# Some New Observations on Generalized Contractive Mappings and Related Results inb-Metric-Like Spaces

Authors: Tatjana Došenović; Manuel De La Sen; Ljiljana Paunović; Dušan Rakić; Stojan Radenović
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we consider, discuss, complement, improve, generalize, and enrich some fixed point results obtained forβ−ψ1−ψ2−contractive conditions in ordered b-metric-like spaces. By using our new approach for the proof that one Picard’s sequence is bbl−Cauchy in the context of b-metric-like spaces, we get much shorter proofs than the ones mentioned in the recent papers. Also, by the use of our method, we complement and enrich some common fixed point results for βq,ϕs,ψ−contraction mappings. Our approach in this paper generalizes and modifies several comparable results in the existing literature.


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## Body

## 1. Introduction

Fixed point theory is one of the most important areas of nonlinear analysis. At the beginning of the development, this part of analysis was related to the use of successive approximation in order to prove the existence and uniqueness of the solution of differential and integral equations. Later on, it is applied in various fields such as economics, physics, chemistry, differential and integral equations, partial differential equations, numerical analysis, and many others. Banach’s contraction principle in metric spaces [1] is one of the most important results in fixed point theory and nonlinear analysis in general. In 1922, when Stefan Banach formulated the concept of contraction and proved the famous theorem, scientists around the world started publishing new results that are related either to the generalization of the contractive mapping such as Kannan, Chatterjea, Hardy–Rogers, Ćirić, and many others or by generalizing space itself. By changing some axioms of ordinary metric space, new classes of so-called generalized metric spaces were obtained such as partial metric space, metric-like space, b−metric space, b−metric-like space, and others. For more details about fixed point theory in metric as well as generalized metric spaces, we encourage readers to see [2–9].In each of them, Banach’s well-known theorem is true inb-metric and b−metric-like spaces regardless of the magnitude of the coefficient s in the triangle relation to each.In [10], Matthews introduced the notion of a partial metric space where nonzero self-distance is considered, which has found great application in computer science. The second important generalization of metric spaces is so-called b−metric spaces. This concept was introduced by Bakhtin [11] and Czerwik [12] where the third axiom of metric spaces, referring to triangular inequality, weakened.Furthermore, Amini Harandi [13] introduced the notion of metric-like space, as a generalization of a partial metric space, where all of the axioms of a metric is satisfied except that self-distance may be positive.In [14], the concept of b−metric-like space which generalizes the notions of partial metric space, metric-like space, and b-metric space is introduced.Relations of the metric spaces and mentioned generalizations are illustrated as follows [15, 16]: metric space⟶partial metric space⟶metric‐like space↓↓↓b‐metric space⟶partial b‐metric space⟶b‐metric‐like space.


## 2. Preliminaries

Firstly, we present some definitions and basic notions of metric-like,b-metric-like spaces, α−admissible mappings, β−ψ1−ψ2−contractive mappings of type-I and type-II, altering distance functions, βq,ϕs,ψ−contraction mappings, and ordered metric-like, that is, ordered b-metric-like space.Definition 1.
(see [13]). Let X be a nonempty set. A mapping bml:X×X⟶0,+∞ is said to be metric-like if the following conditions hold for all x,y,z∈X:(i)
bl1bmlx,y=0 implies x=y(ii)
bl2bmlx,y=bmly,x(iii)
bl3bmlx,z≤bmlx,y+bmly,z
In this case, the pairX,bml is called a metric-like space.Definition 2.
(see [14]). Let X be a nonempty set and s≥1 be a given real number. A b−metric-like space on a nonempty set X is a function bbl:X×X⟶0,+∞ if the following conditions hold for all x,y,z∈X: (i)
bbl1bblx,y=0 implying x=y(ii)
bbl2bblx,y=bbly,x(iii)
bbl3bblx,z≤sbblx,y+bbly,z
In this case, the pairX,bbl is called a b−metric-like space with the coefficient constant s≥1.For some examples, see [17–19] and [13, 20–25].It is clear that each metric-like space isb−metric-like space, while the converse is not true. For more such examples and details see [9, 13, 14, 20, 24–27]. Also, for various metrics, but in the context of complex domain, see [28, 29].Definition 3.
A tripleX,≾¯,bbl is said to be ordered b-metric-like space if X,≾¯ is a partially ordered set and bbl is a b−metric-like on X.Now, we give the definition of convergence of the sequences inb-metric-like space.Definition 4.
(see [14]). Let xn be a sequence in a b−metric-like space X,bbl with the coefficient s.(i)
The sequencexn is said to be convergent to x if limn⟶∞bblxn,x=bblx,x(ii)
The sequencexn is said to be bbl−Cauchy in X,bbl if limn,m⟶∞bblxn,xm exists and is finite(iii)
One says thatb−metric-like space X,bbl is a bbl−complete if for every bbl−Cauchy sequence xn in it there exists an x∈X, such that limn,m⟶∞bblxn,xm=bblx,x=limn⟶∞bblxn,xProperties such as convergence, completeness, and Cauchyness are introduced in the same way for metric and b-metric spaces. And, in these two types of spaces, the limits of the sequence if exists, is unique, as well as the convergent sequence is a Cauchy. Otherwise, for the other 4 types of space (partial metric, partialb-metric, metric-like, and b-metric like), this is not the case.Remark 1.
In ab−metric-like space, the limit of a sequence need not be unique and a convergent sequence need not be a bbl−Cauchy sequence (see Example 7 in [20]). However, if the sequence xn is bbl−Cauchy such that limn,m⟶∞bblxn,xm=0 in the bbl−complete b−metric-like space X,bbl with coefficient s≥1, then the limit of such sequence is unique. Indeed, in such a case, if xn⟶xbblxn,x⟶bblx,x as n⟶∞, we get that bblx,x=0. Now, if xn⟶x and xn⟶y, where x≠y, we obtain that(1)1sbblx,y≤bblx,xn+bblxn,y⟶bblx,x+bbly,y=0+0=0.
Bybbl1, it follows that x=y. A contradiction.Definition 5.
LetX be a nonempty set and suppose f:X⟶X and β:X×X⟶0,1 are mappings. f is called β−admissible if for all u,v∈X,(2)βu,v≥1impliesβfu,fv≥1.
Also, we say thatf is Lβ−admissible (or Rβ−admissible) if for u,v∈X,(3)βu,v≥1impliesβfu,v≥1orβu,fv≥1.The next definition and the corresponding proposition are important in the context of fixed point theory.Definition 6.
(see [30]). A function ψ:0,∞⟶0,∞ is called an altering distance function if it satisfies the following properties:(i)
ψ is continuous and nondecreasing(ii)
ψt=0 iff t=0Definition 7.
(see [31]). The self-mappings f,g:X⟶X are weakly compatible if fgx=gfx, whenever fx=gx.Proposition 1.
(see [31]). Let f and g be weakly compatible self-maps of a nonempty set X. If they have a unique point of coincidence w=fu=gu, then w is the unique common fixed point of f and g.In this paper, we shall use the following result for the proof that some Picard’s sequence isbbl−Cauchy. The proof is completely identical with the corresponding in [32] (see also [33–35]).Lemma 1.
Letxn be a sequence in a b−metric-like space X,bbl with the coefficient s>1 such that(4)bblxn,xn+1≤λbblxn−1,xn,for some λ,0≤λ<1/s, and each n=1,2,….Then, xn is a bbl−Cauchy sequence in X,bbl such that limn,m⟶∞bblxn,xm=0.Remark 2.
It is worth to notice that the previous lemma holds in the context ofb−metric-like spaces for each λ∈0,1. For more details, see [27].Otherwise, many authors for the proof that some sequence inb−metric-like space is bbl−Cauchy use the next lemma.Lemma 2.
(see [26]). Let X,bbl be a b−metric-like space with the coefficient s>1 and assume that un⟶u and vn⟶v as n⟶∞. Then, we have(5)1s2bblu,u−1sbblv,v≤liminfn⟶∞bblun,vn≤limsubn⟶∞bblun,vn≤sbblu,u+s2bblv,v+s2bblu,v.In the case that the coefficients=1, the given b−metric like space X,bbl becomes the metric-like space X,bml. If xn is a given sequence in the metric-like space X,bml, then we have the following very useful result.Lemma 3.
(see [23, 36]). Let X,bml be a metric-like space and let xn be a sequence in it such that limn⟶∞bmlxn,xn+1=0. If xn is not a bml−Cauchy sequence in X,bml, then there exist ε>0 and two sequences mk and nk of positive integers such that mk>nk>k and the following four sequences tend to ε+ when k⟶∞:(6)bmlxmk,xnk,bmlxmk,xnk+1,bmlxmk−1,xnk,bmlxmk−1,xnk+1.


## 3. Main Results

In [37], the authors introduced two new types of contractive mappings, namely, β−ψ1−ψ2 contractive mappings of type-I and of type-II in ordered b−metric-like spaces.Definition 8.
(see [37], Definition 1 and Definition 4). Let X,≾¯,bbl be a partially ordered b−metric-like space with the coefficient s≥1. A mapping f:X⟶X is said to be β−ψ1−ψ2−contractive mapping of type-I (resp., of type II), if there exist two altering distance functions ψ1,ψ2 and β:X×X⟶0,∞ such that(7)βu,fuβv,fvψ1sbblfu,fv≤ψ1△sfu,v−ψ2△sfu,v,for all comparable u,v∈X, where(8)△sfu,v=maxbblu,v,bblu,fu,bblv,fv,bblu,fv+bblv,fu4s,bblu,fubblv,fv1+bblu,v,bblu,fubblv,fv1+bblfu,fv,respectively,(9)△sfu,v=maxbblu,v,bblu,fu,bblv,fv,bblu,fv+bblv,fu4s,bblu,fubblu,fv+bblv,fvbblv,fu1+sbblu,fu+bblu,fu,bblu,fubblu,fv+bblv,fvbblv,fu.In [37], the authors proved the following results for β−ψ1−ψ2−contractive mapping of type-I (resp., of type II).Theorem 1.
(see [37], Theorem 2.2). Let X,≾¯,bbl be a bbl-complete ordered b−metric-like space with the coefficient s≥1. Let f:X⟶X be a β−ψ1−ψ2−contractive mapping of type-I. Assume that the following assertions hold:(1)
f is β−admissible and Lβ−admissible (or Rβ-admissible)(2)
There existsu1∈X such that u1≾¯fu1 and βu1,fu1≥1(3)
f is continuous, non-decreasing with respect to ≾¯, and if fnu1⟶z, then βz,z≥1Then,f has a fixed point.Theorem 2.
(see [37], Theorem 2.3). Let X,≾¯,bbl be a bbl-complete ordered b−metric-like space with the coefficient s≥1. Let f:X⟶X be a β−ψ1−ψ2−contractive mapping of type-I. Suppose that the following conditions hold:(1)
f is β−admissible and Lβ−admissible (or Rβ-admissible)(2)
There existsu1∈X such that u1≾¯fu1 and βu1,fu1≥1(3)
f is nondecreasing with respect to ≾¯(4)
Ifun is a sequence in X such that un≾¯un+1 and βun,un+1≥1 for all n∈ℕ, and un⟶u∈X, as n⟶∞, then un≾¯u and βun,u≥1 for all n∈ℕThen,f has a fixed point.Theorem 3.
(see [37], Theorem 2.5). Instead of the β−ψ1−ψ2−contractive mapping condition of type-I in Theorem 1, assume that β−ψ1−ψ2−contractive mapping condition of type-II is satisfied. Then, f has a fixed point.Similar to Theorem2, the authors proved the following result.Theorem 4.
Let all the conditions of Theorem2 are satisfied, apart from condition (8) which is replaced by (9). Then, f has a fixed point.Now, we give the shorter proofs of Theorems1–4.First, lets>1. Now, by applying (5) and (6) from [37], we get(10)ψ1sbblun+1,un+2≤βun,funβun+1,fun+1ψ1sbblfun,fun+1≤ψ1△sfun,un+1−ψ2△sfun,un+1,which is according to (7) from. △sfun,un+1≤maxbblun,un+1,bblun+1,un+2. It further follows that sbblun+1,un+2≤maxbblun,un+1,bblun+1,un+2. Since s>1, we get that bblun+1,un+2≤1/sbblun,un+1, that is, the sequence un is a bbl−Cauchy by to Remark 2 of Lemma 1. The proof is further the same as in [37].Ifs=1, then the given contractive conditions in all Theorems in [37] imply that the corresponding Picard’s sequence is bml−Cauchy according to Lemma 1.In [20], the authors introduced the so-called βq,ϕs,ψ−contraction mappings and obtained some common fixed point theorems for such contractions in the context of b−metric-like spaces.First, letΨ,Φ denote the class of functions ψ,ϕ:0,+∞⟶0,+∞, respectively, satisfying the following conditions:(i)
ψ is nondecreasing, continuous function, and ψt=0, if and only if t=0(ii)
ϕ is lower semicontinuous and ϕt=0, if and only if t=0Definition 9.
(see [20], Definition 5). Let X,bbl be a b−metric-like space with the coefficient s≥1. Let the constant q≥2 and β∈0,1. The nonlinear self-mappings f,g:X⟶X are called βq,ϕs,ψ−contraction mappings if for all x,y∈X.(11)ψ2sqbblfx,fy≤βψMblx,y−ϕMblx,y,where ψ∈Ψ,ϕ∈Φ, and(12)Mblx,y=maxbblgx,gy,bblgx,fy,bblgy,fx,bblgx,fx+bblgy,fy4s.Theorem 5.
(see [20], Theorem 1). Let X,bbl be a bbl−complete b−metric-like space with the coefficient s≥1, and f,g:X⟶X be mapping satisfying the following conditions:(i)
fX⊂gX(ii)
The pairf,g is a βq,ϕs,ψ−contraction(iii)
Then,f and g have a point of coincidence in X(iv)
Moreover, iff and g are weakly compatible, then f and g have a unique common fixed point in XRemark 3.
Some important remarks regarding the previous definition and theorem, namely, it is worth to notice that, for the setMblx,y of real numbers from (11), we have(13)Mblx,y≤maxbblgx,gy,bblgx,fx,bblgy,fy.
Indeed, this follows from the known estimation:bblgx,fx+bblgy,fy/4s≤1/2smaxbblgx,fx,bblgy,fy.
Therefore, instead of the setMblx,y in [20], we introduce the next set:(14)Nblx,y=maxbblgx,gy,bblgx,fx,bblgy,fy,bblgx,fy+bblgy,fx4s.Now, we give the new formulation and the proof of Theorem1 from [20] taking the set Nblx,y instead of Mblx,y.Theorem 6.
LetX,bbl be a bbl−complete b−metric-like space with the coefficient s≥1, and f,g:X⟶X be a self-mapping satisfy the following conditions:(i)
fX⊂gX and at least one of fX, gX is closed subset in the X,bbl(ii)
the pairg,f is an βq,ϕs,ψ− contraction(iii)
Thenf and g have a unique point of coincidence in X(iv)
Moreover, iff and g are weakly compatible, then f and g have a unique common fixed point in XProof.Step 1.
Uniqueness of point of coincidence for pairf,g. First, suppose that the pair f,g has at least one point of coincidence ω1. If it has other point of coincidence for example ω2≠ω1, this means that there are two points u1≠u2 from X such that ω1=fu1=gu1 and ω2=fu2=gu2. According to (11) where Nblx,y is given with (14), we have(15)ψbblfu1,fu2≤ψ2sqbblfu1,fu2≤βψNblu1,u2−ϕNblu1,u2,where(16)Nblu1,u2=maxbblgu1,gu2,bblgu1,fu1,bblgu2,fu2,bblgu1,fu2+bblgu2,fu14s=maxbblω1,ω2,0,0,bblω1,ω2+bblω1,ω24s=bblω1,ω2.
Now, (15) becomes(17)ψbblω1,ω2≤ψ2sqbblω1,ω2≤βψbblω1,ω2−ϕbblω1,ω2<ψbblω1,ω2−ϕbblω1,ω2,that is, ϕbblω1,ω2=0 which is a contradiction because ω1≠ω2.Step 2.
Existence of the point of coincidence.
Letx0 be an arbitrary point in X. Since fX⊂gX, there exists x1∈X such that fx0=gx1. By continuing this process inductively, we get two sequences xn and zn in X such that(18)zn=fxn=gxn+1.
Ifbblzk,zk+1=0 for some k∈ℕ, then fxk+1=gxk+1 is a unique point of coincidence for the pair f,g. Furthermore, let bblzn,zn+1>0, for all n∈ℕ. Now, we shall prove that(19)bblzn,zn+1≤λbblzn−1,zn,for some λ∈0,1/s. Indeed, according to (11) where instead of Mbl, we take Nbl, if x=xn,y=xn+1, we have(20)bblzn,zn+1≤2sqbblzn,zn+1≤Nblxn,xn+1,where(21)Nblxn,xn+1=maxbblgxn,gxn+1,bblgxn,fxn,bblgxn+1,fxn+1,bblgxn,fxn+1+bblgxn+1,fxn4s=maxbblzn−1,zn,bblzn−1,zn,bblzn,zn+1,bblzn−1,zn+1+04s=maxbblzn−1,zn,bblzn,zn+1,bblzn−1,zn+14s≤maxbblzn−1,zn,bblzn,zn+1,bblzn−1,zn+bblzn,zn+14≤maxbblzn−1,zn,bblzn,zn+1,bblzn−1,zn+bblzn,zn+12≤maxbblzn−1,zn,bblzn,zn+1.
From (20) and (21), we get(22)bblzn,zn+1≤12sqmaxbblzn−1,zn,bblzn,zn+1.
It is clear thatmaxbblzn−1,zn,bblzn,zn+1=bblzn−1,zn. Putting 1/2sq=λ∈0,1/s, we obtain (19). Then, by Lemma 1, the sequence zn is a bbl−Cauchy sequence in the bbl−complete b−metric-like space X,bbl such that limn,m⟶∞bblzn,zm.
Let, for example, the subsetfX be closed. The proof if gX is closed is similar. Then, in the first case, there is a unique point u∈fX such that zn converges to u. Since fX⊂gX, there exists v∈X such that gv=u. We shall show that fv=gv=u. For this proof, we firstly have(23)1sbblu,fv≤bblu,zn+bblfxn,fv≤bblu,zn+12sqNblxn,v,(24)Nblxn,v=maxbblgxn,gv,bblgxn,fxn,bblgv,fv,bblgv,fxn+bblgxn,fv4s=maxbblzn−1,u,bblzn−1,zn,bblu,fv,bblu,zn+bblzn−1,fv4s≤maxbblzn−1,u,bblzn−1,zn,bblu,fv,bblu,zn+sbblzn−1,u+sbblu,fv4s⟶bblu,fvasn⟶∞.
Letting the limit in (23) as n⟶∞, we obtain(25)1sbblu,fv≤0+12sq⋅bblu,fv=12sq⋅bblu,fv.
(25) is possible only if fv=u. Hence, u=fv=gv is a unique point of coincidence. The result further follows by Proposition 1 from [31]. This completes the proof of Theorem 6.As corollaries of our Theorem6, we obtain the next results:Corollary 1.
LetX,bbl be a bbl−complete b−metric-like space with the coefficient s≥1, and f,g:X⟶X be a weakly compatible mappings satisfying the following conditions:(i)
fX⊂gX and at least one of fX,gX is a closed subset in the space X,bb(ii)
Assume thatψ∈Ψ,ϕ∈Φ,β∈0,1 and q≥2 such that the condition(26)ψ2sqbblfx,fy≤βψNblx,y1+ϕNblx,y,holds for all x,y∈XThen,f and g have a unique common fixed point in X.Proof.
By (26), it follows that(27)bblfx,fy≤12sqNblx,y,for all x,y∈X. The result further follows from Theorem 6.Corollary 2.
LetX,bbl be a bbl−complete b−metric-like space with the coefficient s≥1, and f,g:X⟶X be a weakly compatible mappings satisfying the following conditions:(i)
fX⊂gX and at least one of fX,gX is a closed subset in X,bb(ii)
Assume thatψ∈Ψ,ϕ∈Φ,β∈0,1 and q≥2 such that the condition(28)ψ2sqbblfx,fy≤βψNblx,yϕNblx,y1+ϕNblx,y,holds for all x,y∈XThen,f and g have a unique common fixed point in X.Proof.
Condition (28) implies that(29)ψ2sqbblfx,fy≤βψNblx,y,that is,(30)bblfx,fy≤12sqNblx,y,for all x,y∈X. Hence, the result follows by the previous corollary.Corollary 3.
LetX,bbl be a bbl−complete b−metric-like space with the coefficient s≥1, and f,g:X⟶X be a weakly compatible mappings satisfying the following conditions:(i)
fX⊂gX and at least one of fX,gX is a closed subset in X,bb(ii)
Assume thatψ∈Ψ,ϕ∈Φ,β∈0,1 and q≥2 such that the condition(31)ψ2sqbblfx,fy≤βψNblx,y−ϕNblx,y1+ϕNblx,y,holds for all x,y∈XThen,f and g have a unique common fixed point in X.Proof.
It is clear that condition (31) implies (30). The proof then follows from Corollary 2.In particular, by takingψt=t and f=g in Theorem 6, we have the following immediate result.Corollary 4.
LetX,bbl be a bbl−complete b−metric-like space with the coefficient s≥1, and f:X⟶X be a given self-mapping that satisfies(32)sqbblfx,fy≤βNblx,y−ϕNblx,y,for all x,y∈X, where fX is a closed subset in X,bb.Then,f has a unique fixed point.Puttingψt=t and ϕt=1/2t in Theorem 6, we can get the next result.Corollary 5.
LetX,bbl be a bbl−complete b−metric-like space with the coefficient s≥1, and f,g:X⟶X be a weakly compatible mappings satisfying the following conditions:(i)
fX⊂gX and at least one of fX,gX is a closed subset in the space X,bb(ii)
Assume thatβ∈0,1 and q≥2 such that the condition(33)sqbblx,y≤βNblx,y,holds for all x,y∈XThen,f and g have a unique common fixed point in X.Proof.
It is an immediate consequence of Theorem6.Remark 4.
By the known relationa+b≤2maxa,b for all real numbers a,b, authors in [20] obtained the next result as an immediate consequence of theirs (Theorem 1).Corollary 6.
LetX,bbl be a bbl−complete b−metric-like space with the coefficient s≥1. If f,g:⟶X⟶X be self-mappings, there exist q≥2 and real constants ci≥0,i=1,…,5 with ∑i=15ci<1 such that(34)sqbblfx,fy≤c1bblgx,gy+c2bblgx,fy+c3bblgy,fx+c4bblgx,fx+c5bblgy,fy,for all x,y∈X.Then,f and g have a unique common fixed point, provided that the pair f,g is a weakly compatible.Remark 5.
Lemmas1 (for s>1) and 3 (for s=1) give the answer to a common question: does the given contractive condition imply that Picards’s sequence initiated by any point x0 is bbl−Cauchy?Now, we have the following interesting question: does the weakly compatible self-mappingsf,g:X⟶X which satisfy condition (34) have a unique common fixed point in X?



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