# The Monotone Contraction Mapping Theorem

Authors: Joseph Frank Gordon
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, the fixed-point theorem for monotone contraction mappings in the setting of a uniformly convex smooth Banach space is studied. This paper provides a version of the Banach fixed-point theorem in a complete metric space.


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## Body

## 1. Introduction

Many problems arising in different areas of mathematics, such as optimization, variational analysis, and differential equations, can be modeled by the following equation:(1)Tx=x,where T is a nonlinear operator on any set X into itself. The solutions to (1) are called fixed points of T. The fixed-point theory is concerned with finding conditions on the structure that the set X must be endowed as well as on the properties of the operator T:X⟶X, in order to obtain results on the existence (and uniqueness) of fixed points, the data dependence of fixed points, and the construction of fixed points. The set or the ambient space X involved in fixed-point theorems covers a variety of spaces such as lattice, metric space, normed linear space, generalised metric space, uniform space, and linear topological space while the conditions imposed on the operator T are generally metrical or compactness type of conditions.Given a complete metric spaceX,d, the most well-studied types of self-maps are referred to as Lipschitz mappings (or Lipschitz maps, for short), which are given by the metric inequality:(2)dTx,Ty≤kdx,y,for all x,y∈X, where k>0 is a real number, usually referred to as the Lipschitz constant of T. The metric inequality (2) can be classified into three categories, contraction mappings for the case where k<1, nonexpansive mappings for the case where k=1, and expansive mappings for the case where k>1. The most important property of (2) is that they are uniformly continuous. Thus, for any sequence xnn≥1 converging to x in X, there is dTxn,Tx=0 as n⟶∞. The following theorem due to Banach and Steinhaus [1] is the first and simplest of the metric fixed-point theory of Lipschitz maps.Theorem 1.
(contraction mapping theorem). LetX,d be a complete metric space and T:X⟶X be a given contraction. Then, T has a unique fixed point p, and(3)Tnx⟶pasn⟶∞,for eachx∈X.Fixed-point problems of contraction mappings always exist, and it is unique due to Theorem1. This is a very useful result, and it has been applied in the determination of the existence and uniqueness of many results in analysis (both pure and applied) and economics (see, for instance, Border [2], Freeman [3], Picard [4], and Lindelöf [5]).In this paper, a version of Theorem1 in the setting of a smooth Banach spaces for monotone contraction mappings is provided. In other words, the fixed-point theorem for monotone contraction mappings in a uniformly convex smooth Banach space is proved.Definition 1.
(normalised duality mapping, see Lumer [6]). Let X be a Banach space with the norm ⋅ and let X∗ be the dual space of X. Denote ⋅,⋅ as the duality product. The normalised duality mapping J from X to X∗ is defined by(4)Jx:=f∈X∗:f∗2=x2=x,f=fx,for all x∈X. Hahn–Banach theorem guarantees that Jx≠∅ for every x∈X. For the purposes in this work, the interest mostly lies on the case when Jx is single valued for all x∈X, which is equivalent to the statement that X is a smooth Banach space. The normalised duality map J of a Banach space X is sequentially weakly continuous if a sequence xnn≥1 in X is weakly convergent to x, and then the sequence Jxnn≥1 in X∗ is weak star convergent to Jx. That is, given that xn⇀x∈X, then Jxnn≥1⇀∗Jx∈X∗.Remark 1.
By virtue of the Riesz representation theorem, it follows thatJx=x (J is the identity map) when it is in a Hilbert space.
Throughout this paper,ℜ denotes the real part of a complex number. Also, FT is used to denote the set of fixed points of the mapping T (that is, FT=x∈C:Tx=x).Definition 2.
(generalised projection functional, see Alber [7]). Let X be a smooth Banach space and let X∗ be the dual space of X. The generalised projection functional ϕ⋅,⋅:X×X⟶ℝ is defined by(5)ϕy,x=y2−2ℜy,Jx+x2,for all x,y∈X, where J is the normalised duality mapping from X to X∗. It is obvious from the definition that the generalised projection functional ϕ⋅,⋅ satisfies the following inequality:(6)y−x2≤ϕy,x≤y+x2,for all x,y∈X.
The definition for the mapping discussed in this paper is introduced in the following.Definition 3.
(monotone contraction mapping). LetX be a smooth Banach space and let C be a closed subset of X. Then, the mapping T:C⟶C is said to be a monotone contraction mapping if there exists 0≤c<1 such that for all x,y∈C, the following two conditions are satisfied:(1)
ℜTx−Ty,JTx−JTy≤cℜx−y,Jx−Jy,(2)
ℜTm+1x−Tmy,JTn+1x−JTny≤0,where J is the normalised duality mapping and for all m,n≥0 with m≠n.
It should be noted here that, monotone contraction mappings reduce to the contraction type of mappings in (2) when in Hilbert spaces because in Hilbert spaces J is the identity mapping.


## 2. Preliminaries

The following proposition and lemmas are introduced that will be used in the proof of the main result. As before, all notations employed remain as defined.Proposition 1.
(see, for instance, Ezearn [8]). Let X be a normed linear space. Then, for any jx∈Jx,jy∈Jy(7)x−y2≤ℜx−y,jx−jy≤x−yx+y.
Thus,ℜx−y,jx−jy≥0. Moreover, if(8)ℜx−y,jx−jy=0,then jx∈Jy and jy∈Jx; in particular, when X is smooth (resp., strictly convex) then equality occurs if and only if jx=jy (resp., x = y).Lemma 1.
(see, for instance, Ezearn [8]). Let X be a uniformly convex smooth Banach space. Suppose unn≥1,vnn≥1⊂X such that(9)limn⟶∞vn=v∈X,limn⟶∞ℜun−vn,Jun−Jvn=0.
Then,limn⟶∞un=v.Lemma 2.
(see Kamimura and Takahashi [9]). Let X be a uniformly convex and smooth Banach space and let xn and yn be two sequences in X such that either xn or yn is bounded. If limn⟶∞ϕxn,yn=0, then limn⟶∞xn−yn=0.


## 3. Main Result

The proof of the main result of this paper is given in this section, which is accomplished in Theorem2. The following lemma and proposition shall aid in arriving at the conclusion of the main result.Lemma 3.
LetC be a closed subset of a uniformly convex smooth Banach space X and let T:C⟶C be a monotone contraction mapping. Then, T is (point-wise) continuous on C.Proof.
Supposexn⟶x∈C as n⟶∞. Then, by Definition 3, the following is obtained:(10)ℜTxn−Tx,JTxn−JTx≤cℜxn−x,Jxn−Jx.
Sincexn⟶x as n⟶∞, (10) reduces to(11)limn⟶∞ℜTxn−Tx,JTxn−JTx≤0,which by Proposition 1, implies that(12)limn⟶∞ℜTxn−Tx,JTxn−JTx=0.
From (12), put vn=Tx. Then, limn⟶∞vn=Tx. By Lemma 1, un=Txn⟶Tx as n⟶∞ which completes the proof.Proposition 2.
LetC be a closed subset of a uniformly convex smooth Banach space X and let T:C⟶C be a monotone contraction mapping. Then, T has at most one fixed point.Proof.
Suppose thatu,v∈FT with u≠v. Then by Definition 3,(13)ℜTu−Tv,JTu−JTv≤cℜu−v,Ju−Jv,ℜu−v,Ju−Jv≤cℜu−v,Ju−Jv,which reduces to(14)1−cℜu−v,Ju−Jv≤0.
Since0≤c<1, then 1−c>0, and it follows from (14) that(15)ℜu−v,Ju−Jv≤0,which by Proposition 1, implies that(16)ℜu−v,Ju−Jv=0.
SinceX is a strictly convex space, by Proposition 1, it is obtained that u=v which completes the proof.The main result of this paper is stated and proved in the following.Theorem 2.
(monotone contraction mapping theorem). LetC be a closed subset of a uniformly convex smooth Banach space X and let T:C⟶C be a monotone contraction mapping. Then, T has a unique fixed point, that is, FT=p, and that the Picard iteration associated to T, that is, the sequence defined by xn=Txn−1=Tnx0 for all n≥1 converges to p for any initial guess x0∈X.Proof.
To prove existence of a fixed point, it is shown that for any givenx0∈X, the Picard iteration xnn≥0 is a Cauchy sequence. For n>m, the following evaluation is obtained:(17)ℜxn−xm,Jxn−Jxm=ℜxn−xm,Jxn−Jxn−1+Jxn−1−Jxn−2+Jxn−2−Jxm+1+Jxm+1−Jxm=∑k=mn−1ℜxn−xm,Jxk+1−Jxk=∑k=mn−1ℜxn−xn−1+xn−1−xn−2+xn−2−xm+1+xm+1−xm,Jxk+1−Jxk=∑k=mn−1∑l=mn−1k≠lℜxl+1−xl,Jxk+1−Jxk=∑k=mn−1ℜxk+1−xk,Jxk+1−Jxk+∑k=mn−1∑l=mn−1k≠lℜxl+1−xl,Jxk+1−Jxk.
By Definition3 (second part), it is obvious that ∑k=mn−1∑l=mn−1k≠lℜxl+1−xl,Jxk+1−Jxk≤0 and as a result,(18)ℜxn−xm,Jxn−Jxm≤∑k=mn−1ℜxk+1−xk,Jxk+1−Jxk.
Applying Definition3 (first part) several times, (18) reduces to(19)ℜxn−xm,Jxn−Jxm≤cm+cm+1+⋯+cn−1ℜx1−x0,Jx1−Jx0,=cm1+c+c2+⋯+cn−m−1ℜx1−x0,Jx1−Jx0,=cm−cn1−cℜx1−x0,Jx1−Jx0.
Since0≤c<1, as m,n⟶∞, then(20)limn,m⟶∞ℜxn−xm,Jxn−Jxm=0.
Since2ℜxn−xm,Jxn−Jxm=ϕxn,xm+ϕxm,xn, by (20), then limn,m⟶∞ϕxn,xm=limn,m⟶∞ϕxm,xn=0.
By Definition2, it is obvious that(21)xn−xm2≤ϕxn,xm.
Sinceϕxn,xm⟶0 as n,m⟶∞, then for any ϵ>0, there exists a natural number Nϵ such that for all m,n≥Nϵ,(22)xn−xm<ϵ,which implies that the sequence xnn≥0 is bounded.
Now, since eitherxnn≥0 or xmm≥0 is bounded and the fact that limn,m⟶∞ϕxn,xm=limn,m⟶∞ϕxm,xn=0, then by Lemma 2,(23)limn,m⟶∞xn−xm=0.
This implies thatxnn≥0 is a Cauchy sequence. Since X is complete, there exists p∈X such that xn⟶p as n⟶∞. By Lemma 3, T is a continuous self-map, and the following is obtained:(24)p=limn⟶∞xn+1=limn⟶∞Txn=Tlimn⟶∞xn=Tp.
Hence,p is a fixed point of T. By Proposition 2, T has at most one fixed point, and it is deduced that for every choice of x0∈X, the Picard iteration converges to the same value p, that is, the unique fixed point of T which completes the proof.



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*Source: 2879283-2020-05-01.xml*