




































©2024 Ada Academica https://adac.eeEur. J. Math. Anal. 4 (2024) 2doi: 10.28924/ada/ma.4.2
Modified Viscosity Iterative Algorithm for Solving Variational Inclusion and Fixed Point

Problems in Real Hilbert Space

Furmose Mendy1,∗ , John T Mendy2,∗

1Department of Mathematics, University of Toledo, USA
furmosemendy111@gmail.com

2Department of Mathematics, Universita Degli Studi Dell’Aquila, Italy, 67010, Coppito, Via Vetoio, Italy
johntgracemendy@gmail.com

∗Correspondence: furmosemendy111@gmail.com

Abstract. This paper introduces a new iterative algorithm, called the Modified Viscosity Iterativealgorithm, designed to solve problems related to Variational Inclusion and Fixed point in real Hilbertspaces. The algorithm is specifically tailored to handle Multivalued Quasi-Nonexpansive and Demi-contractive operators. The convergence properties of the algorithm are analyzed and established,ensuring its effectiveness in finding solutions for complex mathematical problems in the field of opti-mization and equilibrium.

1. Introduction
Variational inclusion and fixed point problems involving multivalued quasi nonexpansive anddemicontractive operators play a crucial role in the field of mathematics, particularly in real Hilbertspaces.The study of variational inclusion and fixed point problems originated from the theory of opti-mization and nonlinear analysis, and in the mid−20th century, mathematicians began investigatingproblems involving finding points that satisfy certain inclusion and fixed point conditions. Over time,research in this area expanded and became an essential part of functional analysis and optimiza-tion theory. They are widely-used in applications in diverse fields such as engineering, economics,physics, and computer science. They provide a framework to model and solve various real-worldproblems, including equilibrium problems, optimization problems, and variational inequalities.Fixed point problems, on the other hand, deal with finding points that remain unchanged underthe action of an operator. The concept of fixed points has profound implications in mathematicsand its applications. A wide range of problems in analysis, differential equations, and optimization
Received: 22 Nov 2023.
Key words and phrases. Iterative algorithm, Quasi-nonexpansive, Multivalued demiclosed mapping, Hilbert space,Variational inequality, Strongly Monotone Mappings. 1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.2
https://orcid.org/0009-0000-3806-629X
https://orcid.org/0000-0002-3774-0761


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 2theory can be reduced to fixed point problems. They serve as powerful tools to prove the exis-tence and uniqueness of solutions, compute approximations, and establish convergence propertiesof iterative algorithms. Whilst multivalued quasi nonexpansive operators play a pivotal role in vari-ational inclusion and fixed point problems. These operators possess certain properties that ensurethe stability and convergence of iterative algorithms. They have applications in image processing,signal estimation, and constrained optimization, among others.Moreover, fixed point theory for multivalued mappings has also contributed to the developmentof related areas of research, such as operator theory, topological degree theory, and convex anal-ysis. By investigating the properties and behavior of fixed points in multivalued mappings, math-ematicians have gained a deeper understanding of these fields and have been able to establishconnections and develop new techniques.(See, [30], [10, 11], [2], [7], [33], [9] and [26]).Demicontractive operators, on the other hand, exhibit properties of both contractive and nonex-pansive operators. They are broadly used in the study of variational inequalities and play a crucialrole in convex analysis and optimization theory. They provide a bridge between nonlinear andlinear problems, enabling the development of efficient numerical methods for solving variationalproblems arising in diverse areas.Viscosity iterative algorithms have been extensively studied in recent years for finding commonfixed points of single-valued nonexpansive mappings and solving variational inequality problems.These investigations have built upon the concepts of viscosity solutions introduced by variousresearchers. (see e.g [6], [25], [5], [29], [23], [19], [28]).Throughout this paper, we denote H to be real Hilbert space with the inner product 〈., .〉 inducedby the norm ‖.‖. Let K, to be a nonempty, closed and convex subset of H.An operator A : H → H is said to be Lipschitz if there exists a constant L > 0 such that
‖Ax − Ay‖ ≤ L‖x − y‖,∀x, y ∈ H (1.1)

A : H → H is said to be strongly positive if there exists a constant k > 0 such that
〈Ax, x〉 ≥ k‖x‖2, ∀x ∈ H (1.2)

A : H → H is said to be k−strongly monotone if there exists a constant k ∈ (0, 1) such that
〈Ax − Ay , x − y〉H ≥ k‖x − y‖2, ∀x, y ∈ H (1.3)

Definition 1.1. A multivalued mapping(1) T : D(T ) ⊆ H → CB(D) is called L−Lipschitzian if there exists L > 0, such that
H(Tx, T y) ≤ L‖x − y‖,∀x, y ∈ D(T )

and T is contraction if L ∈ (0, 1) and noneaxpansive if L = 1.(2) T is called quasi-nonexpansive if H(Tx, Tp) ≤ ‖x − p‖, ∀x ∈ D(T ), p ∈ F ix(T )

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Eur. J. Math. Anal. 10.28924/ada/ma.4.2 3(3) T : D(T ) ⊆ H → CB(D) is said to be k−stritly pseudo-contractive, if there exists
k ∈ (0, 1) such that for all x, y ∈ D(T ), the following holds;(

H(Tx, T y)
)2
≤ ‖x − y‖2 + k‖(I − T )x − (I − T )y‖2,

If k = 1, the map T is said to be pseudocontractive.(4) [26] T : D(T ) ⊆ E → 2E is said to be demicontractive if F ix(T ) 6= ∅ and for all
p ∈ F ix(T ), x ∈ D(T ) there exists k ∈ (0, 1) such that(

H(Tx, Tp)
)2
≤ ‖x − p‖2 + kd(x, T x)2.

If k = 1, the map T is said to be hemicontractive.
Let (X, d) be a metric space, K be a nonempty subset of X and T : K → 2K be a multivaluedmapping. An element x ∈ K is called a fixed point of T if x ∈ Tx . The fixed point set of T isdenoted by F ix(T ) := {x ∈ D(T ) : x ∈ Tx} where D(T ) := {x ∈ X : Tx 6= ∅}. It is easy to seethat single-valued mapping is a particular case of multivalued mapping.Let D be a nonempty suset of a normed linear space E. The set D is called proximinal (see [13])if for each ψ ∈ E, there exists u ∈ D such that

d(x, u) := inf{‖x − y‖ : y ∈ D},∀x, y ∈ E (1.4)
where d(x, y) := ‖x − y‖ for all x, y ∈ E. Every closed, nonempty and convex set of real Hilbertspace is proximinal. The family of nonempty closed bounded subsets, nonempty compact subsets,and nonempty proximinal bounded subsets be donated as CB(D), K(D) and P (D) respectively.Let A, B ∈ CB(D). Then the Hausdorff metric in H is defined by

H(A, B) = max
{

sup
a∈A

d(a,B), sup
b∈B

d(b,A)
}
. (1.5)

Let A : D(A) ⊂ H → 2H be a multivalued operator. Then A is monotone if (x, u), (y , v) ∈ D(A)such that
G(A) := {x, u) : x ∈ D(A), u ∈ Ax} (1.6)

A monotone mapping A : H → 2H is said to be maximal if its graph G(A) is not properly containedin the graph of any other monotone mapping.A mapping A : H → H is said to be α−inverse strongly if there exits a constant α > 0 such that
〈Ax − Ay , x − y〉H ≥ α‖Ax − Ay‖2, ∀x, y ∈ H (1.7)

Remark 1.2. It can be seen that every α−inverse strongly monotone mapping is 1

α
−Lipschitzmonotone.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.2 4

Let A : H → H be a single-valued nonlinear mapping and Π : H → 2H be a set-valued mapping.Then the variational inclusion problem is as follows: Find x ∈ H, such that
ω ∈M(x) + A(x) (1.8)

where ω is the zero vector in H. We denote the solution of the problem (1.8) by S(M,A). If ω = Athen, problem (1.8) becomes the inclusion problem by Rockafellar [16].Further readings on Zeros ofinclusion problem (See [17], [18], [19], [8], [12])
Let a set value mapping M : H → 2H be maximal monotone. We define a resolvent operator JM

λgenerated by Π and λ as follows
JM
λ = (I − λM)−1(x),∀x ∈ H (1.9)

where λ is a positive number. It is easily to see that the resolvent operator JM
λ is single - valuednonexpensive and 1−inverse strongly monotone , and moreover, a solution of the problem (1.8) isa fixed point of the operator JM

λ (I − λA), ∀λ > 0( See [4]).Let T : H → P (H) be multivalued map and PT : H → CB(H) be defined by
PT (x) = {y ∈ Tx : ‖y − x‖ = d(x, T x)} (1.10)

See examples of a multivalued mapping T with F ix(T ) 6= ∅, T p = {q} for all q ∈ Tp which PTis a demicontractive-type but not a k−strictly pseudocontractive-type mapping in Mendy et al [?]To prove that a multivalued mapping T with F ix(T ) 6= ∅ and Tp = {q} for all q ∈ Tp is ademicontractive-type but not a k-strictly pseudocontractive-type mapping, we need to demonstratethe following three steps:
Step 1.3. Show that T is demicontractive-type.To prove that T is demicontractive-type, we need to show that for all p ∈ F ix(T ) and x ∈ D(T ),there exists k ∈ (0, 1) such that(

H(Tx, Tp)
)2
≤ ‖x − p‖2 + kd(x, T x)2.

Since Tp = {q} for all q ∈ Tp, we have Tp = {p} for all p ∈ F ix(T ). Thus, for any x ∈ D(T ),
Tx = {φ} for some φ ∈ F ix(T ).Now, consider the case when x = φ. In this case, we have H(Tx, Tp) = H(Tφ, Tp) =

H({φ}, {p}) = 0. Therefore, the inequality holds for any k ∈ (0, 1).
Step 1.4. Show that T is not k-strictly pseudocontractive-type.To prove that T is not k-strictly pseudocontractive-type, we need to show that there does notexist a constant k ∈ (0, 1) such that(

H(Tx, Tp)
)2
≤ k‖x − p‖2 + kd(x, T x)2

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Eur. J. Math. Anal. 10.28924/ada/ma.4.2 5for all p ∈ F ix(T ) and x ∈ D(T ).From step 1.3, we know that H(Tx, Tp) = 0 for any p ∈ F ix(T ) and x ∈ D(T ). Therefore, theinequality reduces to
0 ≤ k‖x − p‖2

for all p ∈ F ix(T ) and x ∈ D(T ). However, this inequality cannot hold for all x 6= p since itimplies k ≥ 1
‖x−p‖2 , which contradicts the requirement that k ∈ (0, 1).

Step 1.5. Show that PT is a demicontractive-type.Let PT denote the projection operator associated with multivalued mapping T . Since Tp = {q}for all q ∈ Tp, PT is the single-valued mapping that assigns each p ∈ F ix(T ) to itself.Consider p, x ∈ F ix(T ), with x 6= p. Then Tx = Tp and ‖x − p‖ > 0. Furthermore, d(x, T x) =

d(p, Tp) = 0 since x, p ∈ F ix(T ).Using these values, let’s rearrange the original inequality:
(
H(Tx, Tp)

)2
≤ ‖x − p‖2 + kd(x, T x)2

0 ≤ ‖x − p‖2 + kd(x, T x)2.

Since ‖x−p‖ > 0, the inequality can only hold if k = 0. However, k ∈ (0, 1) by definition, so PTcannot satisfy the inequality for any k ∈ (0, 1). Hence, PT is not a k-strictly pseudocontractive-type mapping.
A popular method for solving problem (1.8) is the well-known forward-backward splitting methodintroduced by Passty [14] and Lions and Mercier [27].The method is formulated as

xn+1 = (I − λnM)−1(I − λnA)x, λn > 0. (1.11)
under the condition that Dom(M) ⊂ Dom(A). It was known in [31], that weak convergence of(1.11) requires quite restrictive assumptions on A and Π, such that the inverse of A is stronglymonotone or Π is Lipschitz continuous and monotone and the operator (A+M) is strongly monotoneon Dom(B). Tseng in [20] and Gibali and Thong in [24] extended and improved results of G.H-G.Chen and R.T. Rockafellar [31].Most recently, Sow [28] introduced and studied a new iterative algorithm and prove convergencetheorems for variation inclusion problem (1.8) and fixed point problem involving multivalued demi-contractive and quasi-nonexpansive mappings in Hilbert spaces. They defined the sequence {ψn}

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Eur. J. Math. Anal. 10.28924/ada/ma.4.2 6as follows 

δn = JM
λn

(I − λnA)xn,

yn = θnδn + (1− θn)vn, vn ∈ Tδn ,

zn = βnyn + (1− βn)un, un ∈ T2yn,

xn+1 = PK(αnγf (xn) + (1− ηαnB)zn)

(1.12)

and prove that under certain conditions, the sequence {xn} converges strongly to a unique fixedpoint that solved the variational inequality.It is our purpose in this paper to construct a new iteration process, that modifies that of Sow [28]and prove that the corresponding sequence {xn} converges strongly to a common point of aninclusion problem and fixed point of a family of multivalued demicontractive and quasi-nonexpansivemappings in Hilbert spaces without any compactness. Our theorems generalize and extend that ofSow [28], and many other results in this directions.
2. Preliminaries

The following lemmas will play a crucial role in the sequel.Let K be a nonempty, closed convex subset of H. The nearest point projection from H to Kdenoted by PK, assigns to each ψ ∈ H the unique point of K, PKψ such that
‖x − PKx‖ ≤ ‖x − y‖,

for all y ∈ K, and for every x ∈ H,
〈x − PKx, y − PKx〉 ≤ 0, ∀y ∈ K (2.1)

Lemma 2.1. [27]. Let Π : H → 2H be a maximal monotone mapping, and Λ : H → H be Lipschitz
and continuous monotone mapping. Then (Π + Λ) : H → 2H is a maximal monotone mapping.

Lemma 2.2. [28]. Let H be real Hilbert space and Λ : H → H be an α−inverse strongly monotone
mapping. Then, (I − θΛ) is nonexpansive mapping for all ψ, π ∈ H and θ ∈ [0, 2α] such that

‖(I − θA)x − (I − θAy‖2 ≤ ‖x − y‖2 + θ(θ − 2α)‖Ax − Ay‖2 (2.2)
Lemma 2.3. [24]. Assume that {an} is a sequence of nonnegative real numbers such that an+1 =

(1 − bn)an + σn for all n ≥ 0, where {αn} is a sequence in (0, 1) and {σn} is a sequence in R
such that

i)

∞∑
n=0

bn =∞, i i) lim
n→∞

sup
σn
bn
≤ 0.

Then lim
n→∞

an = 0

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Eur. J. Math. Anal. 10.28924/ada/ma.4.2 7

Lemma 2.4. (Wang [1]). Let H be a real Hilbert space. Let K be a nonempty closed convex subset
of H. A : H → H be k− strongly monotone and L− Lipschitzian operator with k > 0 and L > 0.

Assume that 0 < η <
2k

L2
and τ = η

(
k −

L2η

2

)
. Then for each t ∈

(
0,min

(
1,

1

τ

))
, we have

‖(I − tηA)x − (I − tηA)y‖ ≤ (I − tτ)‖x − y‖, ∀x, y ∈ H (2.3)
Lemma 2.5. [29]. Let H be a real Hilbert space. Then for every x, y ∈ H, and every λ ∈ (0, 1),
the following holds:

I): ‖x − y‖2 ≤ ‖x‖2 + 2〈y , x + y〉
II: ‖λx + (1− λ)y‖2 ≤ λ‖x‖2 + (1− λ)‖y‖2 − (1− λ)λ‖x − y‖2.

3. Main Results
In this section, we study the convergence properties of the iterative algorithm which is based onviscosity algorithm and forward - backward splitting Method. We now prove the following theorem.
Theorem 3.1. Let H be a real Hilbert space and K be a nonempty, closed convex subset of
H. Let A : K → H be an α−inverse strongly monotone operator and let B : H → H be an
k−strongly monotone and L−Lipschitzian operator. Let f : K→ H be an b−Lipschitzian mapping
and M : H → 2H be a maximal monotone mapping such that the domain of M is included in K.
Let T1, T2 : K → CB(K) be a multivalued β− demicontractive mapping and T3 : K → CB(K)

be a multivalued quasi-nonexpansive mapping. Assume that 0 < η <
2k

L2
, 0 < γb < τ , where

τ = η
(
k −

L2η

2

)
, and I − T1, I − T2 and I − T3 are demiclosed at origin, such that Ω :=

F ix(T1) ∩ F ix(T2) ∩ F ix(T3) ∩ S(M,A) 6= ∅ and T1q = T2q = T3q = {q},∀q ∈ Ω. For given
x0 ∈ K, let {xn} be generated by the algorithm:

δn = JMλn(I − λnA)xn;

yn = θnδn + (1− θn)vn, vn ∈ T1δn;

zn = βnyn + (1− βn)un, un ∈ T2yn;

tn = γnzn + (1− γn)wn, wn ∈ T3zn;

xn+1 = PK(αnγf (xn) + (I − ηαnB)tn)

(3.1)

where {βn}, {γn}, {θn}, {µn}, {λn} and {αn} are real sequence in (0, 1) satisfying the following
conditions

i): lim
n→∞

αn = 0

∞∑
n=0

αn <∞, λn ∈ [a, b] ⊂ (0,min{1, 2α})

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Eur. J. Math. Anal. 10.28924/ada/ma.4.2 8

ii): lim
n→∞

inf(1− βn)(βn − β) > 0 and lim
n→∞

inf(1− θn)(θn − β) > 0, (βn, θn) ∈ (β, 1)

iii): limn→∞ inf(1− γn)γn) > 0

Then, the sequences defined in (3.1), that is {xn} and {δn} converge strongly to unique solution
x∗ ∈ Ω, which also solve the following variational inequality:

〈ηBx∗ − γf (x∗), x∗ − q〉 ≤ 0, ∀q ∈ Ω (3.2)
Proof. From the choice of η and γ from [?], (ηΦ − γψ) is strongly monotone, then the variationalinequality (3.2) has a unique solution. We will first show that there is only one solution.Lets assume that by contradiction that there exist two points x∗, y∗ ∈ Ω which are two solutionof the given inequality, and x∗ 6= y∗, then we have

〈ηBx∗ − γf (x∗), x∗ − y∗〉 ≤ 0 (3.3)
and

〈ηBy∗ − γf (y∗), y∗ − x∗〉 ≤ 0 (3.4)
Therefore from (3.3) and (3.4), we have

〈ηBy∗ − ηBx∗ + γf (x∗)− γf (y∗), y∗ − x∗〉 ≤ 0 (3.5)
Now from the assumption that

L2η

2
> 0 ⇔ α−

L2η

2
< α

⇔ η
(
α−

L2η

2

)
< αη

⇔ τ < αη

So that 0 < γ < τ < αη

〈ηBy∗ − ηBx∗ + γf (x∗)− γf (y∗), y∗ − x∗〉 = 〈ηBy∗ − ηBx∗, y∗ − x∗〉

− 〈γf (y∗)− γf (x∗), y∗ − x∗〉

= 〈ηBy∗ − ηBx∗, y∗ − x∗〉

− γ‖f (y∗)− f (x∗)‖‖y∗ − x∗‖

≥ αη‖x∗ − y∗‖2 − γρ‖x∗ − y∗‖2

= (αη − γρ)‖x∗ − y∗‖2

and this is a contradiction to (3.5), and hence x∗ = y∗, which is required. Again, we note that theoperator PK[I + (αγf − ηαB)] is a contradiction. Now for any fixed point α0 ∈ (0,min
{

1, 1
τ

}
),and ∀x, y ∈ H, we have, by lemma (2.4), and letting Φ = [I + (α0γf − ηα0B)]x and Θ =

[I + (α0γf − ηα0B)]y , we have

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Eur. J. Math. Anal. 10.28924/ada/ma.4.2 9

‖PKΦ− PKΘ‖ ≤ ‖[I + (α0γf − ηα0B)]x − ([I + (α0γf − ηα0B)]y)‖

≤ α0γ‖f (x)− f (y)‖+ ‖(I − ηα0B)x − (I − ηα0B)y‖

≤ α0γρ‖x − y‖+ (I − ατ)‖x − y‖

≤ (I − α0(τ − γρ))‖x − y‖

Thus, by Banach contraction principle, the mapping PK[I + (αγf − ηαB)] has a fixed point, say
x̂ = PK[I+ (αγf −ηαB)] and as such, from (2.1), it is similar in value to the variational inequalitybelow

〈ηBx̂ − γf (x̂), x̂ − q〉 ≤ 0,∀q ∈ Ω

Now we continue with the proof of theorem (3.1)Let q ∈ Ω with the fact that JΠ
λn is 1−inverse strongly monotone,and from [?], we have thefollowing
‖δn − q‖2 ≤ ‖xn − q‖2

Therefore from (3.6), we have
‖δn − q‖ ≤ ‖xn − q‖ (3.6)

From lemma (2.5), with (3.1), and, for the fact that T1q = {q}, T1 is β−demicontrative, we have
‖yn − q‖2 = ‖θn(δn − q) + (1− θn)(vn − q)‖2

= θn‖δn − q‖2 + (1− θn)‖vn − q‖2 − (1− θn)θn‖vn − δn‖2

≤ θn‖δn − q‖2 + (1− θn)H(T1δn, T1q)2 − (1− θn)θn‖vn − δn‖2

≤ θn‖δn − q‖2 + (1− θn)[‖δn − q‖2 + βd(δn, T1δn)2]− (1− θn)θn‖vn − δn‖2

≤ ‖δn − q‖2 − (1− θn)(θn − β)‖vn − δn‖2

(3.7)
Thus, we have

‖yn − q‖2 ≤ ‖δn − q‖2 − (1− θn)(θn − β)‖vn − δn‖2

Since θn ∈ (β, 1), we have
‖yn − q‖2 ≤ ‖δn − q‖2

Again from lemma (2.5), with (3.1), and, for the fact that T2q = {q}, T2 is β−demicontrative, wehave

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Eur. J. Math. Anal. 10.28924/ada/ma.4.2 10

‖zn − q‖2 = ‖βn(yn − q) + (1− βn)(un − q)‖2

= βn‖yn − q‖2 + (1− βn)‖un − q‖2 − (1− βn)βn‖un − yn‖2

≤ βn‖yn − q‖2 + (1− βn)H(T2yn, T2q)2 − (1− βn)βn‖un − yn‖2

≤ βn‖yn − q‖2 + (1− βn)[‖yn − q‖2 + βd(yn, T2yn)2]− (1− βn)βn‖un − yn‖2

≤ ‖yn − q‖2 − (1− βn)(βn − β)‖un − yn‖2 (3.8)
Thus, we have

‖zn − q‖2 ≤ ‖yn − q‖2 − (1− βn)(βn − β)‖un − yn‖2

Since βn ∈ (β, 1), we have
‖zn − q‖2 ≤ ‖yn − q‖2

Now, using the fact that T3q = q, we have the following estimates
‖tn − q‖ = ‖γnzn + (1− γn)wn − q‖

≤ γn‖zn − q‖+ (1− γn)‖wn − q‖

≤ γn‖zn − q‖+ (1− γn)H(T3z, T3q)

≤ γn‖zn − q‖+ (1− γn)‖zn − q‖

≤ ‖zn − q‖ (3.9)
Hence, we can see that

‖tn − q‖ ≤ ‖zn − q‖ ≤ ‖yn − q‖ ≤ ‖δn − q‖ ≤ ‖xn − q‖ (3.10)
Using (3.1), inequality (3.10) and lemma (2.4)

‖xn+1 − q‖ ≤ ‖(αnγf (xn) + (I − ηαnB)tn)− q‖

≤ ‖αnγ(f (xn)− f (q))‖+ (1− τα)‖tn − q)‖+ α‖γf (q)− ηαB‖

≤ αnγ‖f (xn)− f (q)‖+ (1− τα)‖xn − q)‖+ α‖γf (q)− ηαB‖

≤ αnbγ‖xn − q‖+ (1− τα)‖xn − q)‖+ α‖γf (q)− ηαB‖

≤ (1− α(τ − bγ))‖xn − q)‖+ α‖γf (q)− ηαB‖

≤ max
{
‖xn − q‖,

‖γf (q)− ηBq‖
τ − bγ

}
.

Therefore, by induction, it is easy to see that
‖xn+1 − q‖ ≤ max

{
‖x0 − q‖,

‖γf (q)− ηBq‖
τ − bγ

}
, ∀n ≥ 1

https://doi.org/10.28924/ada/ma.4.2


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 11Hence {xn}, {f (xn)} and {Bxn} are bounded.Secondly, we now have the following estimates. From (3.1) and lemma (2.5), we have
‖xn+1 − q‖2 ≤ ‖αn(γf (xn)− ηBq) + (I − ηαnB)(tn − q)‖2

≤ α2
n‖γf (xn)− ηBq‖2 + (1− ταn)2‖tn − q‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

≤ α2
n‖γf (xn)− ηBq‖2 + (1− ταn)2‖zn − q‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

≤ α2
n‖γf (xn)− ηBq‖2 + (1− ταn)2‖yn − q‖2

− (1− ταn)2(1− βn)(βn − β)‖un − yn‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

≤ α2
n‖γf (xn)− ηBq‖2 + (1− ταn)2‖δn − q‖2

− (1− ταn)2(1− θn)(θn − β)‖vn − δn‖2

− (1− ταn)2(1− βn)(βn − β)‖un − yn‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

≤ α2
n‖γf (xn)− ηBq‖2 + (1− ταn)2‖xn − q‖2

− (1− ταn)2(1− θn)(θn − β)‖vn − δn‖2

− (1− ταn)2(1− βn)(βn − β)‖un − yn‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖xn − q‖

≤ ‖xn − q‖2 + α2
n‖γf (xn)− ηBq‖2 − αn(2τ − τ2αn)‖xn − q‖2

− (1− ταn)2(1− θn)(θn − β)‖vn − δn‖2

− (1− ταn)2(1− βn)(βn − β)‖un − yn‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖xn − q‖

Therefore
(1− ταn)2

[
(1− θn)(θn − β)‖vn − δn‖2 + (1− βn)(βn − β)‖un − yn‖2

]
≤ ‖xn − q‖2 − ‖xn+1 − q‖2 − αn(2τ − τ2αn)‖xn − q‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖xn − q‖

+ α2
n‖γf (xn)− ηBq‖2

https://doi.org/10.28924/ada/ma.4.2


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 12Due to the boundedness of {f (xn)} and {xn}, and for some constant M > 0, we have
(1− ταn)2

[
(1− θn)(θn − β)‖vn − δn‖2 + (1− βn)(βn − β)‖un − yn‖2

]
≤ ‖xn − q‖2 − ‖xn+1 − q‖2 + αnM (3.11)

We now show that xn → x . We then consider two cases.
Case 1: Assuming that the sequence {‖xn−q‖} is monotonically decreasing. Then {‖xn−q‖}must be a convergent sequence. Therefore, we have

lim
n→∞

[‖xn − q‖2 − ‖xn+1 − q‖2] = 0, (3.12)
This implies that from (3.11), that

lim
n→∞

(1− θn)(θn − β)‖vn − δn‖2 = 0 (3.13)
and

lim
n→∞

(1− βn)(βn − β)‖un − yn‖2 = 0 (3.14)
Since lim

n→∞
inf(1 − θn)(θn − β) > 0 and lim

n→∞
inf(1 − βn)(βn − β) > 0, with the fact that

vn ∈ T1δn and un ∈ T2yn, it follows that
lim
n→∞

d(δn, T1δn) = 0 (3.15)
and

lim
n→∞

d(yn, T2yn) = 0 (3.16)
observing that

‖yn − δn‖ = ‖θnδn + (1− θn)vn − δn‖

= ‖θnδn + (1− θn)vn − δn + θnδn − θnδn‖

= (1− θn)‖vn − δn‖

≤ ‖vn − δn‖ (3.17)
Taking the limits and from (3.13), we can see that

lim
n→∞

‖yn − δn‖ = 0

https://doi.org/10.28924/ada/ma.4.2


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 13

‖zn − yn‖ = ‖βnyn + (1− βn)un − yn‖

= ‖βnyn + (1− βn)un − yn + βnyn − βnyn‖

= (1− βn)‖un − yn‖

≤ ‖un − yn‖ (3.18)
Again, from (3.14), we can see that

lim
n→∞

‖zn − yn‖ = 0

‖zn − δn‖ = ‖zn − yn + yn − δn‖

≤ ‖zn − yn‖+ ‖yn − δn‖

Hence lim
n→∞

‖zn − δn‖ = 0Now from lemma (2.2), lemma (2.4) and (3.1), we have the following
‖xn+1 − q‖2 ≤ ‖αn(γf (xn)− ηBq) + (I − ηαnB)(tn − q)‖2

≤ α2
n‖γf (xn)− ηB‖2 + (1− τα)2‖tn − q‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

≤ α2
n‖γf (xn)− ηB‖2 + (1− ταn)2‖δn − q‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

≤ α2
n‖γf (xn)− ηB‖2 + (1− ταn)2‖JMλn(1− λA)xn − JMλn(1− λA)q‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

≤ α2
n‖γf (xn)− ηB‖2 + (1− ταn)2

[
‖xn − q‖2 + a(b − 2α)‖Axn − Aq‖2

]
+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖xn − q‖

≤ α2
n‖γf (xn)− ηB‖2 + ‖xn − q‖2 − αn(2τ − τ2αn)‖xn − q‖2

− (1− τα)2a(2α− b)‖Axn − Aq‖2 (3.19)
+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖xn − q‖

Therefore, from (3.19),and with a constant D > 0, we have
(1− τα)2a(2α− b)‖Axn − Aq‖2 ≤ ‖xn − q)‖2 − ‖xn+1 − q)‖2 + αnD

https://doi.org/10.28924/ada/ma.4.2


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 14Since, lim
n→∞

αn = 0, and from the inequality (3.12), with the fact that {xn} is bounded, wehave
lim
n→∞

‖Axn − Aq‖2 = 0 (3.20)
Since JMλn is 1−inverse strongly monotone,and ‖tn−q‖ ≤ ‖δn−q‖, we have the following

‖tn − q‖2 = ‖J(M)
λn

(I − λnA)xn − JMλn(I − λnA)q‖2

≤ 〈tn − q, (I − λnA)xn − (I − λnA)q〉

=
1

2

[
‖(I − λnA)xn − (I − λnA)q‖2

+ ‖tn − q‖2 − ‖(I − λnA)xn − (I − λnA)q − (tn − q)‖2
]

≤
1

2

[
‖xn − q‖2 + ‖tn − q‖2 − ‖xn − tn‖2

+ 2λn〈tn − q,Axn − Aq〉 − λ2
n‖Axn − A)q)‖2

]
≤ ‖xn − q‖2 − ‖xn − tn‖2 + 2λn〈xn − q,Axn − Aq〉 − λ2

n‖Axn − Aq‖2

This gives us
‖tn − q‖2 ≤ ‖xn − q‖2 − ‖xn − tn‖2 + 2λn〈tn − q,Axn − Aq〉 − λ2

n‖Axn − Aq‖2 (3.21)
Therefore
‖xn+1 − q‖2 ≤ ‖αn(γf (xn)− ηBq) + (I − ηαnB)(tn − q)‖2

≤ α2
n‖γf (xn)− ηB‖2 + (1− ταn)2‖tn − q‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

≤ α2
n‖γf (xn)− ηB‖2 + (1− ταn)2

[
‖xn − q‖2 − ‖xn − tn‖2

+ 2λn〈tn − q,Axn − Aq〉 − λ2
n‖Axn − Aq‖2

]
+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖xn − q‖

≤ α2
n‖γf (xn)− ηB‖2 + (1− ταn)2‖xn − q‖2 − (1− ταn)2‖xn − tn‖2

+ 2λn(1− ταn)2〈tn − q,Axn − Aq〉 − λ2
n(1− ταn)2‖Axn − Aq‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖xn − q‖ (3.22)

https://doi.org/10.28924/ada/ma.4.2


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 15Thus, from (3.25), we have
(1− ταn)2‖xn − tn‖2 ≤ α2

n‖γf (xn)− ηB‖2 + ‖xn − q‖2 − ‖xn+1 − q‖2 (3.23)
− αnτ(2− ταn)‖xn − q‖2 + 2λn(1− ταn)2〈tn − q,Axn − Aq〉

− λ2
n(1− ταn)2‖Axn − Aq‖2 + 2αn(1− ταn)‖γf (xn)− ηBq‖‖xn − q‖

Therefor, since αn → 0 as n →∞ with inequalities (3.12) and (3.20), we have
lim
n→∞

‖xn − tn‖ = 0

From (3.10) and lemma (2.5) with the fact that T3 is quasi-nonexpansive, we have thefollowing estimate
‖tn − q‖2 = ‖γnzn + (1− γn)wn − q‖2

= γn‖zn − q‖2 + (1− γn)‖wn − q‖2 − (1− γn)γn‖wn − zn‖2

= γn‖zn − q‖2 + (1− γn)H(T3zn, T3q)2 − (1− γn)γn‖wn − zn‖2

= γn‖zn − q‖2 + (1− γn)‖zn − q‖2 − (1− γn)γn‖wn − zn‖2

≤ ‖xn − q‖2 − (1− γn)γn‖wn − zn‖2 (3.24)
Therefore

‖xn+1 − q‖2 ≤ ‖αn(γf (xn)− ηBq) + (I − ηαnB)(tn − q)‖2

≤ α2
n‖γf (xn)− ηB‖2 + (1− ταn)2‖tn − q‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

≤ α2
n‖γf (xn)− ηB‖2 + (1− ταn)2

[
‖xn − q‖2

− (1− γn)γn‖wn − zn‖2
]

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

≤ α2
n‖γf (xn)− ηB‖2 + (1− ταn)2‖xn − q‖2

− (1− ταn)2(1− γn)γn‖wn − zn‖2 + 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

Hence, we have the following
(1− ταn)2(1− γn)γn‖wn − zn‖2 ≤ α2

n‖γf (xn)− ηB‖2 − ταn(2− ταn)‖xn − q‖2

+ ‖xn − q‖2 − ‖xn+1 − q‖2

+ 2αn(1− ταn)‖γf (xn)− ηBq‖‖tn − q‖

https://doi.org/10.28924/ada/ma.4.2


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 16Therefor, since αn → 0 as n →∞ with inequalities (3.12) and (3.20), we have
(1− ταn)2(1− γn)γn‖wn − zn‖2 ≤ 0

From this we have
lim
n→∞

(1− γn)γn‖wn − zn‖2 = 0 (3.25)
Since lim

n→∞
inf((1− γn)γn) > 0

lim
n→∞

‖wn − zn‖ = 0 (3.26)
Again, with w ∈ T3zn

lim
n→∞

d(zn, T3zn) = 0 (3.27)
Moreover, since H is reflexive and {xn} is bounded, we then prove that lim

n→+∞
sup〈ηBx∗ −

γf (x∗), x∗ − xn〉 ≤ 0. We let the subsequence {xni} of {xn} to converge weakly to x∗∗ in
K, and

lim
n→+∞

〈ηBx∗ − γf (x∗), x∗ − xn〉 = lim
n→+∞

〈ηBx∗ − γf (x∗), x∗ − xni 〉

Again, since I−T1, I−T2 and I−T3 satisfies the demiclosed principle and from (3.32), (3.16)and (3.27), we obtain x∗∗ ∈ F ix(T1)∩F ix(T2)∩F ix(T3). We now show that x∗∗ ∈ S(M,A).Since A is α−inverse strongly monotone, and Lipschitz continuous mapping. Then Fromlemma (2.1), it follows that (M+ A) is maximal monotone.Let (ν, g) ∈ G(M + A), that is g − Aν ∈ M(ν). Since δni = JMλni
(xni ) − λniAxni ), wehave xni − λni xni ∈ (I + λniM)δni , that is 1

λni
(xni − δni − λniAxni ) ∈ M(δni ). By maximalmonotonocity of (M+ A), gives

〈ν − δni , g − Aν −
1

λni
(xni − δni − λniAxni ) ≥ 0

and, therefore
〈ν − δni , g〉 ≥ 〈ν − δni ,Aν −

1

λni
(xni − δni − λniAxni )〉

= 〈ν − δni ,Aν − Aδni + Aδni +
1

λni
(xni − δni − λniAxni )〉

≥ 〈ν − δni ,Aν − Axni 〉+ 〈ν − δni ,
1

λni
(xni − δni 〉

It then follows from ‖δn − xn‖ → 0, ‖Aδn − Axn‖ → 0 and δni → x∗∗ weakly that
lim
n→∞
〈ν − δni , g〉 = 〈ν − x∗∗, g〉 and hence x∗∗ ∈ S(Π,A). Therefore, x∗∗ ∈ Ω,On the other hand, for the fact that x∗ solves the variational inequality (3.30).

https://doi.org/10.28924/ada/ma.4.2


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 17

lim
n→+∞

sup〈ηBx∗ − γf (x∗), x∗ − xn〉 = lim
n→+∞

sup〈ηBx∗ − γf (x∗), x∗ − xni 〉

= 〈ηBx∗ − γf (x∗), x∗ − x∗∗〉 ≤ 0 (3.28)
Lastly, we now prove that lim

n→∞
‖xn − x∗‖ = 0, that is xn → x∗ as n →∞.

‖xn+1 − x∗‖2 ≤ ‖αnγf (xn) + (I − ηαnB)tn − x∗‖2

≤ ‖αn(γf (xn)− γf (x∗)) + (I − ηαnB)tn − x∗‖2

+ 2αn〈ηBx∗ − γf (x∗), x∗ − xn+1〉

≤
[
αnγ‖f (xn)− f (x∗)‖+ ‖(I − ηαnB)(tn − x∗)‖

]2

+ 2αn〈ηBx∗ − γf (x∗), x∗ − xn+1〉

≤
[
αnγb‖xn − x∗‖+ (1− ταn)‖xn − x∗‖

]2

+ 2αn〈ηBx∗ − γf (x∗), x∗ − xn+1〉

≤
[

1− αn(τ − γb)
]2
‖xn − x∗‖2 + 2αn〈ηBx∗ − γf (x∗), x∗ − xn+1〉

≤
[

1− αn(τ − γb)
]
‖xn − x∗‖2 + 2αn〈ηBx∗ − γf (x∗), x∗ − xn+1〉

Thus, from lemma (2.3), it follows that ψn → ψ∗ as n →∞, where bn = αn(τ −
γb), an = ‖xn − x∗‖2 and σn = 2αn〈ηBx∗ − γf (x∗), x∗ − xn+1〉

Case 2: Suppose that the sequence {‖xn − x∗‖} is monotonically increasing. Set Wn :=

‖xn − x∗‖2 and τ := N→ N be a mapping for all n ≥ n0 (for some n0 sufficient large), by
τn := max{k ∈ N : k ≤ n,Wk ≤Wk+1}. Then, τ is a nondecreasing sequence, such that
τn →∞ as n →∞ and Wτ(n) ≤Wτ(n)+1} for all n ≥ n0. Now, from (3.11), we have

(1− ατ(n)τ)
[

(1− θn)(θn − β)‖vτ(n) − δτ(n)‖2 + (1− βn)(βn − β)‖uτ(n) − yτ(n)‖2
]

≤ 2ατ(n)M (3.29)
lim

n→+∞
τ(1−ατ(n))

[
(1−θτ(n))(θτ(n)−β)‖vτ(n)−δτ(n)‖2+(1−βτ(n))(βτ(n)−β)‖uτ(n)−yτ(n)‖2

]
= 0

Since (βτ(n), θτ(n)) ∈ (β, 1) and lim
n→∞

inf γτ(n)(1− γτ(n)) > 0, we have
lim
n→∞

‖uτ(n) − yτ(n)‖ = 0 and lim
n→∞

‖vτ(n) − δτ(n)‖ = 0

With vτ(n) ∈ T1δτ(n) and uτ(n) ∈ T2yτ(n), it follows that
lim
n→∞

d
(
δτ(n), T1δτ(n)

)
= 0 and lim

n→∞
d
(
yτ(n), T2yτ(n)

)
= 0

https://doi.org/10.28924/ada/ma.4.2


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 18Following the same argument in case 1, we conclude that
lim

τ(n)→+∞
sup〈ηBx∗ − γf (x∗), x∗ − xτ(n)+1〉 ≤ 0

Therefore, for all n ≥ n0,and from 3.29, we have
0 ≤ ‖xτ(n)+1 − x∗‖2 − ‖xτ(n) − x∗‖2 ≤ ατ(n)[−αn(τ − γb)‖xτ(n) − x∗)‖2

+ 2ατ(n)〈ηBx∗ − γf (x∗), xτ(n)+1 − x∗〉

‖xτ(n) − x∗‖2 ≤
2

τ − γb 〈ηBx
∗ − γf (x∗), xτ(n)+1 − x∗〉

Then we have lim
n→∞

‖xτ(n) − x∗‖2 = 0. Therefore lim
n→∞

Wτ(n) = lim
n→∞

Wτ(n)+1 = 0.Furthermore, for all n ≥ n0, we have Wτ(n) ≤ Wτ(n)+1 if n 6= τ(n) (that is n > τ(n),because Wj >Wj+1, f or τ(n + 1) ≤ j ≤ n.Hence, 0 ≤Wτ(n) ≤ max
{
Wτ(n),Wτ(n)+1

}
=Wτ(n)+1. Therefore,Wn → 0, as n →

∞ and this implies that xn → x∗ as n →∞. This complete the proof.
�

Now using theorem (3.1), and multivalued mappings are nonexpansive mappings with convexvalues without demiclosed assumptions in the following theorem.
Theorem 3.2. Let H be a real Hilbert space and K be a nonempty, closed convex subset of H. Let
A : K → H be an α−inverse strongly monotone operator and let B : H → H be an k−strongly
monotone and L−Lipschitzian operator. Let f : K → H be an b−Lipschitzian mapping and
M : H → 2H be a maximal monotone mapping such that the domain of M is included in K. Let
T1, T2 : K → CB(K) be a multivalued β− demicontractive mapping and T3 : K → CB(K) be a
multivalued quasi-nonexpansive mapping such that Ω := F ix(T1)∩F ix(T2)∩F ix(T3)∩S(Π,Λ) 6= ∅
and T1q = T2q = T3q = {q},∀q ∈ Ω. For given x0 ∈ K, let {xn} be generated by the algorithm:

δn = JMλn(I − λnA)xn;

yn = θnδn + (1− θn)vn, vn ∈ T1δn;

zn = βnyn + (1− βn)un, un ∈ T2yn;

tn = γnzn + (1− γn)wn, wn ∈ T3zn;

xn+1 = PK(αnγf (xn) + (I − ηαnB)tn)

(3.30)

where {βn}, {γn}, {θn}, {λn} and {αn} are real sequence in (0, 1) satisfying the following condi-
tions

https://doi.org/10.28924/ada/ma.4.2


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 19

i): lim
n→∞

αn = 0

∞∑
n=0

αn <∞

ii): lim
n→∞

inf(1− βn)(βn − β) > 0 and lim
n→∞

inf(1− θn)(θn − β) > 0, (βn, θn) ∈ (β, 1)

iii): limn→∞ inf(1− γn)γn) > 0

Assume that 0 < η <
2k

L2
, 0 < γb < τ , where τ = η

(
k −

L2η

2

)
, and the sequences defined in

(3.30), that is {xn} and {δn} converge strongly to unique solution x∗ ∈ Ω, which also solve the
following variational inequality:

〈ηBx∗ − γf (x∗), x∗ − q〉 ≤ 0, ∀q ∈ Ω (3.31)
Proof. Since every multivalued nonexpansive mapping is quasi-nonexpansive and demicontractive,then, the proof follows Theorem 3.10

�

Now using the same argument of the proof in theorem (3.1) in theorem (3.3), we achieved thedesired results. In theorem (3.3), we let T1 = PT1
, T2 = PT2

and T3 = PT3
without the assumptionsthat T1q = T2q = T3q = {q},∀q ∈ Ω

Theorem 3.3. Let H be a real Hilbert space and K be a nonempty, closed convex subset of H. Let
A : K → H be an α−inverse strongly monotone operator and let B : H → H be an k−strongly
monotone and L−Lipschitzian operator. Let f : K → H be an b−Lipschitzian mapping and
M : H → 2H be a maximal monotone mapping such that the domain of M is included in K. Let
T1, T2 : K → CB(K) be a multivalued β− demicontractive mapping and T3 : K → CB(K) be a
multivalued quasi-nonexpansive mapping such that Ω := F ix(T1)∩F ix(T2)∩F ix(T3)∩S(M,A) 6=
∅. For given x0 ∈ K, let {xn} be generated by the algorithm:

δn = JMλn(I − λnA)xn;

yn = θnδn + (1− θn)vn, vn ∈ T1δn;

zn = βnyn + (1− βn)un, un ∈ T2yn;

tn = γnzn + (1− γn)wn, wn ∈ T3zn;

xn+1 = PK(αnγf (xn) + (I − ηαnB)tn)

(3.32)

where {βn}, {γn}, {θn}, {λn} and {αn} are real sequence in (0, 1) satisfying the following condi-
tions

i): lim
n→∞

αn = 0

∞∑
n=0

αn <∞

https://doi.org/10.28924/ada/ma.4.2


Eur. J. Math. Anal. 10.28924/ada/ma.4.2 20

ii): lim
n→∞

inf(1− βn)(βn − β) > 0 and lim
n→∞

inf(1− θn)(θn − β) > 0, (βn, θn) ∈ (β, 1)

iii): limn→∞ inf(1− γn)γn) > 0

Assume that 0 < η <
2k

L2
, 0 < γb < τ , where τ = η

(
k−

L2η

2

)
, and I−PT1

, I−PT2
and I−PT3

are demiclosed at origin. Hence, the sequences defined in (3.32), that is {xn} and {δn} converge
strongly to unique solution x∗ ∈ Ω, which also solve the following variational inequality:

〈ηBx∗ − γf (x∗), x∗ − q〉 ≤ 0, ∀q ∈ Ω (3.33)
4. Conclusion

The modified general viscosity iterative process presented in this research offers a powerful toolfor solving variational inclusion and fixed point problems involving and and fixed point problem withrespectively set-valued maximal monotone mapping and inverse strongly monotone and multivaluedquasi-nonexpansive and demicontractive operators. Our Theorem presents a new and a modifiedalgorithm for solving simultaneously variational inclusion problem and fixed point problem withrespectively set-valued maximal monotone mapping and inverse strongly monotone and multival-ued demicontractive and quasi-nonexpansive mappings. The result we show here improves andextends the corresponding results of some authors and many other recent results using forward-backward splitting method and general iterative algorithm that gives a strong convergence to aunique solution.
References

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https://doi.org/10.22075/ijnaa.2023.27501.3626
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https://doi.org/10.28924/ada/ma.4.2

	1. Introduction
	2. Preliminaries
	3. Main Results
	4. Conclusion
	References

