# Subordinate Semimetric Spaces and Fixed Point Theorems

Authors: José Villa-Morales
Journal: Journal of Mathematics
 (2018)

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## Abstract

We introduce the concept of subordinate semimetric space. Such notion includes the concept of RS-space introduced by Roldán and Shahzad; therefore the concepts of Branciari’s generalized metric space and Jleli and Samet’s generalized metric space are particular cases. For such spaces we prove a version of Matkowski’s fixed point theorem, and introducing the concept ofq-contraction we get a fixed point theorem of Kannan-Ćirić type. Moreover, using such result we characterize complete subordinate semimetric spaces.


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## Body

## 1. Introduction

Fixed point theory has an enormous number of applications in ordinary (or partial) differential equations, game theory, functional analysis, calculus of several variables (the classical implicit function theorem), etc. This is one of the reasons why we always try to have a fixed point result in a general context, since it could be applied more broadly.Generalizations of the concept of metric space are currently one of the most active branches of functional analysis (see [1–4] or [5], where the notion of quasimetric space is studied). In fact, in Table 1 we present a nonexhaustive review of some generalizations.Table 1 Year Author Concept 1931 W.A Wilson [7] quasi-metric space 1959 H. Nakano [8] modular space 1993 S. Czerwik [9] b-metric space 2000 P. Hitzler, A. K. Seda [10] dislocated metric space 2000 A. Branciari [11] generalized metric space 2015 M. Jleli, B. Samet [1] generalized metric space 2016 A. Roldán, N. Shahzad [2] RS-spaceOur work is related to the notion of generalized metric space introduced by Jleli and Samet [1]. Such concept was immediately generalized by Roldán and Shahzad [2] as follows.Definition 1.
An RS-space is a pair(E,d) where E is a nonempty set and d:E×E→[0,∞] is a function such that the following properties are fulfilled:(i)
ifd(x,y)=0 then x=y,(ii)
d(x,y)=d(y,x) for every x,y∈E,(iii)
there existsc>0 such that if x,y∈E are two points and (xn) is an infinite sequence with limn,m→∞⁡d(xn,xm)=0 and limn→∞⁡d(xn,x)=0 then(1)dx,y≤climsupn→∞⁡dxn,y.The notions of modular space, quasimetric space, dislocated metric space, and generalized metric space (in the Branciari sense) are particular cases of the notion of RS-space (see [1] or [2]). In this paper we introduce the notion of subordinate semimetric space; such concept is a generalization of the notion of RS-space. On the other hand, we give in Example 5 a pair (E,d) which is a subordinate semimetric space that is not an RS-space. In the context of subordinate semimetric spaces we prove a version of Matkowski’s fixed point theorem and introducing the notion of q-contraction we obtain a version of the fixed point theorem of Kannan-Ćirić. Moreover, we prove that if in a subordinate semimetric space every q-contraction has a fixed point then this must be complete.


## 2. Subordinate Metric Spaces

We start with a generalization of the common notion of semimetric space (see [6]).Definition 2.
A semimetric space is a pair(E,d) where E is a nonempty set and d:E×E→[0,∞] is a function that satisfies the following: (i)
for every(x,y)∈E×E, we have(2)dx,y=0⟶x=y,(ii)
for every(x,y)∈E×E, we have(3)dx,y=dy,x.Some important concepts can be introduced with this general notion.Definition 3.
Let(E,d) be a semimetric space.(i)
A sequence(xn) in E converges to x∈E if(4)limn→∞⁡dxn,x=0.(ii)
A sequence(xn) in E is a Cauchy sequence if(5)limm,n→∞⁡dxn,xm=0.(iii)
(E,d) is complete if every Cauchy sequence in E is convergent.In order to get a rich mathematical structure we introduce a substitute of triangle inequality (for a metric).Definition 4.
We say that a semimetric space(E,d) is subordinate if there is a function ζ:[0,∞]→[0,∞] such that (i)
ζ is nondecreasing; limx→0⁡ζx=0,(ii)
for every(x,y)∈E×E, with x≠y, and (xn) being an infinity Cauchy sequence in E such that (xn) converges to x we have(6)dx,y≤ζlimsupn→∞⁡dxn,y.In this case we say that (E,d) is subordinate to ζ or that (E,d) is a subordinate semimetric space.It is clear that every RS-space is a subordinate semimetric space (just takeζ(x)=cx), and we give an example of a subordinate semimetric space that is not an RS-space.Example 5.
LetE0={0}∪N∪(N×N) and define d0:E0×E0→[0,∞] as(7)d0r,s,n,m=d0n,m,r,s=∞,m≠s,1n+r,n≠r,m=s,0,n=r,m=s,d0k,n,m=d0n,m,k=∞,k∈N,m≠k,1n,m=k,m,k=0,d0m,n=d0n,m=∞,m,n∈N,m≠n,n2,n∈N,m=0,0,m=n.Since(8)limn,r→∞⁡d0n,m,r,m=limn,r→∞⁡1n+r=0,limn→∞⁡d0n,m,m=limn→∞⁡1n=0,the sequence {(n,m)}n∈N is an infinite Cauchy sequence that converges to m. Suppose that there is a constant c>0 such that(9)m2=d0m,0≤climsupn→∞⁡d0n,m,0=cm,then c≥m, for all m∈N. In this way (E0,d0) is not an RS-space. However, the semimetric space (E0,d0) is subordinate, for example, to the function ζ(x)=x2.With the subordination concept we will see that we are able to prove some important fixed point theorems. Then a natural question arises:What conditions implies that a semimetric space is subordinate?The answer will give us fixed point theorems in a more general scheme.


## 3. Fixed Point Theorems

We now introduce a type of Kannan-Ćirić contraction condition.Definition 6.
Let(E,d) be a semimetric space. A mapping f:E→E is said to be a q-contraction, with q∈(0,1), if(10)dfx,fy≤qmax⁡dx,fx,dy,fyholds for every (x,y)∈E×E.Letf:E→E be a function. For each x∈E we define f[n](x) recursively as f[0](x)=x and f[n+1](x)=f(f[n](x)), for each n∈N∪{0}. With the above notation we have a version of the Kannan-Ćirić’s fixed point theorem.Theorem 7.
Letf:E→E be a q-contraction on a complete semimetric space (E,d).
( i ) If there is an x0∈E such that(11)limsupn→∞⁡dfnx0,fn+1x0<∞,then (f[n](x0)) converge to some x^∈E.
( i i ) Suppose that (E,d) is subordinate to ζ and(12)ζt<tq,for all0<t<∞.If d(x^,f(x^))<∞ then x^ is the unique fixed point of f.Proof.
( i ) Let us set xn=f[n](x0), for each n∈N. The q-contraction property of f implies(13)limsupn→∞⁡dxn,xn+1=limsupn→∞⁡dfxn-1,fxn≤limsupn→∞⁡qmax⁡dxn-1,fxn-1,dxn,fxn=qlimsupn→∞⁡max⁡dxn-1,xn,dxn,xn+1≤qlimsupn→∞⁡dxn,xn+1.The hypothesis (i) yields limsupn→∞⁡d(xn,xn+1)=0.
Now let us see that(xn) is a Cauchy sequence. Let ε>0, then there is k∈N such that(14)dxn,xn+1<εq,for alln≥k.Therefore, if m,n≥k+1, then(15)dxn,xm=dfxn-1,fxm-1≤qmax⁡dxn-1,xn,dxm-1,xm≤qεq=ε.Then there is x^∈E such that limn→∞⁡d(xn,x^)=0.
( i i ) Let us see that x^ is a fixed point for f. If the set {xn:n∈N} is finite, the Cauchy property of (xn) implies that there exists n0∈N such that xn=xn0=x^, for all n≥n0, then f(x)=f(xn0)=xn0+1=x^. On the other hand, if {xn:n∈N} is an infinite set then there is an infinite Cauchy subsequence (xnj) of (xn) such that limj→∞⁡d(xnj,x^)=0. If d(x^,f(x^))>0, then there is n0∈N such that(16)dxn,xm<12dx^,fx^,for alln,m≥n0,and in this way(17)dx^,fx^≤ζlimsupj→∞⁡dxnj,fx^≤ζqlimsupj→∞⁡max⁡dxnj-1,xnj,dx^,fx^≤ζqmax⁡12dx^,fx^,dx^,fx^=ζqdx^,fx^.If t≔qd(x^,f(x^)), then t≤qζ(t); therefore t=0 or t=∞, but this is not possible, because 0<d(x^,f(x^))<∞. Thus x^ is a fixed point. If y^ is other fixed point, then(18)dx^,y^=dfx^,fy^≤qmax⁡dx^,fx^,dy^,fy^=0.From this the uniqueness of x^ follows.Now let us give an example where condition (12) is necessary to obtain a fixed point.Example 8.
Letζ:0,∞→0,∞ be defined as ζx=xα, with 0<α<1. Given q0∈(0,1) there is t0=(q0/2)1/(1-α)∈(0,1) such that ζ(t0)≥t0/q0. Let us consider the set E={0}∪N with the semimetric(19)dm,n=dn,m=t0min⁡n,m,n,m∈N×N,ζt0m,n,m∈0×N.Since the sequence (n)n∈N is an infinite Cauchy sequence that converge to 0 (this follows from limm→∞⁡dm,0=limm→∞⁡ζt0m=0) we have(20)d0,m≤ζt0m=ζlimsupn→∞⁡t0min⁡n,m=ζlimsupn→∞⁡dn,m.Therefore, the semimetric space (E,d) is subordinate to ζ. The function f:E→E, defined as(21)fn=n+1,does not have fixed points. On the other hand, for m≥1,(22)t0m≤t0≤q0ζt0≤ζt0,this implies(23)df0,fm=d1,m+1=t0≤q0ζt0=q0max⁡ζt0,t0m=q0max⁡d0,f0,dm,fmand, for n≥1,(24)dfn,fm=dn+1,m+1=t0min⁡n+1,m+1=t0t0min⁡n,m≤q0t0min⁡n,m=q0max⁡t0n,t0m=q0max⁡dn,fn,dm,fm.Therefore, f is a q-contraction on the complete semimetric space (E,d) subordinated to ζ without fixed point.Right away we will try a version of Matkowski’s theorem in the context of subordinate semimetric spaces.Theorem 9.
Let(E,d) be a complete semimetric space subordinated to ζ and f:E→E. Suppose that there exists a nondecreasing function φ:[0,∞]→[0,∞] such that limn→∞⁡φ[n](t)=0, for all t∈[0,∞), and(25)dfx,fy≤φdx,y,for allx,y∈E.If there is an x0∈E such that(26)δd,f,x0≔sup⁡dx0,fnx0:n∈N<∞,then (fn(x0)) converges to some x^∈E. Moreover, x^ is the unique fixed point of f.Proof.
Let us takexn=f[n](x0). Suppose m<n, then (27)dxn,xm<φdxn-1,xm-1<φ2dxn-2,xm-2⋮<φmdxn-m,x0≤φmδd,f,x0.The hypothesis implies that (xn) is a Cauchy sequence, then there is x^∈E such that limn→∞d(xn,x^)=0. Suppose that xm0=xn0 for some m0<n0, then xm0=f[n0-m0](xm0), and this yields(28)dxm0,fxm0=dfn0-m0xm0,fn0-m0fxm0≤φn0-m0dxm0,fxm0<φn0-m0-1dxm0,fxm0≤dxm0,fxm0.Hence all the terms in the Cauchy sequence (xn) are different. Moreover(29)dx^,fx^≤ζlimsupn→∞⁡dxn,fx^≤ζlimsupn→∞⁡φdxn-1,x^≤ζlimsupn→∞⁡dxn-1,x^≤ζ0=0.Thus x^ is a fixed point of f. If y^ is other fixed point, then(30)dx^,y^=dfx^,fy^≤φdx^,y^=0.From this the uniqueness of x^ follows.It is clear that a semimetric space(E,d) for which δ(d,f,x)=∞, for each x∈E, does not have fixed points, then (26) is a necessary condition in order to have a function f:E→E and a fixed point.Proposition 10.
Let(E,d) be a semimetric space subordinated to ζ. Let (xn) be a Cauchy sequence in (E,d) with xn≠xm whenever n≠m. If there is a subsequence (xnj) of (xn) such that limj→∞⁡xnj=x, then (xn) converge to x.Proof.
Letε>0, then there is δ>0 such that ζ(t)<ε, for each 0<t<δ. On the other hand, there exists n0∈N for which(31)dxn,xm<δ2,for allm,n≥n0.Therefore,(32)limsupj→∞⁡dxnj,xm≤δ2,for allm≥n0.Since ζ is nondecreasing(33)dx,xm≤ζlimsupj→∞⁡dxnj,xm≤ζδ2<ε,for allm≥n0.Hence limn→∞⁡d(x,xn)=0.With the next result we characterize when a subordinate semimetric space is complete; the corresponding result for metric spaces is due to Subrahmanyam [12].Theorem 11.
Let(E,d) be a semimetric space subordinated to ζ. Let q∈(0,1); if every q-contraction has a fixed point then (E,d) is complete.Proof.
Let us suppose that(E,d) is not complete, then there is a nonconvergent Cauchy sequence (xn). The Cauchy property implies that we can take, if necessary, a subsequence with all the elements different, so we suppose that xn are distinct (see the second part in the proof of Theorem 7). Let us define the sets An={xm:m≥n}, for each n∈N. If z∉An then Proposition 10 implies d(z,An)>0. The Cauchy property of (xn) implies that the sets Mn(z)={k≥n:d(xr,xs)<qd(z,An),  for all r,s≥k}, are not empty. Let us define kn(z)=min⁡Mn(z), then(34)dxr,xknz<qdz,An≤qdz,xt,for allr≥knzand allt≥n.For each z∈E there is n∈N such that z∉An; thus n(z)=min⁡{n∈N:z∉An} is well defined. The function f:E→E, defined as f(z)=xkn(z)(z), does not have fixed points. Indeed, by definition xkn(z)(z)∈Akn(z)(z) and z∉Akn(z)(z), since Akn(z)(z)⊂An(z). Moreover, for z,y∈E, we assume, without loss of generality, that kn(y)(y)>kn(z)(z), then kn(z)(z)≥n(z) and (34) implies (35)dfz,fy=dxknyy,xknzz<qdz,xknzz≤qmax⁡dz,xknzz,dy,xknyy=qmax⁡dz,fz,dy,fy.Thus f is a q-contraction without fixed points; this contradicts the hypothesis.



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