CUBO, A Mathematical Journal Vol. 24, no. 02, pp. 343–368, August 2022 DOI: 10.56754/0719-0646.2402.0343 Fixed point results of (φ,ψ)-weak contractions in ordered b-metric spaces N. Seshagiri Rao 1, B K. Kalyani 1 1Department of Mathematics, School of Applied Science, Vignan’s Foundation for Science, Technology & Research, Vadlamudi-522213, Andhra Pradesh, India. seshu.namana@gmail.com B kalyani.namana@gmail.com ABSTRACT The purpose of this paper is to prove some results on fixed point, coincidence point, coupled coincidence point and cou- pled common fixed point for the mappings satisfying gen- eralized (φ,ψ)-contraction conditions in complete partially ordered b-metric spaces. Our results generalize, extend and unify most of the fundamental metrical fixed point theorems in the existing literature. A few examples are illustrated to support our findings. RESUMEN El propósito de este artículo es demostrar algunos resulta- dos sobre puntos fijos, puntos de coincidencia, puntos de co- incidencia acoplados y puntos de coincidencia acoplados co- munes para aplicaciones que satisfacen condiciones de (φ,ψ)- contracción generalizadas en b-espacios métricos completos parcialmente ordenados. Nuestros resultados generalizan, extienden y unifican la mayoría de los teoremas de punto fijo métricos fundamentales en la literatura existente. Se ilustran algunos ejemplos para apoyar nuestros resultados. Keywords and Phrases: Fixed point, coupled coincidence point, coupled common fixed point, partially ordered b-metric space, compatible, mixed f-monotone. 2020 AMS Mathematics Subject Classification: 47H10, 54H25. Accepted: 03 August, 2022 Received: 02 June, 2021 c©2022 N. Seshagiri Rao et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.56754/0719-0646.2402.0343 https://orcid.org/0000-0003-2409-6513 https://orcid.org/0000-0002-4531-5976 mailto:seshu.namana@gmail.com mailto:kalyani.namana@gmail.com 344 N. Seshagiri Rao & K. Kalyani CUBO 24, 2 (2022) 1 Introduction The usual metric space has been generalized and enhanced in many different directions, one of such generalizations is a b-metric space which was first coined by Czerwik in [16] and is also known as metric type space (Khamsi and Hussain [35] used recently the term “metric type space”)‘. Indeed, in some papers it is considered that this concept has been introduced by Bourbaki [14] in 1974, or that it has been introduced by Bakhtin [12] in 1989, or by Czerwik [16] in 1993 or even by Czerwik [17] in 1998. After extensive searches in zbMATH and Mathematical Reviews, it appears that the first fixed point theorem in a quasimetric space (b-metric spaces) has been established in 1981 by Vulpe et al. [55], who transposed the Picard-Banach contraction mapping principle from metric spaces to the framework of a quasimetric space. Some important information on the introduction of a b-metric spaces can be found from the article “The early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects” by Berinde and Pacurar [13]. Later, a series of papers have been dedicated to the improvement of fixed point results for single valued and multi-valued operators on b-metric spaces by following various topological properties, some of such are from [1, 3, 6, 5, 9, 20, 22, 28, 29, 30, 32, 34, 36, 39, 40, 41, 43, 53]. The concept of coupled fixed points for certain mappings in ordered spaces was first introduced by Bhaskar et al. [23] and applied their results to study the existence and uniqueness of the solutions for boundary valued problems. While the concept of coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings with monotone property in complete partially ordered metric spaces was first introduced by Lakshmikantham et al. [37]. Since then, several authors have carried out further generalizations and improvements in various spaces (see [10, 18, 21, 24, 44, 48]). Aghajani et al. [2] proved some coupled coincidence and coupled fixed point results for mappings satisfying generalized (ψ,φ,θ)-contractive conditions in partially ordered complete b-metric spaces. Later, the results of [2] have been improved and generalized by Huaping Huang et al. [27] in the same space. More works on coupled coincidence and coupled fixed point results for generalized contraction mappings in ordered spaces can be seen from [4, 7, 8, 11, 15, 19, 25, 26, 31, 38, 42, 45, 46, 47, 49, 50, 51, 52]. Recently, some results on fixed point, coincidence point and coupled coincidence points for the mappings satisfying generalized weak contraction contractions in partially ordered b-metric spaces have been discussed by Belay Mituku et al. [39], Seshagiri Rao et al. [53, 54] and Kalyani et al. [33]. The aim of this work is to provide some results on fixed point and coincidence point, coupled coincidence point for the mappings satisfying generalized (φ,ψ)-contractive conditions in an ordered b-metric space. Our results are the variations and the generalizations of the results of [25, 26, 31, 38, 42, 45, 52] and several comparable results in the existing literature. A few numerical examples are illustrated to support the findings. CUBO 24, 2 (2022) Fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 345 2 Mathematical Preliminaries The following definitions and results will be needed in what follows. Definition 2.1 ([39, 53]). A mapping d : P ×P → [0,+∞), where P is a non-empty set is said to be a b-metric, if it satisfies the properties given below for any υ,ξ,µ ∈ P and for some real number s ≥ 1, (a) d(υ,ξ) = 0 if and only if υ = ξ, (b) d(υ,ξ) = d(ξ,υ), (c) d(υ,ξ) ≤ s(d(υ,µ) + d(µ,ξ)). Then (P,d,s) is known as a b-metric space. If (P,�) is still a partially ordered set, then (P,d,s,�) is called a partially ordered b-metric space. Definition 2.2 ([39, 53]). Let (P,d,s) be a b-metric space. Then (1) a sequence {υn} is said to converge to υ, if lim n→+∞ d(υn,υ) = 0 and written as lim n→+∞ υn = υ. (2) {υn} is said to be a Cauchy sequence in P, if lim n,m→+∞ d(υn,υm) = 0. (3) (P,d) is said to be complete, if every Cauchy sequence in it is convergent. Definition 2.3. If the metric d is complete then (P,d,s,�) is called complete partially ordered b-metric space. Definition 2.4 ([39]). Let (P,�) be a partially ordered set and let f,g : P → P be two mappings. Then (1) g is called monotone non-decreasing, if gυ � gξ for all υ,ξ ∈ P with υ � ξ. (2) an element υ ∈ P is called a coincidence (common fixed) point of f and g, if fυ = gυ (fυ = gυ = υ). (3) f and g are called commuting, if fgυ = gfυ, for all υ ∈ P. (4) f and g are called compatible, if any sequence {υn} with lim n→+∞ fυn = lim n→+∞ gυn = µ, for µ ∈ P then lim n→+∞ d(gfυn,fgυn) = 0. (5) a pair of self maps (f,g) is called weakly compatible, if fgυ = gfυ, when gυ = fυ for some υ ∈ P. (6) g is called monotone f-non-decreasing, if fυ � fξ implies gυ � gξ, for any υ,ξ ∈ P. 346 N. Seshagiri Rao & K. Kalyani CUBO 24, 2 (2022) (7) a non empty set P is called well ordered set, if every two elements of it are comparable i.e., υ � ξ or ξ � υ, for υ,ξ ∈ P. Definition 2.5 ([2, 37]). Let (P,�) be a partially ordered set and, let h : P × P → P and f : P → P be two mappings. Then (1) h has the mixed f-monotone property, if h is non-decreasing f-monotone in its first argu- ment and is non-increasing f-monotone in its second argument, that is for any υ,ξ ∈ P υ1,υ2 ∈ P, fυ1 � fυ2 implies h(υ1,ξ) � h(υ2,ξ) and ξ1,ξ2 ∈ P, fξ1 � fξ2 implies h(υ,ξ1) � h(υ,ξ2). Suppose, if f is the identity mapping then h is said to have the mixed monotone property. (2) an element (υ,ξ) ∈ P × P is called a coupled coincidence point of h and f, if h(υ,ξ) = fυ and h(ξ,υ) = fξ. Note that, if f is the identity mapping then (υ,ξ) is said to be a coupled fixed point of h. (3) an element υ ∈ P is called a common fixed point of h and f, if h(υ,υ) = fυ = υ. (4) h and f are commutative, if for all υ,ξ ∈ P, h(fυ,fξ) = f(hυ,hξ). (5) h and f are said to be compatible, if lim n→+∞ d(f(h(υn,ξn)),h(fυn,fξn)) = 0 and lim n→+∞ d(f(h(ξn,υn)),h(fξn,fυn)) = 0, whenever {υn} and {ξn} are any two sequences in P such that lim n→+∞ h(υn,ξn) = lim n→+∞ fυn = υ and lim n→+∞ h(ξn,υn) = lim n→+∞ fξn = ξ, for any υ,ξ ∈ P. We know that a b-metric is not continuous and then we use frequently the following lemma in the proof of our results for the convergence of sequences in b-metric spaces. Lemma 2.6 ([2]). Let (P,d,s,�) be a b-metric space with s > 1 and suppose that {υn} and {ξn} are b-convergent to υ and ξ respectively. Then we have 1 s2 d(υ,ξ) ≤ lim n→+∞ inf d(υn,ξn) ≤ lim n→+∞ supd(υn,ξn) ≤ s 2d(υ,ξ). In particular, if υ = ξ, then lim n→+∞ d(υn,ξn) = 0. Moreover, for each τ ∈ P, we have 1 s d(υ,τ) ≤ lim n→+∞ inf d(υn,τ) ≤ lim n→+∞ supd(υn,τ) ≤ sd(υ,τ). CUBO 24, 2 (2022) Fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 347 3 Main Results The following distance functions are used throughout the paper. A self mapping φ defined on [0,+∞) is said to be an altering distance function, if it satisfies the following conditions: (i) φ is non-decreasing and continuous function, (iii) φ(t) = 0 if and only if t = 0. Let us denote the set of all altering distance functions on [0,+∞) by Φ. Similarly, Ψ denotes the set of all functions ψ : [0,+∞) → [0,+∞) satisfying the following condi- tions: (i) ψ is lower semi-continuous, (ii) ψ(t) = 0 if and only if t = 0. Let (P,d,s,�) be a partially ordered b-metric space with parameter s > 1 and, let g : P → P be a mapping. Set M(υ,ξ) = max { d(ξ,gξ) [1 + d(υ,gυ)] 1 + d(υ,ξ) , d(υ,gυ) d(υ,gξ) 1 + d(υ,gξ) + d(ξ,gυ) ,d(υ,ξ) } , (3.1) and N(υ,ξ) = max { d(ξ,gξ) [1 + d(υ,gυ)] 1 + d(υ,ξ) ,d(υ,ξ) } . (3.2) Let φ ∈ Φ and ψ ∈ Ψ. The mapping g is a generalized (φ,ψ)-contraction mapping if it satisfies the following condition φ(sd(gυ,gξ)) ≤ φ(M(υ,ξ)) − ψ(N(υ,ξ)), (3.3) for any υ,ξ ∈ P with υ � ξ and M,N are same as above. Now, we prove some results for the existence of fixed point, coincidence point, coupled coincidence point and coupled common fixed point of the mappings satisfying a generalized (φ,ψ)-contraction condition in the context of partially ordered b-metric space. We begin with the following fixed point theorem in this paper. Theorem 3.1. Suppose that (P,d,s,�) is a complete partially ordered b-metric space with pa- rameter s > 1. Let g : P → P be a generalized (φ,ψ)-contractive mapping, and be continuous, non-decreasing mapping with respect to �. If there exists υ0 ∈ P with υ0 � gυ0, then g has a fixed point in P. 348 N. Seshagiri Rao & K. Kalyani CUBO 24, 2 (2022) Proof. For some υ0 ∈ P such that gυ0 = υ0, then we have the result. Assume that υ0 ≺ gυ0, then construct a sequence {υn} ⊂ P by υn+1 = gυn, for n ≥ 0. Since g is non-decreasing, then by induction we obtain that υ0 ≺ gυ0 = υ1 � · · · � υn � gυn = υn+1 � · · · . (3.4) If for some n0 ∈ N such that υn0 = υn0+1 then from (3.4), υn0 is a fixed point of g and we have nothing to prove. Suppose that υn 6= υn+1, for all n ≥ 1. Since υn > υn−1 for all n ≥ 1 and then by condition (3.3), we have φ(d(υn,υn+1)) = φ(d(gυn−1,gυn)) ≤ φ(sd(gυn−1,gυn)) ≤ φ(M(υn−1,υn)) − ψ(N(υn−1,υn)). (3.5) From (3.5), we get d(υn,υn+1) = d(gυn−1,gυn) ≤ 1 s M(υn−1,υn), (3.6) where M(υn−1,υn) = max { d(υn,gυn) [1 + d(υn−1,gυn−1)] 1 + d(υn−1,υn) , d(υn−1,gυn−1) d(υn−1,gυn) 1 + d(υn−1,gυn) + d(υn,gυn−1) , 1 11 d(υn−1,υn) } = max { d(υn,υn+1), d(υn−1,υn) d(υn−1,υn+1) 1 + d(υn−1,υn+1) ,d(υn−1,υn) } ≤ max{d(υn,υn+1),d(υn−1,υn)}. (3.7) If max{d(υn,υn+1),d(υn−1,υn)} = d(υn,υn+1) for some n ≥ 1, then from (3.6) follows d(υn,υn+1) ≤ 1 s d(υn,υn+1), (3.8) which is a contradiction. This means that max{d(υn,υn+1),d(υn−1,υn)} = d(υn−1,υn) for n ≥ 1. Hence, we obtain from (3.6) that d(υn,υn+1) ≤ 1 s d(υn−1,υn). (3.9) Since, 1 s ∈ (0,1) then the sequence {υn} is a Cauchy sequence by [1, 6, 41, 22]. But P is complete, then there exists µ ∈ P such that υn → µ. Also, the continuity of g implies that gµ = g( lim n→+∞ υn) = lim n→+∞ gυn = lim n→+∞ υn+1 = µ. (3.10) CUBO 24, 2 (2022) Fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 349 Therefore, µ is a fixed point of g in P . Last result is still valid for g not necessarily continuous, assuming an additional hypothesis on P . Theorem 3.2. In Theorem 3.1 assume that P satisfies, if a non-decreasing sequence {υn} → µ in P, then υn � µ for all n ∈ N, i.e., µ = supυn. Then a non-decreasing mapping g has a fixed point in P. Proof. From Theorem 3.1, we take the same sequence {υn} in P such that υ0 � υ1 � · · · � υn � υn+1 � · · · , that is, {υn} is non-decreasing and converges to some µ ∈ P . Thus from the hypotheses, we have υn � µ, for any n ∈ N, implies that µ = supυn. Next, we prove that µ is a fixed point of g in P , that is gµ = µ. Suppose that gµ 6= µ. Let M(υn,µ) = max { d(µ,gµ) [1 + d(υn,gυn)] 1 + d(υn,µ) , d(υn,gυn) d(υn,gµ) 1 + d(υn,gµ) + d(µ,gυn) ,d(υn,µ) } , (3.11) and N(υn,µ) = max { d(µ,gµ) [1 + d(υn,gυn)] 1 + d(υn,µ) ,d(υn,µ) } . (3.12) Letting n → +∞ and from the fact that lim n→+∞ υn = µ, we get lim n→+∞ M(υn,µ) = max{d(µ,gµ),0,0} = d(µ,gµ), (3.13) and lim n→+∞ N(υn,µ) = max{d(µ,gµ),0} = d(µ,gµ). (3.14) We know that υn � µ for all n, then from contraction condition (3.3), we get φ(d(υn+1,gµ)) = φ(d(gυn,gµ) ≤ φ(sd(gυn,gµ) ≤ φ(M(υn,µ)) − ψ(N(υn,µ)). (3.15) Letting n → +∞ and use of (3.13) and (3.14), we get φ(d(µ,gµ)) ≤ φ(d(µ,gµ)) − ψ(d(µ,gµ)) < φ(d(µ,gµ)), (3.16) which is a contradiction under (3.16). Thus, gµ = µ, that is g has a fixed point µ in P . Now we give a sufficient condition for the uniqueness of the fixed point that exists in Theorem 3.1 and Theorem 3.2. every pair of elements has a lower bound or an upper bound. (3.17) 350 N. Seshagiri Rao & K. Kalyani CUBO 24, 2 (2022) This condition is equivalent to, for every υ,ξ ∈ P, there exists w ∈ P which is comparable to υ and ξ. Theorem 3.3. In addition to the hypotheses of Theorem 3.1 (or Theorem 3.2), condition (3.17) provides the uniqueness of a fixed point of g in P. Proof. From Theorem 3.1 (or Theorem 3.2), we conclude that g has a nonempty set of fixed points. Suppose that υ∗ and ξ∗ be two fixed points of g then, we claim that υ∗ = ξ∗. Suppose that υ∗ 6= ξ∗, then from the hypotheses we have φ(d(gυ∗,gξ∗)) ≤ φ(sd(gυ∗,gξ∗)) ≤ φ(M(υ∗,ξ∗)) − ψ(N(υ∗,ξ∗)). (3.18) Consequently, we get d(υ∗,ξ∗) = d(gυ∗,gξ∗) ≤ 1 s M(υ∗,ξ∗), (3.19) where M(υ∗,ξ∗) = max { d(ξ∗,gξ∗) [1 + d(υ∗,gυ∗)] 1 + d(υ∗,ξ∗) , d(υ∗,gυ∗) d(υ∗,gξ∗) 1 + d(υ∗,gξ∗) + d(ξ∗,gυ∗) ,d(gυ∗,gξ∗) } = max { d(ξ∗,ξ∗) [1 + d(υ∗,υ∗)] 1 + d(υ∗,ξ∗) , d(υ∗,υ∗) d(υ∗,ξ∗) 1 + d(υ∗,ξ∗) + d(ξ∗,υ∗) ,d(υ∗,ξ∗) } = max{0,0,d(υ∗,ξ∗)} = d(υ∗,ξ∗). (3.20) From (3.19), we obtain that d(υ∗,ξ∗) ≤ 1 s d(υ∗,ξ∗) < d(υ∗,ξ∗), (3.21) which is a contradiction. Hence, υ∗ = ξ∗. This completes the proof. Let (P,d,s,�) be a partially ordered b-metric space with parameter s > 1, and let g,f : P → P be two mappings. Set Mf(υ,ξ) = max { d(fξ,gξ) [1 + d(fυ,gυ)] 1 + d(fυ,fξ) , d(fυ,gυ) d(fυ,gξ) 1 + d(fυ,gξ) + d(fξ,gυ) ,d(fυ,fξ) } , (3.22) and Nf(υ,ξ) = max { d(fξ,gξ) [1 + d(fυ,gυ)] 1 + d(fυ,fξ) ,d(fυ,fξ) } . (3.23) CUBO 24, 2 (2022) Fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 351 Now, we introduce the following definition. Definition 3.4. Let (P,d,s,�) be a partially ordered b-metric space with s > 1. The mapping g : P → P is called a generalized (φ,ψ)-contraction mapping with respect to f : P → P for some φ ∈ Φ and ψ ∈ Ψ, if φ(sd(gυ,gξ)) ≤ φ(Mf(υ,ξ)) − ψ(Nf(υ,ξ)), (3.24) for any υ,ξ ∈ P with fυ � fξ, where Mf(υ,ξ) and Nf(υ,ξ) are given by (3.22) and (3.23) respectively. Theorem 3.5. Suppose that (P,d,s,�) is a complete partially ordered b-metric space with s > 1. Let g : P → P be a generalized (φ,ψ)-contractive mapping with respect to f : P → P and, g and f are continuous such that g is a monotone f-non-decreasing mapping, compatible with f and gP ⊆ fP. If for some υ0 ∈ P such that fυ0 � gυ0, then g and f have a coincidence point in P. Proof. By following the proof of Theorem 2.2 in [8], we construct two sequences {υn} and {ξn} in P such that ξn = gυn = fυn+1 for all n ≥ 0, (3.25) for which fυ0 � fυ1 � · · · � fυn � fυn+1 � · · · . (3.26) Again from [8], we have to show that d(ξn,ξn+1) ≤ λd(ξn−1,ξn), (3.27) for all n ≥ 1 and where λ ∈ [0, 1 s ). Now from (3.24) and using (3.25) and (3.26), we get φ(sd(ξn,ξn+1)) = φ(sd(gυn,gυn+1)) ≤ φ(Mf(υn,υn+1)) − ψ(Nf(υn,υn+1)), (3.28) where Mf(υn,υn+1) = max { d(fυn+1,gυn+1) [1 + d(fυn,gυn)] 1 + d(fυn,fυn+1) , d(fυn,gυn) d(fυn,gυn+1) 1 + d(fυn,gυn+1) + d(fυn+1,gυn) , 1 11 d(fυn,fυn+1) } = max { d(ξn,ξn+1) [1 + d(ξn−1,ξn)] 1 + d(ξn−1,ξn) , d(ξn−1,ξn) d(ξn−1,ξn+1) 1 + d(ξn−1,ξn+1) + d(ξn,ξn) ,d(ξn−1,ξn) } = max{d(ξn−1,ξn),d(ξn,ξn+1)} 352 N. Seshagiri Rao & K. Kalyani CUBO 24, 2 (2022) and Nf(υn,υn+1) = max { d(fυn+1,gυn+1) [1 + d(fυn,gυn)] 1 + d(fυn,fυn+1) ,d(fυn,fυn+1) } = max { d(ξn,ξn+1) [1 + d(ξn−1,ξn)] 1 + d(ξn−1,ξn) ,d(ξn−1,ξn) } = max{d(ξn−1,ξn),d(ξn,ξn+1)}. Therefore from equation (3.28), we get φ(sd(ξn,ξn+1)) ≤ φ(max{d(ξn−1,ξn),d(ξn,ξn+1)}) − ψ(max{d(ξn−1,ξn),d(ξn,ξn+1)}). (3.29) If 0 < d(ξn−1,ξn) ≤ d(ξn,ξn+1) for some n ∈ N, then from (3.29) we get φ(sd(ξn,ξn+1)) ≤ φ(d(ξn,ξn+1)) − ψ(d(ξn,ξn+1)) < φ(d(ξn,ξn+1)), (3.30) or equivalently sd(ξn,ξn+1) ≤ d(ξn,ξn+1). (3.31) This is a contradiction. Hence from (3.29) we obtain that sd(ξn,ξn+1) ≤ d(ξn−1,ξn). (3.32) Thus equation (3.27) holds, where λ ∈ [0, 1 s ). Therefore from (3.27) and Lemma 3.1 of [32], we conclude that {ξn} = {gυn} = {fυn+1} is a Cauchy sequence in P and then converges to some µ ∈ P as P is complete such that lim n→+∞ gυn = lim n→+∞ fυn+1 = µ. Thus by the compatibility of g and f, we obtain that lim n→+∞ d(f(gυn),g(fυn)) = 0, (3.33) and from the continuity of g and f, we have lim n→+∞ f(gυn) = fµ, lim n→+∞ g(fυn) = gµ. (3.34) Further, from the triangular inequality of a b-metric and, from equations (3.33) and (3.34) , we get 1 s d(gµ,fµ) ≤ d(gµ,g(fυn)) + sd(g(fυn),f(gυn)) + sd(f(gυn),fµ). (3.35) Finally, we arrive at d(gv,fv) = 0 as n → +∞ in (3.35). Therefore, v is a coincidence point of g CUBO 24, 2 (2022) Fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 353 and f in P . Relaxing the continuity of the mappings f and g in Theorem 3.5, we obtain the following result. Theorem 3.6. In Theorem 3.5, assume that P satisfies for any non-decreasing sequence {fυn} ⊂ P with lim n→+∞ fυn = fυ in fP, where fP is a closed subset of P implies that fυn � fυ,fυ � f(fυ) for n ∈ N. If there exists υ0 ∈ P such that fυ0 � gυ0, then the weakly compatible mappings g and f have a coincidence point in P. Furthermore, g and f have a common fixed point, if g and f commute at their coincidence points. Proof. The sequence, {ξn} = {gυn} = {fυn+1} is a Cauchy sequence from the proof of Theorem 3.5. Since fP is closed, then there is some µ ∈ P such that lim n→+∞ gυn = lim n→+∞ fυn+1 = fµ. Thus from the hypotheses, we have fυn � fµ for all n ∈ N. Now, we have to prove that µ is a coincidence point of g and f. From equation (3.24), we have φ(sd(gυn,gυ)) ≤ φ(Mf(υn,υ)) − ψ(Nf(υn,υ)), (3.36) where Mf(υn,µ) = max { d(fµ,gµ) [1 + d(fυn,gυn)] 1 + d(fυn,fµ) , d(fυn,gυn) d(fυn,gµ) 1 + d(fυn,gµ) + d(fµ,gυn) ,d(fυn,fµ) } → max{d(fµ,gµ),0,0} = d(fµ,gµ) as n → +∞, and Nf(υn,µ) = max { d(fµ,gµ) [1 + d(fυn,gυn)] 1 + d(fυn,fµ) ,d(fυn,fµ) } → max{d(fµ,gµ),0} = d(fµ,gµ) as n → +∞. Therefore equation (3.36) becomes φ(s lim n→+∞ d(gυn,gυ)) ≤ φ(d(fµ,gµ)) − ψ(d(fµ,gµ)) < φ(d(fµ,gµ)). (3.37) 354 N. Seshagiri Rao & K. Kalyani CUBO 24, 2 (2022) Consequently, we get lim n→+∞ d(gυn,gυ) < 1 s d(fµ,gµ). (3.38) Further by triangular inequality, we have 1 s d(fµ,gµ) ≤ d(fµ,gυn) + d(gυn,gµ), (3.39) then (3.38) and (3.39) lead to contradiction, if fµ 6= gµ. Hence, fµ = gµ. Let fµ = gµ = ρ, that is g and f commute at ρ, then gρ = g(fµ) = f(gµ) = fρ. Since fµ = f(fµ) = fρ, then by equation (3.36) with fµ = gµ and fρ = gρ, we get φ(sd(gµ,gρ)) ≤ φ(Mf(µ,ρ)) − ψ(Nf(µ,ρ)) < φ(d(gµ,gρ)), (3.40) or equivalently, sd(gµ,gρ) ≤ d(gµ,gρ), which is a contradiction, if gµ 6= gρ. Thus, gµ = gρ = ρ. Hence, gµ = fρ = ρ, that is ρ is a common fixed point of g and f. Definition 3.7. Let (P,d,s,�) be a complete partially ordered b-metric space with s > 1, φ ∈ Φ and ψ ∈ Ψ. A mapping h : P × P → P is said to be a generalized (φ,ψ)-contractive mapping with respect to f : P → P such that φ(skd(h(υ,ξ),h(ρ,τ))) ≤ φ(Mf(υ,ξ,ρ,τ)) − ψ(Nf(υ,ξ,ρ,τ)), (3.41) for all υ,ξ,ρ,τ ∈ P with fυ � fρ and fξ � fτ, k > 2 where Mf(υ,ξ,ρ,τ) = max { d(fρ,h(ρ,τ)) [1 + d(fυ,h(υ,ξ))] 1 + d(fυ,fρ) , d(fυ,h(υ,ξ)) d(fυ,h(ρ,τ)) 1 + d(fυ,h(ρ,τ)) + d(fρ,h(υ,ξ)) , 1 11 d(fυ,fρ) } , and Nf(υ,ξ,ρ,τ) = max { d(fρ,h(ρ,τ)) [1 + d(fυ,h(υ,ξ))] 1 + d(fυ,fρ) ,d(fυ,fρ) } . Theorem 3.8. Let (P,d,s,�) be a complete partially ordered b-metric space with s > 1. Suppose that h : P × P → P be a generalized (φ,ψ)- contractive mapping with respect to f : P → P and, h and f are continuous functions such that h has the mixed f-monotone property and commutes with f. Also assume that h(P × P) ⊆ f(P). Then h and f have a coupled coincidence point in P, if there exists (υ0,ξ0) ∈ P × P such that fυ0 � h(υ0,ξ0) and fξ0 � h(ξ0,υ0). CUBO 24, 2 (2022) Fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 355 Proof. From the hypotheses and following the proof of Theorem 2.2 of [8], we construct two sequences {υn} and {ξn} in P such that fυn+1 = h(υn,ξn), fξn+1 = h(ξn,υn), for all n ≥ 0. In particular, {fυn} is non-decreasing and {fξn} is non-increasing sequences in P . Now from (3.41) by replacing υ = υn,ξ = ξn,ρ = υn+1,τ = ξn+1, we get φ(skd(fυn+1,fυn+2)) = φ(s kd(h(υn,ξn),h(υn+1,ξn+1))) ≤ φ(Mf(υn,ξn,υn+1,ξn+1)) − ψ(Nf(υn,ξn,υn+1,ξn+1)), (3.42) where Mf(υn,ξn,υn+1,ξn+1) ≤ max{d(fυn,fυn+1),d(fυn+1,fυn+2)} (3.43) and Nf(υn,ξn,υn+1,ξn+1) = max{d(fυn,fυn+1),d(fυn+1,fυn+2)}. (3.44) Therefore from (3.42), we have φ(skd(fυn+1,fυn+2)) ≤ φ(max{d(fυn,fυn+1),d(fυn+1,fυn+2)}) − ψ(max{d(fυn,fυn+1),d(fυn+1,fυn+2)}). (3.45) Similarly by taking υ = ξn+1,ξ = υn+1,ρ = υn,τ = υn in (3.41), we get φ(skd(fξn+1,fξn+2)) ≤ φ(max{d(fξn,fξn+1),d(fξn+1,fξn+2)}) − ψ(max{d(fξn,fξn+1),d(fξn+1,fξn+2)}). (3.46) From the fact that max{φ(c),φ(d)} = φ{max{c,d}} for all c,d ∈ [0,+∞). Then combining (3.45) and (3.46), we get φ(skδn) ≤ φ(max{d(fυn,fυn+1),d(fυn+1,fυn+2),d(fξn,fξn+1),d(fξn+1,fξn+2)}) − ψ(max{d(fυn,fυn+1),d(fυn+1,fυn+2),d(fξn,fξn+1),d(fξn+1,fξn+2)}), (3.47) where δn = max{d(fυn+1,fυn+2),d(fξn+1,fξn+2)}. (3.48) Let us denote, ∆n = max{d(fυn,fυn+1),d(fυn+1,fυn+2),d(fξn,fξn+1),d(fξn+1,fξn+2)}. (3.49) 356 N. Seshagiri Rao & K. Kalyani CUBO 24, 2 (2022) Hence from equations (3.45)-(3.48), we obtain s k δn ≤ ∆n. (3.50) Next, we prove that δn ≤ λδn−1, (3.51) for all n ≥ 1 and where λ = 1 sk ∈ [0,1). Suppose that if ∆n = δn then from (3.50), we get s kδn ≤ δn which leads to δn = 0 as s > 1 and hence (3.51) holds. If ∆n = max{d(fυn,fυn+1),d(fξn,fξn+1)}, i.e., ∆n = δn−1 then (3.50) follows (3.51). Now from (3.50), we obtain that δn ≤ λ nδ0 and hence, d(fυn+1,fυn+2) ≤ λ n δ0 and d(fξn+1,fξn+2) ≤ λ n δ0. (3.52) Therefore from Lemma 3.1 of [32], the sequences {fυn} and {fξn} are Cauchy sequences in P . Hence, by following the remaining proof of Theorem 2.2 of [2], we can show that h and f have a coincidence point in P . Corollary 3.9. Let (P,d,s,�) be a complete partially ordered b-metric space with s > 1, and h : P × P → P be a continuous mapping such that h has a mixed monotone property. Suppose there exists φ ∈ Φ and ψ ∈ Ψ such that φ(skd(h(υ,ξ),h(ρ,τ))) ≤ φ(Mf(υ,ξ,ρ,τ)) − ψ(Nf(υ,ξ,ρ,τ)), for all υ,ξ,ρ,τ ∈ P with υ � ρ and ξ � τ, k > 2 where Mf(υ,ξ,ρ,τ) = max { d(ρ,h(ρ,τ)) [1 + d(υ,h(υ,ξ))] 1 + d(υ,ρ) , d(υ,h(υ,ξ)) d(υ,h(ρ,τ)) 1 + d(υ,h(ρ,τ)) + d(ρ,h(υ,ξ)) ,d(υ,ρ) } , and Nf(υ,ξ,ρ,τ) = max { d(ρ,h(ρ,τ)) [1 + d(υ,h(υ,ξ))] 1 + d(υ,ρ) ,d(υ,ρ) } . Then h has a coupled fixed point in P, if there exists (υ0,ξ0) ∈ P × P such that υ0 � h(υ0,ξ0) and ξ0 � h(ξ0,υ0). Proof. Set f = IP in Theorem 3.8. Corollary 3.10. Let (P,d,s,�) be a complete partially ordered b-metric space with s > 1, and h : P × P → P be a continuous mapping such that h has a mixed monotone property. Suppose CUBO 24, 2 (2022) Fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 357 there exists ψ ∈ Ψ such that d(h(υ,ξ),h(ρ,τ)) ≤ 1 sk Mf(υ,ξ,ρ,τ) − 1 sk ψ(Nf(υ,ξ,ρ,τ)), for all υ,ξ,ρ,τ ∈ P with υ � ρ and ξ � τ, k > 2 where Mf(υ,ξ,ρ,τ) = max { d(ρ,h(ρ,τ)) [1 + d(υ,h(υ,ξ))] 1 + d(υ,ρ) , d(υ,h(υ,ξ)) d(υ,h(ρ,τ)) 1 + d(υ,h(ρ,τ)) + d(ρ,h(υ,ξ)) ,d(υ,ρ) } , and Nf(υ,ξ,ρ,τ) = max { d(ρ,h(ρ,τ)) [1 + d(υ,h(υ,ξ))] 1 + d(υ,ρ) ,d(υ,ρ) } . If there exists (υ0,ξ0) ∈ P × P such that υ0 � h(υ0,ξ0) and ξ0 � h(ξ0,υ0), then h has a coupled fixed point in P. Theorem 3.11. In addition to Theorem 3.8, if for all (υ,ξ),(r,s) ∈ P × P, there exists (c∗,d∗) ∈ P × P such that (h(c∗,d∗),h(d∗,c∗)) is comparable to (h(υ,ξ),h(ξ,υ)) and to (h(r,s),h(s,r)), then h and f have a unique coupled common fixed point in P × P. Proof. From Theorem 3.8, we know that there exists at least one coupled coincidence point in P for h and f. Assume that (υ,ξ) and (r,s) are two coupled coincidence points of h and f, i.e., h(υ,ξ) = fυ, h(ξ,υ,) = fξ and h(r,s) = fr, h(s,r) = fs. Now, we have to prove that fυ = fr and fξ = fs. From the hypotheses, there exists (c∗,d∗) ∈ P × P such that (h(c∗,d∗),h(d∗,c∗)) is comparable to (h(υ,ξ),h(ξ,υ)) and to (h(r,s),h(s,r)). Suppose that (h(υ,ξ),h(ξ,υ)) ≤ (h(c∗,d∗),h(d∗,c∗)) and (h(r,s),h(s,r)) ≤ (h(c∗,d∗),h(d∗,c∗)). Let c∗0 = c ∗ and d∗0 = d ∗ and then choose (c∗1,d ∗ 1) ∈ P × P as fc∗1 = h(c ∗ 0,d ∗ 0), fd ∗ 1 = h(d ∗ 0,c ∗ 0) (n ≥ 1). By repeating the same procedure above, we can obtain two sequences {fc∗n} and {fd ∗ n} in P such that fc ∗ n+1 = h(c ∗ n,d ∗ n), fd ∗ n+1 = h(d ∗ n,c ∗ n) (n ≥ 0). Similarly, define the sequences {fυn}, {fξn} and {frn}, {fsn} as above in P by setting υ0 = υ, ξ0 = ξ and r0 = r, s0 = s. Further, we have that fυn → h(υ,ξ), fξn → h(ξ,υ), frn → h(r,s), fsn → h(s,r) (n ≥ 1). (3.53) Since, (h(υ,ξ),h(ξ,υ)) = (fυ,fξ) = (fυ1,fξ1) is comparable to (h(c ∗,d∗),h(d∗,c∗)) = (fc∗,fd∗) = 358 N. Seshagiri Rao & K. Kalyani CUBO 24, 2 (2022) (fc∗1,fd ∗ 1) and hence we get (fυ1,fξ1) ≤ (fc ∗ 1,fd ∗ 1). Thus, by induction we obtain that (fυn,fξn) ≤ (fc ∗ n,fd ∗ n) (n ≥ 0). (3.54) Therefore from (3.41), we have φ(d(fυ,fc∗n+1)) ≤ φ(s kd(fυ,fc∗n+1)) = φ(s kd(h(υ,ξ),h(c∗n,d ∗ n))) ≤ φ(Mf(υ,ξ,c ∗ n,d ∗ n)) − ψ(Nf(υ,ξ,c ∗ n,d ∗ n)), (3.55) where Mf(υ,ξ,c ∗ n,d ∗ n) = max { d(fc∗n,h(c ∗ n,d ∗ n)) [1 + d(fυ,h(υ,ξ))] 1 + d(fυ,fc∗n) , d(fυ,h(υ,ξ)) d(fυ,h(c∗n,d ∗ n)) 1 + d(fυ,h(c∗n,d ∗ n)) + d(fc ∗ n,h(υ,ξ)) ,d(fυ,fc∗n) } = max{0,0,d(fυ,fc∗n)} = d(fυ,fc∗n) and Nf(υ,ξ,c ∗ n,d ∗ n) = max { d(fc∗n,h(c ∗ n,d ∗ n)) [1 + d(hυ,h(υ,ξ))] 1 + d(fυ,fc∗n) ,d(fυ,fc∗n) } = d(fυ,fc∗n). Thus from (3.55), φ(d(fυ,fc∗n+1)) ≤ φ(d(fυ,fc ∗ n)) − ψ(d(fυ,fc ∗ n)). (3.56) As by the similar process, we can prove that φ(d(fξ,fd∗n+1)) ≤ φ(d(fξ,fd ∗ n)) − ψ(d(fξ,fd ∗ n)). (3.57) From (3.56) and (3.57), we have φ(max{d(fυ,fc∗n+1),d(fξ,fd ∗ n+1)}) ≤ φ(max{d(fυ,fc ∗ n),d(fξ,fd ∗ n)}) − ψ(max{d(fυ,fc∗n),d(fξ,fd ∗ n)}) < φ(max{d(fυ,fc∗n),d(fξ,fd ∗ n)}). (3.58) Hence by the property of φ, we get max{d(fυ,fc∗n+1),d(fξ,fd ∗ n+1)} < max{d(fυ,fc ∗ n),d(fξ,fd ∗ n)}, which shows that max{d(fυ,fc∗n),d(fξ,fd ∗ n)} is a decreasing sequence and by a result there exists CUBO 24, 2 (2022) Fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 359 γ ≥ 0 such that lim n→+∞ max{d(fυ,fc∗n),d(fξ,fd ∗ n)} = γ. From (3.58) taking upper limit as n → +∞, we get φ(γ) ≤ φ(γ) − ψ(γ), (3.59) from which we get ψ(γ) = 0, implies that γ = 0. Thus, lim n→+∞ max{d(fυ,fc∗n),d(fξ,fd ∗ n)} = 0. Consequently, we get lim n→+∞ d(fυ,fc∗n) = 0 and lim n→+∞ d(fξ,fd∗n) = 0. (3.60) By similar argument, we get lim n→+∞ d(fr,fc∗n) = 0 and lim n→+∞ d(fs,fd∗n) = 0. (3.61) Therefore from (3.60) and (3.61), we get fυ = fr and fξ = fs. Since fυ = h(υ,ξ) and fξ = h(ξ,υ), then by the commutativity of h and f, we have f(fυ) = f(h(υ,ξ)) = h(fυ,fξ) and f(fξ) = f(h(ξ,υ)) = h(fξ,fυ). (3.62) Let fυ = a∗ and fξ = b∗ then (3.62) becomes f(a∗) = h(a∗,b∗) and f(b∗) = h(b∗,a∗), (3.63) which shows that (a∗,b∗) is a coupled coincidence point of h and f. It follows that f(a∗) = fr and f(b∗) = fs that is f(a∗) = a∗ and f(b∗) = b∗. Thus from (3.63), we get a∗ = f(a∗) = h(a∗,b∗) and b∗ = f(b∗) = h(b∗,a∗). Therefore, (a∗,b∗) is a coupled common fixed point of h and f. For the uniqueness, let (u∗,v∗) be another coupled common fixed point of h and f, then we have u∗ = fu∗ = h(u∗,v∗) and v∗ = fv∗ = h(v∗,u∗). Since (u∗,v∗) is a coupled common fixed point of h and f, then we get fu∗ = fυ = a∗ and fv∗ = fξ = b∗. Thus, u∗ = fu∗ = fa∗ = a∗ and v∗ = fv∗ = fb∗ = b∗. Hence the result. Theorem 3.12. In addition to the hypotheses of Theorem 3.11, if fυ0 and fξ0 are comparable, then h and f have a unique common fixed point in P. 360 N. Seshagiri Rao & K. Kalyani CUBO 24, 2 (2022) Proof. From Theorem 3.11, h and f have a unique coupled common fixed point (υ,ξ) ∈ P . Now, it is enough to prove that υ = ξ. From the hypotheses, we have fυ0 and fξ0 are comparable then we assume that fυ0 � fξ0. Hence by induction we get fυn � fξn for all n ≥ 0, where {fυn} and {fξn} are from Theorem 3.8. Now by use of Lemma 2.6, we get φ(sk−2d(υ,ξ)) = φ ( sk 1 s2 d(υ,ξ) ) ≤ lim n→+∞ supφ(skd(υn+1,ξn+1)) = lim n→+∞ supφ(skd(h(υn,ξn),h(ξn,υn))) ≤ lim n→+∞ supφ(Mf(υn,ξn,ξn,υn)) − lim n→+∞ inf ψ(Nf(υn,ξn,ξn,υn)) ≤ φ(d(υ,ξ)) − lim n→+∞ inf ψ(Nf(υn,ξn,ξn,υn)) < φ(d(υ,ξ)), which is a contradiction. Thus, υ = ξ, i.e., h and f have a common fixed point in P . Remark 3.13. It is well known that b-metric space is a metric space when s = 1. So, from the result of Jachymski [31], the condition φ(d(h(υ,ξ),h(ρ,τ))) ≤ φ(max{d(fυ,fρ),d(fξ,fτ)}) − ψ(max{d(fυ,fρ),d(fξ,fτ)}) is equivalent to, d(h(υ,ξ),h(ρ,τ)) ≤ ϕ(max{d(fυ,fρ),d(fξ,fτ)}), where φ ∈ Φ, ψ ∈ Ψ and ϕ : [0,+∞) → [0,+∞) is continuous, ϕ(t) < t for all t > 0 and ϕ(t) = 0 if and only if t = 0. So, in view of above our results generalize and extend the results of [15, 23, 25, 31, 37, 38] and several other comparable results. Corollary 3.14. Suppose (P,d,s,�) be a complete partially ordered b-metric space with parameter s > 1. Let g : P → P be a continuous, non-decreasing mapping with regards to � such that there exists υ0 ∈ P with υ0 � gυ0. Suppose that φ(sd(gυ,gξ)) ≤ φ(M(υ,ξ)) − ψ(M(υ,ξ)), (3.64) where M(υ,ξ) and the conditions upon φ,ψ are same as in Theorem 3.1. Then g has a fixed point in P. Proof. Set N(υ,ξ) = M(υ,ξ) in a contraction condition (3.3) and apply Theorem 3.1, we have the required proof. CUBO 24, 2 (2022) Fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 361 Note 1. Similarly by removing the continuity of a non-decreasing mapping g and taking a non- decreasing sequence {υn} as above in Theorem 3.2, we can obtain a fixed point for g in P. Also one can obtain the uniqueness of a fixed point of g by using condition (3.17) in P as by following the proof of Theorem 3.3. Note 2. By following the proofs of Theorems 3.5 - 3.6, we can find the coincidence point for the mappings g and f in P. Similarly, from Theorem 3.8, Theorem 3.11 and Theorem 3.12, one can obtain a coupled coincidence point and its uniqueness, and a unique common fixed point for the mappings h and f in P × P and on P satisfying an almost generalized contraction condition (3.64), where M(υ,ξ), Mf(υ,ξ), Mf(υ,ξ,ρ,τ) and the conditions upon φ,ψ are same as above defined. Corollary 3.15. Suppose that (P,d,s,�) be a complete partially ordered b-metric space with s > 1. Let g : P → P be a continuous, non-decreasing mapping with regards to �. If there exists k ∈ [0,1) and for any υ,ξ ∈ P with υ � ξ such that d(gυ,gξ) ≤ k s max { d(ξ,gξ) [1 + d(υ,gυ)] 1 + d(υ,ξ) , d(υ,gυ) d(υ,gξ) 1 + d(υ,gξ) + d(ξ,gυ) ,d(υ,ξ) } . (3.65) If there exists υ0 ∈ P with υ0 � gυ0, then g has a fixed point in P. Proof. Set φ(t) = t and ψ(t) = (1 − k)t, for all t ∈ (0,+∞) in Corollary 3.14. Note 3. Relaxing the continuity of a map g in Corollary 3.15, one can obtains a fixed point for g on taking a non-decreasing sequence {υn} in P by following the proof of Theorem 3.2. Example 3.16. Define a metric d : P × P → P as below and ≤ is an usual order on P, where P = {1,2,3,4,5,6} d(υ,ξ) = d(ξ,υ) = 0, if υ,ξ = 1,2,3,4,5,6 and υ = ξ, d(υ,ξ) = d(ξ,υ) = 3, if υ,ξ = 1,2,3,4,5 and υ 6= ξ, d(υ,ξ) = d(ξ,υ) = 12, if υ = 1,2,3,4 and ξ = 6, d(υ,ξ) = d(ξ,υ) = 20, if υ = 5 and ξ = 6. Define a map g : P → P by g1 = g2 = g3 = g4 = g5 = 1,g6 = 2 and let φ(t) = t 2 , ψ(t) = t 4 for t ∈ [0,+∞). Then g has a fixed point in P. Proof. It is apparent that, (P,d,s,�) is a complete partially ordered b-metric space for s = 2. Consider the possible cases for υ, ξ in P : Case 1 Suppose υ,ξ ∈ {1,2,3,4,5}, υ < ξ then d(gυ,gξ) = d(1,1) = 0. Hence, φ(2d(gυ,gξ)) = 0 ≤ φ(M(υ,ξ)) − ψ(M(υ,ξ)). 362 N. Seshagiri Rao & K. Kalyani CUBO 24, 2 (2022) Case 2 Suppose that υ ∈ {1,2,3,4,5} and ξ = 6, then d(gυ,gξ) = d(1,2) = 3, M(6,5) = 20 and M(υ,6) = 12, for υ ∈ {1,2,3,4}. Therefore, we have the following inequality, φ(2d(gυ,gξ)) ≤ M(υ,ξ) 4 = φ(M(υ,ξ)) − ψ(M(υ,ξ)). Thus, condition (3.64) of Corollary 3.14 holds. Furthermore, the remaining assumptions in Corol- lary 3.14 are fulfilled. Hence, g has a fixed point in P as Corollary 3.14 is appropriate to g,φ,ψ and (P,d,s,�). Example 3.17. A metric d : P × P → P, where P = {0,1, 1 2 , 1 3 , 1 4 , . . . 1 n , . . .} with usual order ≤ is defined as follows d(υ,ξ) =                  0, if υ = ξ 1, if υ 6= ξ ∈ {0,1} |υ − ξ|, if υ,ξ ∈ { 0, 1 2n , 1 2m : n 6= m ≥ 1 } 3, otherwise. A map g : P → P be such that g0 = 0,g 1 n = 1 12n for all n ≥ 1 and let φ(t) = t, ψ(t) = 4t 5 for t ∈ [0,+∞). Then, g has a fixed point in P. Proof. It is obvious that for s = 12 5 , (P,d,s,�) is a complete partially ordered b-metric space and also by definition, d is discontinuous b-metric space. Now for υ,ξ ∈ P with υ < ξ, we have the following cases: Case 1 If υ = 0 and ξ = 1 n , n ≥ 1, then d(gυ,gξ) = d(0, 1 12n ) = 1 12n and M(υ,ξ) = 1 n or M(υ,ξ) = {1,3}. Therefore, we have φ ( 12 5 d(gυ,gξ) ) ≤ M(υ,ξ) 5 = φ(M(υ,ξ)) − ψ(M(υ,ξ)). Case 2 If υ = 1 m and ξ = 1 n with m > n ≥ 1, then d(gυ,gξ) = d ( 1 12m , 1 12n ) and M(υ,ξ) ≥ 1 n − 1 m or M(υ,ξ) = 3. Therefore, φ ( 12 5 d(gυ,gξ) ) ≤ M(υ,ξ) 5 = φ(M(υ,ξ)) − ψ(M(υ,ξ)). Hence, condition (3.64) of Corollary 3.14 and remaining assumptions are satisfied. Thus, g has a fixed point in P . CUBO 24, 2 (2022) Fixed point results of (φ,ψ)-weak contractions in ordered b-metric... 363 Example 3.18. Let P = C[a,b] be the set of all continuous functions. Let us define a b-metric d on P by d(θ1,θ2) = sup t∈C[a,b] {|θ1(t) − θ2(t)| 2} for all θ1,θ2 ∈ P with partial order � defined by θ1 � θ2 if a ≤ θ1(t) ≤ θ2(t) ≤ b, for all t ∈ [a,b], 0 ≤ a < b. Let g : P → P be a mapping defined by gθ = θ 5 ,θ ∈ P and the two altering distance functions by φ(t) = t, ψ(t) = t 3 , for any t ∈ [0,+∞]. Then g has a unique fixed point in P. Proof. From the hypotheses, it is clear that (P,d,s,�) is a complete partially ordered b-metric space with parameter s = 2 and fulfill all the conditions of Corollary 3.14 and Note 1. Furthermore for any θ1,θ2 ∈ P , the function min(θ1,θ2)(t) = min{θ1(t),θ2(t)} is also continuous and the conditions of Corollary 3.14 and Note 1 are satisfied. 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