Communications on Applied Nonlinear Analysis 

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Convergence Analysis of 𝐒-iteration Process of Generalized Nonlinear 

Variational Inclusion Problem 

 

𝐒𝐲𝐞𝐝 𝐒𝐡𝐚𝐤𝐚𝐢𝐛𝐈𝐫𝐟𝐚𝐧∗, 𝐈𝐪𝐛𝐚𝐥 𝐀𝐡𝐦𝐚𝐝∗∗, 𝐌𝐨𝐧𝐢𝐫𝐮𝐥 𝐈𝐬𝐥𝐚𝐦∗, 𝐌𝐨𝐡𝐝.  𝐅𝐚𝐥𝐚𝐡𝐚𝐭 𝐊𝐡𝐚𝐧∗, 

𝐌𝐝.𝐇𝐢𝐟𝐳𝐮𝐫 𝐑𝐚𝐡𝐚𝐦𝐚𝐧∗ 

* Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, U.P., India  

 **Department of Mechanical Engineering, College of Engineering, Qassim University, Buraidah, Al-Qassim, Saudi 

Arabia  

 E-mail: ssirfan.mm@amu.ac.in, i.ahmad@qu.edu.sa , monirul.amu@gmail.com, gi3635@myamu.ac.in, 

mdhifzurrahaman98@gmail.com 

 

Article History: 

Received: 28-10-2024 

Revised:28-11-2024 

Accepted:28-12-2024 

Abstract:  

To obtain the solution of generalized variational inclusion involving A(. , . ) co-coercive 

operators, a  proposal for E-iteration has also been proposed and analyzed. Existence 

theorems for the solution of generalized variational inclusion are proved by using co-

coercive and relaxed co-coercive mappings. Also, certain particular cases, along with 

their comparison with some methods, have been studied. Finally, we present a  numerical 

example to exemplify and show the convergence of the suggested algorithm in support 

of our main result, which has been formulated by using MATLAB programming. 

Keywords: Algorithm, S -iterative process, A(. , . ) -co-coercive operator, Resolvent 

operator, Sequence analysis 

 

1. Introduction 

Variational inclusions represent an extended category of problems beyond variational inequalities, 

and they hold a significant and elegant position in the fields of optimization and nonlinear analysis. 

Variational inclusions/inequalities involve applications in different fields like mechanics, physics, 

non-linear programming, optimization, and control theory. For details, see [1, 4–11, 13–15, 17–19] 

and the references therein. To solve variational inclusion many iterative techniques have been 

developed; See for example, [6,8,10,12,15,16]. 

 In 2016, Buong et al. [6] proposed an explicit iterative algorithm to find out the solution for 

variational inequalities with a uniformly Gâteaux differentiable norm. To make a clear 

understanding, some examples have been illustrated. In 2017, Sahu et al. [15] proposed a system of 

generalized variational inequalities. In their research, they introduced two parallel iterative methods, 

namely the parallel S-iteration process and the parallel Mann iteration process, to address a particular 

problem. They also examined the convergence of the sequences produced by these parallel iteration 

methods using a numerical example. Their analysis demonstrated that the recommended parallel S-

iteration process outperforms the parallel Mann iteration process. Later Ha et al. [10] suggested a 

simple parallel iterative method in finding out the solution to variational inequalities. It has been 

claimed [10] that the parallel iterative method is more straightforward the one proposed by Buong et 

al. [6].  



Communications on Applied Nonlinear Analysis 

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In addition to this, numerical examples have been [10] to illustrate the effectiveness and superiority 

of the proposed algorithm. Recently Gursoy et al. [9] proposed and analyzed an S-iteration process 

for solving a class of variational inclusion H-monotone operator. A comparison of the suggested 

method has been performed along with some existing methods considered by Fang and Huang [7] 

and Zeng et al. [19]. 

 Motivated by ongoing research in this direction, we have designed a S-iteration for finding the 

solution of generalized variational inclusion problem. Also, existence theorems are proved by using 

cocoercive and relaxed co-coercive mappings. A numerical example has been presented as well to 

illustrate convergence results. 

2. Preliminaries 

We represent the sets of nonnegative real numbers and nonnegative integers as R+ and N0 

respectively. Consider a real Hilbert space denoted as X , where its inner product and norm are 

symbolized as and ∥. ∥ respectively. 

Let S,T, g:ℋ → ℋ  be three single-valued functions and N:ℋ → 2ℋ  be a multi-valued function. 

Consider the generalized variational inclusion problem (GVIP): for some real number ρ and find w ∈

ℋ such as 

     ρ ∈ S(x) − T(x) + τN(g(x)).              (2.1) 

Some exceptional cases of (2.1) are as follows: 

a) If ρ = 0, τ = 1, S = 0 and N is a single-valued function, then (2.1) becomes the problem of 

finding w ∈ ℋ such as  

                                                           0 ∈ N(g(w))− T(w).              (2.2)    

Problem (2.2) was proposed by Noor et al. [14]. 

b) If ρ = 0, τ = 1,T = 0 and g = I (identity function), then (2.1) becomes the problem of 

finding w ∈ ℋ such as 

        0 ∈ 𝑆(𝑤) +𝑁(𝑤).                               (2.3) 

      Problem (2.3) was considered by Fang and Huang [7]. 

It's evident that by appropriately selecting the functions used in equation (2.1), one can identify 

numerous variational inclusion or inequality problems that have been investigated in recent studies, 

as observed in references such as [5, 11, 13]. 

Now, we provide certain definitions and outcomes to reach the primary conclusion of this paper. 

Definition 2.1 ([2,15]) Consider a mapping P:ℋ → ℋ that takes one value at a time. A mapping 

R:ℋ→ ℋ is termed 

a) monotone (in short MT) if   ⟨Rw−Ry,w− y⟩ ≥ 0, ∀w,y ∈ ℋ, 

b) strictly MT if R is MT and 

⟨Rw− Ry,w − y⟩ = 0, ifand only if w= y, 



Communications on Applied Nonlinear Analysis 

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c) strongly MT if there exists r > 0 such as 

⟨Rw− Ry,w − y⟩ ≥ r‖w − y‖2 ,∀w,y ∈ ℋ, 

d) strongly MT with respect to P if there exists γ > 0 such as 

⟨Rw−Ry, Pw− Py⟩ ≥ γ‖w− y‖2 , ∀w,y ∈ ℋ, 

e) Lipschitz continuous if there exists λR > 0 such as 

‖Rw− Ry‖ ≤ λR
‖w− y‖,∀w, y ∈ ℋ, 

f) α-expansive if there exists α > 0 such as 

‖Rw− Ry‖ ≥ α‖w− y‖, ∀w,y ∈ ℋ, 

if α = 1, then it is expansive. 

g) co-coercive if there exists μ′ > 0 such as 

⟨Rw− Ry,w− y⟩ ≥ μ′‖Rw−Ry‖2 , ∀w,y ∈ ℋ, 

h) relaxed co-coercive if there exists a constant γ′ > 0 such as 

 ⟨Rw− Ry,w − y⟩ ≥ (−γ′)‖Rw− Ry‖2 ,∀w, y ∈ ℋ. 

Definition 2.2 ([2]) A set-valued function 𝑁:ℋ → 2ℋ is termed: 

a)  𝑀𝑇 if 

⟨𝑤 − 𝑦,𝑢 − 𝑣⟩ ≥ 0, ∀𝑢, 𝑣 ∈ ℋ, 𝑤 ∈ 𝑁𝑢, 𝑦 ∈ 𝑁𝑣, 

b) strongly 𝑀𝑇 if there exists 𝜂 > 0 such as 

⟨𝑤 − 𝑦,𝑢 − 𝑣⟩ ≥ 𝜂‖𝑢 − 𝑣‖2 ,∀𝑢, 𝑣 ∈ ℋ, 𝑤 ∈ 𝑁𝑢, 𝑦 ∈ 𝑁𝑣, 

c)  maximal 𝑀𝑇 if 𝑁 is 𝑀𝑇 and (𝐼 + 𝜆𝑁)(ℋ) = ℋ hold for all 𝜆 > 0, where 𝐼 stands the 

identity function on ℋ; 

d)  maximal strongly 𝑀𝑇 if 𝑁 is strongly MT and (𝐼 + 𝜆𝑁)(ℋ) = ℋ hold for all 𝜆 > 0; 

e) cocoercive if there exists  𝜇′′  such as 

⟨w− y, u − v⟩ ≥ μ′′‖u − v‖2, ∀u, v ∈ ℋ,w ∈ Nu, y ∈ Ny. 

Definition 𝟐. 𝟑([𝟐,𝟑]) Let A:ℋ ×ℋ → ℋ and P, R:ℋ → ℋ are the the functions. 

a) A(P,.) is termed co-coercive with respect to 𝑃 if there exists μ > 0 such as 

⟨A(Pw, u) − A(Py, u), w − y⟩ ≥ μ2‖Pw − Py‖2 ,∀w, y ∈ ℋ. 

b) A(. , R) is termed relaxed co-coercive with respect to R if there exists μ > 0 such as 

⟨A(u, Rw) − A(u,Ry), w − y⟩ ≥ μ2‖Rw − Ry‖2, ∀w, y ∈ ℋ. 

c) A(P,.) is termed 𝑟1 -Lipschitz continuous with respect to P if there exists r1 > 0 such as 

‖A(Pw, . ) − A(Py, . )‖ ≤ t1‖w− y‖,∀w, y ∈ ℋ. 



Communications on Applied Nonlinear Analysis 

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d)  A(. , R) is termed r2-Lipschitz continuous with respect to R if there exists r2 > 0 such as 

‖A(. , Rw) − A(. , Ry)‖ ≤ t2‖w− y‖,∀w, y ∈ ℋ. 

Definition 2.4 ([2]) Let function A:ℋ ×ℋ → ℋ and P, R:ℋ → ℋ are the singlevalued functions. 

Let N:ℋ → 2ℋ  be a multi-valued function. N is termed A(.,.) − coocercive with respect to the 

functions P and R (or simply A(.,.) − cocoerciveinthesequel) if N is cocoercive with respect to P 

and R and [A(P,R) + λN](ℋ) = ℋ, for every λ > 0. 

Definition 2.5 ([2]) Let A(P, R) be μ-cocoercive with respect to P and γ-relaxed cocoercive with 

respect to R, P is α-expansive, R-Lipschitz continuous, and μ > γ, α > β. Let N be an A(.,.) −

cocoerciveoperatorwithrespectP  and R. The resolvent operator J
λ,N

A(…)
:ℋ → ℋ is defined by 

Jλ,N
A(….)(w) = [A(P,R) + λN]−1(w),∀w ∈ ℋ, λ > 0. (2.4) 

Lemma 2.1 ([2]). Let A(P, Q) be μ-cocoercive with respect to P, γ-relaxed coocoercive with respect 

to R, P is α-expansive, R is β-Lipschitz continuous, and μ > γ, α > β. Let N be an A(.,.) −

cocoerciveoperatorwithtoP  and R. Then the resolvent operator J
λ,N

A(….)
:ℋ → ℋ is 

1

μα2−γβ2
-Lipschitz 

continuous, that is 

‖J
λ,N

A(…)
(w)− J

λ,N

A(…)
(y)‖ ≤

1

μα2 − γβ2
‖w − y‖, ∀w, y ∈ ℋ (2.5) 

3. 𝑺-iteration Algorithms and Convergence Analysis 

The under mentioned lemma ensures the equivalence between fixed point problem and (2.1). This 

serves as the inspiration for the upcoming outcome we will present. 

Lemma 3.1. Let 𝐴:ℋ ×ℋ → ℋ and 𝑃, 𝑅, 𝑆, 𝑇, 𝑔:ℋ →ℋ are single-valued functions with 𝑔(ℋ) ∩

𝑑𝑜𝑚(𝑃) ≠ ∅ and 𝑔(ℋ) ∩ 𝑑𝑜𝑚(𝑅) ≠ ∅, and 𝑁:ℋ → 2ℋ be a multi-valued function such as 𝐴(. , . )  

co-coercive with respect to  and 𝑃, 𝑅 and 𝑔 . Then 𝑤 ∈ ℋ is a solution of (2.1) if and only if 

  

                                   𝑔(𝑤) = 𝐽𝜆,𝑁
𝐴(.,.)[𝐴(𝑃𝑜𝑔(𝑤),𝑅𝑜𝑔(𝑤))− 𝜆(𝑆(𝑤) − 𝑇(𝑤))+ 𝜆𝜌]              (3.1)

 where 𝜆 > 0. 

Algorithm 3.1. The iterative sequence {𝑤𝑛} for all 𝑛 ∈ 𝑁0 is s stated as 

{
 

 
𝑤0 ∈ ℋ

𝑤𝑛+1 = (1− 𝛼𝑛)𝑤𝑛 +𝛼𝑛[𝑤𝑛 −𝑔(𝑤𝑛) + 𝐽𝜆 ,𝑁
𝐴(….)

[𝐴(𝑃𝑜𝑔(𝑤𝑛),𝑅𝑜𝑔(𝑤𝑛))

−𝜆(𝑆(𝑤𝑛) − 𝑇(𝑤𝑛)) + 𝜆𝜌]]

                   (3.2) 

where {𝛼𝑛} is a sequence in [0,1] satisfying the condition∑𝑛=0
∞ 𝛼𝑛 = ∞ . 

 

 

 



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Algorithm 3.2. The iterative sequence {𝑞𝑛} for all 𝑛 ∈ 𝑁0 is s stated as 

{
 
 
 

 
 
 

𝑞0 ∈ ℋ

𝑞𝑛+1 = (1− 𝜉𝑛)𝑞𝑛 + 𝜉𝑛[𝑟𝑛 −𝑔(𝑟𝑛) + 𝐽𝜆 ,𝑁
𝐴(….)[𝐴(𝑃𝑜𝑔(𝑟𝑛),𝑅𝑜𝑔(𝑟𝑛))

−𝜆(𝑆(𝑟𝑛) − 𝑇(𝑟𝑛)) + 𝜆𝜌]]

𝑟𝑛+1 = (1−𝜇𝑛)𝑞𝑛 + 𝜇𝑛[𝑞𝑛 −𝑔(𝑞𝑛) + 𝐽𝜆 ,𝑁
𝐴(….)[𝐴(𝑃𝑜𝑔(𝑞𝑛),𝑅𝑜𝑔(𝑞𝑛))

−𝜆(𝑆(𝑞𝑛)− 𝑇(𝑞𝑛))+ 𝜆𝜌]]

                                    (3.3) 

where {𝜉𝑛} and {𝜇𝑛} are sequences in [0,1] satisfying the condition ∑ 𝜉𝑛
∞
𝑛=0 = ∞. 

Theorem 3.1. Let ℋ be a real Hilbert space and 𝐴:ℋ ×ℋ → ℋ and 𝑃, 𝑅, 𝑆, 𝑇, 𝑔:ℋ → ℋ are single-

valued functions and 𝑁:ℋ → 2ℋ be a multi-valued function such as 𝐴(. , . ) co- MT with respect to 

𝑃, 𝑅, 𝑔 operator. Assume that 𝐴(. , . ) is Lipschitz continuous  with constant 𝑡 > 0, mixed strongly 

MT with respect to 𝑃 and 𝑅 with constant 𝛿 > 0, 𝑔 is strongly MT with constant 𝛿𝑔 > 0 and 

𝑔, 𝑃, 𝑅, 𝑆, 𝑇 are Lipschitz continuous with constants 𝜆𝑔, 𝜆𝑃,𝜆𝑅, 𝜆𝑆 and 𝜆𝑇 respectively. Let {𝑤𝑛} be a 

iterative sequences generated by (3.1) with the sequence {𝛼𝑛} ⊂ [0,1] and satisfying the condition 

∑ 𝛼𝑛
∞
𝑛=0 = ∞, and there exists a constant 𝜆 > 0 such as 

{
(𝜇𝛼2 − 𝛾𝛽2)2(1− 2𝛿𝑔 + 𝜆𝑔

2 ) < [𝜇𝛼2 − 𝛾𝛽2 − 𝑡1𝜆𝑃𝜆𝑔 − 𝑡2𝜆𝑅𝜆𝑔 − 𝜆(𝜆𝑆 + 𝜆𝑇)]
2
,

𝜇 > 𝛾 𝑎𝑛𝑑 𝛼 > 𝛽.
                  (3.4) 

Then, the following statements hold: 

a) There exists 𝜆 > 0 such as 

𝜅 = √1− 2𝛿𝑔 + 𝜆𝑔
2 +

𝑡1𝜆𝑃𝜆𝑔+𝑡2𝜆𝑅𝜆𝑔+𝜆𝜆𝑆+𝜆𝜆𝑇

𝜇𝛼2−𝛾𝛽2 < 1.                                                             (3.5) 

b) The operator 𝐹:ℋ → ℋ defined by 

𝐹(𝑤) = 𝑤 −𝑔(𝑤) + 𝐽𝜆 ,𝑁
𝐴(….)

[𝐴(𝑃𝑜𝑔(𝑤),𝑅𝑜𝑔(𝑤)) − 𝜆(𝑆(𝑤) − 𝑇(𝑤))+ 𝜆𝜌], ∀𝑤 ∈ 𝑋   (3.6) 

is 𝜅-contraction, that is 

‖𝐹(𝑤) − 𝐹(𝑦)‖ ≤ 𝜅‖𝑤− 𝑦‖,∀𝑤, 𝑦 ∈ ℋ.                                                                       (3.7) 

where  𝜅 satisfies (3.5). 

c) The iterative sequence {𝑤𝑛} stated as (3.1) converges strongly to a unique solution 𝑤∗ ∈ ℋ 

of (2.1). 

{
 

 
𝑦0 ∈ ℋ

𝑦𝑛+1 = (1 − 𝜉𝑛)𝑦𝑛 + 𝜉𝑛[𝑦𝑛 −𝑔(𝑦𝑛) + 𝐽𝜆 ,𝑁
𝐴(….)

[𝐴(𝑃𝑜𝑔(𝑦𝑛),𝑅𝑜𝑔(𝑦𝑛))

−𝜆(𝑆(𝑦𝑛)− 𝑇(𝑦𝑛))+ 𝜆𝜌]] ,

                 (3.8) 

converges strongly to 𝑤∗. 

Proof. Using Algorithm 3.1 and the Lipschitz continuity of the resolvent operator, we have 



Communications on Applied Nonlinear Analysis 

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‖wn+1 − wn
‖= ‖(1 − αn

)wn + αn[wn − g(wn
) + J

λ,N

A(….)
[A(Pog(wn

),Rog(wn
))− λ(S(wn

) 

                     −T(wn
))+ λρ]] − [(1− αn

)wn−1 + αn
[wn−1 − g(wn−1

) 

                     +J
λ,N

A(….)
[A(Pog(wn−1

),Rog(wn−1
))− λ(S(wn−1

)− T(wn−1
)) + λρ]] ‖ 

     ≤ (1 − αn
)‖wn− wn−1

‖+ αn‖wn −wn−1 − (g(wn
)− g(wn−1

))‖ 

        +αn‖J
λ ,N

A(….)
[A(Pog(wn

),Rog(wn
))− λ(S(wn

) − T(wn
)) + λρ]] 

        −J
λ,N

A(….)
[A(Pog(wn−1

),Rog(wn−1
)) − λ(S(wn−1

)− T(wn−1
)) + λρ]] ‖ 

     ≤ (1 − αn
)‖wn− wn−1

‖+ αn‖wn −wn−1 − (g(wn
)− g(wn−1

))‖ 

       +
αn

μα
2−γβ

2 ‖A(Pog(wn
),Rog(wn

))− λ(S(wn
)− T(wn

)) 

       −A(Pog(wn−1
),Rog(wn−1

)) − λ(S(wn−1
)− T(wn−1

))‖                  (3.9) 

Since 𝑔 is strongly MT with 𝛿𝑔 and Lipschitz continuous with 𝜆𝑔, we have 

∥ 𝑤𝑛 − 𝑤𝑛−1 − (𝑔(𝑤𝑛) − 𝑔(𝑤𝑛−1)) ∥
2≤∥ 𝑤𝑛 −𝑤𝑛−1 ∥

2− 2⟨𝑔(𝑤𝑛) − 𝑔(𝑤𝑛−1),𝑤𝑛 −𝑤𝑛−1  

      +∥ 𝑔(𝑤𝑛) − 𝑔(𝑤𝑛−1) ∥
2  

               ≤ (1 − 2𝛿𝑔 + 𝜆𝑔
2 ) ∥ 𝑤𝑛 − 𝑤𝑛−1 ∥

2  

which implies that 

∥ wn − wn−1 − (g(wn
) − g(wn−1

)) ∥≤ √1− 2δg + λg
2 ∥ wn −wn−1 ∥.             (3.10) 

Since 𝐴(. , . ) is Lipschitz continuous 𝑃 and 𝑅, and Lipschitz continuous of 𝑃 and 𝑔, we have 

‖𝐴(𝑃𝑜𝑔(𝑤𝑛),𝑅𝑜𝑔(𝑤𝑛))− 𝜆(𝑆(𝑤𝑛)− 𝑇(𝑤𝑛))− (𝐴(𝑃𝑜𝑔(𝑤𝑛−1),𝑅𝑜𝑔(𝑤𝑛−1)) −𝜆(𝑆(𝑤𝑛−1) −

𝑇(𝑤𝑛−1))‖ 

= ‖𝐴(𝑃𝑜𝑔(𝑤𝑛),𝑅𝑜𝑔(𝑤𝑛))− 𝐴(𝑃𝑜𝑔(𝑤𝑛−1),𝑅𝑜𝑔(𝑤𝑛−1))− 𝜆(𝑆(𝑤𝑛) − 𝑆(𝑤𝑛−1)) − 𝜆(𝑇(𝑤𝑛) −

      𝑇(𝑤𝑛−1))‖  

≤ ‖𝐴(𝑃𝑜𝑔(𝑤𝑛),𝑅𝑜𝑔(𝑤𝑛))− 𝐴(𝑃𝑜𝑔(𝑤𝑛−1),𝑅𝑜𝑔(𝑤𝑛−1))‖+ 𝜆‖𝑆(𝑤𝑛) − 𝑆(𝑤𝑛−1)‖ +𝜆‖𝑇(𝑤𝑛)−

      𝑇(𝑤𝑛−1)‖ 

≤ ‖𝐴(𝑃𝑜𝑔(𝑤𝑛),𝑅𝑜𝑔(𝑤𝑛))− 𝐴(𝑃𝑜𝑔(𝑤𝑛−1),𝑅𝑜𝑔(𝑤𝑛))+ 𝐴(𝑃𝑜𝑔(𝑤𝑛−1),𝑅𝑜𝑔(𝑤𝑛))  

      −𝐴(𝑃𝑜𝑔(𝑤𝑛−1),𝑅𝑜𝑔(𝑤𝑛−1))‖+ 𝜆‖𝑆(𝑤𝑛)− 𝑆(𝑤𝑛−1)‖+ 𝜆‖𝑇(𝑤𝑛) − 𝑇(𝑤𝑛−1)‖ 

≤ ‖𝐴(𝑃𝑜𝑔(𝑤𝑛),𝑅𝑜𝑔(𝑤𝑛))− 𝐴(𝑃𝑜𝑔(𝑤𝑛−1),𝑅𝑜𝑔(𝑤𝑛))‖+ ‖𝐴(𝑃𝑜𝑔(𝑤𝑛−1),𝑅𝑜𝑔(𝑤𝑛))           

      −𝐴(𝑃𝑜𝑔(𝑤𝑛−1),𝑅𝑜𝑔(𝑤𝑛−1))‖+ 𝜆‖𝑆(𝑤𝑛) − 𝑆(𝑤𝑛−1)‖+ 𝜆‖𝑇(𝑤𝑛) − 𝑇(𝑤𝑛−1)‖ 

≤ 𝑡1𝜆𝑃𝜆𝑔‖𝑤𝑛− 𝑤𝑛−1‖+ 𝑡2𝜆𝑅𝜆𝑔‖𝑤𝑛 −𝑤𝑛−1‖+ 𝜆𝜆𝑆‖𝑤𝑛 − 𝑤𝑛−1‖+ 𝜆𝜆𝑇‖𝑤𝑛 −𝑤𝑛−1‖ 



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≤ (𝑡1𝜆𝑃𝜆𝑔 + 𝑡2𝜆𝑅𝜆𝑔 + 𝜆𝜆𝑆+ 𝜆𝜆𝑇)‖𝑤𝑛 −𝑤𝑛−1‖                

(3.11) 

On using Equations (3.10) and (3.11), Equation (3.9) becomes 

∥ wn+1 − wn ∥≤ (1− αn) ∥ wn −wn−1 ∥ +αn√1 − 2δg + λg
2 ∥ wn −wn−1 ∥ 

+
αn

μα2 − γβ2
√1− 2λ(δS+ δT)+ λλS

2 + λλT
2 ∥ wn − wn−1 ∥ 

                         ≤ [1 − αn + αnκ] ∥ wn − wn−1 ∥ 

    = [1 − αn(1− κ) ∥ wn −wn−1 ∥,             (3.12) 

where 

κ = √1 − 2δg + λg
2 +

t1λPλg + t2λRλg + λλS + λλT

μα2 − γβ2
. 

By condition (3.4), we have 0 ≤ 𝜅 < 1, thus the sequence {𝑤𝑛} is a Cauchy sequence in ℋ and as ℋ 

is complete, there exists 𝑤∗ ∈ ℋ such as 𝑤𝑛 → 𝑤∗ , as 𝑛 → ∞ . By using the continuity of the 

functions 𝑔, 𝑃, 𝑅, 𝑆, 𝑇, 𝐴, 𝐽
𝜆 ,𝑁

𝐴(….)
, and Algorithm 3.1, we have 

      𝑔(𝑤) = 𝐽
𝜆,𝑁

𝐴(.,.)
[𝐴(𝑃𝑜𝑔(𝑤), 𝑅𝑜𝑔(𝑤))− 𝜆(𝑆(𝑤) − 𝑇(𝑤)) + 𝜆𝜌]. 

From Lemma 3.1, we conclude that 𝑤∗ is a solution of (2.1). 

Theorem 3.2. Let 𝑃, 𝑅, 𝑆, 𝑇, 𝐴, 𝑁, 𝑔, 𝜅  and 𝑤∗ be the same as in Theorem 3.1, and let {𝑤𝑛}, {𝑞𝑛}, {𝑟𝑛} 

be the sequences defined by (3.2), (3.3) and (3.8), respectively with the sequences 𝜉𝑛 ⊂ [0,1] and 

{𝜇𝑛} ⊂ [0,1] satisfying the conditions 𝑙𝑖𝑚
𝑛→∞

𝜉𝑛 = 0 and ∑ 𝜉𝑛
∞
𝑛=0 = ∞ . Then the following assertions 

are identical 

a) {𝑤𝑛} converges to 𝑤∗ ∈ ℋ; 

b) {𝑞𝑛} converges to 𝑤∗ ∈ ℋ; 

c) {𝑟𝑛} converges to 𝑤∗ ∈ ℋ. 

Algorithm 3.3. The iterative sequence {𝑠𝑛} for all 𝑛 ∈ 𝑁0 is stated as 

{
 
 

 
 

s0 ∈ ℋ

sn+1 = tn − g(tn) + Jλ,N

A(….)
[A(Pog(tn),Rog(tn))− λ(S(tn)− T(tn))+ λρ]

tn = (1 − μ
n
)sn + μ

n
[sn − g(sn

)+ J
λ,N

A(….)
[A(Pog(sn

),Rog(sn
))

−λ(S(sn
)− T(sn

))+ λρ]] ,

                           (3.13) 

where {𝜇𝑛} is a sequence in (0,1) satisfying certain control conditions. 

Definition 3.1([4]). Consider two real sequences {𝛼𝑛}𝑛=0
∞  and {𝜃𝑛}𝑛=0

∞  with limits 𝛼 and 𝜃, 

respectively. Assume there exists 



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𝑙𝑖𝑚
𝑛→∞

|𝛼𝑛−𝛼|

|𝜃𝑛−𝜃|
= 𝑙. 

a) If 𝑙 = 0, in such instances, it can be expressed that {𝛼𝑛}𝑛=0
∞  converges faster to 𝛼  than 

{𝜃𝑛}𝑛=0
∞  to 𝜃. 

b) If 𝑙 ∈ (0,∞), in such scenarios, we can affirm {𝛼𝑛}𝑛=0
∞  and {𝜃𝑛 }𝑛=0

∞  have the same rate of 

convergence. 

Remark 3.1.    

a) When 𝑙 = ∞, it indicates that the sequence {𝜃𝑛}𝑛=0
∞  converges more rapidly than {𝛼𝑛}𝑛=0

∞ . 

b) In situations where both sequences {𝑤𝑛}𝑛=0
∞  and {𝑦𝑛}𝑛=0

∞  within the space ℋ converge to the 

same point 𝑝, the ensuing error predictions apply: 

‖𝑤𝑛 − 𝑝‖ ≤ 𝛼𝑛, ∀𝑛 ∈ 𝑁0 ,             (3.14)  

‖𝑦𝑛 −𝑝‖ ≤ 𝜃𝑛, ∀𝑛 ∈ 𝑁0 ,             (3.15)  

where {𝛼𝑛}𝑛=0
∞  and {𝜃𝑛}𝑛=0

∞  are sequences consisting of positive numbers (converges to zero). 

Definition 3.2 ([4]). Suppose we have two sequences, {𝑤𝑛}𝑛=0
∞  and {𝑦𝑛}𝑛=0

∞  both within the space ℋ, 

converging to the same point 𝑝 and satisfying conditions (3.14) and (3.15), respectively. When the 

sequence {𝛼𝑛}𝑛=0
∞  converges more rapidly than {𝜃𝑛}𝑛=0

∞ , we characterise {𝑤𝑛}𝑛=0
∞  as converging 

faster than {𝑦𝑛}𝑛=0
∞  to 𝑝. 

Lemma 3.1 ([18]). Consider two sequences, {𝜎𝑛}𝑛=0
∞  and {𝜌𝑛}𝑛=0

∞ , consisting of positive real 

numbers, which satisfy the inequality given as: 

σn+1 ≤ (1− ϵn)σn + ρ
n ,               (3.16) 

where 𝜖𝑛 ∈ (0,1) for all 𝑛 ≥ 𝑛0 , 𝛴𝑛=1
∞ 𝜖𝑛 = ∞, and 𝜌𝑛 = 𝑜(𝜖𝑛). Then, 𝑙𝑖𝑚

𝑛→∞
𝜎𝑛 = 0. Now, we are ready 

to establish the strong convergence of S-iteration process (3.13) to a unique solution 𝑤∗ of (2.1). 

Theorem 3.3. Let ℋ be a real Hilbert space and 𝐴:ℋ ×ℋ → ℋ and 𝑃, 𝑅, 𝑆, 𝑇, 𝑔:ℋ → ℋ are single-

valued functions and 𝑁:ℋ → 2ℋ be a multi-valued function such as 𝐴(.,.)𝑐𝑜 − MT with respect to 

𝑃, 𝑅, 𝑔 operator. Assume that 𝐴(. , . ) is Lipschitz continuous constant 𝑡 > 0, mixed strongly MT with 

respect to 𝑃 and 𝑅 with constant 𝛿 > 0, 𝑔 is strongly MT with constant 𝛿𝑔 > 0 and 𝑔, 𝑃, 𝑅, 𝑆, 𝑇 are 

Lipschitz continuous with 𝜆𝑔, 𝜆𝑃, 𝜆𝑅, 𝜆𝑆 and 𝜆𝑇 respectively such as 

{(μα2 − γβ2)
2
(1− 2δg + λg

2) < [μα2 − γβ2 − t1λPλg − t2λRλg − λ(λS+ λT)]
2
 

μ > γ and α > β
          (3.17) 

Let {𝑠𝑛} be an iterative sequence in ℋ defined by (3.1) with the sequence {𝜇𝑛} ⊂ (0,1) satisfying 

∑ 𝜇𝑛
∞
𝑛=0 = ∞. Then, the sequence {𝑠𝑛} demonstrates converges strongly towards a unique solution 

𝑤∗ of equation (2.1), and this convergence is associated with the following estimate: 

   ∥ sn −w∗ ∥≤ κn ∏
i=0

n−1

[1− μ
i
(1− κ)] ∥ s0 −w∗ ∥, for n ∈ N 



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Proof. Utilizing (2.3), (3.6) and (3.13), we obtain 

∥ sn+1 −w∗ ∥=∥ tn − g(tn)+ Jλ,NA(…)[A(Pog(tn),Rog(tn))− λ(S(tn)− T(tn))+ λρ] 

                              −(w∗ − g(w∗)+ Jλ,NA(…)[A(Pog(w∗),Rog(w∗))− λ(S(w∗)− T(w∗))+ λρ]) ∥ 

   =∥ tn − w∗ − g(tn) − g(w∗)+ Jλ,NA(…)[A(Pog(tn),Rog(tn))−A(Pog(w∗),Rog(w∗)) 

                −λ(S(tn)− S(w∗))+ λ(T(tn)− T(w∗))]+ λρ ∥.            (3.18) 

We have 

‖J
λ,N
A(.,.)[A(Pog(tn),Rog(tn))− λ(S(tn) − T(tn))+ λρ]  

−J
λ,N

A(….,)
[A(Pog(w∗),Rog(w∗))− λ(S(w∗)− T(w∗))+ λρ]] ‖ 

≤
1

μα2 − γβ2
‖A(Pog(tn),Rog(tn))− λ(S(tn)− T(tn)) 

− (A(Pog(w∗),Rog(w∗))− λ(S(w∗)− T(w∗)))‖ 

               ≤
1

μα
2−γβ

2‖A(Pog(tn),Rog(tn)) −A(Pog(w∗),Rog(w∗)) 

                                       −λ(S(tn)− S(w∗) + T(tn) − T(w∗))‖.               (3.19) 

Since 𝐴(. , . ) is Lipschitz continuous with respect to 𝑃 and 𝑅, and Lipschitz continuous of  𝑃 and 𝑔, 

we have  

‖A(Pog(tn),Rog(tn))− λ(S(tn)− T(tn))− (A(Pog(w∗),Rog(w∗))− λ(S(w∗)− T(w∗))‖ 

= ‖A(Pog(tn),Rog(tn))−A(Pog(w∗),Rog(w∗))− λ(S(tn)− S(w∗)) 

                  −λ(T(tn)− T(w∗))‖ 

≤ ‖A(Pog(tn),Rog(tn)) − A(Pog(w∗),Rog(w∗))‖+ λ‖S(tn)− S(w∗)‖ 

                 +λ‖T(tn) − T(w∗)‖ 

≤ ‖A(Pog(tn),Rog(tn)) − A(Pog(w∗),Rog(tn)) +A(Pog(w∗),Rog(w∗)) 

    −A(Pog(w∗),Rog(w∗))‖+ λ‖S(tn) − S(w∗)‖+ λ‖T(tn)− T(w∗)‖ 

            ≤ ‖A(Pog(tn),Rog(tn))− A(Pog(w∗),Rog(w∗))‖+ ‖A(Pog(w∗),Rog(tn)) 

                     −A(Pog(w∗),Rog(w∗))‖+ λ‖S(tn) − S(w∗)‖+ λ‖T(tn)− T(w∗)‖ 

Because 𝑔 exhibits strongly MT with parameter 𝛿𝑔 and is also Lipschitz continuous with constant 𝜆𝑔, 

it follows that 

∥ 𝑡𝑛 − 𝑤
∗ − 𝑔(𝑡𝑛) + 𝑔(𝑤

∗) ∥2≤∥ 𝑡𝑛 − 𝑤
∗ ∥2− 2⟨𝑔(𝑡𝑛) − 𝑔(𝑤

∗), 𝑡𝑛 − 𝑤
∗⟩ 

              +∥ 𝑔(𝑡𝑛) − 𝑔(𝑤
∗) ∥2 

           ≤ (1 − 2𝛿𝑔 + 𝜆𝑔
2 ) ∥ 𝑡𝑛 −𝑤

∗ ∥2 .            (3.20) 



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which implies that 

  ∥ tn − w∗ − (g(tn) − g(w∗)) ∥≤ √1 − 2δg + λg
2 ∥ tn −w∗ ∥.            (3.21) 

Using (3.19) and (3.21), (3.18) becomes 

∥ 𝑡𝑛 −𝑤
∗ − (𝑔(𝑡𝑛) − 𝑔(𝑤

∗))+ 𝐽
𝜆 ,𝑁

𝐴(.,.)[𝐴(𝑃𝑜𝑔(𝑡𝑛),𝑅𝑜𝑔(𝑡𝑛))− 𝜆(𝑆(𝑡𝑛) − 𝑇(𝑡𝑛)) + 𝜆𝜌] 

     −𝐽
𝜆 ,𝑁

𝐴(.,.)[𝐴(𝑃𝑜𝑔(𝑤∗),𝑅𝑜𝑔(𝑤∗))− 𝜆(𝑆(𝑤∗) − 𝑇(𝑤∗))+ 𝜆𝜌] ∥  

≤ √1 − 2𝛿𝑔 + 𝜆𝑔
2 +

𝑡1𝜆𝑃𝜆𝑔 + 𝑡2𝜆𝑅𝜆𝑔 + 𝜆𝜆𝑆 + 𝜆𝜆𝑇
𝜇𝛼2 −𝛾𝛽2

∥ 𝑡𝑛 −𝑤
∗ ∥ 

                    = 𝜅 ∥ 𝑡𝑛 −𝑤
∗ ∥,                 (3.22) 

Where, 

   𝜅 = √1− 2𝛿𝑔 + 𝜆𝑔
2 +

𝑡1𝜆𝑃𝜆𝑔+𝑡2𝜆𝑅𝜆𝑔+𝜆𝜆𝑆+𝜆𝜆𝑇

𝜇𝛼2−𝛾𝛽2 . 

Using (2.3), (3.6) and (3.13), we have 

∥ tn −w∗ ∥=∥ (1 − μ
n
)(sn − w∗) + μ

n
(sn − g(sn) − (w

∗ − g(w∗)) 

                       +J
λ,N

A(., .)
[A(Pog(sn),Rog(sn)) − λ(S(sn)− T(sn))+ λρ] 

                       −J
λ,N

A(., .)
[A(Pog(w∗),Rog(w∗))− λ(S(w∗)− T(w∗)) + λρ] ∥ 

         ≤ (1 − μ
n
) ∥ sn − w∗ ∥ +μ

n
∥ sn −w∗ − (g(sn)− g(w∗)) 

                      +μ
n
∥ J

λ,N

A(.,.)
[A(Pog(sn),Rog(sn))− λ(S(sn)− T(sn))+ λρ] 

          −J
λ,N

A(., .)
[A(Pog(w∗),Rog(w∗))− λ(S(w∗)− T(w∗)) + λρ] ∥ 

                     ≤ (1 − μ
n
) ∥ sn − w∗ ∥ +μ

n
√1− 2δg + λg

2 ∥ sn − w∗ ∥ 

         +
μn

μα2−γβ2 (t1λPλg + t2λRλg + λλS + λλT) ∥ sn −w∗ ∥ 

         ≤ [1 − μ
n
(1− κ)] ∥ sn −w∗ ∥,                      (3.23) 

where 

   κ = √1− 2δg + λg
2 +

t1λP λg+t2 λRλg+λλS+λλT

μα
2−γβ

2 . 

By (3.22), (3.2) becomes 

  ∥ 𝑠𝑛+1 −𝑤
∗ ∥≤ 𝜅[1 −𝜇𝑛(1− 𝜅)] ∥ 𝑠𝑛 −𝑤

∗ ∥ 

 

 



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inductively, we have 

   ∥ 𝑠𝑛+1 −𝑤
∗ ∥≤ 𝜅𝑛+1 ∏

𝑖=0

𝑛

[1− 𝜇𝑖(1−𝜅) ∥ 𝑠0 − 𝑤
∗ ∥.          (3.24) 

As per classical analysis, it's a widely recognized fact that for any value of 𝑎 ∈ [0,1], the inequality 

1 −𝑎 ≤ 𝑒−𝑎 holds true. Hence, from (3.24), we have 

   ∥ 𝑠𝑛+1 −𝑤
∗ ∥≤∥ 𝑠0 −𝑤

∗ ∥ 𝜅𝑛+1𝑒−(1−𝜅)𝛴𝑖=1
𝑛 𝜇𝑖 .          (3.25) 

It follows that from the assumption ∑ 𝜇𝑖
∞
𝑖=0 = ∞ that 𝑒−(1−𝜅)∑ 𝜇𝑖

𝑛
𝑖=1 → 0 as 𝑛 → ∞, which implies 

that 𝑙𝑖𝑚
𝑛→∞

‖𝑠𝑛 − 𝑤
∗‖ = 0. 

The underneath results shows that convergence rate of the sequences generated by (3.13) is faster 

than (3.2). Therefore, this result has a great importance both from numerical and theoretical aspects. 

Theorem 3.4. Let ℋ, 𝑆, 𝑇, 𝑁, 𝐴, 𝑔, 𝜅  and 𝑤∗  be defined as Theorem 3.3 and suppose {𝜇𝑛}  be a 

sequence in (0,1) such as 𝜇 ≤ 𝜇𝑛  for all 𝑛 ∈ 𝑁0  and for some 𝜇 > 0. For given 𝑤0 = 𝑠0 ∈ ℋ, let 

{𝑤𝑛} and {𝑠𝑛}  be the iterative sequences generated by (3.2) and (3.12), respectively. Then, the 

sequence {𝑠𝑛} converges to 𝑤∗ at a rate faster than {𝑤𝑛} does. 

Proof. From Theorem 3.1, we have 

‖𝑤𝑛 −𝑤
∗‖ ≤ 𝜅𝑛‖𝑤0 −𝑤

∗‖. 

From (3.25), we have 

‖𝑠𝑛+1 −𝑤
∗‖ ≤ 𝜅𝑛+1 ∏[1− (1−𝜅)𝜇𝑖]‖𝑠0 −𝑤

∗‖

𝑛

𝑖=0

, 

or equivalently 

‖sn −w∗‖ ≤ κn ∏[1− (1 − κ)μ
i
]‖s0 − w∗‖.

n−1

i=0

 

It follows from the assumption that 

‖sn −w∗‖ ≤ κn ∏ [1 − (1− κ)μ
i
]

n

i=1

‖s0 −w∗‖ = κn[1− (1 − κ)μ]n‖s0 −w∗‖. 

Set 

𝛼𝑛 = 𝜅
𝑛[1 − (1− 𝜅)𝜇]𝑛‖𝑠0 −𝑤

∗‖, 

𝜃𝑛 = 𝜅𝑛‖𝑢0 −𝑤
∗‖. 

Given that 𝑙𝑖𝑚
𝑛→∞

𝛩𝑛 = 0, 𝑙𝑖𝑚
𝑛→∞

𝛼𝑛 = 0 and 𝑙𝑖𝑚𝜃𝑛 = 0, which means that both sequences {𝛼𝑛} and {𝜃𝑛} 

converge to zero, as stipulated in Definition 3.2. Define 

πn =
αn − 0

θn − 0
=

κn[1− (1− κ)μ]n‖s0 − w∗‖

κn‖u0 − w∗‖
 



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= [1− (1− κ)μ]n. 

Note that 1 − (1− 𝜅)𝜇 ∈ (0,1). This allows us to conclude that 

lim
n→∞

πn = lim
αn − 0

θn − 0
= 0 

Thus, according to Definition 3.1(a), we can infer that the convergence of {𝛼𝑛} is faster than that of 

{𝜃𝑛}, and as a consequence, {𝑠𝑛} converges faster than {𝑤𝑛}. 

In Theorem 3.3, we have discussed that 𝑆-iteration algorithm (3.13) is a better algorithm with a more 

efficient convergence rate. Now, we establish new convergence implications between iterative 

sequences generated by (3.3) and (3.15). 

Theorem 3.5. Let ℋ, 𝑆, 𝑇, 𝑁, 𝐴, 𝑔, 𝜅 and 𝑤∗ be defined as Theorem 3.3 and suppose {𝑞𝑛} and {𝑠𝑛} be 

iterative sequences generated by (2.4) and (3.1), respectively, with the sequences {𝜉𝑛 } and {𝜇𝑛} in ⊂

(0,1). Then the subsequent claims are applicable: 

a) If {
1−𝜉𝑛

𝜇𝑛
}  is bounded, ∑ 𝜇𝑛

∞
𝑛=0 = ∞  and {𝑞𝑛}  converges strongly to 𝑤∗ , then {𝑠𝑛 −  𝑞𝑛} 

converges strongly to 0 with the following estimate :  

‖𝑞𝑛+1 − 𝑠𝑛+1
‖ ≤ 𝜅[1− (1− 𝜅)𝜇𝑛]‖𝑞𝑛 − 𝑠𝑛‖ + (1− 𝜉𝑛){1+ 𝜅}‖𝑞𝑛 −𝑤

∗‖,∀𝑛 ∈ 𝑁0, and {𝑠𝑛} 

converges strongly to 𝑤∗. 

b) If {
1−𝜉𝑛

𝜇𝑛𝜉𝑛
} is bounded and ∑ 𝜉𝑛𝜇𝑛

∞
𝑛=0 = ∞, then {𝑠𝑛 − 𝑞𝑛} converges strongly to 0 with the 

following estimate : 

‖q
n+1

− sn+1‖ ≤ [1− (1− κ)ξ
n
μ

n
]‖sn − q

n
‖ + (1+ κ)(1− ξ

n
)‖sn −w∗‖,∀n ∈ N0 

and {𝑞𝑛} converges strongly to 𝑤∗. 

Proof.  

a) Suppose that {
1−𝜉𝑛

𝜇𝑛
‖ is bounded, ∑ 𝜇𝑛

∞
𝑛=0 = ∞ and {𝑞𝑛} converges strongly to 𝑤∗. We show 

that {𝑠𝑛 −𝑞𝑛} converges strongly to 0. It derives from (3.3), (3.6), (3.7) and (3.15) that 

∥ 𝑞𝑛+1 − 𝑠𝑛+1 ∥ 

=∥ (1− 𝜉𝑛)𝑞𝑛 + 𝜉𝑛 [𝑟𝑛 −𝑔(𝑟𝑛) + 𝐽𝜆 ,𝑁
𝐴(.,.)[𝐴(𝑃𝑜𝑔(𝑟𝑛),𝑅𝑜𝑔(𝑟𝑛))− 𝜆(𝑆(𝑟𝑛) − 𝑇(𝑟𝑛)) 

    +𝜆𝜌]] − 𝑡𝑛 −𝑔(𝑡𝑛) + 𝐽𝜆 ,𝑁
𝐴(.,.)

[𝐴(𝑃𝑜𝑔(𝑡𝑛),𝑅𝑜𝑔(𝑡𝑛))− 𝜆(𝑆(𝑡𝑛) − 𝑇(𝑡𝑛))+ 𝜆𝜌] ∥ 

=∥ (1− 𝜉𝑛)𝑞𝑛 + 𝜉𝑛 [𝑟𝑛 −𝑔(𝑟𝑛) + 𝐽𝜆 ,𝑁
𝐴(.,.)

[𝐴(𝑃𝑜𝑔(𝑟𝑛),𝑅𝑜𝑔(𝑟𝑛))− 𝜆(𝑆(𝑟𝑛) − 𝑇(𝑟𝑛)) 

    +𝜆𝜌]] − [𝑟𝑛 − 𝑔(𝑟𝑛) + 𝐽𝜆 ,𝑁
𝐴(.,.)

[𝐴(𝑃𝑜𝑔(𝑟𝑛),𝑅𝑜𝑔(𝑟𝑛)) − 𝜆(𝑆(𝑟𝑛) − 𝑇(𝑟𝑛)) 

    +𝜆𝜌]]+ [𝑟𝑛 − 𝑔(𝑟𝑛) + 𝐽𝜆 ,𝑁
𝐴(.,.)

[𝐴(𝑃𝑜𝑔(𝑟𝑛),𝑅𝑜𝑔(𝑟𝑛)) − 𝜆(𝑆(𝑟𝑛) − 𝑇(𝑟𝑛)) 

    +𝜆𝜌]]− [𝑡𝑛 − 𝑔(𝑡𝑛) + 𝐽𝜆,𝑁
𝐴(.,.)

[𝐴(𝑃𝑜𝑔(𝑡𝑛),𝑅𝑜𝑔(𝑡𝑛))− 𝜆(𝑆(𝑡𝑛) − 𝑇(𝑡𝑛)) + 𝜆𝜌]] ∥ 



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=∥ (1− 𝜉𝑛)𝑞𝑛 − (1− 𝜉𝑛)(𝑟𝑛 − 𝑔(𝑟𝑛) + 𝐽𝜆,𝑁
𝐴(.,.)[𝐴(𝑃𝑜𝑔(𝑟𝑛),𝑅𝑜𝑔(𝑟𝑛)) − 𝜆(𝑆(𝑟𝑛) 

   −𝑇(𝑟𝑛))+ 𝜆𝜌]] + [𝑟𝑛 − 𝑔(𝑟𝑛) + 𝐽𝜆 ,𝑁
𝐴(.,.)[𝐴(𝑃𝑜𝑔(𝑟𝑛),𝑅𝑜𝑔(𝑟𝑛)) − 𝜆(𝑆(𝑟𝑛) 

   −𝑇(𝑟𝑛))+ 𝜆𝜌]] + [𝑟𝑛 − 𝑔(𝑟𝑛) + 𝐽𝜆 ,𝑁
𝐴(.,.)[𝐴(𝑃𝑜𝑔(𝑟𝑛),𝑅𝑜𝑔(𝑟𝑛)) − 𝜆(𝑆(𝑟𝑛) 

   −𝑇(𝑟𝑛))+ 𝜆𝜌]] − [𝑡𝑛 − 𝑔(𝑡𝑛) + 𝐽𝜆,𝑁
𝐴(.,.)[𝐴(𝑃𝑜𝑔(𝑡𝑛),𝑅𝑜𝑔(𝑡𝑛))− 𝜆(𝑆(𝑡𝑛) − 𝑇(𝑡𝑛)) + 𝜆𝜌]] ∥ 

= (1 − 𝜉𝑛) ∥ 𝑞𝑛 − (𝑟𝑛 − 𝑔(𝑟𝑛) + 𝐽𝜆,𝑁
𝐴(.,.)

[𝐴(𝑃𝑜𝑔(𝑟𝑛),𝑅𝑜𝑔(𝑟𝑛)) − 𝜆(𝑆(𝑟𝑛) − 𝑇(𝑟𝑛)) 

   +𝜆𝜌]) ∥ +𝜉𝑛 ∥ 𝑟𝑛 − 𝑔(𝑟𝑛) + 𝐽𝜆,𝑁
𝐴(.,.)

[𝐴(𝑃𝑜𝑔(𝑟𝑛),𝑅𝑜𝑔(𝑟𝑛)) − 𝜆(𝑆(𝑟𝑛) − 𝑇(𝑟𝑛)) 

   +𝜆𝜌] ∥ −(𝑡𝑛 −𝑔(𝑡𝑛) + 𝐽𝜆 ,𝑁
𝐴(.,.)[𝐴(𝑃𝑜𝑔(𝑡𝑛),𝑅𝑜𝑔(𝑡𝑛))− 𝜆(𝑆(𝑡𝑛) − 𝑇(𝑡𝑛))+ 𝜆𝜌]) 

≤ (1 − 𝜉𝑛) ∥ 𝑞𝑛 −𝐹(𝑟𝑛) ∥ +𝜉𝑛 ∥ 𝐹(𝑟𝑛) − 𝐹(𝑡𝑛) ∥ 

≤ (1 − 𝜉𝑛) ∥ 𝑞𝑛 −𝑤
∗ +𝐹(𝑤∗) − 𝐹(𝑟𝑛) ∥ +𝜉𝑛 ∥ 𝐹(𝑟𝑛) − 𝐹(𝑡𝑛) ∥ 

≤ (1 − 𝜉𝑛)(∥ 𝑞𝑛 −𝑤
∗ ∥ +∥ 𝐹(𝑤∗)− 𝐹(𝑟𝑛) ∥) + 𝜉𝑛 ∥ 𝐹(𝑟𝑛) − 𝐹(𝑡𝑛) ∥ 

≤ (1 − 𝜉𝑛)(∥ 𝑞𝑛 −𝑤
∗ ∥ +𝜅 ∥ 𝑤∗ − 𝑟𝑛 ∥) + 𝜉𝑛𝜅 ∥ 𝑟𝑛 − 𝑡𝑛 ∥ 

≤ (1 − 𝜉𝑛) ∥ 𝑞𝑛 −𝑤
∗ ∥ +(1− 𝜉𝑛)𝜅 ∥ 𝑟𝑛 − 𝑤

∗ ∥ −(1− 𝜉𝑛)𝜅 ∥ 𝑟𝑛 − 𝑡𝑛 ∥ +𝜉𝑛𝜅 ∥ 𝑟𝑛 − 𝑡𝑛 ∥ 

≤ (1 − 𝜉𝑛) ∥ 𝑞𝑛 −𝑤
∗ ∥ +𝜅 ∥ 𝑡𝑛 − 𝑟𝑛 ∥ +(1− 𝜉𝑛)𝜅 ∥ 𝑟𝑛 − 𝑤

∗.             (3.26) 

We have 

∥ rn − w∗ ∥=∥ (1− μ
n
)q

n
+ μ

n
[q

n
− g(q

n
) + J

λ,N

A(.,.)
[A(Pog(q

n
),Rog(q

n
))− λ(S(q

n
)− T(q

n
)) 

+λρ]]− w∗ ∥ 

       =∥ (1− μ
n
)q

n
+ μ

n
[q

n
− g(q

n
) + J

λ,N

A(.,.)
[A(Pog(q

n
),Rog(q

n
))− λ(S(q

n
)− T(q

n
)) 

+λρ]]− [(1− μ
n
)w∗ + μ

n
[w∗− g(w∗)+ Jλ,N

A(.,.)
[A(Pog(w∗),Rog(w∗)) 

−λ(S(w∗) − T(w∗))+ λρ]] ∥ 

                    ≤∥ (1− μ
n
)q

n
− w∗ + μ

n
(F(q

n
)− F(w∗)) ∥ 

                    ≤ (1 − μ
n
) ∥ q

n
−w∗ ∥ +μ

n
κ ∥ q

n
−w∗ ∥ 

                    = [1 − μ
n
(1− κ)] ∥ q

n
− w∗ ∥,             (3.27) 

And 

∥ rn − tn ∥=∥ (1− μ
n
)q

n
+ μ

n
[q

n
− g(q

n
)+ J

λ,N

A(.,.)
[A(Pog(q

n
),Rog(q

n
))− λ(S(q

n
)− T(q

n
)) 

+λρ]]− (1− μ
n
)sn + μ

n
[sn − g(sn)+ J

λ ,N

A(.,.)
[A(Pog(sn),Rog(sn)) 

−λ(S(sn) − T(sn))+ λρ]] ∥ 

                  ≤ (1 − μ
n
) ∥ q

n
− sn ∥ +μ

n
∥ (F(q

n
)− F(sn)) ∥ 



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                  ≤ (1 − μ
n
) ∥ q

n
− sn ∥ +μ

n
κ ∥ q

n
− sn ∥ 

                  = [1 − μ
n
(1− κ)] ∥ q

n
− sn ∥.             (3.28) 

 

Combining (3.26), (3.27) and (3.25) and using the fact 𝜅 ∈ (0,1), we have 

∥ q
n+1

− sn+1 ∥≤ (1− ξ
n
) ∥ q

n
−w∗ ∥ +(1 − ξ

n
)κ[1− (1− κ)μ

n
] ∥ q

n
− w∗ ∥ 

      +κ[1− (1− κ)μ
n
] ∥ q

n
− sn ∥ 

  ≤ κ[1 − (1− κ)μ
n
] ∥ q

n
− sn ∥ 

      +(1− ξ
n
){1+ κ[1− (1− κ)μ

n
]} ∥ q

n
− w∗ ∥ 

  ≤ [1 − (1− κ)μ
n
] ∥ q

n
− sn ∥ +(1+ κ)(1− ξ

n
) ∥ q

n
−w∗ ∥.            (3.29) 

Set 

𝜎𝑛: = ‖𝑞𝑛− 𝑠𝑛‖, 

𝜖𝑛 : = (1− 𝜅)𝜇𝑛, 

𝜌𝑛 : = (1+ 𝜅)(1− 𝜉𝑛)‖𝑞𝑛 −𝑤
∗‖.  

Then, (3.29) becomes 

𝜎𝑛+1 ≤ (1− 𝜖𝑛)𝜎𝑛 + 𝜌𝑛, 𝑛 ≥ 0.                  (3.30) 

Since {
1−𝜉𝑛

𝜇𝑛
} is bounded. we have 𝜌𝑛 = 𝑜(𝜖𝑛). Therefore, an application of Lemma 3.1 to (3.30) 

yields 𝑙𝑖𝑚
𝑛→∞

𝜎𝑛 = 𝑙𝑖𝑚
𝑛→∞

‖𝑞𝑛− 𝑠𝑛‖ = 0. Since 𝑙𝑖𝑚
𝑛→∞

‖𝑞𝑛− 𝑤
∗‖ = 0, it follows that 

‖𝑠𝑛 −𝑤
∗‖ ≤ ‖𝑞𝑛 − 𝑠𝑛‖ + ‖𝑞𝑛− 𝑤

∗‖ → 0,𝑎𝑠   𝑛 → ∞. 

b) Suppose that {
1−𝜉𝑛

𝜉𝑛𝜇𝑛
} is bounded and 𝛴𝑛=0

∞ 𝜇𝑛 = ∞ . Then,  from Theorem (3.2), {𝑠𝑛} strongly 

converges to 𝑤∗. 

We now show that {𝑞𝑛} converges strongly to 𝑤∗. Utilizing (3.3), (3.6), (3.7) and (3.13), we have 

∥ sn+1 − q
n+1

∥=∥ tn − g(tn) + J
λ ,N

A(.,.)
[A(Pog(tn),Rog(tn))− λ(S(tn)− T(tn))+ λρ] 

      −(1− ξ
n
)q

n
+ ξ

n
[rn − g(rn)+ Jλ ,N

A(.,.)
[A(Pog(rn),Rog(rn)) 

                   −λ(S(rn)− T(rn))+ λρ]] ∥  

   ≤ (1 − ξ
n
) ∥ q

n
− F(tn) ∥ +ξ

n
∥ F(rn)− F(tn) ∥ 

   ≤ (1 − ξ
n
){∥ F(tn) − F(w∗) ∥ +∥ F(w∗) − sn ∥ +∥ sn − q

n
∥} 

       +ξ
n
∥ F(rn)− F(tn) ∥ 

   ≤ (1 − ξ
n
){κ ∥ tn − w∗ ∥ +∥ sn −w∗ ∥ +∥ sn − q

n
∥}+ ξ

n
κ ∥ tn − rn ∥,      (3.31) 



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Now, we obtain  

∥ tn − w∗ ∥=∥ (1− μ
n
)sn + μ

n
[sn − g(sn)+ J

λ,N

A(.,.)
[A(Pog(sn),Rog(sn)) 

           −λ(S(sn) − T(sn))+ λρ]] −w∗ ∥ 

       ≤ (1 − μ
n
) ∥ sn − w∗ ∥ +μ

n
∥ F(sn)− F(w∗) ∥ 

       ≤ (1 − μ
n
) ∥ sn − w∗ ∥ +μ

n
κ ∥ sn −w∗ ∥ 

        ≤ [1 − (1− κ)μ
n
] ∥ sn − w∗ ∥.             (3.32) 

And, 

∥ tn − rn ∥=∥ (1− μ
n
)(sn − q

n
)+ μ

n
[sn − g(sn)+ J

λ,N

A(.,.)
[A(Pog(sn),Rog(sn)) 

         −λ(S(sn) − T(sn))+ λρ]] − [q
n
− g(q

n
) + J

λ,N

A(.,.)
[A(Pog(q

n
),Rog(q

n
))    

         −λ(S(q
n
)− T(q

n
)) + λρ]] ∥ 

       ≤ (1 − μ
n
) ∥ sn − q

n
∥ +μ

n
∥ F(sn) − F(w∗) ∥ 

       ≤ [1 − (1− κ)μ
n
] ∥ sn − q

n
∥.                  (3.33) 

Substituting (3.32) and (3.33) into (3.31), we obtain 

∥ sn+1 − q
n+1

∥= ξ
n
κ([1− (1− κ)μ

n
] ∥ sn − q

n
∥ 

                 +(1− ξ
n
){κ[1− (1− κ)μ

n
] ∥ sn − w∗ ∥ +∥ sn − w∗ ∥ +∥ sn − q

n
∥}) 

 ≤ [1 − (1− κ)ξ
n
μ

n
] ∥ sn − q

n
∥ +(1+ κ)(1− ξ

n
) ∥ sn −w∗ ∥.           (3.34) 

Set 𝜎𝑛 : =∥ 𝑠𝑛 − 𝑞𝑛 ∥, 𝜖𝑛 := 𝜉𝑛𝜇𝑛,  and 𝜌𝑛: = (1+𝜅)(1− 𝜉𝑛) ∥ 𝑠𝑛 −𝑤
∗ ∥ . Note that  {

1−𝜉𝑛

𝜇𝑛𝜉𝑛
}  is 

bounded. Also, lim𝑛→∞ ∥ 𝑠𝑛 − 𝑤
∗ ∥= 0, 𝜌𝑛 = 𝑜(𝜖𝑛)  and 𝛴𝑛=0

∞ 𝜇𝑛 = ∞ . Therefore, an application 

Lemma 3.1 to (3.34) yield lim𝑛→∞𝜎𝑛 = lim𝑛→∞ ∥ 𝑠𝑛 − 𝑞𝑛 ∥= 0. When lim𝑛→∞ ∥ 𝑠𝑛 −𝑤
∗ ∥= 0 and 

∥ 𝑞𝑛 − 𝑤
∗ ∥≤∥ 𝑠𝑛 − 𝑞𝑛 ∥ +∥ 𝑠𝑛 −𝑤

∗ ∥, 

we deduce that lim𝑛→∞ ∥ 𝑞𝑛 − 𝑤
∗ ∥= 0.  

Theorem 3.5 (a) establishes the strong convergence of {𝑠𝑛} to 𝑤∗ under convergence of {𝑞𝑛} and the 

condition𝛴𝑛=0
∞ 𝜇𝑛 = ∞. Theorem 3.5 (b) establishes a new convergence theorem for {𝑞𝑛}  under 

boundedness of {
1−𝜉𝑛

𝜇𝑛𝜉𝑛
} and divergence of 𝛴𝑛=0

∞ 𝜉𝑛𝜇𝑛. 

4. Numerical Results   

  In this section, we provide an illustrative example and numerical results that serve to exemplify the 

algorithm's applicability, demonstrating not only the primary outcomes of our paper but also the 

efficiency and convergence of the sequence generated through the iterative approach. 



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Example 4.1.  Let ℋ = ℛ  and define 𝐻,𝑀,𝑁, 𝑆, 𝑇, 𝑃, 𝑅, 𝑔:ℛ → ℛ  and 𝐴:ℛ×ℛ → ℛ   by 

𝐻(𝑤) =
27𝑤

4
, 𝑀(𝑤) =

𝑤

4
+ 1, 𝑁(𝑤) = 𝑤− 1, 𝑆(𝑤) =

9𝑤

2
, 𝑇(𝑤) =

7𝑤

2
, , 𝑃(𝑤) = 3𝑤, 𝑅(𝑤) =

𝑤

2
, 𝑔(𝑤) =

2𝑤

3
,𝐴(𝑤) =

3𝑤

4
− 1, 𝐴(𝑃(𝑤),𝑅(𝑤)) = 𝑃(𝑤) + 𝑅(𝑤),for all𝑤 ∈ ℋ.For  

𝜆 = 1 and = −2 , we have  𝐽
𝜆 ,𝑁

𝐴(.,.)
(𝑤) =

2(𝑤+2)

9
 and 𝐹(𝑤) = 3𝑤, for all 𝑤 ∈ ℋ 

From (3.2) and (3.13) can be written as, 𝑤𝑛+1 = 𝑇𝑤𝑛,and 𝑠𝑛+1 = 𝑇[(1− 𝜇𝑛)𝑠𝑛 +𝜇𝑛𝑇𝑠𝑛] for  𝑛 ∈

𝒩0 ,respectively, where 

     𝑇(𝑤):= 𝐹(𝑤) = 3𝑤, for all 𝑤 ∈ ℋ, 

and {𝜇𝑛} is a sequence in (0,1). We consider 𝛼𝑛 =
1

𝑛
 and 𝜇𝑛 =

1

𝑛2+1
 , 𝑛 ∈ 𝒩0 . Since assumptions of  

Theorem 3.3 are satisfied, therefore the sequence {𝑠𝑛}  converges to a unique solution of (2.1). 

Similarity, the sequence {𝑤𝑛}  convereges to a unique solution of (2.1) by Theorem 3.2. The 

graphical presentation of the convergence of sequences {𝑠𝑛} generated from 𝑠0 = 5,8,11 are given in 

Figure 1. Numerical values of {𝑠𝑛} are given in Tables 1. From Figure 2 and Table 2, we see that the 

sequence {𝑠𝑛} converges faster than the sequence {𝑤𝑛}.  

Table 1: The values of  𝑠𝑛 with initial values 𝑠0 = 5, 𝑠0 = 10 𝑎𝑛𝑑 𝑠0 = 15 

No. of 

Iterations 

For 𝑠0 = 5 

𝑠𝑛 

For 𝑠0 = 10 

𝑠𝑛 

For 𝑠0 = 15 

𝑠𝑛 

n=1 5 8 11 

n=2 13.8993517972893 22.2389628756629 30.5785739540365 

n=3 15.2588266954200 24.4141227126719 33.5694187299239 

n=4 9.32628502653346 14.9220560424535 20.5178270583736 

n=5 3.70219158018539 5.92350652829662 8.14482147640786 

n=6 1.03787208523604 1.66059533637766 2.28331858751928 

n=7 0.216827962271176 0.346924739633882 0.477021516996588 

n=8 0.0350752760961080 0.0561204417537728 0.0771656074114377 

n=9 0.00452286820450332 0.00723658912720530 0.00995031004990729 

n=10 0.000475687588453968 0.000761100141526349 0.00104651269459873 

n=11 4.15749986892055 6.65199979027288e-05 9.14649971162521e-05 

n=12 3.06677733294322e-06 4.90684373270916e-06 6.74691013247509e-06 

n=13 1.93448288920915e-07 3.09517262273463e-07 4.25586235626012e-07 

n=14 1.05525450594488e-08 1.68840720951181e-08 2.32155991307874e-08 

n=17 7.85059315299877e-13 1.25609490447980e-12 1.72713049365973e-12 

n=20 2.10651237441896e-17 3.37041979907034e-17 4.63432722372172e-17 

n=25 7.70878569921844e-26 1.23340571187495e-25 1.69593285382806e-25 

n=27 0 0 0 

n=28 0 0 0 

 

 



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Table2: The values of {𝑠𝑛} and {𝑤𝑛} with initial values 𝑠0 = 𝑤0 = 5. 

Number of Iterations Proposed S-iteration Algorithm 

3.3 (𝑠0 = 5) 

𝑠𝑛 

Proposed Algorithm 3.1 

(𝑤0 = 5) 

𝑤𝑛 

n=1 5 5 

n=2 13.8993517972893 19.4879571810883 

n=3 15.2588266954200 27.0958573369905 

n=4 9.32628502653346 23.1275375716004 

n=5 3.70219158018539 14.3521314352151 

n=6 1.03787208523604 7.22777560406802 

n=7 0.216827962271176 3.37773785150181 

n=8 0.0350752760961080 1.75006635630060 

n=9 0.00452286820450332 1.12929969254448 

n=10 0.000475687588453968 0.870676048112462 

n=13 1.93448288920915e-07 0.572413040490093 

n=17 7.85059315299877e-13 0.404900857028524 

n=20 2.10651237441896e-17 0.333600636560647 

n=25 7.70878569921844e-26 0.258848165970660 

n=100 0 0.000606691920380 

n=200 0 2.36268998330960e-22 

 

 

Figure1: The convergence of  𝑠𝑛 with initial 

values 𝑠0 = 5, 𝑠0 = 10 𝑎𝑛𝑑 𝑠0 = 15. 

 

Figure 2: The convergence of  𝑠𝑛 and 𝑤𝑛with 

initial values 𝑠0 = 𝑤0 = 5. 

5. Conclusion  

In conclusion, within the scope of this study, we have explored a broader variational inclusion 

problem that encompasses 𝐴(. , . )--co-coercive operators within the context of real Hilbert spaces. 

Through the utilization of the resolvent operator technique, we have established an equivalence 

between the generalized variational inclusion problem and its associated fixed-point problem. 

Leveraging this equivalence, we have demonstrated both the existence and uniqueness of a solution 

for the generalized variational inclusion problem, employing co-coercive and relaxed co-coercive 

functions. Also, we have proposed the algorithms involving 𝑆-iteration and 𝐻-MT operators under 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 8s (2025) 

 

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some suitable conditions. Lastly, we provide a numerical example to substantiate our primary 

finding. The results we've obtained serve to expand and provide a more comprehensive framework 

compared to many existing outcomes found in the literature for various systems. 

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