Int. J. Anal. Appl. (2022), 20:67 Existence Suzuki Type Fixed Point Results in Ab-Metric Spaces With Application P. Naresh1,2,∗, G. Upender Reddy2, B. Srinuvasa Rao3 1Department of Mathematics, Sreenidhi Institute of Science and Technology, Ghatkesar, Hyderabad-501301, Telangana, India 2Department of Mathematics, Mahatma Gandhi University, Nalgonda, Telangana, India 3Department of Mathematics, Dr. B.R. Ambedkar University, Srikakulam, Etcherla-532410, Andhra Pradesh, India ∗Corresponding author: parakala2@gmail.com Abstract. In this paper, we give some applications to integral equations as well as homotopy theory via Suzuki contractive type common coupled fixed point results in complete Ab-metric space. We also furnish an example which supports our main result. 1. Introduction The study of fixed points is aexquisite synthesis of analysis, topology, and geometry. Its numerous applications in areas such as homotopy theory, integral, integro-differential, and impulsive differen- tial equations, finding solutions to optimization problems, approximation theory, non-linear analysis, biomechanics, and algorithms have made it an essential tool. Suzuki [1] recently established expanded versions of Edelstein’s and Banach’s fundamental conclu- sions, sparking a great deal of research in this area (See [2–6]). The b-metric space was first introduced by I.A. Bakhtin [7] in 1989. Numerous generalizations of metric spaces were created as a result of the development of b-metric space.The n-tuple metric space was first introduced and its topological features were examined by M.Abbas et al. in 2015 [8]. Ab-metric spaces were first described by M. Ughade et al. [9] as a generalized version of n-tuple metric space. Then, in partially ordered Ab-metric Received: Oct. 21, 2022. 2010 Mathematics Subject Classification. 47H10, 54H25, 58C30, 58J20. Key words and phrases. Suzuki type contraction; ω-compatible; Ab-completeness; coupled common fixed points. https://doi.org/10.28924/2291-8639-20-2022-67 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-67 2 Int. J. Anal. Appl. (2022), 20:67 spaces, N.Mlaiki et al. [19] and K.Ravibabu et al. [11,12] discovered original coupled common fixed point theorems. The idea of coupled fixed point was first suggested in 1987 by Guo and Lakshmikantham [13]. Later, employing a weak contractivity type assumption, Bhaskar and Lakshmikantham [14] derived a novel fixed point theorem for a mixed monotone mapping on a metric space powered with partial ordering. See study findings in [15–20] and related sources for additional results on coupled fixed point outcomes. In the context of Ab-metric spaces, the purpose of the current research is to establish original common coupled fixed point theorems for Suzuki contractive type mapping. We can also provide examples that are appropriate and relevant applications to homotopy and integral equations. Before we can demonstrate the primary findings, we need certain fundamental definitions, results and examples from the literature. 2. Preliminaries Definition 2.1. [9] Let P be a non-empty set and b ≥ 1 be given real number. A mapping Ab : Pn → [0,∞) is called an Ab-metric on P if and only if for all λi,ν ∈ P i = 1,2,3, ..n; the following conditions hold : (Ab1) Ab(λ1,λ2, ........,λn−1,λn)≥ 0, (Ab2) Ab(λ1,λ2, ........,λn−1,λn)=0⇔ λ1 = λ2 = · · · · · ·= λn−1 = λn, (Ab3) Ab(λ1,λ2, ........,λn−1,λn)≤ b   Ab (λ1,λ1, ........,(λ1)n−1,ν) +Ab (λ2,λ2, ........,(λ2)n−1,ν) + · · · · · ·+Ab (λn−1,λn−1, ........,(λn−1)n−1,ν) +Ab (λn,λn, ........,(λn)n−1,ν)   Then the pair (P,Ab) is called an Ab-metric space. Remark 2.1. [9] It should be noted that, the class of Ab-metric spaces is effectively larger than that of A-metric spaces. Indeed each A-metric space is a Ab-metric space with b = 1.However, the converse is not always true. Also Ab-metric space is an "n-dimensional Sb-metric space. Therefore the Sb-metric are special cases of an Ab-metric with n =3. Following example shows that a Ab-metric on P need not be a A-metric on P. Example 2.1. [9] Let P = [0,+∞), define Ab :Pn → [0,+∞) as Ab (λ1,λ2, ........,λn−1,λn) = ∑n i=1 ∑ i 1. Definition 2.2. [9] A Ab- metric space (P,Ab) is said to be symmetric if Ab (λ,λ, · · · ,(λ)n−1,%)= Ab (%,%, · · · ,(%)n−1,λ) for all λ,% ∈P. Int. J. Anal. Appl. (2022), 20:67 3 Definition 2.3. [9] Let (P,Ab) be a Ab-metric space. Then, for λ ∈P, r > 0 we defined the open ball BAb(λ,r) and closed ball BAb[λ,r] with center λ and radius r as follows respectively: BAb(λ,r)= {% ∈P : Ab(%,%, · · · ,(%)n−1,λ) < r}, and BAb[λ,r] = {% ∈P : Ab(%,%, · · · ,(%)n−1,λ)≤ r}. Lemma 2.1. [9] In a Ab-metric space, we have (1) Ab(λ,λ, · · · ,(λ)n−1,%)≤ bAb(%,%, · · · ,(%)n−1,λ); (2) Ab(λ,λ, · · · ,(λ)n−1,ζ)≤ b(n−1)Ab(λ,λ, · · · ,(λ)n−1,%)+b2Ab(%,%, · · · ,(%)n−1,ζ). Definition 2.4. [9] If (P,Ab) be a Ab-metric space. A sequence {λk} in P is said to be: (1) Ab-Cauchy sequence if, for each � > 0, there exists n0 ∈ N such that Ab(λk,λk, · · · · · ·(λk)n−1,λm) < � for each m,k ≥ n0. (2) Ab-convergent to a point λ ∈P if, for each � > 0, there exists a positive integer n0 such that Ab(λk,λk, · · · · · ·(λk)n−1,λ) < � for all n ≥ n0 and we denote by lim k→∞ λk = λ. (3) A Ab-metric space (P,Ab) is called complete if every Ab-Cauchy sequence is Ab-convergent in P. Lemma 2.2. [9] If (P,Ab) be a Ab-metric space with b ≥ 1 and suppose that {λk} is a Ab-convergent to λ and {ζk} is a Ab-convergent to ζ, then we have (i) 1 b2 Ab(λ,λ, · · · ,(λ)n−1,ζ) ≤ lim k→∞ inf Ab(λk,λk, · · · ,(λk)n−1,ζk) ≤ lim k→∞ supAb(λk,λk, · · · ,(λk)n−1,ζk) ≤ b2Ab(λ,λ, · · · ,(λ)n−1,ζ). In particular, if ζk = ζ is constant, then (ii) 1 b2 Ab(λ,λ, · · · ,(λ)n−1,ζ) ≤ lim k→∞ inf Ab(λk,λk, · · · ,(λk)n−1,ζ) ≤ lim k→∞ supAb(λk,λk, · · · ,(λk)n−1,ζ) ≤ b2Ab(λ,λ, · · · ,(λ)n−1,ζ). Theorem 2.1. [1] Let (P;d) be a complete metric space, let T :P →P be a mapping and define a non increasing function θ : [0;1)→ (0;1] by θ(t)=   1, 0≤ t ≤ √ 5−1 2 (1− t)t−2, √ 5−1 2 ≤ t ≤ 1√ 2 (1+ t)−1, 1√ 2 < t ≤ 1 . 4 Int. J. Anal. Appl. (2022), 20:67 Assume that there exists t ∈ [0;1) such that θ(t)d(ρ,Tρ)≤ d(ρ;%) implies d(Tρ;T%)≤ td(ρ;%) for all ρ,% ∈P. Then there exists a unique fixed point a of T . Moreover, lim n→∞ Tnρ = a for all ρ ∈P. In order to obtain our results we need to consider the followings. 3. Main Results Definition 3.1. Let (P,Ab) be a Ab-metric spaces and suppose F : P2 → P be a mapping. If F (ρ,%)= ρ, F (%,ρ)= % for ρ,% ∈P then (ρ,%) is called a coupled fixed point of F. Definition 3.2. Let (P,Ab) be a Ab-metric spaces and suppose F :P2 →P and f :P →P be two mappings. An element (ρ,%) is said to be a coupled coincident point of F and f if F (ρ,%) = f ρ, F (%,ρ)= f %. Definition 3.3. Let (P,Ab) be a Ab-metric spaces and suppose F : P2 → P, f : P → P be two mappings. An element (ρ,%) is said to be a coupled common point of F and f if F (ρ,%) = f ρ = ρ, F (%,ρ)= f % = %, Definition 3.4. Let (P,Ab) be a Ab-metric space. A pair (F,f ) is called weakly compatible if f (F(ρ,%))= F(f ρ,f %) whenever for all ρ,% ∈P such that F (ρ,%)= f ρ, F (%,ρ)= f %. Theorem 3.1. Let (P,Ab) be a Ab-metric space. Suppose that T : P2 → P and f : P → P be a two mappings satisfying the following: η(θ)Ab (f λ,f λ, · · · ,(f λ)n−1,T(λ,ζ))≤max   Ab (f λ,f λ, · · · ,(f λ)n−1, f ρ) , Ab (f ζ, f ζ, · · · ,(f ζ)n−1, f %) Ab (f λ,f λ, · · · ,(f λ)n−1,T(λ,ζ)) , Ab (f ζ, f ζ, · · · ,(f ζ)n−1,T(ζ,λ)) ,   implies Ab (T(λ,ζ),T(λ,ζ), · · · ,(T(λ,ζ))n−1,T(ρ,%))≤ θmax   Ab (f λ,f λ, · · · ,(f λ)n−1, f ρ) , Ab (f ζ, f ζ, · · · ,(f ζ)n−1f %) , Ab (f λ,f λ, · · · ,(f λ)n−1,T(λ,ζ)) , Ab (f ζ, f ζ, · · · ,(f ζ)n−1,T(ζ,λ)) , Ab (f ρ,f ρ, · · · ,(f ρ)n−1,T(ρ,%)) , Ab (f %,f %, · · · ,(f %)n−1,T(%,ρ)) , Ab (f ρ,f ρ, · · · ,(f ρ)n−1,T(λ,ζ)) , Ab (f %,f %, · · · ,(f %)n−1T(ζ,λ))   . (3.1) Int. J. Anal. Appl. (2022), 20:67 5 For all λ,ζ,ρ,% ∈P, where θ ∈ [0,1) and η : [0,1)→ (0,1] defined as η(θ)= 1 b2((n−1)+θ)is a strictly decreasing function, a) T(P2)⊆ f (P) and f (P) is complete, b) pair (T,f ) is ω-compatible. Then there is a unique common coupled fixed point of T and f in P. Proof. Let λ0,ζ0 ∈P be arbitrary, and from (a), we construct the sequences {λp} ,{ζp} , in P as T (λp,ζp)= f λp+1, T (ζp,λp)= f ζp+1, where p =0,1,2, . . . . Case (i): Assume that f λp 6= f λp+1 or f ζp 6= f ζp+1∀ p. (3.2) Since η(θ)Ab (f λ0, f λ0, · · · ,T(λ0,ζ0)) = η(θ)Ab (f λ0, f λ0, · · · , f λ1) ≤ Ab (f λ0, f λ0, · · · , f λ1) ≤ max   Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) , Ab (f λ0, f λ0, · · · ,T(λ0,ζ0)) , Ab (f ζ0, f ζ0, · · · ,T(ζ0,λ0))   . Then from (3.1), we can get Ab(f λ1, f λ1, · · · , f λ2) = Ab (T(λ0,ζ0),T(λ0,ζ0), · · · ,T(λ1,ζ1)) ≤ θmax   Ab (f λ0, f λ0, · · · , f λ1) ,Ab (f ζ0, f ζ0, · · · , f ζ1) , Ab (f λ0, f λ0, · · · ,T(λ0,ζ0)) ,Ab (f ζ0, f ζ0, · · · ,T(ζ0,λ0)) , Ab (f λ1, f λ1, · · · ,T(λ1,ζ1)) ,Ab (f ζ1, f ζ1, · · · ,T(ζ1,λ1)) , Ab (f λ1, f λ1, · · · ,T(λ0,ζ0)) ,Ab (f ζ1, f ζ1, · · · ,T(ζ0,λ0))   ≤ θmax { Ab (f λ0, f λ0, · · · , f λ1) ,Ab (f ζ0, f ζ0, · · · , f ζ1) , Ab (f λ1, f λ1, · · · , f λ2) ,Ab (f ζ1, f ζ1, · · · , f ζ2) } (3.3) Similarly, we can prove that Ab(f ζ1, f ζ1, · · · , f ζ2)≤ θmax   Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) , Ab (f λ1, f λ1, · · · , f λ2) , Ab (f ζ1, f ζ1, · · · , f ζ2)   . (3.4) 6 Int. J. Anal. Appl. (2022), 20:67 Due to (3.3)− (3.4), we conclude that max { Ab(f λ1, f λ1, · · · , f λ2), Ab(f ζ1, f ζ1, · · · , f ζ2) } ≤ θmax   Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) , Ab (f λ1, f λ1, · · · , f λ2) , Ab (f ζ1, f ζ1, · · · , f ζ2)   . (3.5) If max { Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) } ≤max { Ab (f λ1, f λ1, · · · , f λ2) , Ab (f ζ1, f ζ1, · · · , f ζ2) } . Then from (3.5), we have f λ1 = f λ2 or f ζ1 = f ζ2. It is contradiction to (3.2). Hence from (3.5), we have max { Ab (f λ1, f λ1, · · · , f λ2) , Ab (f ζ1, f ζ1, · · · , f ζ2) } ≤ θmax { Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) } . Continuing in this way, we get max { Ab (f λp, f λp, · · · , f λp+1) , Ab (f ζp, f ζp, · · · , f ζp+1) } ≤ θmax { Ab (f λp−1, f λp−1, · · · , f λp) , Ab (f ζp−1, f ζp−1, · · · , f ζp) } ≤ θ2max { Ab (f λp−2, f λp−2, · · · , f λp−1) , Ab (f ζp−2, f ζp−2, · · · , f ζp−1) } ... ≤ θp max { Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) } . Thus Ab (f λp, f λp, · · · , f λp+1)≤ θp max { Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) } , and Ab (f ζp, f ζp, · · · , f ζp+1)≤ θp max { Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) } Now for q > p, by use of (Ab3), we have Ab (f λp, f λp, · · · ,(f λp)n−1, f λq)≤ b   Ab (f λp, f λp, ........,(f λp)n−1, f λp+1) +Ab (f λp, f λp, ........,(f λp)n−1, f λp+1) + · · · · · ·+Ab (f λp,λp, ........,(f λp)n−1, f λp+1) +Ab (f λq, f λq, ........,(f λq)n−1, f λp+1)   ≤ b(n−1)Ab (f λp, f λp, ........,(f λp)n−1, f λp+1) +bAb (f λq, f λq, ........,(f λq)n−1, f λp+1) Int. J. Anal. Appl. (2022), 20:67 7 ≤ b(n−1)Ab (f λp, f λp, ........,(f λp)n−1, f λp+1) +b2Ab (f λp+1, f λp+1, ........,(f λp+1)n−1, f λq) ≤ b(n−1)Ab (f λp, f λp, ........,(f λp)n−1, f λp+1) +b3(n−1)Ab (f λp+1, f λp+1, ........,(f λp+1)n−1, f λp+2) +b4Ab (f λp+2, f λp+2, ........,(f λp+2)n−1, f λq) ≤ b(n−1)Ab (f λp, f λp, ........,(f λp)n−1, f λp+1) +b3(n−1)Ab (f λp+1, f λp+1, ........,(f λp+1)n−1, f λp+2) +b5(n−1)Ab (f λp+2, f λp+2, ........,(f λp+2)n−1, f λp+3) +b7(n−1)Ab (f λp+3, f λp+3, ........,(f λp+3)n−1, f λp+4) + . . .+b2q−2p−2(n−1)Ab (f ζq−2, f ζq−2, ........,(f ζq−2)n−1, f λq−1) +b2q−2p−3Ab (f λq−1, f λq−1, ........,(f λq−1)n−1, f λq) ≤ (n−1) ( bθp +b3θp+1 +b5θp+2 + . . .+b2q−2p−2θq−2 ) max { Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) } +b2q−2p−3θq−1max { Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) } ≤ (n−1)bθp ( 1+b2θ+b4θ2 + . . .+b2q−2p−4θq−p−2 ) max { Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) } +b2q−2p−3θq−p−1max { Ab (f λ0, f λ0, · · · , f λ1) ,Ab (f ζ0, f ζ0, · · · , f ζ1) } ≤ (n−1)bθp ( 1+b2θ+b4θ2 +b6θ3 + . . . ) max { Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) } ≤ (n−1)bθp 1−b2θ max { Ab (f λ0, f λ0, · · · , f λ1) , Ab (f ζ0, f ζ0, · · · , f ζ1) } → 0 as p,q →∞. Hence {f λp} is a Cauchy sequence in f (P) . Similarly we can show that {f ζp}, is Cauchy sequence in f (P). Since f (P) is complete, there exist α,β and a,b in P such that lim p→∞ f λp = α = f a lim p→∞ f ζp = β = f b Since f λp → α, f ζp → β, we may assume that f λp 6= α, f ζp 6= β for infinitely many p. We claim that max { Ab (f a,f a, · · · ,T(λ,ζ)) , Ab (f b,f b, · · · ,T(ζ,λ)) , } ≤ θmax   Ab (f a,f a, · · · , f λ) ,Ab (f b,f b, · · · , f ζ) , Ab (f λ,f λ, · · · ,T(λ,ζ)) , Ab (f ζ, f ζ, · · · ,T(ζ,λ))   8 Int. J. Anal. Appl. (2022), 20:67 for all λ,ζ ∈P with f a 6= f λ,f b 6= f ζ. Let λ,ζ ∈P with f a 6= f λ,f b 6= f ζ. Then there exists a positive integer p0 such that for p ≥ p0, we have Ab (f a,f a, · · · , f λp)≤ 12b2(n−1)Ab (f a,f a, · · · , f λ), Ab (f b,f b, · · · , f ζp)≤ 12b2(n−1)Ab (f b,f b, · · · , f ζ). Now for p ≥ p0, η(θ)Ab (f λp, f λp, · · · ,(f λp)n−1,T(λp,ζp)) ≤ Ab (f λp, f λp, · · · ,(f λp)n−1,T(λp,ζp)) = Ab (f λp, f λp, · · · ,(f λp)n−1, f λp+1) ≤ b(n−1)Ab (f λp, f λp, · · ·(f λp)n−1, f a) +b2Ab (f a,f a, · · · ,(f a)n−1, f λp+1) ≤ b2(n−1)Ab (f a,f a · · · ,(f a)n−1, f λp) +b2Ab (f a,f a, · · · ,(f a)n−1, f λp+1) ≤ b2(n−1)Ab (f a,f a · · · ,(f a)n−1, f λp) +b2(n−1)Ab (f a,f a, · · · ,(f a)n−1, f λp+1) ≤ 1 2 Ab (f a,f a · · · ,(f a)n−1, f λ) + 1 2 Ab (f a,f a, · · · ,(f a)n−1, f λ) ≤ Ab (f a,f a · · · ,(f a)n−1, f λ) ≤ Ab (f λ,f λ, · · · , f λp) ≤ max { Ab (f λ,f λ, · · · , f λp) ,Ab (f ζ, f ζ, · · · , f ζp) , Ab (f λp, f λp, · · · ,T(λp,ζp)) ,Ab (f ζp, f ζp, · · · ,T(ζp,λp)) } . From (3.1), we have Ab (T(λp,ζp),T(λp,ζp), · · · ,T(λ,ζ)) ≤ θmax   Ab (f λp, f λp, f λ) ,Ab (f ζp, f ζp, f ζ) , Ab (f λp, f λp, f λp+1) ,Ab (f ζp, f ζp, f ζp+1) , Ab (f λ,f λ, · · · ,T(λ,ζ)) ,Ab (f ζ, f ζ, · · · ,T(ζ,λ)) , Ab (f λ,f λ, · · · , f λp+1) ,Ab (f ζ, f ζ, · · · , f ζp+1)   . Letting p →∞, we get Ab (f a,f a, · · · ,T(λ,ζ))≤ θmax { Ab (f a,f a, · · · , f λ) ,Ab (f b,f b, · · · , f ζ) , Ab (f λ,f λ, · · · ,T(λ,ζ)) ,Ab (f ζ, f ζ, · · · ,T(ζ,λ)) } . Int. J. Anal. Appl. (2022), 20:67 9 Similarly we can show that Ab (f b,f b, · · · ,T(ζ,λ))≤ θmax { Ab (f b,f b, · · · , f ζ) ,Ab (f a,f a, · · · , f λ) , Ab (f λ,f λ, · · · ,T(λ,ζ)) ,Ab (f ζ, f ζ, · · · ,T(ζ,λ)) } . Thus max { Ab (f a,f a, · · · ,T(λ,ζ)) , Ab (f b,f b, · · · ,T(ζ,λ)) , } ≤ θmax   Ab (f a,f a, · · · .f λ) ,Ab (f b,f b, · · · , f ζ) , Ab (f λ,f λ, · · · ,T(λ,ζ)) , Ab (f ζ, f ζ, · · · ,T(ζ,λ))   . Hence the claim. Now consider Ab (f λ,f λ, · · · ,T(λ,ζ)) ≤ (n−1)bAb (f λ,f λ, · · · , f a)+b2Ab (f a,f a, · · · ,T(λ,ζ)) ≤ (n−1)b2Ab (f a,f a, · · · , f λ) +b2θmax { Ab (f a,f a, · · · , f λ) ,Ab (f b,f b, · · · , f ζ) , Ab (f λ,f λ, · · · ,T(λ,ζ)) ,Ab (f ζ, f ζ, · · · ,T(ζ,λ)) , } ≤ b2 ((n−1)+θ)max   Ab (f a,f a, · · · , f λ) ,Ab (f b,f b, · · · , f ζ) , Ab (f λ,f λ, · · · ,T(λ,ζ)) , Ab (f ζ, f ζ, · · · ,T(ζ,λ))   . Thus η(θ)Ab (f λ,f λ, · · · ,T(λ,ζ))≤max { Ab (f a,f a, · · · , f λ) ,Ab (f b,f b, · · · , f ζ) , Ab (f λ,f λ, · · · ,T(λ,ζ)) ,Ab (f ζ, f ζ, · · · ,T(ζ,λ)) } . Hence from (3.1), we have Ab (T(λ,ζ),T(λ,ζ), · · · ,T(a,b)) ≤ θmax   Ab (f λ,f λ, · · · , f a) ,Ab (f ζ, f ζ, · · · , f b) , Ab (f λ,f λ, · · · ,T(λ,ζ)) ,Ab (f ζ, f ζ, · · · ,T(ζ,λ)) , Ab (f a,f a, · · · ,T(a,b)) ,Ab (f b,f b, · · · ,T(b,a)) , Ab (f a,f a, · · · ,T(λ,ζ)) ,Ab (f b,f b, · · · ,T(ζ,λ))   . Now Ab (f a,f a, · · · ,T(a,b))= lim p→∞ Ab (f λp+1, f λp+1, · · · ,T(a,b)) = lim p→∞ Ab (T(λp,yp),T(λp,ζp), · · · ,T(a,b)) ≤ lim p→∞ θmax   Ab (f λp, f λp, · · · , f a) ,Ab (f ζp, f ζp, · · · f b) , Ab (f a,f a, · · · ,T(a,b)) ,Ab (f b,f b, · · · ,T(b,a)) , Ab (f λp, f λp, · · · ,T(λp,ζp)) ,Ab (f ζp, f ζp, · · · ,T(ζp,λp)) , Ab (f a,f a, · · · ,T(λp,ζp)) ,Ab (f b,f b, · · · ,T(ζp,λp)) ,   ≤ θmax { Ab (f a,f a, · · · ,T(a,b)) ,Ab (f b,f b, · · · ,T(b,a)) } . 10 Int. J. Anal. Appl. (2022), 20:67 Similarly, we can have Ab (f b,f b, · · · ,T(b,a))≤ θmax { Ab (f a,f a, · · · ,T(a,b)) ,Ab (f b,f b, · · · ,T(b,a)) } Thus max { Ab (f a,f a, · · · ,T(a,b)) , Ab (f b,f b, · · · ,T(b,a)) , } ≤ θmax { Ab (f a,f a, · · · ,T(a,b)) , Ab (f b,f b, · · · ,T(b,a)) } . So that T(a,b)= f a and T(b,a)= f b. Thus (a,b) is a coupled coincidence point of T and f . Since the pair (T,f ) is ω-compatible, we have f α = f 2a = f (T(a,b))= T(f a,f b)= T (α,β) f β = f 2b = f (T(b,a))= T(f b,f a)= T (β,α) (3.6) Now η(θ)Ab (f α,f α, · · · ,T (α,β))=0≤max { Ab (f a,f a, · · · , f α) ,Ab (f b,f b, · · · , f β) , Ab (f α,f α, · · · ,T (α,β)) ,Ab (f β,f β, · · · ,T (β,α)) } . Hence from (3.1), we have Ab(f α,f α, · · · , f a)= Ab (T(α,β),T(α,β), · · · ,T(a,b)) ≤ θmax   Ab (f α,f α, · · · , f a) ,Ab (f β,f β, · · · , f b) , Ab (f α,f α, · · · ,T(α,β)) ,Ab (f β,f β, · · · ,T(β,α)) , Ab (f a,f a, · · · ,T(a,b)) ,Ab (f b,f b, · · · ,T(b,a)) , Ab (f a,f a, · · · ,T(α,β)) ,Ab (f b,f b, · · · ,T(β,α))   . ≤ θmax { Ab (f α,f α, · · · , f a) ,Ab (f β,f β, · · · , f b) } . Similarly, we have Ab(f β,f β, · · · , f b)≤ θmax { Ab (f α,f α, · · · , f a) ,Ab (f β,f β, · · · , f b) } . Thus max { Ab (f α,f α, · · · , f a) ,Ab (f β,f β, · · · , f b) , } ≤ θmax { Ab (f α,f α, · · · , f a) , Ab (f β,f β, · · · , f b) } . Hence α = f a = f α and β = f b = f β,. Hence from (3.6), we have (α,β) is a common coupled fixed point of T and f . In the following we will show the uniqueness of common coupled fixed point in P. For this purpose, assume that there is another coupled fixed point (α′,β′) of T,f . Now consider, η(θ)Ab (f α,f α, · · · ,T (α,β))=0≤max { Ab (f α,f α, · · · , f α′) ,Ab (f β,f β, · · · , f β′) , Ab (f α,f α, · · · ,T (α,β)) ,Ab (f β,f β, · · · ,T (β,α)) } Int. J. Anal. Appl. (2022), 20:67 11 by (3.1), we have Ab(α,α, · · · ,α′) = Ab ( T(α,β),T(α,β), · · · ,T(α′,β′) ) ≤ θmax { Ab (α,α, · · · ,α′) ,Ab (β,β, · · · ,β′) } . Similarly, we can show that Ab(β,β, · · · ,β′) ≤ θmax { Ab (α,α, · · · ,α′) ,Ab (β,β, · · · ,β′) } . Thus max { Ab (α,α, · · · ,α′) ,Ab (β,β, · · · ,β′) } ≤ θmax { Ab (α,α, · · · ,α′) ,Ab (β,β, · · · ,β′) } . Hence α = α′,β = β′. Thus (α,β) is the unique common coupled fixed point of T and f . case(ii): If f λp = f λp+1, f ζp = f ζp+1 for some p then f λp = T (λp,ζp), f ζp = T (ζp,λp) so that (λp,ζp) is a coupled coincidence point of T and f . Now proceeding as in case (i) with f λp = α, f ζp = β, we can show that (α,β) is the unique common coupled fixed point of T and f . � Example 3.1. Let P = [0,+∞), define Ab :Pn → [0,+∞) as Ab (λ1,λ2, ........,λn−1,λn) = ∑n i=1 ∑ i p, by use of (Ab3), we have Ab (λp,λp, · · · ,λq)≤ b(n−1)Ab (λp,λp, · · · ,λp+1)+b2Ab (λp+1,λp+1, · · · ,λq) Letting p →∞, we get lim p→∞ Ab (λp,λp, · · · ,λq) ≤ lim p→∞ b2Ab (H(λp,ζp,κp),H(λp,ζp,κp), · · · ,H(λq−1,ζq−1,κq−1)) ≤ lim p→∞ b2q−2p−3Ab (H(ζq−2,ζq−2,κq−2),H(ζq−2,ζq−2,κq−2), · · · ,H(λq−1,ζq−1,κq−1)) 16 Int. J. Anal. Appl. (2022), 20:67 ≤ lim p→∞ b2q−2p−3θmax   Ab (ζq−2,ζq−2, · · · ,λq−1) ,Ab (ζq−2,ζq−2, · · · ,ζq−1) , Ab (ζq−2,ζq−2, · · · ,H(ζq−2,ζq−2,κq−2)) , Ab (ζq−2,ζq−2, · · · ,H(ζq−2,ζq−2,κq−2)) , Ab (λq−1,λq−1, · · · ,H(λq−1,yq−1,κq−1)) , Ab (ζq−1,ζq−1, · · · ,H(ζq−1,λq−1,κq−1)) , Ab (λq−1,λq−1, · · · ,H(ζq−2,ζq−2,κq−2)) , Ab (ζq−1,ζq−1, · · · ,H(ζq−2,ζq−2,κq−2))   . ≤ lim p→∞ b2q−2p−3θq−p−1max { Ab (λ0,λ0, · · · ,λ1) ,Ab (ζ0,ζ0, · · · ,ζ1) } → 0 as q →∞. Hence {λp} is a Cauchy sequence in Ab metric spaces (P,Ab). Similarly we can show that {ζp}, is Cauchy sequence in (P,Ab) and by the completeness of (P,Ab), there exist a,b ∈P with lim p→∞ λp+1 = a lim p→∞ ζp+1 = b (5.2) Since η(θ)Ab (a,a, · · · ,H(a,b,κ))≤max { Ab (a,a, · · · ,λp) ,Ab (b,b, · · · ,ζp) , Ab (a,a, · · · ,H(a,b,κ)) ,Ab (b,b, · · · ,H(b,a,κ)) } we have Ab (a,a, · · · ,H(a,b,κ)) ≤ lim p→∞ Ab (H(λp,ζp,κ),H(λp,ζp,κ), · · · ,H(a,b,κ)) ≤ lim n→∞ θmax   Ab (λp,λp, · · · ,a) ,Ab (ζp,ζp, · · · ,b) , Ab (a,a, · · · ,H(a,b,κ)) ,Ab (b,b, · · · ,H(b,a,κ)) , Ab (λp,λp, · · · ,H(λp,ζp,κ)) ,Ab (ζp,ζp, · · · ,H(ζp,λp,κ)) , Ab (a,a, · · · ,H(λp,ζp,κ)) ,Ab (b,b, · · · ,H(ζp,λp,κ))   ≤ θmax { Ab (a,a, · · · ,H(a,b,κ)) ,Ab (b,b, · · · ,H(b,a,κ)) } . Therefore, max { Ab (a,a, · · · ,H(a,b,κ)) , Ab (b,b, · · · ,H(b,a,κ)) } ≤ θmax { Ab (a,a, · · · ,H(a,b,κ)) , Ab (b,b, · · · ,H(b,a,κ)) } . It follows that H(a,b,κ) = a,H(b,a,κ) = b. Thus κ ∈ P. Hence P is closed in [0,1]. Let κ0 ∈ P, then there exist λ0,ζ0 ∈ U with λ0 = H(λ0,ζ0,κ0), ζ0 = H(ζ0,λ0,κ0). Since U is open, then there exist r > 0 such that BAb(λ0, r)⊆ U. Choose κ ∈ (κ0 − �,κ0 + �) such that |κ−κ0| ≤ 1 Mp < � 2 , then for λ ∈ BAb(λ0, r)= {λ ∈P/Ab(λ,λ, · · · ,λ0)≤ r +Ab(λ0,λ0, · · · ,λ0)}. Also η(θ)Ab (λ,λ, · · · ,H(λ0,ζ0,κ))≤max { Ab (λ,λ, · · · ,λ0) ,Ab (ζ,ζ, · · · ,ζ0) , Ab (λ,λ, · · · ,H(λ0,ζ0,κ)) ,Ab (ζ,ζ, · · · ,H(ζ0,λ0,κ)) } Int. J. Anal. Appl. (2022), 20:67 17 Now we have Ab (H(λ,ζ,κ),H(λ,ζ,κ), · · · ,λ0) = Ab (H(λ,ζ,κ),H(λ,ζ,κ), · · · ,H(λ0,ζ0,κ0)) ≤ (n−1)bAb (H(λ,ζ,κ),H(λ,ζ,κ), · · · ,H(λ,ζ,κ0)) +b2Ab (H(λ,ζ,κ0),H(λ,ζ,κ0), · · · ,H(λ0,ζ0,κ0)) ≤ b(n−1)M|κ−κ0|+b2Ab (H(λ,ζ,κ0),H(λ,ζ,κ0), · · · ,H(λ0,ζ0,κ0)) ≤ b(n−1) 1 Mp−1 +b2Ab (H(λ,ζ,κ0),H(λ,ζ,κ0), · · · ,H(λ0,ζ0,κ0)) . Letting p →∞, we obtain Ab (H(λ,ζ,κ),H(λ,ζ,κ), · · · ,λ0) ≤ b2Ab (H(λ,ζ,κ0),H(λ,ζ,κ0), · · · ,H(λ0,ζ0,κ0)) . ≤ b2θmax   Ab (λ,λ, · · · ,λ0) ,Ab (ζ,ζ, · · · ,ζ0) , Ab (λ,λ, · · · ,H(λ,ζ,κ)) ,Ab (ζ,ζ, · · · ,H(ζ,λ,κ)) , Ab (λ0,λ0, · · · ,H(λ0,ζ0,κ)) ,Ab (ζ0,ζ0, · · · ,H(ζ0,λ0,κ)) , Ab (λ0,λ0, · · · ,H(λ,ζ,κ)) ,Ab (ζ0,ζ0, · · · ,H(ζ,λ,κ))   . ≤ b2θmax { Ab (λ,λ, · · · ,λ0) ,Ab (ζ,ζ, · · · ,ζ0) } . Therefore, we have max { Ab (H(λ,ζ,κ),H(λ,ζ,κ), · · · ,λ0) Ab (H(ζ,λ,κ),H(ζ,λ,κ), · · · ,ζ0) } ≤ b2θmax { Ab (λ,λ, · · · ,λ0) ,Ab (ζ,ζ, · · · ,ζ0) } ≤ b2θmax { r +Ab (λ0,λ0, · · · ,λ0) , r +Ab (ζ0,ζ0, · · · ,ζ0) } . Thus for each fixed κ ∈ (κ0 − �,κ0 + �), H(.,κ) : BAb(λ0, r)→ BAb(λ0, r), H(.,κ) : BAb(ζ0, r)→ BAb(ζ0, r). Then all conditions of Theorem 5.1 are satisfied. Thus we conclude that H(.,κ) has a coupled fixed point in U 2 . But this must be in U2. Since (τ0) holds. Thus, κ ∈P for any κ ∈ (κ0−�,κ0+�). Hence (κ0−�,κ0+�)⊆P. Clearly P is open in [0, 1]. 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Anal. 4 (2010), 2381-2388. https://www.jstor.org/stable/20535364 https://doi.org/10.1155/2011/736063 https://doi.org/10.1155/2011/736063 https://doi.org/10.5120/7239-0073 https://doi.org/10.1186/1687-1812-2012-126 https://doi.org/10.1016/j.na.2009.04.017 https://doi.org/10.1186/s13663-015-0309-2 https://doi.org/10.22436/jnsa.010.04.35 https://doi.org/10.30495/maca.2022.1949822.1046 https://doi.org/10.30495/maca.2022.1949822.1046 https://doi.org/10.1016/0362-546x(87)90077-0 https://doi.org/10.1016/j.na.2005.10.017 https://doi.org/10.1016/j.amc.2010.05.042 https://doi.org/10.1016/j.amc.2010.05.042 https://hrcak.srce.hr/93280 https://dergipark.org.tr/en/download/article-file/83133 https://dergipark.org.tr/en/download/article-file/83133 https://doi.org/10.1186/1687-1812-2012-66 1. Introduction 2. Preliminaries 3. Main Results 4. Application to Integral Equations 5. Application to Homotopy 6. Conclusion References