sequence then there exist ε > 0 and two sequences {mk} and {nk} of positive integers such that ε ≤ lim k→+∞ inf d(xmk , xnk ) ≤ lim k→+∞ sup d(xmk , xnk ) ≤ bε, ε ≤ lim k→+∞ inf d(xmk , xmk+1 ) ≤ lim k→+∞ sup d(xnk , xmk+1 ) ≤ bε, ε ≤ lim k→+∞ inf d(xmk , xnk+1 ) ≤ lim k→+∞ sup d(xmk , xnk+1 ) ≤ bε, ε b ≤ lim k→+∞ inf d(xmk+1 , xnk+1 ) ≤ lim k→+∞ sup d(xmk+1 , xnk+1 ) ≤ b2ε. Zheng et al. [12] introduced a new type of contractions called θ − ϕ−contractions in metric spaces and proved a new fixed point theorems for such mapping. Definition 2. [6] We denote by Θ the set of functions θ : (0,+∞) → [1,+∞) satisfying the following conditions: 1) θ is increasing, 2) For each sequence {xn} ∈ (0,+∞), lim n→+∞ θ(xn) = 1 if and only if lim n→+∞ xn = 0, 3) θ is continuous on (0,+∞). Definition 3. [13] We denote by Φ the set of functions ϕ : [1,+∞) → [1,+∞) satisfying the following conditions: H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 3 of 19 1) ϕ is nondecreasing, 2) For each λ > 1, lim n→+∞ ϕn(λ) = 1, 3) ϕ is continuous on [1,+∞). Lemma 2. [13] If ϕ ∈ Φ, then ϕ(λ) < λ for all λ ∈ (1,+∞) and ϕ(1) = 1. Definition 4. [18] Let (U , d) be a rectangular b−metric space. Let T : U → U and α : U × U → [0,+∞) be two mappings. A mapping T is said to be α−admissible if α(x, y) ≥ 1 implies α(Tx, Ty) ≥ 1. Definition 5. [19] Let T : U → U and α : U ×U → [0,+∞) be two mappings such that T is α−admissible. T is said to be triangular α−admissible if α(x, y) ≥ 1 and α(y, z) ≥ 1 implies α(x, z) ≥ 1. Definition 6. [11] Let (U , d) be a rectangular b−metric space with b > 1 and T : U → U be a mapping. (1) T is called θ − ϕ−contraction if there are θ ∈ Θ and ϕ ∈ Φ such that d(Tx, Ty) > 0 implies θ[b2d(Tx, Ty)] ≤ ϕ[θ(M(x, y))], (1) where M(x, y) = max{d(x, y), d(x, Tx), d(y, Ty), d(y, Tx)}. (2) T is called θ − ϕ− Kannan-type contraction if there are θ ∈ Θ and ϕ ∈ Φ such that d(Tx, Ty) > 0 implies θ[b2d(Tx, Ty)] ≤ ϕ [ θ ( d(x, Tx) + d(y, Ty) 2 )] . (2) (3) T is called θ − ϕ−Reich-type contraction if there are θ ∈ Θ and ϕ ∈ Φ such that d(Tx, Ty) > 0 implies θ[b2d(Tx, Ty)] ≤ ϕ [ θ ( d(x, y) + d(x, Tx) + d(y, Ty) 3 )] . (3) Kari et al. [11] recently obtained the following result. Theorem 1. [11] Let (U , d) be a complete rectangular b−metric space and T : U → U be a θ − ϕ−contraction. Then, T has a unique fixed point. In 2014, Hussain et al. [8] introduced a notion of α−completeness for metric spaces. H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 4 of 19 Definition 7. [8] Let (U , d) be a rectangular b−metric space and α : U ×U → [0,+∞[ be a mapping. The space U is said to be α−complete, if every Cauchy sequence {xn} in U with α(xn, xn+1) ≥ 1 for all n ∈ N, converges in U . Remark 1. (i) In this paper, using Definition 7, we generalize Theorem 1 in several directions. (ii) We also give a generalized version of Definition 7, which opens up new possibilities for further research. In this section, in the end, we list some concepts regarding the multivalued mapping. Let (U , d) be a rectangular b−metric space, we will denote by CB(U) the set of non-empty bounded closed subsets of U . For M,N ∈ CB(U) and x ∈ U , we define d(x,M) = inf a∈M d(x, a) and d(M,N) = sup a∈M d(a,N). The mapping H : CB(U)× CB(U) → [0,+∞), given by H(M,N) = max{ sup a∈M d(a,N), sup b∈N d(b,M)}, is the Hausdorff distance between M and N in CB(U). We define B(U) the set of non- empty compact subsets of U . A point x is said to be a fixed point of multivalued mapping T : U → CB(U) provided x ∈ T (x). 2. Main result First, we introduce the concept of α−admissible θ − ϕ−multivalued contraction in rectangular b−metric spaces. Definition 8. Let (U , d) be a rectangular b−metric space and T : U → B(U) be a mapping and W (x, y) = min{d(x, Tx), d(x, Ty), d(y, Ty), d(y, Tx)}. (i) T is called an α−admissible θ−multivalued contraction if exist θ ∈ Θ, K ≥ 0 and s ∈ (0, 1) such that H(Tx, Ty) > 0 implies θ[α(x, y)b3H(Tx, Ty)] ≤ θ[M(x, y)]s +KW (x, y), (4) for all x, y ∈ U , where M(x, y) = max{d(x, y), d(x, Tx), d(y, Ty), d(y, Tx)}. (ii) T is called an α−admissible θ−ϕ−multivalued contraction if exist θ ∈ Θ and K ≥ 0 such that H(Tx, Ty) > 0 implies θ[α(x, y)b3H(Tx, Ty)] ≤ ϕ[θ(M(x, y))] +KW (x, y), (5) for all x, y ∈ U , where M(x, y) = max{d(x, y), d(x, Tx), d(x, Ty), d(y, Tx)}. H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 5 of 19 (iii) T is called an α−admissible θ − ϕ−multivalued Kannan-type if there are θ ∈ Θ, ϕ ∈ Φ and K ≥ 0 such that H(Tx, Ty) > 0 implies θ[α(x, y)b3H(Tx, Ty)] ≤ ϕ [ θ ( d(x, Tx) + d(y, Ty) 2 )] +KW (x, y), (6) for all x, y ∈ U . (iv) T is called an α−admissible θ− ϕ−multivalued Reich-type if exist θ ∈ Θ, ϕ ∈ Φ and K ≥ 0 such that H(Tx, Ty) > 0 implies θ[α(x, y)b3H(Tx, Ty)] ≤ ϕ [ θ ( d(x, y) + d(x, Tx) + d(y, Ty) 3 )] +KW (x, y), (7) for all x, y ∈ U . (v) T is called α-continuous multivalued mapping if, for all sequences {xn} with α(xn, xn+1) ≥ 1 for every n ∈ N and limn→+∞ xn = x ∈ U , we have limn→+∞ Txn = Tx so that limn→+∞ d(xn, x) = 0 and α(xn, xn+1) ≥ 1 for every n ∈ N, means that limn→+∞H(Txn, Tx) = 0. Theorem 2. Let (U , d) be a rectangular b−metric space and T : U → B(U) be an α−admissible θ−multivalued contraction satisfying: (i) (U , d) is an α−complete metric space, (ii) α(x0, x1) ≥ 1 for x0 ∈ U and x1 ∈ T (U), (iii) T is triangular α−admissible, (iv) T is an α−continuous multivalued mapping. Then, T has a fixed point. Proof. Let {xn} be a sequence in U such that xn+1 ∈ Txn with α(xn, xn+1) ≥ 1, for all k ∈ N ∪ {0}. By (iv), we have θ[H(Txn−1, Txn)] ≤ θ[b3H(Txn−1, Txn)] ≤ θ[α(xn−1, xn)b 3H(Txn−1, Txn)] ≤ θ[M(xn−1, xn)] s +KW (xn−1, xn), for all n ∈ N, where M(xn−1, xn) = max{d(xn−1, xn), d(xn−1, Txn−1), d(xn, Txn), d(xn, Txn−1)} = max{d(xn−1, xn), d(xn, Txn)} H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 6 of 19 and W (xn−1, xn) = min{d(xn−1, Txn−1), d(xn, Txn), d(Txn−1, xn), d(xn−1, Txn)} = min{d(xn−1, Txn−1), d(xn, Txn), 0, d(xn−1, Txn)} = 0. If M(xn−1, xn) = d(xn, Txn), we have d(xn+1, xn) ≤ H(Txn−1, Txn). Since xn+1 ∈ Txn this implies that d(xn, Txn) ≤ d(xn, xn+1). Now, we obtain θ(d(xn+1, xn)) ≤ θ(H(Txn−1, Txn)) ≤ [θ(M(xn−1, xn))] s +KN(xn−1, xn) ≤ [θ(M(xn−1, xn))] s < θ(d(xn, Txn)) = θ(d(xn, xn+1)), which is a contradiction, so M(xn−1, xn) = d(xn−1, xn) and we have θ(d(xn+1, xn)) ≤ θ(H(Txn−1, Txn)) ≤ [θ(M(xn−1, xn))] s +KN(xn−1, xn) ≤ [θ(M(xn−1, xn))] s = [θ(d(xn−1, xn))] s < θ(d(xn−1, xn)). By the properties of θ we have, d(xn, xn+1) < d(xn−1, xn). This implies that the sequence{d(xn, xn+1)}n is strictly decreasing, this implies that there exists α > 0 such that lim n→+∞ d(xn, xn+1) = α. Suppose that α > 0, we can conclude that d(xn, xn+1) ≥ α, for all n ∈ N. We get θ(d(xn+1, xn)) ≤ [θ(d(xn−1, xn))] s ≤ [θ(d(xn−2, xn−1))] s2 ... ≤ [θ(d(x0, x1))] sn . Using the property of θ, we obtain 1 < θ(α) ≤ [θ(d(x0, x1))] sn . (8) H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 7 of 19 Letting n→ +∞ in (8), we get 1 < θ(α) ≤ 1. This a contradiction. Now, we conclude that lim n→+∞ d(xn, xn+1) = 0. Next, we show that {xn}n is a Cauchy sequence in U , there exists an ε > 0 for which we can find sequences of positive integers {xnk } and {xmk } of {xn} such that, for all positive integers k, nk > mk > k, d(xmk , xnk ) ≥ ε (9) d(xmk , xnk−1 ) < ε (10) we get ε ≤ d(xmk , xnk ) ≤ bd(xmk , xmk+1 ) + bd(xmk+1 , xnk+1 ) + bd(xnk+1 , xnk ), (11) letting k → +∞, we get ε b lim n→+∞ sup d(xmk+1 , xnk+1 ) (12) and lim n→+∞ sup d(xmk , xnk ) ≤ bε. (13) Since α(xmk , xnk ) ≥ 1, we have M(xmk , xnk ) = max{d(xmk , xnk ), d(xmk , Txmk ), d(xmk , Txmk ), d(xnk , Txmk )} ≤ max{d(xmk , xnk ), d(xmk , xmk+1 ), d(xmk , xmk+1 ), d(xnk , xmk+1 )} = max{d(xmk , xnk ), d(xmk , xmk+1 ), d(xnk , xmk+1 )} and W (xmk , xnk ) = min{d(xmk , Txmk ), d(xnk , Txnk ), d(Txmk , xnk ), d(xmk , Txnk )} ≤ min{d(xmk , xmk+1 ), d(xnk , xnk+1 ), d(xmk+1 , xnk ), d(xmk , xnk+1 )}, letting n→ +∞, we obtain lim k→+∞ M(xmk , xnk ) ≤ lim k→+∞ max{d(xmk , xnk ), d(xmk , xmk+1 ), d(xnk , xmk+1 )} ≤ max{bε, 0, b2ε} = b2ε and lim k→+∞ W (xmk , xnk ) ≤ lim k→+∞ min{d(xmk , xnk ), d(xmk , xmk+1 ), d(xnk , xmk+1 )} ≤ min{bε, 0, b2ε} H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 8 of 19 = 0. So, we have θ[d(xmk+1 , xnk+1 )] ≤ θ[b3H(Txmk , Txnk )] ≤ θ[α(xmk , xnk )b3H(Txmk , Txnk )] ≤ [θ(M(xmk , xnk ))]s +KW (xmk , xnk ). Letting k → +∞, we obtain θ(εb) ≤ θ[b3 lim k→+∞ d(xmk+1 , xnk+1 )] ≤ [θ( lim k→+∞ M(xmk , xnk ))]s +K lim k→+∞ W (xmk , xnk ) = [θ( lim k→+∞ M(xmk , xnk ))]s ≤ [θ(bε)]s < θ(bε). This implies that bε < bε, which is a contradiction. Consequently, {xn} is a Cauchy sequence in U , so there exists z ∈ U such that lim n→+∞ d(xn, z) = 0. Since T is α−continuous multivalued mapping, we have lim n→+∞ H(Txn, T z) = 0. We now conclude that it is lim n→+∞ d(xn+1, T z) ≤ lim n→+∞ H(Txn, T z) = 0. Therefore, z ∈ Tz i.e. T has a fixed point. Example 1. Let U = [−1, 1]. Define d : U × U → [0,+∞) by d(x, y) = (x − y)2. Then (U , d) is a rectangular b−metric space with parameter b = 2. Define a mapping T : U → B(U) by Tx = { [0, x4 ], if x, y ∈ [0, 14 ] [x, x2], otherwise α(x, y) =  1, if x, y ∈ [0, 14 ] 0, otherwise H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 9 of 19 and the function θ : [0,+∞) → [1,+∞) by θ(x) = 1+x. Then T is triangular α−admissible and H(Tx, Ty) = 1 4(x− y)2. Case 1 If x, y ∈ [0, 14 ] we have α(x, y) ≥ 1 and θ[α(x, y)b3H(Tx, Ty)] ≤ 1 2 (x− y)2 + 1 ≤ d(x, y) + 1 ≤ θ[M(x, y)] +KW (x, y). Case 2 If x, y ∈ (14 ,+∞) we have α(x, y) = 0 and θ[α(x, y)b3H(Tx, Ty)] = θ(0) ≤ θ((x− y)2) ≤ d(x, y) + 1 ≤ θ[M(x, y)] +KW (x, y). Then, T has a fixed point. Theorem 3. Let (U , d) be a complete rectangular b−metric space and T : U → B(U) be an α−admissible θ−multivalued contraction satisfying: (i) (U , d) is an α−complete metric space, (ii) α(x0, x1) ≥ 1 for x0 ∈ U and x1 ∈ T (U), (iii) T is triangular α−admissible, (iv) there exists a sequence {xn} in U such that α(xn, xn+1) ≥ 1 for all n ∈ N ∪ {0} and lim n→+∞ d(xn, z) = 0, for some z ∈ U , then there exists a subsequence {xn(k)} of {xn} such that α(xn(k), z) ≥ 1, for all k ∈ N ∪ {0}. Then, T has a fixed point. Proof. Let {xn} be a sequence in U such that xn+1 ∈ Txn with α(xn, xn+1) ≥ 1, for all n ∈ N ∪ {0} and xn → z ∈ U . By (iv), we show that z ∈ Tz. Suppose that z ̸∈ Tz, we have lim n→+∞ d(Txn, z) = 0 and 1 b2 d(z, Tz) ≤ lim n→+∞ infH(Txn, T z) ≤ lim n→+∞ supH(Txn, T z) ≤ b2d(z, Tz). H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 10 of 19 So, we have θ[b3H(Txn, T z)] ≤ θ[α(xn, z)b 3H(Txn, T z)] ≤ θ[M(xn, z)] s +KW (xn, z) for all n ∈ N, where M(xn, z) = max{d(xn, z), d(xn, Txn), d(z, Tz), d(z, Txn)} and W (xn, z) = min{d(xn, Txn), d(z, Tz), d(Txn, z), d(xn, T z)}. Letting n→ +∞ we obtain lim n→+∞ supM(xn, z) = lim n→+∞ supmax{d(xn, z), d(xn, Txn), d(z, Tz), d(z, Txn)} ≤ lim n→+∞ supmax{d(xn, z), d(xn, xn+1), d(z, Tz), d(z, xn+1)} ≤ d(z, Tz) and lim n→+∞ supW (xn, z) = lim n→+∞ supmin{d(xn, Txn), d(z, Tz), d(Txn, z), d(xn, T z)} ≤ lim n→+∞ supmin{d(xn, xn+1), d(z, Tz), d(xn+1, z), d(xn, T z)} = 0. Then, θ(bd(z, Tz)) ≤ θ[b3 lim n→+∞ H(Txn, T z)] lim n→+∞ θ[b3H(Txn, T z)] ≤ lim n→+∞ θ[α(xn, z)b 3H(Txn, T z)] ≤ θ[ lim n→+∞ M(xn, z)] s +K lim n→+∞ W (xn, z) ≤ [θ(d(z, Tz))]s < θ(d(z, Tz)). This implies that bd(z, Tz) < d(z, Tz), this a contradiction, so z ∈ Tz. Corollary 1. Let (U , d) be a complete rectangular b−metric space and T : U → B(U) be a mapping. If exist θ ∈ Θ and s ∈ (0, 1) such that H(Tx, Ty) > 0 implies θ[b3H(Tx, Ty)] ≤ [θ(d(x, y))]s for all x, y ∈ U , then, T has a fixed point. H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 11 of 19 Theorem 4. Let (U , d) be a complete rectangular b−metric space and T : U → B(U) be an α−admissible θ − ϕ−multivalued contraction satisfying: (i) (U , d) is an α−complete metric space, (ii) α(x0, x1) ≥ 1 for x0 ∈ U and x1 ∈ T (U), (iii) T is triangular α−admissible. (iv) T is an α−continuous multivalued mapping. Then, T has a fixed point. Proof. Let {xn} be a sequence in U such that xn+1 ∈ Txn with α(xn, xn+1) ≥ 1, By (iv), we have θ[H(Txn−1, Txn)] ≤ θ[b3H(Txn−1, Txn)] ≤ θ[α(xn−1, xn)b 3H(Txn−1, Txn)] ≤ ϕ[θ(M(xn−1, xn))] +KW (xn−1, xn) ∀n ∈ N where M(xn−1, xn) = max{d(xn−1, xn), d(xn−1, Txn−1), d(xn, Txn), d(xn, Txn−1)} = max{d(xn−1, xn), d(xn, Txn)} and W (xn−1, xn) = min{d(xn−1, Txn−1), d(xn, Txn), d(Txn−1, xn), d(xn−1, Txn)} = min{d(xn−1, Txn−1), d(xn, Txn), 0, d(xn−1, Txn)} = 0. If M(xn−1, xn) = d(xn, Txn), we have d(xn+1, xn) ≤ H(Txn−1, Txn). Since xn+1 ∈ Txn we have d(xn, Txn) ≤ d(xn, xn+1). Now, we obtain θ(d(xn+1, xn)) ≤ θ(H(Txn−1, Txn)) ≤ ϕ[θ(M(xn−1, xn))] +KN(xn−1, xn) ≤ ϕ[θ(M(xn−1, xn))] < ϕ[θ(d(xn, Txn))] < θ(d(xn, Txn)), which is a contradiction, so, M(xn−1, xn) = d(xn−1, xn) and θ(d(xn+1, xn)) ≤ ϕ[θ(d(xn−1, xn))] < θ(d(xn−1, xn)). H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 12 of 19 By the properties of θ we have, d(xn, xn+1) < d(xn−1, xn). This implies that the sequence {d(xn, xn+1)}n is strictly decreasing, this implies that there exists α > 0 such that lim n→+∞ d(xn, xn+1) = α. Suppose that α > 0, we can conclude that d(xn, xn+1) ≥ α, for all n ∈ N. We get θ(d(xn+1, xn)) ≤ ϕ[θ(d(xn−1, xn))] ≤ ϕ2[θ(d(xn−2, xn−1))] ... ≤ ϕn[θ(d(x0, x1))]. Using the property of θ, we get 1 < θ(α) ≤ ϕn[θ(d(x0, x1))]. (14) Letting n→ +∞ in (14), we get 1 < θ(α) ≤ 1. This a contradiction, so we obtain lim n→+∞ d(xn, xn+1) = 0. Next, we show that {xn}n is a Cauchy sequence in U , there exists an ε > 0 for which we can find sequences of positive integers {xnk } and {xmk } of {xn} such that, for all positive integers k, nk > mk > k, d(xmk , xnk ) ≥ ε (15) d(xmk , xnk−1 ) < ε (16) we get ε ≤ d(xmk , xnk ) ≤ bd(xmk , xmk+1 ) + bd(xmk+1 , xnk+1 ) + bd(xnk+1 , xnk ) (17) Letting k → +∞, we get ε b lim n→+∞ sup d(xmk+1 , xnk+1 ) (18) and lim n→+∞ sup d(xmk , xnk ) ≤ bε. (19) Since α(xmk , xnk ) ≥ 1, we have M(xmk , xnk ) = max{d(xmk , xnk ), d(xmk , Txmk ), d(xmk , Txmk ), d(xnk , Txmk )} ≤ max{d(xmk , xnk ), d(xmk , xmk+1 ), d(xmk , xmk+1 ), d(xnk , xmk+1 )} = max{d(xmk , xnk ), d(xmk , xmk+1 ), d(xnk , xmk+1 )} H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 13 of 19 and W (xmk , xnk ) = min{d(xmk , Txmk ), d(xnk , Txnk ), d(Txmk , xnk ), d(xmk , Txnk )} ≤ min{d(xmk , xmk+1 ), d(xnk , xnk+1 ), d(xmk+1 , xnk ); d(xmk , xnk+1 )} Letting n→ +∞ we obtain lim k→+∞ M(xmk , xnk ) ≤ lim k→+∞ max{d(xmk , xnk ), d(xmk , xmk+1 ), d(xnk , xmk+1 )} ≤ max{bε, 0, b2ε} = b2ε. Now, we obtain lim k→+∞ W (xmk , xnk ) ≤ lim k→+∞ min{d(xmk , xnk ), d(xmk , xmk+1 ), d(xnk , xmk+1 )} ≤ min{bε, 0, b2ε} = 0. So, we have θ[d(xmk+1 , xnk+1 )] ≤ θ[b3H(Txmk , Txnk )] ≤ θ[α(xmk , xnk )b3H(Txmk , Txnk )] ≤ ϕ[θ(M(xmk , xnk ))] +KW (xmk , xnk ). Letting k → +∞, we obtain θ(εb) ≤ θ[b3 lim k→+∞ d(xmk+1 , xnk+1 )] ≤ ϕ[θ( lim k→+∞ M(xmk , xnk ))] +K lim k→+∞ W (xmk , xnk ) = ϕ[θ( lim k→+∞ M(xmk , xnk ))] ≤ ϕ[θ(bε)]. By Lemma 2 we have θ(bε) ≤ ϕ[θ(bε)] < θ(bε). This implies that bε < bε, which is a contradiction. Consequently, {xn} is a Cauchy sequence in U . Therefore, there exists z ∈ U such that lim n→+∞ d(xn, z) = 0. H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 14 of 19 Since T is α−continuous multivalued mapping, we have lim n→+∞ H(Txn, T z) = 0. Now, we obtain, lim n→+∞ d(xn+1, T z) ≤ lim n→+∞ H(Txn, T z) = 0. Therefore, z ∈ Tz i.e. T has a fixed point. Example 2. Let U = A ∪B, where A = { 0, 12 , 1 3 , 1 4 } and B = [1, 2]. Define d : U × U → [0,+∞) by d ( 0, 1 2 ) = d ( 1 2 , 1 3 ) = 0.16, d ( 0, 1 3 ) = d ( 1 3 , 1 4 ) = 0.04, d ( 0, 1 4 ) d ( 1 2 , 1 4 ) = 0.25, d (x, y) = (x− y)2 , for x, y ∈ [1, 2]. Then (U , d) is a rectangular b−metric space with parameter b = 3. Let T : U → B(U) defined by Tx = { A, if x ∈ A[ 0, x2 ] , if x ∈ B, and α(x, y) = { 1 if x, y ∈ [ 0, 14 ] . 0, otherwise and the functions θ : [0,+∞) → [1,+∞) defined by θ(z) = 1 + z and ϕ : [1,+∞) → [1,+∞) defined by ϕ(z) = 1 + z 2 . Then a mapping T is triangular α−admissible and H(Tx, Ty) = 1 4 (x− y)2. Corollary 2. Let (U , d) be a complete rectangular b−metric space and T : U → B(U) be a mapping. If θ ∈ Θ and ϕ ∈ Φ we have H(Tx, Ty) > 0 implies θ[b3H(Tx, Ty)] ≤ ϕ[θ(M(x, y))], for all x, y ∈ U , where M(x, y) = max{d(x, y), d(x, Tx), d(x, Ty), d(y, Tx)}. Then, T has a fixed point. H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 15 of 19 Theorem 5. Let (U , d) be a complete rectangular b−metric space and T : U → B(U) be an α−admissible θ − ϕ−multivalued contraction satisfying (i) (U , d) is an α−complete metric space (ii) α(x0, x1) ≥ 1 for x0 ∈ U and x1 ∈ T (U) (iii) T is triangular α−admissible. (iv) exist {xn} is a sequence in U such that α(xn, xn+1) ≥ 1 for all n ∈ N ∪ {0} and lim n→+∞ d(xn, z) = 0, for some z ∈ U then there exists a subsequence {xn(k)} of {xn} such that α(xn(k), z) ≥ 1 for all k ∈ N ∪ {0}. Then, T has a fixed point. Proof. Let {xn} be a sequence in U such that xn+1 ∈ Txn with α(xn, xn+1) ≥ 1, for all n ∈ N ∪ {0} and xn → z ∈ U . By (iv), we show that z ∈ Tz. Suppose that z ̸∈ Tz, we have lim n→+∞ d(Txn, z) = 0 and 1 b2 d(z, Tz) ≤ lim n→+∞ infH(Txn, T z) ≤ lim n→+∞ supH(Txn, T z) ≤ b2d(z, Tz). So, we have θ[b3H(Txn, T z)] ≤ θ[α(xn, z)b 3H(Txn, T z)] ≤ ϕθ[M(xn, z)] +KW (xn, z) for all n ∈ N, where M(xn, z) = max{d(xn, z), d(xn, Txn), d(z, Tz), d(z, Txn)} and W (xn, z) = min{d(xn, Txn), d(z, Tz), d(Txn, z), d(xn, T z)}. Letting n→ +∞ we obtain lim n→+∞ supM(xn, z) = lim n→+∞ supmax{d(xn, z), d(xn, Txn), d(z, Tz), d(z, Txn)} ≤ lim n→+∞ supmax{d(xn, z), d(xn, xn+1), d(z, Tz), d(z, xn+1)} H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 16 of 19 ≤ d(z, Tz) and lim n→+∞ supW (xn, z) = lim n→+∞ supmin{d(xn, Txn), d(z, Tz), d(Txn, z), d(xn, T z)} ≤ lim n→+∞ supmin{d(xn, xn+1), d(z, Tz), d(xn+1, z), d(xn, T z)} = 0. Now, we obtain θ(bd(z, Tz)) ≤ θ[b3 lim n→+∞ H(Txn, T z)] ≤ lim n→+∞ θ[b3H(Txn, T z)] ≤ lim n→+∞ θ[α(xn, z)b 3H(Txn, T z)] ≤ ϕ(θ[ lim n→+∞ M(xn, z)]) +K lim n→+∞ W (xn, z) ≤ ϕ[θ(d(z, Tz))] < θ(d(z, Tz)). This implies that bd(z, Tz) < d(z, Tz), this a contradiction, then z ∈ Tz. The following corollaries are immediate results of Theorem 4 and Theorem 5. Corollary 3. Let (U , d) be a complete rectangular b−metric space and T : U → B(U) be an α−admissible θ − ϕ−multivalued contraction Kannan type satisfying: (i) (U , d) is an α−complete metric space, (ii) α(x0, x1) ≥ 1 for x0 ∈ U and x1 ∈ T (U), (iii) T is triangular α−admissible, (iv) T is an α−continuous multivalued mapping or exists a sequence {xn} in U such that α(xn, xn+1) ≥ 1 for all n ∈ N∪{0} and lim n→+∞ d(xn, z) = 0, for some z ∈ U then there exists a subsequence {xn(k)} of {xn} such that α(xn(k), z) ≥ 1 for all k ∈ N ∪ {0}. Then, T has a fixed point. Corollary 4. Let (U , d) be a complete rectangular b−metric space and T : U → B(U) be an α−admissible θ − ϕ−multivalued contraction Reich-type satisfying: (i) (U , d) is an α−complete metric space, H. Massit et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6086 17 of 19 (ii) α(x0, x1) ≥ 1 for x0 ∈ U and x1 ∈ T (U), (iii) T is triangular α−admissible, (iv) there exists {xn} is a sequence in U such that α(xn, xn+1) ≥ 1 for all n ∈ N ∪ {0} and lim n→+∞ d(xn, z) = 0, then there exists a subsequence {xn(k)} of {xn} such that α(xn(k), z) ≥ 1 for all k ∈ N ∪ {0}. Then, T has a fixed point. Conclusion We obtain some fixed point theorems for θ−ϕ−multivalued contractions in α−complete rectangular b−metric spaces. We establish some fixed point theorems including the α−admi- ssible θ − ϕ−multivalued Kannan type and Reich type. Our results improve and gener- alize some results from the literature. We believe that our paper may be interesting to researchers in fixed point theory, because using the methods presented in this paper, the following problems remain open: 1. Prove the Hardy-Rogers result for θ − ϕ-multivalued contractions in α-complete rect- angular b-metric spaces. 2. 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