# Common Fixed Points of Two Mappings regarding a Generalizedc-Distance over a Banach Algebra

Authors: Ladan Aryanpour; Hamidreza Rahimi; Ghasem Soleimani Rad
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, applying the concept of a generalizedc-distance in cone b-metric spaces over Banach algebra with a nonnormal solid cone therein, we establish several common fixed point theorems for two noncontinuous mappings satisfying the Han-Xu-type contraction. Our results are interesting, since they are not equivalent to former well-known results regarding a wt-distance in b-metric spaces while they contain recent results corresponding to a generalized c-distance in cone b-metric spaces.


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## Body

## 1. Introduction and Preliminaries

In 2015, Bao et al. [1] suggested a generalized c-distance, which extended many former definitions in [2–10]) and references therein. Moreover, with regard to a survey on fixed point theory corresponding to this distance, see [11–13]).In 2013, Liu and Xu [14] offered a cone metric space over Banach algebra by replacing a Banach space E with a Banach algebra E. After this definition, some other researchers suggested new several various spaces over a Banach algebra and extended available results in [15–17] and their references. In 2015, Huang et al. [18] proposed a c-distance in cone metric spaces over a Banach algebra E. In 2018, Han and Xu [19] proved some common fixed point results by removing the assumption of continuity of the mappings and by deleting the hypothesis of the normality of the cone. Recently, Arabnia et al. [20] suggested a generalized c-distance in cone b-metric spaces over a Banach algebra.Here, we review some basic definitions and preliminary lemmas which are needed to continue.LetE be a Banach algebra with a unit element e, a zero element θ, a norm ∥x∥, and a cone P therein. Define a partial order ≼ with respect to P by x≼y iff y−x∈P. Also, x≼y if x≼y and x≠y, and x≪y iff y−x∈intP (intP is the same as the interior of P). If intP≠∅, then the cone P is named solid. Further, if there is M>0 so that θ≼x≼y deduced that ∥x∥≤M∥y∥ for every x,y∈E, then the cone P is named a normal cone.Definition 1 (see [17]).
LetX be a nonempty set, s≥1 be a constant, and E be a Banach algebra. For all x,y,z∈X, assume that d:X×X→E satisfies the following items: 2
(d1)θ≼dx,y and dx,y=θ iff x=y
(d2)dx,y=dy,x
(d3)dx,z≼sdx,y+dy,zThen,d is named a cone b-metric on X, and X,d is named a cone b-metric space over the Banach algebra E.For definitions such as convergent and Cauchy sequences,c-sequence, completeness, continuity, and examples, see [16, 17]. In the sequel, assume that X,d is a cone b-metric space with the coefficient s≥1 over a Banach algebra E and P is a solid cone therein.Lemma 2 (see [17, 21]).
LetE be a Banach algebra and u,v∈E. Then, the following items hold:
(l1) if ρu<∣w∣ where ρu is the spectral radius and w is a complex constant, then we−u is invertible in E and
(1)we−u−1=∑i=0∞uiwi+1,ρwe−u−1≤1∣w∣−ρu
(l2)if u commutes with v, then ρu+v≤ρu+ρv and ρuv≤ρuρv
(l3)if u,k1,k2∈P with k1≼k2, u≼k1u, and ρk2<1, then u=θ
(l4)if ρu<1, then un is a c-sequence. Furthermore, kun is a c-sequence for each arbitrarily vector k∈PDefinition 3 (see [20]).
LetX,d be a cone b-metric space over a Banach algebra E with a constant s≥1. A function ν:X×X→E is named a generalized c-distance on X if, for all x,y,z∈X, it satisfies in the following properties:
(ν1)θ≼νx,y
(ν2)νx,y≼sνx,z+νz,y
(ν3) for x∈X and a sequence yn converges to y in X, if νx,yn≼u for some u=ux∈P and all n≥1, then νx,y≼su
(ν4)for all c∈E with θ≪c, there exists e∈E with θ≪e so that νz,x≪e and νz,y≪e imply dx,y≪cNotice that a generalizedc-distance contains both wt-distance and c-distance. Further, νx,y=νy,x is not necessarily true and the νx,y=θ dose not imply that x=y for every x,y∈X.Example 4.
TakeX=0,1, E=Cℝ10,1 with ∥f∥=∥f∥∞+∥f′∥∞. Let multiplication in E be just pointwise multiplication. Then, E is a Banach algebra with a unit et=1 for all t∈0,1. Also, let P=f∈A∣ft≥0,∀t∈0,1 be a solid cone. Now, define d:X×X→P⊂A by da,bt=a−b22t for all a,b∈X, where 2t∈P. Then, X,d is a cone b-metric space over a Banach algebra E. Consider a mapping ν:X×X→E by νa,bt=a2+b22t for all a,b∈X. Then, ν is a generalized c-distance on X.Lemma 5 (see [20]).
Consider a generalizedc-distance ν on X with two sequences tn and sn in X, α,β,γ∈X, and pn and qn be two c-sequences. Then, the following cases hold:
(1)
ifνtn,β≼pn and νtn,γ≼qn for n∈ℕ, then β=γ. In particular, if να,β=θ and να,γ=θ, then β=γ(2)
ifνtn,sn≼pn and νtn,γ≼qn for n∈ℕ, then sn converges to γ(3)
ifνtn,tm≼pn for m>n, then tn is a Cauchy sequence in X(4)
ifνβ,tn≼pn for n∈ℕ, then tn is a Cauchy sequence in XLemma 6 (see [20]).
Letν be a generalized c-distance on X. If να,β=νβ,α=θ for α,β∈X, then α=β.In this work, we establish several common fixed point theorems regarding a generalizedc-distance over a Banach algebra by removing the normality of the cone and the continuity of the mappings.


## 2. Main Results

The following theorem is the principal result of this paper using Han-Xu-type contraction [19].Theorem 7.
Consider a generalizedc-distance ν on a complete cone b-metric space X,d over a Banach algebra E. Assume that two mappings F,G:X→X for every a,b∈X satisfy the following relations:
(2)νFa,Gb≼h1νa,b+h2νa,Fa+h3νa,Gb,(3)νGa,Fb≼h1νa,b+h2νa,Ga+h3νa,Fb,
whereh1,h2,h3∈P so that sh3 commutes with h1+h2+sh3 and
(4)ρsh3+ρsh1+sh2+s2h3<1.
Then,F and G have a unique common fixed point.Proof.
Assume thata0∈X is an arbitrary point with Fa0≠a0. Consider the sequence an by putting a2n+1=Fa2n and a2n+2=Ga2n+1 for all n∈ℕ. Applying relation (2) by a=a2n and b=a2n+1, we get
(5)νa2n+1,a2n+2=νFa2n,Ga2n+1≼h1νa2n,a2n+1+h2νa2n,Fa2n+h3νa2n,Ga2n+1≼h1νa2n,a2n+1+h2νa2n,a2n+1+sh3νa2n,a2n+1+νa2n+1,a2n+2,
for alln∈ℕ, which induces that
(6)e−sh3νa2n+1,a2n+2≼h1+h2+sh3νa2n,a2n+1.
Similarly, applying relation (3) by a=a2n+1 and b=a2n+2, we get
(7)νa2n+2,a2n+3=νGa2n+1,Fa2n+2≼h1νa2n+1,a2n+2+h2νa2n+1,a2n+2+sh3νa2n+1,a2n+2+νa2n+2,a2n+3,
for alln∈ℕ, which induces that
(8)e−sh3νa2n+2,a2n+3≼h1+h2+sh3νa2n+1,a2n+2.
Now, the inequalities (6) and (8) show that
(9)e−sh3νan,an+1≼h1+h2+sh3νan−1,an.
Sinceρsh3<1 (by relation (4)), it follows from Lemma 2, (l1), that e−sh3 is invertible and e−sh3−1=∑i=0∞sh3i. Let h=e−sh3−1h1+h2+sh3. Since sh3 commutes with h1+h2+sh3, we obtain
(10)e−sh3−1h1+h2+sh3=h1+h2+sh3e−sh3−1.
Now, seth=e−sh3−1h1+h2+sh3. Then, by Lemma 2, (l1) and (l2), we obtain
(11)ρh=ρe−sh3−1h1+h2+sh3≤11−ρsh3ρh1+h2+sh3<1s,
which implies thate−sh−1=∑i=0∞shi. Moreover, by multiplying e−sh3−1 in relation (9), we get
(12)νan,an+1≼e−sh3−1h1+h2+sh3νan−1,an=hνan−1,an≼⋯≼hnνa0,a1.
Considerm,n∈ℕ with m>n≥1. Using relation (12) and (ν2), we deduce by a simple computation that
(13)νan,am≼e−sh−1shnνa0,a1.
Sinceρh<1/s and s≥1, we have ρh<1 which means that hn is a c-sequence by Lemma 2, (l4). Using Lemma 5, (3), an is a Cauchy sequence. Due to the completeness of the space X, there is a u∈X so that an→u as n→∞. Using relation (13) and (ν3), we have
(14)νan,u≼e−sh−1s2hnνa0,a1,which shows that
(15)νa2n+1,u≼e−sh−1s2h2n+1νa0,a1,(16)νa2n,u≼e−sh−1s2h2nνa0,a1.
Now, we establish thatFu=Gu=u. In relation (2), set a=a2n and b=u. Then, we get
(17)νa2n+1,Gu=νFa2n,Gu≼h1νa2n,u+h2νa2n,a2n+1+sh3νa2n,a2n+1+νa2n+1,Gu,
which induces thate−sh3νa2n+1,Gu≼h1νa2n,u+h2+sh3νa2n,a2n+1. Note that e−sh3 is invertible. Thus, by the inequalities (12) and (16), we get
(18)νa2n+1,Gu≼e−sh3−1h1νa2n,u+h2+sh3νa2n,a2n+1≼e−sh3−1h1e−sh−1s2h2nνa0,a1+h2+sh3sh2nνa0,a1=e−sh3−1h1e−sh−1s2+sh2+sh3h2nνa0,a1.
By considering the inequalities (15) and (18), Lemma 2, (l4), and Lemma 5, (1), we conclude that Gu=u. Now, in relation (3), set a=a2n+1 and b=u. Then, we get
(19)νa2n+2,Fu=νGa2n+1,Fu≼h1νa2n+1,u+h2νa2n+1,a2n+2+sh3νa2n+1,a2n+2+νa2n+2,Fu,
which induces thate−sh3νa2n+2,Fu≼h1νa2n+1,u+h2+sh3νa2n+1,a2n+2. Note that e−sh3 is invertible. Thus, by the inequalities (12) and (15), we get
(20)νa2n+2,Fu≼e−sh3−1h1νa2n+1,u+h2+sh3νa2n+1,a2n+2≼e−sh3−1h1e−sh−1s2h2n+1νa0,a1+h2+sh3sh2n+1νa0,a1=e−sh3−1h1e−sh−1s2+sh2+sh3h2n+1νa0,a1.
By considering the inequalities (16) and (20), Lemma 2, (l4), and Lemma 5, (1), we conclude that Fu=u. Consequently, Fu=Gu=u; that is, u is a common fixed point of F and G. Also, by using the relation (2), we have
(21)νu,u=νFu,Gu≼h1νu,u+h2νu,Fu+h3νu,Gu=h1+h2+h3νu,u,
which induces thate−h1−h2−h3νu,u≼θ. Now, notice that h1+h2+h3≼sh1+sh2+s2h3. Thus, by relation (4), e−h1−h2−h3 is invertible. Hence, by Lemma 2, (l3), we have νu,u=θ. Next, we prove that the common fixed point of F and G is unique. Assume that v is another common fixed point F and G. It follows from relation (2) that
(22)νu,v=νFu,Gv≼h1νu,v+h2νu,Fu+h3u,Gv=h1+h3νu,v.
Sinceh1+h3≼sh1+sh2+s2h3 and by using relation (4), we have νu,v=θ by Lemma 2, (l3). Also, it follows from relation (3) that
(23)νv,u=νGv,Fu≼h1νv,u+h2νv,Gv+h3v,Fu=h1+h3νv,u,
which implies by the above procedure thatνv,u=θ. Now, by Lemma 6, we obtain u=v. Consequently, the common fixed point of F and G is unique. Here, the proof ends.Corollary 8.
Consider a generalizedc-distance ν on a complete cone b-metric space X,d over a Banach algebra E. Assume that two mappings F,G:X→X for every a,b∈X satisfy the following relations:
(24)νFa,Gb≼h1νa,b+h2νa,Fa,νGa,Fb≼h1νa,b+h2νa,Ga,
whereh1,h2∈P with ρsh1+sh2<1. Then, F and G have a unique common fixed point.Proof.
It is sufficient to seth3=θ in Theorem 7.Corollary 9.
Consider a generalizedc-distance ν on a complete cone b-metric space X,d over a Banach algebra E. Assume that a mapping F:X→X for every a,b∈X satisfies the following relation:
(25)νFa,Fb≼h1νa,b+h2νa,Fa,
whereh1,h2∈P with ρsh1+sh2<1. Then, F has a unique fixed point.Proof.
It follows by takingF=G in Corollary 8.Corollary 10.
Consider a generalizedc-distance ν on a complete cone b-metric space X,d over a Banach algebra E. Assume that a mapping F:X→X for every a,b∈X satisfies the following relation:
(26)νFa,Fb≼h1νa,b,
whereh1∈P with ρh1<1/s. Then, F has a unique fixed point.Proof.
It is sufficient to seth2=θ in Corollary 9.Example 11.
LetX=0,1, E=Cℝ10,1 with the norm ∥f∥=∥f∥∞+∥f′∥∞ and multiplication in E be just pointwise multiplication. Then, E is a real Banach algebra with a unit et=1 for all t∈0,1. Take a solid cone P=f∈E∣ft≥0forallt∈0,1 and define the cone b-metric d:X×X→P⊆E by da,b=a−bs2t, where 2t∈P⊂E and s=2. Consider a mapping ν:X×X→E by νa,bt=b22t for all a,b,t∈X. Then, ν is a generalized c-distance in cone b-metric space d over Banach algebra E. Take h1=2/121+3/121t and define the mapping F:X→X by
(27)Fa=211a,a∈ℚ∩X,212a,otherwise.
Clearly,F is not continuous. Also,
(28)ρsh1=10121<1.
On the other hand, we have the following two cases:(i)
for alla∈X and b∈ℚ∩X, we get(29)νFa,Fbt=Fb22t=2121b22t≼h1νa,bt(ii)
for alla∈X and b∈ℚ∩X, we get(30)νFa,Fbt=Fb22t=2144b22t≼h1νa,btThat is, all hypotheses of Corollary10 are held. Thus, F has a unique fixed point at a=0.Corollary 12.
Consider a generalizedc-distance ν on a complete cone b-metric space X,d. Assume that two mappings F,G:X→X for every a,b∈X satisfy the following relations:
(31)νFa,Gb≼h1νa,b+h2νa,Fa+h3νa,Gb,νGa,Fb≼h1νa,b+h2νa,Ga+h3νa,Fb,
whereh1,h2,h3∈P so that sh1+h2+s2+sh3<1. Then, F and G have a unique common fixed point.Proof.
In Theorem7, put ρh1=h1, ρh2=h2, and ρh3=h3. The proof is evident.Remark 13.
In Theorem7 and its corollaries, we take s=1. Then, we obtain the same Theorem 16 and its next corollaries from Han and Xu [19] regarding a c-distance ν over a Banach algebra E. Also, these results generalize some main theorems and its next corollaries in [1, 3, 12, 13, 15, 18, 20].


## 3. Conclusions

In this paper, we established several fixed point results for two mappingsF and G regarding a generalized c-distance ν over a Banach algebra E. Notice that the class of these distances is bigger than the class of usual c-distances over the same Banach algebra. Also, this class is not equivalent to the class of wt-distances in b-metric spaces. Further, we removed the continuity condition of the mappings F and G in expressing our results.



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