The Monotone Contraction Mapping Theorem

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.

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:where is a nonlinear operator on any set into itself. The solutions to (1) are called fixed points of . The fixed-point theory is concerned with finding conditions on the structure that the set must be endowed as well as on the properties of the operator , 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 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 are generally metrical or compactness type of conditions.

Given a complete metric space , 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:for all , where is a real number, usually referred to as the Lipschitz constant of . The metric inequality (2) can be classified into three categories, contraction mappings for the case where , nonexpansive mappings for the case where , and expansive mappings for the case where . The most important property of (2) is that they are uniformly continuous. Thus, for any sequence converging to in , there is as . 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). Let be a complete metric space and be a given contraction. Then, has a unique fixed point , and

Fixed-point problems of contraction mappings always exist, and it is unique due to Theorem 1. 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 Theorem 1 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 be a Banach space with the norm and let be the dual space of . Denote as the duality product. The normalised duality mapping from to is defined byfor all . Hahn–Banach theorem guarantees that for every . For the purposes in this work, the interest mostly lies on the case when is single valued for all , which is equivalent to the statement that is a smooth Banach space. The normalised duality map of a Banach space is sequentially weakly continuous if a sequence in is weakly convergent to , and then the sequence in is weak star convergent to . That is, given that , then .

Remark 1. By virtue of the Riesz representation theorem, it follows that ( is the identity map) when it is in a Hilbert space.
Throughout this paper, denotes the real part of a complex number. Also, is used to denote the set of fixed points of the mapping (that is, ).

Definition 2. (generalised projection functional, see Alber [7]). Let be a smooth Banach space and let be the dual space of . The generalised projection functional is defined byfor all , where is the normalised duality mapping from to . It is obvious from the definition that the generalised projection functional satisfies the following inequality:for all .
The definition for the mapping discussed in this paper is introduced in the following.

Definition 3. (monotone contraction mapping). Let be a smooth Banach space and let be a closed subset of . Then, the mapping is said to be a monotone contraction mapping if there exists such that for all , the following two conditions are satisfied:(1),(2),where is the normalised duality mapping and for all with .
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 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 be a normed linear space. Then, for any Thus, . Moreover, ifthen and ; in particular, when is smooth (resp., strictly convex) then equality occurs if and only if (resp., x = y).

Lemma 1. (see, for instance, Ezearn [8]). Let be a uniformly convex smooth Banach space. Suppose such thatThen, .

Lemma 2. (see Kamimura and Takahashi [9]). Let be a uniformly convex and smooth Banach space and let and be two sequences in such that either or is bounded. If , then .

3. Main Result

The proof of the main result of this paper is given in this section, which is accomplished in Theorem 2. The following lemma and proposition shall aid in arriving at the conclusion of the main result.

Lemma 3. Let be a closed subset of a uniformly convex smooth Banach space and let be a monotone contraction mapping. Then, is (point-wise) continuous on .

Proof. Suppose as . Then, by Definition 3, the following is obtained:Since as , (10) reduces towhich by Proposition 1, implies thatFrom (12), put . Then, . By Lemma 1, as which completes the proof.

Proposition 2. Let be a closed subset of a uniformly convex smooth Banach space and let be a monotone contraction mapping. Then, has at most one fixed point.

Proof. Suppose that with . Then by Definition 3,which reduces toSince , then , and it follows from (14) thatwhich by Proposition 1, implies thatSince is a strictly convex space, by Proposition 1, it is obtained that which completes the proof.

The main result of this paper is stated and proved in the following.

Theorem 2. (monotone contraction mapping theorem). Let be a closed subset of a uniformly convex smooth Banach space and let be a monotone contraction mapping. Then, has a unique fixed point, that is, , and that the Picard iteration associated to , that is, the sequence defined by for all converges to for any initial guess .

Proof. To prove existence of a fixed point, it is shown that for any given , the Picard iteration is a Cauchy sequence. For , the following evaluation is obtained:By Definition 3 (second part), it is obvious that and as a result,Applying Definition 3 (first part) several times, (18) reduces toSince , as , thenSince , by (20), then .
By Definition 2, it is obvious thatSince as , then for any , there exists a natural number such that for all ,which implies that the sequence is bounded.
Now, since either or is bounded and the fact that , then by Lemma 2,This implies that is a Cauchy sequence. Since is complete, there exists such that as . By Lemma 3, is a continuous self-map, and the following is obtained:Hence, is a fixed point of . By Proposition 2, has at most one fixed point, and it is deduced that for every choice of , the Picard iteration converges to the same value , that is, the unique fixed point of which completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The author thanks the colleagues for their proof reading and other helpful suggestions.