@ Appl. Gen. Topol. 23, no. 2 (2022), 287-301 doi:10.4995/agt.2022.16783 © AGT, UPV, 2022 C∗-algebra valued quasi metric spaces and fixed point results with an application Mohammad Asim a , Santosh Kumar b , Mohammad Imdadc and Reny George d a Department of Mathematics, Faculty of Science, SGT University, Gurugram (Haryana) - 122505, India (mailtoasim27@gmail.com) b Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, Tanzania (drsengar2002@gmail.com) c Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India (mhimdad@gmail.com) d Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia (renygeorge02@yahoo.com) Communicated by J. Galindo Abstract In this paper, we introduce the notion of C∗-algebra valued quasi metric space to generalize the notion of C∗-algebra valued metric space and investigate the topological properties besides proving some core fixed point results. Finally, we employ our one of the main results to examine the existence and uniqueness of the solution for a system of Fredholm integral equations. 2020 MSC: 47H10; 54H25; 46L07. Keywords: C∗-algebra; C∗-algebra valued quasi metric spaces; fixed point; integral equations. 1. Introduction The classical Banach contraction principle [8] continues to be one of the most motivating fixed point results, which has inspired several generations of mathe- maticians working in this domain. This principle is not merely an existence and uniqueness result but also offers a very effective computational procedure to Received 30 November 2021 – Accepted 8 June 2022 http://dx.doi.org/10.4995/agt.2022.16783 https://orcid.org/0000-0002-2209-4488 https://orcid.org/0000-0003-2121-6428 https://orcid.org/0000-0003-3270-1365 https://orcid.org/0000-0003-2314-0412 M. Asim, S. Kumar, M. Imdad and R. George compute the fixed point of the underlying contraction map. Several researchers attempted to improve this principle by enlarging the class of spaces. To ac- complish this, the authors introduced various classes of metric spaces namely: (see [1–7, 9–11, 14, 15]. In 1931, W. A. Wilson [16] introduced the notion of quasi metric space. A quasi metric d on a non-empty set A is a function d : A × A → R+ which satisfies d(a,b) ≤ d(a,c) + d(c,b) and d(a,b) = d(b,a) = 0 if and only if a = b, for all a,b ∈ A. A quasi metric satisfies all the conditions of metric with the possible exception of symmetry (i.e., the distance of a point ‘a′ to a point ‘b′ may not equal to the distance of a point ‘b′ to a point ‘a′). On the other hand, Ma et al. [12] set up the class of C∗-algebra valued metric spaces (in short C∗-avMS) by interchanging R (the range set) with a unital C∗-algebra in 2014, which is a more broad class than the class of metric spaces, and used the equivalent to make some fixed point results in such spaces. After one year, Ma et al. [13] again presented the idea of C∗-algebra valued b- metric spaces as a generalization of C∗-algebra valued metric space and proved some fixed point results likewise utilized the of their work for an integral type operator as an application. Enlivened by prior perceptions, we expand the class of C∗-algebra valued metric space by presenting the class of C∗-algebra valued quasi metric space and using the equivalent to make a fixed point result. Additionally, we concentrate on some topological properties of the C∗-algebra valued quasi metric space. Besides, we furnish some examples which show the utility of our main result. 2. Preliminaries Recall some definitions, examples and useful results which are needed in our subsequent discussions. Now, we give the following definition of C∗-algebra valued metric space which is introduced by Ma et al. [12] in 2014. Definition 2.1. Let A be a non-empty set. A mapping d : A×A →A is called a C∗-algebra valued metric on A, if it satisfies the following (for all a,b,c ∈ A): (i) d(a,b) < 0A and d(a,b) = 0A iff a = b; (ii) d(a,b) = d(b,a); (iii) d(a,b) 4 d(a,c) + d(c,b). The triplet (A,A,d) is called a C∗-algebra valued metric space. Now, we introduce yet different type of generalized C∗-algebra valued metric space and quasi metric space, which we refer as C∗-algebra valued quasi metric space. Definition 2.2. Let A be a non-empty set. A mapping d : A × A → A is called a C∗-algebra valued quasi metric on A, if it satisfies the following (for all a,b,c ∈ A): (i) d(a,b) < 0A and d(a,b) = d(b,a) = 0A iff a = b; (ii) d(a,b) 4 d(a,c) + d(c,b). The triplet (A,A,d) is called a C∗-algebra valued quasi metric space. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 288 C∗-avQMS and fixed point results with an application Example 2.3. Let A = R and A = M2(C), the class of bounded and linear operators on a Hilbert space C2. Define d : A×A →A by (for all a,b ∈ A): d(a,b) =   [ a− b 0 0 a− b ] if a ≥ b I2×2 if a < b where I2×2 = [ 1 0 0 1 ] . Then (A,A,d) be a C∗-algebra valued quasi metric space. It is clear that d(a,b) = 0A ⇔ a = b. Now, we consider the following two cases: Case I: For a ≥ b, we have d(a,b) = (a− b)I2×2. • If c < b, then d(a,c) = (a− c)I2×2 and d(c,b) = I2×2. • If b ≤ c < a, then d(a,c) = (a− c)I2×2 and d(c,b) = (c− b)I2×2. • If a ≤ c, then d(a,c) = I2×2 and d(c,b) = (c− b)I2×2. Therefore, we have d(a,b) 4 d(a,c) + d(c,b). Case 2: For a < b, we have d(a,b) = I2×2. • If c < a, then d(a,c) = (a− c)I2×2 and d(c,b) = I2×2. • If a ≤ c < b, then d(a,c) = I2×2 and d(c,b) = I2×2. • If b ≤ c, then d(a,c) = I2×2 and d(c,b) = (c− b)I2×2. Therefore, we have d(a,b) 4 d(a,c) + d(c,b). By the above calculations, we can say that (A,A,d) is a C∗-algebra valued quasi metric space. Example 2.4. Let A = R and A = M2(C). Define d : A×A →A by (for all a,b ∈ A): d(a,b) =   [ b−a 0 0 k(b−a) ] if a ≤ b α [ a− b 0 0 k(a− b) ] if a > b where k,α > 0. Then (A,A,d) is a C∗-algebra valued quasi metric space. Example 2.5. Let A = [1,∞) and A = M2(C). Define d : A×A →A by (for all a,b ∈ A): d(a,b) =   [ ln b− ln a 0 0 ln b− ln a ] if a ≤ b 1 3 [ ln a− ln b 0 0 ln a− ln b ] if a > b where ‘ ln′ is natural logarithmic function. Then (A,A,d) is a C∗-algebra valued quasi metric space. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 289 M. Asim, S. Kumar, M. Imdad and R. George Now, we give some definitions of convergent, left-Cauchy, right-Cauchy and completeness of the quasi metric space as follows. Definition 2.6. Let (A,A,d) be a C∗-algebra valued quasi metric space and {an} a sequence in A. We say that (i) The sequence {an} is called convergent to a ∈ A, written lim n→∞ an = a, if lim n→∞ d(an,a) = lim n→∞ d(a,an) = 0A. (ii) The sequence {an} is called left-Cauchy if for each � � 0A there exists a positive integer N such that d(an,am) ≺ � for all n ≥ m ≥ N. (iii) The sequence {an} is called right-Cauchy if for each � � 0A there exists a positive integer N such that d(an,am) ≺ � for all m ≥ n ≥ N. (iv) The sequence {an} is called Cauchy if for each � � 0A there exists a positive integer N such that d(an,am) ≺ � for all m,n ≥ N, i.e., lim n,m→∞ d(an,am) = 0A. (v) The triplet (A,A,d) is left-complete if every left-Cauchy sequence in (A,A,d) is convergent. (vi) The triplet (A,A,d) is right-complete if every right-Cauchy sequence in (A,A,d) is convergent. (vii) The triplet (A,A,d) is complete if every Cauchy sequence in (A,A,d) is convergent. Remark 2.7. (1) Every C∗-algebra valued metric space is C∗-algebra valued quasi metric space but the converse is not true in general. (2) In a C∗-algebra valued quasi metric space a sequence {an} is Cauchy iff it is left-Cauchy and right-Cauchy. Definition 2.8. Let (A,A,d) C∗-algebra valued quasi metric space. The con- jugate (or dual) C∗-algebra valued quasi metric space is denoted by dc and define by as follows: dc(a,b) = d(b,a), for all a,b ∈ A. A C∗-algebra valued quasi metric space is a C∗-algebra valued metric space iff it coincides with its conjugate dc, for all a,b ∈ A. Let wc be a C ∗-algebra positive valued function on A. The quadruplet (A,A,d,wc) is called a C∗-algebra valued weighted quasi metric space, if for all a,b ∈ A d(a,b) + wc(a) = d(b,a) + wc(b). Then (A,A,d,wc) is said to be C∗-algebra valued generalized weighted quasi metric space. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 290 C∗-avQMS and fixed point results with an application Proposition 2.9. Let (A,A,d) C∗-algebra valued quasi metric space. The associated C∗-algebra valued metric ds is define by: ds(a,b) = 1 2 [d(a,b) + d(b,a)]. The associated C∗-algebra valued metric space ds is the smallest C∗-algebra valued metric space majorising d. Proof. To verify condition (i), for each a,b ∈ A, we have d(a,b) < 0A. Also ds(a,b) = 0A ⇔ 1 2 [d(a,b) + d(b,a)] = 0A ⇔ d(a,b) + d(b,a) = 0A ⇔ d(a,b) = d(b,a) = 0A ⇔ a = b. Now, for condition (ii), for each a,b ∈ A, we have ds(a,b) = 1 2 [d(a,b) + d(b,a)] = 1 2 [d(b,a) + d(a,b)] = ds(b,a). Finally, we show that condition (iii), for each a,b,c ∈ A, we have ds(a,b) = 1 2 [d(a,b) + d(b,a)] = 1 2 [d(a,c) + d(c,b) + d(b,c) + d(c,a)] = 1 2 [d(a,c) + d(c,a)] + 1 2 [d(b,c) + d(c,b)] = ds(a,b) + ds(b,a). Thus, (A,A,ds) is C∗-algebra valued metric space. � Definition 2.10. Let (A,A,d) be a C∗-algebra valued quasi metric space, a ∈ A, N,M ⊆ A and 0A ≺ � ∈A. Denoted by: • The diameter of set N diam(N) = sup{d(a,b) : a,b ∈ N}. • The left-open ball of radius � centered at a BL� (a) = {b ∈ A : d(a,b) ≺ �}. • The right-open ball of radius � centered at a BR� (a) = {b ∈ A : d(b,a) ≺ �}. • The associated C∗-algebra valued quasi metric space open ball of radius � centered at a B� = {b ∈ A : ds(a,b) ≺ �}. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 291 M. Asim, S. Kumar, M. Imdad and R. George • The left-distance from a to N distd(a,N) = inf{d(a,b) : b ∈ N}. • The right-distance from a to N distd(N,a) = inf{d(b,a) : b ∈ N}. • The left-�-neighbourhood of N NL� = inf{a ∈ A : distd(N,a) ≺ �}. • The right-�-neighbourhood of N NR� = inf{a ∈ A : distd(a,N) ≺ �}. • The associated metric �-neighbourhood of N NL� = inf{a ∈ A : dist s d(N,a) ≺ �}. • The distance between N and M d(N,M) = inf{d(a,b) : a ∈ N,b ∈ M}. Proposition 2.11. Let (A,A,d) be a C∗-algebra valued quasi metric space. Then the collection of all open left-balls BL� (a) (right-balls B R � (a)) on A, ULA = {B L � (a) : a ∈ A,� � 0A} forms a left-basis (right-basis) on A. Proof. Take a,b ∈ A and �1,�2 � 0A such that BL�1 (a) ∩ B L �2 (b) 6= ∅. Now, choose c ∈ BL�1 (a) ∩B L �2 (b) and set �3 = min{�1 −d(a,c),�2 −d(b,c)}. Observe that BL�3 (c) ⊆ B L �1 (a) ∩BL�2 (b). Therefore, U L A forms a left-basis on A. Similarly, the collection of all open right-balls BL� (a), URA = {B R � (a) : a ∈ A,� � 0A} forms a right-basis on A. � Every C∗-algebra valued quasi metric space d naturally induces a T0 topol- ogy T LA , where a set N is open if for each a ∈ N there exists � � 0A such that BL� (a) ⊆ N. Similarly, the topology T RA can be define by using the right-balls (that is, BR� (a)) as its base and hence a C ∗-algebra valued quasi metric space (A,A,d) can be naturally associated with a bi-topological space (A,A,T LA ,T R A ). Moreover, if the map d satisfies d(a,b) = 0 ⇔ a = b instead of condition (i) in Definition (2.2) then d induces a T1 topology. Proposition 2.12. Let (A,A,d) be a C∗-algebra valued quasi metric space and associated (A,A,ds) a C∗-algebra valued metric space. Then (1) A sequence {an} is convergent to a in (A,A,d) if and only if {an} is convergent to a in (A,A,ds). (2) A sequence {an} is Cauchy in (A,A,d) if and only if {an} is Cauchy in (A,A,ds). (3) The C∗-algebra valued quasi metric space (A,A,d) is complete if and only if C∗-algebra valued metric space (A,A,ds) is complete. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 292 C∗-avQMS and fixed point results with an application Proof. (1) Suppose that {an} is convergent to a in (A,A,d), that is, lim n→∞ d(an,a) = lim n→∞ d(a,an) = 0A. Which is equivalent to lim n→∞ ds(an,a) = 1 2 [ lim n→∞ d(an,a) + lim n→∞ d(a,an)] = 0A. Hence, the sequence {an} is convergent to a in (A,A,ds). (2) Suppose that {an} is Cauchy in (A,A,d), that is, lim n→∞ d(an,am) = lim n→∞ d(am,an) = 0A. Which is equivalent to lim n→∞ ds(an,am) = 1 2 [ lim n→∞ d(an,am) + lim n→∞ d(am,an)] = 0A. Therefore, the sequence {an} is Cauchy in (A,A,ds). (3) It is a direct consequence of (1) and (2). � Proposition 2.13. Let (A,A,dA) and (B,A,dB) be two C∗-algebra valued quasi metric spaces. Then (1) d(a,b) = dA(a1,a2) + dB(b1,b2), for all a = (a1,b1), b = (a2,b2) ∈ A×B is a C∗-algebra valued quasi metric on A×B. (2) lim n→∞ (an,bn) = (a,b) in (A × B,A,d) if and only if lim n→∞ an = a in (A,A,dA) and lim n→∞ bn = b in (B,A,dB). Particularly, the topology induced by d coincides the product topology on A×B. (3) {(an,bn)} is a Cauchy sequence in (A × B,A,d) if and only if {an} is a Cauchy sequence in (A,A,dA) and {bn} is a Cauchy sequence in (B,A,dB). (4) (A × B,A,d) is complete if and only if (A,A,dA) and (B,A,dB) are complete. Proof. (1) Assume that a = (a1,b1), b = (a2,b2), c = (c1,c2) ∈ A × B. Then, we have d(a,b) = 0A if and only if dA(a1,b1) + dB(a2,b2) = 0A, that is, dA(a1,b1) = dB(a2,b2) = 0A, which implies that a1 = b1 and a2 = b2, that is, a = b. Now, to show the triangular inequality, we have d(a,b) = dA(a1,a2) + dB(b1,b2) 4 dA(a1,c1) + dA(c1,a2) + dB(b1,c2) + dB(c2,b2) = dA(a1,c1) + dB(b1,c2) + dA(c1,a2) + dB(c2,b2) = d(a,c) + d(c,b). Therefore, (A×B,A,d) is a C∗-algebra valued quasi metric space. (2) Let lim n→∞ (an,bn) = (a,b) in (A×B,A,d) if and only if lim n→∞ d ( (an,bn), (a,b) ) = lim n→∞ [ dA(an,a) + dB(bn,b) ] = 0A and lim n→∞ d ( (a,b), (an,bn) ) = lim n→∞ [ dA(a,an) + dB(b,bn) ] = 0A © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 293 M. Asim, S. Kumar, M. Imdad and R. George which is equivalent to lim n→∞ dA(an,a) = lim n→∞ dB(bn,b) = lim n→∞ dA(a,an) = lim n→∞ dB(b,bn) = 0A. Therefore, lim n→∞ an = a in (A,A,dA) and lim n→∞ bn = b in (B,A,dB). Therefore, (3) Suppose the sequence {(an,bn)} is a Cauchy in (A×B,A,d) if and only if lim n,m→∞ d ( (an,bn), (am,bm) ) = lim n,m→∞ [ dA(an,am) + dB(bn,bm) ] = 0A which is equivalent to lim n,m→∞ dA(an,am) = lim n,m→∞ dB(bn,bm) = 0A. Therefore, {an} is a Cauchy sequence in (A,A,dA) and {bn} is a Cauchy se- quence in (B,A,dB). (4) It is a direct consequence of (2) and (3). � 3. Fixed Point Results Now, we present our main result as follows: Theorem 3.1. Let (A,A,d) be complete C∗-algebra valued quasi metric space and f : X → X a mapping satisfies the following (for λ ∈A with ‖λ‖ < 1): (3.1) d(fa,fb) 4 λ∗d(a,b)λ, ∀ a,b ∈ A. Then f has a unique fixed point. Proof. Firstly, select a0 ∈ A and extract an iterative sequence {an} as: an = fan−1 = f na0, ∀ n ∈ N. Now, we want to show that lim n,m→∞ d(an+1,an) = 0A. By choosing a = an+1 and b = an in 3.1, we wet d(an+1,an) = d(fan,fan−1) = λ ∗d(an,an−1)λ 4 (λ∗)2d(an−1,an−2)λ 2 4 . . . 4 (λ∗)nd(a1,a0)λ n. Similarly, we can have d(an,an+1) 4 (λ ∗)nd(a0,a1)λ n. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 294 C∗-avQMS and fixed point results with an application Now, we assert that {an} is Cauchy sequence. For any n,m ∈ N such that n + 1 > m, we have d(an+1,am) 4 d(an+1,an) + d(an,an−1) + · · · + d(am+1,am) 4 (λ∗)nd(a1,a0)λ n + · · · + (λ∗)md(a1,a0)λm = n∑ i=m (λ∗)id(a1,a0)λ i = n∑ i=m (λ∗)i(d(a1,a0)) 1 2 (d(a1,a0)) 1 2 λi = n∑ i=m ( (d(a1,a0)) 1 2 λi )∗( d(a1,a0) 1 2 λi ) = n∑ i=m ∣∣∣(d(a1,a0)) 12 λi∣∣∣2 4 ∥∥∥∥∥ n∑ i=m ∣∣∣(d(a1,a0)) 12 λi∣∣∣2 ∥∥∥∥∥I 4 n∑ i=m ∥∥∥(d(a1,a0)) 12 ∥∥∥2 ∥∥λi∥∥2 I 4 ‖d(a1,a0)‖ n∑ i=m ‖λ‖2i I 4 ‖d(a1,a0)‖ ‖λ‖2m 1 −‖λ‖ I → 0A (as m →∞). Thus, {an} is left-Cauchy sequence, that is lim m→∞ d(an,am) = 0A ∀ n ≥ m ≥ N. Similarly, we can show that {an} is right-Cauchy sequence, that is (for m > n) lim n→∞ d(am,an) = 0A ∀ m ≥ n ≥ N. Therefore, {an} is Cauchy sequence. Since, (A,A,d) is complete C∗-algebra valued quasi metric space, then there exists a point a in A such that lim n→∞ an = a, that is, lim n→∞ d(an,a) = lim n→∞ d(a,an) = 0A. Then, we get d(fa,a) 4 d(fa,an+1) + d(an+1,a) 4 d(fa,fan) + d(an+1,a) 4 λ∗d(a,an)λ + d(an+1,a) 4 ‖λ2‖‖d(a,an)‖I + d(an+1,a). On taking limit as n →∞, we get fa = a. Hence, a is fixed point of f. Now, to show that the fixed point is unique, we assume that there are two fixed points, say a1,a2 ∈ A such that fa1 = a1 and fa2 = a2. Then by using © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 295 M. Asim, S. Kumar, M. Imdad and R. George 3.1, we have ‖d(a1,a2)‖ = ‖d(fa1,fa2)‖ ≤ ‖λ∗d(a1,a2)λ‖ ≤ ‖λ∗‖‖d(a1,a2)‖‖λ‖ = ‖λ‖2‖(a1,a2)‖ deals a contradiction. Hence, a1 = a2, that is, a1 is a unique fixed point of f. This completes the proof. � Example 3.2. In the Example 2.4, we define a self-mapping f : A → A by: fa = a 5 , ∀ a ∈ A. Notice that, d(fa,fb) 4 λ∗d(a,b)λ, (for each a,b ∈ A) satisfies and λ = [√ 5 5 0 0 √ 5 5 ] ∈ A and ‖λ‖ = √ 5 5 = 1 √ 5 < 1. Hence, all the assumptions of Theorem 3.1 are fulfilled and f unique fixed point, namely a = 0A. Before presenting the next theorem we recall the following lemma which is needed is the sequel. Lemma 3.3. Let A be a unital C∗-algebra with a unit I. We have (1) If a ∈ A+ with ‖a‖ < 12 , then I −a is invertible and ‖a(I −a) −1‖ < 1; (2) if a,b ∈ A+ with ab = ba, then ab ∈A+; (3) we denote A ′ = {a ∈ A : ab = ba, ∀b ∈ A}. Let a ∈ A ′ , if b,c ∈ A with b < c < 0A and I − a ∈ A ′ + is an invertible operator, then (I −a)−1b < (I −a)−1c. Theorem 3.4. Let (A,A,d) be complete C∗-algebra valued quasi metric space and f : X → X a continuous mapping satisfies that the following (for λ ∈ A with ‖λ‖ < 1 2 ): (3.2) d(fa,fb) 4 λ[d(fa,b) + d(a,fb)], ∀ a,b ∈ A. Then f has a unique fixed point. Proof. Firstly, select a0 ∈ A and extract an iterative sequence {an} as: an = fan−1 = f na0, ∀ n ∈ N. Now, we want to show that lim n,m→∞ d(an+1,an) = 0A. By choosing a = an+1 and b = an in 3.2, we wet d(an+1,an) = d(fan,fan−1) = λ[d(fan,an−1) + d(an,fan−1)] = λ[d(an+1,an−1) + d(an,an)] 4 λ[d(an+1,an) + d(an,an−1)] = λd(an+1,an) + λd(an,an−1). © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 296 C∗-avQMS and fixed point results with an application Thus, (I −λ)d(an+1,an) 4 λd(an,an−1). Since, λ ∈A with ‖λ‖ < 1 2 , then we have (I−λ)−1 ∈A and also λ(I−λ)−1 ∈A with ‖λ(I −λ)−1‖ < 1 (by Lemma 3.3). Then, by assuming u = λ(I −λ)−1, we obtain d(an+1,an) 4 λ(I −λ)−1d(an,an−1) = ud(an,an−1). Similarly, we can have d(an,an+1) 4 ud(an−1,an). Now, we show that the sequence {an} is Cauchy. Suppose n+ 1 > m, ∀ n,m ∈ N, so we have d(an+1,am) 4 d(an+1,an) + d(an,an−1) + · · · + d(am+1,am) 4 (un + un−1 + · · · + um)d(a1,a0) = n∑ i=m u i 2 u i 2 (d(a1,a0)) 1 2 (d(a1,a0)) 1 2 = n∑ i=m (d(a1,a0)) 1 2 u i 2 u i 2 (d(a1,a0)) 1 2 λi = n∑ i=m ( u i 2 (d(a1,a0)) 1 2 )∗( u i 2 d(a1,a0) 1 2 ) = n∑ i=m ∣∣∣u i2 (d(a1,a0)) 12 ∣∣∣2 4 ∥∥∥∥∥ n∑ i=m ∣∣∣u i2 (d(a1,a0)) 12 ∣∣∣2 ∥∥∥∥∥I 4 n∑ i=m ∥∥∥(d(a1,a0)) 12 ∥∥∥2 ∥∥∥u i2 ∥∥∥2 I 4 ‖d(a1,a0)‖ n∑ i=m ∥∥∥u i2 ∥∥∥i I 4 ‖d(a1,a0)‖ ‖u‖m 1 −‖u‖ I → 0A (as m →∞). Thus, {an} is left-Cauchy sequence, that is lim n,m→∞ d(an,am) = 0A. Similarly, we can have {an} is right-Cauchy sequence, that is (for m > n) lim n,m→∞ d(am,an) = 0A. Therefore, the sequence {an} is Cauchy. Since, (A,A,d) is complete C∗-algebra valued quasi metric space, then there exists a point a in A such that lim n→∞ an = © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 297 M. Asim, S. Kumar, M. Imdad and R. George a, that is, lim n→∞ d(an,a) = lim n→∞ d(a,an) = 0A. Now, by using the continuity of f, we have d(fa,a) 4 d(fa,an+1) + d(an+1,a) 4 d(fa,fan) + d(an+1,a) On taking limit as n →∞, we get fa = a. Hence, a is fixed point of f. Now, to show that the fixed point is unique, we assume that there are two fixed points, say a1,a2 ∈ A such that fa1 = a1 and fa2 = a2. Then by employing 3.4, we have ‖d(a1,a2)‖ = ‖d(fa1,fa2)‖ ≤ ‖λ[d(fa1,a2) + d(a1,fa2)]‖ ≤ ‖λ‖‖d(a1,a2) + d(a1,a2)‖ ≤ ‖λ‖[‖d(a1,a2)‖ + ‖d(a1,a2)‖] = 2‖λ‖‖(a1,a2)‖ a contradiction (since 2‖λ‖ < 1). Hence, a1 = a2, that is, a1 is a unique fixed point of f. This completes the proof. � Now, we obtain following corollaries: Remark 3.5. By taking d(a,b) = d(b,a), for all a,b ∈ A in Theorem 3.1, we obtain Theorem 2.1 of Z. Ma et al. [12]. Remark 3.6. By taking d(a,b) = d(b,a), for all a,b ∈ A in Theorem 3.4, we obtain Theorem 2.3 of Z. Ma et al. [12]. 4. Application To find the existence and uniqueness results of a contractive mapping on complete C∗-algebra valued metric space for the integral type equation is car- ried out by Z. Ma et al. [12] in 2014 whose lines are as under: Example 4.1 ( [12]). Consider the integral equation (4.1) a(ξ) = ∫ ∆ G(ξ,ω,a(ω))dω + h(ξ), ∀ ξ,ω ∈ ∆, where ∆ is a Lebesgue measurable set. Suppose that (1) h is an essentially bounded measurable function defined on ∆ and G : ∆2 ×R → R, (2) there exists a continuous function η : ∆ × ∆ → R and λ ∈ (0, 1) such that | G(ξ,ω,a(ω)) −G(ξ,ω,b(ω)) |≤ λ | η(ξ,ω) | (| a(ω) − b(ω) |) , for all ξ,ω ∈ ∆ and a,b ∈ R. (3) supξ∈∆ ∫ ∆ | η(ξ,ω) | dω ≤ 1. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 298 C∗-avQMS and fixed point results with an application Then the integral equation has a unique solution in A, where A stands for the space of essentially bounded measurable functions defined on ∆. Now, we will utilize Theorem 3.1 to find the solution of following integral equation: (4.2) a(ξ) = ∫ ∆ G(ξ,ω,a(ω))dω + h(ξ), ∀ ξ,ω ∈ ∆, where, ∆ is a Lebesgue measurable set with m(∆) < ∞, G : ∆ × ∆ × R → R and h ∈ A. Define d : A×A →A by (for all a,b ∈ A),: d(a,b) = { π|a−b|+|a| if a 6= b 0A if a = b. where L(H) = A, H stand for the set of square integrable functions defined on ∆, and πa : H → H is the multiplicative operator defined by: πa(θ) = a.θ, for all θ ∈ H. Now, we present our following theorem: Theorem 4.2. Assume that (for all a,b ∈ A) (1) ∃ a continuous function η : ∆ × ∆ → R and λ ∈ (0, 1) such that | G(ξ,ω,a(ω)) −G(ξ,ω,b(ω)) |≤ λ | η(ξ,ω) | (| a(ω) − b(ω) | + | a(ω) |) , for all ξ,ω ∈ ∆. (2) supξ∈∆ ∫ ∆ | η(ξ,ω) | dω ≤ 1. Then the integral equation (4.2) has a unique solution in A. Proof. Define f : A → A by: fa(ξ) = ∫ ∆ G(ξ,ω,a(ω))dω + h(ξ), ∀ ξ,ω ∈ ∆. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 299 M. Asim, S. Kumar, M. Imdad and R. George Set k = λI, then k ∈A and ‖k‖ = λ < 1. For any point u in H, we have ‖d(fa,fb)‖ = sup ‖u‖=1 (π|fa−fb|+|fa|(u),u) = sup ‖u‖=1 ∫ ∆ [∣∣∣∣ ∫ ∆ G(ξ,ω,a(ω)) −G(ξ,ω,b(ω))dω ∣∣∣∣ ] u(ξ) ¯u(ξ)dξ + sup ‖u‖=1 ∫ ∆ (∫ ∆ G(ξ,ω,a(ω)) ) u(ξ) ¯u(ξ)dξ ≤ sup ‖u‖=1 ∫ ∆ [∫ ∆ ∣∣G(ξ,ω,a(ω)) −G(ξ,ω,b(ω))∣∣dω]|u(ξ)|2dξ + sup ‖u‖=1 ∫ ∆ ∣∣∣∣ ∫ ∆ G(ξ,ω,a(ω)) ∣∣∣∣ |u(ξ)|2dξ ≤ sup ‖u‖=1 ∫ ∆ [∫ ∆ ∣∣λη(ξ,ω)(a(ω) − b(ω)+ | a(ω) |)∣∣dω]|u(ξ)|2dξ ≤ sup ‖u‖=1 ∫ ∆ [∫ ∆ ∣∣λ | η(ξ,ω) | (| a(ω) − b(ω) | + | a(ω) |)∣∣dω]|u(ξ)|2dξ ≤ λ sup ‖u‖=1 ∫ ∆ [∫ ∆ |η(ξ,ω)|dω ] |u(ξ)|2dξ‖a− b‖∞ ≤ λ sup ξ∈E ∫ ∆ |η(ξ,ω)|dω sup ‖u‖=1 ∫ ∆ |u(ξ)|2dξ‖a− b‖∞ ≤ λ‖a− b‖∞ = ‖k‖‖d(a,b)‖. Since, ‖k‖ < 1, so one can easily seen that the mapping f satisfies all the assumptions of Theorem 3.1. Hence, (4.2) has a unique solution, means that f has a unique fixed point. � Acknowledgements. The authors are thankful to the learned reviewer for his valuable comments. References [1] A. Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory Appl. 2012, Article ID 204. [2] M. Asim and M. Imdad, C∗-algebra valued extended b-metric spaces and fixed point results with an application, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 82, no. 1 (2020), 207–218. [3] M. Asim and M. Imdad, C∗-algebra valued symmetric spaces and fixed point results with an application, Korean J. Math. 28, no. 1 (2020), 17–30. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 300 C∗-avQMS and fixed point results with an application [4] M. Asim, M. Imdad and S. Radenovic, Fixed point results in extended rectangular b- metric spaces with an application, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81, no. 2 (2019), 43–50. [5] M. Asim, A. R. Khan and M. Imdad, Rectangular Mb-metric spaces and fixed point results, Journal of Mathematical Analysis 10, no. 1 (2019), 10–18. [6] M. Asim, A. R. Khan and M. Imdad, Fixed point results in partial symmetric spaces with an application, Axioms 8, no. 1 (2019): 13. [7] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., Gos. Ped. Inst. Unianowsk 30 (1989), 26–37. [8] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equa- tions integrals, Fund. Math. 3 (1922), 133–181. [9] A. Branciari, A fixed point theorem of BanachÔÇôCaccioppoli type on a class of gener- alized metric spaces, Publ. Math. 57 (2000), 31–37. [10] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Os- traviensis 1, no. 1 (1993), 5–11. [11] H. Long-Guang and Z. Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1468–1476. [12] Z. H. Ma, L. N. Jiang and H. K. Sun, C∗-algebra valued metric spaces and related fixed point theorems, Fixed Point Theory Appl. 2014, Article ID 206. [13] Z. H. Ma and L. N. Jiang, C∗-algebra valued b-metric spaces and related fixed point theorems, Fixed Point Theory Appl. 2015, Article ID 222. [14] S. G. Matthews, Partial metric topology, Annals of the New York Academy of Sciences 728 (1994), 183–197. [15] S. Shukla, Partial b-metric spaces and fixed point theorems, Mediterranean Journal of Mathematics 11, no. 2 (2014), 703–711. [16] W. A. Wilson, On quasi-metric spaces, American Journal of Mathematics 53, no. 3 (1931), 675–684. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 301