EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6173
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Inertial Iterative Method for Generalized Mixed
Equilibrium Problem and Fixed Point Problem

Vahid Darvish1, Grace Nnennaya Ogwo2,∗, Olawale Kazeem Oyewole3,4,
Hammed Anuoluwapo Abass5,6, Amirbek Aminovich Ikramov6

1 Department of Mathematics and Statistics, Nanjing University of Information Science and
Technology, Nanjing China
2 School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China
3 Department of Mathematics and Statistics, Tshwane University of Technology, PMB 007,
Arcadia, Pretoria, South Africa
4 Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical
and Technical Sciences, Saveetha University, Chennai 602 105, Tamil Nadu, India
5 Department of Mathematics and Applied Mathematics, Sefako Makgato Health Science
University, P.O. Box 94, Pretoria 0204, South Africa
6Center of Research and Innovation, Asia International University, Yangiobod MFY, G‘ijduvon
Street, House 74, Bukhara, Uzbekistan

Abstract. In this paper, we study the generalized mixed equilibrium problem and the fixed point
problem. We propose an inertial iterative method for approximating the common solution of a
generalized mixed equilibrium problem of a monotone mapping and a fixed point problem for a
Bregman strongly nonexpansive mapping in the framework of real reflexive Banach spaces. Under
certain mild conditions, we obtain a strong convergence result of the proposed method. Finally,
we present numerical examples to illustrate the applicability of our method.

2020 Mathematics Subject Classifications: 47H05, 47H09, 47J05, 49J25

Key Words and Phrases: Generalized mixed equilibrium problem, fixed point problem, inertial
technique, Bregman strongly nonexpansive mapping

1. Introduction

Let E be a real reflexive Banach space and E∗ be its dual space. Let C be a nonempty,
closed and convex subset of E, Θ : C × C → R a bifunction, Ψ : C → E∗ a nonlinear

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6173

Email addresses: vahid.darvish@mail.com, vdarvish@nuist.edu.cn (V. Darvish),
graceogwo@zjnu.edu.cn (G. N. Ogwo), oyewoleolawalekazeem@gmail.com (O. K. Oyewole),
hammedabass548@gmail.com,hammed.abass@smu.ac.za (H. A. Abass),
amirbekikramov@oxu.uz (A. A. Ikramov)

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



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 2 of 32

mapping and φ : C → R a real valued function. The generalized mixed equilibrium problem
(GMEP) is defined as follows: Find x ∈ C such that

Θ(x, y) + ⟨Ψx, y − x⟩+ φ(y) ≥ φ(x), for all y ∈ C. (1.1)

The set of solutions of (1.1) is denoted by GMEP(Θ,Ψ, φ), that is

GMEP(Θ,Ψ, φ) = {x ∈ C : Θ(x, y) + ⟨Ψx, y − x⟩+ φ(y) ≥ φ(x), for all y ∈ C}.

In particular, if Ψ ≡ 0, the problem (1.1) is reduced to the mixed equilibrium problem
(MEP) [1] defined as follows: Find x ∈ C such that

Θ(x, y) + φ(y) ≥ φ(x), for all y ∈ C. (1.2)

The set of solutions of (1.2) is denoted by MEP (Θ, φ).
If φ ≡ 0, the problem (1.1) is reduced to the generalized equilibrium problem (GEP) [2]
defined as follows: Find x ∈ C such that

Θ(x, y) + ⟨Ψx, y − x⟩ ≥ 0, for all y ∈ C. (1.3)

The set of solution (1.3) is denoted by GEP (Θ,Ψ).
If Θ ≡ 0, the problem (1.1) is reduced to the mixed variational inequality of Browder type
(MVI) [3] defined as follows: Find x ∈ C such that

⟨Ψx, y − x⟩+ φ(y) ≥ φ(x), for all y ∈ C. (1.4)

The set of solution of (1.4) is denoted by MV I(φ,Ψ).
If Ψ ≡ 0 and φ ≡ 0, the problem (1.1) is reduced to the equilibrium problem (EP) [4] for
finding x ∈ C such that

Θ(x, y) ≥ 0, for all y ∈ C. (1.5)

The set of solutions of (1.5) is denoted by EP (Θ).
We observe that (1.1) generalizes (1.2)-(1.5).
The equilibrium problem was introduced by Blum and Oettli [4] and Noor and Oettli [5]
in 1994, and has had a great impact and influence in the development of several branches
of pure and applied sciences. It has been shown that equilibrium problem theory provides
a novel and unified treatment of a wide class of problems that arise in economics, finance,
image reconstruction, ecology, transportation, networks, elasticity, and optimization. The
EP was shown in [4] to cover monotone inclusion problems, saddle point problems, vari-
ational inequality problems, minimization problems, optimization problems, variational
inequality problems, vector equilibrium problems, Nash equilibrium problems in noncoop-
erative games. Some methods have been proposed to solve the equilibrium problem. These
methods include the penalty and gap functions, regularization, extragradient methods and
splitting methods (see [6–10] and other references therein).
For solving the generalized mixed equilibrium problem, let us assume that the bifunction
Θ : C × C → R satisfies the following conditions:



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 3 of 32

(Q1) Θ(x, x) = 0 for all x ∈ C;
(Q2) Θ is monotone, i.e., Θ(x, y) + Θ(y, x) ≤ 0 for any x, y ∈ C;
(Q3) for each y ∈ C, x 7→ Θ(x, y) is upper-hemicontinuous, i.e., for each x, y, z ∈ C,

lim sup
t↘0

Θ(tz + (1− t)x, y) ≤ Θ(x, y);

(Q4) for each x ∈ C, y 7→ Θ(x, y) is convex and lower semicontinuous (see [11]).
Several authors have studied and proposed various iterative methods for studying (1.1).
Tuyen [12] introduced a hybrid projection method for solving systems of GMEP in a
reflexive Banach space and defined it as follows:

yin = ResfΘi,Ψi,φi
xn, i = 1, 2, · · · , N

in := argmaxi=1,2··· ,N{Df (y
i
n, xn)}, ȳn = yinn

Cn := {z ∈ E : Df (z, ȳn ≤ Df (z, xn))}
Qn := {z ∈ E : ⟨∇f(x0)−∇f(xn), z − xn⟩ ≤ 0}
xn+1 = ProjCn∩Qn

(x0), n ≥ 0.

The author obtained a strong convergence result of the proposed method. The limitation
OF this method is the fact that it requires the computation of subsets of Cn and Qn which
can be computationally expensive.
On the other hand, another problem of interest is the fixed point problem (FPP). Let
T : C → C be a nonlinear mapping. A point x ∈ C is a fixed point of T if Tx = x. Let
F (T ) denote the set of fixed points, that is

F (T ) = {x ∈ C : Tx = x}.

Many problems in sciences and engineering can be transformed into a problem of finding
the solution of a fixed point problem (FPP) of a nonlinear mapping. For more information
on fixed point see [13–17]. Moudafi [18] introduced the viscosity approximation method
for a nonexpansive mapping T and defined it as follows:

xn+1 = αnf(xn) + (1− αn)Txn, n ≥ 1,

where {αn} ⊂ (0, 1) and f is a contraction mapping.
Recently, several authors have studied iterative algorithms for finding a common solution
of the FFP and GMEP. In particular, several authors have considered the following
problem (see [19–21] and other references therein):

Find x ∈ C such that
x ∈ F (T ) ∩GMEP(Θ,Ψ, φ).

The motivation for studying a common solution problem lies in its application to problems
whose constraints can be reformulated as FPPs and GMEPs. For instance, in signal
processing, network resource allocation, among others.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 4 of 32

Recently, Takahashi and Takahashi [22] introduced the following iterative scheme for solv-
ing GEP and FPP of a nonexpansive mapping T in a Hilbert space. They defined the
proposed method as follows: Find x1, z ∈ C and

zn ∈ C such that

Θ(zn, y) + ⟨Bxn, y − zn⟩+ 1
rn
⟨y − zn, zn − xn⟩ ≥ 0, y ∈ C

xn+1 = βnxn + (1− βn)T
[
αnz + (1− αn)zn

]
, n ≥ 1

{αn} ⊂ (0, 1), {βn} ⊂ (0, 1), {rn} ⊂ (0,+∞) and B is an α-inverse strongly monotone
mapping. The authors obtained a strong convergent result under certain conditions. Es-
kandari and Raeisi [23] introduced an iterative method for approximating a common so-
lution of FPP of Bregman quasi-nonexpansive mappings and zeros of maximal monotone
operators. They defined the algorithm as follows:

x1 ∈ E,

zn = Resf
λN
n Bn

◦ . . . ◦Resf
λ1
nB1

xn,

yn = βn∇f(zn) + (1− βn)∇f(Tn(zn)),

xn+1 = ∇f∗(αn∇f(qn) + (1− αn)yn),

where {αn} ⊂ (0, 1). Under certain standard conditions, the authors obtained a strong
convergence result.
The ultimate aim of every researcher is to construct effective iterative methods with high
convergence rates to the solutions of the optimization problem under consideration. To
achieve this high rate of convergence, authors employ the inertial technique. Polyak [24]
introduced the inertial extrapolation as an acceleration process to solve smooth convex
minimization problems. It has been shown by several authors that the inertial term
improves the performance of iterative algorithms numerically in terms of the number of
iterations and CPU time. Several authors have studied and proposed iterative algorithms
with the inertial technique for solving optimization problems (see to [25–27] and other
references therein).
Motivated by the above mentioned methods in the literature and the ongoing research in
this area, we introduce a new inertial iterative method for approximating the solutions of
a generalized mixed equilibrium problem with a maximal monotone mapping and fixed
point of a Bregman strongly nonexpansive mapping in the framework of a reflexive Banach
space. Our method does not require us to compute subsets of Cn and Qn. Under mild
condition, we establish a strong convergence result for the proposed method. Finally, we
present numerical examples to illustrate the applicability of our proposed method.
The rest of the paper is organized as follows: In Section 2, we present some basic def-
initions, concepts, lemmas, and results which will be required to obtain the convergence
analysis of the proposed method. In Section 3, we present some required assumptions and
introduce our proposed method. In Section 4, we present our convergence analysis. In
Section 5, we present numerical experiments in comparisons with other related methods
to illustrate the effectiveness of our proposed method. In Section 6, we present a brief
summary of our result.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 5 of 32

2. Preliminaries

Let f : E → (−∞,+∞] be a proper, lower semi-continuous and convex function. We
denote by domf , the domain of f , that is the set {x ∈ E : f(x) < +∞}. For a sequence
{xn} in E, we denote the strong and weak convergence of {xn} to x ∈ E by xn → x and
xn ⇀ x, respectively.
Let x ∈ int(domf), the subdifferential of f at x is the convex set defined by

∂f(x) = {x∗ ∈ E∗ : f(x) + ⟨x∗, y − x⟩ ≤ f(y), for all y ∈ E},

where the Fenchel conjugate of f is the function f∗ : E∗ → (−∞,+∞] defined by

f∗(x∗) = sup{⟨x∗, x⟩ − f(x) : x ∈ E, x∗ ∈ E∗}.

For any x ∈ int(domf), the right-hand derivative of f at x in the derivation y ∈ E is
defined by

f
′
(x, y) := lim

t↘0

f(x+ ty)− f(x)

t
.

The function f is called Gâteaux differentiable at x if limt↘0
f(x+ty)−f(x)

t exists for all

y ∈ E. In this case, f
′
(x, y) coincides with ∇f(x), the value of the gradient (∇f) of f at

x. The function f is called Gâteaux differentiable if it is Gâteaux differentiable for any
x ∈ int(domf) and f is called Fréchet differentiable at x if this limit is attain uniformly
for all y which satisfies ∥y∥ = 1. The function f is uniformly Fréchet differentiable on a
subset C of E if the limit is attained uniformly for any x ∈ C and ∥y∥ = 1. It is known
that if f is Gâteaux differentiable (resp. Fréchet differentiable) on int(domf), then f is
continuous and its Gâteaux derivative ∇f is norm-to-weak∗ continuous (resp. continuous)
on int(domf) (see [28]).
Let f : E → (−∞,+∞] be a Gâteaux differentiable function. The function Df : domf ×
int(domf) → [0,+∞) defined as follows:

Df (x, y) := f(x)− f(y)− ⟨∇f(y), x− y⟩, for all x ∈ dom(f), y ∈ int(dom(f)) (2.1)

is called the Bregman distance with respect to f , [29].

Remark 1. The Bregman distance has the following properties:

(i) the three-point identity, for any x ∈ domf and y, z ∈ int(domf),

Df (x, y) +Df (y, z)−Df (x, z) = ⟨∇f(z)−∇f(y), x− y⟩; (2.2)

[30]

(ii) the four-point identity, for any y, w ∈ domf and x, z ∈ int(domf),

Df (y, x)−Df (y, z)−Df (w, x) +Df (w, z) = ⟨∇f(z)−∇f(x), y − w⟩. (2.3)



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 6 of 32

Definition 1. A Gâteaux differentiable function f is said to be γ-strongly convex if there
exists a constant γ > 0 such that

f(x) ≥ f(y) + ⟨x− y,∇f(y)⟩+ γ

2
∥x− y∥2, for all x ∈ dom(f), y ∈ int(dom(f)).

Lemma 1. [31] Let f be a strongly convex function with constant γ > 0. Then for all
y ∈ dom(f) and x ∈ int(dom(f)), we have:

Df (x, y) ≥
γ

2
∥x− y∥2, (2.4)

where Df (x, y) is the Bregman distance with respect to f.

The Legendre function f : E → (−∞,+∞] is defined in [32]. It is well known that in
reflexive spaces, f is Legendre function if and only if it satisfies the following conditions:
(L1) The interior of the domain of f , int(domf), is nonempty, f is Gâteaux differentiable
on int(domf) and domf = int(domf);
(L2) The interior of the domain of f∗, int(domf∗), is nonempty, f∗ is Gâteaux differentiable
on int(domf∗) and domf∗ = int(domf∗).
Since E is reflexive, we know that (∂f)−1 = ∂f∗ (see [28]). This , with (L1) and (L2),
imply the following equalities:

∇f = (∇f∗)−1, ran∇f = dom∇f∗ = int(domf∗)

and
ran∇f∗ = dom(∇f) = int(domf),

where ran∇f denotes the range of ∇f .
When the subdifferential of f is single-valued, it coincides with the gradient ∂f = ∇f ,
[33]. By Bauschke et al. [32] the conditions (L1) and (L2) also yields that the function f
and f∗ are strictly convex on the interior of their respective domains.
If E is a smooth and strictly convex Banach space, then an important and interesting
Legendre function is f(x) := 1

p∥x∥
p(1 < p < +∞). In this case the gradient ∇f of f

coincides with the generalized duality mapping of E, i.e., ∇f = Jp(1 < p < +∞). In
particular, ∇f = I, the identity mapping in Hilbert spaces. From now on we assume
that the convex function f : E → (−∞,+∞] is Legendre. In connection with Legendre
functions, see also the recent paper [34].

Definition 2. Let f : E → (−∞,+∞] be a convex and Gâteaux differentiable function.
The Bregman projection of x ∈ int(domf) onto the nonempty, closed and convex subset

C ⊂ domf is the necessary unique vector projfC(x) ∈ C satisfying

Df (proj
f
C(x), x) = inf{Df (y, x) : y ∈ C}.

Remark 2. If E is a smooth and strictly convex Banach space and f(x) = ∥x∥2 for all
x ∈ E, then we have that ∇f(x) = 2Jx for all x ∈ E, where J is the normalized duality



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 7 of 32

mapping from E in to 2E
∗
, and hence Df (x, y) reduced to ϕ(x, y) = ∥x∥2−2⟨x, Jy⟩+∥y∥2,

for all x, y ∈ E, which is the Lyapunov function introduced by Alber [35] and Bregman

projection P f
C(x) reduces to the generalized projection ΠC(x) which is defined by

ϕ(ΠC(x), x) = min
y∈C

ϕ(y, x).

If E = H, a Hilbert space, J is the identity mapping and hence Bregman projection P f
C(x)

reduced to the metric projection of H onto C, PC(x).

Definition 3. [36, 37] Let f : E → (−∞,+∞] be a convex and Gâteaux differentiable
function. f is called:

(i) totally convex at x ∈ int(domf) if its modulus of total convexity at x, that is, the
function νf : int(domf)× [0,+∞) → [0,+∞) defined by

νf (x, t) := inf{Df (y, x) : y ∈ domf, ∥y − x∥ = t},

is positive whenever t > 0;

(ii) totally convex if it is totally convex at every point x ∈ int(domf);

(iii) totally convex on bounded sets if νf (B, t) is positive for any nonempty bounded subset
B of E and t > 0, where the modulus of total convexity of the function f on the set
B is the function νf : int(domf)× [0,+∞) → [0,+∞) defined by

νf (B, t) := inf{νf (x, t) : x ∈ B ∩ domf}.

The set levf≤(r) = {x ∈ E : f(x) ≤ r} for some r ∈ R is called a sublevel of f .

Definition 4. [37, 38] The function f : E → (−∞,+∞] is called;

(i) cofinite if domf∗ = E∗;

(ii) coercive [39] if the sublevel set of f is bounded; equivalently,

lim
∥x∥→+∞

f(x) = +∞;

(iii) strongly coercive if lim∥x∥→+∞
f(x)
∥x∥ = +∞;

(iv) sequentially consistent if for any two sequences {xn} and {yn} in E such that {xn}
is bounded,

lim
n→+∞

Df (yn, xn) = 0 ⇒ lim
n→+∞

∥yn − xn∥ = 0.

Lemma 2. [40] The function f is totally convex on bounded subsets if and only if it is
sequentially consistent.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 8 of 32

Lemma 3. [38, Proposition 2.3] If f : E → (−∞,+∞] is Fréchet differentiable and totally
convex, then f is cofinite.

Lemma 4. [40] Let f : E → (−∞,+∞] be a convex function whose domain contains at
least two points.Then the following statements hold:

(i) f is sequentially consistent if and only if it is totally convex on bounded sets;

(ii) If f is lower semicontinuous, then f is sequentially consistent if and only if it is
uniformly convex on bounded sets;

(iii) If f is uniformly strictly convex on bounded sets, then it is sequentially consistent and
the converse implication holds when f is lower semicontinuous, Fréchet differentiable
on its domain and Fréchet derivative ∇f is uniformly continuous on bounded sets.

Lemma 5. [41, Proposition 2.1] Let f : E → R be uniformly Fréchet differentiable and
bounded on bounded subsets of E. Then ∇f is uniformly continuous on bounded subsets
of E from the strong topology of E to the strong topology of E∗.

Lemma 6. [38, Lemma 3.1] Let f : E → R be a Gâteaux differentiable and totally convex
function. If x0 ∈ E and the sequence {Df (xn, x0)} is bounded, then the sequence {xn} is
also bounded.

A mapping T is said to be nonexpansive if ∥Tx − Ty∥ ≤ ∥x − y∥ for all x, y ∈ C. T is
said to be quasi-nonexpansive if F (T ) ̸= ∅ and ∥Tx − p∥ ≤ ∥x − p∥, for all x ∈ C and
p ∈ F (T ). A point p ∈ C is called an asymptotic fixed point of T (see [42]) if C contains
a sequence {xn} which converges weakly to p such that limn→+∞ ∥xn − Txn∥ = 0. We
denote by F̂ (T ) the set of asymptotic fixed points of T .
A mapping T : C → int(domf) with F (T ) ̸= ∅ is called:

(i) quasi-Bregman nonexpansive [38] with respect to f if

Df (p, Tx) ≤ Df (p, x), for all x ∈ C, p ∈ F (T ).

(ii) Bregman relatively nonexpansive [38, 43] with respect to f if,

Df (p, Tx) ≤ Df (p, x), for all x ∈ C, p ∈ F (T ), and F̂ (T ) = F (T ).

(iii) Bregman strongly nonexpansive (see [38, 44]) with respect to f and F̂ (T ) if,

Df (p, Tx) ≤ Df (p, x), for all x ∈ C, p ∈ F̂ (T )

and, if whenever {xn} ⊂ C is bounded, p ∈ F̂ (T ), and

lim
z→+∞

(Df (p, xn)−Df (p, Txn)) = 0,

it follows that
lim

n→+∞
Df (xn, Txn) = 0.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 9 of 32

(iv) Bregman firmly nonexpansive (for short BFNE [45]) with respect to f if, for all
x, y ∈ C,

⟨∇f(Tx)−∇f(Ty), Tx− Ty⟩ ≤ ⟨∇f(x)−∇f(y), Tx− Ty⟩

equivalently,

Df (Tx, Ty)+Df (Ty, Tx)+Df (Tx, x)+Df (Ty, y) ≤ Df (Tx, y)+Df (Ty, x). (2.5)

The existence and approximation of Bregman firmly nonexpansive mappings was studied
in [42]. It is also known that if T is Bregman firmly nonexpansive and f is Legendre
function which is bounded, uniformly Fréchet differentiable and totally convex on bounded
subset of E, then F (T ) = F̂ (T ) and F (T ) is closed and convex. It also follows that every
Bregman firmly nonexpansive mapping is Bregman strongly nonexpansive with respect to
F (T ) = F̂ (T ).

Lemma 7. [40] Let C be a nonempty, closed and convex subset of E. Let f : E → R be
a Gâteaux differentiable and totally convex function. Let x ∈ E, then
1) z = projfC(x) if and only if

⟨∇f(x)−∇f(z), y − z⟩ ≤ 0, for all y ∈ C.

2) Df (y, proj
f
C(x)) +Df (proj

f
C(x), x) ≤ Df (y, x), for all x ∈ E, y ∈ C.

Let f : E → R be a convex, Legendre and Gâteaux differentiable function. Following [35]
and [29], we make use of the function Vf : E × E∗ → [0,+∞) associated with f , which is
defined by

Vf (x, x
∗) = f(x)− ⟨x∗, x⟩+ f∗(x∗), for all x ∈ E, x∗ ∈ E∗. (2.6)

Then Vf is nonexpansive and Vf (x, x
∗) = Df (x,∇f∗(x∗)) for all x ∈ E and x∗ ∈ E∗.

Moreover, by the subdifferential inequality,

Vf (x, x
∗) + ⟨y∗,∇f∗(x∗)− x⟩ ≤ Vf (x, x

∗ + y∗) (2.7)

for all x ∈ E and x∗, y∗ ∈ E∗ [46]. In addition, if f : E → (−∞,+∞] is a proper lower
semicontinuous function, then f∗ : E∗ → (−∞,+∞] is a proper weak∗ lower semicontinu-
ous and convex function (see [47]). Hence, Vf is convex in the second variable. Thus, for
all z ∈ E,

Df

(
z,∇f∗

(
N∑
i=1

ti∇f(xi)

))
≤

N∑
i=1

tiDf (z, xi), (2.8)

where {xi}Ni=1 ⊂ E and {ti}Ni=1 ⊂ (0, 1) with
∑N

i=1 ti = 1.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 10 of 32

Definition 5. Let Br = {x ∈ E : ∥x∥ ≤ r} for all r > 0. Then a function g : E −→ R
is said to be uniformly convex on bounded sets of E [48] if ρr(t) > 0 for all r, t > 0 where
ρr : [0,+∞) −→ [0,+∞] is defined by

ρr(t) = inf
x,y∈Br,∥x−y∥=t,α∈(0,1)

αg(x) + (1− α)g(y)− g[αx+ (1− α)y]

α(1− α)

for all t > 0. The function ρr is called the gauge of uniform convexity of g. The function
g is said to be uniformly convex if the function δg : [0,+∞) −→ [0,+∞], defined by

δg(t) = sup
∥x−y∥=t

{
1

2
g(x) +

1

2
g(y)− g

(
x+ y

2

)}

satisfies that limt↓0
δg(t)

t
= 0.

Lemma 8. [49] Let C be a nonempty, closed and convex subset of int(domf) and T : C →
C be a quasi-Bregman nonexpansive mappings with respect to f . Then F (T ) is closed and
convex.

Definition 6. Let C be a nonempty, closed and convex subsets of a real reflexive Banach
space and let φ be a lower semicontinuous and convex functional from C to R and Ψ : C →
E∗ be a continuous monotone mapping. Let Θ : C × C → R be a bifunctional satisfying
(A1)-(A4). The mixed resolvent of Θ is the operator ResfΘ,φ,Ψ : E → 2C

ResfΘ,φ,Ψ(x) = {z ∈ C : Θ(z, y) + φ(y) + ⟨Ψz, y − z⟩+ ⟨∇f(z)−∇f(x), y − z⟩
≥ φ(z), for all y ∈ C}. (2.9)

Lemma 9. [50] Let f : E → (−∞,+∞] be a coercive and Gâteaux differentiable function.
Let C be a closed and convex subset of E. Assume that φ : C → R be a lower semicon-
tinuous and convex functional, Ψ : C → E∗ be a continuous monotone mapping and the
bifunctional Θ : C × C → R satisfies conditions (A1)-(A4), then dom(ResfΘ,φ,Ψ) = E.

Lemma 10. [50] Let f : E → (−∞,+∞] be a Legendre function. Let C be a closed and
convex subset of E. If the bifunction Θ : C × C → R satisfies conditions (A1)-(A4), then

(i) ResfΘ,φ,Ψ is single-valued;

(ii) ResfΘ,φ,Ψ is a BFNE operator;

(iii) F
(
ResfΘ,φ,Ψ

)
= GMEP(Θ, φ,Ψ);

(iv) GMEP(Θ, φ,Ψ) is closed and convex;

(v) Df

(
p,ResfΘ,φ,Ψ(x)

)
+Df

(
ResfΘ,φ,Ψ(x), x

)
≤ Df (p, x), for all p ∈ F

(
ResfΘ,φ,Ψ

)
, x ∈

E.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 11 of 32

Let Θ : C × C → R be a bifunction and define the mapping BΘ : E → 2E
∗
in the

following way:

BΘ(x) :=


{x∗ ∈ E∗ : Θ(x, y) + φ(y) + ⟨Ψx, y − x⟩ ≥ ⟨x∗, y − x⟩+ φ(x) for all y ∈ C},
x ∈ C,
∅ x /∈ C.

(2.10)

Lemma 11. [51] Let E be a Banach space and f : ER be a Gâteaux differentiable function
which is uniformly convex on bounded subsets of E. Suppose that {xn} and {yn} are two
sequenecs in E. Then, lim

n→+∞
Df (xn, yn) = 0 if and only if lim

n→+∞
∥xn − yn∥ = 0.

Lemma 12. [52] Let {an} be a sequence of nonnegative real numbers, {αn} be a sequence

in (0, 1) with
+∞∑
n=1

αn = +∞ and {bn} be a sequence of real numbers. Assume that

an+1 ≤ (1− αn)an + αnbn, for all n ≥ 1.

If lim sup
k→+∞

bnk
≤ 0 for every subsequence {ank

} of {an} satisfying lim inf
k→+∞

(ank+1
− ank

) ≥ 0,

then lim
n→+∞

an = 0.

3. Main result

In this section, we present the assumptions under which our convergence analysis will be
obtained. Furthermore, we present our proposed algorithm.

Assumption 3.1. Assumption A:
(A1) E is a reflexive Banach space with dual E∗ and C is a nonempty, closed and convex
subset of int(dom(f)).
(A2) T : C → C is a Bregman strongly nonexpansive mapping such that F (T ) = F̂ (T )
and T is uniformly continuous.
(A3) f : E → R is a super coercive Legendre function that is bounded, uniformly Fréchet
differentiable and totally convex on bounded subsets of E.
(A4) Bθj : E → 2E

∗
, j = 1, 2, . . . N is a maximal monotone mapping with dom(Bθj ) ⊂ C.

(A5) The solution set Ω = F (T ) ∩
(⋂N

j=1B
−1
θj

(0∗)
)
̸= ∅.

Assumption B:
(B1) Let {αn} and {βn} be sequences in [0, 1] satisfying the following lim

n→+∞
βn = 0 and

+∞∑
n=1

βn = +∞. 0 < lim inf
n→+∞

αn ≤ lim sup
n→+∞

αn < 1.

(B2) Let θ > 0 and {ξn} be a positive sequence such that lim
n→+∞

ξn
βn

= 0.

(B3) Let {qn} ⊂ E such that lim
n→+∞

qn = q ∈ E.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 12 of 32

Algorithm 3.2. Common solution of generalized mixed equilibrium problem and fixed
point problem
Step 0 : Let x0, x1 ∈ E be arbitrary initial points and set n = 1.
Step 1: Given the (n − 1)th and nth iterates, choose θn such that 0 ≤ θn ≤ θ̂n with θ̂n
defined by

θ̂n =

{
min{θ, ξn

∥xn−xn−1∥}, If xn ̸= xn−1,

θ, Otherwise.
(3.1)

Step 2: Compute
wn = ∇f∗ (∇f(xn) + θn (∇f(xn−1)−∇f(xn)))

zn = ResfBθN
◦ · · · ◦ ResfBθ1

(wn)

yn = ∇f∗ (βn∇f (qn) + (1− βn)∇f (T (zn)))
xn+1 = ∇f∗ (αn∇f (wn) + (1− αn)∇f (T (yn))) .

Set n := n+ 1 and return to Step 1.

Remark 3. The inertial technique used in step 1 of Algorithm 1 can be easily imple-
mented since the value of ∥xn − xn−1∥ is known before choosing θn. We also note that

the restrictive summability condition
+∞∑
n=1

∥xn − xn−1∥ < +∞ often used by several authors

when constructing initial algorithms is dispensed with in our proposed algorithm.

4. Convergence Analysis

First, we present some lemmas which will be needed in obtaining our convergence result.
In the following lemma, we obtain some results for the maximal operator BΘ from the
bifunction Θ. The main idea of the following lemma is from [53].

Lemma 13. Let f : E → (−∞,+∞] be a supercoercive, Legendre, Fréchet differentiable
and totally convex function. Let C be a closed and convex subset of E and assume that
the bifunction Θ : C × C → R satisfies conditions (A1)-(A4) and Ψ is monotone. Then

(1) GMEP(Θ, φ,Ψ) = B−1
Θ (0∗);

(2) BΘ is a maximal monotone mapping;

(3) ResfΘ,φ,Ψ = ResfBΘ
.

Proof. (1) If x ∈ C then from the definition of the mapping BΘ (2.10) we have

x ∈ B−1
Θ (0∗) ⇔ Θ(x, y)+φ(y)+ ⟨Ψx, y−x⟩ ≥ φ(x) for all y ∈ C ⇔ x ∈ GMEP(Θ, φ,Ψ).

(2) We show that BΘ is monotone mapping. Let (x1, x
∗
1) and (x2, x

∗
2) belong to the graph

of BΘ. By the definition of the mapping BΘ, we have

Θ(x1, z) + φ(z) + ⟨Ψx1, z − x1⟩ ≥ ⟨x∗1, z − x1⟩+ φ(x1)



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 13 of 32

and
Θ(x2, z) + φ(z) + ⟨Ψx2, z − x2⟩ ≥ ⟨x∗2, z − x2⟩+ φ(x2)

for any z ∈ C. In particular we have that

Θ(x1, x2) + φ(x2) + ⟨Ψx1, x2 − x1⟩ ≥ ⟨x∗1, x2 − x1⟩+ φ(x1) (4.1)

and
Θ(x2, x1) + φ(x1) + ⟨Ψx2, x1 − x2⟩ ≥ ⟨x∗2, x1 − x2⟩+ φ(x2). (4.2)

Adding equations (4.1) and (4.2) together, we obtain

Θ(x1, x2) + Θ(x2, x1) + φ(x2) + φ(x1) + ⟨Ψx1, x2 − x1⟩+ ⟨Ψx2, x1 − x2⟩
≥ ⟨x∗1, x2 − x1⟩+ ⟨x∗2, x1 − x2⟩+ φ(x1) + φ(x2).

By (A2), it is equivalent to write

0 ≥ Θ(x1, x2) + Θ(x2, x1) + ⟨Ψx1 −Ψx2, x2 − x1⟩
≥ ⟨x∗1 − x∗2, x2 − x1⟩.

It means that ⟨x∗1 − x∗2, x1 − x2⟩ ≥ 0 which follows that BΘ is a monotone mapping.
To show that BΘ is maximal monotone mapping, it is enough to show that ran(BΘ +

∇f) = E∗ ([54, Corollary 2.3]). Let x∗ ∈ E∗ from [55, Proposition 2.3] and [48, Theorem
3.5.10], we have that f is cofinite and therefore ran∇f = intdomf∗ = E∗ which follows
that ∇f is surjective. So, there exists x ∈ E such that ∇f(x) = x∗. From Lemma 9 we

know that dom(ResfΘ,φ,Ψ) = E and from the definition of ResfΘ,φ,Ψ we obtain

Θ
(
ResfΘ,φ,Ψ(x1), x2

)
+ φ(x2) + ⟨Ψ(ResfΘ,φ,Ψ(x1)), x2 −ResfΘ,φ,Ψ(x1)⟩

+⟨∇f(ResfΘ,φ,Ψ(x1))−∇f(x1), x2 −ResfΘ,φ,Ψ(x1)⟩

≥ φ(ResfΘ,φ,Ψ(x1))

for any x2 ∈ C. It follows that

Θ
(
ResfΘ,φ,Ψ(x1), x2

)
+ φ(x2) + ⟨Ψ(ResfΘ,φ,Ψ(x1)), x2 −ResfΘ,φ,Ψ(x1)⟩

≥ ⟨∇f(x1)−∇f(ResfΘ,φ,Ψ(x1)), x2 −ResfΘ,φ,Ψ(x1)⟩

+φ(ResfΘ,φ,Ψ(x1))

for any x2 ∈ C. This shows that ∇f(x1)−∇f(ResfΘ,φ,Ψ(x1)) ∈ BΘ(ResfΘ,φ,Ψ(x1)). Hence

x∗ = ∇f(x1) ∈ (∇f +BΘ)
(
ResfΘ,φ,Ψ(x1)

)
. (4.3)

It follows that x∗ ∈ ran(BΘ +∇f).

(3) It is easy to show that ResfBΘ
is single valued. From Lemma 9 we know that ResfΘ,φ,Ψ

is single valued too. From (4.3) we have

ResfBΘ
= (BΘ +∇f)−1 ◦ ∇f = ResfΘ,φ,Ψ.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 14 of 32

Lemma 14. Let {xn} be a sequence generated by Algorithm 3.2 satisfying Assumption
3.1 (A and B). Then, {xn} is bounded.

Proof. Let p ∈ Ω. Then, from Lemma 8 we have that F (T ) is closed and convex.
From Lemma 10 and the definition of zn we have

Df (p, zn) = Df

(
p,ResfBθN

◦ · · · ◦ResfBθ2
◦ResfBθ1

(wn)
)

≤ Df

(
p,ResfBθ1

(wn)
)

≤ Df (p, wn) . (4.4)

Also, from the definition of wn and (2.8), we obtain

Df (p, wn) ≤ Df (p,∇f∗ (∇f (xn) + θn (∇f (xn−1)−∇f (xn)))

= Df (p,∇f∗ ((1− θn)∇f (xn) + θn∇f (xn−1)))

≤ (1− θn)Df (p, xn) + θnDf (p, xn−1) . (4.5)

Also,

Df (p, yn) = Df (p,∇f∗ (βn∇f (qn) + (1− βn)∇f (T (zn)))

≤ βnDf (p, qn) + (1− βn)Df (p, T (zn))

≤ βnDf (p, qn) + (1− βn)Df (p, zn)

≤ βnDf (p, qn) + (1− βn)Df (p, wn) (4.6)

From the definition of xn+1 and (2.8), we obtain

Df (p, xn+1) = Df (p,∇f∗ (αn∇f (wn) + (1− αn)∇f (T (yn)))

≤ αnDf (p, wn) + (1− αn)Df (p, T (yn))

≤ αnDf (p, wn) + (1− αn)Df (p, yn)

≤ αnDf (p, wn) + (1− αn) [βnDf (p, qn) + (1− βn)Df (p, wn)]

= βn (1− αn)Df (p, qn) + [αn + (1− αn) (1− βn)]Df (p, wn)

≤ βn (1− αn)Df (p, qn) + [1− βn (1− αn)]

[(1− θn)Df (p, xn) + θnDf (p, xn−1)]

≤ max {Df (p, qn) , Df (p, xn) , Df (p, xn−1)} . (4.7)

Since {qn} is bounded and ∇f is bounded on bounded subset of E, there exists a real
number d > 0 such that Df (p, qn) ≤ d, for all n ∈ N. Thus, by induction, we have

Df (p, xn+1) ≤ max {d,Df (p, xn) , Df (p, xn−1)}
...

≤ max
{
d,Df (p, xN0) , Df

(
p, xN0−1

)}
.

This implies that {Df (p, xn)} is bounded. Hence, from Lemma 6 we have that {xn} is
bounded. Consequently, {wn} , {zn} and {yn} are all bounded.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 15 of 32

Lemma 15. Let {xn} be a sequence generated by Algorithm 3.2 satisfying Assumption
3.1 (A and B). Suppose that p ∈ Ω. Then, the following holds:

(i) lim
n→+∞

θn [Df (p, xn−1)−Df (p, xn)) = 0.

(ii) lim
n→+∞

θn
βn

[Df (p, xn−1)−Df (p, xn)] = 0.

Proof.

(i) Let p ∈ Ω. From (3.1), we have

θn ∥xn − xn−1∥ ≤ ξn, for each n ≥ 1. (4.8)

From Assumption 3.1(B2), we have that lim
n→+∞

ξn
βn

= 0 and lim
n→+∞

βn = 0. It follows

that lim
n→+∞

ξn = 0. Hence, we have that

lim
n→+∞

θn ∥xn − xn−1∥ ≤ lim
n→+∞

ξn = 0. (4.9)

Since ∇f is norm-to-norm continuous on subsets of E, we have that

lim
n→+∞

θn ∥∇f (xn)−∇f (xn−1)∥ = 0. (4.10)

Using the three-point identity,

Df (p, xn−1)−Df (p, xn) = −Df (xn−1, xn) + ⟨∇f(xn)−∇f(xn−1), xn−1 − p⟩.
(4.11)

Multiplying (4.11) by θn, we have

θn[Df (p, xn−1)−Df (p, xn)] = −θnDf (xn−1, xn) + θn⟨∇f(xn)−∇f(xn−1), xn−1 − p⟩.
(4.12)

Since ∇f is bounded on bounded sets (Assumption A3), there exists L > 0 such
that Df (xn−1, xn) ≤ L∥xn−1 − xn∥. Thus,

θnDf (xn−1, xn) ≤ Lθn∥xn−1 − xn∥ → 0.

By Cauchy-Schwarz and boundedness of {xn}, we have

|θn⟨∇f(xn)−∇f(xn−1), xn−1 − p⟩| ≤ θn∥∇f(xn)−∇f(xn−1)∥ · ∥xn−1 − p∥ → 0.

Hence, lim
n→∞

θn[Df (p, xn−1)−Df (p, xn)] = 0.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 16 of 32

(ii) Also, since lim
n→+∞

ξn
βn

= 0, we obtain from (4.8) that

lim
n→+∞

θn
βn

∥xn − xn−1∥ ≤ lim
n→+∞

ξn
βn

= 0. (4.13)

Since ∇f is norm-to-norm continuous on subsets of E, we obtain

lim
n→+∞

θn
βn

∥∇f (xn)−∇f (xn−1)∥ = 0. (4.14)

Multiplying (4.11) by θn
βn

, we have

θn
βn

[Df (p, xn−1)−Df (p, xn)] = − θn
βn

Df (xn−1, xn) +
θn
βn

⟨∇f(xn)−∇f(xn−1), xn−1 − p⟩.

From (4.11), (4.13), and (4.14), we have limn→∞
θn
βn

[Df (p, xn−1) − Df (p, xn)] = 0,
which completes the proof.

Lemma 16. Let {xn} be a sequence generated by Algorithm 3.2 satisfying Assumption
3.1(A and B). Suppose that p ∈ Ω. Then, the following holds:

Df (p, xn+1) ≤ [1− βn(1− αn)]Df (p, xn) + β(1− αn)bn,

where bn = 1−βn(1−αn)
1−αn

· θn
βn

[Df (p, xn−1)−Df (p, xn)] + ⟨yn − p,∇f (qn)−∇f(p)⟩ .



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 17 of 32

Proof. Let p ∈ Ω. From (2.7), (4.4), (4.5), we have

Df (p, xn+1) = ∇f (p,∇f∗ (αn∇f (wn) + (1− αn)∇f (T (yn))))

≤ αnDf (p, wn) + (1− αn)Df (p, T (yn))

≤ αnDf (p, wn) + (1− αn) [Df (p,∇∗βn∇f (qn) + (1− βn)∇f (T (zn))]

= αnDf (p, wn) + (1− αn) [Vf (p, βn∇f (qn) + (1− βn)∇f (T (zn)))]

≤ αnDf (p, wn) + (1− αn) [Vf (p, βn∇f (qn) + (1− βn)∇f (T (zn))

− βn (∇f (qn)−∇f(p))− ⟨yn − p,−βn (∇f (qn)−∇f(p)⟩]
= αnDf (p, wn) + (1− αn) [Vf (p, βn∇f(p) + (1− βn)∇f (T (zn)))

+ βn ⟨yn − p,∇f (qn)−∇f(p)⟩
≤ αnDf (p, wn) + (1− αn) [βnDf (p, p) + (1− βn)Df (p, T (zn))

+ βn ⟨yn − p,∇f (qn)−∇f(p)⟩
= αnDf (p, wn) + (1− αn) [(1− βn)Df (p, T (zn))

+ βn ⟨yn − p,∇f (qn)−∇f(p)⟩
≤ αnDf (p, wn) + (1− αn) [(1− βn)Df (p, zn)

+ βn ⟨yn − p,∇f (qn)−∇f(p)⟩
≤ [αn + (1− αn) (1− βn)]Df (p, wn)

+ (1− αn)βn ⟨yn − p,∇f (qn)−∇f(p)⟩
= [1− βn (1− αn)]Df (p, wn) + (1− αn)βn ⟨yn − p,∇f (qn)−∇f(p)⟩
≤ [1− βn (1− αn)] [(1− θn)Df (p, xn) + θnDf (p, xn−1)]

+ (1− αn)βn ⟨yn − p,∇f (qn)−∇f(p)⟩
= [1− βn (1− αn)]Df (p, xn)

+ [1− βn (1− αn)] θn [Df (p, xn−1)−Df (p, xn)]

+ βn (1− αn) ⟨yn − p,∇f (qn)−∇f(p)⟩
= [1− βn (1− αn)]Df (p, xn)

+ βn (1− αn)
[1− βn (1− αn)

(1− αn)
· θn
βn

(Df (p, xn−1)−Df (p, xn)

+ ⟨yn − p,∇f(qn)−∇f (p)⟩
]
,

which completes the proof.

Theorem 1. Let {xn} be a sequence generated by Algorithm 3.2 satisfying Assumption
3.1 (A and B). Then, {xn} converges strongly to p∗ in Ω.

Proof. Let p∗ ∈ Ω. Then, from Lemma 16, we have

Df (p, xn+1) ≤ [1− βn (1− αn)]Df (p, xn)

+βn (1− αn)
[1− βn(1− αn)

(1− αn)
· θn
βn

[Df (p, xn−1)−Df (p, xn)](4.15)



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 18 of 32

+ ⟨yn − p,∇f(qn)−∇f (p)⟩
]

= [1− βn (1− αn)]Df (p, xn) + βn (1− αn) bn, (4.16)

where bn = 1−βn(1−αn)
1−αn

· θn
βn

[Df (p, xn−1)−Df (p, xn)] + ⟨yn − p,∇f (qn)−∇f(p)⟩ . Next,
we show that {Df (p

∗, xn)} converges to zero. To show this, by Lemma 12, we need to
show that lim sup

k→+∞
bnk

≤ 0 for every subsequence {Df (p
∗, xnk

)} of {Df (p
∗, xn)} satisfying

lim inf
k→+∞

(
Df (p

∗, xnk+1
)−Df (p

∗, xnk
)
)
≥ 0. (4.17)

Suppose {Df (p
∗, xnk

)} is a subsequence of {Df (p
∗, xn)} such that (4.17) holds.

From Lemma 7, Lemma 15 and the definition of znk
, we have

lim
k→+∞

Df (xnk
, znk

) = lim
k→+∞

Df

(
xnk

, ResfBθN
◦ . . . ◦ResfBθ1

(wnk
)
)

≤ lim
k→+∞

Df

(
xnk

, ResfBθN−1
◦ · · · ◦ResfBθ1

(wnk
)
)

≤ lim
k→+∞

Df

(
xnk

, ResfBθ1
(wnk

)
)

(4.18)

≤ lim
k→+∞

[
Df

(
p∗, ResfBθ1

(wnk
)
)
−Df (p

∗, xnk
)
]

≤ lim
k→+∞

[Df (p
∗, wnk

)−Df (p
∗, xnk

)] (4.19)

≤ lim
k→+∞

[
(1− θnk

)Df (p
∗, xnk

) + θnk
Df

(
p∗, xnk−1

)
−Df (p

∗, xnk
)
]

= lim
k→+∞

[
θnk

Df

(
p∗, xnk−1

)
− θnk

Df (p
∗, xnk

)
]

= lim
k→+∞

θnk

[
Df

(
p∗, xnk−1

)
−Df (p

∗, xnk
)
]

= 0. (4.20)

From Lemma 1, we obtain
lim

k→+∞
∥xnk

− znk
∥ = 0. (4.21)

Since f is uniformly Fréchet differentiable on bounded subsets of E, by Lemma 5, ∇f is
norm-to-norm uniformly continuous on bounded subsets of E. Hence,

lim
k→+∞

∥∇f (xnk
)−∇f (znk

)∥⋆ = 0. (4.22)

Also, since f is uniformly Fréchet differentiable, it is also uniformly continuous, hence we
obtain that

lim
k→+∞

∥f (xnk
)− f (znk

)∥ = 0. (4.23)

Using the Bregman distance, we obtain

Df (p
∗, xnk

)−Df (p
∗, znk

)



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 19 of 32

= f(p∗)− f (xnk
)− ⟨∇f (xnk

) , p∗ − xnk
⟩ − f(p∗) + f (znk

) + ⟨∇f (znk
) , p∗ − znk

⟩
= f (znk

)− f (xnk
) + ⟨∇f (znk

) , p∗ − znk
⟩ − ⟨∇f (xnk

) , p∗ − xnk
⟩

= f (znk
)− f (xnk

) + ⟨∇f (znk
) , xnk

− znk
⟩ − ⟨∇f (znk

)−∇f (xnk
) , p∗ − xnk

⟩ ,

for p∗ ∈ F (T ).
From (4.21) and (4.23), we obtain

lim
k→+∞

[Df (p, xnk
)−Df (p, znk

)] = 0. (4.24)

Also, since βnk
→ 0 as k → +∞, we obtain

Df (znk
, ynk

) = Df (p
∗, ynk

)−Df (p
∗, znk

)

= Df (p
∗,∇f∗ (βnk

∇f (qnk
) + (1− βnk

)∇f (T (znk
))))−Df (p

∗, znk
)

≤ βnk
Df (p

∗, qnk
) + (1− βnk

)Df (p
∗, T (znk

))−Df (p
∗, znk

)

≤ βnk
Df (p

∗, qnk
) + (1− βnk)Df (p

∗, znk
)−Df (p

∗, znk
)

= βnk
[Df (p

∗, qnk
)−Df (p

∗, znk
)]

→ 0, as k → +∞. (4.25)

Hence, lim
k→+∞

Df (znk
, ynk

) = 0. From 2.4, we obtain

lim
k→+∞

∥znk
− ynk

∥ = 0. (4.26)

Consequently, we have

lim
k→+∞

∥∇f(znk
)−∇f(ynk

)∥ = lim
k→+∞

∥f(znk
)− f(ynk

)∥ = 0.

From (4.21) and (4.26), we obtain

∥xnk
− ynk

∥ = ∥xnk
− znk

+ znk
− ynk

∥
≤ ∥xnk

− znk
∥+ ∥znk

− ynk
∥

= 0,

as k → +∞.
Therefore,

lim
k→+∞

∥xnk
− ynk

∥ = 0. (4.27)

Since f is uniformly Fréchet differentiable on bounded subsets of E, by Lemma 5, ∇f is
norm-to-norm uniformly continuous on bounded subsets of E. Hence

lim
k→+∞

∥∇f (xnk
)−∇f (ynk

)∥∗ = 0. (4.28)

On the other hand, since f is uniformly Fréchet differentiable, we have that f is also
uniformly continuous. Hence,

lim
k→+∞

∥f (xnk
)− f (ynk

)∥ = 0. (4.29)



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 20 of 32

Applying the Bregman distance, we obtain

Df (p
∗, wnk

)−Df (p
∗, ynk

)

= f(p∗)− f (wnk
)− ⟨∇f (wnk

) , p∗ − wnk
⟩ − f(p∗) + f (ynk

) + ⟨∇f (ynk
) , p∗ − ynk

⟩
= f (ynk

)− f (wnk
) + ⟨∇f (ynk

) , p∗ − ynk
⟩ − ⟨∇f (wnk

) , p∗ − wnk
⟩

= f (ynk
)− f (wnk

) + ⟨∇f (ynk
) , wnk

− ynk
⟩+ ⟨∇f (ynk

)−∇f (wnk
) , p∗ − wnk

⟩ ,(4.30)

for p∗ ∈ F (T ). From the definition of wnk
and (4.10), we have

∥∇f (wnk
)−∇f (xnk

)∥ =
∥∥∇f (xnk

) + θnk

(
∇f

(
xnk−1

)
−∇f (xnk

)
)
−∇f (xnk

)
∥∥

= θnk

∥∥∇f
(
xnk−1

)
−∇f (xnk

)
∥∥→ 0 as k → +∞. (4.31)

From (4.28) and (4.31), we can write

lim
k→+∞

∥∇f (wnk
)−∇f (ynk

)∥ ≤ lim
k→+∞

[∥∇f (wnk
)−∇f (xnk

)∥+ ∥∇f(xnk
)−∇f(ynk

)] = 0

(4.32)
Combining (4.30) and (4.32), we have

lim
k→+∞

(Df (p
∗, wnk

)−Df (p
∗, ynk

)) = 0. (4.33)

Also, from (4.27) and (4.32), we have that

lim
k→+∞

∥wnk
− xnk

∥ = 0. (4.34)

From (4.33) and the definition of xnk+1
, we have

Df

(
ynk

, xnk+1

)
= Df

(
p∗, xnk+1

)
−Df (p

∗, ynk
)

= Df (p
∗,∇f∗ (αnk

∇f (wnk
) + (1− αnk

)∇f (T (ynk
))))−Df (p

∗, ynk
)

≤ Df (p
∗,∇f∗ (αnk

∇f (wnk
) + (1− αnk

)∇f (T (ynk
))−Df (p, ynk

)))

≤ αnk
Df (p

∗, wnk
) + (1− αnk

)Df (p
∗, T (ynk

))−Df (p
∗, ynk

)

≤ αnk
Df (p

∗, wnk
) + (1− αnk

)Df (p
∗, ynk

)−Df (p
∗, ynk

)

= αnk
[Df (p

∗, wnk
)−Df (p

∗, ynk
)]

→ 0 as k → +∞.

From Lemma 1, he have
lim

k→+∞

∥∥ynk
− xnk+1

∥∥ = 0. (4.35)

From (4.25), and the fact that βnk
→ 0 as k → +∞, we have

Df (Tynk
, ynk

) = Df (p
∗, ynk

)−Df (p
∗, T ynk

)

≤ βnk
Df (p

∗, qnk
) + (1− βnk

)Df (p
∗, T (znk

))−Df (p
∗, T ynk

)

≤ βnk
Df (p

∗, qnk
) + (1− βnk

)Df (p
∗, znk

)−Df (p
∗, ynk

)

= βnk
[Df (p

∗, qnk
)−Df (p

∗, znk
)] + [Df (p

∗, znk
)−Df (p

∗, ynk
)]

→ 0 as k → +∞.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 21 of 32

Therefore, lim
k→+∞

Df (Tynk
, ynk

) = 0.

From Lemma 1, we obtain
lim

k→+∞
∥Tynk

− ynk
∥ = 0. (4.36)

From (4.35) and (4.36), we have∥∥xnk+1
− Tynk

∥∥ ≤
∥∥xnk+1

− ynk

∥∥+ ∥ynk
− Tynk

∥ → 0, as k → +∞.

In other words,
lim

k→+∞

∥∥xnk+1
− Tynk

∥∥ = 0. (4.37)

From (4.27) and (4.36) we have

∥xnk
− Txnk

∥ ≤ ∥xnk
− ynk

∥+ ∥ynk
− Tynk

∥+ ∥Tynk
− Txnk

∥ → 0

as k → +∞. Hence,
lim

k→+∞
∥xnk

− Txnk
∥ = 0. (4.38)

From (4.27)-(4.38) we have

lim
n→+∞

∥xnk+1
− xnk

∥ = 0. (4.39)

Since {xn} is bounded, there exists a subsequence {xnk
} of {xn} such that {xnk

} ⇀ p∗.
From (4.38), we have ∥xnk

− T (xnk
)∥ → 0 as k → +∞. Hence, p∗ ∈ F (T ). For any

w ∈
(
F (T ) ∩

(⋂N
j=1B

−1
θj

(0∗)
))

, it follows from the there point identity that

|Df (w,wnk
)−Df (w, ynk

)|
= |Df (w, ynk

) +Df (ynk
, wnk

) + ⟨w − ynk
,∇f (ynk

)−∇f (wnk
)⟩ −Df (w, ynk

)|
= |Df (ynk

, wnk
) + ⟨w − ynk

,∇f (ynk
)−∇f (wnk

)⟩|
≤ Df (ynk

, wnk
) + ∥w − ynk

∥ ∥∇f (ynk
)−∇f (wnk

)∥
≤ ∥ynk

− wnk
∥+ ∥w − ynk

∥ ∥∇f (ynk
)−∇f (wnk

)∥
→ 0 as k → +∞ by (4.32).

Hence,
lim

k→+∞
|Df (w,wnk

)−Df (w, ynk
)| = 0.

Since ResfBθ
is BQFNE, we have

Df

(
ResfBθj

◦ · · · ◦ResfBθ1
(wnk

) , ResfBθj−1
◦ . . . ◦ResfBθ1

(wnk
)
)

= Df

(
ResfBθj

◦ResfBθj−1
◦ . . . ◦ResfBθ1

(wnk
) , ResfBθj−1

◦ . . . ◦ResfBθ1
(wnk

)
)

≤ Df (w,wnk
)−Df (w,wnk

)

→ 0



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 22 of 32

as k → +∞ for all j ∈ {1, 2, . . . , N}. It then follows that

lim
k→+∞

Df

(
(ResfBΘj

◦ . . . ◦ResfBΘ1
(wnk

), wnk
)
)
= 0

for all j ∈ {1, 2, . . . , N}. So,

lim
k→+∞

∥ResfBΘj
◦ . . . ◦ResfBΘ1

(wnk
)− wnk

∥ = 0 (4.40)

for all j ∈ {1, 2, . . . , N}.
From the definition of the f -resolvent, we have

∇f
(
ResfBΘj−1

◦ . . . ◦ResfBΘ1
(wnk

)
)
∈
(
∇f + λj

nk
BΘj

) (
ResfBΘj

◦ . . . ◦ResfBΘ1
(wnk

)
)
.

Hence

ζjnk
:=

1

λj
nk

(
∇f(ResfBΘj−1

◦ . . . ◦ResfBΘ1
(wnk

))−∇f(ResfBΘj
◦ . . . ◦ResfBΘ1

(wnk
))
)

for all j ∈ {1, 2, . . . , N}. It follows from the above equations that lim
k→+∞

∥ζjnk∥ = 0 for any

j ∈ {1, 2, . . . , N}. Since xnk
⇀ p∗, we obtain from (4.34) that wnk

⇀ p∗. From (4.40) and
the fact that wnk

⇀ p∗, we obtain

ResfBΘj
◦ . . . ◦ResfBΘ1

(wnk
) ⇀ p∗

for any j ∈ {1, 2, . . . , N}.
Consequently, we have

ResfBΘj
◦ . . . ◦ResfBΘ1

(xnk
) ⇀ p∗

for any j ∈ {1, 2, . . . , N}. From the monotonicity of BΘ, we have

⟨η − ζjnk
, x−ResfBΘj

◦ . . . ◦ResfBΘ1
(xnk

)⟩ ≥ 0.

for all (x, η) ∈ graph(BΘj ). This implies that ⟨η, x − p∗⟩ ≥ 0 for all (x, η) ∈ graph(BΘj )

and for all j ∈ {1, 2, . . . , N}. So, by the maximal monotonicity of BΘj we have p
∗ ∈ B−1

Θj
(0)

for all j ∈ {1, 2, . . . , N}. Therefore p∗ ∈ ∩N
j=1B

−1
Θj

(0).

Hence, we have shown that p∗ ∈
(
F (T ) ∩

(
∩N
j=1B

−1
Θj

(0∗)
))

.

Since E is reflexive and {xnk
} is bounded, there exists a subsequence {xnkj

} of {xnk
} such

that {xnkj
} ⇀ u ∈ C and

lim
j→+∞

〈
∇f(qnk

)−∇f(p̂), xnkj
− p̂
〉

= lim sup
k→+∞

⟨∇f(qnk
)−∇f(p̂), xnk

− p̂⟩

= lim sup
k→+∞

⟨∇f(qnk
)−∇f(p̂), ynk

− p̂⟩.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 23 of 32

It follows from the definition of the Bregman projection that

lim sup
k→+∞

⟨∇f(qnk
)−∇f(p̂), ynk

− p̂⟩ = lim
j→+∞

〈
∇f(qnk

)−∇f(p̂), xnkj
− p̂
〉

(4.41)

= ⟨∇f(q)−∇f(p̂), u− p̂⟩ ≤ 0. (4.42)

By Lemma 15, (4.41) and the condition on α, we can conclude that lim sup
k→+∞

bnk
≤ 0. From

Lemma 4.17 and (4.15) we have limn→+∞Df (p̂, xn) = 0. Therefore, by Lemma 11, we
have lim

n→+∞
xn = p̂. This completes the proof.

Using Theorem 1 and Lemma 13, we have the following result. Let φ = Ψ = θn = 0 and
T = I then we have the following corollary which was obtained in [56].

Corollary 1. Let E be a real reflexive Banach space, C be a nonempty, closed and convex
subset of int(domf). Let f : E → R be a super coercive Legendre function which is
bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let
BΘj : E → 2E

∗
, j = 1, 2, . . . , N be N maximal monotone mapping with dom(B) ⊂ C.

Assume that
(
∩N
j=1B

−1
Θj

(0∗)
)

̸= ∅, {αn} and {βn} be sequences in [0, 1] satisfying the

following conditions:

(i) lim
n→+∞

βn = 0;

(ii)
+∞∑
n=1

βn = +∞;

(iii) 0 < lim inf
n→+∞

αn ≤ lim supn→+∞ αn < 1.

Let {xn} be a sequence generated by

u ∈ E, x1 ∈ E chosen arbitrarily,

zn = ResfBΘN
◦ . . . ◦ResfBΘ1

(xn),

yn = ∇f∗ (βn∇f(qn) + (1− βn)∇f(zn))

xn+1 = ∇f∗ (αn∇f(xn) + (1− αn)∇f(yn)) , (4.43)

where ∇f is the gradient of f . Then the sequence {xn} generated by (4.43) converges to

projf∩N
j=1B

−1
Θj

(0∗)
x as n → +∞.

5. Numerical Experiment

In this section, we present numerical experiments to illustrate the performance of our
proposed method. In all our experiments, we use ∥xn+1 − xn∥ < 10−4 as our stopping
criterion. All the numerical computations were carried out using using Matlab version
R2024(b).
Being a non-accelerated version of our method, we made a comparison with Algorithm
(1.5) in [23] with a short name ”EsRa”.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 24 of 32

Example 1. Let E = R and C = [−1, 1]. Let Θi(x, y) = −9ix2+xy+(9i−1)y2,Ψi(x, y) =

(9i−3)x, φi(x, y) = (9i−6)x, i = 1, 2, 3, · · · , N, we have ResfBΘi
(x) = x

5(9i−3) . Let f = ∥x∥2

and T (x) = PC(x) where

PC(x) =


−1, x < −1

x, x ∈ [−1, 1]

1, x > 1.

Clearly, we observe that the bifunction Θ satisfies (A1)-(A4), Ψ is monotone and T is
Bregman strongly nonexpansive mapping. In this example, we select αn = 1

2n+3 , βn =
n

n+1 , qn = 1
n+1 , ϵ = 1

n3.1 , θ = 0.1 and set N = 100. As a stopping criterion, we use

∥xn+1 − xn∥ ≤ ϵ where ϵ = 10−4. The experiment was conducted for the following initial
values of x0 and x1 given as Cases I-IV:

(Case I) : x0 = 0.96 and x1 = 0.59;

(Case II) : x0 = 1.1 and x1 = 1.3;

(Case III) : x0 = 2.1 and x1 = 1.7;

(Case IV) : x0 = 0.69 and x1 = 0.09.

The report appears in the form of Figures 1- 4 showing that our method converges faster
in terms of number iteration than the non-accelerated version.

0 2 4 6 8 10 12 14

Number of iterations

10-4

10-3

10-2

10-1

100

T
o

l

Case I

Our Algorithm

EsRa

Figure 1: Example 1. Case I.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 25 of 32

0 2 4 6 8 10 12 14

Number of iterations

10-4

10-3

10-2

10-1

100

101

T
o

l

Case II

Our Algorithm

EsRa

Figure 2: Example 1. Case II.

0 2 4 6 8 10 12 14

Number of iterations

10-4

10-3

10-2

10-1

100

101

T
o

l

Case III

Our Algorithm

EsRa

Figure 3: Example 1. Case III.

Example 2. Let E = (ℓ2(R), ∥ · ∥) , where ℓ2(R) :=

{
x : x = {xi}+∞

i=1 ,
+∞∑
i=1

|xi|2 < +∞
}
,

with an inner product ⟨·, ·⟩ : ℓ2 × ℓ2 → R given by ⟨x, y⟩ =
+∞∑
i=1

xiyi where x = {xi}+∞
i=1 ,

y = {yi}+∞
i=1 and the norm ∥ · ∥ : ℓ2 → R is given by ∥x∥2 =

√(
+∞∑
i=1

|xi|2
)
. Let f(x) = x2

2 ,

then f satisfies Assumption 3.1 Let C := {x ∈ ℓ2(R) : ∥x∥2 ≤ 1} and Θi : E × E → R
be defined by Θi(x, y) = −3ix2 + 2ixy + iy2 for all i and x, y ∈ ℓ2. Let Ψi : ℓ2 → R and
φi : ℓ2 → R for all i and x ∈ ℓ2 be given by Ψi = ix2 and φi = ix, respectively. We have
that ResfBΘi

(x) = x
1+7j . Let T = x+2

2 . Clearly, we observe that the bifunction Θ satisfies



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 26 of 32

0 2 4 6 8 10 12 14

Number of iterations

10-4

10-3

10-2

10-1

T
o

l

Case IV

Our Algorithm

EsRa

Figure 4: Example 1. Case IV.

(A1)-(A4), Ψ is monotone and T is Bregman strongly nonexpansive mapping. For example
2, we choose αn = 1

2n+3 , βn = 1
n+1 , qn = 1

5 , ϵ = 1
n3.1 , θ = 0.1 and set N = 100. As a

stopping criterion, we use ∥xn+1−xn∥ ≤ ϵ where ϵ = 10−4. The experiment was conducted
for the following initial values of x0 and x1 given as Cases A-D:

(Case A) : x0 = [0.25, 0.25, 0.33, · · · , 0, 0, · · · ] and x1 = [0.2, 0.2, 0.3, · · · , 0, 0, · · · ];

(Case B) : x0 = [1, 0.5, 0.25, · · · , 0, 0, · · · ] and x1 = [0.2, 0.25, 0.125, · · · , 0, 0, · · · ];

(Case C) : x0 = [0.91, 0.55, 0.53, · · · , 0, 0, · · · ] and x1 = [0.85, 0.65, 0.65, · · · , 0, 0, · · · ];

(Case D) : x0 = [1.2, 0, 0.38, · · · , 0, 0, · · · ] and x1 = [0.8, 0, 1.2, · · · , 0, 0, · · · ].

The report of this experiment is displayed Figures 5- 8 showing that our method converges
faster in terms of number iteration than the non-accelerated version presented in [23].

6. Conclusion

In this paper, we studied the generalized mixed equilibrium problem and the fixed point
problem in the framework of real reflexive Banach spaces. We introduce an inertial method
for approximating the common solution of the above mentioned problems. Under mild con-
ditions, we obtained the strong convergence of our proposed method. Finally, we present
numerical experiments in comparison with an algorithm in the literature to illustrate the
applicability of our proposed method.



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 27 of 32

0 2 4 6 8 10 12 14 16 18

Number of iterations

10-4

10-3

10-2

10-1

100

101

T
o

l

Case A

Our Algorithm

EsRa

Figure 5: Example 2. Case A.

0 2 4 6 8 10 12 14 16 18

Number of iterations

10-4

10-3

10-2

10-1

100

101

T
o

l

Case B

Our Algorithm

EsRa

Figure 6: Example 2. Case B.

References

[1] L. C. Ceng and J. C. Yao. A hybrid iterative scheme for mixed equilibrium prob-
lems and fixed point problems. Journal of Computational and Applied Mathematics,
214:186–201, 2008.

[2] W. Takahashi and M. Toyoda. Weak convergence theorems for nonexpansive map-
pings and monotone mappings. Journal of Optimization Theory and Applications,
118:417–428, 2003.

[3] F. E. Browder. Existence and approximation of solutions of nonlinear variational
inequalities. Proceedings of the National Academy of Sciences of the United States of



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 28 of 32

0 2 4 6 8 10 12 14 16

Number of iterations

10-4

10-3

10-2

10-1

100

101

T
o

l

Case C

Our Algorithm

EsRa

Figure 7: Example 2. Case C.

0 5 10 15

Number of iterations

10-4

10-3

10-2

10-1

100

101

T
o

l

Case D

Our Algorithm

EsRa

Figure 8: Example 2. Case D.

America, 56:1080–1086, 1966.
[4] E. Blum and W. Oettli. From optimization and variational inequalities to equilibrium

problems. Mathematical Student, 63:123–145, 1994.
[5] M. Aslam Noor and W. Oettli. On general nonlinear complementarity problems and

quasi equilibria. Matematiche (Catania), 49:313–331, 1994.
[6] P. L. Combettes and S. A. Hirstoaga. Equilibrium programming in hilbert spaces.

Journal of Nonlinear and Convex Analysis, 6:117–136, 2005.
[7] S. D. Flam and A. S. Antipin. Equilibrium programming using proximal-link algo-

rithms. Mathematical Programming, 78:29–41, 1997.
[8] G. Kassay, S. Reich, and S. Sabach. Iterative methods for solving systems of vari-

ational inequalities in reflexive banach spaces. SIAM Journal on Optimization,



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 29 of 32

21:1319–1344, 2011.
[9] S. Reich and S. Sabach. Three strong convergence theorems regarding iterative meth-

ods for solving equilibrium problems in reflexive banach spaces. Contemporary Math-
ematics, 568:225–240, 2012.

[10] S. Takahashi and W. Takahashi. Viscosity approximation methods for equilibrium
problems and fixed point problems in hilbert spaces. Journal of Mathematical Analysis
and Applications, 331:506–515, 2007.

[11] J. W. Peng and J. C. Yao. Strong convergence theorems of iterative scheme based on
the extragradient method for mixed equilibrium problems and fixed point problems.
Mathematical and Computer Modelling, 49:1816–1828, 2009.

[12] T. M. Tuyen. Parallel iterative methods for solving systems of generalized mixed
equilibrium problems in reflexive banach spaces. Optimization, 66(4):623–637, 2017.

[13] I. Beg, M. Gunaseelan, and G. A. Joseph. Best proximity point of generalized f-
proximal non-self contraction. Journal of Fixed Point Theory and Applications, 23:49,
2021.

[14] A. J. Gnanaprakasam, G. Nallaselli, A. U. Haq, G. Mani, I. A. Baloch, and K. Non-
laopon. Common fixed-points technique for the existence of a solution to frac-
tional integro-differential equations via orthogonal branciari metric spaces. Symmetry,
14:1859, 2022.

[15] Z. Gu, G. Mani, A. J. Gnanaprakasam, and Y. Li. Solving a system of nonlinear in-
tegral equations via common fixed point theorems on bicomplex partial metric space.
Mathematics, 9:1584, 2021.

[16] G. Nallaselli, A. S. Baazeem, A. J. Gnanaprakasam, G. Mani, K. Javed, E. Ameer,
and N. Mlaiki. Fixed point theorems via orthogonal convex contraction in orthogonal
b-metric spaces and applications. Axioms, 12:143, 2023.

[17] R. Ramaswamy, G. Mani, A. J. Gnanaprakasam, O. A. A. Abdelnaby, V. Stojiljković,
S. Radojevic, and S. Radenović. Fixed points on covariant and contravariant maps
with an application. Mathematics, 10:4385, 2022.

[18] A. Moudafi. Viscosity approximation methods for fixed points. Journal of Mathe-
matical Analysis and Applications, 204(1):45–56, 1997.

[19] H. A. Abass, M. Aphane, and M. O. Olayiwola. An inertial method for solving systems
of generalized mixed equilibrium and fixed point problems in reflexive banach spaces.
Asian-European Journal of Mathematics, 16(11):2350207, 2023.

[20] O. K. Oyewole and O. T. Mewomo. Existence results for new generalized mixed
equilibrium and fixed point problems in banach spaces. Nonlinear Functional Analysis
and Applications, pages 273–301, 2020.

[21] N. Petrot, K. Wattanawitoon, and P. Kumam. A hybrid projection method for
generalized mixed equilibrium problems and fixed point problems in banach spaces.
Nonlinear Analysis: Hybrid Systems, 4(4):631–643, 2010.

[22] S. Takahashi and W. Takahashi. Strong convergence theorem for a generalized equi-
librium problem and a nonexpansive mapping in a hilbert space. Nonlinear Analysis,
69:1025–1033, 2008.

[23] G. Z. Eskandani and M. Raeisi. A new algorithm for finding fixed points of bregman



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 30 of 32

quasi-nonexpansive mappings and zeros of maximal monotone operators by using
products of resolvents. Results in Mathematics, 71:1307–1326, 2017.

[24] B. T. Polyak. Some methods of speeding up the convergence of iteration methods.
USSR Computational Mathematics and Mathematical Physics, 4(5):1–17, 1964.

[25] D. V. Hieu. New inertial algorithm for a class of equilibrium problems. Numerical
Algorithms, 80:1413–1436, 2019.

[26] D. V. Hieu. An inertial-like proximal algorithm for equilibrium problems. Mathemat-
ical Methods of Operations Research, 88:399–415, 2018.

[27] N. T. Vinh and L. D. Muu. Inertial extragradient algorithms for solving equilibrium
problems. Acta Mathematica Vietnamica, 44:639–663, 2019.

[28] J. F. Bonnans and A. Shapiro. Perturbation analysis of optimization problems.
Springer, New York, NY, 2000.

[29] Y. Censor and A. Lent. An iterative row-action method for interval convex program-
ming. Journal of Optimization Theory and Applications, 34:321–353, 1981.

[30] M. Teboulle. A simplified view of first order methods for optimization. Mathematical
Programming, 170(1):67–96, 2018.

[31] G. B. Wega and H. Zegeye. Convergence results of forward-backward method for a
zero of the sum of maximally monotone mappings in banach spaces. Computational
and Applied Mathematics, 39(3):223, 2020.

[32] H. H. Bauschke, J. M. Borwein, and P. L. Combettes. Essential smoothness, essen-
tial strict convexity, and legendre functions in banach spaces. Communications in
Contemporary Mathematics, 3:615–647, 2001.

[33] R. R. Phelps. Convex Functions, Monotone Operators, and Differentiability, volume
1364 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, second edition, 1993.

[34] D. Reem and S. Reich. Solutions to inexact resolvent inclusion problems with appli-
cations to nonlinear analysis and optimization. Rendiconti del Circolo Matematico di
Palermo, 67:337–371, 2018.

[35] Y. I. Alber. Metric and generalized projection operators in banach spaces: properties
and applications. In A. G. Kartsatos, editor, Theory and Applications of Nonlinear
Operators of Accretive and Monotone Type, pages 15–50. Marcel Dekker, New York,
1996.

[36] D. Butnariu, S. Reich, and A. J. Zaslavski. There are many totally convex functions.
Journal of Convex Analysis, 13:623–632, 2006.

[37] D. Butnariu and A. N. Iusem. Totally Convex Functions for Fixed Points Computation
and Infinite Dimensional Optimization, volume 40 of Applied Optimization. Kluwer
Academic, Dordrecht, 2000.

[38] S. Reich and S. Sabach. Two strong convergence theorems for a proximal method in
reflexive banach spaces. Numerical Functional Analysis and Optimization, 31:22–44,
2010.

[39] J. B. Hiriart-Urruty and C. Lemaréchal. Convex Analysis and Minimization Algo-
rithms II, volume 306 of Grundlehren der mathematischen Wissenschaften. Springer-
Verlag, 1993.

[40] D. Butnariu and E. Resmerita. Bregman distances, totally convex functions and



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 31 of 32

a method for solving operator equations in banach spaces. Abstract and Applied
Analysis, 2006:1–39, 2006. Article ID 84919.

[41] S. Reich and S. Sabach. A strong convergence theorem for a proximal-type algorithm
in reflexive banach spaces. Journal of Nonlinear and Convex Analysis, 10:471–485,
2009.

[42] S. Reich. A weak convergence theorem for the alternating method with bregman
distances. In A. G. Kartsatos, editor, Theory and Applications of Nonlinear Operators
of Accretive and Monotone Type, pages 313–318. Marcel Dekker, New York, 1996.

[43] D. Butnariu, S. Reich, and A. J. Zaslavski. Asymptotic behavior of relatively nonex-
pansive operators in banach spaces. Journal of Applied Analysis, 7:151–174, 2001.

[44] R. E. Bruck and S. Reich. Nonexpansive projections and resolvents of accretive
operators in banach spaces. Houston Journal of Mathematics, 3:459–470, 1977.

[45] S. Reich and S. Sabach. Two strong convergence theorems for bregman strongly
nonexpansive operators in reflexive banach spaces. Nonlinear Analysis, 73:122–135,
2010.

[46] F. Kohsaka and W. Takahashi. Proximal point algorithms with bregman functions
in banach spaces. Journal of Nonlinear and Convex Analysis, 6:505–523, 2005.

[47] V. Mart́ın-Márquez, S. Reich, and S. Sabach. Iterative methods for approximating
fixed points of bregman nonexpansive operators. Discrete and Continuous Dynamical
Systems - Series S, 6:1043–1063, 2013.

[48] C. Zălinescu. Convex analysis in general vector spaces. World Scientific, River Edge,
NJ, 2002.

[49] S. Reich and S. Sabach. Existence and approximation of fixed points of bregman
firmly nonexpansive mappings in reflexive banach spaces. In Fixed-Point Algorithms
for Inverse Problems in Science and Engineering, volume 49 of Optimization and Its
Applications, pages 301–316. 2011.

[50] V. Darvish. Strong convergence theorem for generalized mixed equilibrium prob-
lems and bregman nonexpansive mapping in banach spaces. Mathematica Moravica,
20(1):69–87, 2016.

[51] E. Naraghirad and J. C. Yao. Bregman weak relatively nonexpansive mappings in
banach space. Fixed Point Theory and Applications, 2013(43):141, 2013.

[52] S. Saejung and P. Yotkaew. Approximation of zeros of inverse strongly monotone
operators in banach spaces. Nonlinear Analysis, 75:742–750, 2012.

[53] S. Sabach. Products of finitely many resolvents of maximal monotone mappings in
reflexive banach spaces. SIAM Journal on Optimization, 21:1289–1308, 2011.

[54] H. H. Bauschke, X. Wang, and L. Yao. General resolvents for monotone operators:
characterization and extension. In Biomedical Mathematics: Promising Directions
in Imaging, Therapy Planning and Inverse Problems, pages 57–74. Medical Physics
Publishing, Madison, WI, USA, 2010.

[55] D. Butnariu, A. N. Iusem, and C. Zălinescu. On uniform convexity, total convexity
and convergence of the proximal point and outer bregman projection algorithms in
banach spaces. Journal of Convex Analysis, 10:35–61, 2003.

[56] S. Timnaka, E. Naraghirad, and N. Hussain. Strong convergence of halpern iteration



V. Darvish et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6173 32 of 32

for products of finitely many resolvents of maximal monotone operators in banach
spaces. Filomat, 31(15):4673–4693, 2017.