# Kaleva-Seikkala’s Type Fuzzyb-Metric Spaces and Several Contraction Mappings

Authors: Shu-Fang Li; Fei He; Shu-Min Lu
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we introduce the concept of Kaleva-Seikkala’s type fuzzyb-metric spaces as a generalization of the notion of b-metric spaces and fuzzy metric spaces. In such spaces, we establish Banach type, Reich type, and Chatterjea type fixed-point theorems, which improve the relevant results in fuzzy metric spaces. Two technical lemmas are employed to ensure that a Picard sequence is a Cauchy sequence. Finally, various applications are given to testify the fact that our main theorems extend the cases of b-metric spaces.


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## Body

## 1. Introduction

In 1965, the theory of fuzzy sets was introduced by Zadeh in [1]. Henceforth, several researchers have discussed and developed this theory and applied the results to various different areas, such as mathematical programming, modeling theory, cybernetics, neural networks, statistics, construction machinery, and image processing (see, e.g., [2–4]). After this pioneering work, some types of fuzzy metric spaces (briefly, FMS) were presented by numerous authors (refer to [5–7]). In particular, Kramosil and Michalek [7], in 1975, gave the notion of FMS as a modification of the notion of probabilistic metric space initiated by Menger [8]. More detailed information about such spaces and various fixed-point theorems in these FMS can be seen in [9–16]. In 1984, another type of fuzzy metric spaces called Kaleva-Seikkala’s type fuzzy metric space (briefly, KS-FMS) was initiated by Kaleva and Seikkala [17], which generalized the metric space by defining a nonnegative fuzzy number as the distance between two points and applying a new triangle inequality which is analogous to the common triangle inequality. Drawing inspiration from [17], much work has been done in KS-FMS (see, e.g., [18–22] and the references therein). Throughout this paper, we denote by ℕ, ℕ+, and ℝ+, the sets of natural numbers, positive integer numbers, and positive real numbers, respectively. All the concepts about KS-FMS not given in this paper are the same as in [23, 24].As a prevalent generalization of the metric spaces, Bakhtin [25] in 1989 introduced the notion of b-metric spaces (briefly, b-MS), which was formally defined by Czerwik [26] in 1993. In the last decades, many authors investigated the existence and uniqueness of the fixed point for various contractions in b-MS (see, e.g., [27–33]). Furthermore, Aghajani et al. [34] generalized the concept of the b-MS to the Gb-metric space and established several fixed-point theorems in such spaces. Very recently, Gupta et al. [35] extended various existing results in Gb-metric spaces.Regarding the concepts of theb-MS and several classical contractions in b-MS, we suggest refer to [25, 26, 32, 33].In 2012, Sedghi and Shobe [36] initiated the definition of b-FMS as a generalization of FMS presented by George and Veeramani in [11]. There are some results in such spaces (see, for example, [36, 37]). Following this trend, Chauhan and Gupta [38] introduced the notion of George and Veeramani’s type fuzzy cone b-MS and established new version of Banach contraction principle. As far as we know, there is no paper devoted to propose Kaleva-Seikkala’s type fuzzy b-metric spaces. Due to the existing results mentioned above and application potential, it is significant to focus on this research topic.In this paper, we introduce the concept of Kaleva-Seikkala’s type fuzzyb-metric spaces (briefly, KS-FbMS) which generalizes the notions of KS-FMS and b-MS. In Section 2, some basic properties and lemmas of KS-FbMS were presented, which will be used later. In Sections 3–5, we establish and prove the fixed-point theorems concerning Banach type contractions, Reich type contractions, and Chatterjea type contractions in such spaces, respectively. It is worth mentioning that the range of all contraction constants in our main results are independent of the space coefficient b. These results improve and generalize the corresponding results in KS-FbMS. Moreover, two techinical lemmas for the proof of Cauchy sequence play a pivotal role in the above theorems. In the final section, we give a lemma to show that a b-MS is a special FbMS. Applying this lemma, some applications are presented to illustrate the fact that our main results extend the cases of b-MS.


## 2. Kaleva-Seikkala’s Type Fuzzyb-Metric Spaces

In this section, we introduce the concept ofKS-FbMS and present some elementary lemmas which will be applied in later sections.Definition 1.
LetS be a nonempty set, d:S×S⟶F+ be a mapping, and b≥1 be a real number. Suppose that L,R:0,1×0,1⟶0,1 be two nondecreasing and symmetric functions such that
(1)L0,0=0,R1,1=1.
Forı∈0,1, define
(2)dx,yı=Iıx,y,℘ıx,y,∀x,y∈S.The following conditions are satisfied:(BM1)dx,y=0¯⇔x=y;(BM2) for eachx,y∈S, dx,y=dy,x;(BM3) for eachx,y,z∈S:(BM3L) dx,ybζ+η≥Ldx,zζ,dz,yη, whenever ζ≤I1x,z,η≤I1z,y and bζ+η≤I1x,y;(BM3R) dx,ybζ+η≤Rdx,zζ,dz,yη, whenever ζ≥I1x,z,η≥I1z,y and bζ+η≥I1x,y.Then,d is called a fuzzy b-metric, and the quintuple S,d,L,R,b is called a fuzzy b-metric space with the coefficient b. If d:S×S⟶F∞+ and S,d,L,R,b satisfies (BM1)-(BM3), then S,d,L,R,b is called a generalized fuzzy b-metric space (briefly, GFbMS).Lemma 2.
LetS,d,L,R,b be a FbMS. For each ı∈0,1 and x,y∈S,
(3)dx,yı=Iıx,y,℘ıx,y.
Then,(1)
limη⟶+∞dx,yη=0 and limη⟶−∞dx,yη=0(2)
For eachı∈0,1, ℘ıx,y is a nonincreasing and left continuous function(3)
dx,yη is a nonincreasing and left continuous function for η∈I1x,y,+∞Lemma 3.
Suppose thatS,d,L,R,b be a FbMS, and if
(R-1) Rx,y≤maxx,y;
(R-2) ∀ı∈0,1, ∃ȷ∈0,ı s.t. Rȷ,r<ı for all r∈0,ı;
(R-3) limı⟶0+Rı,ı=0.
Then, (R-1)⇒(R-2)⇒(R-3).Lemma 4.
LetS,d,L,R,b be a FbMS. Then,
(R-1)⇒℘ıx,y≤b℘ıx,z+℘ız,yforallı∈0,1andx,y,z∈S.
(R-2)⇒ for each ı∈0,1, there exists ȷ=ȷı∈0,ı such that for all x,y,z∈S,
(4)℘ıx,y≤b℘ıx,z+℘ȷz,y.
(R-3)⇒ for each ı∈0,1, there exists ȷ=ȷı∈0,ı such that for all x,y,z∈S(5)℘ıx,y≤b℘ȷx,z+℘ȷz,y.Proof.
(1)
Suppose that, on the contrary, for someı∈0,1 and x0,y0,z0∈S, ℘ıx0,y0>b℘ıx0,z0+℘ız0,y0. We can find ζ,η∈ℝ+ such that bζ+η=℘ıx0,y0≥I1x0,y0, ζ>℘ıx0,z0≥I1x0,z0, and η>℘ız0,y0≥I1z0,y0, which implies that(6)dx0,z0ζ<ıanddz0,y0η<ı.
From (BM3R) and the condition (R-1), we obtain that
(7)ı≤dx0,y0℘ıx0,y0=dx0,y0bζ+η≤Rdx0,z0ζ,dz0,y0η≤maxdx0,z0ζ,dz0,y0η<ı,which is a contradiction.
(2)
Assume that (R-2) holds, i.e., for every ı∈0,1, there is ȷ=ȷı∈0,ı such that Rȷ,r<ı for all r∈0,ı. Since ℘ı is left continuous and nonincreasing, it is sufficient to prove that ℘ıx,y≤b℘ȷx,z+℘rz,y for all r∈0,ı. If for some ı∈0,1 and r∈0,ı, we have ℘ıx0,y0>b℘ȷx0,z0+℘rz0,y0 for some x0,y0,z0∈S. Then, we can find ζ and η such that bζ+η=℘ıx0,y0≥I1x0,y0, ζ>℘ȷx0,z0≥I1x0,z0 and η>℘rz0,y0≥I1z0,y0. It follows that(8)dx0,z0ζ<ȷanddz0,y0η<r.
By means of (BM3R), we have
(9)ı≤dx0,y0℘ıx0,y0=dx0,y0bζ+η≤Rdx0,z0ζ,dz0,y0η≤Rȷ,r<ı,which is a contradiction.
(3)
Suppose that (R-3) holds, i.e., for each ı∈0,1, we have limi⟶0+Rı,ı=0. Then, there is ȷ0∈0,ı such that Rȷ,ȷ<ı for all ȷ<ȷ0. Assume that, on the contrary, for some ı∈0,1 and ȷ∈0,ı, we have ℘ıx0,y0>b℘ȷx0,z0+℘ȷz0,y0 for some x0,y0,z0∈S. Then, we can find ζ,η∈ℕ+ such that bζ+η=℘ıx0,y0≥I1x0,y0, ζ>℘ȷx0,z0≥I1x0,z0 and η>℘ȷz0,y0≥I1z0,y0, which deduces that(10)dx0,z0ζ<ȷanddz0,y0η<ȷ.
On account of (BM3R) and the condition (R-3),
(11)ı≤dx0,y0℘ıx0,y0=dx0,y0bζ+η≤Rdx0,z0ζ,dz0,y0η≤Rȷ,ȷ<ı,which is a contradiction.Definition 5.
LetS,d,L,R,b be a FbMS and xn be a suquence in S.
(1)
Iflimn⟶∞dxn,x=0¯, i.e., limn⟶∞℘ıxn,x=0 for each ı∈0,1, xn is said to converge to x∈S (xn⟶x as n⟶∞ or limn⟶∞xn=x)(2)
Iflimn,m⟶∞dxn,xm=0¯, equivalently, for any given ϵ>0 and ı∈0,1, there exists N=Nϵ,ı∈ℕ+ such that ℘ıxn,xm<ϵ, whenever n,m≥N, xn is said to be a Cauchy sequence(3)
If every Cauchy sequence inS converges, S,d,L,R,b is called completeLemma 6.
LetS,d,L,R,b be a FbMS with (R-2) and xn⊆S be a sequence. If the sequence xn converges to both x∈S and y∈S, then x=y.We end this section by giving an example to illustrate that aFbMS is obviously not a FMS or a b-MS.Example 1.
LetS=0,+∞ and dx,y:ℝ⟶ℝ a mapping. If x=y∈S, we define dx,yη=0¯η for all η∈ℝ. If x,y∈S with x≠y, dx,y is defined by
(12)dx,yη=0,η<0,11+η/x−y4,η≥0.
Define that(13)La,b=mina,b,Ra,b=maxa,b.
Then, the assertions hold:(1)
S,d,L,R,b is complete, and the coefficient is b=8;(2)
S,d,L,R,b is not a FMS.Proof.
(1)
First, we show that (BM1) in Definition1 is satisfied for all x,y∈S. From the definition of dx,y, it is sufficient to verify that dx,yη=0¯η implies x=y. Note that(14)0¯η=1,η=0,0,η≠0.
Suppose that there existx0≠y0 such that dx0,y0η=0¯η. Taking η=1, we have dx0,y0η=1/1+1/x0−y04≠0, which is a contradiction. It is easy to verify that for each x,y∈S, (BM2) in Definition 1 holds. Next, for each x,y,z∈S, we prove that the condition (BM3) is satisfied. By a simple calculation, we get I1x,y=℘1x,y=0 for all x,y∈S.
(i)
We prove (BM3L) with b=8. Let ζ,η∈ℝ satisfying(15)ζ≤I1x,z,η≤I1z,y,8ζ+η≤I1x,y.
SinceLa,b=mina,b, if ζ<I1x,z=0 or η<I1z,y=0, then
(16)mindx,zζ,dz,yη=0≤dx,y8ζ+η.
Assume thatζ=I1x,z=0 and η=I1z,y=0, we obtain that
(17)8ζ+η=80+0=0=I1x,y.
Thus, we conclude that(18)dx,y8ζ+η=1=mindx,zζ,dz,yη=Ldx,zζ,dz,yη.
That completes the proof of (BM3L).We prove (BM3R) with b=8. Let ζ,η∈ℝ satisfying
(19)ζ≥I1x,z,η≥I1z,y,8ζ+η≥I1x,y.Now, we consider the following three cases.Case 1.
Assume that(20)ζ=I1x,z=0,η=I1z,y=0,we have
(21)dx,y8ζ+η=dx,y80+0=1=maxdx,z0,dz,y0=Rdx,zζ,dz,yη.Case 2.
If(22)ζ>I1x,z=0,η=I1z,y=0,or
(23)ζ=I1x,z=0,η>I1z,y=0,(Here, we discuss the previous assumption), then
(24)8ζ+η>0=I1x,y,dx,y8ζ+0≤maxdx,zζ,1=maxdx,zζ,dz,y0=Rdx,zζ,dz,y0.Case 3.
Suppose that(25)ζ>I1x,z=0,η>I1z,y=0.
For eachx,y,z∈X,
(a)
Ifx=y and z∈S, then(26)dx,y8ζ+η=0¯8ζ+η=0≤maxdx,zζ,dz,yη=Rdx,zζ,dz,yη.(b)
Ifx=z≠y or x≠y=z, without loss of generality, let x=z≠y, then(27)dx,y8ζ+η=11+8ζ+η/x−y4=11+8ζ+η/z−y4≤11+η/z−y4=dz,yη=max0,dz,yη=max0¯ζ,dz,yη=Rdx,zζ,dz,yη.(c)
Ifx≠y≠z. Note that(28)minζx−z4,ηz−y4≤ζ+ηx−z4+z−y4.
Might as well setminζ/x−z4,η/z−y4=ζ/x−z4, which implies that ζ/x−z4≤ζ+η/x−z4+z−y4 and
(29)maxdx,zζ,dz,yη=max11+ζ/x−z4,11+η/z−y4=11+ζ/x−z4=dx,zζ.
Thus, we conclude that(30)dx,y8ζ+η=11+8ζ+η/x−y4=11+8ζ+η/x−z+z−y4≤11+8ζ+η/2x−z2+2z−y22≤11+8ζ+η/8x−z4+z−y4=11+ζ+η/x−z4+z−y4≤11+ζ/x−z4=dx,zζ=maxdx,zζ,dz,yη=Rdx,zζ,dz,yη.
The part of (BM3R) is completed. Therefore, S,d,L,R,b is a FbMS.
Finally, we proveS,d,L,R,b is complete. Let xn⊆S be a Cauchy sequence in S,d,L,R,b. For any ε>0, there exists N1>0 such that
(31)℘1/2xm,xn=1−1/2xm−xn41/2=xm−xn4<ϵ4,for all m,n>N1. It implies that xm−xn<ϵ. Thus, xn is a Cauchy sequence in 0,+∞,·. Due to 0,+∞,· is complete, we can find x∗∈0,+∞ such that limn⟶∞xn=x∗. For any ϵ>0 and ı∈0,1, by virtue of xn⟶x∗ as n⟶∞, there is N2>0 such that
(32)xn−x∗<ıϵ1−ı1/4,foralln>N2.
Then,(33)℘ıxn,x∗=1−ıxn−x∗4ı<1−ıı·ıϵ1−ı1/44=ϵ.
Thus,xn⟶x∗ as n⟶∞ in FbMS. Therefore, S,d,L,R,b is complete.
Letx=3, y=1, z=2, and 0<ζ=η. Obviously, ζ>I1x,z=0, η>I1z,y=0 and ζ+η>I1x,y=0. By the definition of R,
(34)Rd3,2ζ,d2,1ζ=maxd3,2ζ,d2,1ζ=max11+ζ/3−24,11+ζ/2−14=11+ζ.d3,12ζ=11+2ζ/3−14=11+ζ/8.
Since0<ζ/8<ζ, we have 1/1+ζ/8>1/1+ζ. Thus,
(35)d3,1ζ+ζ>Rd3,2ζ,d2,1ζ.
Therefore,S,d,L,R,b is not a FMS.


## 3. Banach Type Contractions inKS-FbMS

In this section, we will state and prove a fixed-point theorem for Banach type contractions inKS-FbMS. This theorem extends Banach’s results in FMS.Theorem 7.
Suppose thatS,d,L,R,b be a complete FbMSb≥1 with (R-2). Let F:S⟶S be a mapping. If there exists k∈0,1 such that
(36)℘ıFx,Fy≤k℘ıx,y,∀x,y∈S,for all ı∈0,1, then F admits a unique fixed point in S.Proof.
Sincek∈0,1, there exists l∈ℕ such that kl<1/b. Suppose that ı∈0,1, applying (36), we derive
(37)℘ıFx,Fy≤k℘ıx,y,for all x,y∈S. Clearly,
(38)℘ıF2x,F2y≤k℘ıFx,Fy≤k2℘ıx,y.
Continuing this process, we deduce(39)℘ıFlx,Fly≤kl℘ıx,y.
Letg=Fl, α=kl∈0,1/b. From the above inequality, for each i∈0,1 we obtain that
(40)℘ıgx,gy≤α℘ıx,y,for all x,y∈S.
Now, we shall prove thatg admits a unique fixed point in S. Taking x0∈S, we construct a sequence xn by xn=gxn−1=gnx0 (∀n∈ℕ). For each ı∈0,1, n∈ℕ, by (40), we deduce
(41)℘ıxn,xn+1=℘ıgxn−1,gxn≤α℘ıxn−1,xn≤α2℘ıxn−2,xn−1≤⋯≤αn℘ıx0,x1.
Thus, for eachı∈0,1, due to S,d,L,R,b satisfies (R-2), it follows that we can find ȷ=ȷı∈0,ı such that
(42)℘ıx,y≤b℘ȷx,z+℘ız,y,∀x,y,z∈S.
Form,n∈ℕ with m<n, using (41) and (42), we derive
(43)℘ıxm,xn≤b℘ȷxm,xm+1+℘ıxm+1,xn≤b℘ȷxm,xm+1+b2℘ȷxm+1,xm+2+b2℘ıxm+2,xn≤⋯≤b℘ȷxm,xm+1+b2℘ȷxm+1,xm+2+⋯+bn−m−1℘ȷxn−2,xn−1+bn−m−1℘ıxn−1,xn≤b℘ȷxm,xm+1+b2℘ȷxm+1,xm+2+⋯+bn−m−1℘ȷxn−2,xn−1+bn−m℘ȷxn−1,xn≤bαm℘ȷx0,x1+b2αm+1℘ȷx0,x1+⋯+bn−mαn−1℘ȷx0,x1≤bαm1+bα+b2α2+⋯℘ȷx0,x1=bαm1−bα℘ȷx0,x1.
Due toα∈0,1/b, we can conclude that xn⊆S is a Cauchy sequence. Note that S,d,L,R,b is a complete FbMS. Thus, there exists v∈S such that limn⟶∞xn=v, equivalently, limn⟶∞℘ıxn,v=0 for all ı∈0,1.
For eachı∈0,1 and n∈ℕ, by virtue of (40), we obtain that
(44)℘ıxn+1,gv=℘ıgxn,gv≤α℘ıxn,v.
Thus,℘ıxn+1,gv⟶0 as n⟶∞, that is, xn+1⟶gv as n⟶∞. From Lemma 6, we deduce that gv=v. Therefore, v∈S is a fixed point of g. This completes the proof of the existence of the fixed point.
Ifg has another fixed point ϖ∈S, i.e., gϖ=ϖ and ϖ≠v, then ℘ı0ϖ,v>0 for some i0∈0,1. Using (40), we have
(45)℘ı0ϖ,v=℘ı0gϖ,gv≤α℘ı0ϖ,v<℘ı0ϖ,v,a contradiction. Therefore, ϖ=v, and the fixed point of g is unique. Hence, Flv=v. Then, FlFv=FFlv=Fv, which deduce that Fv is a fixed point of Fl. By the uniqueness of fixed point of Fl=g, we get Fv=v. Furthermore, if there exists v¯∈S such that Fv¯=v¯, then v¯ is a fixed point of Fl. Again, by the uniqueness of the fixed point of Fl, we obtain that v¯=v. Thus, F admits a unique fixed point in S.The following corollary is an immediate consequence of Theorem7.Corollary 8.
Suppose thatS,d,L,R,b be a complete FMS with (R-2), F:S⟶S be a mapping. For each ı∈0,1 and x,y∈S,
(46)℘ıFx,Fy≤k℘ıx,y,where k∈0,1. Then, F has a unique fixed point in S.Proof.
Obviously, aFbMS is a generalized FMS, and thus, taking b=1 in Theorem 7, we obtain the result.To support our results, we give an illustrative example inFbMS.Example 2.
Assume thatS=0,+∞, dx,y:ℝ⟶ℝ a mapping. If x=y∈S, we define dx,yη=0¯η, ∀η∈ℝ. If x,y∈S with x≠y, dx,y is defined by
(47)dx,yη=0,η<0,11+η/x−y2,η≥0.
DefineL,R by
(48)La,b=mina,b,Ra,b=maxa,b.LetY:S⟶S defined by
(49)Yx=3x4,x∈0,1,x2+14,x∈1,+∞.Then,Y is a Banach type contraction with the contraction constant k=5/8.Proof.
We can obtain thatS,d,L,R,b is a complete FbMS with the coefficient b=2, which is similar to Example 1. By the definition of d, it is easy to prove that Y has the unique fixed point 0. Note that
(50)dx,y℘ıx,y=ı,for all ı∈0,1. Then,
(51)℘ıx,y=ı−1x−y2ı.
Now, we divide it into three cases.Case 1.
Ifx,y∈0,1, we can see that
(52)℘ıYx,Yy=ı−1Yx−Yy2ı=ı−13x/4−3y/42ı=ı−1ı·916x−y2=916·ı−1x−y2ı=916℘ıx,y≤58℘ıx,y.Case 2.
Suppose thatx,y∈1,+∞, we can derive that
(53)℘ıYx,Yy=ı−1Yx−Yy2ı=ı−1x/2+1/4−y/2+1/42ı=ı−1ı·14x−y2=14·ı−1x−y2ı=14℘ıx,y≤58℘ıx,y.Case 3.
Assume thatx∈0,1 and y∈1,+∞, we get
(54)℘ıYx,Yy=ı−1Yx−Yy2ı=ı−13x/4−y/2−1/42ı=ı−1ı·1163x−2y−12.
Note thatx−12≤x−y2. Thus, we deduce
(55)3x−2y−12=x−1+2x−y2≤2x−12+4x−y2≤10x−y2,then
(56)℘ıYx,Yy≤ı−1ı·1016x−y2=58·ı−1x−y2ı=58℘ıx,y.
In conclusion,Y is a Banach type contraction, and the contraction constant is k=5/8.


## 4. Reich Type Contractions inKS-FbMS

In this section, our main contribution is to establish a fixed-point theorem concerning Reich type contraction. Firstly, we introduce a crucial lemma to show that a Picard sequence is a Cauchy sequence.Lemma 9.
Suppose thatS,d,L,R,b be a FbMS (b≥1) with (R-2) and xn⊆S a sequence. Assume that there exists Q∈0,1 such that for each ı∈0,1, we can find Pı≥0 satisfying
(57)℘ıxn,xn+1≤PıQn+1,∀n∈ℕ,then xn⊆S is a Cauchy sequence.Proof.
Owing toQ<1, we can find l∈ℕ+ such that Ql<1/b. For each ı∈0,1 and m,n∈ℕ+ with m<n, we will show that
(58)limm,n⟶∞℘ıxm,xn=0.
SinceS,d,L,R,b is with (R-2), there exists ȷ=ȷı∈0,ı such that
(59)℘ıx,y≤b℘ȷx,z+℘ız,y,∀x,y,z∈S.(i)
Ifn−m≤l, then taking account of (57) and (59), we get(60)℘ıxm,xn≤b℘ȷxm,xm+1+℘ıxm+1,xn≤b℘ȷxm,xm+1+b2℘ȷxm+1,xm+2+b2℘ıxm+2,xn≤⋯≤b℘ȷxm,xm+1+b2℘ȷxm+1,xm+2+⋯+bn−m−1℘ȷxn−2,xn−1+bn−m−1℘ıxn−1,xn≤b℘ȷxm,xm+1+b2℘ȷxm+1,xm+2+⋯+bn−m−1℘ȷxn−2,xn−1+bn−m℘ȷxn−1,xn=∑i=1n−mbi℘ȷxm+i−1,xm+i≤∑i=1n−mbiPȷQm+i≤∑i=1lbiPȷQm+i=QmCı,where Cı=∑i=1lbQiPȷ.
(ii)
Ifn−m>l, take θ=n−m/l−1, where n=minκ≥n:κ∈ℕ+, then θ<n−m/l≤θ+1. Thus, 0<n−m+θl≤l. Then, making the most of (5) and (6), we get(61)℘ıxm,xn≤b℘ȷxm,xm+l+℘ıxm+l,xn≤b℘ȷxm,xm+l+b2℘ȷxm+l,xm+2l+b2℘ıxm+2l,xn≤b℘ȷxm,xm+l+b2℘ȷxm+l,xm+2l+⋯+bθ℘ȷxm+θ−1l,xm+θl+bθ℘ıxm+θl,xn≤b℘ȷxm,xm+l+b2℘ȷxm+l,xm+2l+⋯+bθ℘ȷxm+θ−1l,xm+θl+bθ+1℘ȷxm+θl,xn≤bQmCȷ+b2Qm+lCȷ+⋯+bθQm+θ−1lCȷ+bθ+1Qm+θlCȷ=bQmCȷ∑i=0θbQli≤bQmCȷ∑i=0∞bQli=bQmCȷ1−bQl.
Therefore, for eachı∈0,1 and m,n∈ℕ+ (m<n), we conclude that
(62)℘ıxm,xn≤QmmaxCı,bCȷ1−bQl⟶0asm⟶∞.
The proof is completed.Remark 10.
In fact, we cannot prove the uniqueness of the fixed point concerning the Reich type contractions without additional conditions. To figure this out, we consider the assumption thatd has the Fatou property.Let us review the definition of the Fatou property.Definition 11.
Suppose thatS,d,L,R,b be a FbMS (b≥1) with (R-2) and xn⊆S be a sequence. We say that d has the Fatou property if, for each ı∈0,1,
(63)℘ıx,y≤liminfn⟶∞℘ıxn,y,whenever xn⟶x as n⟶∞ and any y∈S.Remark 12.
It is obvious thatd has Fatou property if S,d,L,R,b is a KS-FMS.Now, we establish and prove our main contents of this section.Theorem 13.
Suppose thatS,d,L,R,b be a complete FbMS (b≥1) with (R-2). Assume that F:S⟶S be a mapping. If there exist a1,a2,a3≥0 and a1+a2+a3<1 such that
(64)℘ıFx,Fy≤a1℘ıx,y+a2℘ıx,Fx+a3℘ıy,Fy,∀x,y∈S,for all i∈0,1. If d has the Fatou property, then F admits a unique fixed point in S.Proof.
Letz0∈S, we can construct a sequence znn=0∞ by zn=Fzn−1=Fnz0. Assume that ı∈0,1. By (64), we have
(65)℘ızn,zn+1≤a1℘ızn−1,zn+a2℘ızn−1,zn+a3℘ızn,zn+1=a1+a2℘ızn−1,zn+a3℘ızn,zn+1.
It immediately follows that(66)℘ızn,zn+1≤a1+a21−a3℘ızn−1,zn=β℘ızn−1,zn,where β=a1+a2/1−a3. Since a1+a2+a3<1, we obtain that β=a1+a2/1−a3∈0,1. Again by (64), we get
(67)℘ızn,zn+1≤β℘ızn−1,zn≤β2℘ızn−2,zn−1≤⋯≤βn℘ız0,z1=βn+1Pı,where Pı=1/β℘ız0,z1≥0. Using Lemma 9, we can conclude that zn⊆S is a Cauchy sequence. Since S,d,L,R,b is complete, we can find u∈S such that limn⟶∞zn=u. Next, we prove that u is a fixed point of F. Suppose that, on the contrary, Fu≠u, that is, ℘ı0u,Fu>0 for some ı0∈0,1. By means of (64), we deduce that
(68)℘ı0zn+1,Fu=℘ı0Fzn,Fu≤a1℘ı0zn,u+a2℘ı0zn,Fzn+a3℘ı0u,Fu.
Using the fact thatzn⟶u as n⟶∞ and d has the Fatou property, we get
(69)℘ı0u,Fu≤liminfn⟶∞℘ı0zn,Fu=liminfn⟶∞℘ı0zn+1,Fu.
Note thatlimn⟶∞℘ı0zn,u=limn⟶∞℘ı0zn,zn+1=0. Combining (68) and (69), we obtain that
(70)℘ı0u,Fu≤liminfn⟶∞℘ı0zn+1,Fu≤liminfn⟶∞a1℘ı0zn,u+a2℘ı0zn,Fzn+a3℘ı0u,Fu=a1limn⟶∞℘ı0zn,u+a2limn⟶∞℘ı0zn,zn+1+a3℘ı0u,Fu=a3℘i0u,Fu<℘ı0u,Fu,which is a contradiction. Thus, Fu=u.
Ifv∈S is a fixed point of F and v≠u, that is, fv=v and ℘ı1v,u>0 for some ı1∈0,1, by virtue of (64), we derive
(71)℘ı1v,u=℘ı1Fv,Fu≤a1℘ı1v,u+a2℘ı1v,Fv+a3℘ı1u,Fu=a1℘ı1v,u+a2℘ı1v,v+a3℘ı1u,u=a1℘ı1v,u<℘ı1v,u,which contradicts the assumption that v≠u. Therefore, v=u and F has a unique fixed point in S.Theorem13 can deduce the following corollary in FMS.Corollary 14.
Suppose thatS,d,L,R,b be a complete FMS with (R-2). Assume that F:S⟶S be a mapping. If there exist a1,a2,a3≥0 and a1+a2+a3<1 such that
(72)℘ıFx,Fy≤a1℘ıx,y+a2℘ıx,Fx+a3℘ıy,Fy,∀x,y∈S,for all ı∈0,1. Then, F has a unique fixed point in S.Remark 15.
Takinga2=a3=0 in Theorem 13, we can obtain a result for Banach contractions in FbMS. If we take a1=0 and a2=a3 in Theorem 13, the fixed-point theorem for Kannan type contractions is derived in the same context.Corollary 16.
LetS,d,L,R,b be a complete FbMS (b≥1) with (R-2) and F:S⟶S. If there exists γ∈0,1 such that
(73)℘ıFx,Fy≤γ℘ıx,Fx+℘ıy,Fy,∀x,y∈S,for all ı∈0,1. If d has the Fatou property, then F has a unique fixed point in S.


## 5. Chatterjea Type Contractions inKS-FbMS

The following lemma is crucial in order to prove a Picard sequence is a Cauchy sequence.Lemma 17.
LetS,d,L,R,b be a complete FbMS (b≥1) with (R-2). Assume that F:S⟶S be a mapping. Define the sequence xnn∈ℕ∪0 by xn=Fxn−1=Fnx0. Suppose that for each ı∈0,1(74)℘ıFx,Fy≤c2℘ıx,Fy+℘ıy,Fx,∀x,y∈S,where c∈0,1. Let q∈ℕ such that bcq<1. Then, the following assertions hold:
(i)
For eachı∈0,1 there exists Hı>0 such that ℘ıxk,xn≤Hı, for all k∈0,1,⋯,q, n∈ℕ(ii)
For eachı∈0,1, there exists ȷ∈0,ı and Hȷ>0 such that ℘ıxm,xn≤2bHȷ, for all m,n∈ℕ(iii)
The sequencexn⊆S is a Cauchy sequenceProof.
(i)
Firstly, we show by induction that, for eachı∈0,1,(75)℘ıxk,xn≤Hı=b1−bcqmax℘ȷxı,xȷ:0≤ı,ȷ≤q,for all n∈ℕ, k∈0,1,⋯,q and for some ȷ∈0,ı.
Clearly, (75) holds for all n≤q. Assume that (75) holds for all n≤m (m∈ℕ). Notice that S,d,L,R,b satisfies the condition (R-2). By Lemma 4 (2), we can find ȷ=ȷı∈0,ı such that
(76)℘ıx,y≤b℘ȷx,z+℘ız,y,∀x,y,z∈S.
Next, in order to prove (75) for n=m+1, we divide it into two cases.
Case 1. Assume thatk=0. Then, by (74) and (76), we have
(77)℘ıx0,xn+1≤b℘ȷx0,xq+℘ıxq,xn+1≤b℘ȷx0,xq+b·c2℘ıxq−1,xn+1+℘ıxq,xn≤b℘ȷx0,xq+b·c22℘ıxq−2,xn+1+2℘ıxq−1,xn+℘ıxq,xn−1≤⋯≤b℘ȷx0,xq+b·c2qCq0℘ıx0,xn+1+Cq1ρtx1,xn+⋯+Cqq−1℘ıxq−1,xn−q+2+Cqq℘ıxq,xn−q+1≤b℘ȷx0,xq+b·c2q℘ıx0,xn+1+b·c2qCq1℘ȷx1,xn+⋯+Cqq−1℘ȷxq−1,xn−q+2+Cqq℘ȷxq,xn−q+1,which implies that
(78)℘ıx0,xn+1≤b℘ȷx0,xq1−bc/2q+b·c/2q1−bc/2qCq1℘ȷx1,xn+⋯+Cqq−1℘ȷxq−1,xn−q+2+Cqq℘ȷxq,xn−q+1≤b℘ȷx0,xq1−bc/2q+b·c/2q1−bc/2qCq1+Cq2+⋯+CqqHı≤b1−bc/2qmax℘ȷxı,xȷ:0≤i,j≤q+b·c/2q1−bc/2qCq1+Cq2+⋯+CqqHı=1−bcq1−bc/2qHi+b·c/2q1−bc/2q2q−1Hı=Hi1−bc/2q1−bcq+bcq−bc2q=Hı.
Case 2. Suppose thatk∈1,2,⋯,q. In this case, taking account of (74), we deduce that
(79)℘ıxk,xn+1≤c2℘ıxk−1,xn+1+c2℘ıxk,xn≤c22℘ıxk−2,xn+1+c22℘ıxk−1,xn+c2Hı≤⋯≤c2kρtx0,xn+1+c2k℘ıx1,xn+c2k−1+c2k−2+⋯+c2Hı≤c2k℘ıx0,xn+1+c2kHı+c2k−1+c2k−2+⋯+c2Hı.
From case 1, we derive(80)℘ıxk,xn+1≤c2kHı+c2kHı+c2k−1+c2k−2+⋯+c2Hı≤12k+12k+12k−1+⋯+12Hı=Hı.
Therefore, from the above two cases, we prove that (75) holds for all n∈ℕ.
(ii)
For eachı∈0,1 and m,n∈ℕ, by means of (74) and (76), we obtain that(81)℘ıxm,xn≤b℘ȷxm,xq+℘ıxq,xn≤bHȷ+Hı≤2bHȷ,where Hȷ=b/1−bcqmax℘ȷ0xi,xj:0≤i,j≤q, for some ȷ0∈0,ȷ.
(iii)
For eachı∈0,1 and m,n∈ℕ (m<n), from (74) and (ii), we have(82)℘ıxm,xn≤c2℘ıxm−1,xn+℘ıxm,xn−1≤c22℘ıxm−2,xn+2℘ıxm−1,xn−1+℘ıxm,xn−2≤⋯≤c2mCm0℘ıx0,xn+Cm1℘ıx1,xn−1+⋯+Cmm℘ıxm,xn−m≤c2mCm0+Cm1+⋯+Cmm·2bHȷ≤c2m·2m·2bHȷ=2bHȷcm.
Thus, we can obtain thatlimm,n⟶∞℘ıxm,xn=0. Therefore, we conclude that xn⊆S is a Cauchy sequence.By Lemma17, we establish a fixed-point theorem for Chatterjea type contraction in FbMS.Theorem 18.
Suppose thatS,d,L,R,b be a complete FbMS (b≥1) with (R-2). Assume that F:S⟶S be a mapping. For each ı∈0,1,
(83)℘ıFx,Fy≤c2℘ıx,Fy+℘ıy,Fx,∀x,y∈S,where c∈0,1. Then, F has a unique fixed point u∈S, and for any x∈S, the sequence of iterates Fnx converges to u.Proof.
Letx0∈S, we construct a sequence xnn∈ℕ∪0 by xn=Fxn−1=Fnx0. Owing to c∈0,1, there exists q∈ℕ such that bcq<1. Using the Lemma 17, we can obtain that xn⊆S is a Cauchy sequence. Due to S,d,L,R,b is complete, there exists u∈S such that limn⟶∞xn=u. Next, we show that u is a fixed point of F. For each ı∈0,1, from Lemma 4 (40), we can find ȷ∈0,ı such that
(84)℘ıx,y≤b℘ȷx,z+℘ız,y,∀x,y,z∈S.
By virtue of (83) and (84), we have
(85)℘ıxn,Fu≤c2℘ıxn−1,Fu+c2℘ıxn,u≤c22℘ıxn−2,Fu+c22℘ıxn−1,u+c2℘ıxn,u≤⋯≤c2q℘ıxn−q,Fu+c2q℘ıxn−q+1,u+c2q−1℘ıxn−q+2,u+⋯+c2℘ıxn,u≤bc2q℘ȷxn−q,xn+℘ıxn,Fu+c2q℘ıxn−q+1,u+c2q−1℘ıxn−q+2,u+⋯+c2℘ıxn,u.
It follows that(86)℘ıxn,Fu≤bc/2q1−bc/2q℘ȷxn−q,xn+11−bc/2qc2q℘ıxn−q+1,u+c2q−1℘ıxn−q+2,u+⋯+c2℘ıxn,u.
Note thatlimn⟶∞℘ȷxn−q,xn=0 and limn⟶∞℘ıxn−i,u=0, i=0,1,⋯,q−1. Therefore, we obtain that limn⟶∞℘ıxn,Fu=0, that is, xn converges to Fu, by Lemma 6, we have Fu=u. This completes the proof of the existence of the fixed point of F.
If there existsv∈S such that Fv=v and v≠u, that is, ℘ı0v,u>0 for some ı0∈0,1. Taking (83) into account, we have
(87)℘ı0v,u=℘ı0Fv,Fu≤c2℘ı0v,Fu+℘ı0u,Fv=c2·2℘ı0v,u=c℘ı0v,u<℘ı0v,u,which contradicts the fact that v≠u. Therefore, v=u and F has a unique fixed point in S.Corollary 19.
Suppose thatS,d,L,R,b be a complete FMS with (R-2), F:S⟶S. If there exists c∈0,1 such that
(88)℘ıFx,Fy≤c2℘ıx,Fy+℘ıy,Fx,∀x,y∈S,for all ı∈0,1. Then, F has a unique fixed point u∈S, and for any x∈S, the sequence of iterates Fnx converges to u.Proof.
Takingb=1 in Theorem 18, the desired result is obtained immediately.


## 6. Applications

The aim of the following lemma is to prove that ab-MS is a special FbMS. And then, we can establish the relevant fixed-point theorems in b-MS as corollaries of our main results presented in Sections 3–5.Lemma 20.
LetS,D be a b-MS and dx,y:ℝ⟶ℝ a mapping defined by
(89)dx,yη=D¯x,yη=0¯η−Dx,y,∀x,y∈S.
Then,S,d,L,R,b is a FbMS, where
(90)La,b=mina,b,Ra,b=maxa,b.Proof.
In view of (89), we have
(91)dx,yη=1,η=Dx,y,0,η≠Dx,y.
It is easy to verify (BM1) and (BM2) in Definition1. Next, for every x,y,z∈S, we prove that (BM3) holds.
First, we prove (BM3L). Let ζ, η∈ℝ satisfying
(92)ζ≤I1x,z,η≤I1z,y,bζ+η≤I1x,y.
Note thatLa,b=mina,b. Suppose that ζ<I1x,z or η<I1z,y, we can obtain that
(93)mindx,zζ,dz,yη=0≤dx,ybζ+η.
Assume thatζ=I1x,z and η=I1z,y, we deduce
(94)bζ+η=bDx,z+Dz,y≥Dx,y.
Note thatbζ+η≤I1x,y=Dx,y. Thus, we conclude that
(95)dx,ybζ+η=dx,yDx,y=1=mindx,zζ,dz,yη=Ldx,zζ,dz,yη.
That completes the proof of (BM3L).
Now, we verify (BM3R). Let ζ,η∈ℝ satisfying
(96)ζ≥I1x,z,η≥I1z,y,bζ+η≥I1x,y.
Assume thatζ=I1x,z or η=I1z,y, we get
(97)maxdx,zζ,dz,yη=1≥dx,ybζ+η.
Ifζ>I1x,z and η>I1z,y, we derive
(98)bζ+η>bDx,z+Dz,y≥Dx,y.
Thus, we conclude that(99)dx,ybζ+η=0=maxdx,zζ,dz,yη=Rdx,zζ,dz,yη.
The part of (BM3R) is completed.Remark 21.
From Lemma20, it is obvious that S,D and S,d,L,R,b are homeomorphic, and for each ı∈0,1, ℘ıx,y=Dx,y. Therefore, we give the following three results as corollaries of Theorem 7, Theorem 13, and Theorem 18, respectively.Corollary 22 (see [32]).
Suppose thatS,d be a complete b-MS (b≥1), F:S⟶S. If x,y∈S,
(100)dFx,Fy≤kdx,y,where k∈0,1. Then, F has a unique fixed point in S.Corollary 23.
LetS,d be a complete b-MS (b≥1), F:S⟶S. Suppose that there exist a1,a2,a3≥0 and a1+a2+a3<1 such that
(101)dFx,Fy≤a1dx,y+a2dx,Fx+a3dy,Fy,∀x,y∈S.
Ifd has the Fatou property, then F has a unique fixed point in S.Corollary 24 (see [33]).
LetS,d be a complete b-MS (b≥1), F:S⟶S. If
(102)dFx,Fy≤c2dx,Fy+dy,Fx,∀x,y∈S,where c∈0,1, then F has a unique fixed point x∗, and for any x∈S, the sequence of iterates Fnx converges to x∗.



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