Rectangular cone b-metric spaces over Banach algebra and contraction principle

Fixed Point Theory and Applications, Sep 2017

Rectangular cone b-metric spaces over a Banach algebra are introduced as a generalization of metric space and many of its generalizations. Some fixed point theorems are proved in this space and proper examples are provided to establish the validity and superiority of our results. An application to solution of linear equations is given which illustrates the proper application of the results in spaces over Banach algebra.

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Rectangular cone b-metric spaces over Banach algebra and contraction principle

George et al. Fixed Point Theory and Applications Rectangular cone b-metric spaces over Banach algebra and contraction principle Reny George 3 4 Hossam A Nabwey 2 4 R Rajagopalan 4 Stojan Radenovic´ 0 1 KP Reshma 5 0 Faculty of Mathematics and Statistics, Ton Duc Thang University , Ho Chi Minh City , Vietnam 1 Nonlinear Analysis Research Group, Ton Duc Thang University , Ho Chi Minh City , Vietnam 2 Department of Basic Engineering Sciences, Faculty of Engineering, Menofia University , Menofia , Egypt 3 Department of Mathematics and Computer Science, St. Thomas College , Bhilai, Chhattisgarh , India 4 Department of Mathematics, College of Science, Prince Sattam bin Abdulaziz University , Al-Kharj, Kingdom of Saudi Arabia 5 Department of Mathematics, Rungta College of Engineering and Technology , Bhilai, Chhattisgarh , India Rectangular cone b-metric spaces over a Banach algebra are introduced as a generalization of metric space and many of its generalizations. Some fixed point theorems are proved in this space and proper examples are provided to establish the validity and superiority of our results. An application to solution of linear equations is given which illustrates the proper application of the results in spaces over Banach algebra. fixed points; cone rectangular b-metric space; rectangular metric space; rectangular b-metric space - For a given cone P ⊂ A and x, y ∈ A, we say that x y if and only if y – x ∈ P. Note that is a partial order relation defined on A. For more details on the basic concepts of a Banach algebra, solid cone, unit element e, zero element θ , invertible elements in Banach algebra etc. the reader may refer to [–]. For basic properties of Banach algebra and spectral radius refer to [, ]. In what follows A will always denote a Banach algebra, P a solid cone in A and e the unit element of A. Definition . ([]) Let P be a solid cone in a Banach space E. A sequence {un} ⊂ P is said to be a c-sequence if for each c θ there exists a natural number N such that un c for all n > N . Remark . For more on c-sequences see [, , , ]. Lemma . ([]) Let E be a Banach space. (i) If a, b, c ∈ E and a b c, then a c. (ii) If θ a c for each c θ , then a = θ . 3 Main results In this section first we introduce the definition of a rectangular cone b-metric space over a Banach algebra (in short RCbMS-BA) and furnish examples to show that this concept is more general than that of CMS-BA and CbMS-BA. We then define convergence and a Cauchy sequence in a RCbMS-BA and then prove fixed point results in this space. Definition . Let χ be a nonempty set and drcb : χ × χ → A be such that for all x, y, u, v ∈ χ , x = u, v = y: (RCbM) θ drcb(x, y) and drcb(x, y) = θ if and only if x = y; (RCbM) drcb(x, y) = drcb(y, x); (RCbM) there exist s ∈ P, e s such that drcb(x, y) s[drcb(x, u) + drcb(u, v) + drcb(v, y)]. Then drcb is called a rectangular cone b-metric on χ and (χ , drcb) is called a rectangular cone b-metric space over a Banach algebra (in short RCbMS-BA) with coefficient s. If s = e we say that (χ , drcb) is a rectangular cone metric space over a Banach algebra (in short RCMS-BA). In the above definition if condition RCbM is replaced with (CbM) drcb(x, y) s[drcb(x, z) + drcb(z, y)] for all x, y, z ∈ χ , then (χ , drcb) is a CbMS-BA as defined in []. Note that every CMS-BA is a CbMS-BA and CbMS-BA is a RCbMS-BA but the converse is not necessarily true. Inspired by [, ] we furnish the following examples, which will establish our claim. Example . Let A = {a = (ai,j)× : ai,j ∈ R,  ≤ i, j ≤ }, a = ≤i,j≤ |ai,j|, P = {a ∈ A : ai,j ≥ ,  ≤ i, j ≤ } be a cone in A. Let χ = B ∪ N, where B = { n : n ∈ N}. otherwise. Then (χ , drcb) is a RCbMS-BA over A with coefficient s =   . But it is not possible to find s ∈ P, e s satisfying condition CbM and so (χ , d) is not a CbMS-BA over a Banach algebra A. Example . Let χ = [, ] and let A = CR (χ ). For α = (f , g) and β = (u, v) in A, we define α.β = (f .u, g.v) and α = max( f , g ) where f = supx∈χ |f (x)|. Then A is a Banach algebra with unit e = (, ), zero element θ = (, )and P = {(f , g) ∈ A : f (t) ≥ , g(t) ≥ , t ∈ χ } a cone in A. Consider drcb : χ × χ → A given by ⎧⎪ drcb(x, y)(t) = (, ) ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ drcb(x, y)(t) = (c + d.t, a + bt) if x = y; if x, y ∈ B = [,  ) and a, b, c, d are some fixed real numbers; ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ddrrccbb((xx,, yy))((tt)) == ((|nx –(cy+|d(c.t+), dn.t()a, |+x b–ty)|)(a + b.t)) iyoft∈xhe{=rw,n(i}ns;e≥. ) ∈ B and Clearly (χ , drcb) is a RCbMS-BA over A with s = (, ). Again it is not possible to find a real number s ∈ P, e s satisfying condition CbM and so (χ , drcb) is not a CbMS-BA over a Banach algebra A. For any a ∈ χ , the open sphere with center a and radius λ θ is given by Bλ(a) = b ∈ χ : drcb(a, b) ≺ λ . Let U = {Y ⊆ χ : ∀x ∈ Y , ∃r θ , such that Br(x) ⊆ Z}. Then U defines the rectangular bmetric topology for the RCbMS-BA (χ , drcb). The definitions of convergent sequence, Cauchy sequence, c-sequence and completeness in RCbMS-BA are along the same lines as for CbMS-BA given in [] and so we omit these definitions. Remark . We refer to Example . for the following: (i) Open balls in RCbMS-BA need not be an open set. For example Bλ(  ) with   λ =   is not open because open balls with center  are not contained in Bλ(  ).   (ii) The limit of a sequence in RCbMS-BA is not unique. For instance { n } converges to  and . (iii) Every convergent sequence in RCbMS-BA need not be Cauchy. For d( n , n+p ) =   θ as n → ∞, so { n } is not a Cauchy sequence. (iv) A RCbMS-BA need not be Hausdorff, as it is impossible to find r, r Br () ∩ Br () = φ.  such that Theorem . Let (χ , drcb) be a complete RCbMS-BA over A with θ there exist λ ∈ P, r(λ) <  such that s and T : χ → χ . If for all x, y ∈ χ , then T has a unique fixed point. Proof Let x ∈ χ be arbitrary. Consider the iterative sequence defined by xn+ = Txn for all n ≥ . We divide the proof into three cases. Case : Let r(λ) ∈ [, s ) (s > ). If xn = xn+ then xn is fixed point of T . Moreover, for any x ∈ X the iterative sequence {T nx} (n ∈ N) converges to the fixed point. So, suppose that xn = xn+ for all n ≥ . Setting drcb(xn, xn+) = dn, it follows from (.) that drcb(xn, xn+) = drcb(Txn–, Txn) dn λdn– ≺ dn–, λd(xn–, xn), dn λnd. Repeating this process we obtain We consider d(xn, xn+p) in two cases. (.) (.) (.) (.) i.e. the sequence {dn} is strictly decreasing and from this it follows that dn = dm whenever n = m. Continuing this process we get Again setting dn∗ = drcb(xn, xn+) for any n ∈ N, using (.) we get Since r(λ) < s , we have r(sλ) ≤ sr(λ) ≤ sr(λ).r(λ) < s <  and so e – sλ is invertible and If p is odd, say m + , then using (.) as well as the fact that dn = dm whenever n = m we obtain sλn e + sλ + sλ  + · · · d + sλn+ e + sλ + sλ  + · · · d + smλn+md = ∞ i= sλ i sλnd + sλ i sλn+d ∞ i= = e – sλ –sλnd + e – sλ –sλn+d = λn e – sλ –sd[e + λ]. Since r(λ) < s < , using Lemma . of [], it is easy to see that λn is a c-sequence. Again using Proposition . of [], λn(e – (sλ))–sd[ + λ] → θ as n → ∞, and so it follows that, for any c ∈ A with θ c, there exists a natural number N such that, for any n > N, we have λn e – sλ –sd[ + λ] If p is even say m, using (.) and (.) as well as the fact that dn = dm whenever n = m we obtain drcb(xn, xn+m) s drcb(xn, xn+) + drcb(xn+, xn+) + drcb(xn+, xn+m) s[dn + dn+] + s drcb(xn+, xn+) + drcb(xn+, xn+) + drcb(xn+, xn+m) s[dn + dn+] + s[dn+ + dn+] + s[dn+ + dn+] + · · · + sm–[dm– + dm–] + sm–drcb(xn+m–, xn+m) s λnd + λn+d + s λn+d + λn+d + s λn+d + λn+d + · · · + sm– λm–d + λm–d + sm–λn+m–d∗  sλn e + sλ + sλ + · · · d + sλn+ e + sλ + sλ + · · · d + sm–λn+m–d∗  ∞ i= ∞ i= sλ i sλnd + sλ i sλn+d + sm–λn+m–d∗  e – sλ –sλnd + e – sλ –sλn++d + sm–λn+m–d∗  e – sλ –sλnd[e + λ] + sm–λn+m–d∗  Note that r(λ) < s <  and so using Lemma . of [], it is easy to see that λn is a c-sequence. Again using Proposition . of [], (e – (sλ))–sd[e + λ]λn + ( es – λ)–d∗λn → θ as n → ∞  and so it follows that, for any c ∈ A with θ c, there exists a natural number N such that, for any n > N, we have e – sλ –sd[e + λ]λn + Let N = Max{N, N}. Then for all n ≥ N we have e – λ s – d∗λn  c. drcb(xn, xn+p) c. nl→im∞ xn = u. Thus {xn} is a Cauchy sequence and since (χ , drcb) is complete, we can find u ∈ χ such that Since dn = dm whenever n = m there exists k ∈ N such that drcb(u, Tu) = {dk, dk+, . . .}. Then for any n > k (.) (.) (.) drcb(u, Tu) s drcb(u, xn) + drcb(xn, xn+) + drcb(xn+, Tu) = s drcb(u, xn) + dn + drcb(Txn, Tu) s drcb(u, xn) + dn + λdrcb(xn, u) s (e + λ)drcb(xn, u) + λnd → θ as n → ∞, i.e. Tu = u. Now if Tv = v and drcb(u, v) = θ then using (.) one can easily deduce that drcb(u, v) = , and so the fixed point is unique. Case : Let r(λ) ∈ [ s , ) (s > ). In this case, we have r(λ)n →  as n → ∞, and so there exists n ∈ N such that r(λ)n < s . Note that r(λn ) ≤ r(λ)n < s . Also by (.), drcb T n x, T n y = drcb T T n–x , T T n–y λdrcb T n–x , T n–y = λdrcb T T n–x , T T n–y λdrcb T n–x , T n–y · · · λno drcb(x, y). Thus by case , T n has a unique fixed point u∗ ∈ X. Now we have T n Tu∗ = T n+ u∗ = T T n u∗ = Tu∗, (.) i.e. Tu∗ is also a fixed point of T n . Hence, by the uniqueness of the fixed point of T n we get Tu∗ = u∗. Now suppose Tu = u and Tv = v. Then T n u = T n–(Tu) = T n–u = · · · = Tu = u and T n v = T n–(Tv) = T n–v = · · · = Tv = v. By the uniqueness of the fixed point of T n we get u = v. Case : s = . The proof follows from case . Remark . In an open problem in [] the authors have asked whether it is possible to increase the range of λ in Theorem . of [] from (, s ) to (, ). Since every rectangular b-metric space is a RCbMS-BA, Theorem . gives a positive answer to the question posed by the authors. Definition . Let (χ , drcb) be a RCbMS-BA, θ s and T : χ → χ . Then T is called a weak Kannan contraction iff there exist L, λ ∈ P such that  ≤ r(λ) < s+ , and drcb(Tu, Tv) λ drcb(u, Tu) + drcb(v, Tv) + L.α(u, v) ∀u, v ∈ χ (.) and α(u, v) = drcb(u, Tv) or drcb(v, Tu). Theorem . Let (χ , drcb) be a complete RCbMS-BA with θ s, and T : χ → χ be a mapping. If T is a weak Kannan contraction mapping then T has a fixed point. Further if L <  or drcb(Tx, Ty) L∗. drcb(x, Tx) + d(y, Ty) (.) for some L∗ ∈ P, then the fixed point is unique. Proof Let x ∈ χ be arbitrary. Consider the iterative sequence defined by xn+ = Txn for all n ≥ . Let drcb(xn, xn+) = dn and suppose α(x, y) = drcb(x, Ty). It follows from (.) that drcb(xn, xn+) = drcb(Txn–, Txn), dn λ[dn– + dn]. If α(x, y) = drcb(x, Ty) drcb(xn, xn+) = drcb(Txn–, Txn), Thus in both cases dn dn dn λ[dn– + dn]. λ[dn– + dn], (e – λ)–λdn– = βdn–, where β = (e – λ)–λ. Repeating this process we obtain dn βnd. (.) Also, for α(x, y) = drcb(x, Ty) drcb(xn, xn+) = drcb(Txn–, Txn+) = drcb(Txn+, Txn–) λ[dn– + dn+] + L.dn λ[dn– + dn+] + L.dn; for α(x, y) = drcb(y, Tx) where η = (λ(e + β) + Lβ)d ∈ P. Thus we have Note that r(λ) <  and so (e – λ)– is invertible and (e – λ)– = i∞= λi. Therefore r(β) = r((e – λ)–λ) = ( i∞= λi) ≤ i∞= r(λ)i = –r(rλ(λ)) (as r(λ) < ). Thus we have r(β) < s . Therefore r sβ = sr β ≤ sr(β)r(β) = s <  so e – sβ is invertible and e – sβ – = sβ i. ∞ i= We will analyze drcb(xn, xn+p) as follows: For some odd p say m +  drcb(xn, xn+m+) s dn + dn+ + drcb(xn+, xn+m+) (.) (.) s[dn + dn+] + s dn+ + dn+ + drcb(xn+, xn+m+) s[dn + dn+] + s[dn+ + dn+] + s[dn+ + dn+] + · · · + smdn+m s βnd + βn+d + s βn+d + βn+d + s βn+d + βn+d + · · · + smβn+md sβn  + sβ + sβ + · · · d + sβn+  + sβ + sβ + · · · d = ∞ Note that r(β) < s <  and using Lemma . of [], βn is a c-sequence. Again using Proposition . of [], (e – (sβ))–sβnd + (e – (sβ))–sβn+d → θ as n → ∞. It follows that, for any c ∈ A with θ c, there exists N ∈ N such that, for any n > N, we have e – sβ –sβnd + e – sβ –sβn+d (.) For some even p, say m, drcb(xn, xn+m) s dn + dn+ + drcb(xn+, xn+m) s[dn + dn+] + s dn+ + dn+ + drcb(xn+, xn+m) s[dn + dn+] + s[dn+ + dn+] + s[dn+ + dn+] + · · · sβn  + sβ + sβ + · · · d + sβn+  + sβ + sβ + · · · d + sm–ηβm–βn–d since r(β) < s . ∞ i= ∞ i= (.) (.) (.) d(xn, xn+p) c. Note that r(β) < s <  and using Lemma . of [], βn– is a c-sequence. Again using Proposition . of [] (e – (sβ))–sββn–d + (e – (sβ))–sββn–d + η( es – β)–βn–d → θ as n → ∞. It follows that, for any c ∈ A with θ c, there exists N ∈ N such that, for any n > N, drcb(xn, xn+m) e – sβ –sββn–d + e – sβ –sββn–d + η e – β s Let N = Max{N, N}. Then for all n ≥ N we have Thus {xn} is a Cauchy sequence and by completeness of (χ , drcb) there exists u ∈ χ such that Since dn = dm whenever n = m there exists k ∈ N such that d(u, Tu) = {dk, dk+, . . .}. Then for any n > k Note that r(sλ) ≤ sr(λ) <  <  and so e – sλ is invertible. Also, r(β) < s < . Hence using Lemma . of [], βn is a c-sequence and use of Proposition . of [] gives s(e – λs)–[drcb(u, xn) + (e + λ)βnd] + L.drnb(u, xn+) → θ as n → ∞. It follows that, for c ∈ A and θ c, there exists N ∈ N, such that, for any n > N, drcb(u, Tu) s(e – λs)– drcb(u, xn) + ( + λ)βnd + L.drcb(u, xn+) c, (.) i.e. Tu = u. Uniqueness follows easily from (.). Theorem . Let (χ , drcb) be a complete RCbMS-BA with s ≥  and T : χ → χ be a mapping. If there exists λ ∈ P such that  ≤ r(λ) < s+ , and for all x, y ∈ χ then T has a unique fixed point. (.) Proof Note that (.) implies (.) and (.). Hence the result follows from Theorem .. Corollary . Theorem . of [] and Theorem . of []. Proof Note that, for k ∈ P with r(k) <  , where r(k) <  and r(k) < . Thus T satisfies conditions (.) and (.) of Theorem ., with s = . Since every CMS-BA is a RCbMS-BA with s = , the proof follows from Theorem .. Corollary . Theorem . of [] and Theorem . of []. Proof Since every CMS-BA is a RCbMS-BA with s = , the proof follows from Theorem .. Example . Let A = {a = (ai,j)× : ai,j ∈ R,  ≤ i, j ≤ }, a = maxi  j= |ai,j|, P = {a ∈ A : ai,j ≥ ,  ≤ i, j ≤ } be a cone in A. Let χ = A ∪ B, where A = [,  ] and B = [, ]. Let drcb : χ × χ → A be given by if u = v; if u, v ∈ A – {,  ,  ,  ,  ,  }; if u = n (n ≥ ) ∈ A and v ∈ {, }; otherwise. drcb(,  ) = drcb(  ,  ) = drcb(  ,  ) = ⎧ drcb(,  ) = drcb(  ,  ) = drcb(  ,  ) = ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ drcb(,  ) = drcb(  ,  ) = drcb(  ,  ) = drcb(,  ) = drcb(  ,  ) = drcb(  ,  ) = drcb(,  ) = drcb(  ,  ) = drcb(  ,  ) = .. .. ; .. .. ; .. .. ; . . ; . . .. .. ;     drcb(u, v) = ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ dddrrrcccbbb(((uuu,,, vvv))) === |u–nnv|nn|u–v| |u–v| |u–v| Then (χ , drcb) is a RCbMS-BA over A with s =   . However, for u = n and v = m it is impossible to find s ∈ P, e s such that drcb( n , m ) N s(drcb( n , ) + drcb(, m )) for all n, m ∈ , drcb(  ,  ) = . . . . drcb(  ,  ) + d(  ,  ) = .. .. . Define T by and so (χ , drcb) is not a CbMS-BA over A. Also (χ , drcb) is not a CMS-BA over A as ⎧  ⎪  , ⎪⎩  , u ∈ B ∪ {  }; u ∈ A – {D ∪  }. Tu = ⎨  – u, u ∈ D = { n : n ≥ , n = }; drcb(Tu, Tv) = drcb(  ,  ) =    drcb(v, Tv) = drcb(  ,  ) = (.). Then T satisfies condition (.) For α(x, y) = θ then it is enough if we take L sufficiently large. If α(u, v) = θ , we proceed as follows. Case (i): u ∈ B ∪ {  }, v ∈ D, drcb(u, Tv) = drcb(u,  – v); drcb(v, Tu) = drcb(v,  ); α(u, v) = θ iff u + v =  or v =  . Since v ∈ D, v =  . u + v =  only at u =  and v =  . Then .. .. ; drcb(u, Tu) = drcb(  ,  ) =  .. .. ; .. .. . Clearly we can find λ = k   k with k ∈ (,  ) satisfying Case (ii): u ∈ B ∪ {  }, v ∈ A – {D ∪  }, drcb(u, Tv) = drcb(u,  ); drcb(v, Tu) = drcb(v,  ); α(u, v) =  only at u =  or v =  . But u =  . Let v =  and u ∈ B ∪ {  }. Then drcb(Tu, Tv) = drcb(  ,  ) = .. .. ; drcb(v, Tv) = drcb(  ,  ) =  .. .. ; drcb(u, Tu) = |x–  | |x–  | drcb(x,  ) = |x–  | |x–  | ; when u =  , drcb(u, Tu) = ist k ∈ (,  ) such that λ = k  satisfying (.).  k Case (iii): u ∈ D, v ∈ A – {D ∪  }, drcb(u, Tv) = drcb(u,  ); drcb(v, Tu) = drcb(v,  – u); α(u, v) =  at u =  and u + v =  . At u =  , drcb(Tu, Tv) = (  ,  ) = . Hence condition (.) is satisfied. At u = n and v =  – n , drcb(Tu, Tv) = drcb(  – n ,  ) = drcb(u, Tu) = drcb( n ,  – n ) = | n –  | | n –  | | n –  | | n –  | ; drcb(v, Tv) = drcb(  – n ,  ) = |  – n | |  – n | |  – n | |  – n | ; |  – n | |  – n | |  – n | |  – n | ; .. .. . Clearly there exNote that drcb(u, Tu) + drcb(v, Tv) = | n –  | + |  – n | = |  – n | = drcb(Tu, Tv). Hence (.) is satisfied with λ =     . Other cases follow similarly. Indeed condition (.) is satisfied. Note that drcb(u, Tu) + drcb(v, Tv) = θ for any u, v ∈ χ and so T satisfies (.) for sufficiently large L∗. Theorem . is thus applicable and Fix(T ) = {  }. However, condition (.) is not satisfied at u =  and v =  as drcb(Tu, Tv) = drcb(  ,  ) = .. ..  [drcb(u, Tu) + drcb(v, Tv)] =  [drcb(  ,  ) + drcb(  ,  )] = .. .. . Hence Theorem . is not applicable. Example . Let χ = [,  ] and drcb(x, y) = |x – y|. Let Tx = x for all x, y ∈ χ . Then Theorem . is applicable on T and Fix(T ) = {}. However, Corollary . is not applicable on T . If we take X = [, ] then T satisfies (.) but neither L <  nor T satisfy (.).  and  are two fixed points of T . Now, we will apply Theorem . to study the existence and uniqueness of solutions of a system of linear equations. Consider the following system of linear equations: ⎧⎪ ax + ax + · · · + anxn = b, ⎪⎪⎨ ax + ax + · · · + anxn = b, . . ⎪⎪⎪⎩ .anx + anx + · · · + annxn = bn, C with aij, bi (i =  · · · n, j =  · · · n) ∈ . Theorem . If n i=,i=j |aij| + | – ajj| < , then (.) has a unique solution. Proof Consider the Banach algebra A = {a = (aij)n×n : aij ∈ C,  ≤ i, j ≤ n}, with e being the identity matrix of order n, multiplication defined as ordinary matrix multiplication and a = in= |aij|. Let P = {a ∈ A : ai,j ≥ ,  ≤ i, j ≤ } be a cone in A. Let χ = C . Define drcb : χ × χ → A by (.) ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ α ⎛⎝ xxxn –––...yyyn xxxn –––...yyyn ......... xxxn–––...yyyn ⎞⎠ drcb(x, y) = ⎨⎪ ⎛ p |x–y| p |x–y| ... p |x–y| ⎞ ⎪⎪⎪⎪⎪⎪⎪ ⎜⎜⎝ p |x...–y| p |x...–y| ... p |x...–y|⎟⎟⎠ p |xn–yn| p |xn–yn| ... p |xn–yn| ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎝⎛ xxxn –––...yyyn xxxn –––...yyyn ......... xxxn–––...yyyn ⎠⎞ if x, y ∈ X, x – y ∈ [ – ,  ] × [ – ,  ] × · · · × [ – ,  ],   α is the largest integer such that α ∈ {, , , , , , , , } and maxi |xi – yi| < α ; if x = (x, x, . . . , xn) ∈ X: x + x + · · · + xn = p ∈ N and y = (y, y, . . . , yn) ∈ N × N × · · · × N; otherwise. Then (χ , drcb) is a RCbMS-BA over A with s = . Let T : χ → X be defined by Tx = (I – A)x + B for all x ∈ χ , where An×n, χsn× and Bn× are the coefficient matrices of (.). Then the system of linear equation (.) is equivalent to x = Tx. We will show that T satisfies (.). Let x, y ∈ χ . Case . x – y ∈ [ – ,  ] × [ – ,  ] × · · · × [ – ,  ] and α is the largest integer such that maxi{|xi – yi| : i = , , . . . , n} < α , α ∈ {, , , , , , , , }. Then Tx = (γ, γ, . . . , γn); γi = jn=,j=i aijxj + ( – aii)xi, Ty = (η, η, . . . , ηn); ηi = jn=,j=i aijxj + ( – aii)xi and Tx – Ty = (λ, λ, . . . , λn) where λi = jn=,j=i aij(xj – yj) + ( – aii)(xi – yi) and |λi| = γi – ηi = n j=,j=i |aij||xj – yj| + | – aii||xi – yi| < β , β ≥ α. Thus we have x – y . . . x – y ⎞ x – y . . . x – y⎟ ... ... ⎟⎟⎟⎠ . xn – yn . . . xn – yn Case . x – y ∈/ [ – ,  ] × [ – ,  ] × · · · × [ – ,  ]. Then we have ⎛ ( – a) ⎜ a drcb(Tx, Ty) = ⎜⎜⎝⎜ ... an ⎛ x – y × ⎜⎜⎜⎜⎝ x –... y xn – yn an Thus in both cases we have d(Tx, Ty) ≤ γ .d(x, y), where . . . an a an . . . an ⎞ the system of linear equations (.) has a unique solution. Theorem . If n j=,j=i |aij| + | – aii| < , then the conclusion of Theorem . still holds. Proof Let A and P be as in the proof of Theorem . and a = Let drcb : χ × χ → A be given by j= |aij|. Let χ = C . n drcb(x, y) = ⎧ ⎛ |x–y| |x–y| ... |x–y| ⎞ ⎨ ⎝ |x –...y| |x –...y| ... |x –...y|⎠ ⎩ |xn–yn| |xn–yn| ... |xn–yn| ∀x, y ∈ X. Tx = (I – A)x + B for all x ∈ X, Then (χ , drcb) is a RCbMS-BA over A with s = . Define the self map T of χ by where An×n, Xn× and Bn× are the coefficient matrices of (.). Then the system of linear equations (.) is the problem x = Tx. We will show that T satisfies (.). Let x, y ∈ χ . Then ⎜ drcb(Tx, Ty) = ⎜⎜ ⎜ ⎝ ⎛ ( – a) a . . . an a . . . an an ⎞ an ⎟ ... ⎟⎟⎟ drcb(x, y) = γ .drcb(x, y) ⎠ . . . ( – ann) with γ = I – A and r(γ ) ≤ γ = n i=,i=j |aij| + | – ajj| < . Thus T satisfies (.) and so by Theorem . the system of linear equations (.) has a unique solution. Acknowledgements This project is supported by Deanship of Scientific research at Prince Sattam bin Abdulaziz University, Al kharj, Kingdom of Saudi Arabia, under International Project Grant No. 2016/01/6714. The authors are thankful to the learned reviewers for their valuable suggestions which helped in bringing this paper in its present form. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 1. Liu , H, Xu, S: Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings . Fixed Point Theory Appl . 2013 , Article ID 320 ( 2013 ) 2. 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Reny George, Hossam A Nabwey, R Rajagopalan, Stojan Radenović, KP Reshma. Rectangular cone b-metric spaces over Banach algebra and contraction principle, Fixed Point Theory and Applications, 2017, 14, DOI: 10.1186/s13663-017-0608-x