EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6154
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Efficient Viscosity Algorithms for Solving the Split
Equality Fixed-Point Problem

L. B. Mohammed1, A. Kılıçman2,∗, D. Bamanga1

1 Department of Mathematics, Faculty of Physical Sciences, Federal University Dutse,
PMB 7156, Dutse, Jigawa State, Nigeria
2 Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah
Alam, Selangor, Malaysia

Abstract. Solving the Split Equality Fixed-Point Problem (SEFPP) often requires computing the
norms of bounded and linear operators, a task that can be computationally demanding. To tackle
this challenge, we investigated the SEFPP for quasi-pseudocontractive mappings in Hilbert spaces
and proposed innovative viscosity algorithms to solve the problem. We established the strong
convergence of these algorithms under appropriate conditions. To validate our theoretical results,
we conducted numerical experiments, which not only confirmed the efficacy of our results but also
demonstrated their advantages over existing methods in the literature. Our work generalizes and
extends several significant results from prior research, contributing to the broader understanding
of fixed-point problems and related problems.

2020 Mathematics Subject Classifications: 47H09, 47H10, 47J25

Key Words and Phrases: fixed point problem, iterative algorithm, nonlinear mappings, weak
and strong convergence

1. Introduction

For j = 1, 2, let Hj be Hilbert spaces, Cj be nonempty, closed, and convex subsets of
Hj , and Aj : H1 → H2 be linear and bounded mappings, with A∗

j denoting the adjoint of
Aj .

The problem of finding
z ∈ C1 such that A1z ∈ C2, (1)

is known as the Split Feasibility Problem (SFP). This problem was introduced by Censor
et al. [1]. To solve problem (1), Byrne [2] proposed the following algorithm, known as the
”CQ algorithm”:

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6154

Email addresses: lawanbulama@gmail.com (L. B. Mohammed), kilicman@uitm.edu.my (A. Kılıçman),
abudawud02@gmail.com (D. Bamanga)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 2 of 18

zn+1 = PC1 (I − λA∗
1(I − PC2)A1) zn, (2)

where λ ∈
(
0, 2

A1A∗
1

)
. This algorithm requires the computation of PCj onto Cj , which is

feasible when these projections have closed-form expressions. More results on the SFP
and its applications can be found in [3–7].

The Split Equality Problem (SEP), which is related to the SFP, was introduced by
Moudafi and Al-Shemas [8]. The SEP involves finding

y ∈ C1 and z ∈ C2 such that A1y = A2z. (3)

By setting A2 = I (identity mapping), the SEP reduces to the SFP. To solve problem
(3), Moudafi and Al-Shemas [8] proposed the following algorithm:{

yn+1 = PC2 (yn + λnA∗
2(A1yn −A2zn)) ;

zn+1 = PC1 (zn − λnA∗
1(A1yn −A2zn)) , n ≥ 0;

(4)

where (y0, z0) ∈ H1 ×H2 are chosen arbitrarily, and PCj , are metric projections onto Cj .
Under certain conditions imposed on {λn}, a weak convergence result was obtained.

Since any nonempty, closed, and convex subset of a Hilbert space can be represented
as the fixed point set of its corresponding projector, see [7], therefore, equation (3) can be
simplified to finding

y ∈ Fix(T1) and z ∈ Fix(T2) such that A1y = A2z, (5)

where Tj : Hj → Hj , j = 1, 2, are nonlinear operators with Fix(Tj) ̸= ∅. Problem (5) is
known as the Split Equality Fixed Point Problem (SEFPP).

Motivated by the results in [8], Moudafi [5] proposed the following algorithm:{
yn+1 = T2 (yn + λnA∗

2(A1yn −A2zn)) ;
zn+1 = T1 (zn − λnA∗

1(A1yn −A2zn)) , n ≥ 0.
(6)

By imposing certain conditions on the parameters and operators involved, a weak
convergence result for algorithm (6) was obtained. However, implementing this algorithm
requires computing the inverse of a bounded linear operator, which is generally a difficult
task. To address this challenge, Byrne [2] introduced an alternative algorithm for solving
the SFP that eliminates the need for such an inverse.

Moudafi’s algorithm in [5] involved firmly quasi-nonexpansive mapping, a class that
includes quasi-nonexpansive mapping. Since quasi-pseudocontractive mapping includes
firmly quasi-nonexpansive, directed, and demicontractive mappings, this motivated Chang
et al., [9] to introduce the following algorithm for solving the SEFPP involving quasi-
pseudocontractive mappings and proved the weak convergence result of the algorithm:

yn+1 = βnyn + (1− βn)
(
(1− η)I + ηT1((1− ζ)I + ζT2)

)
vn;

vn = yn + λnA∗
2(A1yn −A2zn);

zn+1 = βnzn + (1− βn)
(
(1− η)I + ηT1((1− ζ)I + ζT1)

)
un;

un = zn − λnA∗
1(A1yn −A2zn), n ≥ 0.

(7)



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 3 of 18

This algorithm relies on prior knowledge of operator norms. Recently, Mohammed
and Kilicman [10] investigated the SEFPP involving quasi-pseudocontractive mappings
in Hilbert spaces. They developed innovative algorithms and demonstrated their conver-
gences, both with and without prior knowledge of the operator norm for bounded and
linear mappings.

Recently, Wang et al., [11], studied the SEFPP for the class of demicontractive opera-
tors in Hilbert spaces and proved the strong convergence results by proposing the following
algorithm: 

yn+1 = βng1(yn) + (1− βn)vn,
vn = yn − λn (yn − T1yn +A∗

1 (A1yn −A2zn)) ,

zn+1 = βng2(yn) + (1− βn)wn,
wn = zn − λn (zn − T2zn + λnA∗

2 (A2zn −A1yn)) , n ≥ 0.

(8)

The results from Chang et al. [9] and Mohammed and Kilicman [10], on the other hand,
only show strong convergence if the operators are thought to be semi-compact. This com-
pactness condition can be limiting, as many nonlinear mappings do not satisfy the com-
pactness condition (see Example 1 for more details). Consequently, it was proposed that
future studies could be focused on establishing strong convergent results without relying
on the compactness assumption. In this paper, we aimed to obtain the strong convergence
result of the proposed algorithm without imposing the semi-compactness condition on the
operators involved, which is a critical consideration in infinite-dimensional spaces.

The paper is organized as follows: The introduction offers an overview of the study’s
background and context. This is followed by the preliminary section, where key definitions
and lemmas are introduced. Section 3 presents the main results of the research, and Section
4 discusses the numerical results.

2. Preliminaries

This section offers a few fundamental findings that support the paper’s primary find-
ings.

Definition 1. A mapping T1 : H1 → H1 is said to be;

(i) Fixed point of T1
(
Fix(T1)

)
if T1z = z, for all z ∈ H1, and we denote the set of

Fix(T1) by {z ∈ Fix(T1) : T1z = z}.

(ii) Nonexpansive if ∥T1y − T1z∥ ≤ ∥y − z∥,∀y, z ∈ H1.

(iii) Quasi-nonexpansive if ∥T1y − z∥ ≤ ∥y − z∥,∀y ∈ H1 and z ∈ Fix(T1).

(iv) Directed if ∥T1y − z∥2 ≤ ∥y − z∥2 − ∥T1y − y∥2,∀y ∈ H1 and z ∈ Fix(T1).

(v) Demicontractive if ∥z − T1y∥2 ≤ ∥z − y∥2 + k∥y − T1y∥2, ∀y ∈ H1, z ∈ Fix(T1)
and k ∈ [0, 1), and it is called a quasi-pseudocontractive if k = 1.



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 4 of 18

(vi) Semi-compact if for any bounded sequence {zn} ⊆ H1 with ∥zn − T1zn∥ → 0, then
there exist {zni} ⊆ {zn} such that zni → z ∈ H1.

Remark 1. From the definitions provided above, we observe that the class of quasi pseu-
docontractive mapping is fundamental. This class encompasses various types of nonlin-
ear mappings, including demicontractive mapping, directed mapping, quasi-nonexpansive
mapping, and strictly pseudocontractive mapping, all of which serve as special cases. For
further details and examples, see [9, 10].

Remark 2. It is obvious that if T1 is quasi-nonexpansive then

∥T1y − y∥ ≤ 2 ⟨y − T1y, y − z⟩ .

Lemma 1. (Chang et al., [9] ) Suppose T1 : H1 → H1 is Lipschitz with L > 0, and
U1 := (1− η)I + ηT1((1− ζ)I + ζT1), then

(i) Fix(T1) = Fix((1− η)I + ηT1((1− ζ)I + ζT1)) = Fix(U1);

(ii) U1 is demiclosed at zero only if T is demiclosed at zero;

(iii) U1 is L2− Lipschitzian;

(iv) U1 is quasi-nonexpansive only if T1 is quasi-pseudocontractive.

Lemma 2. (Xu, [12]) Let {an}, {bn} ⊆ R+ such that
∑∞

n=0 bn < ∞. If

an+1 ≤ (1 + bn)an or an+1 ≤ an + bn, ∀n ≥ 0,

then lim
n→∞

an exists.

Lemma 3. (Xu, [12]) Let bn > 0 for n ∈ N, and suppose the following recurrence holds:

bn+1 ≤ (1− αn)bn + αnΘn + εn, n ≥ 0,

where {αn} ⊂ (0, 1) and {Θn} ⊂ R satisfy the conditions:

(i)
∑∞

n=0 αn = ∞;

(ii) εn ≥ 0 for all n ≥ 0, and
∑∞

n=0 εn < ∞;

(iii) lim sup
n→∞

Θn ≤ 0 or
∑∞

n=1 αn|Θn| < ∞.

Then, lim
n→∞

bn = 0.



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 5 of 18

3. Main Results

In what follows, S will denote the solution set of equation (1), that is,

S := {y ∈ Fix(T1) and z ∈ Fix(T2) such that A1y = A2z}. (9)

Suppose that for j = 1, 2, the following assumptions hold:

(K1) Tj : Hj → Hj , are two quasi-pseudocontractive operators with Fix(Tj) ̸= ∅, in
addition, T1 is also L- Lipschitz.

(K2) Aj : Hj → Hj are linear and bounded operators with their adjoints A∗
j .

(K3) (Tj − I) are demiclosed at origin.

(K4) Fj : Hj → Hj are contraction mappings with contraction constant ρ ∈ (0, 1];

(K5) Let Uj = (1− η)I + ηTj((1− ζ)I + ζTj), and define (yn, zn) ⊆ H1 ×H2 by
yn+1 = αnF1(vn) + (1− αn)U1vn;
vn = (1− τn)yn + τnU1yn + τnA∗

1(A1yn −A2zn);

zn+1 = αnF2(wn) + (1− αn)U2wn;
wn = (1− τn)zn + τnU2zn + τnA∗

2(A2zn −A1yn), ∀n ≥ 0;

(10)

where (y0, z0) ∈ H1 × H2 are chosen arbitrary, 0 < αn < 1 such that
∑
n≥1

αn = ∞,∑
n≥1

(1− αn)αn < ∞, and lim
n→∞

αn = 0, and 0 < τn < 1 such that

inf
n≥1

τn
(
(1− αn(1− 2ρ2))λ− τn

)
≥ β > 0,

where λ = 1
2max{1,∥A1∥2,∥A2∥2}

, and 0 < η < ζ < 1
1+

√
1+L2

.

Lemma 4. Let {(yn, zn)} be the sequence generated by algorithm (10), then

(i) {(yn, zn)} is bounded;

(ii) lim sup
n→∞

∥U1yn − yn +A∗
1(A2zn −A1yn)∥2 = 0, and

lim sup
n→∞

∥U2zn − zn +A∗
2(A1yn −A2zn)∥2 = 0;

(iii) lim sup
n→∞

∥U1yn − yn∥ = 0, lim sup
n→∞

∥U2zn − zn∥ = 0, and lim sup
n→∞

∥A1yn −A2zn∥ = 0.

Proof. Let (y, z) ∈ S. By Lemma 1, it follows that U1 is quasi-nonexpansive. By
algorithm (10) we have

∥zn+1 − z∥2 = ∥αnF2(wn) + (1− αn)U2wn − z∥2



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 6 of 18

= ∥αn(F2(wn)− z) + (1− αn)(U2wn − z)∥2

=αn ∥F2(wn)−F2(z) + F2(z)− z∥2 + (1− αn) ∥U2wn − z∥2

− (1− αn)αn ∥U2wn −F2(wn)∥2

≤(1− αn(1− 2ρ2))
∥∥wn − z∥2 + 2αn∥F2(z)− z

∥∥2 , and (11)

∥wn − z∥2 =
∥∥∥(1− τn)(zn − z) + τn

(
U2zn +A∗

2(A2zn −A1yn)− z
)∥∥∥2

=(1− τn) ∥zn − z∥2 + τn∥U2zn − zn +A∗
2(A2zn −A1yn) + zn − z∥2

− (1− τn)τn∥U2zn − zn +A∗
2(A2zn −A1yn)∥2

≤∥zn − z∥2 + τ2n∥U2zn − zn +A∗
2(A2zn −A1yn)∥2

− 2τn ⟨zn − z, zn − U2zn −A∗
2(A2zn −A1yn)⟩ . (12)

Thus, by equations (11) and (12), we have

∥zn+1 − z∥2 ≤(1− αn(1− 2ρ2))
∥∥zn − z∥2 + 2αn∥F2(z)− z

∥∥2
+ τ2n∥U2zn − zn +A∗

2(A2zn −A1yn)∥2

− 2(1− αn(1− 2ρ2))τn ⟨zn − z, zn − U2zn −A∗
2(A2zn −A1yn)⟩ . (13)

Similarly,

∥yn+1 − y∥2 ≤(1− αn(1− 2ρ2))
∥∥yn − y∥2 + 2αn∥F1(y)− y

∥∥2
+ τ2n ∥U1yn − yn +A∗

1(A1yn −A2zn)∥2

− 2(1− αn(1− 2ρ2))τn ⟨yn − y, yn − U1yn −A∗
1(A1yn −A2zn)⟩ . (14)

By equations (13) and (14), we deduce that

Γn+1 ≤(1− αn(1− 2ρ2))Γn + 2αn

(
∥F2(z)− z∥2 + ∥F1(y)− y∥2

)
+ τ2nkn

− 2τn(1− αn(1− 2ρ2))
(
⟨zn − z, zn − U2zn −A∗

2(A2zn −A1yn)⟩

+ ⟨yn − y, yn − U1yn −A∗
1(A1yn −A2zn)⟩

)
, (15)

where Γn := ∥yn − y∥2 + ∥zn − z∥2 , and

kn := ∥U2zn − zn +A∗
2(A2zn −A1yn)∥2 + ∥U1yn − yn +A∗

1(A1yn −A2zn)∥2 .

On the other hand,

⟨zn − z, zn − U2zn −A∗
2(A2zn −A1yn)⟩ = ⟨zn − z, zn − U2zn⟩

− ⟨A2zn −A2z,A2zn −A1yn⟩

≥1

2
∥zn − U2zn∥2



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 7 of 18

− ⟨A2zn −A2z,A2zn −A1yn⟩ . (16)

Similarly,

⟨yn − y, yn − U1yn −A∗
1(A1yn −A2zn)⟩ ≥

1

2
∥yn − U1yn∥2

− ⟨A1yn −A1y,A1yn −A2zn⟩ . (17)

By (16) and (17), and the fact that A1y = A2z, we have

⟨zn − z, zn − U2zn −A∗
2(A2zn −A1yn)⟩+ ⟨yn − y, yn − U1yn −A∗

1(A1yn −A2zn)⟩

≥1

2
∥zn − U2zn∥2 − ⟨A2zn −A2z,A2zn −A1yn⟩

+
1

2
∥yn − U1yn∥2 − ⟨A1yn −A1y,A1yn −A2zn⟩

=
1

2

(
∥zn − U2zn∥2 + ∥A1yn −A2zn∥2

)
+
1

2

(
∥yn − U1yn∥2 + ∥A1yn −A2zn∥2

)
≥1

2

(
∥zn − U2zn∥2 +

1

∥A2∥2
∥A∗

2(A1yn −A2zn)∥2
)

+
1

2

(
∥yn − U1yn∥2 +

1

∥A1∥2
∥A∗

1(A1yn −A2zn)∥2
)

≥ 1

2max{1, ∥A2∥2}

(
∥zn − U2zn∥2

+ ∥A∗
2(A1yn −A2zn)∥2

)
+

1

2max{1, ∥A1∥2}

(
∥yn − U1yn∥2

+ ∥A∗
1(A1yn −A2zn)∥2

)
≥ 1

4max{1, ∥A1∥2 , ∥A2∥2}

((
∥yn − U1yn∥

+ ∥A∗
1(A1yn −A2zn)∥

)2

+
(
∥zn − U2zn∥2

+ ∥A∗
2(A1yn −A2zn)∥

)2)
≥λ

2

(
∥zn − U2zn −A∗

2(A2zn −A1yn)∥2

+∥yn − U1yn −A∗
1(A1yn −A2zn)∥2

)
. (18)

By equations (15) and (18), and noticing that (1− αn(1− 2ρ2)) > 0, we have



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 8 of 18

Γn+1 ≤ (1− αn(1− 2ρ2))Γn + 2αn

(
∥F2(z)− z∥2 + ∥F1(y)− y∥2

)
− τn

(
(1− αn(1− 2ρ2))λ− τn

)
kn (19)

≤ (1− αn(1− 2ρ2))Γn + 2αn

(
∥F2(z)− z∥2 + ∥F1(y)− y∥2

)
.

Thus,

Γn+1 ≤(1− αn(1− 2ρ2))Γn +
αn(1− 2ρ2)

(1− 2ρ2)

(
2∥F2(z)− z∥2 + 2∥F1(y)− y∥2

)
.

≤max
{
Γn, 2∥F2(z)− z∥2 + 2∥F1(y)− y∥2

}
.

By induction, we deduce that

Γn+1 ≤max
{
Γ0, 2∥F2(z)− z∥2 + 2∥F1(y)− y∥2

}
.

Thus Γn := ∥zn − z∥2 + ∥yn − y∥2 is bounded, which further implies that {(yn, zn)} is
also bounded.

By equation (19), we have

τn
(
(1− αn(1− 2ρ2))λ− τn

)(
∥zn − U2zn −A∗

2(A2zn −A1yn)∥2

+ ∥yn − U1yn −A∗
1(A1yn −A2zn)∥2

)
≤ ∥zn − z∥2 − ∥zn+1 − z∥2

+ ∥yn − y∥2 − ∥yn+1 − y∥2

+ 2αn

(
∥F2(z)− z∥2 + ∥F1(y)− y∥2

)
.

Since inf
n≥1

τn
(
(1− αn(1− 2ρ2))λ− τn

)
≥ β, we therefore deduce that

lim sup
n→∞

(
∥zn − U2zn −A∗

2(A2zn −A1yn)∥2 + ∥yn − U1yn −A∗
1(A1yn −A2zn)∥2

)
= 0.

This turns to implies that

lim sup
n→∞

∥yn − U1yn −A∗
1(A1yn −A2zn)∥2 = 0, and

lim sup
n→∞

∥zn − U2zn −A∗
2(A2zn −A1yn)∥2 = 0. (20)

Since U1 is quasi-nonexpansive mapping, by Remark 2, we have

1

2
∥yn − U1yn∥2 + ⟨A1yn −A2zn,A1yn −A1y⟩ ≤ ⟨yn − U1yn, yn − y⟩



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 9 of 18

+ ⟨A1yn −A2zn,A1yn −A1y⟩
= ⟨yn − U1yn −A∗

1(A1yn −A2zn), yn − y⟩
≤ ∥yn − U1yn −A∗

1(A1yn −A2zn)∥∥yn − y∥. (21)

Since {∥yn − y∥} is bounded, thus, by equation (20) we deduce that

lim sup
n→∞

∥yn − U1yn∥ = 0. (22)

Similarly,

lim sup
n→∞

∥zn − U2zn∥ = 0. (23)

On the other hand,

∥A1yn −A2zn∥2 = ⟨A1yn −A2zn,A1yn −A2zn⟩
= ⟨A1yn −A2zn,A1yn −A1y⟩+ ⟨A2zn −A1yn,A2zn −A2z⟩ ,
= ⟨A∗

1(A1yn −A2zn), yn − y⟩+ ⟨A∗
2(A2zn −A1yn), zn − z⟩

≤∥A∗
1(A1yn −A2zn) + (yn − U1yn)− (yn − U1yn)∥∥yn − y∥∥

+ ∥A∗
2(A2zn −A1yn) + (zn − U2zn)− (zn − U2zn)∥∥zn − z∥

≤∥A∗
1(A1yn −A2zn)− (yn − U1yn)∥∥yn − y∥

+ ∥yn − U1yn∥∥yn − y∥
+ ∥A∗

2(A2zn −A1yn)− (zn − U2zn)∥∥zn − z∥
+ ∥zn − U2zn∥∥zn − z∥.

Thus, by (20) and the fact that {∥zn − z∥} and {∥yn − y∥} are bounded, we see that

lim sup
n→∞

∥A1yn −A2zn∥ = 0.

This completes the proof of this lemma.

We are now in a position to prove that (yn, zn) → (y, z).

Theorem 1. Suppose conditions (K1)–(K5) are satisfied, and that S ̸= ∅. Then the
sequence {(yn, zn)} generated by algorithm (10) converges strongly to (y, z) ∈ S.

Proof. Let (y, z) ∈ S, then by Lemma 1, we see that U1 is quasi-nonexpansive, and by
algorithm (10), we have

∥yn+1 − y∥2 = ∥αn(F1(vn)− y) + (1− αn)(U1vn − y)∥2

=α2
n ∥F1(vn)− y∥2 + (1− αn)

2∥U1vn − y∥2

+ 2(1− αn)αn ⟨F1(vn)− y,U1vn − y⟩
=(α2

nρ
2 + (1− αn)

2)∥vn − y∥2 + α2
n ∥F1(y)− y∥2



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 10 of 18

+ 2(1− αn)αn ⟨F1(vn)− y,U1vn − y⟩ , and (24)

2 ⟨F1(vn)− y,U1vn − y⟩ =2 ⟨F1(vn)−F1(y),U1vn − y⟩+ 2 ⟨F1(y)− y,U1vn − y⟩
≤(1 + ρ2)∥vn − y∥2 + 2 ⟨F1(y)− y,U1vn − y⟩
≤(1 + ρ2)∥vn − y∥2 + ∥F1(y)− y∥2 + ∥U1vn − vn∥2

+ 2 ⟨F1(y)− y, vn − y⟩ . (25)

By equations (24) and (25), we have

∥yn+1 − y∥2 ≤(α2
nρ

2 + (1− αn)
2)∥vn − y∥2 + α2

n ∥F1(y)− y∥2

+ (1− αn)αn

(
(1 + ρ2)∥vn − y∥2 + ∥F1(y)− y∥2 + ∥U1vn − vn∥2

+ 2 ⟨F1(y)− y, vn − y⟩
)

≤(1− (1− ρ2)αn)∥vn − y∥2 + αn∥F1(y)− y∥2

+ (1− αn)αn

(
∥U1vn − vn∥2 + 2 ⟨F1(y)− y, vn − y⟩

)
. (26)

By equations (12) and (26), we see that

∥yn+1 − y∥2 ≤(1− (1− ρ2)αn) ∥yn − y∥2 + αn∥F1(y)− y∥2

+ (1− (1− ρ2)αn)τ
2
n∥U1yn − yn +A∗

1(A1yn −A2zn)∥2

− 2(1− (1− ρ2)αn)τn ⟨yn − y, yn − U1yn −A∗
1(A1yn −A2zn)⟩

+ (1− αn)αn

(
∥U1vn − vn∥2 + 2 ⟨F1(y)− y, vn − y⟩

)
. (27)

Similarly,

∥zn+1 − z∥2 ≤(1− (1− ρ2)αn)∥zn − z∥2 + αn∥F2(z)− z∥2

+ (1− (1− ρ2)αn)τ
2
n∥U2zn − zn +A∗

2(A2zn −A1yn)∥2

− 2(1− (1− ρ2)αn)τn ⟨zn − z, zn − U2zn −A∗
2(A2zn −A1yn)⟩

+ (1− αn)αn

(
∥U2wn − wn∥2 + 2 ⟨F2(z)− z, wn − z⟩

)
. (28)

Equations (18), (27), and (28) give

Γn+1 ≤(1− (1− ρ2)αn)Γn + αn

(
∥F1(y)− y∥2 + ∥F2(z)− z∥2

)
− τn((1− (1− ρ2)αn)λ− τn)kn

+ (1− αn)αn

(
∥U1vn − vn∥2 + ∥U2wn − wn∥2

+ 2 ⟨F1(y)− y, vn − z⟩+ 2 ⟨F2(z)− z, wn − y⟩
)

≤(1− (1− ρ2)αn)Γn + αn

(
∥F1(y)− y∥2 + ∥F2(z)− z∥2

)



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 11 of 18

+ (1− αn)αn

(
∥U1vn − vn∥2 + ∥U2wn − wn∥2

+ 2∥F1(y)− y∥∥vn − y∥+ 2∥F2(z)− z∥∥wn − z∥
)
. (29)

Since (yn, zn) is bounded, we see that ∥wn − z∥ is bounded, and similarly, ∥vn − y∥ is
also bounded.

On the other hand,

∥U1vn − vn∥ ≤ ∥U1vn − y∥+ ∥vn − y∥
≤ 2∥vn − y∥. (30)

Similarly,

∥U2wn − wn∥ ≤ 2∥wn − z∥. (31)

Therefore, ∥U1vn−vn∥ and ∥U2wn−wn∥ are also bounded. Since (yn, zn) is bounded, it
follows that there exist (y, z) ∈ S for which zn ⇀ z and yn ⇀ y. Thus wn = zn−τn(U2zn−
zn) + τnA∗

1(A2zn −A1yn) ⇀ z and vn = yn − τn(U1yn − yn) + τnA∗
2(A1yn −A2zn) ⇀ y.

On the other hand, we see that

lim sup
n→∞

(
∥U1vn − vn∥2 + ∥U2wn − wn∥2

+ 2∥F2(z)− z∥∥wn − z∥+ 2∥F1(y)− y∥∥vn − y∥
)
= 0. (32)

By equation (29), we deduce that

Γn+1 ≤(1− (1− ρ2)αn)Γn + (1− ρ2)αn
αn(∥F1(y)− y∥2 + ∥F2(z)− z∥2)

(1− ρ2)

+ (1− αn)αn

(
∥U1vn − vn∥2 + ∥U2wn − wn∥2 + ∥F1(y)− y∥2 + ∥F2(z)− z∥2

+ 2∥F1(y)− y∥∥vn − y∥+ 2∥F2(z)− z∥∥wn − z∥
)
.

Thus, we see that

(i)
∑
n≥0

(1− ρ2)αn = ∞;

(ii) lim
n→∞

αn(∥F1(y)−y∥2+∥F2(z)−z∥2)
(1−ρ2)

= 0; and
∑
n≥0

(1− ρ2)α
2
n(∥F1(y)−y∥2+∥F2(z)−z∥2)

(1−ρ2)
< ∞;

(iii) ηn ≥ 0 and
∑
n≥0

ηn < ∞;

where ηn = (1 − αn)αn

(
∥U1vn − vn∥2 + ∥U2wn − wn∥2 + ∥F1(y) − y∥2 + ∥F2(z) −

z∥2 + 2∥F1(y)− y∥∥vn − y∥+ 2∥F2(z)− z∥∥wn − z∥
)
.



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 12 of 18

therefore, by Lemma 3, we deduce that lim
n→∞

Γn = 0. That is

lim
n→∞

(
∥yn − y∥2 + ∥zn − z∥2

)
= 0.

Finally, we show that (y, z) ∈ S.
Since, U1 = (1 − η)I − ηT1((1 − ζ)I + ζT1)I, where 0 < η < ζ < 1

1+
√
1+L2

and T1 is

Lipschitz, we have

η∥yn − T1yn∥ = ∥yn − (1− η)yn − ηT1yn∥
= ∥yn − (1− η)yn − ηT1((1− ζ)I + ζT1)yn + ηT1((1− ζ)I + ζT1)yn − ηT1yn∥
≤ ∥yn − (1− η)yn − ηT1((1− ζ)I + ζT1)yn∥
+ ∥ηT1((1− ζ)I + ζT1)yn − ηT1yn∥
≤ ∥yn − U1yn∥+ ηL∥((1− ζ)I + ζT1)yn − yn∥
= ∥yn − U1yn∥+ ηζL∥yn − T1yn∥.

Therefore,

∥yn − T1yn∥ ≤ 1

(1− ζL)η
∥yn − U1yn∥. (33)

Similarly,

∥zn − T2zn∥ ≤ 1

(1− ζL)η
∥zn − U2zn∥. (34)

By (22) and (23), we see that

lim
n→∞

∥yn − T1yn∥ = 0, and lim
n→∞

∥zn − T2zn∥ = 0. (35)

Now that yn → y and lim
n→∞

∥T1yn − yn∥ = 0 couple with the demiclosedness of (T1− I)

at origin, we have y ∈ Fix(T1).
Similarly, zn → z and lim

n→∞
∥T2zn − zn∥ = 0 together with the demiclosedness of (T2−I)

at origin, we see that z ∈ Fix(T2).
On the other hand, since yn → y, it is not difficult to see that

vn = (1− τn)yn + τnU1yn + τnA∗
1(A1yn −A2zn) → 0.

Similarly, wn → z.
Since A1 and A2 are continuous mappings, we have

A1vn → A1y, and A2wn → A2z.

This implies that
A1vn −A2wn → A1y −A2z,



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 13 of 18

which further implies that

∥A1y −A2z∥ ≤ lim inf
n→∞

∥A1vn −A2wn∥ = 0.

Thus, A1y = A2z, and noticing that (y, z) ∈ Fix(T1) × Fix(T2), we conclude that
(y, z) ∈ S, which completes the proof.

Corollary 2. Suppose conditions (K1) − (K4) are satisfied, and that S ≠ ∅. Then the
sequence {(yn, zn)} generated by

yn+1 = αnvn + (1− αn)U1vn;
vn = (1− τn)yn + τnU1yn + τnA∗

1(A1yn −A2zn);

zn+1 = αnwn + (1− αn)U2wn;
wn = (1− τn)zn + τnU2zn + τnA∗

2(A2zn −A1yn), ∀n ≥ 0;

(36)

where Uj = (1−η)I+ηTj((1−ζ)I+ζTj), j = 1, 2, (y0, z0) ∈ H1×H2 are chosen arbitrary,
0 < αn < 1 such that

∑
n≥1

αn = ∞,
∑
n≥1

(1− αn)αn < ∞, and lim
n→∞

αn = 0, 0 < τn < 1

such that inf
n≥1

τn
(
(1− αn(1− 2ρ2))λ− τn

)
≥ β > 0, where λ = 1

2max{1,∥A1∥2,∥A2∥2}
, and

0 < η < ζ < 1
1+

√
1+L2

. Then the sequence {(yn, zn)} converges to (y, z) ∈ S.
Proof. This proof is a direct consequence of Theorem 1 by setting F = I. Algorithm

36 was studied by Mohammed and Kilicman [10] in their work.

Corollary 3. Suppose conditions (K1) − (K4) are satisfied, and that S ≠ ∅. Then the
sequence {(yn, zn)} generated by

yn+1 = αnvn + (1− αn)U1vn;
vn = yn + λnA∗

1(A1yn −A2zn);

zn+1 = αnwn + (1− αn)U2wn;
wn = zn + λnA∗

2(A2zn −A1yn),∀n ≥ 0;

(37)

where Uj = (1−η)I+ηTj((1−ζ)I+ζTj), j = 1, 2, (y0, z0) ∈ H1×H2 are chosen arbitrary,
0 < αn < 1, such that

∑
n≥1

αn = ∞,
∑
n≥1

(1− αn)αn < ∞, and lim
n→∞

αn = 0, 0 < λn < 1, and

0 < η < ζ < 1
1+

√
1+L2

. Then the sequence {(yn, zn)} converges to (y, z) ∈ S.
Proof. This proof follows directly from Theorem 1 by taking F = I, and τn = 0.

Algorithm 37 was proposed by (Chang et al., [9]).

Corollary 4. Suppose conditions (K1) − (K4) are satisfied, and that S ≠ ∅. Then the
sequence {(yn, zn)} generated by



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 14 of 18


yn+1 = U1vn;
vn = yn + λnA∗

1(A1yn −A2zn);

zn+1 = U2wn;
wn = zn + λnA∗

2(A2zn −A1yn),∀n ≥ 0;

(38)

where Uj = (1−η)I+ηTj((1− ζ)I+ ζTj), j = 1, 2 (y0, z0) ∈ H1×H2 are chosen arbitrary,
0 < αn < 1, such that

∑
n≥1

αn = ∞,
∑
n≥1

(1− αn)αn < ∞, and lim
n→∞

αn = 0, and 0 < λn < 1,

and 0 < η < ζ < 1
1+

√
1+L2

. Then the sequence {(yn, zn)} converges to (y, z) ∈ S.
Proof. This proof follows directly from Theorem 1 by taking F = I, and τn = αn = 0.

Algorithm 38 was proposed and studied by (Moudafi and Al-Shemas [8]).

Corollary 5. Suppose conditions (K1) − (K4) are satisfied, and that S ≠ ∅. Then the
sequence {(yn, zn)} generated by

yn+1 = βnF1(yn) + (1− βn)vn,
vn = yn − λn (yn − T1yn +A∗

1 (A1yn −A2zn)) ,

zn+1 = βnF2(yn) + (1− βn)wn,
wn = zn − λn (zn − T2zn +A∗

2 (A2zn −A1yn)) , n ≥ 0;

(39)

where (y0, z0) ∈ H1 × H2 are chosen arbitrary, 0 < βn < 1, such that
∑
n≥1

βn = ∞,∑
n≥1

(1− βn)βn < ∞, and lim
n→∞

βn = 0, 0 < λn < 1 such that inf
n≥1

λn (λ− λn) ≥ β > 0,

where λ = 1
2max{1,∥A1∥2,∥A2∥2}

. Then the sequence {(yn, zn)} converges to (y, z) ∈ S.
Proof. This proof is a direct consequence of Theorem 1 by setting Uj = (1 − η)I +

ηTj((1− ζ)I + ζTj) = I. Algorithm 39 was studies by Wang et al., [11] in their work.

4. Numerical Examples

This section presents numerical results that demonstrate our theoretical findings and
compares them with some existing results from the literature.

The following example is an example of a nonlinear mapping that is not semi-compact

Example 1. Let H1 = ℓ2(N), and define a mapping T : ℓ2 → ℓ2 by

T (x) =

{(
1− 1

n

)
en, if x = en for some n,

0, otherwise.

then T is not semi-compact.



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 15 of 18

Proof. Clearly, T is not linear. L et {xn} ⊆ ℓ2 defined by xn = en, where en is
the standard basis vector with 1 in the n-th position and 0 elsewhere. This sequence is
bounded since ∥en∥ = 1 for all n.

On the other hand, ∥xn − T (xn)∥ = 1
n → 0

However, no subsequence of {xn} converges strongly in ℓ2 since

∥xn − xm∥ = ∥en − em∥ =
√
2

does not converge to 0. Thus, T is not semi-compact.

Example 2. The mapping F : R → R define by F(z) = 1
2 sin(z), for all z ∈ R is a

contraction mapping.

Proof. Clearly, the function is continuous for all z ∈ R. By the Mean Value Theorem
(MVT), there exists c ∈ R such that

|F(y)−F(z)| = 1

2
| sin(y)− sin(z)|

≤ cos(c)|y − z|
≤ |y − z|.

Thus we see that F is a contraction mapping with a contraction constant cos(c).

The following are examples of quasi pseudocontractive mapping.

Example 3. The mapping F : R → R define by F(z) = z
2 for all z ∈ R then F is quasi

pseudocontractive with Fix(F) = 0.

Example 4. The mapping F : R → R define by F(x) = x+ sin(x) for all x ∈ R then F
is quasi pseudocontractive.

Proof. It is not difficult to see that with Fix(F) = 0, F is quasi pseudocontractive
mapping.

Corollary 6. In Theorem (9), let H1 = R, C,Q ⊆ (0,∞), and define the operators
A1y = y and A2y = z

5 . It follows that A1 = A∗
1 = 1 and A2 = A∗

2 = 1
5 , respectively. Let

F : R → R be defined by

F(z) =
1

2
sin(z), for all z ∈ R,

which is a contraction mapping. Define the mappings T1 : C → R and T2 : Q → R as

T1y =
y

2
, ∀y ∈ C,

and
T2z = z + sin(z), ∀z ∈ Q.

Clearly, T1 and T2 are quasi-pseudocontractive mappings with fixed points Fix(T1) = 0 and
Fix(T2) = 0. Let η = 1

5 , ξ = 1
7 , τn = 1

9 , and αn = 1
11 . These parameters satisfy the

hypotheses of Theorem 9. By setting the number of iterations to 150 and using Maple, we
obtain the following results:



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 16 of 18

n
Algorithm 10

{yn} {zn}
1 1.0000000000 1.0000000000

2 0.8958712178 0.9167912023

3 0.8037675809 0.8418283630

4 0.7219891992 0.7740135339

. . .

. . .

. . .

148 0.0000000258 0.0000110780

149 0.0000000233 0.0000102608

150 0.0000000210 0.0000095038

Table 1: The numerical results of algorithm (10), starting with the initial values y1 = 1 and z1 = 1, showed
how the sequence (yn, zn) converges to (0, 0).

Figure 1: Graphical results presentation of algorithm (10), starting with the initial values y1 = 1 and z1 = 1,
demonstrated how the sequence (yn, zn) converges to (0, 0).

The following Table provides a comparison of algorithm (10) as presented in Table 1
with some of the recent results published in the literature.



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 17 of 18

n
Algorithm 36 Algorithm 37 Algorithm 39
{yn} {zn} {yn} {zn} {yn} {zn}

1 1.00000 1.00000 1.00000 1.00000 1.00000 1.0000

2 0.94774 0.96880 0.95916 0.97832 1.07480 1.0935

3 0.89822 0.93872 0.91999 0.95721 1.14970 1.1725

. . . . . . .

. . . . . . .

. . . . . . .

148 0.00036 0.01277 0.00209 0.05188 1.93210 2.0360

149 0.00034 0.01240 0.00200 0.05088 1.93210 2.0360

150 0.00032 0.012046 0.00192 0.04989 1.93210 2.0360

Table 2: This table presents the convergence rates of Algorithms 36, 37, and 39, as studied by Mohammed
and Kilicman [10], Chang et al. [9], and Wang et al. [11], respectively. By comparing Algorithm 10 with
these algorithms, it is clear that the proposed in this paper converges more rapidly. This demonstrates that
the method proposed in this paper is more efficient in terms of convergence speed compared to the existing
methods.

Figure 2: Graphical results presentation of algorithms (10), (36), (37) and (39) starting with the initial values
y1 = 1 and z1 = 1, demonstrated how the sequence (yn, zn) converges to (0, 0).

5. Conclusion

This study focused on the Split Equality Fixed-Point Problem (SEFPP) within the
context of quasi-pseudocontractive mappings in Hilbert spaces. We introduced new vis-
cosity algorithms for solving this problem, which indicated that the proposed algorithms
converged strongly in an infinite-dimensional Hilbert space. Our findings not only broad-
ened several key results from the existing literature but also addressed the computational
challenges associated with calculating operator norms. To support our theoretical results,
numerical experiments were performed, and a comparison with existing methods confirmed
the superior efficiency and effectiveness of our proposed algorithms.

In summary, the algorithms developed in this paper provide a reliable solution to the
SEFPP, achieving strong convergence without the need for the compactness assumption on



L. B. Mohammed, A. Kılıçman, D. Bamanga / Eur. J. Pure Appl. Math, 18 (4) (2025), 6154 18 of 18

the operators involved. Our research advances the field of fixed-point theory by extending
prior results and offering more practical solutions to the SEFPP, especially in scenar-
ios where computing operator norms is challenging. The algorithms’ strong convergence
makes them well-suited for practical applications.

Acknowledgements

The authors wish to express their gratitude to the Tertiary Education Trust Fund
(TETFund) for the financial support provided for the conduct of this research under its
Institutional-Based Research (IBR) scheme

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