Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 513 https://internationalpubls.com Fixed Points of Generalized - Geraghty Ciric -Rational Type Contraction in B- Metric Spaces Dr. P.Harikrishna1, Dr. Kusuma Tummala2 , Dr. V.Sree Ramani3, Dr. Y.Jayababu4, Dr. T. Nageswara Rao5 1 Associate Professor, Vignan’s Institute of Information Technology, Visakhapatnam,, Andhrapradesh , India. Email: phk.2003@gmail.com 2Assistant Professor, Department of Humanities and Sciences, VNR Vignana Jyothi Institute of Engineering and Technology, Bachupally, Kukatpally, Hyderabad-500090, Telangana State, India. Email: kusumatummala9@gmail.com 3 Assistant Professor, Department of Mathematics, Chaitanya Bharathi Institute of Technology,Gandipet, Hyderabad- 500075, Telangana State, India. Email: sreeramani_maths@cbit.ac.in 4Professor, Department of CSE, Pragathi Enginnering College, Surampalem. Kakinada. India , Email: yjbabu4166@gmail.com 5Associate Professor, Department of Mathematics, Koneru Lakshmaih Education Foundation (KLEF), Guntur, Andhrapradesh, India, E mail: tnraothota@kluniversity.in. Corresponding Author:Email: phk.2003@gmail.com1 Article History: Received: 30-09-2024 Revised: 28-11-2024 Accepted: 09-12-2024 Abstract: In this paper we prove the existence and uniqueness of the fixed points generalized ciric type Geraghty rational contractions in b-metric spaces, our results extend some of the known theorems. Keywords: Fixed point; b-metric space; Geraghty –ciric type contraction. AMS(2010) Mathematics Subject Classification: 54H25, 74H10. Introduction One of the most important development of nonlinear analysis is fixed point theory. This idea is useful in science and engineering fields. Banach [2] first proposed the principle, one of the fundamental conclusions of conventional functional analysis, in 1922. One well-known and generally accepted outcome of fixed point theory is this idea. In 1973, Geraghty [13] demonstrated the existence of fixed point solutions in the context of full metric spaces [MS] and provided an important expansion of the Banach contraction principle [BCP] by substituting a function with certain qualities for a constant. As you can see from [8, 9, 12, 13] and the references therein, numerous researchers have since expanded and broadened the Geraghty conclusion in different ways. In metric spaces, Ćirić [4,5] demonstrated the Ćirić-type fixed point theorem, which is thought to be one of the most important generalizations of the BCP Definition 1.1 [15]. Let H be a nonempty set and let t ≥1. A mapping 𝑑: 𝐻x𝐻 →R is said to be a b-metric space if ∀ a,b.c in H, the following conditions are satisfied. (b1) 𝑑 (a, b)=0 if and only if a=b, (b2) 𝑑 (a, b) = 𝑑 (b, a) mailto:kusumatummala9@gmail.com mailto:sreeramani_maths@cbit.ac.in mailto:yjbabu4166@gmail.com mailto:tnraothota@kluniversity.in mailto:phk.2003@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 514 https://internationalpubls.com (b3) 𝑑 (a, c) ≤ t[𝑑 (a, b)+ 𝑑 (b, c)]. In this case, the pair (H, 𝑑) is called a b-metric space (with constant s). Note that every metric space is b-metric for t=1, but the converse is not true. Let S be the class of functions of non –decreasing functions 𝛽: [0, ∞) → [1, 1 t ) which satisfy the condition lim n→∞ 𝛽(𝑡𝑛) = 1 𝑡 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 lim n→∞ 𝑡𝑛 = 0 for some 𝑡 ≥ 1. Geraghty [14] proved the following theorem. Theorem 1.2. [14] Let (K, d) be a CMS(complete metric space). Let H : K → K be a self map. If ∃, β ∈ S such that d(H(u), H(v)) ≤ β(d(u, v))d(u, v) for all u, v ∈ H, then f has a unique common fixed point in H . Definition 1.3. [14] A self map H : K → K is said to be a generalized Geraghty contraction If ∃β ∈ S such that d(H(r), H(s)) ≤ β (M (r, s))M (r, s) (1.13.1) 𝑀(𝑟, 𝑠) = max { 𝑑(𝑟, 𝑠), 𝑑(𝑟, 𝐻𝑟), 𝑑(𝑠, 𝐻𝑠), (𝑑(r, Hs) + 𝑑(s, Hr)/2 } for all r, s ∈ H. Definition :1.4 [1] A mapping H: K → 𝐾 on a b metric space (K,d, t) with t ≥ 1 is called a Ciric - type Geraghty contraction mapping If ∃, 𝛽 ∈ F such that 𝑑(𝐻𝑢, 𝐻𝑣) ≤ 𝑀(𝑟, 𝑠), 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢, 𝑣 ∈ 𝐾 Where 𝑀(𝑟, 𝑠) = 𝑚𝑎𝑥 {𝛽(𝑑(𝑟, 𝑠))𝑑(𝑟, 𝑠), 𝛽(𝑑(𝑟, 𝐻𝑟))𝑑(𝑟, 𝐻𝑟), 𝛽(𝑑(𝑠, 𝐻𝑠))𝑑(𝑠, 𝐻𝑠), 𝛽(𝑑(𝑟, 𝐻𝑠))𝑑(𝑟, 𝐻𝑠), 𝛽(𝑑(𝑠, 𝐻𝑟))𝑑(𝑟, 𝐻𝑠)} Definition1.5 : [1] A mapping H: K → 𝐾 on a b metric space (K,d, t) with t ≥ 1 is called a Ciric - type Geraghty contraction mapping if there exists 𝛽 ∈ S such that 𝑑(𝐻𝑢, 𝐻𝑣) ≤ 𝑀(𝑢, 𝑣), 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢, 𝑣 ∈ 𝐾 Where 𝑀(𝑢, 𝑣) = 𝑚𝑎𝑥 {𝛽(𝑑(𝑢, 𝑣))𝑑(𝑢, 𝑣), 𝛽(𝑑(𝑢, 𝐻𝑢))𝑑(𝑢, 𝐻𝑢), 𝛽(𝑑(𝑣, 𝐻𝑣))𝑑(𝑣, 𝐻𝑣), 𝛽(𝑑(𝑢, 𝐻𝑣))𝑑(𝑢, 𝐻𝑣), 𝛽(𝑑(𝑣, 𝐻𝑢))𝑑(𝑣, 𝐻𝑢)} Theorem 1.6 :[ 8,9] Let ( M,d) be a CMS and 𝑇: 𝑀 → 𝑀 be a Geraghty – ciric –contraction with some 𝛽 ∈ S , Then T has fixed point and unique . In 2019, Faraji et.al [ 8] proved a fixed point theorem with Geraghty –type contractive maps in b- metric spaces . Theorem 1.7 [ 8] Let (M,d,v) be a complete- b metric space with 𝑣 ≥ 1 and let T : M → 𝑀 , be a self –mapping and if there exist 𝛽 ∈ S , 𝑑(𝑇𝑢, 𝑇𝑣) ≤ 𝛽(𝐿(𝑢, 𝑣))𝐿(𝑢, 𝑣), 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢, 𝑣 ∈ 𝑀, where 𝐿(𝑢, 𝑣) = max {𝑑(𝑢, 𝑣), 𝑑(𝑢, 𝑇𝑢), 𝑑(𝑣, 𝑇𝑣), 1 2𝑣 [𝑑(𝑢, 𝑇𝑣) + 𝑑(𝑣, 𝑇𝑢)]} then T has a unique fixed point. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 515 https://internationalpubls.com In 1975, Dass and Gupta [6] extended the BCP type of rational terms. Theorem 1.8. ( [3]). Let (H, d) be a CMS and H : K → K be a mapping such that there exist α, β ≥ 0 with α + β < 1 satisfying 𝑑(𝐻𝑢, 𝐻𝑣) ≤ 𝛼 𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑢,𝐻𝑣)] 1+𝑑(𝑢,𝑣) + β𝑑(𝑢, 𝑣) for all u, v ∈ H. Then H has a unique fixed point. Lemma [ 9] Let (M, d,v) be a metric space with 𝑣 ≥ 1 and let 𝑇: 𝑀 → 𝑀 be a self mapping . Let 𝑥0 ∈ 𝑀 be given and {𝑥𝑛} be a sequence in M such that 𝑥𝑛 = 𝑇𝑥𝑛−1 for all n in N, the sequence defined by An =max {𝑑(𝑥𝑝 , x𝑞)|0 ≤ 𝑝, 𝑞 ≤ 𝑛 𝑎𝑛𝑑 𝑝, 𝑞 ∈ N0}, for n ∈ N0. If T satisfies the contractivity condition in (1), then {An} is bounded . In 2024, kalo.et.al [1 ] proved fixed point theorems in Geraghty-ciric-type contraction mapping in b- metric spaces Theorem 1.10[1 ] Let (M, d, v) be a complete b-metric space with t ≥ 1 and let T: M → M be a self- mapping Ciric -type Geraghty contraction (1), Then T has a unique fixed point x* in K. This Lemma can use to prove results. Lemma [ 1.11]. Let (M, d,v) be a b-metric space with t ≥ 1 and let {an} and {bn} be b- convergent to x, y in M, then we have 1 𝑣2 𝑑(𝑎, 𝑏) ≤ liminf 𝑛→∞ 𝑑(𝑎𝑛, 𝑏𝑛) ≤ limsup 𝑛→∞ 𝑑(𝑎𝑛, 𝑏𝑦𝑛) ≤ 𝑣2 𝑑(𝑎, 𝑏) In particular, if a=b, we have lim 𝑛→∞ 𝑑(𝑎𝑛, 𝑏𝑛) = 0. and for any c in M, 1 𝑣 𝑑(𝑎, 𝑐) ≤ liminf 𝑛→∞ 𝑑(𝑎𝑛, 𝑐) ≤ limsup 𝑛→∞ 𝑑(𝑎𝑛, 𝑐𝑧) ≤ 𝑣 𝑑(𝑎, 𝑐). MAIN RESULTS: Now, we define ciric type Geraghty contraction with rational type of expressions in b metric spaces. Definition 2.1: A mapping H: K → 𝐾 on a b- metric space (K,d, t) with t ≥ 1 is called a– Ciric - type Geraghty contraction with rational mapping , if there exists 𝛽 ∈ S such that 𝑑(𝐻𝑢, 𝐻𝑣) ≤ 𝑀(𝑢, 𝑣), 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢, 𝑣 ∈ 𝐾,……2.1.1 Where 𝑀(𝑢, 𝑣) = 𝑚𝑎𝑥 {𝛽 ( 𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢)] 1+𝑑(𝑣,𝐻𝑣) ) ( 𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢)] 1+𝑑(𝑣,𝐻𝑣) ), 𝛽 ( 𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣)] 1+𝑑(𝑢,𝐻𝑣) ) ( 𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣)] 1+𝑑(𝑢,𝐻𝑣) ), 𝛽 ( 𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢)] 1+𝑑(𝑢,𝐻𝑣) ) ( 𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢)] 1+𝑑(𝑢,𝐻𝑣) )} First, we prove that sequence is { Bn} is bounded. Theorem 2. 2: Let (K, d,t) be a b-metric space with t ≥ 1 and let H : K → K be a self- mapping. Let s0 ∈ K be given and {sn} be a sequence in K , sn = H sn–1 for all n ∈ N. The sequence Bn =max {d(sp , sq)|0 ≤ p, q ≤ n and p, q ∈ N0} (2.2.1) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 516 https://internationalpubls.com for n ∈ N0. If H satisfies the contractivity condition in (2.1.1), then {Bn} is bounded {d(sp , sq)/ 0 ≤ p, q ≤ n and p, q ∈ N0} for n ∈ N0. If K satisfies condition of (2.1.1) , then {Bn} is bounded. Proof: Let n ∈ N. Then for any p, q ∈ N with 1 ≤ 𝑝, 𝑞 ≤ 𝑛, from condition (2.1.1), we have 𝑑(𝑠𝑝 , s𝑞) = 𝑑(𝐻𝑠𝑝−1, 𝐻𝑠𝑞−1) ≤ 𝑀(𝑠𝑝−1, 𝑠𝑞−1) = max {𝛽 ( 𝑑(𝑠𝑝−1,𝐻𝑠𝑝−1)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠𝑝−1)] 1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1) ) ( 𝑑(𝑠𝑝−1,𝐻𝑠𝑝−1)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠𝑝−1)] 1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1) ), 𝛽 ( 𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1)] 1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1) ) ( 𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1)] 1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1) ), 𝛽 ( 𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1)] 1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1) ) ( 𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1)[1+𝑑(𝑠𝑞−1,𝐻𝑠𝑝−1)] 1+𝑑(𝑠𝑝−1,𝐻𝑠𝑞−1) )} < 1 𝑡 max{ ( 𝑑(𝑠𝑝−1,𝐻𝑠𝑝−1)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠𝑝−1)] 1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1) ), ( 𝑑(𝑠𝑞−1, 𝐻𝑠𝑝−1)[1 + 𝑑(𝑠𝑞−1, 𝐻𝑠𝑞−1)] 1 + 𝑑(𝑠𝑝−1, 𝐻𝑠𝑞−1) ) , 𝑑(𝑠𝑞−1, 𝐻𝑠𝑞−1)[1 + 𝑑(𝑠𝑞−1, 𝐻𝑠𝑝−1)] 1 + 𝑑(𝑠𝑝−1, 𝐻𝑠𝑞−1) } ≤ B𝑛, so that max {𝑑(𝑠𝑝 , s𝑞)| 0 ≤ 𝑝, 𝑞 ≤ 𝑛 𝑎𝑛𝑑 𝑝, 𝑞 ∈ N0} < Bn. Consequently, there is 𝑤𝑛 ∈ 𝑁, 𝑤𝑖𝑡ℎ 1 ≤ 𝑤𝑛 ≤ 𝑛 such that Bn =𝑑(𝑠𝑝 , s𝑤𝑛} Here, we can see that 0 ≤ B𝑛 ≤ 𝐵𝑛+1 for all 𝑛 ∈ 𝑁. Now we have to prove sequence {𝐵𝑛} is bounded. On the contrary, we assume that {𝐵𝑛} is not bounded. Since {𝐵𝑛} is non decreasing sequence of non negative reals, we have lim 𝑛→∞ 𝐵𝑛 = ∞. Now, by using b- triangular inequality on 𝑑(𝑠𝑝 , s𝑤𝑛} and using the inequality (1) , 𝐵𝑛 = 𝑑(𝑠𝑝 , s𝑤𝑛} ≤ 𝑡[𝑑(𝑠0, 𝑠𝑤𝑛) ≤ 𝑣[𝑑(𝑠0, 𝑠1) + 𝑑(𝑠1, 𝑠𝑤𝑛)] ---(i) = 𝑡[𝑑(𝑠0, 𝑠1) + 𝑣𝑀(𝑠0, 𝑠𝑤𝑛−1)] Where , 𝑀(𝑠0, 𝑠𝑤𝑛−1)= max {𝛽 ( 𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠0)] 1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1) ) ( 𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠0)] 1+𝑑(𝑠𝑞−1,𝐻𝑠𝑞−1) ), 𝛽 ( 𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1) ) ( 𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1) ), 𝛽 ( 𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1) ) ( 𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1) )} Here observe that the sequences {𝛽 ( 𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞−1,𝐻𝑠0)] 1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1) )}, 𝛽 ( 𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1) )}, { 𝛽 ( 𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1) )} are the sequences of real Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 517 https://internationalpubls.com numbers and the sub sequences {𝛽 ( 𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0)] 1+𝑑(𝑣,𝐻𝑣) )}, 𝛽 ( 𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠𝑤𝑛𝑘−1)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛𝑘−1) )}, { 𝛽 ( 𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠𝑤𝑛𝑘−1)[1+𝑑(𝑠𝑤𝑛𝑘−1,𝐻𝑠0)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛𝑘−1) )} Case (i): Suppose that 𝑀(𝑠0, 𝑠𝑤𝑛−1) = {𝛽 ( 𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0)] 1+𝑑(𝑣,𝐻𝑣) )} ------(ii) then from (i) and (ii) , we get 𝐵𝑛𝑘 ≤ 𝑑(𝑠𝑝 , s𝑤𝑛} ≤ 𝑡[𝑑(𝑠0, 𝑠𝑤𝑛) ≤ 𝑣[𝑑(𝑠0, 𝑠1) + 𝑑(𝑠1, 𝑠𝑤𝑛)] ---(i) ≤ 𝑡[𝑑(𝑠0, 𝑠1) + 𝑣𝑀(𝑠0, 𝑠𝑤𝑛−1)] ≤ 𝑡 [𝑑(𝑠0, 𝑠1) + 𝑣 {𝛽 ( 𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0)] 1+𝑑(𝑣,𝐻𝑣) )} 𝐵𝑛𝑘]----(iii) so that, 1 𝑡 − 𝑑(𝑠0,𝑠1) 𝐵𝑛𝑘 ≤ 𝛽 ( 𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0)] 1+𝑑(𝑣,𝐻𝑣) ) < 1 𝑡 Since as 𝑘 → ∞, 𝐵𝑛𝑘 → ∞, so that lim 𝑛→∞ ( 1 𝑡 − 𝑑(𝑠0,𝑠1) 𝐵𝑛𝑘 ) = 1 𝑡 . Therefore lim sup 𝑘→∞ 𝛽 ( 𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0)] 1+𝑑(𝑣,𝐻𝑣) ) = 1 𝑡 . Hence lim 𝑛→∞ 𝑑(𝑠0, 𝑠𝑤𝑛𝑘−1) = 0. Since 𝛽 is the class of functions 𝑆, taking limit on (iii), we get lim Bnk ≤ 𝑘→∞ 𝑡 [𝑑(𝑠0, 𝑠1) + 𝑡 {𝛽 ( 𝑑(𝑠0, 𝐻𝑠0)[1 + 𝑑(𝐻𝑠𝑞𝑘−1, 𝐻𝑠0)] 1 + 𝑑(𝑣, 𝐻𝑣) )}] 𝑡. 𝑑(𝑠0, 𝑠1), contradiction, 𝐵𝑛𝑘 → ∞, as 𝑘 → ∞. Case (ii) Suppose that 𝑀(𝑠0, 𝑠𝑤𝑛−1) = {𝛽 ( 𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1) )} ------(v) then from (i) and (v) , we get 𝐵𝑛𝑘 ≤ 𝑡 [𝑑(𝑠0, 𝑠1) + 𝑡 {𝛽 ( 𝑑(𝑠0,𝐻𝑠0)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠0)] 1+𝑑(𝑣,𝐻𝑣) )} 𝐵𝑛𝑘]----(iii) < (t+1) 𝑑(𝑠0, 𝑠1), which is contradiction to , 𝐵𝑛𝑘 → ∞, as 𝑘 → ∞. Case (iii) Suppose that 𝑀(𝑠0, 𝑠𝑤𝑛−1) = 𝛽 ( 𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1) ) ------(vi) then from (i) and (vi) , we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 518 https://internationalpubls.com 𝐵𝑛𝑘 ≤ 𝑡 [𝑑(𝑠0, 𝑠1) + 𝑡 {𝛽 ( 𝑑(𝑠𝑤𝑛−1,𝐻𝑠𝑤𝑛−1)[1+𝑑(𝑠𝑤𝑛−1,𝐻𝑠0)] 1+𝑑(𝑠0,𝐻𝑠𝑤𝑛−1) )} 𝐵𝑛𝑘] < (t+1) 𝑑(𝑠0, 𝑠1), which is contradiction to , 𝐵𝑛𝑘 → ∞, as 𝑘 → ∞. so that we are getting contradictions in all the cases and hence the sequence {𝐵𝑛} not bounded. Now we prove the existence of the fixed point with unique in – Ciric -type Geraghty contraction with rational terms in b –metric spaces. Theorem 2.3: Let (K, d, t) be a CbMS (Complete b- metric space) space with t ≥ 1 and let H : K → K be a map – Ciric -type Geraghty contraction with rational terms (2.11), Then H has a unique fixed point s in K. Proof: Let𝑠0 ∈ 𝐾. Now consider a sequence {𝑠𝑛} in K by defining 𝑠𝑛 = 𝐾𝑠𝑛−1 = 𝐾𝑛𝑠0 ∀ 𝑛 ∈ 𝑁. Now we show that {𝑠𝑛}𝑛 ∈ 𝑁 is a b- Cauchy sequence in K. We take a sequence as 𝐵𝑛 = 𝑚𝑎𝑥{𝑑(𝑠𝑝, 𝑠𝑞)|0 ≤ 𝑝, 𝑞 ≤ 𝑛, 𝑝, 𝑞 ∈ 𝑁} By using the above theorem 2. 2 , ∃M >0 such that 𝐵𝑛 ≤ 𝑀 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ∈ 𝑁 . As {𝐵𝑛} is an increasing sequence, we have lim 𝑛→∞ 𝐵𝑛 ≤ 𝑀. We take a sequence {𝜆𝑛} on a b- metric space (K, d,t) by {𝜆𝑛} = sup { 𝑑(𝑠𝑝, 𝑠𝑞)|𝑝, 𝑞 ≥ 𝑛, 𝑝, 𝑞 ∈ 𝑁}, then we have 0 ≤ 𝜆𝑛 ≤ 𝜆𝑛−1 ≤ 𝜆𝑛−2 ≤ ⋯ ≤ 𝜆0 = lim 𝑛→∞ 𝐵𝑛 ≤ 𝑀 ∀ 𝑛 ∈ 𝑁. The sequence {𝜆𝑛} is a decreasing and bounded sequence of non negative real numbers, and hence it is converges to some 𝑙 ≥0, that is lim 𝑛→∞ 𝜆𝑛 = 𝑙. Then there exist two sub sequences { 𝑠𝑝𝑘 } and { 𝑠𝑞𝑘 } of { 𝑠𝑛} with 𝑞𝑘 > 𝑝𝑘 ≥ 𝑘 for 𝑘 ∈ 𝑁 such that 𝑑(𝑠𝑝𝑘 , 𝑠𝑞𝑘 ) → 𝑙 𝑎𝑠 𝑘 → ∞. 2.3.1 Now we have to prove that𝑙 = 0. On the contrary, assume that 𝑙 > 0. Put 𝑢 = 𝑠𝑝𝑘−1, 𝑎𝑛𝑑 𝑣 = 𝑠𝑞𝑘−1 in the inequality. We have 𝑑(𝑠𝑝𝑘, 𝑠𝑞𝑘) = 𝑑(𝐻𝑠𝑝𝑘−1 ,𝐻𝑠𝑞𝑘−1) ≤ 𝑀(𝑠𝑝𝑘−1 ,𝑠𝑞𝑘−1) 2.3.2 Where (𝑠𝑝𝑘−1 ,𝑠𝑞𝑘−1) = max{ 𝛽 ( 𝑑(𝑠𝑝𝑘−1 ,,𝐻𝑠𝑝𝑘−1 ,)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1 ,)] 1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1) ) ( 𝑑(𝑠𝑝𝑘−1 ,,𝐻𝑠𝑝𝑘−1 ,)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1 ,)] 1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1) ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 519 https://internationalpubls.com 𝛽 ( 𝑑(𝑠𝑞𝑘−1, 𝐻𝑠𝑝𝑘−1 ,)[1 + 𝑑(𝑠𝑞𝑘−1, 𝐻𝑠𝑞𝑘−1)] 1 + 𝑑(𝑠𝑝𝑘−1 ,, 𝐻𝑠𝑞𝑘−1) ) ( 𝑑(𝑠𝑞𝑘−1, 𝐻𝑠𝑝𝑘−1 ,)[1 + 𝑑(𝑠𝑞𝑘−1, 𝐻𝑠𝑞𝑘−1)] 1 + 𝑑(𝑠𝑝𝑘−1 ,, 𝐻𝑠𝑞𝑘−1) ) 𝛽 ( 𝑑(𝑠𝑞𝑘−1, 𝐻𝑠𝑞𝑘−1)[1 + 𝑑(𝑠𝑞𝑘−1, 𝐻𝑠𝑝𝑘−1 ,)] 1 + 𝑑(𝑠𝑝𝑘−1 ,, 𝐻𝑠𝑞𝑘−1) ) ( 𝑑(𝑠𝑞𝑘−1, 𝐻𝑠𝑞𝑘−1)[1 + 𝑑(𝑠𝑞𝑘−1, 𝐻𝑠𝑝𝑘−1 ,)] 1 + 𝑑(𝑠𝑝𝑘−1 ,, 𝐻𝑠𝑞𝑘−1) )} Now the maximum is one of the terms of R.H.S of 𝑀(𝑠𝑝𝑘−1 ,𝑠𝑞𝑘−1) Now we consider three as cases as the possibility of each one term of R H S. Suppose that 𝑀(𝑠𝑝𝑘−1 ,𝑠𝑞𝑘−1)= 𝛽 ( 𝑑(𝑠𝑝𝑘−1 ,,𝐻𝑠𝑝𝑘−1 ,)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1 ,)] 1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1) ) ( 𝑑(𝑠𝑝𝑘−1 ,,𝐻𝑠𝑝𝑘−1 ,)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1 ,)] 1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1) ) for all k in N. Therefore , from 2.32𝑑(𝑠𝑝𝑘 , 𝑠𝑞𝑘) ≤ 𝛽 ( 𝑑(𝑠𝑝𝑘−1 ,,𝐻𝑠𝑝𝑘−1 ,)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1 ,)] 1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1) ) ( 𝑑(𝑠𝑝𝑘−1 ,,𝐻𝑠𝑝𝑘−1 ,)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1 ,)] 1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1) ) ≤ 𝛽 ( 𝑑(𝑠𝑝𝑘−1 ,,𝐻𝑠𝑝𝑘−1 ,)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1 ,)] 1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1) ) 𝜆𝑘−1 …..2.3.3 Taking limit suprimum 𝑎𝑠 𝑘 → ∞ on both sides of (2.3.2), we get lim 𝑘→∞ 𝑠𝑢𝑝 𝑑(𝑠𝑝𝑘, 𝑠𝑞𝑘) ≤ lim 𝑘→∞ 𝑠𝑢𝑝 𝛽 ( 𝑑(𝑠𝑝𝑘−1 ,,𝐻𝑠𝑝𝑘−1 ,)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1 ,)] 1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1) ) 𝜆𝑘−1 From 2.3.1 , we get 𝑙 ≤ 𝛽 ( 𝑑(𝑠𝑝𝑘−1 ,, 𝐻𝑠𝑝𝑘−1 ,)[1 + 𝑑(𝐻𝑠𝑞𝑘−1, 𝐻𝑠𝑝𝑘−1 ,)] 1 + 𝑑(𝑠𝑞𝑘−1, 𝐻𝑠𝑞𝑘−1) ) 𝑙, 1 𝑡 ≤ 1 ≤ lim 𝑘→∞ 𝑠𝑢𝑝 𝛽 ( 𝑑(𝑠𝑝𝑘−1 ,,𝐻𝑠𝑝𝑘−1 ,)[1+𝑑(𝐻𝑠𝑞𝑘−1,𝐻𝑠𝑝𝑘−1 ,)] 1+𝑑(𝑠𝑞𝑘−1,𝐻𝑠𝑞𝑘−1) ) < 1 𝑡 Since 𝛽 is the class of functions S have lim 𝑘→∞ 𝑑(𝑠𝑝𝑘−1, 𝑠𝑞𝑘−1) = 0 Using 2.3.1 and 2.3.3 we get 𝑙 = lim 𝑘→∞ 𝑑(𝑠𝑝𝑘, 𝑠𝑞𝑘) = 0, Which is contradiction to our assumption 𝑙 > 0. Hence, lim 𝑛→∞ 𝜆𝑛 = 𝑙 = 0. Similarly, we can show that in the remaining two cases 𝑙 = lim 𝑛→∞ 𝜆𝑛 = 0. Now, let 𝑚, 𝑛 ∈ 𝑁 , 𝑤𝑖𝑡ℎ 𝑚 > 𝑛, We get lim 𝑛→∞ 𝑑(𝑠𝑛, 𝑠𝑚) ≤ lim 𝑛→∞ 𝜆𝑛 = 0. Hence, the sequence {𝑠𝑛} is a b- Cauchy sequence in K. Since K is complete, the sequence {𝑠𝑛} is convergent to some s* in M. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 520 https://internationalpubls.com Now we prove that s* is a fixed point of H. Assume that Hs* ≠s*, (Hs∗, s∗) > 0 , taking u= sn, v = s∗ 𝑑(𝑠𝑛+1, Hs∗) = 𝑑(𝐻𝑠𝑛, Hs∗) ≤ M(sn, s∗)……2.3.4 Where M(sn, s∗) = 𝑚𝑎𝑥 {𝛽 ( 𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn)] 1+𝑑(s∗,𝐻s∗) ) ( 𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn)] 1+𝑑(s∗,𝐻s∗) ), 𝛽 ( 𝑑(s∗,𝐻sn)[1+𝑑(s∗,𝐻s∗)] 1+𝑑(sn,𝐻s∗) ) ( 𝑑(s∗,𝐻sn)[1+𝑑(s∗,𝐻s∗)] 1+𝑑(sn,𝐻s∗) ), 𝛽 ( 𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn)] 1+𝑑(sn,𝐻s∗) ) ( 𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn)] 1+𝑑(sn,𝐻s∗) )} = 𝑚𝑎𝑥 {𝛽 ( 𝑑(sn,sn+1)[1+𝑑(𝐻s∗,sn+1)] 1+𝑑(s∗,𝐻s∗) ) ( 𝑑(sn,sn+1)[1+𝑑(𝐻s∗,sn+1)] 1+𝑑(s∗,𝐻s∗) ), 𝛽 ( 𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗)] 1+𝑑(sn,𝐻s∗) ) ( 𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗)] 1+𝑑(sn,𝐻s∗) ), 𝛽 ( 𝑑(s∗,𝐻s∗)[1+𝑑(s∗,sn+1)] 1+𝑑(sn,𝐻s∗) ) ( 𝑑(s∗,𝐻s∗)[1+𝑑(s∗,sn+1)] 1+𝑑(sn,𝐻s∗) )} Again, three cases will arise : Case (i) : Suppose that M(sn, s∗) = 𝛽 ( 𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn)] 1+𝑑(s∗,𝐻s∗) ) ( 𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn)] 1+𝑑(s∗,𝐻s∗) ) then from 2.3.4, we have 𝑑(𝑠𝑛+1, Hs∗) = 𝛽 ( 𝑑(sn, 𝐻sn)[1 + 𝑑(𝐻s∗, 𝐻sn)] 1 + 𝑑(s∗, 𝐻s∗) ) ( 𝑑(sn, 𝐻sn)[1 + 𝑑(𝐻s∗, 𝐻sn)] 1 + 𝑑(s∗, 𝐻s∗) ) as limit 𝑛 → ∞ on both sides of the above inequality and from the theorem (2.2.1) , we get lim 𝑛→∞ ( 𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn)] 1+𝑑(s∗,𝐻s∗) ) = 0, so that 1 𝑡 𝑑(𝑠∗, 𝐻𝑠∗) ≤ lim 𝑛→∞ sup 𝑑(𝑠𝑛+1, Hs∗) ≤ lim 𝑛→∞ 𝑠𝑢𝑝 𝛽 ( 𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn)] 1+𝑑(s∗,𝐻s∗) ) lim 𝑛→∞ 𝑠𝑢𝑝( 𝑑(sn,𝐻sn)[1+𝑑(𝐻s∗,𝐻sn)] 1+𝑑(s∗,𝐻s∗) ) = 0 And hence consequently, we get 𝑑(𝑠∗, 𝐻𝑠∗) = 0, which contradicts our assumption 𝑑(Hs∗, s∗) > 0 . Case (ii) : Suppose that M(sn, s∗) = 𝛽 ( 𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗)] 1+𝑑(sn,𝐻s∗) ) ( 𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗)] 1+𝑑(sn,𝐻s∗) ) then from 2.3.4, we have 𝑑(𝑠𝑛+1, Hs∗) = 𝛽 ( 𝑑(s∗, sn+1)[1 + 𝑑(s∗, 𝐻s∗)] 1 + 𝑑(sn, 𝐻s∗) ) ( 𝑑(s∗, sn+1)[1 + 𝑑(s∗, 𝐻s∗)] 1 + 𝑑(sn, 𝐻s∗) ) as limit 𝑛 → ∞ on both sides of the above inequality and from the theorem (i) , we get lim 𝑛→∞ ( 𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗)] 1+𝑑(sn,𝐻s∗) ) = 0, so that 1 𝑡 𝑑(𝑠∗, 𝐻𝑠∗) ≤ lim 𝑛→∞ sup 𝑑(𝑠𝑛+1, Hs∗) ≤ lim 𝑛→∞ 𝑠𝑢𝑝 𝛽 ( 𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗)] 1+𝑑(sn,𝐻s∗) ) ( 𝑑(s∗,sn+1)[1+𝑑(s∗,𝐻s∗)] 1+𝑑(sn,𝐻s∗) ) = 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 521 https://internationalpubls.com And hence consequently, we get 𝑑(𝑠∗, 𝐻𝑠∗) = 0, which contradicts our assumption 𝑑(Hs∗, s∗) > 0 . Case (iii) : Suppose that M(sn, s∗) = 𝛽 ( 𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn)] 1+𝑑(sn,𝐻s∗) ) ( 𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn)] 1+𝑑(sn,𝐻s∗) ) then from 2.3.4, we have 𝑑(𝑠𝑛+1, Hs∗) = 𝛽 ( 𝑑(s∗, 𝐻s∗)[1 + 𝑑(s∗, 𝐻sn)] 1 + 𝑑(sn, 𝐻s∗) ) ( 𝑑(s∗, 𝐻s∗)[1 + 𝑑(s∗, 𝐻sn)] 1 + 𝑑(sn, 𝐻s∗) ) as limit 𝑛 → ∞ in the above and from the theorem (i) , we get lim 𝑛→∞ ( 𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn)] 1+𝑑(sn,𝐻s∗) ) = 0, so that 1 𝑡 𝑑(𝑠∗, 𝐻𝑠∗) ≤ lim 𝑛→∞ sup 𝑑(𝑠𝑛+1, Hs∗) ≤ lim 𝑛→∞ 𝑠𝑢𝑝 𝛽 ( 𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn)] 1+𝑑(sn,𝐻s∗) ) ( 𝑑(s∗,𝐻s∗)[1+𝑑(s∗,𝐻sn)] 1+𝑑(sn,𝐻s∗) ) = 0 And hence consequently, we get 𝑑(𝑠∗, 𝐻𝑠∗) = 0, This is contradiction to our assumption, 𝑑(Hs∗, s∗) > 0 . Therefore from all these cases , we can get 𝑠∗ = 𝐻𝑠∗. Hence 𝑠∗ is a fixed point of H. Now, our aim is to prove that the uniqueness of the fixed point. Assume that 𝑟 ∈ 𝐻 is other fixed point of H such that𝑠∗ ≠ 𝑟, 𝑑(𝑠∗, 𝑟) > 0. From (2.3.1) 𝑑(𝑠∗, 𝑟) = 𝑑(𝐻𝑠∗, 𝑟) ≤ 𝑀(𝑠∗, 𝑟) = 𝑚𝑎𝑥 {𝛽 ( 𝑑(𝑠∗,𝐻𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗)] 1+𝑑(𝑟,𝐻𝑟) ) ( 𝑑(𝑠∗,𝐻𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗)] 1+𝑑(𝑟,𝐻𝑟) ), 𝛽 ( 𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝐻𝑟)] 1+𝑑(𝑠∗,𝐻𝑟) ) ( 𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝐻𝑟)] 1+𝑑(𝑠∗,𝐻𝑟) ), 𝛽 ( 𝑑(𝑟,𝐻𝑟)[1+𝑑(𝑟,𝐻𝑠∗)] 1+𝑑(𝑠∗,𝐻𝑟) ) ( 𝑑(𝑟,𝐻𝑟)[1+𝑑(𝑟,𝐻𝑠∗)] 1+𝑑(𝑠∗,𝐻𝑟) )} 𝑚𝑎𝑥 {𝛽 ( 𝑑(𝑠∗,𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗)] 1+𝑑(𝑟,𝑟) ) ( 𝑑(𝑠∗,𝑠∗)[1+𝑑(𝐻𝑟,𝐻𝑠∗)] 1+𝑑(𝑟,𝑟) ), 𝛽 ( 𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝑟)] 1+𝑑(𝑠∗,𝐻𝑟) ) ( 𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝑟)] 1+𝑑(𝑠∗,𝐻𝑟) ), 𝛽 ( 𝑑(𝑟,𝑟)[1+𝑑(𝑟,𝐻𝑠∗)] 1+𝑑(𝑠∗,𝐻𝑟) ) ( 𝑑(𝑟,𝑟)[1+𝑑(𝑟,𝐻𝑠∗)] 1+𝑑(𝑠∗,𝐻𝑟) )} = 𝛽 ( 𝑑(𝑟, 𝐻𝑠∗)[1 + 𝑑(𝑟, 𝑟)] 1 + 𝑑(𝑠∗, 𝐻𝑟) ) ( 𝑑(𝑟, 𝐻𝑠∗)[1 + 𝑑(𝑟, 𝑟)] 1 + 𝑑(𝑠∗, 𝐻𝑟) ) ≤ ( 𝑑(𝑟,𝐻𝑠∗)[1+𝑑(𝑟,𝑟)] 1+𝑑(𝑠∗,𝐻𝑟) ) < 1 𝑡 𝑑(𝑠∗, 𝑟) , which is contradiction. So that 𝑠∗ = 𝑟 hence 𝑠∗ is the only fixed point of H in M. We derive corollaries from Theorem2.3 Corollaries: Corollary 2.4. Let (𝐾, 𝑑, 𝑡) be a CbMS with 𝑡 ≥ 1 . Suppose that 𝐻: 𝐾 → 𝐾 be a self map and 𝛽 is the class of Geraghty functions 𝑆 then for any 𝑢, 𝑣 ∈ 𝐾 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 522 https://internationalpubls.com 𝑑(𝐻𝑢, 𝐻𝑣) ≤ 𝛽( 𝑁(𝑢, 𝑣))𝑁(𝑢, 𝑣) Where 𝑁(𝑢, 𝑣)= 𝑚𝑎𝑥 {( 𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢)] 1+𝑑(𝑣,𝐻𝑣) ), ( 𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣)] 1+𝑑(𝑢,𝐻𝑣) ),( 𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢)] 1+𝑑(𝑢,𝐻𝑣) )} . Then H has a unique fixed point 𝑢∗ ∈ 𝐾. 𝑃𝑟𝑜𝑜𝑓: For any u, v in K, N(u,v) one of the term of 𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢)] 1+𝑑(𝑣,𝐻𝑣) ), ( 𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣)] 1+𝑑(𝑢,𝐻𝑣) ),( 𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢)] 1+𝑑(𝑢,𝐻𝑣) ) Then it follows that 𝑑(𝐻𝑢, 𝐻𝑣) ≤ 𝛽( 𝑁(𝑢, 𝑣))𝑁(𝑢, 𝑣) ≤ 𝛽( 𝑁(𝑢, 𝑣))𝑁(𝑢, 𝑣) ≤ 𝑚𝑎𝑥 {𝛽 ( 𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢)] 1+𝑑(𝑣,𝐻𝑣) ) ( 𝑑(𝑢,𝐻𝑢)[1+𝑑(𝐻𝑣,𝐻𝑢)] 1+𝑑(𝑣,𝐻𝑣) ), 𝛽 ( 𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣)] 1+𝑑(𝑢,𝐻𝑣) ) ( 𝑑(𝑣,𝐻𝑢)[1+𝑑(𝑣,𝐻𝑣)] 1+𝑑(𝑢,𝐻𝑣) ), 𝛽 ( 𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢)] 1+𝑑(𝑢,𝐻𝑣) ) ( 𝑑(𝑣,𝐻𝑣)[1+𝑑(𝑣,𝐻𝑢)] 1+𝑑(𝑢,𝐻𝑣) )} = M(u, v), and observe that all the hypothesis of Theorem 2 are satisfies, and hence we can have that H has unique fixed point. The following is an example in support our main result. Examples. Example 2.5: Let 𝐾 = {1, 1 2 , 1 4 } 𝑈{0} and define the function from 𝑑: 𝐾𝑋𝐾 → 𝑅+ by 𝑑(𝑥, 𝑦) = (𝑥 − 𝑦)2, 𝐻(𝑥) = 1 𝑡2+1 , 𝑖𝑓 𝑥 = 1 𝑡 , 0 if x=0. We define𝛽(𝛼) = 𝑒−𝛼, 𝛼 > 0, 𝛽(0) = 0 , all the conditions holds and ‘0’ fixed point and is unique. Conclusions: In this we proved the existence and uniqueness of the fixed points generalized ciric type Geraghty rational contractions in b-metric spaces, our results extend some of the known theorems, kalo.et.al [1 ] proved fixed point theorems in Geraghty-ciric-type contraction mapping in b- metric spaces. We derived some corollaries and given examples in support our main result. References [1] Albray Gebremariam Kalo., Kidane Tola and Haider Ebrahim Yesuf: Fixed point results of Geraghty- ciric –type contraction mappings in b-metric space with applications, Fixed Point Theory Algorithms Sci Eng, 2024:8, doi.org/10.1186/s 1336-024-00764-3. 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