EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5826
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Common Fixed Points of Asymptotically Regular
Mappings in Convex Metric Spaces with Application

Abdul Rahim Khan1, Godwin Chidi Ugwunnadi2,3,∗, Maggie Aphane3,
Faizan Yousaf1, Amir Abbas1

1 Department of Mathematics and Statistics, University of Southern Punjab, Multan,
Pakistan
2 Department of Mathematics, Faculty of Science and Engineering, University of Eswatini,
Private Bag 4, Kwaluseni M201, Eswatini
3 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences
University, Medunsa, P.O. Box 94, Pretoria 0204, South Africa

Abstract. In this paper, based upon Górnicki’s work on fixed points of a continuous asymptoti-
cally regular self-mapping on a metric space, we establish common fixed point results for similar
self-mappings and their average mappings on a convex metric space by using various types of con-
tractive conditions. The closedness and convexity of the set of fixed points of a non- self- mapping
is obtained here in the context of a uniformly convex hyperbolic space. We also apply our findings
to solve Volterra type integral equations, demonstrating practical use of our work in mathematical
analysis and its related fields.

2020 Mathematics Subject Classifications: 47H05, 47J20, 47J25, 65K15

Key Words and Phrases: Górnicki Type Contraction Mapping; Common Fixed Point; Convex
Metric Space, Uniformly Convex Hyperbolic Space; Asymptotically Regular Mapping; Nonlinear
Integral Equations

1. Introduction

Fixed point theory is an integral part of modern mathematics, offering essential tools and
techniques for resolving various problems in nonlinear analysis, optimization, economics,
and engineering. Over the past two decades, the development of this theory in metric-
type spaces has garnered significant attention from researchers, especially its usefulness
for solving many existence problems in nonlinear differential and integral equations with
applications in engineering and applied sciences [1]. The relevance and applicability of

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5826

Email addresses: abdulrahimkhan@isp.edu.pk (A. R. Khan), gcugwunnadi@uniswa.sz (G. C.
Ugwunnadi), maggie.aphane@smu.ac.za (M. Aphane), faizanyousef967@gmil.com ( F. Yousaf),
aamirbuzdar112@gmail.com (A. Abbas)

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



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 2 of 23

fixed point theory continues to expand as new results and methodologies emerge.
Recently, Górnicki [2] generalized a well-known result of Reich [3] related to contractions
on a complete metric space. This new result has been further extended by Bisht [4],
Karapinar et al. [5], and Panja et al. [6] for discontinuous mappings and by Khan and
Oyetunbi [7] for two discontinuous mappings satisfying a Lipschitz-Kannan type condition
considered in [8]. These advancements highlight the dynamic nature of fixed point theory.
Applications of fixed point theory have been widely explored in various settings. Specif-
ically, the theory has been instrumental in developing iterative methods for solving exis-
tence and uniqueness problems in differential and integral equations, as well as systems of
linear equations. These applications demonstrate the versatility of fixed point theory in
addressing complex mathematical challenges. The theory’s ability to ensure the existence
and uniqueness of solutions makes it a valuable tool for tackling a wide range of real-world
problems in mathematics, engineering and applied sciences [1, 9, 10].
In this paper, we aim to contribute to this growing body of knowledge by proving fixed
point and common fixed point results for asymptotically regular mappings in hyperbolic
spaces and convex metric spaces. Our work is built on a basic concept introduced by
Browder and Petryshyn [11], namely, asymptotically regular mapping and explores its
implications in more general spaces.

2. Preliminaries

Let (Q, d) be a metric space and E : Q → Q be a mapping. A point q0 ∈ Q is called
fixed point of E if Eq0 = q0. The set of fixed points of E is denoted and defined as
Fix(E) := {q0 ∈ Q : Eq0 = q0}.

The concept of asymptotically regular mappings was introduced by Browder and
Petryshyn [11]. A mapping E : Q → Q is called asymptotically regular at x0 ∈ Q, if
limn→∞ d(Enx0, E

n+1x0) = 0. If E is asymptotically regular at each point of Q, then E is
said to be asymptotically regular on Q. E is non-expansive if d(Ex,Ey) ≤ d(x, y) for all
x, y ∈ Q. Consider Q = R , the set of real numbers with its usual metric d(q, p) = |q − p|.
Suppose E : Q → Q is given by Eq = q

2 . This function is both asymptotically regu-
lar and non-expansive. If E is a function on a metric space Q into itself, then the set
O(E, e) := {Ene : n = 0, 1, 2, 3, · · · } is called the orbit of E at the point e ∈ Q. E is
called orbitally continuous at p ∈ Q if for any sequence {qk} ⊂ O(E, q) for some q ∈ Q,
limk→∞ qk = p entails that limk→∞Eqk = Ep. We say that E is orbitally continuous on
Q if E is orbitally continuous at each point p ∈ Q. Clearly, continuity implies orbital
continuity, but the converse is not true [4, 12, 13].
A self-mapping E on a metric space Q, is called k-continuous, k = 1, 2, · · ·

if lim
n→∞

Ek−1qn = z implies that lim
n→∞

Ekqn = Ez.

Note that 1-continuity is equivalent to continuity, and for any k = 1, 2, ..., k-continuity
implies (k + 1)-continuity, while the converse is not true. Moreover, continuity of E and
k-continuity of E are independent conditions when k > 1([13], Examples 1.2-1.5).
Górnicki [2] has obtained the following result.



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 3 of 23

Theorem 1. ([2], Theorem 2.6) Suppose that (Q, d) is a complete metric space and E is
a continuous asymptotically regular self-mapping on Q satisfying the following condition:

d(Eq,Ep) ≤ µd(q, p) + ν{d(q, Eq) + d(p,Ep)},∀q, p ∈ Q (1)

where 0 ≤ µ < 1, 0 ≤ ν < ∞. Then E has a unique fixed point x ∈ Q and Enq → x for
each q ∈ Q .

Khan and Oyetunbi [7] have obtained the following generalization of Theorem 1.

Theorem 2. ([7],Theorem 2.2) Suppose that (Q, d) is a complete metric space, E and F
are asymptotically regular self-mappings on Q satisfying the following condition:

d(Eq, Fp) ≤ µd(q, p) + ν{d(q, Eq) + d(p, Fp)}, ∀q, p ∈ Q (2)

where 0 ≤ µ < 1, 0 ≤ ν < ∞. Suppose further that E and F are either k-continuous for
some k ≥ 1 or orbitally continuous. Then E and F have a unique common fixed point b.
Furthermore, limn→∞Enq = b = limn→∞ Fnq for any q ∈ Q.

Here is a result of Khan and Oyetunbi [7] without asymptotically regularity of the
mappings.

Theorem 3. [7, Theorem 2.6] Suppose that (Q, d) is a complete metric space and E,F :
Q → Q are continuous mappings satisfying (2). Suppose that E and F have a com-
mon approximate fixed point sequence (i.e. there exists a sequence {qn} ⊂ Q such that
d(qn, Eqn) → 0 and d(qn, F qn) → 0 as n → ∞). Then E and F have a unique common
fixed point q. In particular, qn → q, as n → ∞.

Lemma 1 ([14],Lemma 2.1). Let {τn} , {ϕn} and {χn} be three real sequences with τn ≥ 0
and ϕn ∈ (0, 1). Suppose that
(i)τn+1 ≤ (1− ϕn)τn + ϕnχn

(ii)
∑∞

n=1 ϕn = ∞
(iii) lim supn→∞χn ≤ 0 or

∑∞
n=1 ϕnχn is convergent.

Then limn→∞ τn = 0.

Definition 1. [14] Let R+ := {f ∈ R | f ≥ 0}. Define a function κ : R+ → [0, 1]
satisfying the following properties:

(i) 0 ≤ κ(f) < 1 for all f > 0.

(ii) limn→∞ κ(fn) = 1 implies limn→∞ fn = 0.

The collection of all functions κ(f) is denoted by S.

Kohlenbouch [10] proposed the concept of a hyperbolic metric space as follows.



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 4 of 23

Definition 2. A hyperbolic space is a triplet (Q, d, η) where (Q, d) is a metric space and
η : Q×Q×[0, 1] → Q satisfies the following conditions, for all ω, ξ, κ, ι ∈ Q and µ, x ∈ [0, 1]
(η1) : d(ι, η(ω, ξ, µ)) ≤ (1− µ)d(ι, ω) + µd(ι, ξ),
(η2) : d(η(ω, ξ, µ), η(ω, ξ, x)) = |µ− x|d(ω, ξ),
(η3) : η(ω, ξ, µ) = η(ξ, ω, 1− µ),
(η4) : d(η(ω, ι, µ), η(ξ, κ, µ)) ≤ (1− µ)d(ω, ξ) + µd(ι, κ).

In case only (η1) is satisfied, then the Definition 2 of hyperbolic space coincides with
the convex metric space introduced by Takahashi [15]. Clearly, a hyperbolic space is a
convex metric space. In the case of a convex metric space, we shall replace η by W (a
convex structure on Q) and denote it by (Q, d,W ). A nonempty subset J of a convex
metric space Q is convex if W (p, q, λ) ∈ J for all p, q ∈ J and λ ∈ [0, 1].
In the context of a normed space Q, the natural convex structure on Q is given by

W (p, q;λ) = λp+ (1− λ)q, p, q ∈ Q and λ ∈ [0, 1].

If J is a convex subset of a normed space and E : J → J, then the average mapping
Eµ : J → J is given by

Eµp = (1− µ)p+ µEp, where µ ∈ (0, 1].

A hyperbolic space (Q, d, η) is said to be uniformly convex if for all x, y, z ∈ Q, r > 0
and ε ∈ (0, 2], there exists δ ∈ (0, 1] such that d (η (x, y, 1/2) , z) ≤ (1 − δ)r whenever
d(x, z) ≤ r and d(y, z) ≤ r and d(x, y) ≥ εr.

A map h : (0,∞) × (0, 2] → (0, 1] which provides in the above definition, a δ = h(r, ε)
for r > 0 and for a fixed ε ∈ (0, 2] , is called modulus of uniform convexity. We call h
monotone if it decreases with r ( for a fixed ε). Let {qn} be a bounded sequence in a
hyperbolic space Q. For q ∈ Q, we define a continuous functional r(., {qn}) : Q → [0,∞)
by

r(q, {qn}) = lim
n→∞

sup d(qn, q).

The aymptotic radius ρ = r({qn}) of {qn} is given by

ρ = inf{r(q, {qn}) : q ∈ Q}.

The asymptotic center of a bounded sequence {qn} with respect to a subset U of Q is
defined as:

AU ({qn}) = {q ∈ Q : r(q, {qn}) ≤ r(p, {qn}) for any p ∈ U}.

If the asymptotic center is taken with respect to Q, then it is simply denoted by A({qn}). It
is known that uniformly convex Banach spaces and even CAT(0) spaces enjoy the property
that ”bounded sequences have unique asymptotic centers with respect to closed convex
subsets”. The following lemma ensures that this property also holds in complete uniformly
convex hyperbolic spaces.



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 5 of 23

Lemma 2. ([16, Lemma 2.2]). Let (Q, d, η) be a complete uniformly convex hyperbolic
space with monotone modulus of uniform convexity. Then every bounded sequence {qn} in
Q has a unique asymptotic center with respect to any nonempty closed convex subset U of
Q.

Recall that a bounded sequence {qn} in Q is known as ∆ − convergent to q ∈ Q if q
is the unique asymptotic center of {un} for every subsequence {un}of {qn}. In this case,
we write

∆− lim
n→∞

{qn} = q.

We include the following lemmas of Berinde and Pacurar [17, 18] for a ready reference.

Lemma 3. Let (Q,d,W) be a convex metric space. For all x, y ∈ Q and any λ ∈ [0, 1], we
have

d(x, y) = d(x,W (x, y;λ)) + d(W (x, y;λ), y).

Proof. By the triangular inequality and (η1), we get

d(x, y) ≤ d(x,W (x, y;λ)) + d(W (x, y;λ), y)

≤ λd(x, x) + (1− λ)d(x, y) + λd(x, y) + (1− λ)d(y, y) = d(x, y).

Lemma 4. Let (Q, d,W ) be a convex metric space. For all x, y ∈ Q and any λ ∈ [0, 1],
we have

d(x,W (x, y;λ)) = (1− λ)d(x, y)

and
d(W (x, y;λ), y) = λd(x, y).

Proof. By (η1), we get

d(x,W (x, y;λ)) ≤ (1− λ)d(x, y),

and
d(W (x, y;λ), y) ≤ λd(x, y).

If we had strict inequality in either of the above two inequalities, then, by Lemma 3, we
would reach the contradiction

d(x, y) = d(x,W (x, y;λ)) + d(W (x, y;λ), y) < d(x, y).

Lemma 5. Let (Q, d,W ) be a convex metric space and E : Q → Q be a mapping. Define
the mapping Eλ : Q → Q by

Eλx = W (x,Ex;λ), x ∈ Q.

Then for any λ ∈ [0, 1),
Fix(E) = Fix(Eλ).



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 6 of 23

Proof. For λ = 0, Eλ = E and the assertion is trivial. Assume λ ∈ (0, 1) and let
a ∈ Fix(E). This means a = Ea and therefore it follows that

d(a,Eλa) = d(a,W (a,Ea;λ)) ≤ d(a, a) + (1− λ)d(a,Ea) = 0,

i.e., a ∈ Fix(Eλ).
Conversely, assume that a ∈ Fix(Eλ). This means that d(a,Ea) = 0, which implies

d(a,W (a,Ea;λ)) = 0.

By Lemma 4,
d(a,W (a,Ea;λ)) = (1− λ)d(a,Ea),

so it follows that
(1− λ).d(a,Ea) = 0,

which, in view of the fact that (1− λ) ̸= 0, implies d(a,Ea) = 0. Hence a ∈ Fix(E).

3. Common fixed points results

Following the notion introduced in Definition 1 by Huang and Qian [14], we extended
Theorem 3 by replacing µ with a function κ ∈ S and weakening the continuity hypothesis.

Theorem 4. Suppose that (Q, d) is a complete metric space and E and F are asymptoti-
cally regular self-mappings on Q. Suppose that there exist a function κ ∈ S and a constant
K ∈ [0,+∞) satisfying the following condition:

d(Eq, Fp) ≤ κ(d(q, p))d(q, p) +K{d(q, Eq) + d(p, Fp)} (3)

for all q, p ∈ Q.
Suppose that E and F have a common approximate fixed point sequence (i.e., there is a
sequence {qh} ⊂ Q, such that d(qh, Eqh) → 0 and d(qh, F qh) → 0 as h → ∞). Then E
and F have a unique common fixed point p provided E and F are either k-continuous or
orbitally continuous.
In particular, {qh} → p as h → ∞.

Proof. Take q ∈ Q. Define qh = Ehq and ph = F hq for all h ∈ N.
By (3), we have

d(Eh+1q, F h+1q) = d(E(Ehq), F (F hq)) (4)

d(Eh+1q, F h+1q) ≤ κ(d(Ehq, F hq))d(Ehq, F hq)

+K[d(Ehq, Eh+1q) + d(F hq, F h+1q)]. (5)

In view of the condition, 0 ≤ κ(d(Ehq, F hq)) ≤ 1, we consider two cases for lim suph→∞ κ(d(Ehq), F hq).
Case 1:

lim sup
h→∞

κ(d(Ehq, F hq)) = 1.



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 7 of 23

In this case, there exists a subsequence {κ(d(Ehkq, F hkq))} of {κ(d(Ehq, F hq))} such that

lim
k→∞

κ(d(Ehkq, F hkq)) = 1. (6)

Now (6) implies on the basis of Definition 1 ,

lim
k→∞

d(Ehkq, F hkq) = 0. (7)

We prove that {Ehkq} is a Cauchy sequence.
Suppose on the contrary that {Ehkq} is not a Cauchy sequence. Then there exists ϵ0 > 0
and two integer sequences {hk̃(i)}, {hk(i)} of {hk} with hk̃(i) > hk(i) > i such that

d(E
hk̃(i)q, Ehk(i)q) ≥ ϵ0, i = 1, 2, 3, · · · (8)

Consequently, we have
ϵ0 ≤ d(E

hk̃(i)q, Ehk(i)q),

ϵ0 ≤ d(E
hk̃(i)q, Ehk(i−1)q) + d(E

hk̃(i−1)q, Ehk(i−1)q) + d(E
hk̃(i−1)q, Ehk(i)q). (9)

If i → ∞ in (9), then by asymptotic regularity of E, we obtain

lim inf
i→∞

d(E
hk̃(i−1)q, Ehk(i−1)q) ≥ ϵ0. (10)

Now by (3), we have

d(E
hk̃(i)q, Ehk(i)q) ≤ d(E

hk̃(i)q, F hk(i)q) + d(F hk(i)q, Ehk(i)q) (11)

d(E
hk̃(i)q, Ehk(i)q) ≤ κ(d(E

hk̃(i−1)q, F hk(i−1)q))d(E
hk̃(i−1)q, F hk(i−1)q)

+K[d(E
hk̃(i−1)q, F

hk̃(i)q) + d(F hk(i−1)q, F hk(i)q)]

+d(F hk(i)q, Ehk(i)q).

and

d(E
hk̃(i)q, Ehk(i)q) ≤ κ(d(E

hk̃(i−1)q, F hk(i−1)q))d(E
hk̃(i−1)q, Ehkĩq)

+d(E
hk̃(i)q, Ehk(i−)q) + d(Ehk(i)q, F hk(i)q)d(F hk(i)q, F hk(i−1)q)

+K[d(E
hk̃(i−1)q, Ehkĩq) + d(F hk(i−1)q, F hk(i)q)

+d(F hk(i)q, Ehk(i)q)]. (12)

Dividing both sides of the inequality (12) by d(E
hk̃(i)q, Ehk(i)q) and comparing the new

inequality with (8), we conclude that



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 8 of 23

1 ≤ κ(d(E
hk̃(i−1)q, F hk(i−1)q))(

d(E
hk̃(i−1)q, Ehk̃(i)q)

d(E
hk̃(i)q, Ehk(i)q)

)

+1 +
d(Ehk(i)q, F hk(i)q)

d(E
hk̃(i)q, Ehk(i)q)

+
d(F hk(i)q, F hk(i−1)q)

d(E
hk̃(i)q, Ehk(i)q)

+K
d(E

hk̃(i−1)q, F
hk̃(i)q) + d(F hk(i−1)q, F hk(i)q)

d(E
hk̃(i)q, Ehk(i)q)

+
d(F hk(i)q, Ehk(i)q)

d(E
hk̃(i)q, Ehk(i)q)

. (13)

Let i → ∞ in (13), using (7), (8), asymptotically regularity of E and F and the fact
0 ≤ κ(.) ≤ 1, we deduce that

lim
i→∞

κ(d(E
hk̃(i)−1

q, F hk(i)−1q)) = 1.

Using Definition 1, we obtain

lim
i→∞

(d(E
hk̃(i)−1

q, F hk(i)−1q)) = 0. (14)

In view of

d(E
hk̃(i)−1

q, Ehk(i)−1q) ≤ d(E
hk̃(i)−1

q, F hk(i)−1q) + d(F hk(i)−1q, Ehk(i)−1q),

a combination of (7) and (14), gives

lim
i→∞

d(E
hk̃(i)−1

q, F hk(i)−1q) = 0.

It contradicts (10). Hence {qhk
} = {Ehkq} is a Cauchy sequence.

Since Q is complete, {qhk
} converges to u in Q. Since

d(F hkq, u) ≤ d(Ehkq, F hkq) + d(Ehkq, u),

therefore by (7), we conclude that {F hkq} converges to u.
Suppose that E is orbitally continuous. Let {Eqh} be a sequence in the orbit of E at the
point q. The orbital continuity of E implies that {Eqhk

} converges to Eu. By asymptoti-
cally regularity of E, we have

lim
k→∞

d(qhk+1
, qhk

) = lim
k→∞

d(Ehk+1q, Ehkq) = 0,

which gives

lim
k→∞

d(qhk+1
, qhk

) ≤ lim
k→∞

(
d(qhk+1

, Eqhk+1
) + d(Eqhk+1

, Eqhk
) + d(Eqhk

, qhk
)
)
.



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 9 of 23

Since
d(qhk+1

, Eqhk+1
) → 0,

d(Eqhk
, qhk

) → 0,

therefore,
lim
k→∞

d(qhk+1
, qhk

) = 0.

Hence
lim
k→∞

Eqhk
= lim

k→∞
qhk+1

= lim
k→∞

qhk
= u

implies that Eu = u by the uniqueness of limit.
Now suppose that E is k-continuous. In view of

d(qhk+j
, qhk

) ≤ d(qhk+j
, qhk+j−1

) + · · ·+ d(qhk+1
, qhk

)

and
d(Ehk+jqhk+j

, Ehkqhk
) = d(Ehk+jq, Ehk+j−1q) + · · ·+ d(Ehk+1q, Ehkq),

for all j = 1, 2, 3, · · · , k by asymptotic regularity of E, we get

lim
k→∞

qhk+j
= lim

k→∞
qhk

= u, j = 1, 2, 3, · · · , k. (15)

In particular,

lim
k→∞

Ek−1qhk
= lim

k→∞
qhk+j−1

= u. (16)

As E is k-continuous, (16) gives

lim
k→∞

Ekqhk
= Eu. (17)

By (15), we have
lim
k→∞

Ekqhk
= lim

k→∞
qhk+j

= u. (18)

A combination of (17) and (18), gives

Eu = u.

Similarly, we can prove that
Fu = u.

So u is a common fixed point of E and F .
Next, assume that v is another common fixed point of E and F with u ̸= v.

Definition 1 (i) applied to (3) gives

d(u, v) = d(Eu,Fv) ≤ κ(d(u, v))d(u, v) +K[d(u,Eu) + d(v, Fv)].



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 10 of 23

d(u, v) ≤ κ(d(u, v))d(u, v).

d(u, v) < d(u, v),

which is a contradiction. Therefore common fixed point of E and F is unique.
Case 2:

lim sup
h→∞

κ(d(Ehq, F hq)) < 1.

In this case, there exists σ ∈ (0, 1) such that

0 < κ(d(Ehq, F hq)) < σ.

By (5) we have

(d(Eh+1q, F h+1q)) ≤ σd(Ehq, F hq) +K(d(Ehq, Eh+1q) + d(F hq, F h+1q)). (19)

Let

uh = d(Ehq, F hq), vh = 1− σ, wh =
K(d(Ehq, Eh+1q) + d(F hq, F h+1q))

1− σ
.

By (19), we have
uh+1 ≤ (1− vh)uh + vhwh,∀h ∈ N.

Since E and F are asymptotically regular on Q, we conclude

lim
h→∞

wh = 0.

Moreover,
∞∑
h=1

vh =
∞∑
h=1

(1− σ) = ∞.

By Lemma 1,
lim
h→∞

(Ehq, F hq) = 0.

Hence for any subsequence {hk(i)} of {hk}, we have

lim
i→∞

(Ehk(i)q, F h(i)q) = 0.

Thus (7) holds. The rest of proof is the same as in Case I.

Next, we present a numerical example to illustrate Theorem 4.

Example 1. Consider Q = [0, 1], equipped with the metric d defined by d(q, p) = |q − p|.
Define self-mappings on Q:

Eq =
q

2
and Fq =

q

3
for all q ∈ Q.



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 11 of 23

For q = 1
2 ∈ Q,

lim
k→∞

d
(
Ek

(1
2

)
, Ek+1

(1
2

))
= lim

k→∞

∣∣∣ 1

2k+1
− 1

2k+2

∣∣∣ = 0

implies that E is asymptotically regular. Similarly, it can be demonstrated that F is asymp-
totically regular. Furthermore, the mappings E and F are orbitally continuous. Next, we
establish that E and F satisfy condition (3) with the parameter K = 1. Now define
κ : R+ → [0, 1] by

κ(t) =

{
1
2 + 1

2 t if t ∈ [0, 1],

1 if t ∈ (1,+∞).

Then κ ∈ S, for any q, p ∈ Q, since |q − p| ≤ 1. Consider (3) in the form

κ(d(q, p))d(q, p) + d(q, Eq) + d(p, Fp)− d(Eq, Fp)

=
1

2
(1 + |q − p|)|q − p|+

∣∣∣q − q

2

∣∣∣+ ∣∣∣p− p

3

∣∣∣− ∣∣∣q
2
− p

3

∣∣∣
=

1

2
(1 + |q − p|)|q − p|+ q

2
+

2p

3
−
∣∣∣q
2
− p

3

∣∣∣. (20)

Now, there are two cases to consider for (20):
Case i. If q ≥ p, then we have:

1

2
(1 + (q − p))(q − p) +

q

2
+

2p

3
−
(q
2
− p

3

)
=

1

2
(q − p) +

1

2
(q − p)2 +

q

2
+

2p

3
− q

2
+

p

3

=
1

2
(q + p) +

1

2
(q − p)2 ≥ 0;

thus
d(Eq, Fp) ≤ κ(d(q, p))d(q, p) + d(q, Eq) + d(p, Fp).

Case ii. if q < p, then we have:

1

2
(1 + (p− q))(p− q) +

q

2
+

2p

3
−
∣∣∣q
2
− p

3

∣∣∣ =

{
1
2(p− q) + 1

2(p− q)2 + p if q
3 ≥ p

3 ,
1
2(p− q) + 1

2(p− q)2 + q + p
3 if q

2 < p
3

≥ 0;

hence
d(Eq, Fp) ≤ κ(d(q, p))d(q, p) + d(q, Eq) + d(p, Fp).

Therefore E and F satisfy (3) with K = 1. Next, we show that E and F have a common
approximate fixed point sequence. Consider the sequence {qk} ⊂ Q where qk = 1

k .

d(qk, Eqk) =

∣∣∣∣1k − 1

2k

∣∣∣∣ = 1

2k
→ 0 as k → ∞,



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 12 of 23

and

d(qk, F qk) =

∣∣∣∣1k − 1

3k

∣∣∣∣ = 1

3k
→ 0 as k → ∞.

Hence, E and F have a common approximate fixed point sequence. Since E and F are both
continuous (and thus k-continuous), therefore by Theorem 4, they have a unique common
fixed point q = 0. In particular, {qk} → 0 as k → ∞.

We now present a partial extension of Theorem 2 for a function which is not asymptotically
regular and is defined on a closed and convex subset of a uniformly convex hyperbolic space.

Theorem 5. Let J be a nonempty closed and convex subset of complete uniformly convex
hyperbolic space Q with monotone modulus of convexity η. Let E : J → Q be a mapping
satisfying

d(Eq,Ep) ≤ µd(q, p) + ν{d(q, Eq) + d(p,Ep)}, for all q, p ∈ J, (21)

where 0 ≤ µ < 1, 0 ≤ ν < ∞ with µ+2νblue<1. Let {qk} be a bounded sequence in J such
that limk→∞ d(qk, Eqk) = 0 and ∆− limk→∞ qk = q∗. Then E has a unique fixed point q∗.

Proof. Let {qk} be any bounded sequence in J . Since J is a subset of Q which is
complete convex hyperbolic space with monotone uniform convexity, therefore by Lemma
2, {qk} has a unique asymptotic center in J .
But ∆− limk→∞ qk = q∗. So A({qk}) = q∗. Using (21), we get

d(qk, Eq∗) ≤ d(qk, Eqk) + d(Eqk, Eq∗).

≤ d(qk, Eqk) + µd(qk, q
∗) + ν[d(qk, Eqk) + d(q∗, Eq∗)].

≤ (1 + ν)d(qk, Eqk) + (µ+ ν)d(qk, q
∗) + νd(qk, Eq∗).

Thus

d(qk, Eq∗) ≤ 1 + ν

1− ν
d(qk, Eqk) +

µ+ ν

1− ν
d(qk, q

∗)

=
1 + ν

1− ν
d(qk, Eqk) +

µ+ 2ν − ν

1− ν
d(qk, q

∗)

≤ 1 + ν

1− ν
d(qk, Eqk) + d(qk, q

∗). (22)

Since limk→∞ d(qk, Eqk) = 0, taking lim sup on both sides of inequality (22), we obtain

r(Eq∗, qk) = lim sup
k→∞

d(qk, Eq∗) ≤ lim sup
k→∞

d(qk, q
∗) = r(q∗, {qk}).

We know that asymptotic center of sequence {qk} is unique, so Eq∗ = q∗; thus q∗ ∈
Fix(E). The uniqueness of fixed point follows by (21).



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 13 of 23

Example 2. Let Q = R2 and d∗ : R2 × R2 → [0,∞) be defined by

d∗(q̄, p̄) =
√

(q1 − p1)2 + (q21 − q2 − p21 + p2)2, (23)

where q̄ = (q1, q2), p̄ = (p1, p2) ∈ R2. Then (R2, d∗) is not a metric space in the classical
sense but is a Hardamard space (see [19]). Hence (R2, d∗) is a complete uniformly convex
hyperbolic space (for details, see Example 2.1 in [20]). Let J = [0, 1]× [0, 1].
Define E : J → Q as Eq̄ = (q1, q

2
1 + q2), where q̄ = (q1, q2).

For any µ ∈
[
2
5 , 1

)
and 0 ≤ ν < ∞ with µ + 2ν < 1, we need to verify the following, for

any q̄, p̄ ∈ J ,

Γ := µd∗(q̄, p̄) + ν[d∗(q̄, Eq̄) + d∗(p̄, Ep̄)]− d∗(Eq̄,Ep̄) ≥ 0. (24)

Thus, for any q̄ = (q1, q2), p̄ = (p1, p2) ∈ J , by using (23) in (24), we get

Γ = d∗((q1, q2), (p1, p2)) + d∗((q1, q2), (q1, q
2
1 + q2)) + d∗((p1, p2), (p1, p

2
1 + p2))

−d∗((q1, q
2
1 + q2), (p1, p

2
1 + p2))

= µ
√

(q1 − p1)2 + (q21 − q1 − p22 + p2)2 + ν
(√

(q1 − q1)2 + (q21 − q2 − q21 + (q21 + q2))2

+
√

(p1 − p1)2 + (p21 − p2 − p21 + (p21 + p2))2
)

−
√
(q1 − p1)2 + (q21 − (q21 + q2)− p21 + (p21 + p2))2

= µ
√
(q1 − p1)2 + ((q2 − p2) + (q21 − p21))

2 + ν(q21 + p21)−
√

(q1 − p1)2 + (q2 − p2)2

≥ 0

Hence (24) holds. Therefore E satisfies (21) with µ ∈
[
2
5 , 1

)
and 0 ≤ ν < ∞ where

µ+ 2ν < 1. For any qk ∈ J , define qk =
(
1
k ,

1
k

)
. Then Eqk =

(
1
k ,

1
k2

+ 1
k

)
, so from (23),

we get that

d∗(qk, Eqk) = d∗(
1

k
,
1

k2
+

1

k
)

=

√(1
k
− 1

k

)2
+
( 1

k2
− 1

k
− 1

k2
+
( 1

k2
+

1

k

))2
=

1

k2
.

It follows that d∗(qk, Eqk) → 0 as k → ∞. All the assumptions of Theorem 5 hold and
hence, the mapping E has a unique fixed point at (0, 0).

Khan [21] has studied some properties of the set of fixed points of a ∗-nonexpansive
mapping in the context of strictly convex Banach spaces. In the following result, we
establish the closedness and convexity of the set of fixed points of a contractive-type
mapping on a convex metric space.

Theorem 6. Let J be a closed and convex subset of a convex metric space Q and E : J →
Q be a mapping satisfying (21). Then Fix(E) is closed and convex.



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 14 of 23

Proof. If Fix(E) = ∅, then nothing to show, since empty set is closed and convex.
Now assume that Fix(E) ̸= ∅. We first show that Fix(E) is closed. Let {qk} be a sequence
in Fix(E) such that {qk} converges to a point u in J . We show that u ∈ Fix(E). By (21)
, we obtain

d(qk, Eu) = d(Eqk, Eu) ≤ µd(qk, u) + ν[d(qk, Eqk) + d(u,Eu)].

Implies
d(qk, Eu) ≤ µd(qk, u) + ν[d(u, qk) + d(qk, Eu)].

Thus

d(u,Eu) ≤ d(u, qk) + d(qk, Eu)

≤ d(u, qk) + µd(qk, u) + ν[d(qk, Eqk) + d(u,Eu)]

= (1 + µ)d(qk, u) + νd(u,Eu);

hence

d(u,Eu) ≤ 1 + µ

1− ν
d(qk, u) (25)

As µ ∈ [0, 1) and µ + 2ν < 1, that is, ν < 1 and since {qk} → u as k → ∞, therefore
limk→∞ d(qk, u) = 0. From (25), we get u ∈ Fix(E) as desired.

Next, we show that Fix(E) is convex. Let q, p ∈ Fix(E) and α ∈ [0, 1]. For any z ∈
J , assume that z = η(q, p, α). Now we show that z ∈ Fix(E), that is, z = Ez or
E(η(q, p, α)) = η(q, p, α). Thus

d(q, Ez) = d(Eq,Ez) ≤ µd(q, z) + ν[d(q, Eq) + d(z, Ez)].

≤ µd(q, z) + ν[d(z, q) + d(q, Ez)].

≤ (µ+ ν)d(q, z) + νd(q, Ez).

Hence

d(q, Ez) ≤ (µ+ ν)

1− ν
d(q, z) ≤ d(q, z). (26)

Similarly,

d(p,Ez) ≤ d(p, z). (27)

Using (26), (27) and (η1), we get

d(q, p) ≤ d(q, Ez) + d(Ez, p).

≤ d(q, z) + d(z, p).

≤ d(q, η(q, p, α)) + d(η(q, p, α), p)

≤ (1− α)d(q, q) + αd(q, p) + (1− α)d(q, p) + d(p, p)



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 15 of 23

= d(p, q)

Thus, we conclude by (26)) and (27) that d(q, Ez) = d(q, z) and d(p,Ez) = d(p, z) because
if d(q, Ez) < d(q, z) or d(p,Ez) < d(p, z), then we will obtain a contradiction d(p, q) <
d(p, q). Hence Ez = z, that is, Eη((q, p, α)) = η(q, p, α) for all q, p ∈ Fix(E) and α ∈ [0, 1].
Therefore Fix(E) is convex.

Based on Lemma 5, we introduce the contractive condition for self-mappings Tλ and
Sλ on Q as follows:

d(Tλx, Sλy) ≤ M(d(x, y)) +K{d(x, Tλx) + d(y, Sλy)} (28)

where 0 ≤ M < 1, 1 < K < ∞ and M + 2K ≤ 1.
An extension of Theorem 2, is obtained by replacing the maps E and F on a metric
space with their, respective, average mappings, in the context of a convex metric space,
as follows.

Theorem 7. Let (Q, d,W ) be a complete convex metric space. Assume that Tλ and Sλ

are asymptotically regular self-mappings on Q satisfying (28). If Tλ and Sλ are orbitally
continuous or k-continuous for some k ≥ 1, then Tλ and Sλ have a unique common fixed
point p. Furthermore, limn→∞ Tn

λ x = p = limn→∞ Sn
λx for any x ∈ Q.

Proof. Step 1: We note that the Picard iterations of Tλ and Sλ form the Krasnoselskij
iterative sequence {xn}∞n=0 defined by

xn+1 = Tλxn = (1− µ)xn + µTλxn and xn = Sλxn−1 n = 0, 1, 2, · · · (29)

Using (28), we get

d(xn+1, xn) ≤ Md(xn, xn−1) +K{d(xn, xn+1) + d(xn−1, xn)};

thus

d(xn+1, xn) ≤
(M +K)

1−K
d(xn, xn−1)

d(xn+1, xn) ≤ αd(xn, xn−1),
(
α :=

M +K

1−K

)
, (30)

which inductively implies that

d(Tn+1
λ x, Sn

λx) = d(xn+1, xn) ≤ αnd(x1, x0). (31)

This implies
d(Tn+1

λ x, Sn
λx) → 0 as n → ∞.



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 16 of 23

Step 2: We now prove that {xn}∞n=0 is a Cauchy sequence. Suppose on contrary that
{xn}∞n=0 is not a Cauchy sequence. Then there exisst ϵ > 0 and sequences of numbers
{m(k)} and {n(k)} with m(k) > n(k) > k for k = 1, 2, 3, · · · such that

d(T
m(k)
λ , T

n(k)
λ ) ≥ ϵ. (32)

Assume that m(k) ≥ n(k). So by (31), we have d(T
m(k)−1
λ x, T

n(k)
λ x) < ϵ.

Hence

ϵ ≤ d(T
m(k)
λ x, T

n(k)
λ x) ≤ d(T

n(k)
λ x, T

n(k)−1
λ x) + d(T

m(k)−1
λ x, T

n(k)
λ x)

< d(T
m(k)
λ x, T

m(k)−1
λ x) + ϵ. (33)

Letting k → ∞ and using asymptotically regularity of Tλ, we get

lim
k→∞

d(T
m(k)
λ x, T

n(k)
λ x) = ϵ.

Now the following inequality and asymptotic regularity of Tλ

d(T
m(k)−1
λ x, T

n(k)−1
λ x) ≤ d(T

m(k)−1
λ x, T

m(k)
λ x) + d(T

m(k)
λ x, T

m(k)
λ x)

+d(T
n(k)
λ x, T

n(k)−1
λ x)

imply

lim
k→∞

d(T
m(k)−1
λ x, T

n(k)−1
λ x) = ϵ.

From (29), we get

d(T
m(k)
λ x, T

n(k)
λ x) ≤ d(T

m(k)
λ x, S

n(k)
λ x) + d(S

n(k)
λ x, T

n(k)
λ x)

and

d(T
m(k)
λ x, T

n(k)
λ x) ≤ d(S

n(k)
λ x, T

n(k)
λ x) +Md(T

m(k)−1
λ x, T

n(k)−1
λ x)

+K[d(T
m(k)−1
λ x, T

m(k)
λ x) + d(S

n(k)−1
λ x, S

n(k)
λ x)]

+Md(T
n(k)−1
λ x, S

n(k)−1
λ x).

Letting k → ∞, it follows by (30) (32) and (33) that ϵ ≤ Mϵ. This is a contradiction.
Therefore {xn} is a Cauchy sequence. Given that Q is complete, {xn} converges to p ∈ Q.
Moreover,

d(Sn
λx, p) ≤ d(Sn

λx, T
n
λ x) + d(Tn

λ x, p),

so it follows from Step 1 that Tn
λ x and Sn

λx converge to p ∈ Q.



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 17 of 23

Step 3: Tλ and Sλ have a unique common fixed point p. Let Tλ be k-continuous. Since
limn→∞ Tn−1

λ x = p. So by k-continuity of Tλ, limn→∞ Tn
λ x = Tλp. By uniqueness of limit

, Tλp = p.
Analogically, suppose that Tλ is orbital continuous . Since limn→∞ xn = p, orbital conti-
nuity of Tλ implies that

lim
n→∞

Tλxn = Tλp.

This yield Tλp = p. Similarly, limn→∞ Sn
λx = p, we have that Sp = p. Hence p ∈

Fix(Tλ) ∩ Fix(Sλ).
Uniqueness: Assume that p, q ∈ Fix(Tλ)

⋂
Fix(Sλ) and q1 ̸= p. Let x = p and y = q1.

Then (28) becomes d(p, q1) ≤ Md(p, q1). It is a contradiction. Hence p is a unique common
fixed point of Tλ and Sλ.

In the light of Lemma 5, we define Zemfirescue mapping in terms of Tλ:

Definition 3. Let (Q, d,W ) be a complete convex metric space and Tλ : Q → Q satisfies
the following conditions.

(i) d(Tλx, Tλy) ≤ ad(x, y), 0 < a < 1,

(ii) d(Tλx, Tλy) ≤ b
(
(d(x, Tλx) + d(y, Tλy)

)
, 0 < b < 1

2

(iii) d(Tλx, Tλy) ≤ c
(
d(x, Tλy) + d(y, Tλx)

)
, 0 < c < 1

2 .

An analogue of Theorem 1 by Zamfiresuc [22] for the mapping Tλ is presented below.

Theorem 8. Let (Q, d,W ) be a complete convex metric space and Tλ : Q → Q be Zam-
firescue asymptotically regular map. Then,

(i) Fix(Tλ) = {p}, and

(ii) the sequence {xn}∞n=0 obtained from the iterative process xn+1 = W (xn, Tλxn;λ),
n ≥ 0 converges to p, for x0 ∈ Q.

Proof. We note that the Picard iterations of Tλ actually form the Kranoselskij iterative
process {xn}∞n=0. Now, choose x0 ∈ Q arbitrarily and fix integer n ≥ 0.
Consider x = Tn

λ x0 and y = Tn+1
λ x0, we obtain by Definition 3(i), with δ < 1

d(Tn+1
λ x0, T

n+2
λ x0) ≤ δd(Tn

λ x0, T
n+1
λ x0)

and by Definition 3(ii)

d(Tn+1
λ x0, T

n+2
λ x0) ≤ b

(
d(Tn

λ x0, T
n+1
λ x0) + d(Tn+1

λ x0, T
n+2
λ x0)

)
which implies

d(Tn+1
λ x0, T

n+2
λ x0) ≤

b

1− b
d(Tn

λ x0, T
n+1
λ x0).



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 18 of 23

Thus

d(Tn+1
λ x0, T

n+2
λ x0) ≤ δd(Tn

λ x0, T
n+1
λ x0) where δ :=

b

1− b
.

Similarly, with condition (iii) of Definition 3, we obtain

d(Tn+1
λ x0, T

n+2
λ x0) ≤ c

(
d(Tn

λ x0, T
n+2
λ x0) + d(Tn+1

λ x0, T
n+1
λ x0)

)
.

Hence
d(Tn+1

λ x0, T
n+2
λ x0) ≤ δd(Tn

λ x0, T
n+1
λ x0).

This inequality is true for every n. So that {Tn
λ (x0)}∞n=0 is a Cauchy sequence and therefore

converges to some point p ∈ Q.

(i) We now prove that p is a fixed point of Tλ. Suppose Tλp ̸= p. We consider the ball

B = {p ∈ Q : d(p, x) ≤ 1

4
d(p, Tλp)}.

Observe that d(x, Tλp) ≥ 3
4d(p, Tλp) for every point p ∈ B. So there exists a number

N such that Tn
λ x0 ∈ B for each n ≥ N . Now taking x = TN

λ x0 and y = p. We must
have one of the following situations:

(a) d(Tn+1
λ x0, Tλp) ≤ ad(TN

λ x0, p), which sets a contradiction as follows

d(Tn
λ x0, p) ≤

1

4
d(p, Tλp) < d(Tn+1

λ x0, Tλp),

(b) d(Tn
λ x0, Tλp) ≤ b

(
d(Tn

λ x0, T
n+1
λ x0) + d(p, Tλp)

)
, contradicting

b
(
d(Tn

λ x0, T
n+1
λ x0) + d(p, Tλp)

)
<

1

2
(d(Tn

λ x, p) + d(p, Tn+1
λ x0) + d(p, Tλp))

≤ 3

4
d(p, Tλp)

≤ d(Tn+1
λ x0, Tλp)

(c) d(Tn+1
λ x0, Tλp) ≤ c

(
d(Tn

λ x0, Tλ(p)) + d(Tn+1
λ x0, p)

)
contradicting

c
(
d(Tn

λ x0, Tλ(p)) + d(Tn+1
λ x0, p)

)
<

1

2
(d(Tn

λ x0, p) + d(p, Tλp) + d(Tn+1
λ x0, p))

≤ 3

4
d(p, Tλ, p)

≤ d(Tn+1
λ x0, Tλp)

Thus Tλp = p. Now we show that this fixed point p is unique. Suppose that
this is not true .
Let Tλp

′
= p

′
for some point p

′ ̸= p ∈ Q. Then

d(Tλp, Tλp
′
) = d(p, p

′
)



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 19 of 23

d(Tλp, Tλp
′
) > d(p, Tλp) + d(p

′
, Tλp

′
)

d(Tλp, Tλp
′
) =

1

2
(d(p, Tλp

′
) + d(p

′
, Tλ(p)))

So that none of the three conditions of Zamfirescue map is satisfied by the
points p and p

′
. This is a contradiction. Hence Tλ has a unique fixed point.

(ii) Obvious.

It is natural to ask, when the assumption that Tλ is asymptotically regular in Theorem 7,
is satisfied. For an affirmative answer to this question, we need the following useful result.

Theorem 9. [9, Theorem 5.2.7] Let J be a nonempty convex subset of a normed space
Q and T be non-expansive map on J . If for x0 ∈ J, {Tn

λ x0} is bounded, then the average
map Tλ is asymptotically regular at x0.

For the existence of fixed points, apart from the other conditions, Huang and Qian ([14,
Theorem 2.5]) have imposed continuity condition on the mappings already satisfying con-
tractive condition similar to (1). In the result to follow, we employ weak requirement of
nonexpansiveness to get nonexpansive version of Górnicki result for the average mapping.

Theorem 10. Let J be a nonempty convex subset of a normed space Q and T : J → J be
non-expansive map satisfying (1). If for x0 ∈ J, {Tn

λ x0} is bounded, then Tλ has a unique
fixed point p ∈ Q. Moreover, {xn}∞n=0, the Krasnoselskij iterative sequence converges to p.

Proof.
The map Tλ is asymptotically regular at x0 by Theorem 9. Now the rest of the proof

is similar to that of Theorem 1.

Remark 1.

(1) Theorem 2.5 of Khan and Oyetuabi [7], holds in a convex metric space with the same
proof.

(2) Theorem 10 provides a non-expansive version of Theorem 1 with a very simple proof.

4. Application: Volterra-Type Integral Equations

In this section, we investigate the existence and uniqueness of the common solution of
Volterra-type integral equations, utilizing the common fixed point result established in
Theorem 4.

Volterra-type integral equations play a significant role in various fields, including physics,
biology, and engineering, in view of their capability to model systems with memory effects.
These equations are crucial for capturing the dynamics of processes where the future state



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 20 of 23

depends on the entire history of the system, such as in viscoelastic materials, population
dynamics, and heat conduction. The importance of Volterra-type integral equations lies
in their ability to provide insights into the existence and uniqueness of solutions to com-
plex mathematical problems. By applying fixed point theorems, researchers can establish
stability and convergence of these solutions; thereby addressing fundamental challenges in
both theoretical and applied mathematics. The reader interested in this matter, is referred
to the recent literature developed in [23–25].

Let Q be the space of continuous functions on [0, T ] equipped with the supremum norm.
That is,

Q := {u : [0, T ] → R : u is continuous}.

Define a metric d : Q×Q → R+ by

d(u, v) = ||u− v||∞ = sup
t∈[0,T ]

|u(t)− v(t)|, u, v ∈ Q.

Then (Q, d) is a complete metric space.
We consider the following Volterra-type integral equations formulated as a common fixed
point problem of the following nonlinear mappings:

u(t) =

∫ t

0
K1(t, s, u(s))ds (34)

v(t) =

∫ t

0
K2(t, s, v(s))ds (35)

for all t, s ∈ [0, T ], where the kernels K1(t, s, u(s)) and K2(t, s, v(s)) are known function.
We will find the solution of (34) and (35).

Now, we prove the following theorem to ensure the existence of common solution of the
integral equations (34) and (35).

Theorem 11. Assume that the following conditions are satisfied:

(a) K1,K2 : [0, T ]× [0, T ]×Q → R+,

(b) define the mappings E and F as follows:

(Eu)(t) =

∫ t

0
K1(t, s, u(s))ds (36)

(Fv)(t) =

∫ t

0
K2(t, s, v(s))ds. (37)

Assume further that



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 21 of 23

(i) there exists a continuous functions τ : [0, T ] × [0, T ] → R+ and a continuous and
nondecreasing function α : R+ → R+ such that α(0) = 0 and α(t) < t for t > 0
satisfying ∣∣∣K1(t, s, u(s))−K2(t, s, v(s))

∣∣∣ ≤ τ(t, s)α(|u− v|)

for t, s ∈ [0, T ] and u(s), v(s) ∈ Q.

(ii) sup
t∈[0,T ]

∫ t

0
τ(t, s)ds ≤ ω, for some ω ∈ (0, 1).

Then E and F have a unique common solution.

Proof. Let u, v ∈ Q. By assumptions (i) and (ii), we have

d(Eu,Fv) = ||Eu− Fv||∞ = sup
t∈[0,T ]

|(Eu)(t)− (Fv)(t)|

= sup
t∈[0,T ]

∣∣∣ ∫ t

0
K1(t, s, u(s))ds−

∫ t

0
K2(t, s, v(s))ds

∣∣∣
= sup

t∈[0,T ]

∣∣∣ ∫ t

0

(
K1(t, s, u(s))−K2(t, s, v(s))

)
ds
∣∣∣

≤ sup
t∈[0,T ]

∫ t

0

∣∣∣K1(t, s, u(s))−K2(t, s, v(s))
∣∣∣ds

≤ sup
t∈[0,T ]

∫ t

0
τ(t, s)α(|u− v|)ds

≤ sup
t∈[0,T ]

∫ t

0
τ(t, s)|u− v|ds

≤ ||u− v||∞ sup
t∈[0,T ]

∫ t

0
τ(t, s)ds

≤ ω||u− v||∞
= ω

(
||u− v + (E(u)− u)− (E(u)− u) + (v − F (v))− (v − F (v))||∞

)
≤ ω

(
2||u− v||∞|+ ||u− E(u)||∞ + ||v − F (v)||∞ + ||E(u)− F (v)||∞

)
;

thus

d(Eu,Fv) ≤ 2ω

1− ω
d(u, v) +

ω

1− ω

(
d(u,Eu) + d(v, Fv)

)
Letting κ(d(u, v)) := 2ω

1−ω , it follows that the mappings E and F satisfy (3). Hence by
Theorem 4, there exists a unique common fixed point of E and F which is the common
solution of the Voterra-type integral equations (34) and (35).



A. R. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5826 22 of 23

5. Conclusions

We have extended the existing body of knowledge on fixed point theory for asymptotically
regular mappings on a metric space by proving new common fixed point results on convex
metric spaces. Our work verifies that under certain contractive conditions, asymptotically
regular self-mappings not only possess unique fixed points but also exhibit properties
of closedness and convexity for their fixed point sets. We have also demonstrated that
replacing standard mappings with their average mappings in a convex metric space context
retains their fixed point properties. Furthermore, we have applied our theoretical findings
to solve integral equations, showcasing the practical utility of our theorems in real-world
problems. These results are pivotal in advancing the understanding of fixed point theory in
more general and complex spaces, offering new avenues for future research and applications
in various mathematical and practical fields.
Looking ahead, several interesting directions arise for future exploration:

(i) Establishing an analogue of Theorem 5 in a convex metric space.

(ii) Finding an analogue of Theorem 6 for a quasi-nonexpansive map.

(iii) Addressing the question posed before Theorem 9 in the context of a convex metric
space.

Acknowledgements

The authors G. C. Ugwunnadi and M. Aphane are grateful to Department of Math-
ematics and Applied Mathematics, Sefako Makgato Health Science University, Pretoria
0204, South Africa for supporting this research work.

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