International Journal of Analysis and Applications

Volume 19, Number 2 (2021), 288-295

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-288

FIXED POINT THEOREM FOR MONOTONE NON-EXPANSIVE MAPPINGS

JOSEPH FRANK GORDON∗

Department of Mathematics,

Nanjing University of Aeronautics and Astronautics,

Nanjing 211106, China

∗Corresponding author: jgordon@aims.edu.gh

Abstract. In this paper, we study the fixed point theorem for monotone nonexpansive mappings in the

setting of a uniformly smooth and uniformly convex smooth Banach space.

1. Introduction

Given a complete metric space (X ,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

d(Tx,Ty) ≤ kd(x,y),(1.1)

for all x,y ∈X , where k > 0 is a real number, usually referred to as the Lipschitz constant of T . The metric

inequality (1.1) can be classified into three categories, thus contraction mappings for the case where k < 1,

non-expansive mappings for the case where k = 1 and expansive mappings for the case where k > 1. The

most important property of (1.1) is that they are uniformly continuous. Thus, for any sequence {xn}n≥1

converging to x in X , we have d(Txn,Tx) = 0 as n → ∞. It is well known that when X is complete

and T is a contraction mapping, then T has unique fixed point and the sequence of Picard iteration Tn(x)

converges to the fixed point of T as n → ∞. Fixed points problems of contraction mappings always exist

Received January 28th, 2021; accepted February 22nd, 2021; published March 17th, 2021.

2010 Mathematics Subject Classification. 47H10, 54H25.

Key words and phrases. monotone nonexpansive mappings; normalised duality mappings; uniformly convex spaces; uni-

formly smooth spaces; fixed points.

©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

288

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-288


Int. J. Anal. Appl. 19 (2) (2021) 289

and it’s unique due to Banach [1]. Edelstein [2] also showed when T is a contractive mapping (that is,

d(Tx,Ty) < d(x,y)) on a compact metric space X , then T has a unique fixed point and the fixed point can

be iteratively approximated by the Picard iteration xn+1 = Txn.

In metric spaces, the only non-trivial thing one can say about nonexpansive mappings is that the Picard

iteration is a bounded sequence, which is as a result of the inequality

d(Tnx,Tny) ≤ d(x,y),∀n ≥ 0, x,y ∈X ,

where T is a nonexpansive mapping. Even in compact metric spaces, with the exception of the contrac-

tive mappings described above, generally one cannot find a fixed point (if it exists) by the Picard iter-

ation. It is therefore imperative that one considers the more specialised complete metric spaces: that

is, the Banach spaces, where linearity and homogeneity affords more structure to the nonexpansive map-

pings and their fixed points. We use Fix(T) to denote the set of fixed points of the mapping T (that is,

Fix(T) = {x ∈C : Tx = x}).

An approximate fixed point sequence of a nonexpansive self-map T on a closed convex subset C of Banach

space X is any sequence {xn}n≥1 ⊂C such that

lim
n→∞

‖xn −Txn‖ = 0.

When C is bounded or Fix(T) 6= ∅, then such a sequence always exists. One of the ways to construct an

approximate fixed point sequence for nonexpansive mappings is to use the Banach contraction mapping

theorem [1] to obtain a sequence {xn} in C such that

xn = αnx0 + (1 −αn)Txn, n ≥ 1

where the initial guess x0 is taken arbitrarily in C and {αn} is a sequence in the interval (0,1) such that

αn → 0 as n →∞. By assuming that Fix(T) 6= ∅, this sequence {xn} is bounded (indeed, ‖xn−p‖≤‖x0−p‖

for all p ∈ Fix(T)). Hence

‖xn −Txn‖ = αn‖x0 −Txn‖→ 0,

and {xn} is an approximate fixed point sequence for T. The immediate conclusion from the above deduction

is the following result on compact star-convex sets.

Theorem 1.1. Let T be a nonexpansive self-mapping on a compact star-convex subset of a Banach space.

Then T has a fixed point.



Int. J. Anal. Appl. 19 (2) (2021) 290

Theorem 1.1 is proved by means of the Banach contraction mapping theorem [1], and it is in this spirit that

we employ the Monotone contraction mapping theorem [8] to prove the following weaker but generalized

version of Theorem 1.1.

Theorem 1.2 (Main theorem). Let X be a uniformly smooth, uniformly convex smooth Banach space with

a sequentially weakly continuous normalized duality mapping, C ⊂X be a weakly-compact star-domain such

that 0 ∈ kerC. Then every monotone nonexpansive mapping, T : C →C has a fixed point.

It is not clear to the author if the ‘sequentially weakly continuity’ condition can be removed, which would

be desirable; however, all attempts to do so presently has not been successful and we hope that we may be

able to remove it in subsequent work. Throughout this paper, < denotes the real part of a complex number.

We also use ker(C) to denote the kernel of a star convex subset C (equivalently, star-domain) of a normed

linear space, that is {x ∈C : ax + (1 −a)y ∈C,∀∈ [0, 1],y ∈C}.

Definition 1.1 (Normalised Duality Mapping, see Lunner [7],1961). 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

Jx := {f ∈X∗ : ‖f‖2∗ = ‖x‖
2 = 〈x,f〉 = fx},

for all x ∈ X . Hahn Banach theorem guarantees that Jx 6= ∅ for every x ∈ X . For our purposes in this

work, our 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. We say that the normalized duality map J of a Banach space

X is sequentially weakly continuous if a sequence {xn}n≥1 in X is weakly convergent to x, then the sequence

{Jxn}n≥1 in X∗ is weak-star convergent to Jx. That is, given that xn ⇀ x ∈X , then {Jxn}n≥1
∗
⇀ Jx ∈X∗.

Remark 1.1. By virtue of the Riesz-Representation theorem, it follows that Jx = x (J is the identity

map) when we are in a Hilbert space.

Definition 1.2 (Monotone Contraction Mapping, see Gordon [8], 2020). Let X 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. <〈Tx−Ty,JTx−JTy〉≤ c<〈x−y,Jx−Jy〉,

2. <〈Tm+1x−Tmy,JTn+1x−JTny〉≤ 0,

where J is the normalised duality mapping and for all m,n ≥ 0 with m 6= n.



Int. J. Anal. Appl. 19 (2) (2021) 291

In this paper, we consider the case where c = 1 in the above definition and introduce the following set of

new mappings.

Definition 1.3 (Monotone Nonexpansive Mapping). Let X 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 nonexpansive mapping if the following

two conditions are satisfied:

1. <〈Tx−Ty,JTx−JTy〉≤<〈x−y,Jx−Jy〉,

2. <〈Tm+1x−Tmy,JTn+1x−JTny〉≤ 0,

where J is the normalised duality mapping and for all m,n ≥ 0 with m 6= n.

We should note here that, monotone nonexpansive mappings reduce to the nonexpansive type of mappings

in (1.1) when in Hilbert spaces because in Hilbert spaces J is the identity mapping.

These references Browder [3], Göhde [4], Alpach [5] and Kirk [6] can be consulted for fixed point problems

on nonexpansive mappings.

2. Preliminaries

We introduce the following theorem, proposition and lemmas that will be used in the proof of our main

result. As before, all notations employed remain as defined.

Theorem 2.1 (Monotone contraction mapping theorem, see Gordon [8], 2020). Let C 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, Fix(T) = {p} and that the Picard iteration associated to T , that is, the

sequence defined by xn = T(xn−1) = T
n(x0) for all n ≥ 1 converges to p for any initial guess x0 ∈X.

Proposition 2.1 (see for instance Ezearn, [9]). Let X be a normed linear space. Then for any jx ∈ Jx,jy ∈

Jy

(‖x‖−‖y‖)2 ≤<〈x−y,jx− jy〉≤ ‖x−y‖(‖x‖ + ‖y‖).

Thus, <〈x−y,jx− jy〉≥ 0. Moreover if

<〈x−y,jx− jy〉 = 0,



Int. J. Anal. Appl. 19 (2) (2021) 292

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).

Proposition 2.2 (see for instance Ezearn, [9]). Let X be a Banach space and let X∗ be the dual space of X.

Denote 〈·, ·〉 the duality product. Now for {xn}n≥1 ⊂ X and {fn}n≥1 ⊂ X∗, suppose either of the following

conditions hold

• {xn} ⇀ x and {fn}→ f

• {xn}→ x and {fn}
∗
⇀ f

Then limn→∞〈xn,fn〉 = 〈x,f〉.

Lemma 2.1 (Uniform Continuity in Uniformly Smooth Spaces). Let X be a uniformly smooth Banach space.

Then the normalised duality map J : X →X∗ is norm-to-norm uniformly continuous.

3. Main Results

In this section, we first give the proof of Theorem 1.1 following the proof of our main result, Theorem 1.2.

Proof of Theorem 1.1. Let C be a compact star convex subset of a Banach space X with a distinguished

point ‘p’. Let T : C →C be a non-expansive mapping on C. For n ≥ 1, define Tn : C →C by,

Tnx =
( n
n + 1

)
Tx +

( 1
n + 1

)
p,∀x ∈C.

Obviously, Tn is a contraction mapping on C. Therefore, by the Banach contraction mapping theorem [1],

Tn has a unique fixed point xn in C. Now consider,

‖Txn −xn‖ = ‖Txn −Tnxn‖,

=
∥∥∥Txn −( n

n + 1

)
Txn −

( 1
n + 1

)
p
∥∥∥,

=
( 1
n + 1

)∥∥∥Txn −p∥∥∥,∀n ≥ 1.
Hence ‖Txn − xn‖ → 0 as n → ∞ since C is bounded. Since C is compact, the sequence {xn}n≥1 has a

convergence subsequence {xnk}k≥1 which converges to some x
∗ ∈ C and by continuity of T , Txnk → Tx

∗.

Then consider,

Txnkxnk = xnk =
( nk
nk + 1

)
Txnk +

( 1
nk + 1

)
p.

By passing k →∞, we have Tx∗ = x∗ and hence x∗ is a fixed point of T in C and that completes the proof.



Int. J. Anal. Appl. 19 (2) (2021) 293

The proof of our theorem uses the ideas of the proof of Theorem 1.1 by creating an internal contraction in

order to obtain an approximate fixed point sequence for these new mappings. The proof of our main result

is as follows.

Proof of Theorem 1.2. Now for every natural number n ≥ 1, define a new mapping Tn : C →C as

Tn(x) =
(

1 −
1

n

)
Tx.

Clearly, Tn is a self-mapping since 0 ∈ kerC. Now we have the following:

Tx =
1(

1 − 1
n

)Tnx and Ty = 1(
1 − 1

n

)Tny.
By substituting Tx and Ty into Definition 1.3, we have the following:

<〈
(

1 −
1

n

)−1
Tnx−

(
1 −

1

n

)−1
Tny,

(
1 −

1

n

)−1
J(Tnx) −

(
1 −

1

n

)−1
J(Tny)〉≤<〈x−y,Jx−Jy〉,(

1 −
1

n

)−2
<〈Tnx−Tny,JTnx−JTny〉≤<〈x−y,Jx−Jy〉.

Multiply the last inequality by
(

1 − 1
n

)2
to obtain

<〈Tnx−Tny,JTnx−JTny〉≤
(

1 −
1

n

)2
<〈x−y,Jx−Jy〉.

Since 0 ≤
(

1 − 1
n

)2
< 1, then by Theorem 2.1, Tn has a unique fixed point say xn, that is, xn = Txn =(

1 − 1
n

)
Txn and therefore ‖xn −Txn‖ = 1n‖Txn‖.

Since C is bounded, then supn≥1 ‖Txn‖ = D < ∞ where D is constant.

Hence

lim
n→∞

‖xn −Txn‖ = 0,(3.1)

where {xn}n≥1 is an approximate fixed point sequence for the monotone nonexpansive mapping T. Clearly,

equation (3.1) implies xn − Txn → 0 as n → ∞. Since C is weakly-compact, then the sequence {xn}n≥1

has a weakly converging subsequence. Without loss of generality, let {xn}n≥1 be the weakly converging

subsequence and x ∈ C be the weak limit of this subsequence, that is, xn ⇀ x as n → ∞. Given that

xn −Txn ⇀ 0 (strong convergence implies weak convergence) and xn ⇀ x, then Txn ⇀ x as n →∞.

We can clearly see that Definition 1.3 is equivalent to the following evaluation:

<〈x−y + Tx−Ty,Jx−Jy −JTx + JTy〉≥<〈Tx−Ty,Jx−Jy〉−<〈x−y,JTx−JTy〉(3.2)



Int. J. Anal. Appl. 19 (2) (2021) 294

for all x,y ∈C. Since {xn}n≥1 and it weak limit are both contained in C, then by replacing y with = xn in

equation (3.2), we obtain the following

<〈x−xn + Tx−Txn,Jx−Jxn −JTx + JTxn〉≥<〈Tx−Txn,Jx−Jxn〉

−<〈x−xn,JTx−JTxn〉
(3.3)

Taking limit as n →∞, the left hand side of equation (3.3) becomes

lim
n→∞

<〈x−xn + Tx−Txn,Jx−Jxn −JTx + JTxn〉.(3.4)

Since xn − Txn → 0, then by Lemma 2.1, we have that Jxn − JTxn → 0. Now with Txn ⇀ x and

Jxn −JTxn → 0, then by Proposition 2.2, as n →∞, equation (3.4) becomes

(3.5) <〈Tx−x,Jx−JTx〉 = −<〈x−Tx,Jx−JTx〉.

Again, taking limit of the right hand side of equation (3.3) as n →∞, we have

(3.6) lim
n→∞

[<〈Tx−Txn,Jx−Jxn〉−<〈x−xn,JTx−JTxn〉].

Given that xn =
(

1 − 1
n

)
Txn for all n ≥ 1, then substituting this sequence into equation (3.6), we obtain

lim
n→∞

[<〈Tx−Txn,Jx− (1 −
1

n
)JTxn〉−<〈x− (1 −

1

n
)Txn,JTx−JTxn〉],

which by expansion gives the following:

lim
n→∞

[<〈Tx,Jx〉− (1 −
1

n
)<〈Tx,JTxn〉−<〈Txn,Jx〉

−<〈x,JTx〉 + <〈x,JTxn〉 + (1 −
1

n
)<〈Txn,JTx〉].

(3.7)

By the sequentially weakly continuity of X , if Txn ⇀ x then JTxn
∗
⇀ Jx so that by Proposition 2.2, we

have <〈Tx,JTxn〉 → <〈Tx,Jx〉 and <〈Txn,Jx〉 → <〈x,Jx〉 as n → ∞. Hence as n → ∞, equation (3.7)

reduces to:

(3.8) <〈Tx,Jx〉−<〈Tx,Jx〉−<〈x,Jx〉−<〈x,JTx〉 + <〈x,Jx〉 + <〈x,JTx〉 = 0.

By equation (3.5) and equation (3.8), equation (3.3) reduces to

<〈x−Tx,Jx−JTx〉≤ 0,

which by Proposition 2.1 leads to

<〈x−Tx,Jx−JTx〉 = 0.

Since X is strictly convex, then by Proposition 2.1, we have x−Tx = 0 which implies x ∈ Fix(T) and that

completes the proof.



Int. J. Anal. Appl. 19 (2) (2021) 295

Availability of data and material: No data and material were used for this research.

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

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

References

[1] S. Banach, H. Steinhaus, Sur le principe de la condensation de singularités. Fundam. Math. 3(9) (1927), 50-61.

[2] M. Edelstein, An extension of Banach’s contraction principle. Proc. Amer. Math. Soc. 12(1) (1961), 7-10.

[3] F.E. Browder, Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA. 54(4) (1965), 1041-1044.

[4] D. Göhde, Zum prinzip der kontraktiven abbildung. Math. Nachr. 30(3-4) (1965), 251-258.

[5] D.E. Alspach, A fixed point free nonexpansive map. Proc. Amer. Math. Soc. 82(3) (1981), 423-424.

[6] W.A. Kirk, M.A. Khamsi, Introduction to Metric Spaces and Fixed Point Theory. John Willy & Sons, Inc., New York,

Chichester, Singapore, Toronto, 2001.

[7] L. Günter, Semi-inner-product spaces. Trans. Amer. Math. Soc. 100(1) (1961), 29-43.

[8] J.F. Gordon, The Monotone Contraction Mapping Theorem. J. Math. 2020 (2020), 2879283.

[9] J. Ezearn, Fixed Point Theory of Some Generalisations of Lipschitz mappings with applications to linear and non-linear

problems, PhD. Thesis, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana, 2017.


	1. Introduction
	2. Preliminaries
	3. Main Results
	References