@
Appl. Gen. Topol. 21, no. 1 (2020), 135-158

doi:10.4995/agt.2020.12220

c© AGT, UPV, 2020

Fejér monotonicity and fixed point theorems

with applications to a nonlinear integral

equation in complex valued Banach spaces

Godwin Amechi Okeke a and Mujahid Abbas b

a Department of Mathematics, School of Physical Sciences, Federal University of Technology,

Owerri, P.M.B. 1526 Owerri, Imo State, Nigeria (godwin.okeke@futo.edu.ng)
b Department of Mathematics, Government College University, 54000 Lahore, Pakistan

Department of Mathematics and Applied Mathematics, University of Pretoria ( Hatfield Campus),

Lynnwood Road, Pretoria 0002, South Africa (abbas.mujahid@gmail.com)

Communicated by S. Romaguera

Abstract

It is our purpose in this paper to prove some fixed point results and Fejér
monotonicity of some faster fixed point iterative sequences generated
by some nonlinear operators satisfying rational inequality in complex
valued Banach spaces. We prove that results in complex valued Ba-
nach spaces are valid in cone metric spaces with Banach algebras. Fur-
thermore, we apply our results in solving certain mixed type Volterra-
Fredholm functional nonlinear integral equation in complex valued Ba-
nach spaces.

2010 MSC: 47H09; 47H10; 49M05; 54H25.

Keywords: complex valued Banach spaces; fixed point theorems; Fejér
monotonicity; iterative processes; cone metric spaces with Ba-
nach algebras; mixed type Volterra-Fredholm functional nonlin-
ear integral equation.

1. Introduction

Fixed point theory, which is famous in sciences and engineering due to its ap-
plications in solving several nonlinear problems in these fields of study became
one of the most interesting area of research in the last sixty years. For example,

Received 16 August 2019 – Accepted 18 December 2019

http://dx.doi.org/10.4995/agt.2020.12220


G. A. Okeke and M. Abbas

it has shown the importance of theoretical subjects, which are directly appli-
cable in different applied fields of science. Other areas of applications includes
optimization problems, control theory, economics and a host of others. In par-
ticular, it plays an important role in the investigation of existence of solutions
to differential and integral equations, which direct the behaviour of several real
life problems for which the existence of solution is critical (see, e.g. [25], [42]).
In 1922, Banach [12] provided a general iterative method to construct a fixed
point result and proved its uniqueness under linear contraction in complete
metric spaces. This famous results of Banach have been generalized in several
directions by many researchers. These generalization were made either by us-
ing the contractive condition or by imposing some additional conditions on the
ambient space. Some of these generalizations of metric spaces includes: rect-
angular metric spaces, pseudo metric spaces, D-metric spaces, partial metric
spaces, G-metric spaces and cone metric spaces (see, e.g. [1], [22], [23]).

The notion of complex valued metric spaces was introduced by Azam et al.
[11] in 2011. They established some fixed point theorems for a pair of map-
pings satisfying rational inequality. Their results is intended to define rational
expressions which are meaningless in cone metric spaces. Although complex
valued metric spaces form a special class of cone metric spaces (see, e.g. [2],
[6]), yet the definition of cone metric spaces rely on the underlying Banach
space which is not a division ring. Consequently, rational expressions are not
meaningful in cone metric spaces, this means that results involving mappings
satisfying rational expressions cannot be generalized to cone metric spaces. In-
view of this deficiency, Azam et al. [11] introduced the concept of complex
valued metric spaces. It is known that in cone metric spaces the underlying
metric assumes values in linear spaces where the linear space may be even in-
finite dimensional, whereas in the case of complex valued metric spaces the
metric values belong to the set of complex numbers which is one dimensional
vector space over the complex field. This instance is the major motivation
for the consideration of complex valued metric spaces independently (see, [6]).
Hence, results in this direction cannot be generalized to cone metric spaces, but
to complex valued metric spaces. It is known that complex valued metric space
is useful in many branches of Mathematics, including number theory, algebraic
geometry, applied Mathematics as well as in physics including hydrodynamics,
mechanical engineering, thermodynamics and electrical engineering (see, e.g.
[41]). Several authors have obtained interesting and applicable results in com-
plex valued metric spaces (see, e.g. [2], [3], [5], [8], [6], [11], [25], [36], [40], [41],
[42]).

It is known that there is a close relationship between the problem of solving
a nonlinear equation and that of approximating fixed points of a corresponding
contractive type operator (see, e.g. [14], [15], [32]). Hence, there is a practical
and theoretical interests in approximating fixed points of several contractive
type operators. Since, the introduction of the notion of complex valued metric
spaces by Azam et al. [11] in 2011, most results obtained in literature by many

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Fejér monotonicity and fixed point theorems

authors are existential in nature (see, e.g. [8], [11], [36], [41], [42]). Conse-
quently, there is a gap in literature with respect to the approximation of the
fixed point of several nonlinear mappings in this type of space. Recently, Okeke
[32] exploited the idea of complex valued metric spaces to define the concept
of complex valued Banach spaces and then initiated the idea of approximating
the fixed point of nonlinear mappings in complex valued Banach spaces.

The theory of integral and differential equations is an important aspect of
nonlinear analysis and the most applied tool for proving the existence of the
solutions of such equations is the fixed point technique (see, e.g. [12], [18], [19],
[33]). One of the most frequent and difficult problems faced by scientists in
mathematical sciences is nonlinear problems. This is because nature is intrinsi-
cally nonlinear (see, e.g. [19]). Solving nonlinear equations is cumbersome but
important to mathematicians and applied mathematicians such as engineers
and physicist. Some authors have used the fixed point iterative methods in
solving such equations (see, e.g. [18], [19], [33]). In this paper, we apply our
results in solving certain mixed type Volterra-Fredholm functional nonlinear
integral equation in complex valued Banach spaces.

It is our purpose in this paper to prove some fixed point results and Fejér
monotonicity of some faster fixed point iterative sequences generated by some
nonlinear operators satisfying rational inequality in complex valued Banach
spaces. We prove that results in complex valued Banach spaces are valid in
cone metric spaces with Banach algebras. Our results validates the fact that
fixed point theorems in the setting of cone metric spaces with Banach algebras
are more useful than the standard results in cone metric spaces and that results
in cone metric spaces with Banach algebras cannot be reduced to corresponding
results in cone metric spaces. Furthermore, we apply our results in solving
certain mixed type Volterra-Fredholm functional nonlinear integral equation in
complex valued Banach spaces. Our results unify, generalize and extend several
known results to complex valued Banach spaces, including the results of [4],
[9], [10], [18], [19], [28], [33]) among others.

2. Preliminaries

The following symbols, notations and definitions which can be found in [11]
will be useful in this study. Let C be the set of complex numbers and z1,z2 ∈ C.
Define a partial order - on C as follows:

z1 - z2 if and only if Re(z1) ≤ Re(z2), Im(z1) ≤ Im(z2).

It follows that z1 - z2 if one of the following conditions is satisfied:
(i) Re(z1) = Re(z2), Im(z1) < Im(z2),
(ii) Re(z1) < Re(z2), Im(z1) = Im(z2),
(iii) Re(z1) < Re(z2), Im(z1) < Im(z2),
(iv) Re(z1) = Re(z2), Im(z1) = Im(z2).
In particular, we will write z1 � z2 if z1 6= z2 and one of (i), (ii), and (iii) is
satisfied and we will write z1 ≺ z2 if only (iii) is satisfied.

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G. A. Okeke and M. Abbas

Note that
(a) a,b ∈ R and a ≤ b =⇒ az - bz for all z ∈ C;
(b) 0 - z1 � z2 =⇒ |z1| < |z2|;
(c) z1 - z2 and z2 ≺ z3 =⇒ z1 ≺ z3.
Definition 2.1 ([11]). Let X be a nonempty set. Suppose that the mapping
d : X ×X → C, satisfies:
1. 0 - d(x,y), for all x,y ∈ X and d(x,y) = 0 if and only if x = y;
2. d(x,y) = d(y,x) for all x,y ∈ X;
3. d(x,y) - d(x,z) + d(z,y), for all x,y,z ∈ X.

Then d is called a complex valued metric on X, and (X,d) is called a complex
valued metric space.

Recently, Okeke [32] defined a complex valued Banach space and proved
some interesting fixed point theorems in the framework of complex valued Ba-
nach spaces.

Definition 2.2 ([32]). Let E be a linear space over a field K, where K = R
(the set of real numbers) or C (the set of complex numbers). A complex valued
norm on E is a complex valued function ‖.‖ : E → C satisfying the following
conditions:
1. ‖x‖ = 0 if and only if x = 0, x ∈ E;
2. ‖kx‖ = |k|.‖x‖ for all k ∈ K, x ∈ E;
3. ‖x + y‖- ‖x‖ + ‖y‖ for all x,y ∈ E.

A linear space with a complex valued norm defined on it is called a complex
valued normed linear space, denoted by (E,‖.‖). A point x ∈ E is called an
interior point of a set A ⊆ E if there exist 0 ≺ r ∈ C such that

B(x,r) = {y ∈ E : ‖x−y‖≺ r}⊆ A.
A point x ∈ E is called a limit point of the set A whenever for each 0 ≺ r ∈ C,
we have

B(x,r) ∩ (AnE) 6= ∅.
The set A is said to be open if each element of A is an interior point of A. A
subset B ⊆ E is said to be closed if it contains each of its limit point. The
family

F = {B(x,r) : x ∈ E, 0 ≺ r}
is a sub-basis for a Hausdorff topology τ on E.

Suppose xn is a sequence in E and x ∈ E. If for all c ∈ C, with 0 ≺ c there
exists n0 ∈ N such that for all n > n0, ‖xn −xn+m‖≺ c, then {xn} is called a
Cauchy sequence in (E,‖.‖). If every Cauchy sequence is convergent in (E,‖.‖),
then (E,‖.‖) is called a complex valued Banach space.
Example 2.3 ([32]). Let E = C be the set of complex numbers. Define
‖.‖ : C×C → C by

‖z1 −z2‖ = |x1 −x2| + i|y1 −y2| ∀z1,z2 ∈ C,
where z1 = x1 +iy1, z2 = x2 +iy2. Clearly, (C,‖.‖) is a complex valued normed
linear space.

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Fejér monotonicity and fixed point theorems

Example 2.4 ([32]). Let E = C be the set of complex numbers. Define a
mapping ‖.‖ : C×C → C by

‖z1−z2‖ = eik|z1−z2|, ∀z1,z2 ∈ C, where k ∈ [0,
π

2
], z1 = x1+iy1, z2 = x2+iy2.

Then (C,‖.‖) is a complex valued normed linear space.

Example 2.5 ([32]). Let (C[a,b],‖.‖∞) be the space of all continuous complex
valued functions on a closed interval [a,b], endowed with the Chebyshev norm

‖x−y‖∞ = max
t∈[a,b]

|x(t) −y(t)|eik, x,y ∈ C[a,b], k ∈ [0,
π

2
].

Then (C[a,b],‖.‖∞) is a complex valued Banach space, since the elements of
C[a,b] are continuous functions, and convergence with respect to the Chebyshev
norm ‖.‖∞ corresponds to uniform convergence. We can easily show that every
Cauchy sequence of continuous functions converges to a continuous function,
i.e. an element of the space C[a,b].

In 1975, Dass and Gupta [21] extended the Banach contraction mapping
principle by proving the following theorem for mappings satisfying contractive
condition of the rational type in the framework of complete metric spaces.

Theorem 2.6 ([21]). Let (X,d) be a complete metric space and let T be a
mapping on X. Assume that there exist α,β ∈ (0, 1) satisfying α + β < 1 and

d(Tx,Ty) ≤ αd(y,Ty)
1 + d(x,Tx)

1 + d(x,y)
+ βd(x,y) (2.1)

for all x,y ∈ X. Then T has a unique fixed point z. Moreover {Tnx} converges
to z for all x ∈ X.

The following theorem for a Meir-Keeler contraction of the rational type was
proved in 2013 by Samet et al. [39] in the framework of complete metric spaces.

Theorem 2.7 ([39]). Let (X,d) be a complete metric space and T be a mapping
from X into itself. We assume that the following hypothesis holds:
given ε > 0, there exists δ(ε) > 0 such that

2ε ≤ d(y,Ty)
1 + d(x,Tx)

1 + d(x,y)
+ d(x,y) < 2ε + δ(ε) =⇒ d(Tx,Ty) < ε. (2.2)

Then T has a unique fixed point ζ ∈ X. Moreover, for any x ∈ X, the sequence
{Tnx} converges to ζ.

In 2007, Agarwal et al. [7] introduced the S iteration process as follows:


x0 ∈ D,
yn = (1 −βn)xn + βnTxn,
xn+1 = (1 −αn)Txn + αnTyn, n ∈ N,

(2.3)

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G. A. Okeke and M. Abbas

In 2014, Gürsoy and Karakaya [20] introduced the Picard-S iterative process
as follows: 


x0 ∈ D,
zn = (1 −βn)xn + βnTxn,
yn = (1 −αn)Txn + αnTzn
xn+1 = Tyn.

(2.4)

In 2015, Thakur et al. [44] introduced the following iterative process:


x0 ∈ D,
zn = (1 −βn)xn + βnTxn,
yn = T((1 −αn)xn + αnzn)
xn+1 = Tyn.

(2.5)

The authors in [44] proved that the Thakur iterative process (2.5) converges
faster than Picard, Mann [30], Ishikawa [26], S [7], Noor [31] and Abbas [4]
iteration processes for Suzuki’s generalized nonexpansive mappings. Recently,
Ullah and Arshad [45] introduced the M-iteration. They proved that this it-
erative process converges faster than all of S [7], Picard-S [19], Picard, Mann
[30], Ishikawa [26], Noor [31], SP [35], CR [17], S∗ [27], Abbas [4] and Normal-S
[38] iteration processes. The following is the M-iteration process introduced by
Ullah and Arshad [45] in 2018.


x0 ∈ D,
zn = (1 −αn)xn + αnTxn,
yn = Tzn
xn+1 = Tyn.

(2.6)

In 2013, Khan [28] introduced the Picard-Mann hybrid iterative process.
The iterative process for one mapping case is given by the sequence {mn}∞n=1.


m1 = m ∈ D,
mn+1 = Tzn,
zn = (1 −αn)mn + αnTmn, n ∈ N,

(2.7)

where {αn}∞n=1 is in (0, 1). Khan [28] proved that this iterative process con-
verges faster than all of Picard, Mann and Ishikawa iterative processes in the
sense of Berinde [15] for contractive mappings.

Recently, Okeke and Abbas [33] introduced the Picard-Krasnoselskii hybrid
iterative process defined by the sequence {xn}∞n=1 as follows:


x1 = x ∈ D,
xn+1 = Tyn,
yn = (1 −λ)xn + λTxn, n ∈ N,

(2.8)

where λ ∈ (0, 1). The authors proved that this new hybrid iteration process
converges faster than all of Picard, Mann, Krasnoselskii and Ishikawa iterative
processes in the sense of Berinde [15]. They also used this iterative process to
find the solution of delay differential equations.

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Fejér monotonicity and fixed point theorems

Definition 2.8 ([15]). Let {an}∞n=0, {bn}∞n=0 be two sequences of positive
numbers that converge to a, respectively b. Assume there exists

l = lim
n→∞

|an −a|
|bn − b|

. (2.9)

1. If l = 0, then it is said that the sequence {an}∞n=0 converges to a faster than
the sequence {bn}∞n=0 to b;
2. If 0 < l < ∞, then we say that the sequences {an}∞n=0 and {bn}∞n=0 have
the same rate of convergence.

Definition 2.9. Let D be a nonempty subset of a complex valued Banach
space (E,‖.‖). The diameter of D is

diamD = sup
(x,y)∈D×D

|‖x−y‖|. (2.10)

The distance to D is the function

‖.‖D : E → C : x → inf |‖x−D‖|. (2.11)

The following lemma will be useful in this study.

Lemma 2.10 ([32]). Let (E,‖.‖) be a complex valued Banach space and let
{xn} be a sequence in E. Then {xn} converges to x if and only if |‖xn −x‖|→ 0
as n →∞.

Lemma 2.11 ([32]). Let (E,‖.‖) be a complex valued Banach space and let
{xn} be a sequence in E. Then {xn} is a Cauchy sequence if and only if
|‖xn −xn+m‖|→ 0 as n →∞.

Lemma 2.12 ([43]). Let {βn}∞n=0 be a nonnegative sequence for which one
assumes there exists n0 ∈ N, such that for all n ≥ n0 one has satisfied the
inequality

βn+1 ≤ (1 −µn)βn + µnγn,
where µn ∈ (0, 1), for all n ∈ N,

∑∞
n=0 µn = ∞ and γn ≥ 0, ∀N. Then the

following inequality holds

0 ≤ lim sup
n→∞

βn ≤ lim sup
n→∞

γn.

3. Fejér monotonicity and fixed point theorems in complex
valued Banach spaces

In this section, we prove some Fejér monotonicity and fixed point results
in the framework of complex valued Banach spaces. Our results improves and
extend some known results in the framework of complex valued Banach spaces,
including the results of Bauschke and Combettes [13], Cegielski [16] and Dass
and Gupta [21] among others. We begin this section by defining the concept
of Fejér monotonicity in the framework of complex valued Banach spaces and
also provide some examples.

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G. A. Okeke and M. Abbas

Definition 3.1. Let D be a nonempty subset of a complex valued Banach space
(E,‖.‖) and let {xn} be a sequence in E. Then {xn}n∈N is Fejér monotone with
respect to D if for each x ∈ D and each n ∈ N,

‖xn+1 −x‖- ‖xn −x‖. (3.1)

Example 3.2. Suppose {xn}n∈N is a bounded sequence in C that is increasing
(respectively decreasing). Then the sequence {xn}n∈N is Fejér monotone with
respect to [sup{xn}n∈N, +∞) (respectively (−∞, inf{xn}n∈N]).

Example 3.3. Let D be a nonempty subset of a complex valued Banach space
(E,‖.‖) and T : D → D be a mapping on D with F(T) := {x ∈ D : Tx = x} 6=
∅. Assume that there exist α,β ∈ (0, 1) satisfying α + β < 1 and

‖Tx−Ty‖- α‖y −Ty‖
1 + ‖x−Tx‖
1 + ‖x−y‖

+ β‖x−y‖ (3.2)

for all y ∈ F(T). Let x0 ∈ D and set xn+1 = Txn, ∀n ∈ N. Then {xn}n∈N is
Fejér monotone with respect to F(T).

Now, using relation (3.2) together with the facts that α,β ∈ (0, 1) and
y ∈ F(T), we have

‖xn+1 −y‖ - α‖y −Ty‖
1+‖xn−Txn‖

1+‖xn−y‖
+ β‖xn −y‖

= α.0
(

1+‖xn−Txn‖
1+‖xn−y‖

)
+ β‖xn −y‖

= β‖xn −y‖
- ‖xn −y‖. (3.3)

Hence, {xn}n∈N if Fejér monotone with respect to F(T).

Proposition 3.4. Let D be a nonempty subset of a complex valued Banach
space (E,‖.‖) and let {xn}n∈N be a sequence in E. Suppose that {xn}n∈N is
Fejér monotone with respect to D. Then the following hold:
(i) {xn}n∈N is bounded.
(ii) For every x ∈ D, (|‖xn −x‖|)n∈N converges.
(iii) {‖.‖D(xn)}n∈N is decreasing and converges.
(iv) Let m ∈ N and let n ∈ N. Then

|‖xn+m −xn‖|≤ 2‖.‖D(xn). (3.4)

Proof. (i) Suppose x ∈ D. It follows from (3.1) that {xn}n∈N lies in B(x, |‖x0−
x‖|). Hence, {xn}n∈N is bounded.
(ii) By (3.1), we have

|‖xn+1 −x‖|≤ |‖xn −x‖|−→ 0 as n →∞. (3.5)

Hence, by Lemma 2.10, we have that {xn}n∈N −→ x as n →∞.
(iii) Suppose x ∈ D, since {xn}n∈N is Fejér monotone, it follows that

{‖.‖D(xn+1)}n∈N = inf |‖xn+1 −x‖|≤ inf |‖xn −x‖| = {‖.‖D(xn)}n∈N (3.6)

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Fejér monotonicity and fixed point theorems

Hence, by Lemma 2.10, we obtain {‖.‖D(xn)}n∈N −→ 0 as n →∞.
(iv) Since {xn}n∈N is Fejér monotone, then by (3.1), we have
|‖xn+m −xn‖| ≤ |‖xn+m −x‖| + |‖xn −x‖|

≤ 2|‖xn −x‖|. (3.7)
By taking infimum over x ∈ D in (3.7), we have the desired result. The proof
of Proposition 3.4 is completed. �

Proposition 3.5. Let D be a nonempty closed convex subset of a complex
valued Banach space (E,‖.‖) and T : D → D be a mapping on D with F(T) :=
{x ∈ D : Tx = x} 6= ∅. Assume that there exist λ,β ∈ (0, 1) satisfying λ+β < 1
and

‖Tx−Ty‖- λ‖y −Ty‖
1 + ‖x−Tx‖
1 + ‖x−y‖

+ β‖x−y‖ (3.8)

for all x,y ∈ D. For arbitrary chosen x0 ∈ D, let the sequence {xn} be generated
by the M-iteration process (2.6), where αn ∈ (0, 1) for each n ∈ N, then {xn}
is Fejér monotone with respect to F(T).

Proof. Suppose p ∈ F(T), then by (2.6), (3.8) and the facts that λ,β ∈ (0, 1)
and αn ∈ (0, 1) for all n ∈ N, we obtain
‖xn+1 −p‖ = ‖Tyn −p‖

- λ‖p−Tp‖1+‖yn−Tyn‖
1+‖yn−p‖

+ β‖yn −p‖
= λ.0

(
1+‖yn−Tyn‖

1+‖yn−p‖

)
+ β‖yn −p‖

= β‖yn −p‖
= β‖Tzn −p‖
- β

[
λ‖p−Tp‖1+‖zn−Tzn‖

1+‖zn−p‖
+ β‖zn −p‖

]
- λ.0

(
1+‖zn−Tzn‖

1+‖zn−p‖

)
+ β‖zn −p‖

= β‖zn −p‖
= β[‖(1 −αn)xn + αnTxn −p‖]
- (1 −αn)‖xn −p‖ + αn‖Txn −p‖
- (1 −αn)‖xn −p‖ + αn

[
λ‖p−Tp‖1+‖xn−Txn‖

1+‖xn−p‖
+ β‖xn −p‖

]
= (1 −αn)‖xn −p‖ + αn

[
λ.0
(

1+‖xn−Txn‖
1+‖xn−p‖

)
+ β‖xn −p‖

]
= (1 −αn)‖xn −p‖ + αnβ‖xn −p‖
- (1 −αn)‖xn −p‖ + αn‖xn −p‖
= ‖xn −p‖. (3.9)

This means that ‖xn+1 − p‖ - ‖xn − p‖ as desired. Therefore, {xn} is Fejér
monotone. The proof of Proposition 3.5 is completed. �

Theorem 3.6. Let D be a nonempty closed convex subset of a complex valued
Banach space (E,‖.‖) and T : D → D be a mapping on D. Assume that there
exist λ,β ∈ (0, 1) satisfying λ + β < 1 and

‖Tx−Ty‖- λ‖y −Ty‖
1 + ‖x−Tx‖
1 + ‖x−y‖

+ β‖x−y‖ (3.10)

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G. A. Okeke and M. Abbas

for all x,y ∈ D. For arbitrary chosen x0 ∈ D, let the sequence {xn} be gener-
ated by the M-iteration process (2.6), where αn ∈ (0, 1) for each n ∈ N, and∑∞
n=0 αn = ∞. Then {xn} converges strongly to a unique fixed point p of T.

Proof. We want to show that xn −→ p as n → ∞. Now, using relation (2.6)
and (3.10), we obtain:

‖xn+1 −p‖ = ‖Tyn −Tp‖
- λ‖p−Tp‖1+‖yn−Tyn‖

1+‖yn−p‖
+ β‖yn −p‖

= λ.0
(

1+‖yn−Tyn‖
1+‖yn−p‖

)
+ β‖yn −p‖

= β‖yn −p‖. (3.11)
Next, we obtain the following estimate:

‖yn −p‖ = ‖Tzn −Tp‖
- λ‖p−Tp‖1+‖zn−Tzn‖

1+‖zn−p‖
+ β‖zn −p‖

= λ.0
(

1+‖zn−Tzn‖
1+‖zn−p‖

)
+ β‖zn −p‖

= β‖zn −p‖
= β‖(1 −αn)xn + αnTxn −p‖
- β(1 −αn)‖xn −p‖ + βαn‖Txn −Tp‖
- β(1 −αn)‖xn −p‖ + βαn

[
λ‖p−Tp‖1+‖xn−Txn‖

1+‖xn−p‖
+ β‖xn −p‖

]
= β(1 −αn)‖xn −p‖ + β2αn‖xn −p‖
= β(1 −αn(1 −β))‖xn −p‖. (3.12)

Using (3.12) in (3.11), we have

‖xn+1 −p‖ - β‖yn −p‖
- β2(1 −αn(1 −β))‖xn −p‖. (3.13)

Continuing this process gives the following relations


‖xn+1 −p‖- β2(1 −αn(1 −β))‖xn −p‖
‖xn −p‖- β2(1 −αn−1(1 −β))‖xn−1 −p‖

‖xn−1 −p‖- β2(1 −αn−2(1 −β))‖xn−2 −p‖
...

‖x1 −p‖- β2(1 −α0(1 −β))‖x0 −p‖.

(3.14)

From relation (3.14), we obtain the followings:

‖xn+1 −p‖- ‖x0 −p‖β2(n+1)
n∏
k=0

(1 −αk(1 −β)). (3.15)

Using the fact that β ∈ (0, 1) and αn ∈ (0, 1) for each n ∈ N, we have

(1 −αn(1 −β)) < 1. (3.16)

In classical analysis, it is known that 1 −x ≤ e−x for all x ∈ [0, 1]. Now using
these facts together with relation (3.15), we obtain

‖xn+1 −p‖- ‖x0 −p‖β2(n+1)e−(1−β)
∑n

k=0
αk. (3.17)

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Fejér monotonicity and fixed point theorems

From (3.17), it follows that

|‖xn+1 −p‖|≤ |‖x0 −p‖|β2(n+1)e−(1−β)
∑n

k=0
αk −→ 0 as n →∞. (3.18)

Hence, by Lemma 2.10, it follows that {xn}−→ p as n →∞.
Next, we show that the fixed point p of T is unique. Now suppose that p∗

is another fixed point of T, then by (3.10), we have

‖p−p∗‖ - λ‖p∗ −Tp∗‖1+‖p−Tp‖
1+‖p−p∗‖ + β‖p−p

∗‖
= λ.0

(
1

1+‖p−p∗‖

)
+ β‖p−p∗‖

= β‖p−p∗‖. (3.19)
Relation (3.19) implies that

|‖p−p∗‖|≤ β|‖p−p∗‖|. (3.20)
Which is a contradiction, since β ∈ (0, 1). Hence, p = p∗ as desired. The proof
of Theorem 3.6 is completed. �

Lemma 3.7. Let D be a nonempty closed convex subset of a complex valued
Banach space (E,‖.‖) and T : D → D be a mapping on D with F(T) 6= ∅.
Assume that there exist λ,β ∈ (0, 1) satisfying λ + β < 1 and

‖Tx−Ty‖- λ‖y −Ty‖
1 + ‖x−Tx‖
1 + ‖x−y‖

+ β‖x−y‖ (3.21)

for all x,y ∈ D. For arbitrary chosen x0 ∈ D, let the sequence {xn} be generated
by the M-iteration process (2.6), then limn→∞ |‖xn − p‖| exists for any p ∈
F(T).

Proof. From relation (3.9), it follows that

|‖xn+1 −p‖|≤ |‖xn −p‖|−→ 0 as n →∞. (3.22)
Hence, by Lemma 2.10 we see that {‖xn −p‖} is bounded and non-increasing
for each p ∈ F(T). Therefore, limn→∞ |‖xn −p‖| exists as desired. The proof
of Lemma 3.7 is completed. �

Lemma 3.8. Let D be a nonempty subset of a complex valued Banach space
(E,‖.‖). Let the sequence {xn}⊆ E be Fejér monotone with respect to D. If at
least one cluster point x∗ of {xn} belongs to D, then xn → x∗.

Proof. Since every Fejér monotone sequence is bounded, it follows that {xn}
has a weak cluster point x∗. Let a subsequence {xkn} of {xn} converge to
x∗ ∈ D. We now prove that {xn} converges to x∗. Suppose x′ ∈ E, x′ 6= x∗ is
another cluster point of {xn} such that a subsequence {xmn} converges to x′.
Suppose ε := 1

2
‖x′ − x∗‖ � 0, let n0 ∈ N be such that ‖xmn − x′‖ ≺ ε and

‖xkn −x∗‖≺ ε, for all n ≥ n0 and let mn > kn0. Using the triangle inequality
and Fejér monotonicity of {xn} with respect to D, we have

2ε = ‖x′ −x∗‖- ‖x′ −xmn‖ + ‖xmn −x
∗‖≺ 2ε. (3.23)

This means that

2ε = |‖x′ −x∗‖|≤ |‖x′ −xmn‖| + |‖xmn −x
∗‖| < 2ε, (3.24)

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G. A. Okeke and M. Abbas

which is a contradiction. Therefore, by Lemma 2.10, it follows that xn → x∗.
The proof of Lemma 3.8 is completed. �

4. Cone metric spaces with Banach algebras

Let A denote a real Banch algebra. This means that A is a real Banach
space in which an operation of multiplication is defined, subject to the following
properties (for each x,y,z ∈A, α ∈ R):
(i) (xy)z = x(yz);
(ii) x(y + z) = xy + xz and (x + y)z = xz + yz;
(iii) α(xy) = (αx)y = x(αy);
(iv) ‖xy‖≤‖x‖‖y‖.

In this paper, we assume that A has a unit; i.e. a multiplicative identity
e such that ex = xe = x for each x ∈ A. The inverse of x is denoted by x−1
(see, e.g. Rudin [37]).

In 2012, Öztürk and Başarir [34] generalized the concept of cone metric
spaces introduced by Huang and Zhang [23] by replacing a Bancach space with
a Banach algebra A in cone metric spaces. They called this new concept BA-
cone metric spaces. Abbas et al. [2] proved that complex valued metric spaces
introduced in [11] is a BA-cone metric space, that is a cone metric space over
a solid cone in commutative division Banach algebra A (see, [2], [34]).

Perhaps unaware of the work of Öztürk and Başarir [34], in 2013 Liu and
Xu [29] introduced the concept of cone metric spaces with Banach algebras, by
replacing Banach spaces with Banach algebras as the underlying space of cone
metric spaces. They proved that fixed point theorems in the setting of cone
metric spaces with Banach algebras are more useful than the standard results in
cone metric spaces and that results in cone metric spaces with Banach algebras
cannot be reduced to corresponding results in cone metric spaces. .

Example 4.1 ([29]). Let A = Mn(R) = {a = (aij)n×n|aij ∈ R for all 1 ≤
i,j ≤ n} be the algebra of all n-square real matrices, and define the norm

‖a‖ =
∑

1≤i,j≤n

|aij|.

Then A is a real Banach algebra with the unit e, the identity matrix.
Let P = {a ∈A|aij ≥ 0 for all 1 ≤ i,j ≤ n}. Then P ⊂A is a normal cone

with normal constant M = 1.
Let X = Mn(R), and define the metric d : X ×X →A by

d(x,y) = d((xij)n×n, (yij)n×n) = (|xij −yij|)n×n ∈A.

Then (X,d) is a cone metric space with a Banach algebra A.

Example 4.2 ([29]). Let A be the Banach space C(K) of all continuous real-
valued functions on a compact Hausdorff topological space K, with multiplica-
tion defined pointwise. Then A is a Banach algebra, and the constant function
f(t) = 1 is the unit of A.

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Fejér monotonicity and fixed point theorems

Let P = {f ∈ A|f(t) ≥ 0 for all t ∈ K}. Then P ⊂ A is a normal cone
with a normal constant M = 1.

Let X = C(K) with the metric mapping d : X ×X →A defined by

d(f,g) = |f(t) −g(t)|, where t ∈K.

Then (X,d) is a cone metric space with a Banach algebra A.

Example 4.3 ([29]). Let A = `1 = {a = (an)n≥0|
∑∞
n=0 |an| < ∞} with

convolution as multiplication:

ab = (an)n≥0(bn)n≥0 =


 ∑
i+j=n

aibj



n≥0

.

Thus A is a Banach algebra. The unit e is (1, 0, 0, · · ·).
Let P = {a = (an)n≥0 ∈ A|an ≥ 0 for all n ≥ 0}, which is a normal cone

in A. And let X = `1 with the metric d : X ×X →A defined by

d(x,y) = d((xn)n≥0, (yn)n≥0) = (|xn −yn|)n≥0.

Then (X,d) is a cone metric space with A.

Motivated by the results above, we now prove that results in complex valued
Banach spaces (see, e.g. Okeke [32]) are true in the context of cone metric
spaces with Banach algebras. Moreover, we show that our results cannot be
deduced in cone metric spaces.

Theorem 4.4. Let D be a nonempty closed convex subset of a complete cone
metric space with Banach algebras (A,‖.‖) and T : D → D be a contraction
mapping satisfying the following contractive condition

‖Tx−Ty‖-
ϕ(‖x−Tx‖) + a‖x−y‖

e + M‖x−Tx‖
, ∀x,y ∈ D, a ∈ [0, 1), M ≥ 0, (4.1)

where ϕ : C+ → C+ is a monotone increasing function such that ϕ(0) = 0. Let
{mn} be an iterative sequence generated by the Picard-Mann hybrid iterative
process (2.7) with real sequence {αn}∞n=0 in [0, 1] satisfying

∑∞
n=0 αn = ∞.

Then {mn} converges strongly to a unique fixed point p of T.

Proof. We now show that xn → p as n →∞. Using (2.7) and (4.1), we obtain:
‖mn+1 −p‖ = ‖Tzn −p‖

- ϕ(‖p−Tp‖)+a‖zn−p‖
e+M‖p−Tp‖

=
ϕ(‖0‖)+a‖zn−p‖

e+M‖0‖
= a‖(1 −αn)mn + αnTmn −p‖
- a(1 −αn)‖mn −p‖ + aαn‖Tmn −p‖
- a(1 −αn)‖mn −p‖ + aαn

[
ϕ(‖p−Tp‖)+a‖mn−p‖

e+M‖p−Tp‖

]
= a(1 −αn)‖mn −p‖ + aαn

[
ϕ(‖0‖)+a‖mn−p‖

e+M‖0‖

]
= a(1 −αn)‖mn −p‖ + a2αn‖mn −p‖
= a(1 −αn(1 −a))‖mn −p‖. (4.2)

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G. A. Okeke and M. Abbas

Using the fact that (1−αn(1−a)) < 1 and a ∈ [0, 1), we obtain the following
inequalities from (4.2).



‖mn+1 −p‖- a(1 −αn(1 −a))‖mn −p‖
‖mn −p‖- a(1 −αn−1(1 −a))‖mn−1 −p‖
‖mn−1 −p‖- a(1 −αn−2(1 −a))‖mn−2 −p‖
...
‖m2 −p‖- a(1 −α1(1 −a))‖m1 −p‖.

(4.3)

From relation (4.3), we derive

‖mn+1 −p‖- ‖m1 −p‖an+1
n∏
k=1

(1 −αk(1 −a)), (4.4)

where (1−αk(1−a)) ∈ (0, 1), since a ∈ [0, 1) and αk ∈ [0, 1] for all k ∈ N. It is
well-known in classical analysis that 1 −x ≤ e−x for all x ∈ [0, 1]. Using these
facts together with relation (4.4), we have

‖mn+1 −p‖-
‖m1 −p‖an+1

e(1−a)
∑

n
k=1 αk

. (4.5)

Therefore,

lim
n→∞

|‖mn+1 −p‖|≤
{
|‖m1 −p‖an+1|
|e(1−a)

∑
n
k=1 αk|

}
−→ 0 as n →∞. (4.6)

Therefore by Lemma 2.1 we have that limn→∞‖mn−p‖ = 0. This means that
mn → p as n →∞ as desired.

Next, we show that T has a unique fixed point p ∈ F(T) := {p ∈ D : Tp =
p}. Assume that p∗ is another fixed point of T, then we have
‖p−p∗‖ = ‖Tp−Tp∗‖

- ϕ(‖p−Tp‖)+a‖p−p
∗‖

e+M‖p−Tp‖
=

ϕ(‖0‖)+a‖p−p∗‖
e+M‖0‖

= a‖p−p∗‖. (4.7)
This implies that

|‖p−p∗‖|≤ |‖p−p∗‖|. (4.8)
Hence, by Lemma 2.10 we have that p = p∗. The proof of Theorem 4.4 is
completed. �

Proposition 4.5. Let D be a nonempty closed convex subset of a complete cone
metric space with Banach algebras (A,‖.‖) and let T : D → D be a mapping
defined as follows

‖Tx−Ty‖-
ϕ(‖x−Tx‖) + a‖x−y‖

e + M‖x−Tx‖
, ∀x,y ∈ D, a ∈ [0, 1), M ≥ 0, (4.9)

where ϕ : C+ → C+ is a monotone increasing function such that ϕ(0) = 0.
Suppose that each of the iterative processes (2.7) and (2.8) converges to the
same fixed point p of T where {αn}∞n=0 and λ are such that 0 < α ≤ λ,αn < 1

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Fejér monotonicity and fixed point theorems

for all n ∈ N and for some α. Then the sequence {xn} generated by the Picard-
Krasnoselskii hybrid iterative process (2.8) have the same rate of convergence
as the sequence {mn} generated by the Picard-Mann hybrid iterative process
(2.7).

Proof. The proof of Proposition 4.5 follows similar lines as in the proofs of
([32], Proposition 2.2) and Theorem 4.4. �

Remark 4.6. Observe that the results in Theorem 4.4 and Proposition 4.5
was proved for mappings satisfying rational inequality, which is meaningless in
cone metric spaces. This means that these results cannot be reduced to some
corresponding results in cone metric spaces.

5. Applications to a nonlinear integral equation

It is our purpose in this section to show that the M- iterative process (2.6)
converges strongly to the solution of a mixed type Volterra-Fredholm functional
nonlinear integral equation in complex valued Banach spaces. Our results gen-
eralize and extend some known results to complex valued Banach spaces, in-
cluding the results of Crăciun and Şerban [18], Gürsoy [19] among others.

In 2011, Crăciun and Şerban [18] considered the following mixed type Volterra-
Fredholm functional nonlinear integral equation:

x(t) = F

(
t,x(t),

∫ t1
a1

· · ·
∫ tm
am

K(t,s,x(s))ds,

∫ b1
a1

· · ·
∫ bm
am

H(t,s,x(s))ds

)
,

(5.1)
where [a1; b1] × ··· × [am; bm] be an interval in Rm, K,H : [a1; b1] × ··· ×
[am; bm] × [a1; b1] × ··· × [am; bm] × R → R continuous functions and F :
[a1; b1] ×···× [am; bm] ×R3 → R. They established the following results.

Theorem 5.1 ([18]). We assume that:
(i) K,H ∈ C([a1,b1] ×···× [am,bm] × [a1,b1] ×···× [am,bm] ×R);
(ii) F ∈ C([a1,b1] ×···× [am,bm] ×R3);
(iii) there exist α,β,γ nonnegative constants such that:

|F(t,u1,v1,w1) −F(t,u2,v2,w2)| ≤ α|u1 −u2| + β|v1 −v2| + γ|w1 −w2|,
for all t ∈ [a1,b1] ×···× [am,bm], u1,u2,v1,v2,w1,w2 ∈ R;
(iv) there exist LK and LH nonnegative constants such that:

|K(t,s,u) −K(t,s,v)| ≤ LK|u−v|,

|H(t,s,u) −H(t,s,v)| ≤ LH|u−v|,
for all t,s ∈ [a1,b1] ×···× [am,bm], u,v ∈ R;
(v) α + (βLK + γLH)(b1 −a1) · · ·(bm −am) < 1.
Then, the equation (5.1) has a unique solution x∗ ∈ C([a1,b1]×···× [am,bm]).

Remark 5.2 ([18]). Let (B, |.|) be a Banach space. Then Theorem 5.1 remains
also true if we consider the mixed type Volterra-Fredholm functional nonlinear
integral equation (5.1) in the Banach space B instead of Banach space R.

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G. A. Okeke and M. Abbas

Let D be a nonempty subset of a complex valued Banach space (E,‖.‖) and
let {mn} be an iterative process defined by the M -iteration associated with F,
which is generated as follows:


m0 ∈ D
zn = (1 −αn)mn + αnTmn
gn = Tzn
mn+1 = Tgn,

(5.2)

where {αn} is a real sequence in (0, 1). Consequently, we now obtain the fol-
lowing analogue of Theorem 5.1 in complex valued Banach spaces.

Theorem 5.3. We consider the complex valued Banach space BC = C([a1,b1]×
···× [am,bm],‖.‖C), where ‖.‖C is the Chebyshev’s norm defined by ‖x−y‖C =
|x−y|i, ∀x,y ∈BC. We assume that:
(i) K,H ∈ C([a1,b1] ×···× [am,bm] × [a1,b1] ×···× [am,bm] ×R);
(ii) F ∈ C([a1,b1] ×···× [am,bm] ×R3);
(iii) there exist α,β,γ nonnegative constants such that:

|F(t,u1,v1,w1) −F(t,u2,v2,w2)| ≤ α|u1 −u2| + β|v1 −v2| + γ|w1 −w2|,

for all t ∈ [a1,b1] ×···× [am,bm], u1,u2,v1,v2,w1,w2 ∈ R;
(iv) there exist LK and LH nonnegative constants such that:

|K(t,s,u) −K(t,s,v)| ≤ LK|u−v|,

|H(t,s,u) −H(t,s,v)| ≤ LH|u−v|,

for all t,s ∈ [a1,b1] ×···× [am,bm], u,v ∈ R;
(v) α + (βLK + γLH)(b1 −a1) · · ·(bm −am) < 1.
Then, the mixed type Volterra-Fredholm functional integral equation (5.1) has
a unique solution p ∈ C([a1; b1] ×···× [am; bm]).

Proof. Since our analysis is in the complex valued Banach space BC = C([a1,b1]×
···×[am,bm],‖.‖C), where ‖.‖C is the Chebyshev’s norm defined by ‖x−y‖C =
|x−y|i, ∀x,y ∈BC, and the operator

A : BC →BC,

defined by

A(x)(t) = F

(
t,x(t),

∫ t1
a1

· · ·
∫ tm
am

K(t,s,x(s))ds,

∫ b1
a1

· · ·
∫ bm
am

H(t,s,x(s))ds

)
(5.3)

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Fejér monotonicity and fixed point theorems

Using conditions (iii) and (iv), we have

|A(u)(t) −A(v)(t)| - α|u(t) −v(t)| + β|
∫ t1
a1
· · ·
∫ tm
am

(K(t,s,u(s))−
K(t,s,v(s)))ds|+
γ
∣∣∣∫ b1a1 · · ·∫ bmam (H(t,s,u(s)) −H(t,s,v(s)))ds∣∣∣

- α|u(t) −v(t)| + β
∫ t1
a1
· · ·
∫ tm
am

LK|u(s) −v(s)|ds+
γ
∫ b1
a1
· · ·
∫ bm
am

LH|u(s) −v(s)|ds
- [α + (βLK + γLH)(b1 −a1) · · ·(bm −am)]‖u−v‖C
= [α + (βLK + γLH)(b1 −a1) · · ·(bm −am)]|u−v|i.

(5.4)
It follows from relation (5.4) that

|‖A(u)(t) −A(v)(t)‖C| ≤ |[α + (βLK + γLH)(b1 −a1) · · ·(bm −am)]|u−v|i|
= [α + (βLK + γLH)(b1 −a1) · · ·(bm −am)]|u−v|.

(5.5)
Using Lemma 2.10 in (5.5) together with condition (v), we see that operator
A is a contraction, so that by the Banach contraction mapping principle, we
have that operator A has a unique fixed point F(A) = {p}. This means that
our equation (5.1) has a unique solution p ∈ C([a1; b1] ×···× [am; bm]). The
proof of Theorem 5.3 is completed. �

Theorem 5.4. Suppose that all the conditions (i) - (v) in Theorem 4.2 are
satisfied. Let the sequence {mn} be generated by the M-iteration process (5.2),
where {αn} ⊂ (0, 1) is a real sequence satisfying

∑∞
n=0 αn = ∞. Then the

mixed type Volterra-Fredholm functional integral equation (5.1) has a unique
solution, say p ∈ C([a1; b1] ×···× [am; bm]) and the sequence {mn} converges
to p.

Proof. We consider the complex valued Banach space BC = C([a1,b1] ×···×
[am,bm],‖.‖C), where ‖.‖C is the Chebyshev’s norm defined by ‖x − y‖C =
|x − y|i, ∀x,y ∈ BC. Let {mn} be the sequence generated by the M-iteration
process (5.2) for the operator A : BC →BC defined by

A(x)(t) = F

(
t,x(t),

∫ t1
a1

· · ·
∫ tm
am

K(t,s,x(s))ds,

∫ b1
a1

· · ·
∫ bm
am

H(t,s,x(s))ds

)
.

(5.6)

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G. A. Okeke and M. Abbas

We want to show that mn −→ p as n →∞. Using (5.2), (5.1) and assumptions
(i) - (v) we obtain

‖mn+1 −p‖C = |A(gn)(t) −A(p)(t)|
= |F

(
t,gn(t),

∫ t1
a1
· · ·
∫ tm
am

K(t,s,gn(s))ds,
∫ b1
a1
· · ·
∫ bm
am

H(t,s,gn(s))ds
)
−

F
(
t,p(t),

∫ t1
a1
· · ·
∫ bm
am

K(t,s,p(s))ds,
∫ b1
a1
· · ·
∫ bm
am

H(t,s,p(s))ds
)
|

- α|gn(t) −p(t)| + β|
∫ t1
a1
· · ·∫ tm

am
K(t,s,gn(s))ds−

∫ t1
a1
· · ·
∫ tm
am

K(t,s,p(s))ds|+
γ|
∫ b1
a1
· · ·
∫ bm
am

H(t,s,gn(s))ds−
∫ b1
a1
· · ·
∫ bm
am

H(t,s,p(s))ds|
- α|gn(t) −p(t)| + β

∫ t1
a1
· · ·
∫ bm
am
|K(t,s,gn(s)) −K(t,s,p(s))|ds+

γ
∫ b1
a1
· · ·
∫ bm
am
|H(t,s,gn(s)) −H(t,s,p(s))|ds

- α|gn(t) −p(t)| + β
∫ t1
a1
· · ·
∫ tm
am

LK|gn(s) −p(s)|ds+
γ
∫ b1
a1
· · ·
∫ bm
am

LH|gn(s) −p(s)|ds
- [α + (βLK + γLH)

∏m
i=1(bi −ai)]‖gn −p‖C.

(5.7)
Next, we have the following estimate

‖gn −p‖C = |A(zn)(t) −A(p)(t)|
= |F

(
t,zn(t),

∫ t1
a1
· · ·
∫ tm
am

K(t,s,zn(s))ds,
∫ b1
a1
· · ·
∫ bm
am

H(t,s,zn(s))ds
)
−

F
(
t,p(t),

∫ t1
a1
· · ·
∫ bm
am

K(t,s,p(s))ds,
∫ b1
a1
· · ·
∫ bm
am

H(t,s,p(s))ds
)
|

- α|zn(t) −p(t)| + β|
∫ t1
a1
· · ·∫ tm

am
K(t,s,zn(s))ds−

∫ t1
a1
· · ·
∫ tm
am

K(t,s,p(s))ds|+
γ
∣∣∣∫ b1a1 · · ·∫ bmam H(t,s,zn(s))ds−∫ b1a1 · · ·∫ bmam H(t,s,p(s))ds∣∣∣

- α|zn(t) −p(t)| + β
∫ t1
a1
· · ·
∫ tm
am
|K(t,s,zn(s)) −K(t,s,p(s))|ds+

γ
∫ b1
a1
· · ·
∫ bm
am
|H(t,s,zn(s)) −H(t,s,p(s))|ds

- α|zn(t) −p(t)| + β
∫ t1
a1
· · ·
∫ tm
am

LK|zn(s) −p(s)|ds+
γ
∫ b1
a1
· · ·
∫ bm
am

LH|zn(s) −p(s)|ds
- [α + (βLK + γLH)

∏m
i=1(bi −ai)]‖zn −p‖C.

(5.8)

‖zn −p‖C - (1 −αn)|mn(t) −p(t)| + αn|A(mn)(t) −A(p)(t)|
= (1 −αn)|mn(t) −p(t)|+

αn|F
(
t,mn(t),

∫ t1
a1
· · ·
∫ tm
am

K(t,s,mn(s))ds,
∫ b1
a1
· · ·
∫ bm
am

H(t,s,mn(s))ds
)
−

F
(
t,p(t),

∫ t1
a1
· · ·
∫ bm
am

K(t,s,p(s))ds,
∫ b1
a1
· · ·
∫ bm
am

H(t,s,p(s))ds
)
|

- (1 −αn)|mn(t) −p(t)| + αnα|mn(t) −p(t)| + αnβ
∫ t1
a1
· · ·∫ tm

am
LK|mn(s) −p(s)|ds

+αnγ
∫ b1
a1
· · ·
∫ bm
am

LH|mn(s) −p(s)|ds
- {1 −αn (1 − [α + (βLK + γLH)

∏m
i=1(bi −ai)])}‖mn −p‖C.

(5.9)

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 152



Fejér monotonicity and fixed point theorems

Using (5.8) and (5.9) in (5.7), together with the fact that [α+(βLK+γLH)
∏m
i=1(bi−

ai)] < 1 in assumption (v) we obtain

‖mn+1 −p‖C - [α + (βLK + γLH)
∏m
i=1(bi −ai)]

2×
{1 −αn (1 − [α + (βLK + γLH)

∏m
i=1(bi −ai)])}‖mn −p‖C

- {1 −αn (1 − [α + (βLK + γLH)
∏m
i=1(bi −ai)])}‖mn −p‖C.

(5.10)
Hence, by induction (5.10) becomes

‖mn+1−p‖C -
n∏
k=0

{
1 −αk

(
1 − [α + (βLK + γLH)

m∏
i=1

(bi −ai)]

)}
‖m0−p‖C.

(5.11)
From the fact that αk ∈ (0, 1) for each k ∈ N, together with assumption (v),
we have {

1 −αk

(
1 − [α + (βLK + γLH)

m∏
i=1

(bi −ai)]

)}
< 1. (5.12)

It is known in analysis that ex ≥ 1 − x for all x ∈ [0, 1]. Therefore (5.11)
becomes

‖mn+1 −p‖C - ‖m0 −p‖Ce−(1−[α+(βLK +γLH )
∏m

i=1
(bi−ai)])

∑n
k=0

αk)

= |m0 −p|ie−(1−[α+(βLK +γLH )
∏m

i=1(bi−ai)])
∑n

k=0 αk).
(5.13)

This means that

|‖mn+1 −p‖C| - |‖m0 −p‖Ce−(1−[α+(βLK +γLH )
∏m

i=1
(bi−ai)])

∑n
k=0

αk)|
= |m0 −p|e−(1−[α+(βLK +γLH )

∏m
i=1(bi−ai)])

∑n
k=0 αk) −→ 0

(5.14)
as k →∞. Therefore, by Lemma 2.10, we have xn −→ p as n →∞ as desired.
The proof of Theorem 5.4 is completed. �

We now turn our attention to proving the data dependence of the solution
for the integral equation (5.1) via the M-iterative process (5.2).

Suppose BC is as in Theorem 5.3 and the operators T,T̃ : BC → BC are
defined by

T(x)(t) = F

(
t,x(t),

∫ t1
a1

· · ·
∫ tm
am

K(t,s,x(s))ds,

∫ b1
a1

· · ·
∫ bm
am

H(t,s,x(s))ds

)
(5.15)

T̃(x)(t) = F

(
t,x(t),

∫ t1
a1

· · ·
∫ tm
am

K̃(t,s,x(s))ds,

∫ b1
a1

· · ·
∫ bm
am

H̃(t,s,x(s))ds

)
,

(5.16)

where K,K̃,H,H̃ ∈ C([a1; b1] ×···× [am; bm] × [a1; b1] ×···× [am; bm] ×R).

Theorem 5.5. Let F,K and H be defined as in Theorem 5.3 and let {mn}
be the iterative sequence generated by the M-iteration process (5.2) associated
with T. Let {m̃n} be an iterative sequence generated by

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 153



G. A. Okeke and M. Abbas




m̃0 ∈ D,
z̃n = (1 −αn)m̃n + αnT̃m̃n,
g̃n = T̃ z̃n
m̃n+1 = T̃ g̃n,

(5.17)

where BC is as defined in Theorem 5.2 and {αn} is a real sequence in (0, 1)
satisfying
(a) 1

2
≤ αn for each n ∈ N, and

(b)
∑∞
n=0 αn = ∞. Furthermore, suppose

(c) there exist nonnegative constants λ1 and λ2 such that |K(t,s,u)−K̃(t,s,u)| ≤
λ1 and |H(t,s,u) − H̃(t,s,u)| ≤ λ2, for all u ∈ R and for all t,s ∈ [a1; b1] ×
···× [am; bm].

If p and p̃ are solutions of corresponding nonlinear equations (5.15) and
(5.16) respectively, then we have

|‖p− p̃‖|≤
4(βλ1 + γλ2)

∏m
i=1(bi −ai)

1 − [α + (βLK + γLH)
∏m
i=1(bi −ai)

. (5.18)

Proof. We consider the complex valued Banach space BC = C([a1,b1] ×···×
[am,bm],‖.‖C), where ‖.‖C is the Chebyshev’s norm defined by ‖x − y‖C =
|x−y|i, ∀x,y ∈BC.

Now using (5.1), (5.2), (5.15), (5.16), (5.17) and assumptions (i) - (v) to-
gether with conditions (a) - (c), we have

‖mn+1 − m̃n+1‖C = ‖Tgn − T̃ g̃n‖C
= |F

(
t,gn(t),

∫ t1
a1
· · ·
∫ tm
am

K(t,s,gn(s))ds,
∫ b1
a1
· · ·
∫ bm
am

H(t,s,gn(s))ds
)
−

F
(
t, g̃n(t),

∫ t1
a1
· · ·
∫ tm
am

K̃(t,s, g̃n(s))ds,
∫ b1
a1
· · ·
∫ bm
am

H̃(t,s, g̃n(s))ds
)
|

- α|gn(t) − g̃n(t)| + β
∫ t1
a1
· · ·
∫ tm
am
|K(t,s,gn(s)) − K̃(t,s, g̃n(s))|ds+

γ
∫ b1
a1
· · ·
∫ bm
am
|H(t,s,gn(s)) − H̃(t,s, g̃n(s))|ds

- α|gn(t) − g̃n(t)|+
β
∫ t1
a1
· · ·
∫ tm
am

(|K(t,s,gn(s)) −K(t,s, g̃n(s))|+
|K(t,s, g̃n(s)) − K̃(t,s, g̃n(s))|)ds+
γ
∫ b1
a1
· · ·
∫ bm
am

(|H(t,s,gn(s)) −H(t,s, g̃n(s))|+
|H(t,s, g̃n(s)) − H̃(t,s, g̃n(s))|)ds

- α|gn(t) − g̃n(t)| + β
∫ t1
a1
· · ·
∫ tm
am

(LK|gn(s) − g̃n(s)| + λ1)ds+
γ
∫ b1
a1
· · ·
∫ bm
am

(LH|gn(s) − g̃n(s)| + λ2)ds
- α‖gn − g̃n‖C + β(LK‖gn − g̃n‖C + λ1)

∏m
i=1(bi −ai)+

γ(LH‖gn − g̃n‖C + λ2)
∏m
i=1(bi −ai)

- [α + (βLK + γLH)
∏m
i=1(bi −ai)]‖gn − g̃n‖C+

(βλ1 + γλ2)
∏m
i=1(bi −ai).

(5.19)

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 154



Fejér monotonicity and fixed point theorems

‖gn − g̃n‖C = ‖Tzn − T̃ z̃n‖C
= |F

(
t,zn(t),

∫ t1
a1
· · ·
∫ tm
am

K(t,s,zn(s))ds,
∫ b1
a1
· · ·
∫ bm
am

H(t,s,zn(s))ds
)
−

F
(
t, z̃n(t),

∫ t1
a1
· · ·
∫ tm
am

K̃(t,s, z̃n(s))ds,
∫ b1
a1
· · ·
∫ bm
am

H̃(t,s, z̃n(s))ds
)
|

- α|zn(t) − z̃n(t)|+
β
∫ t1
a1
· · ·
∫ tm
am

(|K(t,s,zn(s)) −K(t,s, z̃n(s))|+
|K(t,s, z̃n(s)) − K̃(t,s, z̃n(s))|)ds+
γ
∫ b1
a1
· · ·
∫ bm
am

(|H(t,s,zn(s)) −H(t,s, z̃n(s))|+
|H(t,s, z̃n(s)) − H̃(t,s, z̃n(s))|)ds

- α|zn(t) − z̃n(t)| + β
∫ t1
a1
· · ·
∫ tm
am

(LK|zn(s) − z̃n(s)| + λ1)ds+
γ
∫ b1
a1
· · ·
∫ bm
am

(LH|zn(s) − z̃n(s)| + λ2)ds
- α‖zn − z̃n‖C + β(LK‖zn − z̃n‖C + λ1)

∏m
i=1(bi −ai)+

γ(LH‖zn − z̃n‖C + λ2)
∏m
i=1(bi −ai)

- [α + (βLK + γLH)
∏m
i=1(bi −ai)]‖zn − z̃n‖C+

(βλ1 + γλ2)
∏m
i=1(bi −ai).

(5.20)

‖zn − z̃n‖C - (1 −αn)|mn(t) − m̃n(t)| + αn|T(mn)(t) − T̃(m̃n)(t)|
- (1 −αn)|mn(t) − m̃n(t)| + αn{α|mn(t) − m̃n(t)|+

β
∫ t1
a1
· · ·
∫ tm
am

(LK|mn(s) − m̃n(s)| + λ1)ds+
γ
∫ b1
a1
· · ·
∫ bm
am

(LH|mn(s) − m̃n(s)| + λ2)ds}
- {1 −αn(1 − [α + (βLK + γLH)

∏m
i=1(bi −ai)])}‖mn − m̃n‖C+

αn(βλ1 + γλ2)
∏m
i=1(bi −ai).

(5.21)
Using (5.21) in (5.20), together with assumption (v), we have:

‖gn − g̃n‖C - {1 −αn(1 − [α + (βLK + γLH)
∏m
i=1(bi −ai)])}‖mn − m̃n‖C+

αn(βλ1 + γλ2)
∏m
i=1(bi −ai) + (βλ1 + γλ2)

∏m
i=1(bi −ai).

(5.22)
Using (5.22) in (5.19) together with assumption (v), we have

‖mn+1 − m̃n+1‖C - {1 −αn(1 − [α + (βLK + γLH)
∏m
i=1(bi −ai)])}×

‖mn − m̃n‖C+
αn(βλ1 + γλ2)

∏m
i=1(bi −ai)+

(βλ1 + γλ2)
∏m
i=1(bi −ai)+

(βλ1 + γλ2)
∏m
i=1(bi −ai)

- {1 −αn(1 − [α + (βLK + γLH)
∏m
i=1(bi −ai)])}×

‖mn − m̃n‖C+
αn (1 − [α + (βLK + γLH)

∏m
i=1(bi −ai)])×(

4(βλ1+γλ2)
∏m

i=1
(bi−ai)

(1−[α+(βLK +γLH )
∏

m
i=1

(bi−ai))

)
.

(5.23)

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 155



G. A. Okeke and M. Abbas

From relation (5.23), we choose the sequences βn, µn and γn as follows:




βn = ‖mn − m̃n‖C,
µn = αn(1 − [α + (βLK + γLH)

∏m
i=1(bi −ai)]) ∈ (0, 1),

γn =
4(βλ1+γλ2)

∏m
i=1

(bi−ai)
(1−[α+(βLK +γLH )

∏
m
i=1(bi−ai))

.
(5.24)

Therefore, from relation (5.23), we see that all the conditions of Lemma 2.3 are
satisfied. Hence, we have

‖p− p̃‖C -
4(βλ1 + γλ2)

∏m
i=1(bi −ai)

(1 − [α + (βLK + γLH)
∏m
i=1(bi −ai))

. (5.25)

This implies that

|‖p− p̃‖C| ≤
4(βλ1 + γλ2)

∏m
i=1(bi −ai)

(1 − [α + (βLK + γLH)
∏m
i=1(bi −ai))

. (5.26)

The proof of Theorem 5.5 is completed. �

Remark 5.6. Theorem 5.4 and Theorem 5.5 generalize, unify and extend sev-
eral known results from real Banach spaces to complex valued Banach spaces,
including the results of Gürsoy [19] among others.

Acknowledgements. The first author’s research is supported by the Ab-
dus Salam School of Mathematical Sciences, Government College University,
Lahore, Pakistan through grant number: ASSMS/2018-2019/452.

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