Some fixed point results are given for a class of functional contractions acting on (reflexive) triangular symmetric spaces. Technical connections with the corresponding theories over (standard) metric and partial metric spaces are also being established.

1. Introduction

Let X be a nonempty set. By a symmetric over X we will mean any map d:X×X→R+:=[0,∞[ with

d(x,y)=d(y,x), ∀x,y∈X.

The couple (X,d) will be referred to as a symmetric space. The introduction of such structures goes back to Wilson [1]; for its “modern” aspects, we refer to Hicks and Rhoades [2].

Call the symmetric d, triangular if

d(x,z)≤d(x,y)+d(y,z), ∀x,y,z∈X,

and reflexive-triangular, provided it fulfills (the stronger condition)

d(x,z)+d(y,y)≤d(x,y)+d(y,z), ∀x,y,z∈X.

The class of such particular (symmetric) spaces has multiple connections with the one of partial metric spaces, due to Matthews [3]. For, as we will see in the following, the fixed point theory for functional contractive maps in (reflexive) triangular symmetric spaces is a common root of both corresponding theories in standard metric spaces and partial metric spaces. This ultimately tells us that, for most of the function contractions taken from the list in Rhoades [4], any such theory over partial metric spaces is nothing but a clone of the corresponding one developed for standard metric spaces. Further aspects will be delineated elsewhere.
2. Preliminaries

Let (X,d) be a symmetric space, where d(·,·) is triangular. Call the subset Y of X, d-singleton provided y1,y2∈Y⇒d(y1,y2)=0.

(A) We introduce a 0d-convergence and a 0d-Cauchy structure on X under the lines in Romaguera [5] (related to the context of partial metric spaces; cf. Section 4). Given the sequence (xn) in X and the point x∈X, we say that (xn), 0d-converges to x (written as xn→0dx) provided d(xn,x)→0 as n→∞; that is,

∀ε>0, ∃i=i(ε):n≥i⇒d(xn,x)<ε.

The set of all such points x will be denoted as 0d-limn(xn); note that it is a d-singleton, because d is triangular. If 0d-limn(xn) is nonempty, then (xn) is called0d-convergent. We stress that the concept (b01) does not match the standard requirements in Kasahara [6], because, for the constant sequence (xn=u;n≥0), we do not have xn→0du if d(u,u)≠0. Further, call the sequence (xn), 0d-Cauchy when d(xm,xn)→0 as m,n→∞, m≠n; that is,

∀ε>0,∃j=j(ε):j≤m<n⇒d(xm,xn)<ε.

As d is triangular, any 0d-convergent sequence is 0d-Cauchy too, but the reciprocal is not in general true. Let us say that (X,d) is 0-complete, if each 0d-Cauchy sequence is 0d-convergent.

(B) Call the sequence (xn;n≥0), 0d-semi-Cauchy provided

d(xn,xn+1)→0, asn→∞.

Clearly, each 0d-Cauchy sequence is 0d-semi-Cauchy, but not conversely. The following auxiliary statement about such objects is useful for us.

Lemma 1.

Let (xn;n≥0) be a 0d-semi-Cauchy sequence in X that is not 0d-Cauchy. There exist then ε>0, j(ε)∈N, and a couple of rank-sequences (m(j);j≥0), (n(j);j≥0) with
(1)j≤m(j)<n(j),d(xm(j),xn(j))≥ε,∀j≥0,(2)n(j)-m(j)≥2,d(xm(j),xn(j)-1)<ε,∀j≥j(ε),(3)limjd(xm(j)+p,xn(j)+q)=ε,∀p,q∈{0,1}.

Proof.

As (xn;n≥0) is not 0d-Cauchy, there exists, via (b02), an ε>0 with
(4)A(j)∶={(m,n)∈N×N;j≤m<n,d(xm,xn)≥ε}≠∅,{(m,n)∈N×N;j≤m<n,d(xm,xn)∅≥ε}≠∅∀j≥0.
Having this precise, denote, for each j≥0,
(5)m(j)=minDom(A(j)),n(j)=minA(m(j)).
As a consequence, the couple of rank-sequences (m(j);j≥0), (n(j);j≥0) fulfills (1). On the other hand, letting the index j(ε)≥0 be such that
(6)d(xk,xk+1)<ε,∀k≥j(ε),
it is clear that (2) holds too. Finally, by the triangular property,
(7)ε≤d(xm(j),xn(j))≤d(xm(j),xn(j)-1)+d(xn(j)-1,xn(j))<ε+d(xn(j)-1,xn(j)),∀j≥j(ε),
and this establishes the case (p=0, q=0) of (3). Combining with
(8)d(xm(j),xn(j))-d(xn(j),xn(j)+1)≤d(xm(j),xn(j)+1)≤d(xm(j),xn(j))+d(xn(j),xn(j)+1),∀j≥j(ε)
yields the case (p=0, q=1) of the same. The remaining situations are deductible in a similar way.

Note finally that the exposed facts do not exhaust the whole completeness theory of such objects applicable to partial metric spaces. Some complementary aspects involving these last objects may be found in Oltra and Valero [7].

(C) Let ℱ(A) stand for the class of all functions from A to itself. For any φ∈ℱ(R+), and any s in R+0:=]0,∞[, put

L+φ(s)=infε>0Φ[s+](ε); where Φ[s+](ε)=sup{φ(t);s≤t<s+ε}.

By this very definition, we have the representation
(9)L+φ(s)=max{limsupt→s+φ(t),φ(s)},∀s∈R+0.
Clearly, the quantity in the right member may be infinite. A basic situation when this cannot hold may be described as follows. Call φ∈ℱ(R+), normal when [φ(0)=0;φ(t)<t,∀t>0]. Note that, under such a property, one has
(10)φ(s)≤L+φ(s)≤s,∀s∈R+0.
The following consequence of this will be useful. Given the sequence (tn;n≥0) in R+ and s∈R+, define tn↓s (as n→∞) provided [tn≥s, ∀n] and tn→s.
Lemma 2.

Let the function φ∈ℱ(R+) be normal, and let s∈R+0 be arbitrarily fixed. Then,

limsupnφ(tn)≤L+φ(s), for each sequence (tn) with tn↓s,

there exists a sequence (rn) with rn↓s and φ(rn)→L+φ(s).

Proof.

(i) Given ε>0, there exists a rank p(ε)≥0 such that s≤tn<s+ε, for all n≥p(ε); hence
(11)limsupnφ(tn)≤sup{φ(tn);n≥p(ε)}≤Φ[s+](ε).
It suffices taking the infimum over ε>0 in this relation to get the desired fact.

(ii) From (b04), L+φ(s)=infε>0Φ[s+](ε), so, by the definition of infimum,
(12)∀ε>0,∃δ∈]0,ε[:L+φ(s)≤Φ[s+](δ)<L+φ(s)+ε.
This, in the case of L+φ(s)=0, gives the written conclusion with (rn=s;n≥0), for, as a direct consequence of (10), one has φ(s)=0. Suppose now that L+φ(s)>0. Again from (b04),
(13)∀ε∈]0,L+φ(s)[,∃δ∈]0,ε[:L+φ(s)-ε<L+φ(s)≤Φ[s+](δ)<L+φ(s)+ε.
This, along with the definition of supremum, tells us that there must be some r in [s,s+δ[ with L+φ(s)-ε<φ(r)<L+φ(s)+ε. Taking a sequence (εn) in R+0 with εn→0, there exists a corresponding sequence (rn) in R+ with rn↓s and φ(rn)→L+φ(s), hence the conclusion.

(The last assertion is a consequence of (9) above.) In particular, the normal function φ∈ℱ(R+) is right limit normal, whenever it is usc at the right on R+0:

limsupt→s+φ(t)≤φ(s), for each s∈R+0.

Note that this property is fulfilled when φ is continuous at the right on R+0, for, in such a case, limsupt→s+φ(t)=φ(s), for all s∈R+0. Another interesting example is furnished by Lemma 2. Let us say that the normal function φ∈ℱ(R+) is Geraghty normal provided (cf. Geraghty [8])

(tn;n≥0)⊆R+0,φ(tn)/tn→1 imply tn→0.

Lemma 3.

Let the normal function φ∈ℱ(R+) be Geraghty normal. Then, φ is necessarily right limit normal.

Proof.

Suppose that the normal function φ is not right limit normal. From (10), there exists some s∈R+0 with L+φ(s)=s. Combining with Lemma 2, there exists a sequence (rn;n≥0) with rn↓s and φ(rn)→s, whence φ(rn)/rn→1; that is, φ is not Geraghty normal.

Remark 4.

The reciprocal of this is not in general true. In fact, for the (continuous) right limit normal function [φ(t)=t(1-e-t),t≥0] and the sequence (tn=n+1;n≥0) in R+0, we have φ(tn)/tn→1, but, evidently, tn→∞.

3. Main Result

Let (X,d) be a symmetric space, with, in addition,

d is triangular and (X,d) is 0-complete.

Further, let T:X→X be a selfmap of X. Call z∈X, d-fixed if and only if d(z,Tz)=0; the class of all such elements will be denoted as Fix(T;d). Technically speaking, the points in question are obtained by a limit process as follows. Let us say that x∈X is a Picard point (modulo(d,T)) if (i) (Tnx;n≥0) is 0d-convergent, and (ii) each point of 0d-limnTnx is in Fix(T;d). If this happens for each x∈X, then T is referred to as a Picard operator (modulo d); if (in addition) Fix(T;d) is d-singleton, then T is called a global Picard operator (modulo d); compare Rus [9, Chapter 2, Section 2.2].

Now, concrete circumstances guaranteeing such properties involve (in addition to (c01)) contractive selfmaps T with the d-asymptotic property:

limnd(Tnx,Tn+1x)=0, ∀x∈X.

Precisely, denote for x,y∈X:
(14)M(x,y)=max{d(x,y),d(x,Tx),d(y,Ty)},K(x,y)=(12)[d(x,Ty)+d(Tx,y)],P(x,y)=max{M(x,y),K(x,y)};
and fix G∈{M,P}. Given φ∈ℱ(R+), we say that T is(d,G;φ)-contractive, if

d(Tx,Ty)≤φ(G(x,y)), ∀x,y∈X.

The main result of this note is the following.
Theorem 5.

Suppose (under (c02)) that T is (d,G;φ)-contractive, for some right limit normal function φ∈ℱ(R+). Then, T is a global Picard operator (modulo d).

Proof.

Let z1,z2∈Fix(T;d) be arbitrary fixed. By this very choice,
(15)M(z1,z2)=max{d(z1,z2),0,0}=d(z1,z2).
In addition (from the triangular property)
(16)d(z1,Tz2)≤d(z1,z2)+d(z2,Tz2)=d(z1,z2),d(z2,Tz1)≤d(z2,z1)+d(z1,Tz1)=d(z1,z2),
so that K(z1,z2)≤d(z1,z2), which tells us that P(z1,z2)=d(z1,z2). On the other hand, again from the choice of our data and the triangular property,
(17)d(z1,z2)≤d(z1,Tz1)+d(z2,Tz2)+d(Tz1,Tz2)=d(Tz1,Tz2).
Combining with the contractive condition yields (for either choice of G)
(18)d(z1,z2)≤φ(d(z1,z2)),
wherefrom d(z1,z2)=0, so that Fix(T;d) is d-singleton. It remains now to establish the Picard property. Fix some x0∈X, and put xn=Tnx0,n≥0; note that by (c02), (xn;n≥0) is 0d-semi-Cauchy.

(I) We claim that (xn;n≥0) is 0d-Cauchy. Suppose this is not true. By Lemma 1, there exist ε>0, j(ε)∈N, and a couple of rank-sequences (m(j);j≥0), (n(j);j≥0), with the properties (1)–(3). For simplicity, we will write (for j≥0), m, n in place of m(j), n(j), respectively. By the contractive condition,
(19)d(xm,xn)≤d(xm,xm+1)+d(xn,xn+1)+φ(G(xm,xn)),∀j≥j(ε).

Denote (rj:=M(xm,xn),sj:=K(xm,xn),tj:=P(xm,xn);j≥0). From (1), tj≥rj≥ε, for all j≥j(ε); moreover, (3) yields rj,sj,tj→ε as j→∞. So, passing to limit as j→∞ in (19) one gets (via Lemma 2) ε≤L+φ(ε)<ε, contradiction, so that our assertion follows.

(II) As (X,d) is 0-complete, this yields xn→0dz as n→∞, for some z∈X. We claim that z is an element of Fix(T;d). Suppose not: that is, ρ:=d(z,Tz)>0. By the previously mentioned properties of (xn;n≥0), there exists k(ρ)∈N such that, for all n≥k(ρ),
(20)d(xn,xn+1),d(xn,z)<ρ2,d(xn,Tz)≤d(xn,z)+ρ<3ρ2.

This gives (again for all n≥k(ρ))
(21)M(xn,z)=ρ,K(xn,z)<ρ;henceP(xn,z)=ρ.
By the contractive condition, we then have (for either choice of G)
(22)ρ≤d(z,xn+1)+φ(G(xn,z))=d(z,xn+1)+φ(ρ),∀n≥k(ρ).
Passing to limit as n→∞ yields ρ≤φ(ρ), contradiction. Hence, z is an element of Fix(T;d), and the proof is complete.

4. Reflexive-Triangular Case

Now, it remains to determine concrete circumstances under which T is d-asymptotic. Let (X,d) be a symmetric space, with

d is reflexive-triangular and (X,d) is 0-complete.

Further, let T be a selfmap of X; and fix G∈{M,P}.

Lemma 6.

Suppose that T is (d,G;φ)-contractive, for some right limit normal function φ∈ℱ(R+). Then, T is d-asymptotic.

Proof.

By definition, we have
(23)M(x,Tx)=max{d(x,Tx),d(Tx,T2x)},∀x∈X.
On the other hand, by the reflexive-triangular property,
(24)K(x,Tx)≤(12)[d(x,Tx)+d(Tx,T2x)]≤max{d(x,Tx),d(Tx,T2x)}.
So, by simply combining these,
(25)P(x,Tx)=max{d(x,Tx),d(Tx,T2x)},∀x∈X.
Fix some x∈X, and put (ρn:=d(Tnx,Tn+1x);n≥0). From the contractive condition, [ρn+1≤φ(max{ρn,ρn+1}),∀n≥0]. As φ is normal, this yields
(26)ρn+1≤φ(ρn),∀n≥0.
In particular, (ρn;n≥0) is descending; hence, ρ:=limnρn exists in R+. Assume that ρ>0. By Lemma 2, we must have ρ≤L+φ(ρ)<ρ; contradiction. Hence, ρ=0, and the conclusion follows.

Now, by simply combining this with Theorem 5, we have (under (d01)) the following.

Theorem 7.

Suppose that T is (d,G;φ)-contractive, for some right limit normal function φ∈ℱ(R+). Then, T is a global Picard operator (modulo d).

A basic particular case of this result is the following. Call the symmetric d on X an almost partial metric provided it is reflexive-triangular and

d(x,y)=0⇒x=y(d is sufficient).

Note that, in such a case,
(27)[∀Y∈𝒫0(X)];Yisd-singleton⇒Yissingleton,(28)Fix(T;d)⊆Fix(T)(=theclassofallfixedpointsofTinX).
Assume in the following that

d is almost partial metric and (X,d) is 0-complete.

Theorem 8.

Let the selfmap T be (d,G;φ)-contractive, for some right limit normal function φ∈ℱ(R+). Then,
(29)Fix(T;d)=Fix(T)={z},whered(z,z)=0,(30)Tnx→0dzasn→∞,∀x∈X.

Proof.

By Theorem 7, we have (taking (27)+(28) into account)
(31)Fix(T;d)={z},withz∈Fix(T),d(z,z)=0,
and, moreover, (30) holds. It remains to establish that Fix(T)={z}. For each w∈Fix(T), we must have (by (30)) Tnw→0dz, which means d(w,z)=0; hence (as d = sufficient) w=z. The proof is complete.

Now, let us give two important examples of such objects.

(A) Clearly, each (standard) metric on X is an almost partial metric. Then, Theorem 8 is just the main result in Jachymski [10]; see also Cho et al. [11]. In fact, its argument mimics the one in those papers. The only “specific” fact to be underlined is related to the reflexive-triangular property of our symmetric d.

(B) According to Matthews [3], call the symmetric d, a partial metric provided it is reflexive-triangular and

[d(x,x)=d(y,y)=d(x,y)]⇒x=y (d is strongly sufficient)

Note that, by the reflexive-triangular property, one has (with z=x)
(32)d(x,x)+d(y,y)≤2d(x,y),∀x,y∈X,
and this, along with (d04), yields (d02); that is, each partial metric is an almost partial metric. Hence, Theorem 8 is applicable to such objects; its corresponding form is just the main result in Romaguera [12]; see also Altun et al. [13]. It is to be stressed here that the Matthews property (d05) was not used in the quoted statement. This forces us to conclude that this property is not effective in most fixed point results based on such contractive conditions. On the other hand, the argument used here is, practically, a clone of that developed for the standard metric setting. Hence—at least for such results—it cannot get us new insights for the considered matter. Clearly, the introduction of an additional order structure on X does not change this conclusion. Hence, the results in the area due to Altun and Erduran [14] are but formal copies of the ones (in standard metric spaces) due to Agarwal et al. [15]. This is also true for the common fixed points question, when, for example, the results in Shobkolaei et al. [16] or Karapınar and Shatanawi [17] are but a translation of the ones (in standard metric spaces) due to Jachymski [18]. Finally, we may ask whether this reduction scheme comprises as well the class of contraction maps in general complete partial metric spaces taken as in Ilić et al. [19]. Formally, such results are not reducible to the previous ones. But, from a technical perspective, this is possible; see Turinici [20] for details.

5. Triangular Symmetrics

Let (X,d) be a symmetric space, taken as in (c01), and let T be a selfmap of X.

Lemma 9.

Suppose that T is (d,M;φ)-contractive, for some right limit normal function φ∈ℱ(R+). Then, T is d-asymptotic.

The argument is based on the evaluation (23); see also Zhu et al. [21].

Now, by simply combining this with Theorem 5, we have (under (c01)) the following.

Theorem 10.

Suppose that T is (d,M;φ)-contractive, for some right limit normal function φ∈ℱ(R+). Then, T is a global Picard operator (modulo d).

A basic particular case of this result is to be stated as below. Call the symmetric d(·,·), a weak almost partial metric provided it is triangular and sufficient (i.e., (d02) holds). Note that, in such a case, relations (27) and (28) are still retainable. Assume in the following that

d is a weak almost partial metric and (X,d) is 0-complete.

Theorem 11.

Let the selfmap T be (d,M;φ)-contractive, for some right limit normal function φ∈ℱ(R+). Then, conclusions of Theorem 8 are holding.

The proof mimics the one of Theorem 8 (if one takes Theorem 10 as starting point), so, it will be omitted.

Now, let us give two important examples of such objects.

(A) Clearly, each (standard) metric on X is a weak almost partial metric. Then, Theorem 11 is comparable with the main result in Jachymski [10].

(B) Remember that the symmetric d is called a partial metric provided it is reflexive-triangular and (d04) + (d05) hold. As before, (32) tells us (via (d04)) that each partial metric is a weak almost partial metric; hence, Theorem 11 is applicable to such objects. In particular, when φ is linear (φ(t)=λt, t∈R+, for some λ∈[0,1[), one recovers the Banach type fixed point result in Aage and Salunke [22], which, in turn, includes the one in Valero [23]. On the other hand, Lemma 3 tells us that Theorem 11 includes as well a related fixed point statement due to Dukić et al. [24]; see also Golubović et al. [25]; moreover, by Remark 4, the converse inclusion is not in general true. It is to be stressed here that the Matthews property (d05) was not effectively used in the quoted statement; in addition, the (stronger) reflexive-triangular property of d was replaced by the triangular property of the same. As before, the argument used here is, practically, a clone of that developed in the standard metric setting; whence, the results we just quoted are technically deductible from the one in Boyd and Wong [26]. Further developments of these results may be performed under the lines in Turinici [27].

Acknowledgment

The author is very indebted to both referees, for a number of useful suggestions.

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