

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6062
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Solving System of Monotone Variational Inclusion
Problems with Multiple Output Sets in Banach Spaces

H. A. Abass1,4,∗, M. Aphane1, O. K. Oyewole2, O. K. Narain3, K. I. Mustafoyev4

1 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences
University, P.O. Box 94 Medunsa 0204, Pretoria, South Africa
2 Department of Mathematics, Tshwane University of Technology, Arcadia, PMB 0007,
Pretoria, South Africa
3 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal,
Durban, South Africa
4 Center of Research and Innovation, Asia International University, Yangiobod MFY, G‘ijduvon
Street, House 74, Bukhara, Uzbekistan

Abstract. In this article, we introduce a self-adaptive method for approximating solutions of
split common fixed point problem of Bregman demigeneralized mappings and system of monotone
variational inclusion problem with multiple output sets in reflexive Banach spaces. By employing
our iterative method, we prove a strong convergence theorem for approximating solutions of the
aforementioned problems. In summary, we state some consequences of our main result. The result
discuss in this paper extends and complements many related results in literature.

2020 Mathematics Subject Classifications: 47H06, 47H09, 47J05, 47J25

Key Words and Phrases: Bregman demigeneralized mapping, monotone operators, self-adapative
method, split common fixed point problem

1. Introduction

For modelling inverse problems which arise from phase retrievals and medical image re-
construction, (see [1]), Censor and Elfving [2] introduced the Split Feasibility Problem
(SFP) in 1994, which is to find

u∗ ∈ C such that Fu∗ ∈ Q, (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6062

Email addresses: hammed.abass@smu.ac.za, hammedabass548@gmail.com (H. A. Abass),
maggie.aphane@smu.ac.za (M. Aphane), oyewoleolawalekazeem@gmail.com (O. K. Oyewole),
Naraino@ukzn.ac.za, (O. K. Narain), k.mustafoyev@oxu.uz (K. I. Mustafoyev)

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


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 2 of 22

where C and Q are nonempty, closed and convex subsets of real Banach spaces E1 and
E2 respectively, and F : E1 → E2 is a bounded linear operator. The SFP have been well
studied in the framework of real Hilbert spaces, uniformly convex and uniformly smooth
Banach spaces, see ([3–5] and other references contained in). Different optimization prob-
lems have been formulated in terms of SFP (1), for instance, If Q = {b} in SFP (1) is a
singleton, then we have the following convexly constrained linear inverse problem (CCLIP)
defined as follows:

Find a point u∗ ∈ C such that Fu∗ = b.

Also, if C = Fix(T ) = {x ∈ E : x = Tx} and Q = Fix(S), then SFP (1) becomes split
common fixed point problem (SCFPP) which is to find a point

u∗ ∈ Fix(T ) such that Fu∗ ∈ Fix(S). (2)

Since the introduction of the SCFPP (2), authors have considered several schematic meth-
ods for approximating its solution. For instance, Censor and Segal [6] introduced the fol-
lowing iterative algorithm for solving the SCFPP (2) in finite dimensional spaces. They
defined the algorithm as follows:

xn+1 = T (xn + τF t(S − I)Fxn),

for each n ≥ 1, where τ ∈ (0, 0γ ) with γ being the largest eigenvalue of the matrix F tF

(F t being the matrix transposition). Also, Moudafi [7] introduced a relaxed algorithm
for approximating a solution of SCFPP (2) and proved some weak convergence results in
Hilbert spaces with the mappings T and S being quasi-nonexpansive mappings.
The variational inclusion problem consists of finding a point x∗ ∈ E such that

0 ∈ (A+B)x∗, (3)

where A : E → E∗ is a single-valued mapping and B : E → 2E
∗
is a multi-valued

mapping on a real Banach space E with dual space E∗. Combining the notions of SFP
and VIP, Moudafi [8] introduced the following Split Variational Inclusion Problem (SVIP):
Let H1 and H2 be real Hilbert spaces, Ai : Hi → Hi, i = 1, 2 be single-valued mappings,
Bi : Hi → 2Hi be multi-valued mappings and F : H1 → H2 be a bounded linear operator.
The SVIP consists of finding x∗ ∈ H1 such that

0 ∈ (A+B)x∗ (4)

and such that

y∗ = Fx∗ solves 0 ∈ (A+B)Fx∗. (5)

We note that since its introduction, the SVIP has been considered in other more general
frameworks than the Hilbert spaces (see [9–19] and the references therein).


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 3 of 22

The several variants of the SFP continue to recieve attention of various authors, notably
because of the many rich applications, (see [6, 20]). There have been attempts at ex-
tending the SFP for more operators to cover the previous studies in the literature. For
instance, Reich and Tuyen [21] introduced the Generalized Split Common Monotone In-
clusion Problem (GSCMIP): Let i = 1, 2, · · · , N, Hi be real Hilbert spaces, Ai : Hi → 2Hi

be maximal monotone operators. Let Fi : Hi → Hi+1 be bounded linear opertors for
i = 1, 2, · · · , N − 1 such that Ti ̸= 0. Then the GSCMIP is to find x∗ ∈ H1 such that

0 ∈ A1(x
∗), 0 ∈ A2(F1(x

∗)), · · · , 0 ∈ AN (TN−1TN−2 · · ·T1(x
∗)). (6)

Very recently, the same authors in [16] introduced and studied a Split Common Mono-
tone Inclusion Problem with Multiple Output sets (SCMIPOS) in Hilbert spaces. Let
H,H1, · · · , HN be real Hilbert spaces, Fi : H → Hi, i = 1, 2, · · · , N be bounded linear
operators. Let B : H → 2H , Bi : Hi → 2Hi , i = i, 2, · · · , N be maximal monotone
operators, then SCMIPOS consists of finding a point x∗ ∈ H such that

x∗ ∈ B−1(0) ∩

(
N⋂
i=1

F−1
i (B−1

i (0))

)
. (7)

In this paper, our motivation is in two folds. First, we combine the notions of SVIP and
the SCMIPOS to introduce a Split Variational Inclusion Problem with Multiple Output
sets (SVIPOS) in the framework of real Banach spaces. Let E = E0, E1, E2, · · · , EN be
real Banach spaces and Fi : E → Ei, i = 0, 1, · · · , N with F0 = IE be bounded linear
operators. For i = 0, 1, · · · , N, let Ai : Hi → Hi with A = A0 be single-valued mappings
and Bi : Hi → 2Hi with B = B0 be multi-valued mappings. Then the SVIPOS is the
problem of finding a point x∗ ∈ E such that

x∗ ∈ (A+B)−1(0)
⋂(

N⋂
i=1

F−1
i ((Ai +Bi)

−1(0))

)
. (8)

On the other hand, the Fixed Point Problem (FPP) for a multi-valued mapping have
been well discussed due to its many applications. For instance, the FPP is used in game
theory, control theory, convex optimization differential inclusion and so on [22–26]. The
problem of obtaining a common solution of a fixed point problem (in short, FPP) and
other optimization problems have been considered in recent articles. We note that these
type of problems become more applicable in real life problems whose constraints can be
modelled as fixed point and optimization problems. In this direction, Izuchukwu et al.
[15] studied the following split monotone variational inclusion and fixed point problem
between Hilbert space and a Banach space which is defined as follows:

Find x∗ ∈ Fix(T ) ∩ (A+B)−1(0) such that Fu∗ ∈ G−1(0),

where H is a Hilbert space, E is a uniformly convex and uniformly smooth Banach space,
T a multivalued quasi-nonexpansive mapping, B : H → 2H and G : E → 2E are maximal
monotone operators, F : H → E is a bounded linear operator. They proposed a viscosity
iterative scheme and under mild conditions and proved a strong convergence theorem.


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 4 of 22

Inspired by the results discussed above, our second motivation is to propose an iterative
algorithm for approximating a common solution of a fixed point problem and split varia-
tional inclusion problem with multiple output sets. The proposed method combines the
Mann iterative, the Halpern technique and a carefully selected step size to avoid the de-
pendence of the method on prior knowledge of the operator norms. Using this method, we
prove a strong convergence method for approximating a common solution of an SVIPOS
and a fixed point problem for a Bregman multi-valued mapping in the framework of real
reflexive Banach spaces. In particular, the following are some of the highlights of the
present study:

(i) The main result in this article generalizes the results in [27] and [14] from p-uniformly
Banach spaces which are also uniformly smooth to reflexive Banach spaces.

(ii) The problem considered in [19] is a special case of the one considered in this article
and generalizes the results in [6, 7, 11, 19, 28, 29] from real Hilbert spaces to a
reflexive Banach spaces.

(iii) It is worth mentioning that the proof of convergence proposed in this paper is differ-
ent from the ones in [6, 14, 27, 29] in the sense that our approach does not distinguish
between whether the sequence generated by our algorithm is Fejer-monotone or not.
Our approach is simple and more elegant.

(iv) We dispensed the sets {Cn, Dn, Qn}n∈N in our algorithm as this gives difficulties
in computation. Lastly, our iterative algorithm is designed in such a way that it
does not require prior knowledge of operator norm as this also gives difficulties in
computation.

2. Preliminaries

We state some known and useful results which will be needed in the proof of our main theo-
rem. In the sequel, we denote strong and weak convergence by ”→” and ”⇀”, respectively.
For any x ∈ E, we denote the value of x∗ ∈ E at x by ⟨x, x∗⟩ .
Let E be a reflexive Banach space with E∗ its dual and Q be a nonempty closed and
convex subset of E. Let g : E → (−∞,+∞] be a proper, lower semicontinuous and
convex function, then the Fenchel conjugate of g is the map g∗ : E∗ → (−∞,+∞] defined
by

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

Let the domain of g be denoted by domg = {x ∈ E : g(x) < +∞}, hence for any y ∈ E,
we define the directional derivative of g at x in the direction of y by

g0(x, y) = lim
t→0+

g(x+ ty)− g(x)

t
.


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 5 of 22

The function g is said to be

(i) Gâteaux differentiable at x if limt→0+
g(x+ty)−g(x)

t exists for any y. At this time, the
gradient of g at x is the linear function ∇g

E(x) satisfying

⟨∇g
E(x), y⟩ := g0(x, y), ∀ y ∈ E.

(ii) Gâteaux differentiable, if it is Gâteaux differentiable for any x ∈ int(domg); where
int(domg) stands for the interior of domain of g.

(iii) Fréchet differentiable at x, if its limit is attained uniformly in ||y|| = 1;

(iv) Uniformly Fréchet differentiable on a subset Q of E, if the above limit is attained
uniformly for x ∈ Q and ||y|| = 1.

Let g : E → (−∞,+∞] be a function, then g is said to be:

(i) essentially smooth, if the subdifferential of g denoted by ∂g is both locally bounded
and single-valued on its domain, where ∂g(x) = {x∗ ∈ E∗ : g(x) + ⟨y − x, x∗⟩ ≤
g(y), y ∈ E};

(ii) essentially strictly convex, if (∂g)−1 is locally bounded on its domain and g is strictly
convex on every convex subset of dom ∂g;

(iii) Legendre, if it is both essentially smooth and essentially strictly convex. See [30, 31]
for more details on Legendre functions.

Alternatively, a function g is said to be Legendre if it satisfies the following conditions:

(i) The int(domg) is nonempty, g is Gâteaux differentiable on int(dom)g and dom∇g =
int(domg);

(ii) The int(domg∗) is nonempty, g∗ is Gâteaux differentiable on int(domg∗) and dom∇g∗

E∗ =
int(domg∗).

Definition 1. [32] Let E be a Banach space. A function g : E → (−∞,∞] is said to
be proper if the interior of its domain dom(g) is nonempty. Let g : E → (−∞,∞] be a
convex and Gâteaux differentiable function. Then the Bregman distance corresponding to
g is the function Dg : dom(g)× intdom(g) → R defined by

Dg(x, y) := g(x)− g(y)− ⟨x− y,∇g
E(y)⟩, ∀ x, y ∈ E. (9)

is called the Bregman distance with respect to g. It is clear that Dg(x, y) ≥ 0 for all
x, y ∈ E.


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 6 of 22

It is well-known that Bregman distance Dg does not satisfy the properties of a metric
because Dg fail to satisfy the symmetric and triangular inequality property. However, the
Bregman distance satisfies the following so-called three point identity: for any x ∈ domg
and y, z ∈ int(domg),

Dg(x, z) = Dg(x, y) +Dg(y, z) + ⟨x− y,∇g
E(y)−∇g

E(z)⟩. (10)

In particular,

Dg(x, y) = −Dg(y, x) + ⟨y − x,∇g
E(y)−∇g

E(x)⟩, ∀ x, y ∈ E.

Let B : E → 2E
∗
be a set-valued mapping. We define the domain and range of B by

domB = {x ∈ E : Bx ̸= ∅} and ranB =
⋃

x∈E Bx, respectively. The graph of B denoted
by G(B) = {(x, x∗) ∈ E × E∗ : x∗ ∈ Bx}. The mapping B ⊂ E × E∗ is said to be
monotone [33] if ⟨x − y, x∗ − y∗⟩ ≥ 0 whenever (x, x∗), (y, y∗) ∈ B. It is also said to be
maximal monotone [34] if its graph is not contained in the graph of any other monotone
operator on E. If B ⊂ E × E∗ is maximal monotone, then we can represent the set
B−1(0) = {z ∈ E : 0 ∈ Bz} is closed and convex.
Let A : E → 2E

∗
be a mapping, then the resolvent associated with A and λ for any λ > 0

is the mapping ResgλA : E → 2E defined by

ResgλA := (∇g
E + λA)−1 ◦ ∇g

E .

It is worth mentioning that a mapping A : E → 2E
∗
is called Bregman inverse strongly

monotone (BISM) on the set C if

C ∩ (domg) ∩ (int dom g) ̸= ∅,

and for any x, y ∈ C ∩ (int dom g), η ∈ Ax and ξ ∈ Ay, we have

⟨η − ξ, (∇g∗

E∗(x)− η)−∇g∗

E∗(∇g
E(y)− ξ)⟩ ≥ 0.

The anti-resolvent Ag
λ : E → 2E associated with the mapping A : E → 2E

∗
and λ > 0 is

defined by

Ag
λ := (∇g

E)
−1 ◦ (∇g

E − λA). (11)

A point p ∈ Q is called an asymptotic fixed point of T if Q contains a sequence {xn} which
converges weakly to p such that lim

n→∞
||Txn − xn|| = 0. We denote by ˆFix(T ) the set of

asymptotic fixed points of T .
Let Q be a nonempty closed and convex subset of int(dom g), then we define an operator
T : Q → int(domg) to be :

(i) Bregman relatively nonexpansive (BRNE), if Fix(T ) ̸= ∅, and

Dg(p, Tx) ≤ Dg(p, x), ∀ p ∈ Fix(T ), x ∈ Q and ˆFix(T ) = Fix(T ).


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 7 of 22

(ii) Bregman quasi-nonexpansive mapping (BQNE), if Fix(T ) ̸= ∅ and

Df (p, Tx) ≤ Df (p, x),∀ x ∈ Q and p ∈ Fix(T ).

(iii) Bregman firmly nonexpansive (BFNE), if

⟨∇g
E(Tx)−∇g

E(Ty), Tx− Ty⟩ ≤ ⟨∇g
E(x)−∇g

E(y), Tx− Ty⟩, ∀ x, y ∈ E.

Definition 2. [35] Let C be a nonempty, closed and convex subset of a reflexive Banach
space E and g : E → (−∞,+∞] be a strongly coercive Bregman function. Let β and
γ be real numbers with β ∈ (−∞, 1) and γ ∈ [0,∞), respectively. Then a mapping T :
C → E with Fix(T ) ̸= ∅ is called Bregman (β, γ)-demigeneralized if for any x ∈ C and
p ∈ Fix(T ),

⟨x− p,∇g
E(x)−∇g

E(Tx)⟩ ≥ (1− β)Dg(x, Tx) + γDg(Tx, x), ∀ x ∈ E and p ∈ F (T ).

Definition 3. A function g : E → R is said to be strongly coercive if

lim
||xn||→∞

g(xn)

||xn||
= ∞.

Lemma 1. [19] Let E be a Banach space, s > 0 be a constant, ρs be the gauge of uniform
convexity of g and g : E → R be a strongly coercive Bregman function. Then,
(i) For any x, y ∈ Bs and α ∈ (0, 1), we have

Dg

(
x,∇g∗

E∗ [α∇g
E∇

g
E(y) + (1− α)∇g

E(z)]
)
≤ αDg(x, y) + (1− α)Dg(x, z)− α(1− α)ρs(||∇g

E(y)−∇g
E(z)||),

(ii) For any x, y ∈ Bs,

ρs(||x− y||) ≤ Dg(x, y).

Lemma 2. [36] Let E be a reflexive Banach space, g : E → R be a strongly coercive
Bregman function and V be a function defined by

V (x, x∗) = g(x)− ⟨x, x∗⟩+ g∗(x∗), x ∈ E, x∗ ∈ E∗.

The following assertions also hold:

Dg(x,∇g∗

E∗(x
∗)) = V (x, x∗), for all x ∈ E and x∗ ∈ E∗.

V (x, x∗) + ⟨∇g∗

E∗(x
∗)− x, y∗⟩ ≤ V (x, x∗ + y∗) for all x ∈ Eand x∗, y∗ ∈ E∗.


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 8 of 22

Lemma 3. [35] Let E1 and E2 be two Banach spaces. Let F : E1 → E2 be a bounded linear
operator and T : E2 → E2 be a Bregman (ϕ, σ)-demigeneralized for some ϕ ∈ (−∞, 1) and
σ ∈ [0,∞). Suppose that K = ran(A) ∩ Fix(T ) ̸= ∅ (where ran(B) denotes the range of
B). Then for any (x, q) ∈ E1 ×K,

⟨x− q, F ∗(∇g2
E2
(T (Fx)))⟩ ≥ (1− ϕ)Dg2(Fx, T (Fx)) + σDg2(T (Fx), Fx)

≥ (1− ϕ)Dg2(Fx, T (Fx)). (12)

So, given any real numbers ξ1 and ξ2, the mapping L1 : E1 → [0,∞) and L2 : E2 → [0.∞)
formulated for x ∈ E1 as

L1(x) =


Dg2 (Fx,TFx)

D∗
g1

(F ∗(∇g2
E2

(Fx)),F ∗(∇g2
E2

(TFx))
, if , (I − T )Fx ̸= 0,

ξ1, otherwise,
(13)

and

L2(x) =


D∗

g1
(∇g1

E1
(x)−γF ∗(∇g2

E2
(Fx)−∇g2

E2
(TFx)),∇g1

E1
(x))

D∗
g1

(F ∗(∇g2
E2

(Fx)),F ∗(∇g2
E2

(TFx))
, if , (I − T )Fx ̸= 0,

ξ2, otherwise,
(14)

are well-defined, where γ is any nonnegative real number. Moreover, for any (x, p) ∈
E1 ×K, we have

Dg1(q, y) ≤ Dg1(q, x)− (γ(1− ϕ)L1(x)− L2(x))Dg∗1
(F ∗(∇g2

E2
(Fx)), F ∗(∇g2

E2
(TFx)),

(15)

where

y = (∇g1
E1
)−1[∇g1

E1
(x)− γF ∗(∇g2

E2
(Fx)−∇g2

E2
(TFx))].

Lemma 4. [36] Let E be a Banach space and g : E → R a Gâteaux differentiable function
which is uniformly convex on bounded subsets of E. Let {x}n∈N and {yn}n∈N be bounded
sequences in E. Then,

lim
n→∞

Dg(yn, xn) = 0 ⇒ lim
n→∞

||yn − xn|| = 0.

Lemma 5. [37] Let g : E → (−∞,+∞] be a Legendre function. Let {Ti}Ni=1 : E → E be a

BQNE such that
N⋂
i=1

Fix(Ti) ̸= ∅ and {βi}Ni=0 ⊂ (0, 1) satisfy
N∑
i=0

βi = 1. Define a mapping

S : E → E by Sx := (∇g
E)

−1(β0∇g
E(x) +

N∑
i=1

βi∇g
E(Tix)) for all x ∈ E. Then S is BQNE

such that Fix(S) =
N⋂
i=1

Fix(Ti).


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 9 of 22

Lemma 6. [38] Let B : E → 2E
∗
be a maximal monotone operator and A : E → E∗ be

a BISM mapping such that (A + B)−1(0∗) ̸= ∅. Let g : E → R be a Legendre function,
which is uniformly Fréchet differentiable and bounded on bounded subset of E. Then,

Dg(u,ResgλB ◦Ag(x)) +Dg(ResgλB(x), x) ≤ Dg(u, x), for any u ∈ (A+B)−1(0∗), x ∈ E and λ > 0.

Lemma 7. [38] Let B : E → 2E
∗
be a maximal monotone operator and A : E → E∗ be

a BISM mapping such that (A + B)−1(0∗) ̸= ∅. Let g : E → R be a Legendre function,
which is uniformly Fréchet differentiable and bounded on bounded subset of E. Then,
(i) (A+B)−1(0∗) = Fix(ResgλB ◦Ag

λ);

(ii) ResgλB ◦Ag
λ is a BSNE operator with Fix(ResgλB ◦Ag

λ) =
ˆFix(ResgλB ◦Ag

λ).

Lemma 8. [39] Let g : E → R be a Gâteaux differentiable and totally convex function. If
x0 ∈ E and the sequence {Dg(xn, x0)} is bounded, then the sequence {xn} is also bounded.

Definition 4. Let C be a nonempty closed and convex subset of a reflexive Banach space
E and g : E → (−∞,+∞] be a strongly coercive Bregman function. A Bregman projection
of x ∈ int(domg) onto C ⊂ int(domg) is the unique vector ProjgC(x) ∈ C satisfying

Dg(ProjgC(x), x) = int{Dg(y, x) : y ∈ C}.

Lemma 9. [40] Let C be a nonempty closed and convex subset of a reflexive Banach space
E and x ∈ E. Let g : E → R be a strongly coercive Bregman function. Then,
(i) z = ProjgC(x) if and only if ⟨∇g

E(x)−∇g
E(z), y − z⟩ ≤ 0, ∀ y ∈ C.

(ii) Dg(y, ProjgC(x)) +Dg(ProjgC(x), x) ≤ Dg(y, x), ∀ y ∈ C.

Lemma 10. [41] Let {an} and {dn} be sequences of nonnegative real numbers such that

an+1 ≤ (1− δn)an + bn + dn, n ≥ 1,

where {δn} is a sequence in (0, 1) and {bn} is a real sequence. Assume that
∞∑
n=1

dn <

∞,
∞∑
n=1

δn = ∞ and lim sup
n→∞

bn
δn

≤ 0, then lim
n→∞

an = 0.

Lemma 11. [42] Let {Γn} be a sequence of real numbers that does not decrease at infinity
in the sense that there exists a subsequence {Γnj} of {Γn} which satisfies Γnj < Γnj+1 for
all j ∈ N. Define the sequence {τ(n)}n≥n0 of integers as follows:

τ(n) := max{k ≤ n : Γk < Γk+1},

where n0 ∈ N such that {k ≤ n0 : Γk < Γk+1} ≠ ∅. Then, the following hold:

(i) τ(n0) ≤ τ(n0 + 1) ≤ · · · and τ(n) → ∞,

(ii) Γτ(n) ≤ Γτ(n)+1 and Γτ(n) ≤ Γτ(n)+1, ∀ n ≥ n0.


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 10 of 22

3. Main Result

Throughout this section, we assume that

Assumption 1.

(i) Let Ei for i = 0, 1, 2, · · · , N be reflexive Banach spaces where E0 = E, g : E →
(−∞,+∞] and gi : Ei → (−∞,+∞] be strongly coercive Legendre functions which
are bounded, uniformly Fréchet differentiable and totally convex on bounded subsets
of E and Ei, i = 1, 2, · · · , N, respectively. Let ∇g

E and ∇gi
Ei

be the gradients of E
dependent on g and Ei dependent on gi respectively.

(ii) Let Fj : E → E∗, j = 1, 2, · · · ,m be BISM mappings and Gj : E → E∗, j =
1, 2, · · · ,m be maximal monotone mappings respectively. Suppose Ai : E → Ei, i =
1, 2, · · · , N be bounded linear operator such that Ai ̸= 0 and A∗

i be the adjoint of Ai.

(iii) Si : Ei → Ei, i = 0, 1, 2, · · · , N be Bregman (ρS , µS)− demigeneralized mapping such

that ρS ∈ (−∞, 1) and µS ∈ [0,∞). Assume that Ω :=
{
x∗ ∈

m⋂
j=1

Fix(T j
σ)∩Fix(S) :

Aix
∗ ∈

N⋂
i=1

Fix(Si)} ≠ ∅,

(iv) Let γ > 0 be a real number and {αn}n∈N, {βj}mj=0 and {λi,n}n∈N be sequences in

(0,1) with
m∑
j=0

βj = 1 and
N∑
i=0

λi,n = 1 respectively, satisfying the following control

condition:

(i) lim
n→∞

αn = 0,
∞∑
n=1

αn = ∞.

Let T j
σ := ResgσGj

◦ F g
j for j = 1, 2, · · · ,m, clearly Fix(T j

σ) = (Fj + Gj)
−1(0) for each

j = 1, 2, · · · ,m and σ > 0. Define the sequence {xn} by the following recursive formula:

Algorithm 1. For fixed u ∈ E, let {xn}∞n=1 be a sequence generated by x1 ∈ E such that
zn = (∇g

E)
−1
[ N∑
i=0

λi,n

(
∇g

E(xn)− γA∗
i (∇

gi
Ei
(Aixn)−∇gi

Ei
(SiAixn))

)]
yn = (∇g

E)
−1
[
(β0∇g

E(zn) +
m∑
j=1

βj∇g
E(T

j
σzn)

]
xn+1 = (∇g

E)
−1
[
αn∇g

E(u) + (1− αn)∇g
E(yn)

]
.

(16)

Suppose {ξ1,n}n∈N and {ξ2,n}n∈N are two sequences, where

ξ1,n =


Dgi (Aixn,SiAixn)

D∗
g(A

∗
i (∇

gi
Ei

(Aixn)),A∗
i (∇

gi
Ei

(SiAixn))
, if , (I − Si)Aixn ̸= 0,

ξ1, otherwise,


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 11 of 22

and

ξ2,n =


D∗

g(∇
g
E(xn)−γA∗

i (∇
gi
Ei

(Aixn)−∇gi
Ei

(SiAixn)),∇g
E(xn))

D∗
g(A

∗
i (∇

gi
Ei

(Aixn)),A∗
i (∇

gi
Ei

(SiAixn))
, if , (I − Si)Aixn ̸= 0,

ξ2, otherwise.

Then, the sequence {xn} defined in (16) converges strongly to v = ProjgΩu, where ProjgΩ
is the Bregman projection of E onto Ω.

Proof. Let Vσ = β0∇g
E+β1∇g

E(ResgσG1
◦F g

1 )+β2∇g
E(ResgσG2

◦F g
2 )+· · ·+βj∇g

E(ResgσGm
◦

F g
m), then yn = Vσzn. By applying Lemma 5 and using the fact that T j

σ is BQNE then we

have that Fix(Vσ) =
m⋂
j=1

Fix(T j
σ) =

m⋂
j=1

(Fj + Gj)
−1(0). Let v ∈ Ω, then we obtain from

Lemma 3 that

Dg(v, zn) = Dg(v, (∇g
E)

−1
[ N∑
i=0

λi,n(∇g
E(xn)− γA∗

i (∇
gi
Ei
(Aixn)−∇gi

Ei
(SiAixn)))

]
≤ Dg(v, xn)−

N∑
i=0

λi,n(γ(1− ρS)ξ1,n − ξ2,n)D
∗
g(A

∗
i (∇

gi
Ei
(Aixn)), A

∗
i (∇

gi
Ei
(SiAixn))

(17)

≤ Dg(v, xn). (18)

It follows from (16) and (18) that

Dg(v, yn) = Dg(v, (∇g
E)

−1
[
β0∇g

E(zn) +
m∑
j=1

βj∇g
E(T

j
σzn)

]
)

≤ β0Dg(v, zn) +

m∑
j=1

βjDg(v, T
j
σzn)

≤ β0Dg(v, zn) +

m∑
j=1

βjDg(v, zn)

= Dg(v, zn) (19)

≤ Dg(v, xn). (20)

Using (16), (18) and (19), we get

Dg(v, xn+1) = Dg(v, (∇g
E)

−1
[
αn∇g

E(u) + (1− αn)∇g
E(yn)

]
)

≤ αnDg(v, u) + (1− αn)Dg(v, yn) (21)

≤ αnDg(v, u) + (1− αn)Dg(v, zn)

≤ αnDg(v, u) + (1− αn)Dg(v, xn)

≤ max{Dg(v, u), Dg(v, xn)}


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 12 of 22

...

≤ max{Dg(v, u), Dg(v, x1)}. ∀ n ≥ 1.

Thus, we obtain that the sequence {Dg(v, xn)}n∈N is bounded. Using Lemma 8, then we
conclude that {xn}n∈N is bounded. Consequently, {yn}n∈N and {zn}n∈N are bounded.
By Lemma 6, (16) and (20), we obtain that

Dg(v, yn) = Dg(v, (∇g
E)

−1[β0∇g
E(zn) +

m∑
j=1

βj∇g
E(T

j
σzn)])

≤ β0Dg(v, zn) +
m∑
j=1

βjDg(v, T
j
σzn)

≤ β0Dg(v, zn) +

m∑
j=1

βj
(
Dg(v, zn)−Dg(T

j
σzn, zn)

)
= Dg(v, zn)−

m∑
j=1

βjDg(T
j
σzn, zn) (22)

≤ Dg(v, xn)−
m∑
j=1

βjDg(T
j
σzn, zn) (23)

From (17), (21) and (22), we get

Dg(v, xn+1) ≤ αnDg(v, u) + (1− αn)Dg(v, yn)

≤ αnDg(v, u) + (1− αn)
(
Dg(v, zn)−

m∑
j=1

βjDg(T
j
σzn, zn)

)
= αnDg(v, u) + (1− αn)Dg(v, xn)− (1− αn)

m∑
j=1

βjDg(T
j
σzn, zn)

− (1− αn)
N∑
i=0

λi,n(γ(1− ρS)ξ1,n − ξ2,n)D
∗
g(A

∗
i (∇

gi
Ei
(Aixn)), A

∗
i (∇

gi
Ei
(SiAixn)).

(24)

We now divide the remaining proof into two cases.
Case 1: Suppose that there exists n0 ∈ N such that {Dg(v, xn)} is non-increasing, then
we obtain that lim

n→∞
Dg(v, xn) exists. Thus,

Dg(v, xn)−Dg(v, xn+1) → 0, n → ∞. (25)

From (24), 25 and condition (i) of Assumption (1), we have that

(1− αn)
[ N∑
i=0

λi,n(γ(1− ρS)ξ1,n − ξ2,n)D
∗
g(A

∗
i (∇

gi
Ei
(Aixn)), A

∗
i (∇

gi
Ei
(SiAixn))


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 13 of 22

+

m∑
j=1

βjDg(T
j
σzn, zn))

]
≤ αnDg(v, u) + (1− αn)Dg(v, xn)−Dg(v, xn+1),

which implies from Lemma 4 that

lim
n→∞

Dg(T
j
σzn, zn) = 0 = lim

n→∞
||T j

σzn − zn||. (26)

Also,

lim
n→∞

N∑
i=0

λi,n(γ(1− ρS)ξ1,n − ξ2,n)D
∗
g(A

∗
i (∇

gi
Ei
(Aixn)), A

∗
i (∇

gi
Ei
(SiAixn)) = 0. (27)

Therefore, we have

lim
n→∞

D∗
g(A

∗
i (∇

gi
Ei
(Aixn)), A

∗
i (∇

gi
Ei
(SiAixn)) = 0. (28)

Hence, by applying Lemma 4, (12) and the properties of D∗
g and A, we get

lim
n→∞

||Aixn − SiAixn|| = 0, i = 0, 1, 2, · · · , N. (29)

In view of (16), (26), (28) and Lemma 4, we obtain that

lim
n→∞

Dg(zn, xn) = 0 = lim
n→∞

||zn − xn||, (30)

and

lim
n→∞

Dg(yn, zn) = 0 = lim
n→∞

||yn − zn|| = 0. (31)

By applying (30) and (31) we obtain

lim
n→∞

Dg(yn, xn) = 0 = lim
n→∞

||yn − xn||. (32)

More so, employing condition (i) of Assumption 1 and Lemma 4, we arrive at

lim
n→∞

Dg(xn+1, yn) = 0 = lim
n→∞

||xn+1 − yn||. (33)

We therefore conclude from (32) and (33) that

lim
n→∞

Dg(xn+1, xn) = 0 = lim
n→∞

||xn+1 − xn||. (34)

Since {xn} is bounded and E is reflexive, there exists a subsequence {xnk
} of {xn} such

that {xnk
} ⇀ x∗. Also, from (30) and (32), there exist subsequences {znk

} of {zn} and
{ynk

} of {yn} which converge weakly to x∗ respectively. Thus, for each i = 0, 1, 2, · · ·N, Ai

is a bounded linear operator, then it follows that Aixnk
⇀ Aix

∗. Hence, using the demi-
closedness principle and (29), we arrive at Aix

∗ ∈ Fix(Si) for all i = 0, 1, 2, · · ·N. Also,


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 14 of 22

from (26), we obtain that x∗ ∈ ˆFix(T j
σ) = Fix(T j

σ) for each j = 1, 2, · · ·m. This implies

from Lemma 7 that x∗ ∈
m⋂
j=1

Fix(T j
σ) =

m⋂
j=1

(Fj + Gj)
−1(0). Therefore, we conclude that

x∗ ∈ Ω.
Next is to show that

⟨∇g
E(u)−∇g

E(z), xn+1 − z⟩ ≤ 0.

Now, from (34), we have

lim sup
n→∞

⟨∇g
E(u)−∇g

E(z), xn+1 − z⟩ = lim
k→∞

⟨∇g
E(u)−∇g

E(z), xnk+1 − z⟩

≤ ⟨∇g
E(u)−∇g

E(z), x
∗ − z⟩.

Hence, we obtain that

lim sup
k→∞

⟨∇g
E(u)−∇g

E(z), xn+1 − z⟩ ≤ ⟨∇g
E(u)−∇g

E(z), x
∗ − z⟩

≤ 0. (35)

Next is to prove that {xn} converges strongly to v ∈ Ω. Using Lemma 2, (18) and (20),

Dg(v, xn+1) = Dg(v, (∇g
E)

−1
[
αn∇g

E(u) + (1− αn)∇g
E(yn)

]
)

= Vg(v, αn∇g
E(u) + (1− αn)∇g

E(yn))

≤ Vg(v, αn∇g
E(u) + (1− αn)∇g

E(yn)− αn(∇g
E(u)−∇g

E(v))

+ ⟨αn(∇g
E(u)−∇g

E(v)), xn+1 − v)⟩
= Vg(v, αn∇g

E(v) + (1− αn)∇g
E(yn)) + αn⟨∇g

E(u)−∇g
E(v), xn+1 − v⟩

≤ αnVg(v,∇g
E(v)) + (1− αn)Vg(v,∇g

E(yn)) + αn⟨∇g
E(u)−∇g

E(v), xn+1 − v⟩
= αnDg(v, v) + (1− αn)Dg(v, yn) + αn⟨∇g

E(u)−∇g
E(v), xn+1 − v⟩

≤ (1− αn)Dg(v, xn) + αn⟨∇g
E(u)−∇g

E(v), xn+1 − v⟩. (36)

In view of Lemma 10 and (35), we conclude that lim
n→∞

Dg(v, xn) = 0. Therefore {xn}
converges strongly to v.
Case 2: Suppose that there exists a subsequence {nk} of {n} such that Dg(v, xnk

) <
Dg(v, xnk+1

) for all k ∈ N. We define a positive integer sequence {τ(n)} by

τ(n) := max{k ∈ n : Dg(v, xk) < Dg(v, xk+1)}

for all n ≥ n0 (for some n0 large enough). Applying Lemma 11, we have {τ(n)} to be
non-decreasing sequence such that τ(n) → ∞ as n → ∞ and

Dg(v, xτ(n))−Dg(v, xτ(n)+1) ≤ 0.


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 15 of 22

Following the same argument to the one used in Case 1 of the proof of (16), we obtain
that 

lim
τ(n)→∞

Dg(T
j
σzτ(n), zτ(n)) = 0, for j = 1, 2, · · · ,m,

lim
τ(n)→∞

||Aixτ(n) − SiAixτ(n)|| = 0, for i = 0, 1, 2, · · · , N, r

lim
τ(n)→∞

Dg(zτ(n), xτ(n)) = 0,

lim
τ(n)→∞

Dg(yτ(n), xτ(n)) = 0,

lim
τ(n)→∞

⟨∇g
E(u)−∇g

E(v), xτ(n)+1 − v⟩ ≤ 0.

(37)

and

Dg(v, xτ(n)+1) ≤ (1− ατ(n))Dg(v, xτ(n)) + ατ(n)⟨∇
g
E(u)−∇g

E(v), xτ(n)+1 − v⟩.

Using Lemma 11, we arrive at

Dg(v, xτ(n)) ≤ Dg(v, xτ(n)+1).

Hence, we conclude that lim
n→∞

Dg(v, xn) = 0. Therefore, {xn} converges strongly to v. This

completes the proof of our theorem.


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 16 of 22

If we put m = 1, then we have the following iterative method which solves Ω := {x∗ ∈

(F +G)−1(0) ∩ Fix(S) : Aix
∗ ∈

N⋂
i=1

Fix(Si)} ≠ ∅.

Corollary 1.

Algorithm 2. For fixed u ∈ E, let {xn}∞n=1 be a sequence generated by x1 ∈ E such that
zn = (∇g

E)
−1
[ N∑
i=0

λi,n

(
∇g

E(xn)− γA∗
i (∇

gi
Ei
(Aixn)−∇gi

Ei
(SiAixn))

)]
yn = (∇g

E)
−1
[
(βn∇g

E(zn) + (1− βn)∇g
E(ResgσG ◦ F g)

]
xn+1 = (∇g

E)
−1
[
αn∇g

E(u) + (1− αn)∇g
E(yn)

]
.

(38)

where 0 < a ≤ βn ≤ b < 1. Suppose {ξ1,n}n∈N and {ξ2,n}n∈N are two sequences, where

ξ1,n =


Dgi (Aixn,SiAixn)

D∗
g(A

∗
i (∇

gi
Ei

(Aixn)),A∗
i (∇

gi
Ei

(SiAixn))
, if , (I − Si)Aixn ̸= 0,

ξ1, otherwise,

and

ξ2,n =


D∗

g(∇
g
E(xn)−γA∗

i (∇
gi
Ei

(Aixn)−∇gi
Ei

(SiAixn)),∇g
E(xn))

D∗
g(A

∗
i (∇

gi
Ei

(Aixn)),A∗
i (∇

gi
Ei

(SiAixn))
, if , (I − Si)Aixn ̸= 0,

ξ2, otherwise.

Then, the sequence {xn} defined in (38) converges strongly to v = ProjgΩu, where ProjgΩ
is the Bregman projection of E onto Ω.

Here we consider the split common fixed point problem of Bregman demigeneralized map-

ping which is defined as Ω := {x∗ ∈ Fix(S) : Aix
∗ ∈

N⋂
i=1

Fix(Si)} ≠ ∅.

Corollary 2.

Algorithm 3. For fixed u ∈ E, let {xn}∞n=1 be a sequence generated by x1 ∈ E such thatzn = (∇g
E)

−1
[ N∑
i=0

λi,n

(
∇g

E(xn)− γA∗
i (∇

gi
Ei
(Aixn)−∇gi

Ei
(SiAixn))

)]
xn+1 = (∇g

E)
−1
[
αn∇g

E(u) + (1− αn)∇g
E(zn)

]
.

(39)

Suppose {ξ1,n}n∈N and {ξ2,n}n∈N are two sequences, where

ξ1,n =


Dgi (Aixn,SiAixn)

D∗
g(A

∗
i (∇

gi
Ei

(Aixn)),A∗
i (∇

gi
Ei

(SiAixn))
, if , (I − Si)Aixn ̸= 0,

ξ1, otherwise,

and

ξ2,n =


D∗

g(∇
g
E(xn)−γA∗

i (∇
gi
Ei

(Aixn)−∇gi
Ei

(SiAixn)),∇g
E(xn))

D∗
g(A

∗
i (∇

gi
Ei

(Aixn)),A∗
i (∇

gi
Ei

(SiAixn))
, if , (I − Si)Aixn ̸= 0,

ξ2, otherwise.

Then, the sequence {xn} defined in (39) converges strongly to v = ProjgΩu, where ProjgΩ
is the Bregman projection of E onto Ω.


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 17 of 22

4. Numerical Example

In this section, we give a numerical example to illustrate the performance of our method.
Example 1: Let E = Ei = R4 for i = 1, 2. We define hm : R → (−∞,+∞] by hm(x) =
1
2x

2, m = 1, 2, 3, 4. Also, let g = gi for i = 1, 2 be defined by g : R2 → (−∞,+∞],
g(x) = h1(x) + h2(x) + h3(x) + h4(x) =

1
2x

2
1 +

1
2x

2
2 +

1
2x

2
3 +

1
2x

2
4. Then, we have

∇g(x) = (∇h(x1)),∇h(x2),∇h(x3),∇h(x4) = (x1, x2, x3, x4) =


1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1



x1
x2
x3
x4

 .

For i = 0, 1, 2, let Ai : R → R be defined by Ai(x) =
x

(i+1) for x = (x1, x2, x3, x4) ∈ R4.

We also define the mapping Si : R → R by Si(x) = −(i + 1)x for each i = 0, 1, 2. Then

the mappings Si are
(
− 1

i+1 , 0
)
-Bregman demigeneralized. Now, define the mappings

F1, F2, F3 : R → R respectively by

F1 =


1 0 0 2
1 0 0 1
1 0 1 1
1 0 0 −1

 , F2 =


1 1 0 2
1 2 0 1
1 2 1 2
1 2 0 3

 , F3 =


1 1 0 2
1 2 0 1
1 0 5 1
1 2 0 3


and the mappings G1, G2, G3 : R → R respectively by

G1 =


1 1 0 −2
1 2 −2 1
−1 0 0 1
0 2 0 3

 , G2 =


1 −2 −1 2
0 0 1 3
−1 2 −3 4
0 3 0 5

 , G3 =


0 2 0 −2
0 0 1 −3
1 2 0 1
1 3 0 2

 .

It is easy to see for any λ > 0, that

T1(x) = (∇g
E + λG1) ◦ ∇g

E ◦ (∇g
E)

−1(∇g
E − λF1)(x)

=



1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

+ λ×


1 1 0 −2
1 2 −2 1
−1 0 0 1
0 2 0 3



−1 


1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

− λ×


1 0 0 2
1 0 0 1
1 0 1 1
1 0 0 −1




x1
x2
x3
x4



=



1 + λ λ 0 −2λ
λ 1 + 2λ −2λ λ
−λ 0 1 λ
0 2λ 0 1 + 3λ



−1 


1− λ 0 0 2λ
−λ 1 0 −λ
−λ 0 1− λ −λ
−λ 0 0 1 + λ




x1
x2
x3
x4

 .

Suppose λ = 1, we obtain

T1(x) =




2 1 0 −2
1 3 −2 1
−1 0 1 1
0 2 0 4



−1 


0 0 0 2
−1 1 0 −1
−1 0 0 −1
−1 0 0 2




x1
x2
x3
x4




H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 18 of 22

=



1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1




x1
x2
x3
x4

 .

Proceeding same way, we obtain

T2(x) =



1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1




x1
x2
x3
x4


and

T3(x) =
1

9999




15554 14443 17776 −2222
−8148 11851 7407 −7407
−2963 2963 11851 1852
−370 370 −1481 1481




x1
x2
x3
x4

 .

For this example, we choose αn = 1
n+1 , β0 =

2
n+15 , β1 =

6
n+15 , β2 =

3+n
n+15 and β3 =

4
n+15 .

We also choose γ = 0.75, λ0,n = 5n
10n+17 , λ1,n = 3n+10

10n+17 and λ2,n = 2n+7
10n+17 . Let En =

∥xn+1 − xn∥2 < 10−4 be the stopping criterion. We illustrate this example with different
initial values of x1.

(Case 1) x1 = (1, 1, 2, 2)′;

(Case 2) x1 = (5, 5, 5, 5)′;

(Case 3) x1 = (0.25, 0.5, 0.25, 0.25)′;

(Case 4) x1 = (10, 5,−5,−20)′.

The results of this experiment are presented in Figure 1.


H. A. Abass et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6062 19 of 22

0 2 4 6 8 10 12 14 16 18

Number of iterations

0

2

4

6

8

10

12

14

16

18

20

E
n

Algorithm 3.2

0 2 4 6 8 10 12 14 16 18

Number of iterations

0

50

100

150

200

250

E
n

Algorithm 3.2

0 2 4 6 8 10 12 14 16 18

Number of iterations

0

0.05

0.1

0.15

0.2

0.25

0.3

E
n

Algorithm 3.2

0 2 4 6 8 10 12 14 16 18

Number of iterations

0

500

1000

1500

2000

2500

E
n

Algorithm 3.2

Figure 1: Example 4. Top left: Case 1, Top right: Case 2, Bottom left: Case 3, Bottom right: Case 4.

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