@
Appl. Gen. Topol. 23, no. 1 (2022), 31-43

doi:10.4995/agt.2022.16165

© AGT, UPV, 2022

Some fixed point results for enriched

nonexpansive type mappings in Banach spaces

Rahul Shukla and Rajendra Pant

Department of Mathematics & Applied Mathematics, University of Johannesburg Kingsway Cam-

pus, Auckland Park 2006, South Africa (rshukla.vnit@gmail.com, pant.rajendra@gmail.com)

Communicated by S. Romaguera

Abstract

In this paper, we introduce two new classes of nonlinear mappings
and present some new existence and convergence theorems for these
mappings in Banach spaces. More precisely, we employ the Kras-
nosel’skĭı iterative method to obtain fixed points of Suzuki-enriched
nonexpansive mappings under different conditions. Moreover, we ap-
proximate the fixed point of enriched-quasinonexpansive mappings via
Ishikawa iterative method.

2020 MSC: 47H10; 47H09.

Keywords: nonexpansive mapping; enriched nonexpansive mapping; Ba-
nach space.

1. Introduction

Let C be a nonempty subset of a Banach space (B,‖.‖). A mapping ξ : C →C
is said to be nonexpansive if

‖ξ(ϑ) − ξ(ν)‖≤‖ϑ−ν‖
for all ϑ,ν ∈ C. Bruck [5] observed that apart from being an obvious general-
ization of the contraction mapping, nonexpansive mappings are important due
to their connection with the monotonicity methods. Perhaps, nonexpansive
mappings belong to the first class of nonlinear mappings for which fixed point
theorems were obtained by using the geometric properties of the underlying Ba-
nach spaces rather than the compactness assumptions (see fixed point theorems

Received 31 August 2021 – Accepted 24 November 2021

http://dx.doi.org/10.4995/agt.2022.16165
https://orcid.org/0000-0002-9835-0935
https://orcid.org/0000-0001-9990-2298


R. Shukla and R. Pant

due to Browder [3], Göhde [12] Kirk [15]). This class of mappings also appears
in applications as transition operators for initial value problems (of differen-
tial inclusion), accretive operators, monotone operators, variational inequality
problems and equilibrium problems. A number of extensions and generaliza-
tions of nonexpansive mappings in different directions have been considered
by many mathematicians in the literature to enlarge the class of nonexpansive
mappings, see [11, 17, 8, 25, 20, 21, 24] (see also the references therein).

In 2008, Suzuki [25] introduced a new type of mapping which is more general
than nonexpansive mapping, as follows.

Definition 1.1 ([25]). Let C be a nonempty subset of a Banach space (B,‖.‖).
A mapping ξ : C →C is said to satisfy condition (C) if for all ϑ,ν ∈C

1

2
‖ϑ− ξ(ϑ)‖≤‖ϑ−ν‖ implies ‖ξ(ϑ) − ξ(ν)‖≤‖ϑ−ν‖.

A mapping satisfying condition (C) is also known as Suzuki type generalized
nonexpansive mapping.

Recently, Berinde [1] introduced the following class of nonlinear mappings.

Definition 1.2. Let (B,‖.‖) be a Banach space. A mapping ξ : B →B is said
to be b-enriched nonexpansive mapping if there exists b ∈ [0,∞) such that for
all ϑ,ν ∈B

(1.1) ‖b(ϑ−ν) + ξ(ϑ) − ξ(ν)‖≤ (b + 1)‖ϑ−ν‖.

It is shown that every nonexpansive mapping ξ is a 0-enriched mapping. It
is interesting to note that both these classes of mappings, Suzuki type nonex-
pansive and b-enriched nonexpansive mappings are independent. A couple of
examples below illustrate these facts.

Example 1.3 ([25]). Let C = [0, 3] be a subset of R endowed with the usual
norm. Define ξ : C →C by

ξ(ϑ) =

{
0, if ϑ 6= 3,
1, if ϑ = 3.

Then ξ satisfies condition (C). However at ϑ = 2.5 and ν = 3

‖b(ϑ−ν) + ξ(ϑ) − ξ(ν)‖ = ‖b(2.5 − 3) + (0 − 1)‖
= b(0.5) + 1 > b(0.5) + 0.5 = (b + 1)|ϑ−ν|

and ξ is not b-enriched nonexpansive mapping for any b ∈ [0,∞).

Example 1.4 ([1]). Let C =
[
1
2
, 2
]
⊂ R and ξ : C → C be a mapping defined

as ξ(ϑ) = 1
ϑ
. Then F(ξ) = {1} and ξ is a 3

2
-enriched nonexpansive mapping.

On the other hand at ϑ = 1 and ν = 1
2
, we have

1

2
‖1 − ξ(1)‖ = 0 ≤

1

2
=

∥∥∥∥1 − 12
∥∥∥∥

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 32



Fixed point results for enriched nonexpansive type mappings

and ∥∥∥∥ξ
(

1

2

)
− ξ(1)

∥∥∥∥ = |2 − 1| = 1 > 12 =
∥∥∥∥12 − 1

∥∥∥∥ .
Thus ξ is not a mapping satisfying condition (C).

Now an interesting question arises that does there exists a class of mappings,
which contains both the b-enriched nonexpansive mappings and Suzuki-type
generalized nonexpansive mappings? Herein, we answer this question, affirma-
tively. Indeed, we introduce a new class of mappings, namely, Suzuki-enriched
nonexpansive mapping.

On the other hand, to check that a given mapping belongs to any of the
classes of nonexpansive type mappings can not be an easy task. Keeping this
point in mind to make task easier, Diaz and Metcalf [7] considered the following
class of mappings known as quasinonexpansive mapping

Definition 1.5. A mapping ξ : C → C is said to be quasinonexpansive if for
all ϑ ∈C and ϑ† ∈ F(ξ) 6= ∅,

‖ξ(ϑ) −ϑ†‖≤‖ϑ−ϑ†‖

where F(ξ) is the set of all fixed points of ξ.

It is well known that a nonexpansive mapping with a fixed point is quasi-
nonexpansive. However the converse need not to be true. Again, it is inter-
esting to see that the classes of b-enriched nonexpansive mappings and that
of quasi-nonexpansive mappings are independent in nature, see [23]. Keeping
this in mind, we generalize the class of quasinonexpansive mappings in the
sense of b-enriched nonexpansive mappings. In particular, we introduce a new
class of mappings namely enriched-quasinonexpansive mappings. This class of
mappings properly contains both quasinonexpansive mappings and b-enriched
nonexpansive mappings.

Motivated by Berinde [1, 2], Suzuki [25], Diaz and Metcalf [7] and others, we
introduce two new nonlinear classes of mappings in the setting of Banach spaces
and establish some existence and convergence theorems for these classes of map-
pings. We ensure the existence of fixed points for Suzuki-enriched nonexpansive
mappings in Banach spaces under certain assuptions. We employ Ishikawa it-
erative method to approximate the fixed points of enriched-quasinonexpansive
mappings and obtain some weak and strong convergence theorems. Our results
complement, extend, and generalize certain results from [1, 2, 25, 7, 18, 9].

2. Preliminaries

Definition 2.1 ([10]). A Banach space B is said to be uniformly convex if for
every ε ∈ (0, 2] there is some δ > 0 so that, for any ϑ,ν ∈B with ‖ϑ‖ = ‖ν‖ = 1,
the condition ‖ϑ−ν‖≥ ε implies that

∥∥ϑ+ν
2

∥∥ ≤ 1 − δ.
© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 33



R. Shukla and R. Pant

Definition 2.2 ([19]). Let (B,‖ · ‖) be a Banach space. A space B satisfies
Opial property if, for every weakly convergent sequence {ϑn} with weak limit
ϑ ∈B it holds:

lim inf
n→∞

‖ϑn −ϑ‖ < lim inf
n→∞

‖ϑn −ν‖

for all ν ∈B with ϑ 6= ν.

All finite dimensional Banach spaces and all Hilbert spaces satisfy the weak-
Opial property. Spaces `p (p ∈ (1,∞)) are Opial spaces but Lp(∈ (1,∞), p 6= 2)
spaces are not. [10].

Definition 2.3 ([22]). The mapping ξ : C → C with F(ξ) 6= ∅ satisfies
Condition (I) if there is a nondecreasing function f : [0,∞) → [0,∞) with
f(0) = 0,f(r) > 0 for r ∈ (0,∞) such that ‖ϑ− ξ(ϑ)‖ ≥ f(d(ϑ,F(ξ))) for all
ϑ ∈C, where d(ϑ,F(ξ)) = inf{‖ϑ−ν‖ : ν ∈ F(ξ)}.

Let C be a convex subset of a Banach space B and ξ : C → C a mapping.
The following iterative method is known as the Krasnosel’skĭı iterative method
(see [16]):

(2.1)

{
ϑ1 ∈C
ϑn+1 = αϑn + (1 −α)ξ(ϑn)

where α ∈ (0, 1).

Lemma 2.4. Let C be a nonempty convex subset a Banach space B. Let ξ :
C →C be a mapping, define S : C →C as follows:

S(ϑ) = (1 −λ)ϑ + λξ(ϑ)
for all ϑ ∈C and λ ∈ (0, 1). Then F(S) = F(ξ).

Definition 2.5. Let C be a nonempty subset of a Banach space B. A mapping
ξ : C →C is said to be compact if ξ(C) has a compact closure.

Lemma 2.6 ([27, p. 484]). Let B be a uniformly convex Banach space. If
two sequences {ϑn}, {νn} in B such that lim sup

n→∞
‖ϑn‖ ≤ θ, lim sup

n→∞
‖νn‖ ≤ θ,

lim
n→∞

‖αnϑn + (1 −αn)νn‖ = θ, where {αn}⊆ [η1,η2] ⊂ [0, 1] and θ ≥ 0. Then
lim
n→∞

‖ϑn −νn‖ = 0.

Lemma 2.7. Let B be a uniformly convex Banach space and C a nonempty
closed convex subset of B. Let S : C →C be a quasinonexpansive mapping with
F(S) 6= ∅. For given ϑ1 ∈C, for all n ∈ N, γn,δn ∈ [c,d] with c,d ∈ (0, 1), we
can define a sequence {ϑn} (Ishikawa iterative method [13]) as follows:

(2.2)

{
νn = (1 −γn)ϑn + γnS(ϑn)
ϑn+1 = (1 − δn)ϑn + δnS(νn),

Then we have the followings:

(1) lim
n→∞

‖ϑn −z‖ exists for all z ∈ F(S).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 34



Fixed point results for enriched nonexpansive type mappings

(2) lim
n→∞

‖ϑn −S(ϑn)‖ = 0.

Proof. From (2.2)

‖ϑn+1 −z‖ ≤ (1 −δn)‖ϑn −z‖ + δn‖S(νn) −z‖
≤ (1 −δn)‖ϑn −z‖ + δn‖νn −z‖
≤ (1 −δn)‖ϑn −z‖ + δn{(1 −γn)‖ϑn −z‖ + γn‖S(ϑn) −z‖}
≤ (1 −δn)‖ϑn −z‖ + δn‖ϑn −z‖ = ‖ϑn −z‖.

Hence the sequence {‖ϑn −z‖} is monotone nonincreasing and lim
n→∞

‖ϑn −z‖
exists. Let

(2.3) lim
n→∞

‖ϑn −z‖ = r > 0.

Since, S is a quasinonexpansive mapping

(2.4) lim sup
n→∞

‖S(νn) −z‖≤ lim sup
n→∞

‖νn −z‖≤ lim
n→∞

‖ϑn −z‖ = r

and

(2.5) lim
n→∞

‖(1 − δn)(ϑn −z) + δn(S(νn) −z)‖ = lim
n→∞

‖ϑn+1 −z‖ = r.

From (2.3), (2.4), (2.5) and Lemma 2.6, we have

(2.6) lim
n→∞

‖ϑn −S(νn)‖ = 0

Again

‖ϑn+1 −z‖ ≤ (1 − δn)‖ϑn −z‖ + δn‖S(νn) −z‖
≤ (1 − δn)‖ϑn −z‖ + δn‖νn −z‖

which implies

‖ϑn+1 −z‖−‖ϑn −z‖
δn

≤‖νn −z‖−‖ϑn −z‖.

Since δn ∈ [c,d]
‖ϑn+1 −z‖−‖ϑn −z‖

d
≤‖νn −z‖−‖ϑn −z‖.

Thus
r ≤ lim inf

n→∞
‖νn −z‖

From (2.4), we get

(2.7) lim
n→∞

‖νn −z‖ = r = lim
n→∞

‖(1 −γn)(ϑn −z) + γn(S(ϑn) −z)‖

From (2.3), (2.7) and Lemma 2.6, we get

lim
n→∞

‖ϑn −S(ϑn)‖ = 0.

�

Lemma 2.8. Let B be a uniformly convex Banach space and C a nonempty
closed convex subset of B. Let S : C →C be a quasinonexpansive mapping with
F(S) 6= ∅. Then F(S) is closed.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 35



R. Shukla and R. Pant

Lemma 2.9 (Demiclosedness principle, [4]). Let B be a uniformly convex Ba-
nach space, C a closed convex subset of B and ξ : C →C a mapping with a fixed
point. Suppose {ϑn} is a sequence in B such that {ϑn} converges weakly to ϑ
and lim

n→∞
‖ϑn − ξ(ϑn)‖ = 0. Then ξ(ϑ) = ϑ. That is, I − ξ is demiclosed at

zero.

3. Suzuki-enriched nonexpansive mapping

In this section, we introduce the following new class of mappings:

Definition 3.1. Let (B,‖.‖) be a Banach space and C a nonempty subset of
B. A mapping ξ : C → C is said to be Suzuki-enriched nonexpansive mapping
if there exists b ∈ [0,∞) such that for all ϑ,ν ∈C

1

2(b + 1)
‖ϑ− ξ(ϑ)‖≤‖ϑ−ν‖ implies(3.1)

‖b(ϑ−ν) + ξ(ϑ) − ξ(ν)‖≤ (b + 1)‖ϑ−ν‖.

It can be seen that every Suzuki-nonexpansive mapping ξ is a Suzuki-
enriched nonexpansive mapping with b = 0.

Theorem 3.2. Let B be a Banach space and C a nonempty compact convex
subset of B. Let ξ : C → C be a mapping satisfying (3.1). For given ϑ1 ∈ C,
define a sequence {ϑn} in C by
(3.2) ϑn+1 = (1 −λ)ϑn + λξ(ϑn)

for all n ∈ N, where λ ∈
[

1
2(b+1)

, 1
b+1

)
. Then F(ξ) 6= ∅ and {ϑn} strongly

converges to a point in F(ξ).

Proof. By the definition of mapping ξ, we have

1

2(b + 1)
‖ϑ− ξ(ϑ)‖≤‖ϑ−ν‖ implies(3.3)

‖b(ϑ−ν) + ξ(ϑ) − ξ(ν)‖≤ (b + 1)‖ϑ−ν‖.
for all ϑ,ν ∈C. Take µ = 1

b+1
∈ (0, 1) and put b = 1−µ

µ
in (3.3) then the above

inequality is equivalent to

1

2
µ‖ϑ− ξ(ϑ)‖≤‖ϑ−ν‖ implies(3.4)

‖(1 −µ)(ϑ−ν) + µ(ξ(ϑ) − ξ(ν))‖≤‖ϑ−ν‖.
Define the mapping S as follows:

S(ϑ) = (1 −µ)ϑ + µξ(ϑ) for all ϑ ∈C.
Thus

(3.5) ‖S(ϑ) −ϑ‖ = µ‖ξ(ϑ) −ϑ‖.
Then from (3.4), we get

1

2
‖ϑ−S(ϑ)‖≤‖ϑ−ν‖ implies ‖S(ϑ) −S(ν)‖≤‖ϑ−ν‖

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 36



Fixed point results for enriched nonexpansive type mappings

for all ϑ,ν ∈ C. Thus S is a mapping satisfying condition (C). Thus all the
assumptions of [25, Theorem 2] are satisfied and S has a fixed point in C. From
Lemma 2.4, F(S) = F(ξ) 6= ∅. Next, for given ϑ1 ∈ C and any λ ∈

[
1
2
, 1
)

consider the sequence

(3.6) ϑn+1 = (1 −λ)ϑn + λS(ϑn).

From [25, Theorem 2], {ϑn} strongly converges to a fixed point of S. But
F(S) = F(ξ) and

(1 −λ)ϑ + λS(ϑ) = (1 −λµ)ϑ + λµξ(ϑ)

for all ϑ ∈C. Since λ ∈
[
1
2
, 1
)

and µ = 1
b+1

. This implies that λµ ∈
[

1
2(b+1)

, 1
b+1

)
.

Therefore for any λ ∈
[

1
2(b+1)

, 1
b+1

)
, the sequence {ϑn} defined by (3.8) strongly

converges to a point in F(ξ). �

Theorem 3.3. Let B be a Banach space with the Opial property. Let C be
a nonempty weakly compact convex subset of B and ξ : C → C a mapping
satisfying (3.1). For given ϑ1 ∈C, define a sequence {ϑn} in C by

(3.7) ϑn+1 = (1 −λ)ϑn + λξ(ϑn)

for all n ∈ N, where λ ∈
[

1
2(b+1)

, 1
b+1

)
. Then F(ξ) 6= ∅ and {ϑn} weakly

converges to a point in F(ξ).

Proof. Following the same proof technique as in Theorem 3.2, we can define a
mapping S : C →C as follows:

S(ϑ) =

(
1 −

1

b + 1

)
ϑ +

1

b + 1
ξ(ϑ) for all ϑ ∈C

and S is a mapping satisfying condition (C). Then all the assumptions of [23,
Theorem 5] are satisfied, hence {ϑn} weakly converges to a fixed point of S.
But F(S) = F(ξ). This completes the proof. �

Theorem 3.4. Let B be a uniformly convex in every direction (or UCED)
Banach space and C a nonempty weakly compact convex subset of B. Let ξ :
C →C be a a mapping satisfying (3.1). Then ξ admits a fixed point in C.

Proof. Following largely the proof of Theorem 3.2, we can define a mapping
S satisfying condition (C). Thus all the assumptions of [25, Theorem 5] are
satisfied and it is guaranteed that S has at least one fixed point. From Lemma
2.4, F(S) = F(ξ) 6= ∅. �

Theorem 3.5. Let B be a UCED Banach space and C a nonempty weakly
compact convex subset of B. Let G be a family of commuting mappings on C
satisfying (3.1). Then G has a common fixed point.

Proof. Following the same proof technique of [25, Theorem 6] one can get the
desired result. �

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 37



R. Shukla and R. Pant

Theorem 3.6. Let B be a uniformly convex Banach space whose dual B∗ has
the Kadec-Klee property. Let C be a nonempty bounded closed convex subset
of B and ξ : C → C a mapping satisfying (3.1). For given ϑ1 ∈ C, define a
sequence {ϑn} in C by
(3.8) ϑn+1 = (1 −λ)ϑn + λξ(ϑn)

for all n ∈ N, where λ ∈
[

1
2(b+1)

, 1
b+1

)
. Then F(ξ) 6= ∅ and {ϑn} weakly

converges to a point in F(ξ).

Proof. Following the same proof technique as in Theorem 3.2, we can define a
mapping S : C →C as follows:

S(ϑ) =

(
1 −

1

b + 1

)
ϑ +

1

b + 1
ξ(ϑ) for all ϑ ∈C

and S is a mapping satisfying condition (C). Then all the assumptions of [14,
Theorem 11] are satisfied, hence {ϑn} weakly converges to a fixed point of S.
But F(S) = F(ξ). This completes the proof. �

We can obtain the following results due to consequence of Theorem 3.6.

Corollary 3.7. Let B be a uniformly convex Banach space having Fréchet
differentiable norm. Let C, ξ and {ϑn} be same as in Theorem 3.6. Then
F(ξ) 6= ∅ and {ϑn} weakly converges to a point in F(ξ).

4. Enriched-quasinonexpansive mapping

Now, we introduce the following new class of mappings:

Definition 4.1. Let (B,‖.‖) be a Banach space and C a nonempty subset of
B. A mapping ξ : C → C is said to be b-enriched quasinonexpansive mapping
if there exists b ∈ [0,∞) such that for all ϑ ∈C and ν ∈ F(ξ) 6= ∅
(4.1) ‖b(ϑ−ν) + ξ(ϑ) −ν‖≤ (b + 1)‖ϑ−ν‖.

Remark 4.2.

• It can be seen that every quasinonexpansive mapping is a 0-enriched
quasinonexpansive mapping.

• Every b-enriched nonexpansive mapping with a fixed point is b-enriched
quasinonexpansive mapping but the converse need not be true.

We consider the following examples, see [6, Example 6.23].

Example 4.3. Let B = `∞ and C := {ϑ ∈ `∞ : ‖ϑ‖∞ ≤ 1}. Define ξ : C → C
by

ξ(ϑ) = (0,ϑ21,ϑ
2
2,ϑ

2
3, . . . )

for ϑ = (ϑ1,ϑ2,ϑ3, . . . ) ∈ C. Then it can be seen that ξ is continuous from C
into C with p = (0, 0, . . . ) and F(ξ) = {p}. Furthermore,
‖ξ(ϑ) −p‖∞ = ‖ξ(ϑ)‖∞ = ‖(0,ϑ21,ϑ

2
2,ϑ

2
3, . . . )‖∞

≤ ‖(ϑ1,ϑ2,ϑ3, . . . )‖∞ = ‖ϑ‖∞ = ‖ϑ−p‖∞

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 38



Fixed point results for enriched nonexpansive type mappings

for all ϑ ∈ C. Thus, ξ is quasi-nonexpansive mapping and hence 0-enriched
quasinonexpansive mapping. However, ξ is not enriched-nonexpansive for any
b ∈ [0,∞). For, if ϑ = ( 3

4
, 3
4
, 3
4
, . . . ) and ν = ( 1

2
, 1
2
, 1
2
, . . . ), it is clear that

ϑ,ν ∈C. Furthermore, for any b ∈ [0,∞)

‖b(ϑ−ν) + ξ(ϑ) − ξ(ν)‖∞ =
∥∥∥∥
(
b

4
,

4b + 5

16
,

4b + 5

16
, . . .

)∥∥∥∥
∞

=
4b + 5

16
>
b + 1

4
= (b + 1)

∥∥∥∥14, 14, 14, . . .
∥∥∥∥
∞

= (b + 1)‖ϑ−ν‖∞.
Proposition 4.4. Let ξ : C → C be a Suzuki-enriched nonexpansive mapping
with any b ∈ [0,∞) and F(ξ) 6= ∅. Then ξ is a b-enriched quasinonexpansive
mapping for any b ∈ [0,∞).
Proof. Let z ∈ F(ξ) and ϑ ∈C. Since 1

2(b+1)
‖z −ξ(z)‖ = 0 ≤‖ϑ−z‖, we have

‖b(ϑ−ν) + ξ(ϑ) −z‖≤ (b + 1)‖ϑ−z‖.
�

In the above proposition the inclusion is strict, the following illustrative
example [25, Example 2] verifies this fact.

Example 4.5. Let C = [0, 3] ⊂ R and ξ : C →C be a mapping defined as

ξ(ϑ) =

{
0, if ϑ 6= 3
2, if ϑ = 3.

Then F(ξ) = {0} and ξ is a b-enriched quasinonexpansive mapping for any
b ∈ [0,∞). On the other hand at ϑ = 3 and ν = 4, 1

2(b+1)
‖3−ξ(3)‖ = 1

2(b+1)
≤

1 = ‖3 − 2‖, we have
‖b(ϑ−ν)+ξ(ϑ)−ξ(ν)‖ = ‖b(3−2)+ξ(3)−ξ(2)‖ = (b+2) > (b+1) = (b+1)‖ϑ−ν‖
and ξ is not a Suzuki-enriched nonexpansive mapping for any b ∈ [0,∞).

For some fix ϑ1 ∈C, the Ishikawa iterative method can be defined as follows
[13]:

(4.2)

{
νn = (1 −βn)ϑn + βnξ(ϑn)
ϑn+1 = (1 −αn)ϑn + αnξ(νn),

where {βn} and {αn} are sequences in [0, 1].
Theorem 4.6. Let B be a uniformly convex Banach space and C a nonempty
closed convex subset of B. Let ξ : C → C be a b-enriched quasinonexpansive
mapping and ξ satisfies Condition I. For given ϑ1 ∈ C, for all n ∈ N, αn ∈
(c,d), βn ∈

(
c
b+1

, d
b+1

)
with c,d ∈ (0, 1), define a sequence {ϑn} as follows:{

νn = (1 −βn)ϑn + βnξ(ϑn)
ϑn+1 = (1 −αn)ϑn + αn

[(
1 − 1

b+1

)
νn +

1
b+1

ξ(νn)
]
.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 39



R. Shukla and R. Pant

Then {ϑn} strongly converges to a point in F(ξ).

Proof. By the definition of b-enriched quasinonexpansive mapping, we have

‖b(ϑ−ν) + ξ(ϑ) −ν‖≤ (b + 1)‖ϑ−ν‖(4.3)

for all ϑ ∈ C and ν ∈ F(ξ). Take µ = 1
b+1
∈ (0, 1) and put b = 1−µ

µ
in (4.3),

then the above inequality is equivalent to

‖(1 −µ)(ϑ−ν) + µ(ξ(ϑ) −ν)‖≤‖ϑ−ν‖.(4.4)

Define the mapping S as follows:

(4.5) S(ϑ) = (1 −µ)ϑ + µξ(ϑ) for all ϑ ∈C.

From Lemma 2.4, F(S) = F(ξ). Then from (4.4), we get

‖S(ϑ) −ν‖≤‖ϑ−ν‖

for all ϑ ∈ C and ν ∈ F(S). Thus S : C → C is a quasinonexpansive mapping.
For given ϑ1 ∈ C, for all n ∈ N, γn,αn ∈ [c,d] with c,d ∈ (0, 1), we can define
a sequence {ϑn} as follows:

(4.6)

{
νn = (1 −γn)ϑn + γnS(ϑn)
ϑn+1 = (1 −αn)ϑn + αnS(νn),

From Lemma 2.7, lim
n→∞

‖ϑn−z‖ exists for all p ∈ F(S). Thus lim
n→∞

d(ϑn,F(S))

exists. Since ξ satisfies Condition I

‖ϑn − ξ(ϑn)‖ = (b + 1)‖ϑn −S(ϑn)‖≥ f(d(ϑn,F(ξ))) = f(d(ϑn,F(S))).

From Lemma 2.7, lim
n→∞

‖ϑn −S(νn)‖ = 0. Thus lim
n→∞

d(ϑn,F(S)) = 0. Follow-

ing largely the proof of [26, Theorem 3], we can choose a subsequence {ϑnj} of
{ϑn} such that

‖ϑnj −pj‖≤
1

2j

for all j ∈ N, where {pj} ⊆ F(ξ). It can be easily seen that {pj} is a Cauchy
sequence and strongly converges to a point p in F(ξ), since F(ξ) is closed.
Therefore {ϑn} strongly converges to p ∈ F(ξ). Using the definition of S, we
have {

νn = (1 −βn)ϑn + βnξ(ϑn)
ϑn+1 = (1 −αn)ϑn + αn

[(
1 − 1

b+1

)
νn +

1
b+1

ξ(νn)
]

where βn =
γn
b+1

. This completes the proof. �

Theorem 4.7. Let B be a uniformly convex Banach space and C a nonempty
closed convex subset of B. Let ξ : C → C be a continuous and b-enriched
quasinonexpansive mapping with F(ξ) 6= ∅. For given ϑ1 ∈C, define a sequence
{ϑn} in C by

ϑn = ξ
n
α,β(ϑ1), ξα,β = (1−α)I + α

[(
1 −

1

b + 1

)
I +

1

b + 1
ξ

]
[(1−β)I + βξ]

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 40



Fixed point results for enriched nonexpansive type mappings

for all n ∈ N, where α ∈ (0, 1), β ∈
[
0, 1

b+1

)
and I is an identity mapping.

Then {ϑn} strongly converges to a point in F(ξ) if and only if d(ϑn,F(ξ))) →∞
as n →∞.

Proof. Following the same proof technique as in Theorem 3.2, we can define a
mapping S : C →C as follows:

S(ϑ) =

(
1 −

1

b + 1

)
ϑ +

1

b + 1
ξ(ϑ) for all ϑ ∈C

and S is a quasinonexpansive mapping. For given ϑ1 ∈C, α ∈ (0, 1), γ ∈ [0, 1),
we can define a sequence {ϑn} as follows:

ϑn = S
n
α,γ(ϑ1), Sα,γ = (1 −α)I + αS[(1 −γ)I + γS]

Using the definition of S, we have

ϑn = ξ
n
α,β(ϑ1), ξα,β = (1−α)I + α

[(
1 −

1

b + 1

)
I +

1

b + 1
ξ

]
[(1−β)I + βξ]

where β = γ
b+1
∈
[
0, 1

b+1

)
. Since ξ is continuous, S is continuous. Then all the

assumptions of [9, Theorem 3.1] are satisfied, hence {ϑn} strongly converges to
a fixed point of S. But F(S) = F(ξ). This completes the proof. �

Theorem 4.8. Let B be a uniformly convex Banach space and C a nonempty
closed convex subset of B. Let ξ : C → C be a b-enriched quasinonexpansive
mapping and I − ξ is demiclosed at zero. For given ϑ1 ∈ C, for all n ∈ N,
αn ∈ (c,d), βn ∈

(
c
b+1

, d
b+1

)
with c,d ∈ (0, 1), define a sequence {ϑn} as

follows: {
νn = (1 −βn)ϑn + βnξ(ϑn)
ϑn+1 = (1 −αn)ϑn + αn

[(
1 − 1

b+1

)
νn +

1
b+1

ξ(νn)
]
.

Then {ϑn} weakly converges to a point in F(ξ).

Proof. We can define a sequence {ϑn} as in (4.6). Since space B is uniformly
convex, B is reflexive. By the reflexiveness of B there exists a subsequence
{ϑnj} of {ϑn} such that {ϑnj} weakly converges to some p ∈ C. By Lemma
2.7, lim

n→∞
‖ϑn −S(νn)‖ = 0 and

lim
n→∞

‖ϑn − ξ(ϑn)‖ = 0.

From the demiclosedness principle of I − ξ we have
p ∈ ωw(ϑn) ⊂ F(ξ).

Thus, to prove that {ϑn} weakly converges to a fixed point of ξ, it suffices
to show the unique weak limit for each subsequences of {ϑn}, that is, ωw(ϑn)
(cluster points (ω-limit) set of a sequence {ϑn}) is a singleton. Arguing by
contradiction, assume that {ϑn} does not converge weakly to p, i.e., take p,q ∈
ωw(ϑn) and let {ϑnj} and {ϑmj} be subsequences of {ϑn} such that ϑnj ⇀ p

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 41



R. Shukla and R. Pant

and ϑmj ⇀ q, respectively. If p 6= q, and standard application of Opial’s
property gives us the following contradiction:

lim
n→∞

‖ϑn −p‖ = lim
j→∞

‖ϑnj −p‖ < lim
j→∞

‖ϑnj −q‖

= lim
n→∞

‖ϑn −q‖ = lim
j→∞

‖ϑnj −q‖

< lim
j→∞

‖ϑnj −p‖ = lim
n→∞

‖ϑn −p‖.

This completes the proof �

Theorem 4.9. Let B be a uniformly convex Banach space and C a nonempty
closed convex subset of B. Let ξ : C → C be a compact b-enriched quasinonex-
pansive mapping and I−ξ is demiclosed at zero. For given ϑ1 ∈C, αn ∈ (c,d),
βn ∈

(
c
b+1

, d
b+1

)
with c,d ∈ (0, 1), define a sequence {ϑn} as follows:{

νn = (1 −βn)ϑn + βnξ(ϑn)
ϑn+1 = (1 −αn)ϑn + αn

[(
1 − 1

b+1

)
νn +

1
b+1

ξ(νn)
]
.

Then {ϑn} strongly converges to a point in F(ξ).

Proof. From the proof of Theorem 3.2, we can define a quasinonexpansive
mapping S (as in (4.5)). We can define a sequence as in (2.2). From Lemma
2.7, lim

n→∞
‖ϑn −S(νn)‖ = 0 and

(4.7) lim
n→∞

‖ϑn − ξ(ϑn)‖ = 0.

Since the range of C under ξ is contained in a compact set, there exists a sub-
sequence {ξ(ϑnj )} of {ξ(ϑn)} strongly converges to ϑ† ∈ C. By (4.7), the sub-
sequence {ϑnj} strongly converges ϑ†. By the demiclosedness principle ξ(ϑ†) =
ϑ†, and {ϑn} strongly converges to a point ϑ† in F(ξ). �

Acknowledgements. The authors are thankful to the reviewer and the editor
for their constructive comments. The first author acknowledges the support
from the GES 4.0 fellowship, University of Johannesburg, South Africa.

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