Ratio Mathematica Volume 47, 2023 Common Fixed Point Theorems for (ϕ,F)- Integral Type Conractive Mapping on C∗-Algebra Valued b-Metrix Space Jahir Hussain Rasheed* Maheshwaran Kanthasamy† Abstract The object of this paper, we establish the concept of integral type of common fixed point theorem for new type of generalized C∗-valued contractive mapping. The main theorem is an existence and unique- ness of common fixed-point theorem for self-mappings with (ϕ,F)- contractive conditions on complete C∗-algebra valued b-metric space. Moreover, some illustrated examples are also provided. Keywords: C∗-algebra valued, 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. 1Received on August 22, 2023. Accepted on February 23, 2023. Published on April 4, 2023. doi: 10.23755/rm.v41i0.834. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 186 R. Jahir Hussain and K. Maheshwaran 1 Introduction In 2002, Branciari [2002] introduced the concept of integral type contraction on fixed point solution. Many writers researched at the presence of fixed points for a variety of integral type contractive mappings, see Liu et al. [2018]. Especially, Liu et al. [2014] several more fixed point theorems for integral type contractive mappings in complete metric spaces. After that Ma et al. [2014] and Ma and Jiang [2015] presented the notion of C∗-algebra-valued metric space, C∗-algebra- valued b-metric space and investigated certain fixed point results for self-mapping under certain contractive conditions. Alsulami et al. [2016] investigated that fixed point theorem in the classical Banach fixed point theorem can be used to produce C∗-algebra-valued b- metric space in fixed point results Kamran et al. [2016]. We symbolize A as an unital C∗-algebra, and Ah= {a ∈A :a =a∗}. Especially, an element a ∈A is a positive factor, if a = a∗. A natural partial order on Ah given by a ≤b if fθ ≤ (b − a), where q signifies the zero element in A. Then, let A+ and A′ symbolize the set {a ∈A :θ ≤ a} and the set {a ∈A :ab = ba, ∀ b∈ A}, respectively and |a| = (a∗a) 1 2 . 2 Preliminaries Definition 2.1 (Ma and Jiang [2015]). Let χ be a non-empty set and ω∈ A such that ω ≥ I. Suppose that the mapping Db : X × X → A is held, the following constraints exist. (i) θ ≤ Db(ζ, η) and Db(ζ, η) = θ iff ζ = η; (ii) Db(ζ, η) = Db(η, ζ); (iii) Db(ζ, η) ≤ ω(Db(ζ, ϑ) + Db(ϑ, η)) for all ζ, η, ϑ ∈ χ. Then, Db is called C∗-algebra-valued b-metric on X and (χ, A, Db) is called C∗-algebra-valued b-metric space. Definition 2.2 (Ma and Jiang [2015]). Let (χ, A, Db) be C∗-algebra valued b- metric space. Assume that {ζn} is a sequence in χ and ζ ∈ χ. If for each ϵ > θ, there exists N such that ∀ n > N , ||d(ζn, ζ)|| ≤ ϵ then {ζn} is alleged to be con- vergent with regard to A, and {ζn} converges to ζ, i.e., we take limn→∞ ζn = ζ. If for each ϵ > θ, there exists N such that ∀ l n,m > N , ||d(ζn, ζm)|| ≤ ϵ, then {ζn} is referred to as a Cauchy sequence in χ. (χ, A, Db) is referred to as a com- plete C∗-algebra-valued b-metric space if every Cauchy sequence is convergent in χ. 187 Common fixed point theorems for (ϕ,F) integral type contractive mapping on C∗-algebra valued b-metrix space Definition 2.3 (Mustafa et al. [2021]). Let the non-decreasing function F : A+ → A+ be positive linear mapping satisfying following constraints: (i) F is continuous; (ii) F (a) = θ iff a = θ; (iii) limn→∞ Fn (a) = θ. Definition 2.4 (Mustafa et al. [2021]). Suppose that A and B are C∗-algebra. A mapping F :A→ B is said to be C∗-homomorphism if : (i) F (aζ + bη) = aF(ζ) + bF(η) for all a,b ∈C and ζ,η ∈ A; (ii) F(ζη) = F(ζ)F(η) for all ζ,η ∈ A; (iii) F (ζ∗) = F(ζ)∗ for all ζ ∈ A; (iv) F maps the unit in A to the unit in B. Lemma 2.1. Let (χ,A,Db) be a C∗-algebra valued b-metric space such that Db(ζ,η) ∈ A, for all ζ,η ∈ χ where ζ ̸= η. Let ϕ : A+ → A+ be a function with the following properties: (i) ϕ(a) = θ if and only if a = θ; (ii) ϕ(a) < a, for all a ∈ A; (iii) either ϕ(a) ≤ Db(ζ,η) or Db(ζ,η) ≤ ϕ(a), where a ∈ A and ζ,η ∈ χ. Corolary 2.1. Every C∗-homomorphism is bounded. Lemma 2.2. Every ∗-homomorphism is positive. Definition 2.5 (Branciari [2002]). The function ξ : χ → χ is called sub-additive integrable function iff ∀ a,b ∈ χ, ∫ a+b 0 ξdt ≤ ∫ a 0 ξdt + ∫ b 0 ξdt. 188 R. Jahir Hussain and K. Maheshwaran 3 Main Results Definition 3.1. Let (χ,A,Db) is a complete C∗-algebra valued b-metric space. Let L,M, :χ → χ be a integral C∗-valued contractive mapping and F (∫ Db(Lζ,Mη) 0 ξdt ) ≤ F (∫ I(ζ,η) 0 ξdt ) − ϕ (∫ Db(ζ,η) 0 ξdt ) (1.1) I(ζ,η) ≤ ( α ∫ Db(ζ,η) 0 ξdt + γ ∫ [Db(ζ,Lζ)+Db(η,Mη)] 0 ξdt +δ ∫ [Db(ζ,Mη)+Db(η,Lζ)] 0 ξdt ) For all ζ,η ∈ χ, where ω ∈ A′+, α + γ + δ ≥ 0 with ωα + γ(ω + 1) + δ (ω(ω + 1)) < 1. F ∈Ψ and ϕ ∈ Φ and ξ : χ → χ is the Lebesgus-integral function. Theorem 3.1. Let (χ,A,Db) is a complete C∗-algebra valued b-metric space. 3.1 are ∗−homomorphisms and with the constraint F ( a) ≤ ϕ( a) and ξ : χ → χ is a Lebesgus-integral mapping which is summable, non-negative and such that for all ε > 0, ∫ ε 0 ξdt > 0. Then L and M have a unique common fixed point in χ. Proof. Let ζ0 ∈ χ and define ζn= Lζn−1, ζn+1= Mζnwe have F (∫ Db(ζn,ζn+1) 0 ξdt ) = F (∫ Db(Lζn−1,Mζn) 0 ξdt ) ≤ F (∫ I(ζn−1,ζn) 0 ξdt ) − ϕ (∫ Db(ζn−1,ζn) 0 ξdt ) = F ( α ∫ Db(ζ,η) 0 ξdt + γ ∫ [Db(ζ,Lζ)+Db(η,Mη)] 0 ξdt +δ ∫ [Db(ζ,Mη)+Db(η,Lζ)] 0 ξdt ) − ϕ (∫ Db(ζn−1,ζn) 0 ξdt ) =  F(α)F (∫ Db(ζn−1, ζn) 0 ξdt ) +F(γ)F (∫ [Db(ζn−1,Lζn−1)+Db(ζn,Mζn)] 0 ξdt ) +F(δ)F (∫ [Db(ζn−1,Mζn)+Db(ζn,Lζn−1)] 0 ξdt ) − ϕ (∫ Db(ζn−1,ζn) 0 ξdt ) .   Therefore, ∥ F (∫ Db(ζn,ζn+1) 0 ξdt ) ∥=∥ F (∫ Db(Lζn−1,Mζn) 0 ξdt ) ∥ ≤   ∥ F(α)∥ ∥ F (∫ Db(ζn−1,ζn) 0 ξdt ) ∥ +∥ F(γ)∥ ∥ F (∫ [Db(ζn−1,Lζn−1)+Db(ζn,Mζn)] 0 ξdt ) ∥ +∥ F(δ)∥ ∥ F (∫ [Db(ζn−1,Mζn)+Db(ζn,Lζn−1)] 0 ξdt ) ∥ − ∥ ϕ (∫ Db(ζn−1,ζn) 0 ξdt ) ∥ → 0 as n → +∞.   189 Common fixed point theorems for (ϕ,F) integral type contractive mapping on C∗-algebra valued b-metrix space Give that ϕ (2.1) and F (2.4) are strongly monotone functions. We have∫ Db(ζn,ζn+1) 0 ξdt = ∫ Db(Lζn−1,Mζn) 0 ξdt ≤ ( α ∫ Db(ζn−1,ζn) 0 ξdt + γ ∫ [Db(ζn−1,Lζn−1)+Db(ζn,Mζn)] 0 ξdt +δ ∫ [Db(ζn−1,Mζn)+Db(ζn,Lζn−1)] 0 ξdt ) = ( α ∫ Db(ζn−1,ζn) 0 ξdt + γ ∫ [Db(ζn−1,ζn)+Db(ζn,ζn+1)] 0 ξdt +δ ∫ [Db(ζn−1,ζn+1)+Db(ζn,ζn)] 0 ξdt ) ≤ (α + γ) ∫ Db(ζn−1, ζn) 0 ξdt + γ ∫ Db(ζn,ζn+1) 0 ξdt + δ ∫ Db(ζn−1,ζn+1) 0 ξdt ≤ (α + γ) ∫ Db(ζn−1, ζn) 0 ξdt + γ ∫ Db(ζn,ζn+1) 0 ξdt + ωδ ∫ (Db(ζn−1,ζn)+Db(ζn,ζn+1)) 0 ξdt ≤ (α + γ + ωδ ) ∫ Db(ζn−1, ζn) 0 ξdt + (γ + ωδ) ∫ Db(ζn,ζn+1) 0 ξdt. This implies that ∫ Db(ζn,ζn+1) 0 ξdt ≤ α + γ + ωδ γ + ωδ ∫ Db(ζn,ζn−1) 0 ξdt ∫ Db(ζn,ζn+1) 0 ξdt ≤ h ∫ Db(ζn,ζn−1) 0 ξdt where, h = α+γ+ωδ γ+ωδ < 1. Thus, we have ∥ ∫ Db(ζn−1,ζn) 0 ξdt ∥ ∥ ∫ Db(ζn,ζn+1) 0 ξdt ∥≤∥ h ∥ ∥ ∫ Db(ζn,ζn−1) 0 ξdt ∥→ 0, asn,m → +∞. If n > m ∫ Db(ζn,ζm) 0 ξdt ≤ ( ω ∫ Db(ζn,ζn−1) 0 ξdt + ω 2 ∫ Db(ζn−1,ζn−2) 0 ξdt + . . . + ωn−m ∫ Db(ζm−1,ζm) 0 ξdt ) . Applying the constraint of theorem then, F (∫ Db(ζn,ζm) 0 ξdt ) ≤  F ( ω ∫ Db(ζn,ζn−1) 0 ξdt ) +F ( ω2 ∫ Db(ζn−1,ζn−2) 0 ξdt ) + . . . + F ( ωn−m ∫ Db(ζm−1,ζm) 0 ξdt )   190 R. Jahir Hussain and K. Maheshwaran ≤  F(ω)F (∫ Db(ζn,ζn−1) 0 ξdt ) +F (ω2) F (∫ Db(ζn−1,ζn−2) 0 ξdt ) + . . . + F (τn−m) F (∫ Db(ζm−1,ζm) 0 ξdt )   ≤   F ( ω ∫ I(ζn,ζn−1) 0 ξdt ) − ϕ ( ω ∫ Db(ζn,ζn−1) 0 ξdt ) +F ( ω2 ∫ I(ζn−1,ζn−2) 0 ξdt ) − ϕ ( ω2 ∫ Db(ζn−1,ζn−2) 0 ξdt ) + . . . + F ( ωn−m ∫ I(ζm−1,ζm) 0 ξdt ) − ϕ ( ωn−m ∫ Db(ζm−1,ζm) 0 ξdt )   =   F   αω ∫ Db(ζn,ζn−1) 0 ξdt +γω ∫ [Db(ζn−1,ζn)+Db(ζn,ζn+1)] 0 ξdt +δω ∫ [Db(ζn−1,ζn+1)+Db(ζn,ζn)] 0 ξdt   −ϕ ( ω ∫ Db(ζn,ζn−1) 0 ξdt ) + . . . +F   αω n−m ∫ Db(ζm−1,ζm) 0 ξdt +γωn−m ∫ [Db(ζm,ζm−1)+Db(ζm−1,ζm−2)] 0 ξdt +δωn−m ∫ [Db(ζm,ζm−2)+Db(ζm−1,ζm−1)] 0 ξdt   −ϕ ( ωn−m ∫ Db(ζm−1,ζm) 0 ξdt )   Therefore F (∫ Db(ζn,ζm) 0 ξdt ) = F(α)F(ω)F (∫ Db(ζn,ζn−1) 0 ξdt ) +F(γ)F(ω)F (∫ [Db(ζn−1,ζn)+Db(ζn,ζn+1)] 0 ξdt ) +F(δ)F(ω)F (∫ [Db(ζn−1,ζn+1)+Db(ζn,ζn)] 0 ξdt ) −ϕ ( ω ∫ Db(ζn,ζn−1) 0 ξdt ) + . . . +F(α)F (ωn−m) F (∫ Db(ζm−1,ζm) 0 ξdt ) +F(γ)F (ωn−m) F (∫ [Db(ζm,ζm−1)+Db(ζm−1,ζm−2)] 0 ξdt ) +F(δ)F (ωn−m) F (∫ [Db(ζm,ζm−2)+Db(ζm−1,ζm−1)] 0 ξdt ) −ϕ ( ωn−m ∫ Db(ζm−1,ζm) 0 ξdt ) . Since the property of ϕ (2.1) and F (2.4) is strongly monotone, we have 191 Common fixed point theorems for (ϕ,F) integral type contractive mapping on C∗-algebra valued b-metrix space ∫ Db(ζn,ζm) 0 ξdt ≤ αω ∫ Db(ζn,ζn−1) 0 ξdt + γω ∫ [Db(ζn−1,ζn)+Db(ζn,ζn+1)] 0 ξdt +δω ∫ [Db(ζn−1,ζn+1)+Db(ζn,ζn)] 0 ξdt + . . . +αωn−m ∫ Db(ζm−1,ζm) 0 ξdt +γωn−m ∫ [Db(ζm,ζm−1)+Db(ζm−1,ζm−2)] 0 ξdt +δωn−m ∫ [Db(ζm,ζm−2)+Db(ζm−1,ζm−1)] ξdt so we get ∫ Db(ζn,ζm) 0 ξdt ≤   ∥ α ∥∥ ω ∥∥ ∫ Db(ζn,ζn−1) 0 ξdt ∥ + ∥ γ ∥∥ ω ∥∥ ∫ [Db(ζn−1,ζn)+Db(ζn,ζn+1)] 0 ξdt ∥ + ∥ δ ∥∥ ω ∥∥ ∫ [Db(ζn−1,ζn+1)+Db(ζn,ζn)] 0 ξdt ∥ + . . . + ∥ α ∥∥ ωn−m ∥∥ ∫ Db(ζm−1,ζm) 0 ξdt ∥ + ∥ γ ∥∥ ωn−m ∥∥ ∫ [Db(ζm,ζm−1)+Db(ζm−1,ζm−2)] 0 ξdt ∥ + ∥ δ ∥∥ ωn−m ∥∥ ∫ [Db(ζm,ζm−2)+Db(ζm−1,ζm−1)] 0 ξdt ∥   → 0, as n.m → +∞. Then {ζn} is Cauchy sequence. Since (χ,A,Db) is a complete C∗-algebra valued b-metric space there exists u ∈ χ such that ζn → u as n → ∞. Now since ∫ Db(u,Mu) 0 ξdt ≤ ω [∫ Db(u,ζn+1) 0 ξdt + ∫ Db(ζn+1,Mu) 0 ξdt ] = ω [∫ Db(ζn+1,Mu) 0 ξdt + ∫ Db(u,ζn+1) 0 ξdt ] = ω [∫ Db(Lζn,Mu) 0 ξdt + ∫ Db(u,ζn+1) 0 ξdt ] F (∫ Db(u,Mu) 0 ξdt ) = ω [ F (∫ Db(Lζn,Mu) 0 ξdt ) +F (∫ Db(u,ζn+1) 0 ξdt )] ≤ ω [ F (∫ I(ζn,u) 0 ξdt ) − ϕ (∫ Db(ζn,u) 0 ξdt )] + ω [ F (∫ Db(u,ζn+1) 0 ξdt )] 192 R. Jahir Hussain and K. Maheshwaran ∥ F (∫ Db(Lζn,Mu) 0 ξdt ) ∥≤   ∥ ω ∥∥ F (∫ Db(u,ζn+1) 0 ξdt ) ∥ + ∥ ω ∥∥ Fα ∥∥ ∫ Db(ζn,u) 0 ξdt ∥ + ∥ ω ∥∥ Fγ ∥ ∥ ∫ [Db(ζn,ζn+1)+Db(u,Mu)] 0 ξdt ∥ + ∥ ω ∥∥ Fδ ∥∥ ∫ [Db(ζn,Mu)+Db(u,Lζn)] 0 ξdt ∥ − ∥ ω ∥∥ ϕ (∫ Db(ζn,u) 0 ξdt ) ∥ .   Using the property of ϕ (2.1), we get ∥ F (∫ Db(Lζn,Mu) 0 ξdt ) ∥ ≤   ∥ ω ∥∥ F (∫ Db(u,ζn+1) 0 ξdt ) ∥ + ∥ ω ∥∥ Fα ∥∥ ∫ Db(ζn,u) 0 ξdt ∥ + ∥ ω ∥∥ Fγ ∥ ∥ ∫ [Db(ζn,ζn+1)+Db(u,Mu)] 0 ξdt ∥ + ∥ ω ∥∥ Fδ ∥∥ ∫ [Db(ζn,Mu)+Db(u,Lζn)] 0 ξdt ∥   . Where F (2.4) is strongly monotone, then ∥ ∫ (Db(Lζn,Mu)) 0 ξdt ∥≤   ∥ ω ∥∥ ∫ (Db(u,ζn+1)) 0 ξdt ∥ + ∥ ω ∥∥ α ∥∥ ∫ Db(ζn,u) 0 ξdt ∥ + ∥ ω ∥∥ γ ∥ ∥ ∫ [Db(ζn,ζn+1)+Db(u,Mu)] 0 ξdt ∥ + ∥ ω ∥∥ δ ∥∥ ∫ [Db(ζn,Mu)+Db(u,Lζn)] 0 ξdt ∥   = ∥ ω ∥∥ ∫ (Db(u,ζn+1)) 0 ξdt ∥ + ∥ ω ∥   ∥ α ∥∥ ∫ Db(ζn,u) 0 ξdt ∥ + ∥ γ ∥ ∥ ∫ [Db(ζn,ζn+1)+Db(u,Mu)] 0 ξdt ∥ + ∥ δ ∥∥ ∫ [Db(ζn,Mu)+Db(u,ζn+1)] 0 ξdt ∥   as ζn → u and ζn+1 → u as n → ∞, we get ∥ 1−ωγ−ωδ ∥ ∥ ∫ Db(u,Mu) 0 ξdt ∥≤ [ ∥ ω ∥∥ α ∥∥ ∫ Db(ζn,u) 0 ξdt ∥ + ∥ ω ∥∥ 1 + δ ∥∥ ∫ Db(u,ζn+1) 0 ξdt ∥ ] → 0 as n → ∞. 193 Common fixed point theorems for (ϕ,F) integral type contractive mapping on C∗-algebra valued b-metrix space Hence ∥ ∫ Db(Mu,u) 0 ξdt ∥= 0 since ∥ 1 − ωγ − ωδ ∥> 0. As a result, Mu = u that is u is a fixed point of M. Similarly we are able to demonstrate that Lu = u. Hence Lu = Mu = u. This demonstrates that u is common fixed point of L and M. Let v be a different fixed point common to L and M. (i.e) Lv = Mv = v such that u ̸= v we have ∫ Db(u,v) 0 ξdt = ∫ Db(Lu,Mv) 0 ξdt then F (∫ Db(u,v) 0 ξdt ) = F (∫ Db(Lu,Mv) 0 ξdt ) ≤ F (∫ I(u,v) 0 ξdt ) − ϕ (∫ Db(u,v) 0 ξdt ) ∥ F (∫ Db(Lu,Mv) 0 ξdt ) ∥≤   ∥ Fα ∥∥ ∫ Db(u,v) 0 ξdt ∥ + ∥ Fγ ∥∥ ∫ [Db(u,Lu)+Db(v,Mv)] 0 ξdt ∥ + ∥ Fδ ∥∥ ∫ [Db(u,Mv)+Db(v,Lu)] 0 ξdt ∥ − ∥ ϕ (∫ Db(u,v) 0 ξdt ) ∥   Using the property of ϕ (2.1), we get ∥ F (∫ Db(Lu,Mv) 0 ξdt ) ∥≤   ∥ Fα ∥∥ ∫ Db(u,v) 0 ξdt ∥ + ∥ Fγ ∥ ∥ ∫ [Db(u,Lu)+Db(v,Mv)] 0 ξdt ∥ + ∥ Fδ ∥∥ ∫ [Db(u,Mv)+Db(v,Lu)] 0 ξdt ∥   where F (2.4) is strongly monotone, then ∥ ∫ (Db(Lu,Mv)) 0 ξdt ∥ ≤   ∥ α ∥∥ ∫ Db(u,v) 0 ξdt ∥ + ∥ γ ∥ ∥ ∫ [Db(u,Lu)+Db(v,Mv)] 0 ξdt ∥ + ∥ δ ∥∥ ∫ [Db(u,Mv)+Db(v,Lu)] 0 ξdt ∥   ≤ ∥ α + 2δ ∥ ∥ ∫ Db(u,v) 0 ξdt ∥ ≤ ∥ ωα + (ω + 1)γ + ω(ω + 1)δ ∥ ∥ ∫ Db(u,v) 0 ξdt ∥ < ∥ ∫ Db(u,v) 0 ξdt ∥ . Which is a contradiction. Hence ∥ ∫ Db(u,v) ξdt ∥= 0 and u = v. Thus u is a unique common fixed point of L and M. Corolary 3.1. Let (χ,A,Db) is a complete C∗-algebra valued b-metric space. Let L :χ → χ be a contractive mapping and F (∫ Db(Lnζ,Lnη) 0 ξdt ) ≤ F (∫ I(ζ,η) 0 ξdt ) − ϕ (∫ Db(ζ,η) 0 ξdt ) 194 R. Jahir Hussain and K. Maheshwaran I(ζ,η) ≤  α ∫ Db(ζ,η) 0 ξdt + β ∫ [1+Db(ζ,Lnζ)]Db(η,Lnη) 1+Db(ζ,η) 0 ξdt +γ ∫ [Db(ζ,Lnζ)+Db(η,Lnη)] 0 ξdt +δ ∫ [Db(ζ,Lnη)+Db(η,Lnζ)] 0 ξdt   for all ζ,η ∈ χ, where ω ∈ A′+, α + β + γ + δ ≥ 0 with ωα + β + γ(ω + 1) + δ (ω(ω + 1)) < 1. F and ϕ are ∗−homomorphisms and with the constraint F ( a) ≤ ϕ( a) and ξ : χ → χ is a Lebesgus-integral mapping which is summable, non-negative and such that for all ε > 0, ∫ ε 0 ξdt > 0. Then L have a unique fixed point in χ. Example 3.1. Let χ = [0,1] and A =R2 with a norm ∥ ζ ∥= |ζ| be a real C∗- algebra. We define p = {(ζ,η ) ∈ R2 : ζ ≥ 0, η ≥ 0}. The partial order ≤ with respect to the C∗-algebra R2. ζ1 ≤ ζ2 and η1 ≤ η2 for all (ζ1,η1) ,(ζ2,η2) ∈ R2. Let Db : χ×χ → R2 suppose that Db(ζ,η) = 2 (|ζ − η|, |ζ − η|) for ζ,η ∈ χ. Then, (χ,A,Db) is a C∗-algebra valued b-metric space where ω = 1 in theo- rem 3.1. Let F,ϕ : p → p be the mappings defined as follows: For T =(ζ,η) ∈ p F (T ) =   (ζ,η), if ζ ≤ 1 and η ≤ 1, (ζ2,η) , if ζ > 1 and η ≤ 1, (ζ,η2) , if ζ ≤ 1 and η > 1, (ζ2,η2) , if ζ > 1 and η > 1. and for S = (S1,S2) ∈ p with V = min {S1,S2} , ϕ(S) = {( V2 2 , V 2 2 ) , if V ≤ 1( 1 2 , 1 2 ) , if V > 2 Then, F and ϕ have the properties mentioned in (2.4) and (2.1). Let L,M :χ → χ be defined as follows: L(ζ) = { 1 32 , if 0 ≤ ζ ≤ 1 2 0, if 1 2 < ζ ≤ 1 ; M(ζ) = 1 32 , for ζ ∈ χ Then, L and M have the required properties mentioned in theorem 3.1. Let α = 1 16 , β = 0 , γ = 1 64 and δ = 1 64 . It can be verified that: F (Db (Lζ,Mη)) ≤ F (N(ζ,η)) − ϕ(Db(ζ,η)) , ∀ ζ,η ∈ χ with η ≤ ζ . Hence, Theorem 3.1 is satisfied. Then demonstrate that 0 is a unique common fixed point of L and M. 195 Common fixed point theorems for (ϕ,F) integral type contractive mapping on C∗-algebra valued b-metrix space 4 Conclusions In Theorem 3.1 we have formulated a contractive conditions to modify and ex- tend the concept of common fixed point theorem for C∗-algebra valued b-metric space via (ϕ,F)-integral type contractive mapping. The existence and uniqueness of the result is presented in this article. We have also given some example which satisfies the contractive condition of our main result. Our result may be the vi- sion for other authors to extend and improve several results in such spaces and applications to other related areas. 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 H. 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