EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6311
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fixed Point Techniques for Graph-Based Mappings
with Image Recovery Approach

Sayan Panma1,2, Raweerote Suparatulatorn2,3, Tanadon Chaobankoh1,2,∗

1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai
50200, Thailand
2 Advanced Research Center for Computational Simulation, Chiang Mai University, Chiang
Mai 50200, Thailand
3 Department of Mathematics, Faculty of Science, Lampang Rajabhat University, Lampang
52100, Thailand

Abstract. This article introduces an algorithm designed to approximate a common fixed point
for a class of graph-based mappings in a real Hilbert space. We show that the sequence generated
by this algorithm weakly converges to a common fixed point. Moreover, we explore the algorithm’s
practical application by investigating its effectiveness in solving image recovery problems related
to fine particulate matter (PM2.5) pollution. Its performance is then evaluated by comparing it
to existing methods.

2020 Mathematics Subject Classifications: 47J25, 47J26 65J15, 65Y05, 68W10

Key Words and Phrases: Fixed point problem, directed graph, G-nonexpansive mapping, image
processing, process innovation

1. Introduction

The fixed point problem demonstrates exceptional versatility in addressing real-world
challenges, particularly signal and image recovery. In this domain, iterative methods are
used to extract solutions from noise-corrupted data. To enhance fixed point analysis,
researchers have incorporated graph theory, as initially proposed by Jachymski [1]. By
combining graph theory with mapping analysis, G-nonexpansive mappings represent a
significant advancement in fixed point theory, contributing to the development of more
powerful and efficient iterative schemes.

Recent studies have introduced algorithms for finding common fixed points of G-
nonexpansive mappings in graph-based metric spaces. Khemphet et al. [2] improved signal

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6311

Email addresses: sayan.panma@cmu.ac.th (S. Panma),
raweerote.s@gmail.com (R. Suparatulatorn), tanadon.c@cmu.ac.th (T. Chaobankoh)

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



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 2 of 20

recovery using inertial and Mann iterations, demonstrating enhanced performance com-
pared to previous methods. In a related work, Jun-On et al. [3] proposed a new inertial
parallel algorithm for solving the common fixed point problem and applied it to signal
recovery tasks. In 2024, a monotone hybrid algorithm based on a parallel inertial SP-
iteration was proposed, with its weak convergence established and effectiveness demon-
strated in solving linear systems, differential equations, and signal recovery problems [4].
Additionally, further research concerning this type of mappings can be found in, for ex-
ample, the works of [5–9]. Through theoretical analysis and numerical experiments, these
studies have collectively broadened the scope of fixed point theory for G-nonexpansive
mappings.

With the growing urgency of environmental concerns, particularly fine particulate mat-
ter (PM2.5) pollution, methods are being developed to accurately estimate ground-level
concentrations. In [10], an image-based PM2.5 estimation method using image processing
and linear regression was proposed by analyzing differences between images with high and
low PM2.5 concentrations. Also, in [11], the authors introduced a CNN-SVR approach
for image-based PM2.5 estimation, enabling real-time analysis from single image captures.
This approach, supported by its ability to handle extensive datasets and complex algo-
rithms (see [12–14]) makes it a significant contribution to PM2.5 prediction, potentially
improving air quality management.

In [15], an algorithm was presented for generating a sequence {xn} in a Banach space
E, achieving faster convergence than Picard, Mann, and Agarwal et al. methods for
nonexpansive mappings T : E → E. The iterative process is given by:

x1 ∈ E,
xn+1 = (1− αn)Tyn + αnTzn,
yn = (1− βn)Txn + βnTzn,
zn = (1− γn)xn + γnTxn, n ∈ N,

(1)

where αn, βn, γn are sequences in the open unit interval.
Inspired by these findings, based on the iterative process (1), we develop Algorithm 1, a

hybrid algorithm that combines the parallel monotone hybrid algorithm with the inertial
method. We establish weak convergence theorems for finding common fixed points of
G-nonexpansive mappings in real Hilbert spaces endowed with directed graphs G under
certain conditions. Finally, we validate our results through a real-world PM2.5-related
image recovery task in Chiang Mai, Thailand, where we process images using multiple
blurring filters and conduct a comparative analysis against existing algorithms presented
in [3] and [4].

2. Preliminaries

In this section, we present some useful notations, definitions and results that will be
used in the next section.

A directed graph G consists of a non-empty vertex set V (G) and an edge set E(G),
where each edge is an ordered pair that specifies a direction from one vertex to another.



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 3 of 20

A non-empty set X is said to be equipped with a directed graph G = (V (G), E(G)) if G
has no parallel edges (i.e., all edges in E(G) are distinct).

Throughout this article, we assume that X := (X, ⟨·, ·⟩) is a real Hilbert space, where
∥ · ∥ denotes the induced norm, and is equipped with a directed graph G = (V (G), E(G)).
We also denote ⇀ and → as weak and strong convergence, respectively.

Definition 1. [16] Let J be a self-mapping on X. Then, J is said to be G-nonexpansive
if

(i) J is edge-preserving, that is, for u, v ∈ V (G),

(u, v) ∈ E(G) ⇒ (Ju, Jv) ∈ E(G);

(ii) for u, v ∈ V (G),
(u, v) ∈ E(G) ⇒ ∥Ju− Jv∥ ≤ ∥u− v∥.

Definition 2. Let J : X → X be a self-mapping. A point x ∈ X is called a fixed point of
J if

J(x) = x.

The set of fixed points of J , denoted Fix(J), is defined as {x ∈ X : J(x) = x}.

Let {J1, J2, . . . , Jn} be a collection of self-mappings on X. A point x ∈ X is called a
common fixed point of the mappings in this collection if it satisfies the condition:

Ji(x) = x for all i = 1, 2, . . . , n.

We list two essential facts in the following proposition. The first is a direct conse-
quence of the definition of the squared norm and the properties of the inner product. The
second, which is frequently used in the analysis of convex combinations, follows from the
parallelogram law and can be found in [17].

Proposition 1. For x, y ∈ X,

∥x+ y∥2 ≤ ∥x∥2 + 2⟨y, x+ y⟩, and (2)

∥γx+ (1− γ)y∥2 = γ∥x∥2 + (1− γ)∥y∥2 − γ(1− γ)∥x− y∥2 (3)

for any γ ∈ R.

The following lemma provides a sufficient condition for the convergence of non-negative
real number sequences and will be used in the next section.

Lemma 1 ([18]). Let {un} and {vn} be sequences in [0,∞) such that un+1 ≤ un + vn for

all n and
∞∑
n=1

vn < ∞. Then, {un} is a convergent sequence in [0,∞).



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 4 of 20

3. Main results

Throughout this section, unless otherwise specified, we assume the following conditions
hold.

(i) E(G) is convex.

(ii) Each mapping Ji : X → X is G-nonexpansive for all i = 1, 2, . . . , N .

(iii) F :=

N⋂
i=1

Fix(Ji) ̸= ∅.

Building upon the preceding discussion, we now present our proposed algorithm in-
spired by the previous findings and based on the algorithm in [15], which combines the
parallel monotone hybrid algorithm and the inertial method.

Algorithm 1

Initialization: Choose x0, x1 ∈ X, and let n := 1.
Iterative Steps: Iteratively construct the sequence {xn} as follows.
Step 1. Calculate

dn = xn + βn(xn − xn−1),

where {βn} is a sequence of real numbers.
Step 2. Calculate

cin = (1− µi
n)dn + µi

nJidn,

bin = (1− ηin)Jidn + ηinJic
i
n,

and

ain = (1− σi
n)Jib

i
n + σi

nJic
i
n,

where {µi
n}, {ηin} and {σi

n} are sequences in [0, 1] for all i = 1, 2, . . . , N .
Step 3. Set

xn+1 = argmax
{∥∥ain − dn

∥∥ : i = 1, 2, . . . , N
}
.

Continue the iterative process by replacing n with n+ 1 and repeating all steps.

For convenience, we provide the following useful results, adapted to our current setting
from [17] and [19], respectively.

Observation 1. Let {xn} be a sequence in X such that its weak sequential cluster points
are in F. If {∥xn − u∥} is convergent for all u ∈ F, then {xn} is weakly convergent in F.



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 5 of 20

Observation 2. For i ∈ {1, 2, 3, ..., N}, suppose that {un} is a sequence in X satisfying
un ⇀ u and (un − Jiun) → 0 for some u ∈ X. If there exists a subsequence {unk

} of {un}
such that (unk

, u) ∈ E(G) for all k ∈ N, then u ∈ F.

We now let {xn} be a sequence generated by Algorithm 1. To facilitate the proof of
our main result, we proceed by stating and proving the following lemmas.

Lemma 2. Let x∗ ∈ F. Suppose that

(i)
∞∑
n=1

|βn|∥xn − xn−1∥ < ∞;

(ii) {(dn, x∗), (x∗, dn)} ∩ E(G) ̸= ∅.

Then, {∥xn − x∗∥} is convergent and {xn} is bounded.

Proof. From (ii), we may assume that (dn, x
∗) ∈ E(G). The edge-preserving property

of Ji implies that (Jidn, x
∗) = (Jidn, Jix

∗) ∈ E(G) for all i = 1, 2, . . . , N. Then, it follows
from the convexity of E(G) that

(cin, x
∗) = ((1− µi

n)dn + µi
nJidn, x

∗) ∈ E(G).

Also, we obtain that (Jic
i
n, x

∗), (bin, x
∗), (Jib

i
n, x

∗), and (ain, x
∗) ∈ E(G) for all i =

1, 2, . . . , N . Now, for any i = 1, 2, . . . , N, from Proposition 1, we have the inequalities:

∥Jicin − x∗∥2 ≤ ∥cin − x∗∥2

= ∥(1− µi
n)dn + µi

nJidn − x∗∥2

= ∥(1− µi
n)(dn − x∗) + µi

n(Jidn − x∗)∥2

= (1− µi
n)∥dn − x∗∥2 + µi

n∥Jidn − x∗∥2 − (1− µi
n)µ

i
n∥dn − Jidn∥2

≤ (1− µi
n)∥dn − x∗∥2 + µi

n∥dn − x∗∥2 − (1− µi
n)µ

i
n∥dn − Jidn∥2

= ∥dn − x∗∥2 − (1− µi
n)µ

i
n∥dn − Jidn∥2,

and

∥Jibin − x∗∥2 ≤ ∥bin − x∗∥2

≤ (1− ηin)∥Jidn − x∗∥2 + ηin∥Jicin − x∗∥2

≤ (1− ηin)∥dn − x∗∥2 + ηin∥dn − x∗∥2 − ηin(1− µi
n)µ

i
n∥dn − Jidn∥2

= ∥dn − x∗∥2 − ηin(1− µi
n)µ

i
n∥dn − Jidn∥2. (4)

Consequently, we have the following.

∥ain − x∗∥2 ≤ (1− σi
n)∥Jibin − x∗∥2 + σi

n∥Jicin − x∗∥2

≤ (1− σi
n)∥dn − x∗∥2 − (1− σi

n)η
i
n(1− µi

n)µ
i
n∥dn − Jidn∥2

+ σi
n∥dn − x∗∥2 − σi

n(1− µi
n)µ

i
n∥dn − Jidn∥2



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 6 of 20

= ∥dn − x∗∥2 − (1− σi
n)η

i
n(1− µi

n)µ
i
n∥dn − Jidn∥2 − σi

n(1− µi
n)µ

i
n∥dn − Jidn∥2.

(5)

This implies that

∥ain − x∗∥ ≤ ∥dn − x∗∥
= ∥xn + βn(xn − xn−1)− x∗∥
≤ ∥xn − x∗∥+ |βn|∥xn − xn−1∥.

Finally, we obtain that

∥xn+1 − x∗∥ ≤ ∥xn − x∗∥+ |βn|∥xn − xn−1∥.

By (i) and Lemma 1, the sequence {∥xn − x∗∥} is convergent and thus {xn} is bounded.

Lemma 3. Assume that for all i = 1, 2, . . . , N , the following conditions are met:

(i)

∞∑
n=1

|βn|∥xn − xn−1∥ < ∞;

(ii) (dn, x
∗), (x∗, dn) ∈ E(G) for all x∗ ∈ F;

(iii) 0 < lim inf
n→∞

µi
n ≤ lim sup

n→∞
µi
n < 1;

(iv) lim inf
n→∞

σi
n > 0;

(v) G is transitive.

Then, lim
n→∞

∥dn − Jidn∥ = 0 for all i = 1, 2, . . . , N .

Proof. Let x∗ ∈ F. We first consider the following calculations.
From inequalities (2) and (5), and the definition of {dn},

∥ain − x∗∥2 ≤ ∥xn − x∗ + βn(xn − xn−1)∥2

− (1− σi
n)η

i
n(1− µi

n)µ
i
n∥dn − Jidn∥2 − σi

n(1− µi
n)µ

i
n∥dn − Jidn∥2

≤ ∥xn − x∗∥2 + 2⟨βn(xn − xn−1), dn − x∗⟩
− (1− σi

n)η
i
n(1− µi

n)µ
i
n∥dn − Jidn∥2 − σi

n(1− µi
n)µ

i
n∥dn − Jidn∥2

≤ ∥xn − x∗∥2 + 2|βn|∥xn − xn−1∥∥dn − x∗∥
− (1− σi

n)η
i
n(1− µi

n)µ
i
n∥dn − Jidn∥2 − σi

n(1− µi
n)µ

i
n∥dn − Jidn∥2. (6)

From (6), there exists some in such that xn+1 = ainn and that

σin
n (1− µin

n )µin
n ∥dn − Jindn∥2 ≤ ∥xn − x∗∥2 − ∥xn+1 − x∗∥2 + 2|βn|∥xn − xn−1∥∥dn − x∗∥.

It is clear that the right handed side of (6) tends to zero when n → ∞. Therefore



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 7 of 20

lim
n→∞

∥dn − Jindn∥ = 0. (7)

Next, we will show that limn→∞ ∥dn − Jidn∥ = 0. By the triangle inequality and the
G-nonexpansivity of Jin , we have that

∥xn+1 − dn∥ ≤ (1− σin
n )∥Jinbinn − dn∥+ σin

n ∥Jincinn − dn∥
≤ (1− σin

n )∥Jinbinn − Jindn∥+ (1− σin
n )∥Jindn − dn∥

+ σin
n ∥Jincinn − Jindn∥+ σin

n ∥Jindn − dn∥
≤ (1− σin

n )∥binn − dn∥+ ∥Jindn − dn∥+ σin
n ∥Jincinn − Jindn∥

≤ (1− σin
n )∥binn − Jindn∥+ (2− σin

n )∥Jindn − dn∥+ σin
n ∥Jincinn − Jindn∥.

We obtain the following calculations by applying the definitions of binn and cinn together
with the G-nonexpansivity of Jin :

∥xn+1 − dn∥ ≤ (1− σin
n )ηinn ∥Jincinn − Jindn∥+ (2− σin

n )∥Jindn − dn∥+ σin
n ∥Jincinn − Jindn∥

= {(1− σin
n )ηinn + σin

n }∥Jincinn − Jindn∥+ (2− σin
n )∥Jindn − dn∥

≤ {(1− σin
n )ηinn + σin

n }∥cinn − dn∥+ (2− σin
n )∥Jindn − dn∥

= {(1− σin
n )ηinn + σin

n }µin
n ∥Jindn − dn∥+ (2− σin

n )∥Jindn − dn∥
= [{(1− σin

n )ηinn + σin
n }µin

n + (2− σin
n )]∥Jindn − dn∥. (8)

By (7), taking n → ∞, we can conclude that

lim
n→∞

∥ain − dn∥ = 0 for all i = 1, 2, . . . , N.

Now, it follows from (5) that there exists a constant C > 0 such that

σi
n(1− µi

n)µ
i
n∥dn − Jidn∥2 ≤ ∥dn − x∗∥2 − ∥ain − x∗∥2

− (1− σi
n)η

i
n(1− µi

n)µ
i
n∥dn − Jidn∥2

≤ C(∥dn − x∗∥ − ∥ain − x∗∥)
≤ C∥dn − ain∥.

Finally, we have that

lim
n→∞

∥dn − Jidn∥ = 0

for all i = 1, 2, . . . , N .

Analogously, we establish the following lemma using a similar approach.

Lemma 4. Assume that for all i = 1, 2, . . . , N , the following conditions are met:



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 8 of 20

(i)

∞∑
n=1

|βn|∥xn − xn−1∥ < ∞;

(ii) (dn, x
∗), (cin, dn) and (bin, dn) ∈ E(G) for all x∗ ∈ F;

(iii) 0 < lim inf
n→∞

µi
n ≤ lim sup

n→∞
µi
n < 1;

(iv) lim inf
n→∞

σi
n > 0.

Then, lim
n→∞

∥dn − Jidn∥ = 0 for all i = 1, 2, . . . , N .

Proof. Here, we replace conditions (ii) and (iv) of Lemma 3 with condition (ii) of
Lemma 4. The proof then proceeds similarly to that of Lemma 3.

Subsequently, after replacing certain assumptions, we present the following lemma.

Lemma 5. Assume that for all i = 1, 2, . . . , N , the following conditions are met:

(i)
∞∑
n=1

|βn|∥xn − xn−1∥ < ∞;

(ii) (dn, x
∗), (x∗, dn) ∈ E(G) for all x∗ ∈ F;

(iii) 0 < lim inf
n→∞

µi
n ≤ lim sup

n→∞
µi
n < 1;

(iv) lim inf
n→∞

ηin > 0 and lim sup
n→∞

σi
n < 1;

(v) G is transitive.

Then, lim
n→∞

∥dn − Jidn∥ = 0 for all i = 1, 2, . . . , N .

Proof. Let x∗ ∈ F. It follows from (6) that

∥ain − x∗∥2 ≤ ∥xn − x∗∥2 + 2|βn|∥xn − xn−1∥∥dn − x∗∥ − (1− σi
n)η

i
n(1− µi

n)µ
i
n∥dn − Jidn∥2.

Based on this, we observe that

(1− σin
n )ηinn (1− µin

n )µin
n ∥dn − Jindn∥2 ≤ ∥xn − x∗∥2 − ∥xn+1 − x∗∥2 + 2|βn|∥xn − xn−1∥∥dn − x∗∥.

This leads to the conclusion that

lim
n→∞

∥dn − Jindn∥ = 0.

Next, by (8), we have that

∥xn+1 − dn∥ ≤
(
{(1− σin

n )ηinn + σin
n }µin

n + (2− σin
n )

)
∥Jindn − dn∥.



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 9 of 20

Accordingly,

lim
n→∞

∥ain − dn∥ = 0 for all i = 1, 2, . . . , N.

By (5), we establish the existence of a positive constant D > 0 such that

(1− σi
n)η

i
n(1− µi

n)µ
i
n∥dn − Jidn∥2 ≤ ∥dn − x∗∥2 − ∥ain − x∗∥2

− σi
n(1− µi

n)µ
i
n∥dn − Jidn∥2

≤ D(∥dn − x∗∥ − ∥ain − x∗∥)
≤ D∥dn − ain∥.

This concludes the proof, as we have shown that

lim
n→∞

∥dn − Jidn∥ = 0 (9)

for all i = 1, 2, . . . , N .

Applying designated modifications, we procure the ensuing lemmas, which are requisite
for the proof of a subsequent theorem.

Lemma 6. Assume that for all i = 1, 2, . . . , N , the following conditions are met:

(i)
∞∑
n=1

|βn|∥xn − xn−1∥ < ∞;

(ii) (dn, x
∗), (cin, dn) and (bin, dn) ∈ E(G) for all x∗ ∈ F;

(iii) 0 < lim inf
n→∞

µi
n ≤ lim sup

n→∞
µi
n < 1;

(iv) lim inf
n→∞

ηin > 0 and lim sup
n→∞

σi
n < 1.

Then, lim
n→∞

∥dn − Jidn∥ = 0 for all i = 1, 2, . . . , N .

Proof. Conditions (ii) and (iv) of Lemma 5 can be replaced by condition (ii) of Lemma
6, and the proof remains valid.

Lemma 7. Assume that for all i = 1, 2, . . . , N , the following conditions are met:

(i)

∞∑
n=1

|βn|∥xn − xn−1∥ < ∞;

(ii) (x∗, dn), (dn, c
i
n) and (dn, b

i
n) ∈ E(G) for all x∗ ∈ F;

(iii) 0 < lim inf
n→∞

µi
n ≤ lim sup

n→∞
µi
n < 1;



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 10 of 20

(iv) either lim inf
n→∞

σi
n > 0 or lim inf

n→∞
ηin > 0 and lim sup

n→∞
σi
n < 1.

Then, lim
n→∞

∥dn − Jidn∥ = 0 for all i = 1, 2, . . . , N .

Proof. The proof of Lemma 5 remains valid when its conditions (ii), (iv) and (v) are
substituted with conditions (ii) and (iv) from Lemma 7.

Building upon the preceding lemmas, we are now ready to present our main result,
which establishes the convergence of our proposed algorithm.

Theorem 1. Suppose the following hold.

(i) All the conditions in Lemma 3 are satisfied.

(ii) (dnk
, u) ∈ E(G) for all k whenever {dnk

} is a subsequence of {dn} such that dnk
⇀ u

for some u ∈ X.

Then, the sequence {xn} is weakly convergent in F.

Proof. From Lemmas 2 and 3, {∥xn − x∗∥} converges for all x∗ ∈ F and lim
n→∞

∥dn −
Jidn∥ = 0 for all i = 1, 2, . . . , N . Now, let u be a weak sequential cluster point of {xn}. It
follows that there exists a subsequence {xnk

} such that xnk
⇀ u. From condition (i), we

have that
∥dn − xn∥ = ∥xn + βn(xn − xn−1)− xn∥ = |βn|∥xn − xn−1∥.

Taking n → ∞, we have that {∥dn − xn∥} converges to 0.
This means that dnk

⇀ u. Then, assumption (ii) implies that (dnk
, u) ∈ E(G).

Applying Observation 2, we have that u ∈ F. Finally, by Observation 1, the sequence
{xn} is weakly convergent in F.

The following results are obtained through the application of the previously established
lemmas.

Theorem 2. Under the hypotheses of either one of Lemma 4, Lemma 5, Lemma 6 or
Lemma 7, and condition (ii) of Theorem 1, the sequence {xn} is weakly convergent in F.

Proof. The proof is a direct adaptation of that of Theorem 1, originally relying on the
conditions in Lemma 3, which equally be substituted with those in Lemma 4, Lemma 5,
Lemma 6, or Lemma 7.

We conclude this section with a special case. Observe that nonexpansive mappings are
a special case of G-nonexpansive mappings when the edge set E(G) contains all possible
ordered pairs (x, y) for x, y ∈ X. Indeed, in this case, the condition “for all x, y ∈ X such
that (x, y) ∈ E(G)” simply reduces to “for all x, y ∈ X.” This yields the following result.

Theorem 3. For each i = 1, 2, . . . , N , let the mapping Ji : X → X be G-nonexpansive.
Assume the following conditions hold:



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 11 of 20

(i)

∞∑
n=1

|βn|∥xn − xn−1∥ < ∞;

(ii) 0 < lim inf
n→∞

µi
n ≤ lim sup

n→∞
µi
n < 1;

(iii) either lim inf
n→∞

σi
n > 0 or lim inf

n→∞
ηin > 0 and lim sup

n→∞
σi
n < 1.

Then, the sequence {xn} is weakly convergent in F.

Proof. It is clear that condition (ii) of Lemma 7 and condition (ii) of Theorem 1 hold.
Therefore, by Theorem 2, the sequence {xn} weakly converges to an element in F.

4. The algorithms proposed for the experimental study

The experimental investigation detailed in the subsequent section aims to assess the
performance characteristics of three algorithms. The first algorithm is derived from Al-
gorithm 1 through the imposition of the constraints σi

n = 0 and ηin = 1 for all iterations
n. For convenience, we will also denote this modification of Algorithm 1 by Algorithm 1
itself.

To facilitate a comprehensive comparative analysis, the study will also incorporate
algorithms previously published in the literature, specifically Algorithm 2 and Algorithm
3 presented in [3] and [4], respectively.

Algorithm 2 [3]

Initialization: Choose x0, x1 ∈ X, and let n := 1.
Iterative Steps: Iteratively construct the sequence {xn} as follows.
Step 1. Calculate

dn = xn + βn(xn − xn−1),

where {βn} ⊂ [0, 1].
Step 2. Calculate

cin = (1− µi
n)dn + µi

nJidn and

ain = (1− ηin)Jidn + ηinJic
i
n,

where {µi
n}, {ηin} ⊂ [0, 1] for all i = 1, 2, . . . , N .

Step 3. Set
xn+1 = arg max

1≤i≤N
∥ain − dn∥.

Then, proceed by updating n to n+ 1 and repeating all steps.



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 12 of 20

Algorithm 3 [4]

Initialization: Choose x0, x1 ∈ X, and let n := 1.
Iterative Steps: Iteratively construct the sequence {xn} as follows.
Step 1. Calculate

dn = xn + βn(xn − xn−1),

where {βn} ⊂ [0, 1].
Step 2. Calculate

cin = (1− µi
n)dn + µi

nJidn,

bin = (1− ηin)c
i
n + ηinJic

i
n and

ain = (1− σi
n)b

i
n + σi

nJib
i
n,

where {µi
n}, {ηin}, {σi

n} ⊂ [0, 1] for all i = 1, 2, . . . , N .
Step 3. Set

xn+1 = arg max
1≤i≤N

∥ain − dn∥.

Then, proceed by updating n to n+ 1 and repeating all steps.

5. Image recovery problem subject to various types of blur

The process of recovering images degraded by various blurring filters demands a pre-
cise mathematical representation. The following equation effectively models this complex
scenario, enabling efficient and accurate recovery solutions.

hi = Mix
∗ + εi,

where hi is the observed image that has been blurred by the i-th blurring filter, Mi is
the matrix that models the blurring effect for the i-th filter, and εi represents the noise
present in the observed image hi for all i = 1, 2, . . . , N .

The objective is to reconstruct the latent image x∗ from degraded observations hi,
which are corrupted by both blur and noise. This inverse problem is addressed through a
sequence of optimization procedures, each corresponding to a different blurring filter:

min
x

ζi∥x∥1 +
1

2
∥Mix− hi∥22

where ζi > 0 is a regularization parameter for all i = 1, 2, . . . , N . To apply our main result
in solving the optimization problems, for all i = 1, 2, . . . , N , we define

Ji(·) = proxsiGi
(I − si∇Fi)(·),

where Fi(·) =
1

2
∥Mi(·) − hi∥22, Gi(·) = ζi∥ · ∥1 and 0 < si <

2

∥Mi∥22
. Consequently,

{Ji : i = 1, 2, . . . , N} is established as a family of G-nonexpansive mappings, given that
the associated graph G is the trivially defined complete graph.



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 13 of 20

(a) Chiang Mai International Airport (CNX)
in April 2024 (b) Hazy Skies in Chiang Mai

Figure 1: Original test images.

Figure 1 shows the original test images: (a) Chiang Mai International Airport, with a
resolution of 962× 1000× 3 pixels, taken with Samsung Galaxy S22 Ultra, and (b) Hazy
Skies in Chiang Mai, with a resolution of 1000× 862× 3 pixels, selected from the NASA
Visible Earth website.

The three blurring filters, M1, M2 and M3, are generated using a motion blur kernel
with a motion length of 29. Specifically, M1 corresponds to an angle of 0◦, M2 to 45◦,
and M3 to 90◦. An additive zero-mean white Gaussian noise with a standard deviation
of 10−3 is then added for all i = 1, 2, 3. The resulting degraded images are presented in
Figure 2.

We investigate the behavior of our algorithm and then compare it with Algorithm 2
and Algorithm 3 under the following assumptions.

(i) For recovering Figure 2 (a) - (c), let x0 = x1 = 1 ∈ X := R962×1000×3.

(ii) For recovering Figure 2 (d) - (f), let x0 = x1 = 1 ∈ X := R1000×862×3.

(iii) Set ζi = 10−3, for all i = 1, 2, 3.

(iv) Set t0 = 1, and define tn =
1 +

√
1 + 4t2n−1

2
for all n ∈ N.



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 14 of 20

(a) M1 (b) M2 (c) M3

(d) M1 (e) M2 (f) M3

Figure 2: Degraded images.

(v) Define the sequence {βn} by

βn =


min

{
1

n2∥xn − xn−1∥2
, 0.5

}
if xn ̸= xn−1 and n > 100;

0.5 if xn = xn−1 and n > 100;
tn−1 − 1

tn
otherwise

for all n ∈ N.

The further control parameters of the three algorithms are detailed in Table 1.
Figure 3 (a) - (c) presents the recovered images from Figure 2 (a) - (c) utilizing Algo-

rithm 2, Algorithm 3 and Algorithm 1, respectively. Similarly, Figure 3 (d) - (f) depicts
the recovered images from Figure 2 (d) - (f) using the same respective methods. Each
image shown in Figure 3 is the result of 100 iterations (n = 100). The performance of the
algorithms in the image recovery process is quantitatively assessed using Gradient Mag-
nitude Similarity Deviation (GMSD) [20] and Multi-Scale Gradient Magnitude Similarity
Deviation (MS-GMSD) [21]. Lower values of GMSD and MS-GMSD indicate greater sim-
ilarity between the restored and original images, reflecting improved image quality. The
GMSD and MS-GMSD values for the recovered images in Figure 3 are presented in Table



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 15 of 20

2. GMSD and MS-GMSD convergence data for the iterative reconstruction of Figure 2
are presented in Figures 4 and 5.

Table 1: The control parameters of the three algorithms.

si µi
n ηin σi

n

Algorithm 2 ∥Mi∥−2
2 0.5 0.5 -

Algorithm 3 ∥Mi∥−2
2 0.5 0.5 0.5

Algorithm 1 ∥Mi∥−2
2 0.5 1 0

Table 2: Performance metrics (GMSD and MS-GMSD) for the recovered images from
Figure 3.

Restored images GMSD MS-GMSD

Fig 3 (a) Algorithm 2 0.0039 0.0036
Fig 3 (b) Algorithm 3 0.0033 0.0033
Fig 3 (c) Algorithm 1 0.0025 0.0030
Fig 3 (d) Algorithm 2 0.0249 0.0210
Fig 3 (e) Algorithm 3 0.0207 0.0179
Fig 3 (f) Algorithm 1 0.0119 0.0115

In this experiment, we conduct a numerical comparison of our proposed Algorithm
1 against two existing methods: Algorithm 2 [3] and Algorithm 3 [4]. We evaluate the
convergence and reconstruction accuracy of these algorithms using the GMSD and MS-
GMSD metrics. Table 2 presents the GMSD and MS-GMSD values for the recovered
images shown in Figure 3, offering a quantitative assessment of the performance of three
different algorithms. Figures 4 and 5 illustrate the performance of each algorithm, display-
ing the GMSD and MS-GMSD values as a function of iteration count for the reconstruction
of Figure 2.

Table 2 reveals that Algorithm 1 generally achieves the lowest GMSD and MS-GMSD
values, indicating better image restoration quality compared to Algorithms 2 and 3, par-
ticularly evident in the results for Figure 3 (c) and (f). While Algorithm 3 demonstrates
comparable performance in Figure 3 (b) and (e), Algorithm 2 consistently exhibits the
highest GMSD and MS-GMSD values across the dataset, suggesting a relatively lower
restoration efficacy. The lower values for both metrics in Algorithm 1 suggest that it is
more effective in preserving image details and structural similarity after the restoration
process.

As seen in Figures 4(a) and 5(a), all three algorithms demonstrate a decreasing trend
in GMSD with increasing iterations, indicating convergence toward a solution. However,
Algorithm 1 consistently exhibits the lowest GMSD values across all iterations, suggesting
better reconstruction accuracy compared to Algorithm 2 and Algorithm 3. In Figure 4,



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 16 of 20

(a) Algorithm 2
(GMSD = 0.0039 and
MS-GMSD = 0.0036)

(b) Algorithm 3
(GMSD = 0.0033 and
MS-GMSD = 0.0033)

(c) Algorithm 1
(GMSD = 0.0025 and
MS-GMSD = 0.0030)

(d) Algorithm 2
(GMSD = 0.0249 and
MS-GMSD = 0.0210)

(e) Algorithm 3
(GMSD = 0.0207 and
MS-GMSD = 0.0179)

(f) Algorithm 1
(GMSD = 0.0119 and
MS-GMSD = 0.0115)

Figure 3: Results of image recovery as performed by the three algorithms.

Algorithm 1 begins with a GMSD value around 0.25, rapidly reducing the error within
the first 15 iterations, then continuing to converge more gradually, ultimately finishing
below 0.05 and maintaining the lowest error throughout. In Figure 5, it again shows a
sharp reduction in error during the first 20 iterations and continues to outperform the
other algorithms. Its MS-GMSD values also remain the lowest across all iterations. This
trend is further corroborated by the MS-GMSD analysis in Figures 4(b) and 5(b), where
Algorithm 1 again achieves the lowest values. The consistent patterns observed in both
GMSD and MS-GMSD metrics confirm the effectiveness of Algorithm 1 in minimizing
reconstruction errors.

Algorithm 1 resulted in the lowest GMSD and MS-GMSD values when compared to
Algorithm 2 and Algorithm 3, indicating its effectiveness in reconstructing the data rep-
resented in Figure 2. While Algorithm 2 and Algorithm 3 also demonstrated convergence,
Algorithm 3 generally yielded slightly lower GMSD and MS-GMSD values compared to
Algorithm 2. The evidence presented suggests that Algorithm 1 emerges as the preferred



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 17 of 20

(a) GMSD (b) MS-GMSD

Figure 4: GMSD and MS-GMSD values by recovering Figure 2 (a) - (c).

(a) GMSD (b) MS-GMSD

Figure 5: GMSD and MS-GMSD values by recovering Figure 2 (d) - (f).

choice for the iterative reconstruction of Figure 2 data, due to its performance in mini-
mizing both metrics. This outcome highlights the importance of selecting an appropriate
algorithm to achieve optimal reconstruction accuracy.

6. Conclusions

We have presented an algorithm for common fixed point approximation of graph-based
mappings in Hilbert spaces and proven its convergence. Its effectiveness in PM2.5-related
image recovery was demonstrated and confirmed through comparative analysis, showing
competitive performance and higher-quality results. The algorithm’s efficiency highlights
its ability to optimize computational resources. Furthermore, the convergence behavior
directly influences the quality of the numerical results.



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 18 of 20

Moreover, it is worth noting that the subsequent list of potential reductions indicates
Algorithm 1, when simplified, might be reducible to algorithms already established in the
literature.

(A) Set σi
n = 1 for all n ∈ N and i = 1, 2, . . . , N in Algorithm 1.

(B) Set ηin = 1 for all n ∈ N and i = 1, 2, . . . , N in Algorithm 1.

(C) Set ηin = 0 for all n ∈ N and i = 1, 2, . . . , N in Algorithm 1.

(D) Set σi
n = 0 for all n ∈ N and i = 1, 2, . . . , N in Algorithm 1.

Acknowledgements

This research project was supported by:

(i) Fundamental Fund 2025, Chiang Mai University, Chiang Mai, Thailand;

(ii) Thailand Science Research and Innovation (TSRI) (FRB680102/0162);

(iii) Chiang Mai University, Chiang Mai, Thailand;

(iv) Centre of Excellence in Mathematics, MHESI, Bangkok, Thailand.

References

[1] J. Jachymski. The contraction principle for mappings on a metric space with a graph.
Proceedings of the American Mathematical Society, 136(4):1359–1373, 2008.

[2] A. Khemphet, R. Suparatulatorn, P. Varnakovida, and P. Charoensawan. A modi-
fied parallel algorithm for a common fixed-point problem with application to signal
recovery. Symmetry, 15(1464), 2023.

[3] N. Jun-On, R. Suparatulatorn, M. Gamal, and W. Cholamjiak. An inertial parallel
algorithm for a finite family of G-nonexpansive mappings applied to signal recovery.
AIMS Mathematics, 7(2):1775–1790, 2022.

[4] D. Yambangwai and T. Thianwan. A parallel inertial SP-iteration monotone hybrid
algorithm for a finite family of G-nonexpansive mappings and its application in linear
system, differential, and signal recovery problems. Carpathian Journal of Mathemat-
ics, 40(2):535–557, 2024.

[5] R. Suparatulatorn, P. Saksuriya, T. Suebcharoen, and K. Chaichana. An iterative
approach to common fixed points of G-nonexpansive mappings with applications in
solving the heat equation. Axioms, 13(11):729, 2024.

[6] P. Sridarat, R. Suparatulatorn, S. Suantai, and Y. J. Cho. Convergence analysis
of SP-iteration for G-nonexpansive mappings with directed graphs. Bulletin of the
Malaysian Mathematical Sciences Society, 42(5):2361–2380, 2019.



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 19 of 20

[7] S. Suantai, M. Donganont, and W. Cholamjiak. Hybrid methods for a countable fam-
ily of G-nonexpansive mappings in hilbert spaces endowed with graphs. Mathematics,
7(10):936, 2019.

[8] S. Suantai, K. Kankam, P. Cholamjiak, and W. Cholamjiak. A parallel monotone
hybrid algorithm for a finite family of G-nonexpansive mappings in hilbert spaces
endowed with a graph applicable in signal recovery. Computational and Applied
Mathematics, 40(4):145, 2021.

[9] P. Charoensawan, D. Yambangwai, W. Cholamjiak, and R. Suparatulatorn. An iner-
tial parallel algorithm for a finite family of G-nonexpansive mappings with application
to the diffusion problem. Advances in Difference Equations, 2021(453), 2021.

[10] J. J. Liaw, Y. F. Huang, C. H. Hsieh, D. C. Lin, and C. H. Luo. PM2.5 concentration
estimation based on image processing schemes and simple linear regression. Sensors,
20(8):2423, 2020.

[11] J. Ma, K. Li, Y. Han, P. Du, and J. Yang. Image-based PM2.5 estimation and its
application on depth estimation. In 2018 IEEE International Conference on Acous-
tics, Speech and Signal Processing (ICASSP), pages 1857–1861, Calgary, AB, Canada,
2018.

[12] T. Zheng, M. H. Bergin, S. Hu, J. Miller, and D. E. Carlson. Estimating ground-level
PM2.5 using micro-satellite images by a convolutional neural network and random
forest approach. Atmospheric Environment, 230:117451, 2020.

[13] K. Gu, J. Qiao, and X. Li. Highly efficient picture-based prediction of PM2.5 con-
centration. IEEE Transactions on Industrial Electronics, 66(4):3176–3184, 2019.

[14] H. Wang, X. Yuan, X. Wang, Y. Zhang, and Q. Dai. Real-time air quality estimation
based on color image processing. In 2014 IEEE Visual Communications and Image
Processing Conference, pages 326–329, Valletta, Malta, 2014.

[15] M. Abbas and T. Nazir. A new faster iteration process applied to constrained mini-
mization and feasibility problems. Matematički Vesnik, 66(3):223–234, 2014.

[16] M. R. Alfuraidan and M. A. Khamsi. Fixed points of monotone nonexpansive map-
pings on a hyperbolic metric space with a graph. Fixed Point Theory and Applications,
2015(1):44, 2015.

[17] H. H. Bauschke and P. L. Combettes. Convex analysis and monotone operator theory
in Hilbert spaces. CMS Books in Mathematics. Springer, New York, NY, USA, 2011.

[18] A. Auslender, M. Teboulle, and S. Ben-Tiba. A logarithmic-quadratic proximal
method for variational inequalities. Computational Optimization and Applications,
12(1):31–40, 1999.

[19] R. Suparatulatorn, S. Suantai, and W. Cholamjiak. Hybrid methods for a finite
family of G-nonexpansive mappings in hilbert spaces endowed with graphs. AKCE
International Journal of Graphs and Combinatorics, 14(2):101–111, 2017.

[20] W. Xue, L. Zhang, X. Mou, and A. C. Bovik. Gradient magnitude similarity devia-
tion: A highly efficient perceptual image quality index. IEEE Transactions on Image
Processing, 23(2):684–695, 2014.

[21] B. Zhang, P. V. Sander, and A. Bermak. Gradient magnitude similarity deviation
on multiple scales for color image quality assessment. In 2017 IEEE International



S. Panma, R. Suparatulatorn, T. Chaobankoh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6311 20 of 20

Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1253–1257,
New Orleans, LA, USA, 2017.