# Ordered Uniform Convexity in Ordered Convex Metric Spaces with an Application to Fixed Point Theory

Authors: Ismat Beg
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we introduce the concept of ordered uniform convexity in ordered convex metric spaces and study some properties of order uniform convexity. Finally, as application, we connect our results with existence of fixed points for monotone non-expansive mappings defined on these spaces.


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## Body

## 1. Introduction

Menger [1] initiated the study of convexity in metric spaces which was further developed by many authors [2–10]. The terms “metrically convex” and “convex metric space” are due to [5]. Beg [11] proved that a closed convex subset of a uniformly convex complete metric space is a Chebyshev set. Afterward, Abdelhakim [12] studied convex functions on these spaces. Subsequently, Beg [13] introduced the idea of ordered convex metric spaces and proved several results on these spaces. On the other hand, in 1936, J.A. Clarkson introduced the class of uniformly convex normed spaces and established Clarkson’s inequality [14]. He also proved that Lp0,1 is a uniformly convex Banach space whenever 1<p<∞. Since the appearance of uniform convexity on the scene, many important results were obtained on uniformly convex spaces by several researchers [15]. Recently, Alfuraidan and Khamsi [16] extended the notion of uniform convexity to order uniform convexity on Banach spaces. The aim of this note is to amalgamate these ideas by giving the concept of (o)-uniformly convex ordered convex metric spaces and obtain sufficient conditions for existence of fixed points for monotone non-expansive mappings defined on these spaces.We conclude with the plan of the paper. Section2 presents some basic notations and definitions from the existing literature on order convex metric spaces. Section 3 deals with the concept of (o)-uniformly convex ordered convex metric spaces. In Section 4, an application to obtain sufficient conditions for existence of fixed points for monotone non-expansive mappings defined on these spaces is given. Conclusion is given in Section 5.


## 2. Preliminaries

This section is devoted to laying out the basic notation pertaining to order and convex structure on metric spaces.Definition 1 (see [17]).
A binary relationa⪯b defined for some pairsa,b of elements of a set X is called an order relation in X if ⪯ is reflexive, transitive, and antisymmetric. A reflexive and transitive relation ⪯ is called pre-order.Remark 2 (see [17]).
Let⪯ be a binary relation on a set X. By a≺b we mean a⪯b and a≠b. Relation ~ is defined as a~bifa⪯bandb⪯a. The inverse of ⪯ is defined as a≽b if b⪯a. Incomparable elements a and b (i.e., a≰b and a≯b) are denoted by a⋈b. Transitivity of order relation ⪯ implies a⊀b⊀c⇒a⊀c for all a,b,c∈X.Definition 3 (see [17]).
An ordered set is called totally ordered if it has no incomparable elements.Definition 4 (see [10]).
LetX,d be a metric space and I=0,1. A mapping ω:X×X×I⟶X is said to be a convex structure on X if for each a,b,λ∈X×X×I and u∈X,
(1)du,ωa,b;λ≤λdu,a+1−λdu,b.
The spaceX having a convex structure ω is said to be convex metric space. A non-empty subset K of X is called convex if ωa,b;λ∈K whenever a,b,λ∈K×K×I.Remark 5 (see [10–12]).
The convex metric spaceX has the properties;
(i)
ωa,b;1=a, ωa,b;0=b, ωa,a;λ=a(ii)
Open spheresBa,r=b∈X:da,b<r and closed spheres Ba,r=b∈X:da,b≤r are convex(iii)
IfKα:α∈A is a family of convex subsets of X, then ∩Kαα∈A is convexAny normed space and a convex subset of a normed space are a convex metric space. There are several examples in the existing literature [4, 9, 10, 12] of convex metric spaces which are not embedded in any normed space.Definition 6 (see [13]).
A convex metric spaceX is said to have property L if for all a,b in X and λ,μ,ν in I, we have
(i)
wa,b;λ=ωb,a;1−λ(ii)
ωωa,b;λ,ωa,b;μ;ν=ωa,b;λν+μ1−νEach normed space has propertyL, if we define ωa,b;t=ta+1−tb. In Definition 6, taking μ=0 and using Remark 2, we obtain
(2)ωωa,b;λ,b;ν=ωa,b;λν.Definition 7 (see [13]).
An ordered relation⪯ on a convex metric space X is said to be continuous if the sets λ:ωa,b;λ⪯c and λ:ωa,b;λ⪰c are closed for a,b,c in X.Definition 8 (see [13]).
A metric spaceX,d with convex structure ω and continuous order relation ⪯ is called ordered convex metric space and denoted by X,d,ω,⪯.ForA⊂X, we denote by co¯A the closed convex hull of A and by diamA the diameter of A.


## 3. (o)-Uniformly Convex Ordered Convex Metric Spaces

In this section, we give the concept of order uniformly convex ordered convex metric spaces and study their properties.An ordered interval in an ordered convex metric spaceX,d,ω,⪯ is any of the following subsets
(3)p,⟶=a∈X:p≤a,⟵,q=a∈X:a≤q,p,q=p,⟶∩⟵,q,for anyp,q in X. It can be easily shown that ordered intervals p,⟶ and ⟵,q are convex and interval p,q is bounded closed and convex. We denote the ordered interval with center at a and radius r in p,q by Ira=b:b∈p,q∩Ba,r for any a in p,q.Definition 9.
An ordered convex metric spaceX,d,ω,⪯ is called ordered uniformly convex (in short (o)-UC) if for any a,b,c in X and ε>0, there exists δ=δε such that, for all r>0 and comparable a,b∈c,⟶ or a,b∈⟵,c with dc,a≤r,dc,b≤r and da,b≥rε,(4)dc,ωa,b;12≤r1−δ<r.Example 1.
LetX be the family of closed intervals αi,βi such that 0≤αi≤βi≤1. Define the Hausdorff metric d on X, i.e., for Ii=αi,βi and Ij=αj,βj(5)dIi,Ij=maxsupζ∈Iidζ,Ij,supη∈Ijdη,Iiwheredζ,Ij=infη∈Ijdζ,η.Now the convex structureω:X×X×I⟶X is defined by ωIi,Ij;λ=λαi+1−λαj,λβi+1−λβj for λ∈I=0,1. Order X by inclusion, i.e., Ii°Ij⇔Ii⊂Ij.NowX,d,ω,⪯ is an (o)-UC.Example 2.
LetX=ℝ×ℝ be a set, a=α,β and b=ζ,η be two points in X. Set X is ordered by a⪯b⇔α≤ζ and β≤η. Next, consider the metric d:X×X⟶ℝ given by
(6)da,b=α−ζ2+β−η22ifaandbarecomparable.maxα−ζ,b−ηifa⋈b,for any a,b in X. Convex structure ω on X is given by ωa,b;λ=dλα+1−λζ,λβ+1−λη. The ordered convex metric space X,d,ω,⪯ is an (o)-UC.Example 3.
Ordered uniformly convex normed spaces are ordered uniformly convex ordered convex metric spaces.Remark 10.
In Goebel, Sekowski and Stachura [18, Theorem 1] δ is continuous function, but we only assume the existence of a positive number δ such that δ is a function of ε. In Alfuraidan and Khamsi [16, Definition 1] δ depends on r and ε, but our δ only depends on ε.Lemma 11.
LetX,d,ω,⪯ be a complete and (o)-UC space with property L. Then, for any ε>0,r>0andt∈u,v where 0<u≤v<1, we have
(7)dc,ωa,b;t≤r1−δ2minu,1−vε,for all a,b,c∈X, for which a,b∈c,⟶ or a,b∈⟵,c and
(8)dc,a≤r,dc,b≤randda,b≥rε.Proof.
Lett∈I. We can assume t<1/2, with out any loss of generality. Then, mint,1−t=t. Fix c in X and let a,b∈c,⟶ or a,b∈⟵,c be selected so that
(9)dc,a≤r,dc,b≤randda,b≥rε,for ε>0 and r>0. Set a¯=ωa,b;2t. Then, using Definition 6ωa,b;t=ωa¯,b;1/2 and
(10)da¯,b=2tda,b≥2trε≥r2minu,1−vε,hold. Since order intervals are convex, the (o)-UC implies
(11)dc,ωa¯,b;12≤r1−δ2minu,1−vε.Theorem 12.
LetX,d,ω,⪯ be a complete and (o)-UC space with property L. Then, every decreasing sequence pn,qn of non-empty intervals of X has non-empty intersection.Proof.
Letpn,qn be a decreasing sequence of non-empty intervals of X. If diampn,qn>0 for every n∈ℕ, then there exist a,b in pn,qn such that da,b≥diampn,qn/2. Since dc,a≤diampn,qn,dc,b≤diampn,qn for all c∈pn,qn and the space is (o)-UC, there exists δ>0 such that
(12)dc,ωa,b;12≤diampn,qn1−δ<diampn,qn.
This implies that there existsln1∈pn,qn such that
(13)dc,ln1≤diampn,qn1−δ,
for allc in pn,qn. Let
(14)Ln1=ln1,ln+11.ln+21,ln+31,ln+41,...
ClearlyLn1≠∅ and Ln1⊃Ln+11 for every n. Suppose diamLn1>0 for every n. Then, there exist a,b∈Ln1 such that da,b≥diamLn1/2. Choose
(15)Bn1=⋂k=0∞IdiamLn1ln+k1.
Then,Bn1⊃co_Ln1 and dc,a≤diamLn1,dc,b≤diamLn1 for every c in co_Ln1. As X is (o)-UC, there exists ln2∈co_Ln1⊂pn,qn such that
(16)dc,ln2≤diampn,qn1−δ2,for all c in co_Ln1. Similarly, we obtain co_Ln2,co_Ln3,⋯ and ln3,ln4,⋯. It is obvious that
(17)pn,qn⊃coLn1⊃coLn2⊃⋯and=diamcoLnm⟶0asm⟶∞.
SinceX is complete, therefore there exists ln in X such that
(18)⋂m=1∞co_Lnm=lnforeveryn.
From(19)⋂m=1∞co_Lnm⊃⋂m=1∞co_Ln+1m,we have l1=l2=l3=⋯. Thus, there exists l with l∈pn,qn for all n. Hence,
(20)⋂n=1∞pn,qn≠∅Definition 13.
An (o)-UC spaceX is said to be (o)-2UC if there exists a constant β>0 such that δε≥βε2.Example 4.
Consider the setXofallrealsequencesξi satisfying, Σ∞i=1ξi2<∞. Letx=ξi and y=ηi be two points in X.Theorder⪯isdefinedbyx⪯yifandonlyifξi≤ηiforalli, and metric d on X is given by
(21)dx,y=Σ∞i=1ξi−ηi21/2ifxandyarecomparablemaxξi−ηiifx⋈y.Convexity structureω on the space X is defined by ωx,y;α=αξi+1−αηi. The space X,d,ω,⪯ is a (o)-2UC.Definition 14.
LetC be a non-empty subset of a complete metric space X,d. A function f:C⟶0,∞ is called a t-function if there exists a bounded sequence an in X such that
(22)fa=limsupdan,an⟶∞.Theorem 15.
Letan be a bounded monotone sequence in a non-empty closed and convex subset C of a complete and (o)-2UC space X,d,ω,⪯ with property L. If
(23)C∗=∩n∈ℕan,⟶∩C=supn∈ℕan,⟶∩Cifaniso−increasing∩n∈ℕ⟵,an∩C=⟵,infn∈ℕan∩Cifaniso−decreasing,then the t-function f:C⟶0,∞, defined by an, has a unique minimum point in C∗.Proof.
Assume thatan is a monotone (o)-increasing sequence and ζn is a sequence of positive real numbers which converges to 0. Then, there exists cn in C∗ such that
(24)fcn≤f0+ζn=inffa:a∈C∗+ζn=inflimsupn⟶∞dan,a:a∈C∗+ζn,
forn∈ℕ. If f0=0, then cn is a Cauchy sequence. The limit of cn is the desired unique minimum point. Otherwise, assume f0>0. Now to show that cn is a Cauchy sequence. We assume otherwise and prove by contradiction. Let cn not be a Cauchy sequence. Then, there exists a subsequence cσn and ε0>0 such that dcσn,cσm≥ε0, for any n≠m∈ℕ. Choose a fix N∈ℕ such that 2ζσn<f0, for every n≥N. Now for n,m≥N, and n≠m, we obtain by using the definition of the t-function f. that there exists k0∈ℕ such that
(25)supi≥k0dai,cσn≤f0+2ζσn≤2f0,supi≥k0dai,cσm≤f0+2ζσm≤2f0.
The (o)-2UC implies the existence of aβ>0 such that δε0≥βε02>0. As
(26)dcσn,cσm≥ε0≥f0+2maxζσn,ζσmε02f0,f0+2maxζσn,ζσm≥f0.
Therefore, we have(27)dai,ωcσn,cσm;12≤f0+2maxcσn,cσm1−βε02,for any i≥k0 because cn are ⪯-bigger than an, by definition of C∗. Thus, the above inequality further implies
(28)fωcσn,cσm;12≤f0+2maxcσn,cσm1−βε02.
Hence,(29)f0≤f0+2maxcσn,cσm1−βε02.
Lettingn,m⟶∞, we obtain
(30)f0≤f01−βε02,thus a contradiction to f0>0 and βε02>0. This proves that cn is Cauchy. Since f. is continuous, we have flimn⟶∞cn=limn⟶∞fcn=f0, i.e., limn⟶∞cn is a minimum point of f. in C∗. It further implies that there is a unique minimum point in C∗.Lemma 16.
LetX,d,ω,⪯ be a complete and (o)-2UC space with property L. Let r>0andtn∈u,v where 0<u≤v<1.Letan and bnbetwosequencesinc,⟶or⟵,c for some c∈X. Assume
(31)limsupn⟶∞dc,an≤r,limsupn⟶∞dc,bn≤randlimn⟶∞dc,wan,bn;tn=r.
Then, we have(32)limn⟶∞dan,bn=0.Proof.
Choose ac∈X and two sequences an and bn in c,⟶ or ⟵,c. Assume that
(33)limsupn⟶∞dc,an≤r,limsupn⟶∞dc,bn≤randlimn⟶∞dc,wan.bn;tn=r,for some r≥0 and tn∈u,v where 0<u≤v<1. We prove that
(34)limn⟶∞dan,bn=0.
In caser=0, it is. Therefore, we assume r>0. Since
(35)dc,ωan,bn;tn≤maxdc,an,dc,bn.
It is obvious in caser=0, (32) is obvious. Next assume that r>0. As
(36)dc,ωan,bn;tn≤maxdc,an,dc,bn,for any n∈ℕ, we have
(37)r=limn⟶∞dc,ωan,bn;tn≤liminfmaxdc,an,dc,bnn⟶∞≤limsupn⟶∞maxdc,an,dc,bn≤maxlimsupn⟶∞dc,an,limsupn⟶∞dc,bn≤r.
Therefore,limn⟶∞rn=r where rn=maxdc,an,dc,bn, for any n∈ℕ. Assume that limn⟶∞dan,bn≠0. Then, there exist ε0>0 and subsequences aσn and bσn such that daσn,bσn≥ε0, for any n∈ℕ. Since limn⟶∞rn=r, there exists n0∈ℕ such thatr/2≤rσn≤3r/2, for any n≥n0. Hence,
(38)daσn,bσn≥ε0=rσnε0rσn≥rσn2ε03r,holds for any n≥n0. Because X is (o)-2UC, there exists ζ>0 such that
(39)δ2minu,1−v2ε03r≥ζ,for any n≥n0. Using Lemma 11, we deduce that
(40)dc,ωaσn,bσn;tσn≤rσn1−ζ,for any n≥n0. If we let n⟶+∞, we get r≤r1−ζ≤r, which is the desired contradiction. Therefore, we have limn⟶∞dan,bn=0.Theorem 17.
LetC be a non-empty closed convex subset of a complete and (o)-2UC space X,d,ω,⪯ with property L.LetainX be such that C⊂a,⟶ or C⊂⟵,a. Then, there exists a unique c∈C such that
(41)da,C=infda,b:b∈C=da,c.Proof.
Letda,C=r. Since C is closed, if r=0, then a=c∈C. Assume r>0. Let cn⊂C be such that limn⟶∞da,cn=r, then cn is a Cauchy sequence. To prove this assume otherwise that cn is not a Cauchy sequence. It implies that there exist ε0>0 and a subsequence cσn such that dcσn,cσn+1≥ε0, for any n∈ℕ. Since C is convex, we have
(42)r≤limsupn⟶∞da,ωcσn,cσn+1;12≤12limsupn⟶∞da,cσn+12limsupn⟶∞da,cσn+1=r.
Now using Lemma16 and the assumption C⊂a,⟶ or C⊂⟵,a, we obtain
(43)limn⟶∞dcσn,cσn+1=0.
It contradicts the fact thatdcσn,cσn+1≥ε0 for n∈ℕ. Hence, cn is Cauchy. Thus, limn⟶∞cn=c exists. As C is closed c∈C. Uniqueness is obvious.


## 4. Application

Ran and Reurings [19] gave a generalization of the Banach’s contraction theorem to monotone mappings (see also [20–22]). In this section, we show that new notion of (o)-2UC is related to the fixed point problem of monotone non-expansive mappings on (o)-2UC spaces.Definition 18.
LetX,d,ω,⪯ be an ordered convex metric space. Let ∅≠C⊂X. A map T:C⟶C is said to be monotone, ifTa⪯Tbwhenevera⪯b.Definition 19.
LetX,d,ω,,⪯ be an ordered convex metric space. Let ∅≠C⊂X. AmapT:C⟶C is said to be monotone K-Lipschitzian, with K≥0, if T is monotone and dTa,Tb≤Kda,b whenever a°b.IfK=1, then T is said to be a monotone non-expansive mapping. A point a∈C is said to be a fixed point of T if Ta=a. We call a sequence an⊂C an approximating sequence for T if limn⟶∞dan,Tan=0 (for details about fixed point, we refer to [23]).Theorem 20.
LetC be a non-empty bounded closed and convex subset of a complete and (o)-2UC space X,d,ω,⪯ with property L. Let T:C⟶C be a monotone non-expansive mapping such that there exists a0∈Cwhich satisfiesa0⪯Ta0 (resp. a0±Ta0). Then, ThasafixedpointainC such that a0⪯aresp.a0±a.Proof.
Assume thata0⪯Ta0. Then, the orbit Tna0 is monotone (o)-increasing. As C is bounded, Theorem 15 implies that the t-function f:C∗⟶C∗ given by Tna0, where C∗=∩n∈ℕTna0,⟶∩C has a unique minimum point a∈C∗. Since T is monotone and a0⪯Ta0, thus T:C∗⟶C∗. Now a is a fixed point of T. To prove this, note that since Ta0⪯a, for any n∈ℕ, and T is monotone non-expansive, therefore dTn+1a0,Ta≤dTna0,a, for each n∈ℕ. Hence. fTa≤fa. It also follows that Ta is a minimum point of T in C∗.
The uniqueness of the minimum point impliesTa=a.LetC be a non-empty convex subset of a complete ordered convex metric space X,d,ω,⪯. Choose any a0∈C. A Mann type iteration process of a map T:C⟶C is the sequence an defined by
(44)an+1=ωTan,an;tn,for eachn∈ℕ, where tn⊂0,1. Next, assume that T is monotone non-expansive and p is a fixed point of T comparable to a0. The fact that order intervals are convex implies that an is a sequence whose terms belong to p,⟶ or ⟵,p. Therefore,
(45)dan+1,p≤tndTan,p+1−tndan,p=tndTan,Tp+1−tndan,p≤tndan,p+1−tndan,p=dan,p,for any n∈ℕ,which implies that dan,p is decreasing. Hence, dan,p has a limit. Set limn⟶∞dan,p=r. If r=0, then dan,p converges to a fixed point of T. If r>0, we will show that an is an approximate fixed point sequence of T, i.e.,
(46)limn⟶∞dan,Tan=0.Theorem 21.
LetC be a non-empty bounded closed and convex subset of a complete and (o)-2UC space X,d,ω,⪯ with property L. Let T:C⟶C be a monotone non-expansive mapping such that there exists a0∈C that is comparable to Ta0. Denote the Mann type iterative sequence generated by a0 by an, with tn∈u,v for some u>0 and v<1. Then, an is an approximating sequence for T.Proof.
Theorem20 implies the existence of a fixed point e of T comparable to a0. Assume =limn⟶∞dan,e>0. Suppose that limn⟶∞dan,Tan≠0. Now limn⟶∞dTan,e=r. To prove this considers a non-trivial ultra-filter ℧ over ℕ. Then, lim℧tn=t∈u,v. Therefore,
(47)r=lim℧dan+1,e≤tlim℧dTan,e+1−tr.
It follows fromt≠0 that lim℧dTan,e>r. Thus, we obtain
(48)lim℧dTan,e≤lim℧dan,e=r.
Therefore,lim℧dTan,e=r. Since ℧ was arbitrary, we have limn⟶∞dTan,e=r. Hence, we obtain
(49)limn⟶∞dan,e=limn⟶∞dTan,e=limn⟶∞dωan,Tan;tn,e=r,
withtn∈u,v for any n∈ℕ. Lemma 16 further implies that
(50)limn⟶∞dan,Tan=0.In [16, 22], similar results are derived for monotone non-expansive mappings on Banach spaces.


## 5. Conclusion

In this study, we introduced ordered uniform convexity in ordered convex metric spaces due to Beg [13] and investigated several of its interesting properties including ordered 2-uniform convexity. As applications, we proved some fixed point theorems. This is the first paper on (o)-UC and (o)-2UC metric spaces. It is this author’s expectation that in the future this will open a new area of research and these results will attract researchers for further exploration. In addition to study further results in metric spaces, there is possibility to study these in the setting of partial metric spaces, b-metric spaces, dislocated metric spaces, CAT (0) spaces, etc.



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