On η , γ f , g -Contractions in Extended b -Metric Spaces

Hindawi Advances in Mathematical Physics Volume 2022, Article ID 9290539, 8 pages https://doi.org/10.1155/2022/9290539 Research Article On ðη, γÞð f ,gÞ-Contractions in Extended b-Metric Spaces Jayashree Patil ,1 Basel Hardan ,2 Ahmed A. Hamoud ,3 Amol Bachhav,4 Homan Emadifar ,5 Afshin Ghanizadeh ,6 Seyyed Ahmad Edalatpanah ,7 and Hooshmand Azizi 8 1 Department of Mathematics, Vasantrao Naik Mahavidyalaya, Cidco, Aurangabad, India Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India 3 Department of Mathematics, Taiz University, Taiz380015, Yemen 4 Naveen Jindal School of Management, University of Texas at Dallas, Dallas 75080, USA 5 Department of Mathematics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran 6 Department of Statistics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran 7 Department of Applied Mathematics, Ayandegan Institute of Higher Education, Tonekabon, Iran 8 Department of Electrical and Computer Engineering, No. 1 Faculty of Kermanshah, Technical and Vocational University (TVU), Kermanshah, Iran 2 Correspondence should be addressed to Homan Emadifar; [email protected] Received 2 June 2022; Revised 17 September 2022; Accepted 19 September 2022; Published 11 October 2022 Academic Editor: Eugen Radu Copyright © 2022 Jayashree Patil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we give a concept of ðη, γÞð f ,gÞ -contraction in the setting of expanded b–metric spaces and discuss the existence and uniqueness of a common fixed point. Introduced results generalize well-known fixed point theorems on contraction conditions and in the given spaces. 1. Introduction and Preliminaries The tremendous applications of fixed point theory had always inspired the growth of this domain. In 1922, Banach formulated his most simple but very natural result which is now popularly referred to as the Banach contraction principle. In the course of the last several decades, this principle has been extended and generalized in many directions with several applications in many branches. Employing simulation functions, Khojasteh et al. [1] initiated the idea of Zcontractions and utilized the same to cover the varied types of nonlinear contractions in the existing literature. Later, Argoubi et al. [2] and Roldán-López-de-Hierro et al. [3] independently sharpened the notion of simulation functions and also proved some coincidences and common fixed point results. Very recently, Lopez et al. [4] introduced the notion of R-contractions in order to extend several nonlinear con- tractions such as Z-contractions, manageable contractions, and Meir-Keeler contractions. Indeed, R-contractions are associated with R-functions that satisfy two independent conditions involving two sequences of nonnegative real numbers. Soon, inspired by R-contractions, Shahzad et al. [5] introduced the notion of ðη, γÞ-contractions which remains an extension of ðR, γÞ-contractions given in [6] by Roldán-López-de-Hierro and Shahzad wherein the authors proved very interesting results. Czerwik [7] established a successful generalization of the metric space concept by introducing the notion of b-metric space. Following this, a number of authors have introduced respective interesting theorems in b-metric, (see [8–14]). Newly, Kamran et al. [15] inspired by the concept of bmetric space, they introduced the concept of extended b space and also developed some fixed point theorems for 2 Advances in Mathematical Physics self-mappings defined in such spaces. Their results extend/ generalize many of the results already available in the literature. In this paper, we shall define a general contraction condition with the help of some auxiliary functions and investigate the existence and uniqueness of a fixed point for such mappings in the frame of b-metric space. Definition 1 (see [7]). Let X be a nonempty set and let d : X × X ⟶ ½0,∞Þ satisfy the following for all u, v, w ∈ X. (i) dðu, vÞ = 0 ⇔ u = v (ii) dðu, vÞ = dðv, uÞ (iii) dðu, wÞ ≤ b½dðu, vÞ + dðv, wފ, where b ≥ 1 The pair ðX, dÞ is called a b-metric space; when b = 1, the b-metric space becomes a usual metric space. Example 2 (see [8, 9]). Let X = ℝ2 . Then, the functional d : X × X ⟶ ½0,∞Þ defined by 8 ju − u2 j + jv1 − v2 j, > > < 1 dððu1 , v1 Þ, ðu2 , v2 ÞÞ ≔ ju1 − u2 j + jv1 − v2 j, > > : 0, is a b-metric space on X with b = 2. Example 3 (see [16]). The space Lp ½0, 1Š (where 0 < p < 1) of Ð all real functions uðtÞ, t ∈ ½0, 1Š such that 10 juðtÞjp dt < ∞, together with the functional δðu, vÞ = ð 1 juðt Þ − vðt Þjp dt 0 1/p ,∀u, v ∈ Lp ½0, 1Š, ð2Þ is a b-metric space with b = 21/p . Example 4 (see [17]). Let X = fa, b, cg and d : X × X ⟶ ℝ+ such that ðu1 , v1 Þ, ðu2 , v2 Þ ∈ ½0, 1Þ × ½0, 1Þ ðu1 , v1 Þ, ðu2 , v2 Þ ∈ ð1,∞Þ × ð1,∞Þ otherwise, The pair ðX, dψ Þ is called an extended b-metric space. If ψðu, vÞ = b, for b ≥ 1, then we reduce to Definition 1. Example 6 (see [17]). Let X = fa, b, cg ∪ ℝ+0 and d : X × X ⟶ ½0,∞Þ be defined by (i) If u, v ∈ ℝ+0 , then dψ ðu, vÞ = ju − vj2 , (ii) If u ∈ fa, b, cg and v ∈ ℝ+0 , then dψ ðu, vÞ = dψ ðv, uÞ = 1 and dψ ðu, uÞ = 0 (iii) If u, v ∈ fa, b, cg, such that dψ ða, bÞ = 1, dψ ða, cÞ = dða, bÞ = dðb, aÞ = dða, cÞ = dðc, aÞ = b, dðb, cÞ = dðc, bÞ = α ≥ c, ð3Þ dða, aÞ = dðb, bÞ = dðc, cÞ = a: α ½dðu, wÞ + dðw, vފ: 2 Definition 5 (see [15]). Let X be a nonempty set and ψ : X × X ⟶ ½1,∞Þ, and let dψ : X × X ⟶ ½0,∞Þ satisfy: (ii) dψ ðu, vÞ = dψ ðv, uÞ (iii) dψ ðu, wÞ ≤ ψðu, wÞ½dψ ðu, vÞ + dψ ðv, wފ ð5Þ with dψ ðu, vÞ = dψ ðv, uÞ and dψ ðu, uÞ = 0. ð4Þ Therefore, ðX, dÞ is a b-metric space for all u, v, w ∈ X. If α > c, then the ordinary triangle inequality does not hold, and ðX, dÞ is not a metric space. (i) dψ ðu, vÞ = 0 ⇔ u = v 1 and dψ ðb, cÞ = 2, 2 Notice that d is not a metric space since dψ ðb, cÞ > dψ ðb, aÞ + dψ ða, cÞ. However, it is easy to see that d is an extended b-metric space for ψ : X × X ⟶ ½1,∞Þ, where Then, dðu, vÞ ≤ ð1Þ ψðu, vÞ ≔ 8 4 > > >3 < 2 > > > : 1 if u, v ∈ fa, b, cg, if u, v ∈ ℝ+0 , ð6Þ if ðu, vÞ or ðv, uÞ ∈ fa, b, cg × ℝ+0 : Example 7 (see [15]). Let X = Cð½a, bŠ, ℝÞ be the space of all continuous real valued functions defined on ½a, bŠ, let ψ : X × X ⟶ ½1,∞Þ where ψðu, vÞ = juðtÞj + jvðtÞj + 2, and note that X is a complete extended b-metric space by considering dψ ðu, vÞ = sup juðt Þ − vðt Þj2 : t∈½a,bŠ ð7Þ Advances in Mathematical Physics 3 In this context, wonderful theorems established by the authors in extended b-metric space, for examples, Fahed et al. and Swapna and Phaneendra [18, 19], got some new fixed point results in an extended b-metric space. Also, Ullah et al. [20] proved fixed point theorems in complex-valued extended b-metric spaces. In this bearing, Mitrović et al. [21] established new results in extended b-metric space, in follows that we recollect some fundamental notions, for example, convergence, the notion of the Cauchy sequence, and completeness in an extended b- metric space. Definition 8 (see [22]). Let ðX, dψ Þ be an extended b-metric space, and then ηðu, vÞ ≥ 1 ⇒ ηðTu, TvÞ ≥ 1: ð10Þ Definition 15 (see [17]). Let Φ be the family of functions ϕ : ½0,∞Þ ⟶ ½0,∞Þ satisfying the following conditions (i) ϑ is nondecreasing (ii) ϑðχÞ < χ, χ > 0 2. Main results Definition 16. Let ðX, dÞ be an extended b-metric space and let η : X × X ⟶ ½0,∞Þ and ψ : X × X ⟶ ½1,∞Þ such that (i) A sequence un ∈ X is said to converge to u0 ∈ X if, ∀ε > 0, there exists N = NðεÞ ∈ ℕ such that dψ ðun , u0 Þ < ε, ∀n ≥ N. We write lim un = u0 ηðu, vÞdψ ðTu, TvÞ ≤ ψð f ðu, vÞÞ, ð11Þ n⟶∞ (ii) A sequence un in X is said to be Cauchy if, ∀ε > 0, there exists N = NðεÞ ∈ ℕ such that dψ ðun , um Þ < ε, ∀n, m ≥ N Definition 9 (see [15]). An extended b -metric space ðX, dψ Þ is complete if every Cauchy sequence in X is convergent. Lemma 10 (see [22]). Let ðX, dψ Þ be a complete extended bmetric space. If dψ is continuous map, then every convergent sequence in X has a unique limit. Theorem 11 (see [15]). Suppose ðX, dψ Þ is an extended bmetric space such that dψ is a continuous mapping. Suppose T : X ⟶ X, it fulfills dψ ðTu, TvÞ ≤ ηdψ ðu, vÞ,∀u, v ∈ X, ð8Þ where η ∈ ½0, 1Þ is such that, for each u0 ∈ X, we have lim ψðun , um Þ < 1/η. Here, T n u0 = un , n = 1, 2, ⋯Then, T n,m⟶∞ has exactly one fixed point u0 , moreover ∀v ∈ X, T n v ⟶ u0 . For our objectives, we recall the definition of orbital admissible maps introduced by Popescu [23]. where ‹ψ ðSu, SvÞ, ‹ψ ðTu, SuÞ, ‹ψ ðSv, TvÞ, 9 > = : f ðu, vÞ = sup ‹ψ ðSu, Tuދψ ðSv, TvÞ ‹ψ ðSu, TvÞ + ‹ψ ðSv, TuÞ > > , ; : ‹ψ ðSu, SvÞ 2 sup ½ψðSv, TuÞ, ψðSu, Tvފ 8 > < ð12Þ Then, S and T are ðη, γÞ f -contraction for all u, v ∈ X, where ψ ∈ Ψ. The headmost principal result of this paper is as follows: Theorem 17. Let ðX, dÞ be a complete extended b - metric space, and let S, T : X ⟶ X be an ðη, γÞ f -contraction mappings. Let lim n,m⟶∞ sup ϑn+1 ðrÞ ψðun , um Þ < 1, ϑn ðrÞ ð13Þ for all u0 ∈ X, r > 0 where T n un−n = Sun = un , n ∈ ℕ. Assume also that (i) S and T are η-orbital admissible Definition 12. Let S be a self-map on X and η : X × X ⟶ ½0,∞Þ. We say that S is an η-orbital admissible if for all u, v ∈ X, we have À Á ηðu, SuÞ ≥ 1 ⇒ η Su, S2 u ≥ 1: ð9Þ Remark 13 (see [23]). Every η-admissible mapping is an α -orbital admissible mapping. Definition 14 (see [24]). For a nonempty set X, suppose T : X ⟶ X and η : X × X ⟶ ½0,∞Þ are mappings. One says that self-mapping T on X is η -admissible if for u, v ∈ X, one has (ii) There exists w ∈ X such that ηðSw, TwÞ ≥ 1 (iii) S and T are continuous Then, S and T possess a unique coincidence fixed point u0 ; that is, Su = Tu = u0 . Proof. By a supposition, for some u0 ∈ X, we have un = Sun = T n u0 , ∀n ∈ ℕ. Suppose that un0 = Tun0 −1 = Sun0 . Let that un ≠ un+1 , ∀n ∈ ℕ. Since T and S are η-admissible, we get ηðu0 , u1 Þ = ηðSu0 , Tu0 Þ ≥ 1 ⇒ ηðSu1 , Tu1 Þ = ηðu1 , u2 Þ ≥ 1: ð14Þ 4 Advances in Mathematical Physics On regard of (15) and (11), we get Repeatedly, we obtain for all n ∈ ℕ ð15Þ ηðun , un+1 Þ ≥ 1: dψ ðun , un+1 Þ = dψ ðSun , Sun+1 Þ = dψ ðTun−1 , Tun Þ ≤ ϑð f ðun−1 , un ÞÞ, ð16Þ where f ðun−1 , un Þ = sup 8 > > < 9 > > = dψ ðSun−1 , Sun Þ, dψ ðTun−1 , Sun−1 Þ, dψ ðSun , Tun Þ, dψ ðSun−1 , Tun−1 Þdψ ðSun , Tun Þ dψ ðSun−1 , Tun Þ + dψ ðSun , Tun−1 Þ > > > > , : dψ ðSun−1 , Sun Þ 2 sup ½ψðSun , Tun−1 Þ, ψðSun−1 , Tun ފ ; 9 8 dψ ðun−1 , un Þ, dψ ðun , un−1 Þ, dψ ðun , un+1 Þ, > > > > = < = sup d ðu , u Þd ðu , u Þ dψ ðun−1 , un+1 Þ + dψ ðun , un Þ > ψ n−1 n ψ n n+1 > > > , : dψ ðun−1 , un Þ 2 sup ½ψðun , un Þ, ψðun−1 , un+1 ފ ;   dψ ðun−1 , un+1 Þ = sup dψ ðun , un−1 Þ, dψ ðun , un+1 Þ, 2 sup ½ψðun , un Þ, ψðun−1 , un+1 ފ   dψ ðun−1 , un Þ + dψ ðun , un+1 Þ ≤ sup dψ ðun , un−1 Þ, dψ ðun , un+1 Þ, 2 È É = sup dψ ðun , un−1 Þ, dψ ðun , un+1 Þ : Now, if f ðun−1 , un Þ = dψ ðun , un+1 Þ for some n ∈ ℕ, thus À Á dψ ðun , un+1 Þ ≤ ϑ dψ ðun , un+1 Þ < dψ ðun , un+1 Þ, ð18Þ We will show that fun g is a Cauchy sequence, as follows:  à dψ ðun , un+m Þ ≤ ψðun , un+m Þ dψ ðun , un+1 Þ + dψ ðun+1 , un+m Þ ≤ ψðun , un+m Þdψ ðun , un+1 Þ + ψðun , un+m Þdψ ðun+1 , un+m Þ: which is a contradiction. On the other hand, if f ðun−1 , un Þ = dψ ðun , un−1 Þ, then for all n ≥ 1, we have À Á dψ ðun , un+1 Þ ≤ ϑ dψ ðun , un−1 Þ < dψ ðun , un−1 Þ: ð17Þ ð19Þ ð24Þ Also,  à dψ ðun+1 , un+m Þ ≤ ψðun+1 , un+m Þ dψ ðun+1 , un+2 Þ + dψ ðun+2 , un+m Þ Sequentially for all n ∈ ℕ, we get ≤ ψðun+1 , un+m Þdψ ðun+1 , un+2 Þ À Á dψ ðun , un+1 Þ ≤ ϑn dψ ðu0 , u1 Þ : ð20Þ ð25Þ Thus, there exists k ≥ 0 such that dψ ðun , un+1 Þ = k, when n ⟶ ∞: And so, until the inequality (24) reaches to ð21Þ Taking n ⟶ ∞ to inequality (19), we obtain k ≤ ϑðkÞ ⇒ k = 0: + ψðun+1 , un+m Þdψ ðun+2 , un+m Þ: ð22Þ Therefore, when n ⟶ ∞, we get  à dψ ðun , un+m Þ ≤ ψðun , un+m Þ dψ ðun , un+1 Þ + dψ ðun+1 , un+m Þ À Á ≤ ψðu1 , un+m Þψðu2 , un+m Þ ⋯ ψðun , un+m Þϑn dψ ðu0 , u1 Þ + ψðu1 , un+m Þψðu2 , un+m Þ ⋯ ψðun , un+m Þψ À Á Á ðun+1 , un+m Þϑn+1 dψ ðu0 , u1 Þ +⋯+⋯+ψðu1 , un+m Þψðu2 , un+m Þ ⋯ ψ À Á Á ðun+m−1 , un+m Þϑn+m−1 dψ ðu0 , u1 Þ n+m−1 i À ÁY À Á = 〠 ϑi dψ ðu0 , u1 Þ ψ u j , un+m = bn+m−1 , i=n dψ ðun , un+1 Þ = 0: ð23Þ j=1 ð26Þ Advances in Mathematical Physics 5 Then, we conclude that ð27Þ dψ ðun , un+m Þ < bn+m−1 − bn : À Á dψ ðSω, Sω∗ Þ = dψ ðTω, Tω∗ Þ = dψ ðω, ω∗ Þ ≤ ϑ dψ ðω, ω∗ Þ : ð35Þ The series Therefore, dψ ðω, ω∗ Þ = 0; thus, ω = ω∗ . Hence, T and S possess a unique coincidence fixed point in X. ∞ i À Á À ÁY ψ u j , un+m : 〠 ϑi dψ ðu0 , u1 Þ i=1 ð28Þ j=1 n Suppose ϑ ðdψ ðu0 , u1 ÞÞ Qn j=1 ψðu j , un+m Þ = Bn , Definition 18. Let ðX, dÞ be an extended quasimetric space, and we say that S and T are ðη, γÞg -contraction such that η : X × X ⟶ ½0,∞Þ and ψ : X × X ⟶ ½1,∞Þ if for all u, v ∈ X fulfilled then À ÁQ À Á ϑn+1 dψ ðu0 , u1 Þ n+1 B j=1 ψ u j , un+m ÁQ n À Á = n+1 , nÀ Bn ϑ dψ ðu0 , u1 Þ j=1 ψ u j , un+m À Á ϑn+1 dψ ðu0 , u1 Þ B Á ψðun+1 , un+m Þ = n+1 < 1, nÀ Bn ϑ dψ ðu0 , u1 Þ To mitigation the continuity case on the given self-mappings, we will modify Definition 16 as follows: ð29Þ by (13) where n, m ⟶ ∞. Then, (28) converges by [25]. As result, in perspective of (27), we have ηðu, vÞdψ ðTu, TvÞ ≤ ϑðgðu, vÞÞ, ψ ∈ Ψ, and gðu, vÞ = sup dψ ðun , un+m Þ = 0, asn, m ⟶ ∞: ð30Þ Then, fun g is a Cauchy sequence, and since ðX, dÞ is a complete extended quasimetric space, there exists ω ∈ X such that dψ ðun , ωÞ = 0, as n ⟶ ∞: ð31Þ By condition (ii), we obtain lim dψ ðTun , TωÞ = lim dψ ðun+1 , TωÞ = dψ ðω, TωÞ = 0, n⟶∞ n⟶∞ lim dψ ðSun , SωÞ = lim dψ ðun , TωÞ = dψ ðω, SωÞ = 0: n⟶∞ n⟶∞ ð32Þ Hence, we deduce that Tω = Sω = ω. Furthermore, let ω∗ ∈ X such that Tω∗ = Sω∗ = ω∗ where ω ≠ ω∗ . So, by (12), we obtain dψ ðSω, Sω∗ Þ = dψ ðTω, Tω∗ Þ ≤ ϑð f ðω, ω∗ ÞÞ, ð33Þ ð36Þ   dψ ðSu, SvÞ dψ ðSu, TuÞ + dψ ðSu, TvÞ dψ ðSu, TvÞ + dψ ðSu, TuÞ : , , 2 2 2 sup ½ψðu, TuÞ, ψðu, Suފ ð37Þ By remove continuity of the given mappings, we get the following major result. Theorem 19. Let ðX, dÞ be a complete extended quasimetric space, and let S, T : X ⟶ X be ðη, γÞg -contraction mappings. Let (13) and conditions (i) and (ii) of Theorem 17 be satisfied. Assume also that (iii) If fun g is a sequence in X such that ηðun , un+1 Þ ≥ 1, ∀n and un ⟶ u ∈ X as n ⟶ ∞, then there exists a subsequence funk g ⊂ fun g such that ηðun , unk Þ ≥ 1, ∀k Then, S and T possess a coincidence fixed point u0 , that is, Su = Tu = u0: Proof. By inequality (11) and condition (iii) in Theorem 19, there exists funk g ⊂ fun g such that ηðunk , uÞ ≥ 1, ∀k. Applying inequality (11), we obtain that where dψ ðunk+1 , TuÞ = dψ ðTunk , TuÞ ≤ ηðunk , uÞdψ ðTunk , TuÞ ≤ ϑðgðunk , uÞÞ: f ðω, ω∗ Þ = sup 8 > > < dψ ðSω, Sω∗ Þ, dψ ðTω, SωÞ, dψ ðSω∗ , Tω∗ Þ, 9 > > = dψ ðSω, TωÞdψ ðSω∗ , Tω∗ Þ dψ ðSω, Tω∗ Þ + dψ ðSω∗ , TωÞ > > > > , : 2 sup ½ψðSω∗ , TωÞ, ψðSω, Tω∗ ފ ; dψ ðSω, Sω∗ Þ   dψ ðω, ω∗ Þ = dψ ðω, ω∗ Þ: = sup dψ ðω, ω∗ Þ, sup ½ψðω∗ , ωÞ, ψðω, ω∗ ފ ð34Þ ð38Þ And dψ ðunk , SuÞ = dψ ðSunk , SuÞ ≤ ηðunk , uÞdψ ðSunk , SuÞ ≤ ϑðgðunk , uÞÞ: ð39Þ 6 Advances in Mathematical Physics Also, we have dψ ðSunk , SuÞ dψ ðSunk , Tunk Þ + dψ ðSunk , TuÞ dψ ðSunk , Tunk Þ + dψ ðSunk , TuÞ gðunk , uÞ = sup , , 2 sup ½ψðunk , Tunk Þ, ψðunk , Sunk ފ 2   dψ ðu, SuÞ dψ ðu, TuÞ dψ ðu, TuÞ = sup : , , 2 2 sup ½ψðSu, uÞ, ψðu, Tuފ  Then,  ð40Þ Thus, gðunk , uÞ = sup   dψ ðu, SuÞ dψ ðu, TuÞ : , 2 2 ð41Þ dψ ðu, SuÞ , 2 ð42Þ 2dψ ðu, SuÞ ≤ dψ ðu, SuÞ, ask ⟶ ∞, ð45Þ When gðunk , uÞ = then neither dψ ðu, SuÞ > 0 nor dψ ðu, SuÞ = 0 for some u ∈ X when k ⟶ ∞. If dψ ðu, TuÞ > 0, then by inequality (39), we obtain gðunk , uÞ > 0. Thus, from Definition 18, we get ψðgðunk , uÞÞ < gðunk , uÞ: Which is an ambivalence. Therefore, dψ ðu, SuÞ = 0 and u = Su. Likewise, we can get that u = Tu. Hence, u is a common fixed point for S and T in X, that is, u = Su = Tu. Let u1 , u2 ∈ X be two common fixed points of S and T such that u1 ≠ u2 then by (38) and (39), we get dψ ðu1 , u2 Þ = dψ ðTu1 , Tu2 Þ ð43Þ ≤ ηðu1 , u2 Þdψ ðTu1 , Tu2 Þ Also, by inequalities (38) and (42), we have dψ ðSunk , SuÞ ≤ ηðunk , uÞdψ ðSunk , SuÞ ≤ ϑðgðunk , uÞÞ < gðunk , uÞ = ≤ ϑðgðu1 , u2 ÞÞ, dψ ðu, SuÞ : 2 ð44Þ where dψ ðSu1 , Su2 Þ dψ ðSu1 , Tu1 Þ + dψ ðSu1 , Tu2 Þ dψ ðSu1 , Tu1 Þ + dψ ðSu1 , Tu2 Þ , , 2 sup ½ψðu1 , Tu1 Þ, ψðu1 , Su1 ފ 2   dψ ðu1 , u2 Þ dψ ðu1 , u2 Þ dψ ðu1 , u2 Þ = : = sup , 2 2 sup ½ψðu1 , u1 ފ gðu1 , u2 Þ = sup  Now, if dψ ðu1 , u2 Þ > 0, then ϑðgðu1 , u2 ÞÞ < ðgðu1 , u2 ÞÞ, which implies by (46) that 2dψ ðu1 , u2 Þ < dψ ðu1 , u2 Þ, ð46Þ ð48Þ but this is a contradiction. Hence, dψ ðu1 , u2 Þ = 0, i.e., u1 = u2 . This proves the uniqueness of the common fixed point of given mappings.     1 1 1 1 3 = dψ , = , , 2 3 4 5 10     1 1 1 1 2 = dψ , = , dψ , 2 5 3 4 10     1 1 1 1 6 dψ , = dψ , = , 2 4 5 3 10     1 1 1 1 = dψ , = ⋯ = 0, dψ , 2 2 3 3 dψ ð47Þ  ð49Þ dψ ðu, vÞ = ju − vj ∈ G 3 , else: Example 20. Suppose that G 1 ∪ G 2 ∪ G 3 = X, where G 1 = ð−∞,0Š, G 2 = ð1/2, 1/3, 1/4, 1/5Þ and G 3 = ½1, 2Š. Consider the extended b-metric space on X as follows: It is apparent that the triangle inequality on G 1 is not fulfilled. Actually, Advances in Mathematical Physics       6 1 1 1 1 1 1 5 = dψ , ≥ dψ , + dψ , = : 10 2 4 2 3 3 4 10 7 ð50Þ No data was used in this study. Observes that the condition (iii) in Definition 5 is satisfied. Suppose that f , g : X ⟶ X are defined as 8 0, if u ∈ G 1 , > > > > > > <1 Tu = 4 , if u ∈ G 2 , > > > > > 1 > : , if u ∈ G 3 , 2 ð51Þ Taking ηðu, vÞ = 1, if u, v ∈ G 2 ∪ G 3 0, if else, ð52Þ T and S are an ðη, γÞ-contractive mappings with γðtÞ = t/2, t ∈ ½0,∞Þ. Furthermore, there exists u0 ∈ X such that ηðTu0 , Su0 Þ ≥ 1. Actually, we have for u = 1/2 ∈ X.        1 1 1 1 ,S =η , = 1: η T 2 2 4 2 Conflicts of Interest The authors declare that they have no conflicts of interest. References Su = u,∀u ∈ X: ( Data Availability ð53Þ Now, suppose that u, v ∈ X with ηðu, vÞ ≥ 1. It implies that u, v ∈ G 2 ∪ G 3 . By Definition 1, we have ηðTu, SvÞ ≥ 1, and then T and S are η-admissible mappings. Also, T and S are clearly not continuous mappings. Otherwise, if un ⊂ X such that, ηðun , un+1 Þ ≥ 1, then un ∈ G 2 ∪ G 3 for all n. Assume the sequence fun g is considered iteratively as f un−1 = un for all n. Considering the arbitrary point u0 ∈ X, it is located in either G 2 or G 3 ; so, we have two cases: In case u0 ∈ G 2 , thus fun g is constant sequence and un ⟶ 1/4 ∈ G 2 . Then, for all n, we have ηðun , un+1 Þ ≥ 1, which implies that ηðun , 1/4Þ = ηðun+1 , 1/4Þ ≥ 1. In case u0 ∈ G 3 , thus fun g is constant sequence and un ⟶ 1/2 ∈ G 2 . 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