Ratio Mathematica Volume 42, 2022 Common Fixed Point for a Contractive Mapping via (α, β, ψ)-Admissibility in b-metric space Jahir Hussain Rasheed* Maheshwaran Kanthasamy† Dhamodharan Durairaj‡ Abstract In this paper, we establish the concept of a common fixed point theo- rem for new type of generalized contractive mappings. Furthermore, we employ our main result to shows a common fixed point theorem for a pair of self-mappings (R,S) in b-metric space via (α,β,ψ)- admissibility type contractive condition. An example is also given to verify the main result. Keywords: α-admissible mapping, common fixed point, b-metric spaces. 2020 AMS subject classifications: 47H10, 54H25, 54M20. 1 *Department of Mathematics, Jamal Mohamed College (Autonomous) (Affiliated to Bharathi- dasan University), Tiruchirappalli-620020, Tamilnadu, India; hssn jhr@yahoo.com. †Department of Mathematics, Jamal Mohamed College (Autonomous) (Affiliated to Bharathi- dasan University), Tiruchirappalli-620020, Tamilnadu, India; mahesksamy@gmail.com. ‡Department of Mathematics, Jamal Mohamed College (Autonomous) (Affiliated to Bharathi- dasan University), Tiruchirappalli-620020, Tamilnadu, India; dharan raj28@yahoo.co.in. 1Received on May 1st, 2022. Accepted on June 28th, 2022. Published on June 30th, 2022. doi: 10.23755/rm.v41i0.781. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 245 R. Jahir Hussain, K. Maheshwaran, D. Dhamodharan 1 Introduction In the last fifty years, fixed point theories lie in finding and proving the unique- ness of solutions for many questions of Applied Sciences such as Physics, Chem- istry, Economics, and Engineering. In 1922, Stefan Banach [S.Banach [1922]] proved a fixed point theorem for contractive mappings in complete metric spaces. In 1969, Nadler [Nadler [1969]] introduced the concept of multi-value function. Later, Czerwik [Czerwik [1993]] and Bakhtin [Bakhtin [1989]] initiate the con- cept of b-metrics metric space. Khan [Khan et al. [1984]] introduced the altering distance mapping to formulate a new contractive condition in fixed point theory in order to extend the Banach fixed point theorem to new forms. For some ex- tension to the Banach contraction theorem. Recently, Abodayeh et al. [Abodayeh et al. [2017]] introduced a new notion, named almost perfect function, to formu- late new contractive conditions to modify and extend some fixed point theorems known in the literature. Now, we mention the notions of altering distance function and almost perfect function. 2 Preliminaries Definition 2.1 (Khan et al. [1984] ). A self-function ψ on R+ ∪ {0} is called an altering distance function if ψ satisfies the following conditions: (1) ψ(s) = 0 ⇐⇒ s = 0. (2) ψ is a nondecreasing and continuous function. Definition 2.2 (Abodayeh et al. [2017]). A nondecreasing self-function ψ on R+∪ {0} is called an almost perfect function if ψ satisfies the following conditions: (1) ψ(s) = 0 ⇐⇒ s = 0. (2) If for all sequence (sn) in R+ ∪ {0} with ψ(sn) → 0 it holds sn → 0. Definition 2.3 (Samet et al. [2012]). Let R be a self-mapping on X and α : X × X → R+ ∪ {0} be a function. Then, R is called α-admissible if for all v, w ∈ X with α(v, w) ≥ 1 it holds α(Rv, Rw) ≥ 1. The definition of triangular α-admissibility for a single mapping Definition 2.4 (Karupinar et al. [2013]). Let R be a self-mapping on X and α : X × X → R+ ∪ {0}. Then, we call R triangular α-admissible if (1) R is α-admissible; and 246 Common fixed point for a contractive mapping via (α, β, ψ)-admissibility in b-metric space (2) For all v, w, u ∈ X with α(v, w) ≥ 1 and α(w, u) ≥ 1 it holds α(v, w) ≥ 1. Definition 2.5 (Abdeljawad [2013]). Let R and S be two self mappings on X and α : X × X → R+ ∪ {0} be a function. Then, the pair (R, S) is called α-admissible if z, w ∈ X and α(z, w) ≥ 1 imply α(Rz, Sw) ≥ 1 and α(Sz, Rw) ≥ 1. Definition 2.6 (Hussain et al. [2014]). Let Db be a metric on a set X and α, β : X × X → R+ ∪ {0} be functions. Then, X is called α, β-complete if and only if {xn} is a Cauchy sequence in X and α(xn, xn+1) ≥ β(xn, xn+1) for all n ∈ N imply (xn) converges to some x ∈ X. Definition 2.7 (Hussain et al. [2014]). Let Db be a metric on a set X and α, η : X × X → R+ ∪ {0} be functions. A self-mapping S on X is called α, β- continuous if {xn} is a sequence in X, xn → x as n → ∞ and α(xn, xn+1) ≥ β(xn, xn+1) for all n ∈ N imply Sxn → Sx as n → ∞. Definition 2.8 (Mehemet and Kiziltunc [2013]). Let X be a non-empty set and let s ≥ 1 be a given real number. A function Db : X × X → R+ ∪ {0} is called a b-metric provide that, for all x, y, z ∈ X, (1) Db(x, y) = 0 if and only if x = y (non-negative axiom) (2) Db(x, y) = Db(y, x) (symmetric axiom) (3) Db(x,z) ≤ s[Db(x, y) + Db(y, z)]. (s-Triangular inequality). A pair (X, Db) is called a b-metric space. Definition 2.9. Let R,S, be two self-mappings on the set X and α, β : X×X → R+ ∪ {0} be functions. We say that (R, S) is a pair of (α, β)-admissibility if z, w ∈ X and α(z, w) ≥ β(z, w) imply α(Rz, Sw) ≥ β(Rz, Sw) and α(Sz, Rw) ≥ β(Sz, Rw). Example 2.1. Define self-mappings R and S on a set of real numbers by Ru = u2 and Su = { −u2, if u < 0; u2, if u ≥ 0. Additionally, define α,β : X × X → R+ ∪ {0} via α(u,v) = eu+v and β(u,v) = eu. Then, (R,S) is a pair of (α,β)-admissibility. 247 R. Jahir Hussain, K. Maheshwaran, D. Dhamodharan 3 Main Results Definition 3.1. Let ψ be a nondecreasing function on R+ ∪ {0} . We call ψ a perfect control function if the following conditions hold: (i) ψ(t) = 0 ⇐⇒ t = 0. (ii) If (tn) is a sequence in R+ ∪ {0} and ψ(tn) → 0 as n → +∞ implies tn → 0 as n → +∞. (iii) ψ(u + v) ≤ ψ(u) + ψ(v) for all u, v ∈ R+ ∪ {0} . (iv) ψn(λx) = λnψ(x). Definition 3.2. Let (X,Db) be a b-metric space with constant s ≥ 1. Let R, S be two self-mappings on X, ψ be a perfect self-mapping on R+ ∪ {0}, α, β : X × X → R+ ∪ {0} be functions. We say that the pair (R, S) is an (α, β, ψ)- Admissibility type contraction if there exists λ ∈ [0, 1) such that z, w ∈ X and α(z, w) ≥ β(z, w) imply ψ(Db(Rz,Sw)) ≤ λψ(Db(z,w)) + λψ(Db(z,Rz)) + λψ(Db(w,Sw)) +λψ(Db(w,Rz)) + λψ(Db(z,Sw)) (1) and ψ(Db(Sz,Rw)) ≤ λψ(Db(z,w)) + λψ(Db(z,Sz)) + λψ(Db(w,Rw)) +λψ(Db(w,Sz)) + λψ(Db(z,Rw)) (2) Theorem 3.1. Let (X,Db) be a b-metric space with constant s ≥ 1. Let α, β : X × X → R+ ∪ {0} be function and (R,S) be a self-mappings on X. Assume following conditions: (i) (X,Db) is an α,β-complete b-metric space. (ii) R and S are α,β-continuous. (iii) (R,S) is pair of (α,β)-admissibility. (iv) If v,w,z are in X, with α(v,w) ≥ β(v,w) and α(w,z) ≥ β(w,z), then α(v,z) ≥ β(v,z). (v) There exists x0 ∈ X such that α(Rx0,SRx0) ≥ β(Rx0,SRx0) and α(SRx0,Rx0) ≥ β(SRx0,Rx0). Then R and S have a common fixed point. 248 Common fixed point for a contractive mapping via (α, β, ψ)-admissibility in b-metric space Proof. In view of condition (v) we start with x0 ∈ X in such away that α(Rx0,SRx0) ≥ β(Rx0,SRx0) and α(SRx0,Rx0) ≥ β(SRx0,Rx0). Now, let x1 = Rx0 and x2 = Sx1. Then α(x0,x1) ≥ β(x0,x1) and α(x1,x0) ≥ β(x1,x0). In view of condition (iii), we have α(x1,x2) = α(Rx0,Sx1) ≥ β(Rx0,Sx1) = β(x1,x2) and α(x2,x1) = α(Sx1,Rx0) ≥ β(Sx1,Rx0) = β(x2,x1) Again we put x3 = Sx2. Then condition (iii) implies that α(x2,x3) = α(Sx1,Rx2) ≥ β(Sx1,Rx2) = β(x2,x3) and α(x3,x2) = α(Rx2,Sx1) ≥ β(Rx2,Sx1) = β(x3,x2) Putting x4 = Sx3 and referring to condition (iii) we conclude α(x3,x4) = α(Rx2,Sx3) ≥ β(Rx2,Sx3) = β(x3,x4) and α(x4,x3) = α(Sx3,Rx2) ≥ β(Sx3,Rx2) = β(x4,x3) Continuing in the same manner, we contract a sequence (xn) in X with x2n+1 = Rx2n and x2n+2 = Sx2n+1 such that α(xn,xn+1) ≥ β(xn,xn+1) ∀ n ∈ N and α(xn+1,xn) ≥ β(xn+1,xn) ∀ n ∈ N From condition (iv) we see that α(xn,xm) ≥ β(xn,xm) ∀ n,m ∈ N If there exists q ∈ N such that x2q = x2q+1, then x2q = Rx2q and hence R has a fixed point. From contractive condition (1), we have ψ(Db(x2q+1,x2q+2)) = ψ(Db(Rx2q,Sx2q+1)) ≤ ψ  λ(Db(x2q,x2q+1)) + λ(Db(x2q,Rx2q)) + λ(Db(x2q+1,Sx2q+1)) +λ(Db(x2q+1,Rx2q)) + λ(Db(x2q,Sx2q+1))   ≤ ψ ( 2λ(Db(x2q,x2q+1)) + λ(Db(x2q+1,x2q+2)) +λs[(Db(x2q,x2q+1) + Db(x2q+1,x2q+2))] ) ≤ ψ ( λ(2 + s)(Db(x2q,x2q+1)) +λ(1 + s)(Db(x2q+1,x2q+2)) ) 249 R. Jahir Hussain, K. Maheshwaran, D. Dhamodharan ≤ ψ ( λ(2+s) 1−λ(1+s)(Db(x2q,x2q+1)) ) (3) The last inequality is correct only if ψ( λ(2+s) 1−λ(1+s)(Db(x2q,x2q+1))) = 0. The prop- erties of ψ and Db imply that x2q+1 = x2q+2. Hence, x2q = Rx2q = Sx2q. Thus, R and S have a common fixed point of R and S. If there exists q ∈ N such that x2q+1 = x2q+2 then x2q+1 = Tx2q+1 and hence S has a fixed point. From contractive condition (2), we have ψ(Db(x2q+2,x2q+3)) = ψ(Db(Sx2q+1,Rx2q+2)) ≤ ψ ( λ(Db(x2q+1,x2q+2)) + λ(Db(x2q+1,Sx2q+1)) + λ(Db(x2q+2,Rx2q+2)) +λ(Db(x2q+2,Sx2q+1)) + λ(Db(x2q+1,Rx2q+2)) ) ≤ ψ ( 2λ(Db(x2q+1,x2q+2)) + λ(Db(x2q+2,x2q+3)) +λs[(Db(x2q+1,x2q+2) + Db(x2q+2,x2q+3))] ) ≤ ψ ( λ(2 + s)(Db(x2q+1,x2q+2)) +λ(1 + s)(Db(x2q+2,x2q+3)) ) ≤ ψ ( λ(2+s) 1−λ(1+s)(Db(x2q+1,x2q+2)) ) (4) The last inequality is correct only if ψ( λ(2+s) 1−λ(1+s)(Db(x2q+1,x2q+2))) = 0. The properties of ψ and Db imply that x2q+2 = x2q+3. Hence, x2q+1 = Rx2q+1 = Sx2q+1. Thus, R and S have a common fixed point of R and S. Now, assume that xn ̸= xn+1 ∀ n ∈ N. For n ∈ N ∪ {0}, we get ψ(Db(x2n+1,x2n+2)) = ψ(Db(Rx2n,Sx2n+1)) ≤ ψ ( λ(Db(x2n,x2n+1)) + λ(Db(x2n,Rx2n)) + λ(Db(x2n+1,Sx2n+1)) +λ(Db(x2n+1,Rx2n)) + λ(Db(x2n,Sx2n+1)) ) ≤ ψ ( 2λ(Db(x2n,x2n+1)) + λ(Db(x2n+1,x2n+2)) +λs[(Db(x2n,x2n+1) + Db(x2n+1,x2n+2))] ) ≤ ψ ( λ(2 + s)(Db(x2n,x2n+1)) +λ(1 + s)(Db(x2n+1,x2n+2)) ) ≤ ψ ( λ(2+s) 1−λ(1+s)(Db(x2n,x2n+1)) ) (5) Let [δ = λ(2+s) 1−λ(1+s)]. Hence ψ(Db(x2n+1,x2n+2)) ≤ ψ(δ(Db(x2n,x2n+1))) 250 Common fixed point for a contractive mapping via (α, β, ψ)-admissibility in b-metric space Using argument similar to the above, we may show that ψ(Db(x2n,x2n+1)) = ψ(Db(Sx2n−1,Rx2n)) ≤ ψ ( λ(Db(x2n−1,x2n)) + λ(Db(x2n−1,Sx2n−1)) + λ(Db(x2n,Rx2n)) +λ(Db(x2n,Sx2n−1)) + λ(Db(x2n−1,Rx2n)) ) ≤ ψ ( 2λ(Db(x2n−1,x2n)) + λ(Db(x2n,x2n+1)) +λs[(Db(x2n−1,x2n) + Db(x2n,x2n+1))] ) ≤ ψ ( λ(2 + s)(Db(x2n−1,x2n)) +λ(1 + s)(Db(x2n,x2n+1)) ) ≤ ψ ( λ(2+s) 1−λ(1+s)(Db(x2n−1,x2n))) ) ≤ ψ ( (δ(Db(x2n−1,x2n))) ) (6) Combining equation (5) and (6) together, we reach ψ(Db(xn,xn+1)) = ψ(Db(Sxn−1,Rxn)) ≤ ψ ( λ(Db(xn−1,xn)) + λ(Db(xn−1,Sxn−1)) + λ(Db(xn,Rxn)) +λ(Db(xn,Sxn−1)) + λ(Db(xn−1,Rxn)) ) ≤ ψ ( 2λ(Db(xn−1,xn)) + λ(Db(xn,xn+1)) +λs[(Db(xn−1,xn) + Db(xn,xn+1))] ) ≤ ψ ( λ(2 + s)(Db(xn−1,xn)) +λ(1 + s)(Db(xn,xn+1)) ) ≤ ψ ( λ(2+s) 1−λ(1+s)(Db(xn−1,xn)) ) ≤ ψ ( δ(Db(xn−1,xn)) ) (7) By recurring equation (7) n-times, we deduce ψ(Db(xn,xn+1)) ≤ ψ(δ(Db(xn−1,xn))) ≤ δψ(Db(xn−2,xn−1)) ≤ δ(δψ(Db(xn−2,xn−1))) = δ2ψ(Db(xn−2,xn−1)) ... 251 R. Jahir Hussain, K. Maheshwaran, D. Dhamodharan ≤ δnψ(Db(x0,x1)). (8) On allowing n → ∞ in equation (8), we get lim n→+∞ ψ(Db(xn,xn+1)) = 0 (9) The properties of ψ implies that lim n→+∞ Db(xn,xn+1) = 0 (10) We intend to prove that (xn) is Cauchy sequence in X, take n,m ∈ N with m > n. We divide the proof into four cases: Case 1: n is an odd integer and m is an even integer. Therefore, there exists t ∈ N and an odd integer h such that n = 2t + 1 and m = 2t + 1 + h. Since α(xn,xm) ≥ β(xn,xm), we have ψ(Db(xn,xm)) = ψ(Db(x2t+1,x2t+1+h)) = ψ(Db(Rx2t,Sx(2t+h))) ≤ ψ ( λ(Db(x2t,x2t+h)) + λ(Db(x2t,Rx2t)) + λ(Db(x2t+h,Sx2t+h)) +λ(Db(x2t+h,Rx2t)) + λ(Db(x2t,Sx2t+h)) ) = ψ ( λ(Db(x2t,x2t+h)) + λ(Db(x2t,x2t+1)) + λ(Db(x2t+h,x2t+1+h)) +λ(Db(x2t+h,x2t+1)) + λ(Db(x2t,x2t+1+h)) ) ≤ ψ ( λ ∑2t+h−1 i=2t (Db(xi,xi+1)) + λ(Db(x2t,x2t+1)) + λ(Db(x2t+h,x2t+1+h)) +λ ∑2t+h−1 i=2t+1 (Db(xi,xi+1)) + λ(Db(x2t,x2t+1+h)) ) ≤ ψ ( λ(2 + s) ∑∞ i=2t(Db(xi,xi+1)) + λ(Db(x2t,x2t+1)) +λ(Db(x2t+h,x2t+1+h)) + λs(Db(x2t,x2t+1)) ) Where,k = λ(2 + s) ≤ ψ ( k ∑∞ i=2t(Db(xi,xi+1)) + λ(Db(x2t,x2t+1)) +λ(Db(x2t+h,x2t+1+h)) + λs(Db(x2t,x2t+1)) ) ≤ ψ ( k2t+1 1−k (Db(x0,x1)) + λ(Db(x2t,x2t+1)) +λ(Db(x2t+h,x2t+1+h)) + λs(Db(x2t,x2t+1)) ) By permitting n,m → ∞ in above inequalities and considering equation (9) lim n→+∞ ψ(Db(xn,xm)) = 0 252 Common fixed point for a contractive mapping via (α, β, ψ)-admissibility in b-metric space The properties of ψ implies that lim n→+∞ Db(xn,xm) = 0 (11) Case 2:n and m are both even integers. Applying the triangular inequality of the b-metric db, we have Db(xn,xm) ≤ s[Db(xn,xn+1) + Db(xn+1,xm)], for m ≥ n Letting n → ∞ and in view of equation (10) and (11),we get lim n→+∞ ψ(Db(xn,xm)) = 0. Case 3: n is an even integer and m is an odd integer. Applying the triangular inequality of the b-metric Db, we have Db(xn,xm) ≤ s[Db(xn,xn+1) + Db(xn+1,xm)] Db(xn,xm) ≤ s[Db(xn,xn+1)+s[Db(xn+1,xm−1)+Db(xm−1,xm)]] , for m ≥ n On permitting m,n → ∞ and considering equation (10) and (11), we get lim n→+∞ ψ(Db(xn,xm)) = 0. Case 4: n and m are both odd integers. Applying the triangular inequality of the b-metric Db, we have Db(xn,xm) ≤ s[Db(xn,xm−1) + Db(xm−1,xm)], for m ≥ n On permitting n → ∞ and in view of equation (10) and (11), we get lim n→+∞ ψ(Db(xn,xm)) = 0. Combining all cases with each other, we conclude that lim n→+∞ ψ(Db(xn,xm)) = 0. Thus, we conclude that (xn) is a Cauchy sequence in X. The α, β-completeness of the b-metric space (X,Db) ensures that there is x ∈ X such that xn → x. Using the α, β-continuity of the mappings R and S, we deduce that x2n+1 = Rx2n → Rx and x2n+2 = Sx2n+1 → Sx . by uniqueness of limit, we obtain Rx = Sx = x. Thus, x is a fixed point of R.2 253 R. Jahir Hussain, K. Maheshwaran, D. Dhamodharan Example 3.1. Define Db : R+0 ×R + 0 → R + 0 by Db(z,w) = |z−w| and let R,S be two self-mappings on R+0 define by Rz = z 2 and Sw = w 4 . In addition, define the function ψ : R+0 → R + 0 by ψ nλx = λnψ(x), where ψ(x) = x 1+x . Furthermore, the functions α,β : X × X → R+0 define by α(p,q) = { ep+q, if p,q ∈ [0,1]; 0, if p > 1 or q > 1. . and β(p,q) = { ep, if p,q ∈ [0,1]; 1, if p > 1 or q > 1. . Then: 1. ψ is a perfect control function. 2. There exists u0 ∈ X such that α(Ru0,Ru1) ≥ β(Ru0,Ru1) and α(Ru1,Ru0)] ≥ β(Ru1,Ru0). 3. (R,S) is a pair of (α,β)-admissibility. 4. R and S are α, β-continuous. 5. (X,Db) is an α, β-complete b-metric space. 6. (R,S) is an (α,β,ψ)-contraction. Proof. It is an easy matter to see equations (1) to (3). To prove (4), let (un) be any sequence in R+0 whenever un → u ∈ R + 0 and α(un,un+1) ≥ β(un,un+1) ∀ n ∈ N. Case 1: If un = u for all n, where un ∈ [0,1] ∀ n ∈ N. We conclude that Run → Ru as n → ∞. Case 2: If un ̸= u, for all n, we notice that u = 0. Hence, un → 0 in ([0,1], |.|). Therefore, |u 2 ,0| → 0 = Ru in (R+0 ,Db); that is R is α,β-continuous. To prove (5), let (un) be a Cauchy sequence in (R + 0 ,Db) such that α(un,un+1) ≥ β(un,un+1). Then, un ∈ [0,1] ∀n ∈ N. If there exists u ∈ [0,1] such that un = u for all n, then, un → u as n → +∞. Now, suppose the elements of (un) are distinct. Give ϵ > 0, since (un) is a Cauchy sequence in (R + 0 ,Db), then there exists n0 ∈ N such that |un,um| < ϵ ∀ m > n ≥ n0. Therefore, |un,0| < 0 ∀ n ≥ n0. So, un → 0 in (R+0 ,Db). Thus, (R + 0 ,Db) is an α,β-complete b-metric space. To prove (6), let z,w ∈ X be such that α(z,w) ≥ β(z,w). Then, z,w ∈ [0,1]. 254 Common fixed point for a contractive mapping via (α, β, ψ)-admissibility in b-metric space So, ψ(Db(Rz,Sw)) = ψ(Db( z 2 , w 4 )) = ψ(|z 2 , w 4 |) = | z 2 , w 4 | 1+| z 2 , w 4 | = 1 2 |z, w 2 | 1+ 1 2 |z, w 2 | = |z, w 2 | 2+|z, w 2 | ≤ 1 6 ( |z,w| 1+|z,w|) ≤ 1 6 ψ(Db(z,w)) ψ(Db(Rz,Sw)) ≤ 16ψ(Db(z,w)) + 1 6 ψ(Db(z,Rz)) + 1 6 ψ(Db(w,Sw))+ 1 6 ψ(Db(w,Rz)) + 1 6 ψ(Db(z,Sw)), (∵ λ = 1 6 ) Similarly, we can show that ψ(Db(Sz,Rw)) ≤ 16ψ(Db(z,w)) + 1 6 ψ(Db(z,Sz)) + 1 6 ψ(Db(w,Rw))+ 1 6 ψ(Db(w,Sz)) + 1 6 ψ(Db(z,Rw)), (∵ λ = 1 6 ) Hence, R and S satisfy definition 3.2. Therefore, R and S satisfy all the condition of theorem. Therefore, R and S have a common fixed point. 4 Conclusions In Theorem 3.1 we have formulated a new contractive conditions to modify and extend some common fixed point theorem for a pair of self-mappings (R,S) in b-metric space via (α,β,ψ)-admissibility type. The existence and uniqueness of the result is presented in this article. We have also given some example which satisfies the condition of our main result. Our result may be the vision for other authors to extend and improve several results in such spaces and applications to other related areas. 255 R. Jahir Hussain, K. Maheshwaran, D. Dhamodharan 5 Acknowledgements The authors thanks the management, Ratio Mathematica for their constant support towards the successful completion of this work. We wish to thank the anonymous reviewers for a careful reading of manuscript and for very useful com- ments and suggestions. References T. Abdeljawad. Meir-keeler α-contractive fixed and common fixed point theo- rems. Fixed Point Theory Appl, 19:1–10, 2013. K. Abodayeh, A. Bataihah, A. Ansari, and W. Shatanawi. Some fixed point and common fixed point results throughw-distance under nonlinear contractions. Gazi University Journal of Science, 30:293–302, 2017. I. Bakhtin. The contraction mapping principle in quasimetric spaces. Functional Analysis, 30:26–37, 1989. S. Czerwik. Contraction mappings in b-metric spaces. Acta Math Inform Univ Ostraviensis, 1:5–11, 1993. N. Hussain, M. Kutbi, and P. Salimi. Fixed point theory in α-complete metric spaces with applications. Abstract and Appled Analysis, 2014:1–11, 2014. E. Karupinar, P. Kumam, and P. Salimi. On α − ψ-meir-keeler contractive map- pings. Fixed Point Theory and Applications, 94:1–13, 2013. M. Khan, M. Swaleh, and S. Sessa. Fixed point theorems by altering distances between the points. Bulletin of the Australian Mathematical Society, 30:1–9, 1984. K. Mehemet and H. Kiziltunc. On some well-known fixed point theorems in b-metrics spaces. Turkish Journal of Analysis and Number Theory, 1:13–16, 2013. S. Nadler. Multi-valued contraction mappings. Pacific Journal of Mathematics, 30:475–488, 1969. B. Samet, C. Vetro, and P. Vetro. Fixed point theorems for a α − ψ-contractive mappings. Nonlinerar Analysis, 75:2154–2165, 2012. S.Banach. Sur les operations dans les ensembles et leur application aux equation sitegrales. Fundamenta Mathematicae, 3:133–181, 1922. 256