Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

NEW CLASSES OF GENERAL

TRIEQUILIBRIUM INCLUSIONS^∗

Muhammad Aslam Noor^†

Khalida Inayat Noor^‡

Communicated by S. Trean¸t˘a

10.56082/annalsarscimath.2026.2.179

DOI

Abstract

Some new classes of general triequilibrium inclusions are introduced

and investigated. We establish the equivalence between the general

triequilibrium inclusions and the ﬁxed point problems, which is used to

discuss the existence of the unique solution. Using various techniques

such as resolvent methods and dynamical systems coupled with ﬁnite

diﬀerence approach, we suggest and analyze a number of new multi step

methods for solving triequilibrium inclusions. Convergence analysis of

these methods is investigated under suitable conditions. Sensitivity

analysis is also investigated. Various special cases are discussed as

applications of the main results. Several open problems are suggested

for future research.

Keywords: equilibrium inclusions, convex functions, ﬁxed points, iter-

ative methods, convergence analysis, dynamical system, sensitivity analysis.

MSC: 26D15, 26D10, 49J40, 65N35,49J40, 90C26, 90C30.

1 Introduction

Equlibrium problems which were introduced by Blum et al. [7] and Noor

et al. [55] provide us with a uniﬁed, natural, novel, innovative and general

^∗Accepted for publication on February 05, 2026

^†noormaslam@gmail.com, Department of Mathematics, University of Wah, Wah Cantt,

Pakistan

^‡khalidan@gmail.com, Department of Mathematics, University of Wah, Wah Cantt,

Pakistan

179

General triequilibrium problems

180

technique to study a wide class of problems arising in diﬀerent branches of

mathematical and engineering sciences. Equilibrium problems can be viewed

as a novel and important generalization of the variational inequalities and

variational principles. By variational principles, we mean maximum and

minimum problems arising in game theory, mechanics, geometrical optics,

general relativity theory, ﬁeld theory, economics, transportation, diﬀerential

geometry and related areas. These are fascinating interesting ﬁelds that a

wide class of unrelated problems can be studied in the general and uniﬁed

framework of variational inequalities and equilibrium problems. For more

details of the applications and generalizations of the equilibrium problems

and variational inequalities, see [5–7, 10 –19, 21–23, 26 –28, 28–47, 49–55, 58–

65,68] and the references therein.

One of the most diﬃcult and important problem is the development

of eﬃcient numerical methods. Lions and Stampacchia [21] and Noor [26]

proved that the quasi variational inequalities are equivalent to the ﬁxed

point problem. This alternative formulation was used to suggest and in-

vestigate three-step iterations for solving the variational inequalities. These

three-step iterations contain Noor (three step) iterations [30–32], Picard

method, Mann (one step) iterations and Ishikawa (two-step) iterations as

special cases. Suantai et. al. [62] have also considered some novel forward-

backward algorithms for optimization and their applications to compressive

sensing and image inpainting. Noor iterations have inﬂuenced the research

in the ﬁxed point theory and will continue to inspire further research in

fractal geometry, chaos theory, coding, number theory, spectral geometry,

dynamical systems, complex analysis, nonlinear programming, graphics and

computer aided design. These three-step schemes are a natural generaliza-

tion of the splitting methods for solving partial diﬀerential equations.

The projected dynamical systems associated with variational inequalities

were considered by Dupuis and Nagurney [13]. The novel feature of the pro-

jected dynamical system is that its set of stationary points corresponds to

the set of the corresponding set of the solutions of the variational inequality

problem. This dynamical system is a ﬁrst order initial value problem. Con-

sequently, equilibrium and nonlinear problems arising in various branches

in pure and applied sciences can now be studied in the setting of dynamical

systems. It has been shown [13,23,44,47,50,51,53,54,66] that the dynami-

cal systems are useful in developing some eﬃcient numerical techniques for

solving variational inequalities and related optimization problems.

The sensitivity analysis provides useful information for designing or plan-

ning various equilibrium systems. Sensitivity analysis can provide new in-

sight and stimulate new ideas and techniques for problem solving. Dafer-

M.A. Noor, K.I. Noor

181

mos [12] studied the sensitivity analysis of the variational inequalities using

the ﬁxed point technique. This approach has strong geometrical ﬂavor and

has been investigated for various classes of variational inequalities and their

variant forms, see [12,29,42,47,53,54,65] and the references therein.

We would like to point out that it is not possible to establish the equva-

lence between the equilibrium problems and the ﬁxed point problems. Due

to these drawback, one can not suggest the multistep iterative methods for

solving the equilibrium problems. To overcome its draw back and facts,

Noor and Noor [24] have introduced and studied some classes of equilibrium

inclusions. They have proved that the equilibrium inclusions are equivalent

to the ﬁxed point problems. These equivalent ﬁxed point formulation have

been used to suggest and analyze some classes of hybrid iterative methods.

Motivated and inspired by the research activities in this direction, we intro-

duce some new classes of extended general triequilibrium inclusions involving

the maximal monotone operator. We establish the equivalence between the

extended general triequilibrium inclusions and ﬁxed point problem exploring

the resolvent operator approach. This alternative equivalent formulation is

used to consider the existence of the solution as well as to analyze some multi

step an iterative method for solving the triequilibrium inclusions. Several

special cases are discussed as applications of the triequilibrium l inclusions

in Section 2. These multi step methods can be viewed as a novel gener-

alization of the Noor (three step) iterations [25 ], which have applications

in ﬁxed point, fractal geometry, information technology, machine learning

and medical sciences and signal processing. In section 3, we discuss the

unique existence of the solution as well as to suggest several inertial itera-

tive method along with the convergence analysis. In Section 4, dynamical

system approach is applied to study the stability of the solution as well as

to suggest some iterative methods for solving the extended general triequi-

librium problems exploring the ﬁnite diﬀerence idea. Our results in this

section can be viewed as signiﬁcant reﬁnement of the known results. Sensi-

tivity analysis for variational inequalities has been studied by many authors

using quite diﬀerent techniques. In Section 5, we obtain some new results

for the sensitivity analysis of the extended general triequilibrium inclusions.

One of the main purposes of this paper is to demonstrate the close con-

nection among various classes of algorithms for the solution of the extended

general equilibrium inclusions and to point out that researchers in diﬀerent

ﬁeld of equilibrium inclusions and optimization. These results may motivate

and bring a large number of novel, innovate potential applications, exten-

sions and interesting topics in these areas. We have given only a brief intro-

duction of this new ﬁeld of triequilibrium inclusions. The interested readers

General triequilibrium problems

182

may explore this ﬁeld further and discover novel and fascinating applications

of the extended general equilibrium inclusions in other areas of sciences such

as fractal geometry, chaos theory, coding, number theory, spectral geome-

try, dynamical systems, complex analysis, nonlinear programming, graphics,

computer aided design and related other optimization problems. It is ex-

pected the techniques and ideas of this paper may be starting point for

further research.

2 Formulations and basic facts

Let Ω be a nonempty closed convex set in a real Hilbert space H. We denote

by h·, ·i and k · k be the inner product and norm, respectively. First of all,

we recall some concepts from convex analysis [1,10,25,54] which are needed

in the derivation of the main results.

We consider the extended general triequilibrium inclusion problem. For

given nonlinear operators T, g, h = H −→ H, a trifunction F(., ., .) : H×H×

H −→ H and maximal monotone operator A(.), we consider the problem of

ﬁnding µ ∈ H, such that

0 ∈ ρF(µ, T(µ), ν) + g(µ) − h(µ) + ρA(g(µ)),

∀ν ∈ H,

(1)

which is called the extended general triequilibrium inclusion.

Special Cases.

1. For g = h, the problem (1) reduces to funding µ ∈ H such that

0 ∈ ρF(µ, T(µ), ν) + ρA(g(µ)),

∀ν ∈ H

(2)

is known as the triequilibrium inclusion.

2. If A(·) = ∂ϕ(·) : H −→ R ∪ {+∞} , the subdiﬀerential of a convex,

proper and lower semi-continuous function ϕ(·), then problem (1) is

equivalent to ﬁnding µ ∈ H such that

hρF(µ, T(µ), ν) + g(µ) − h(µ), h(ν) − g(µ)i

+ρ(ϕ(h(ν)) − ϕ(g(µ)) ≥ 0,

∀ν ∈ H,

(3)

which is called the mixed general triequilibrium variational inequality.

M.A. Noor, K.I. Noor

183

3.

If the function ϕ(·) is the indicator function of a closed convex set

Ω in H, that is,

ꢀ

0,

if µ ∈ Ω

ϕ(µ) =

+∞, otherwise ,

then problem (3) is equivalent to ﬁnding µ ∈ Ω, such that

hF(µ, T(µ), ν) + g(µ) − h(µ), h(ν) − g(µ)i ≥ 0,

∀ν ∈ Ω,

(4)

is called the general triequilibrium variational inequality.

4. For F(µ, T(µ), ν) = hT(µ), g(µ) − g(ν)i, the problem (4) reduces to

ﬁnding µ ∈ Ω such that

hT(µ), g(µ) − g(ν)i ≥ 0,

∀ν ∈ Ω,

(5)

is called the general variational inequality, introduced and studied by

Noor [27] in 1988. For applications, modiﬁcation and numerical as-

pects of the general variational inequalities, see [27 ,32 ,53 ,54].

5.

If Ω^∗= {µ ∈ H, hµ, νi ≥ 0,

∀ν ∈ Ω} is a polar cone of the convex

cone Ω in H and h = g, then the problem (4) is equivalent to ﬁnding

µ ∈ H, such that

g(µ) ∈ Ω,

F(µ, T(µ), ν) ∈ Ω^∗,

hF(µ, T(µ), ν), g(µ)i = 0,

(6)

is called the triequilibrium complementarity problem, which appears

to be a new one. For F(µ, T(µ), ν) = T(µ), the triequilibrium problem

(6) reduces to ﬁnding µ ∈ H such that

g(µ) ∈ Ω,

T(µ) ∈ Ω^∗,

hT(µ), g(µ)i = 0,

is known as the general complementarity problem, introduced and

studied by Noor [27] in 1988, which include the nonlinear comple-

mentarity problem as a special case. For applications, formulations

and generalizations of the complementarity problems, see [9,27 ,32 ,41,

53,54].

For special choices of the single valued operators T, h, g, A(., .) the continu-

ous trifunction F(., ., .) and the closed convex set Ω, one can obtain a wide

class of complementarity problems and variational inequality problems as

special cases of the extended general triequilibrium problem (1). Thus, it is

clear that the problem (1) is very general and unifying one and has numerous

applications in pure and applied sciences.

We now recall some well known results and notions.

General triequilibrium problems

184

Deﬁnition 1. If A is a set valued maximal monotone operator on H, then,

for a constant ρ > 0, the resolvent operator is deﬁned by

J_A= (I + ρA)⁻¹(µ),

∀µ ∈ H,

where I is the identity operator.

It is known that the resolvent operator J_Ais single-valued deﬁned on all

of H by Minty’s theorem [58] and satisﬁes the following assumption.

Assumption 1. [58] The resolvent operator J_Ais nonexpansive.

kJ_A(µ) − J_A(ν)k ≤ kµ − νk, ∀µ, ν ∈ H.

(7)

Assumption 1 is used to prove the existence of a solution of extended

general triequilibrium inclusions as well as in analyzing convergence of the

iterative methods.

Deﬁnition 2. The trifunction F(., ., .) : H × H → H with respect to an

arbitrary operator T is said to be:

1. Strongly jointly monotone, if there exist a constant α > 0, such that

2

F(µ, T(µ), ν) − F(η, T(η), ν) ≥ αkµ − ηk ,

∀η, µ, ν ∈ H.

2. jointly Lipschitz continuous, if there exists a constant β > 0, such that

kF(µ, T(µ), ν) − F(η, T(η), ν)k ≤ βkµ − ηk,

∀η, µ, ν ∈ H.

3. jointly monotone, if

hF(µ, T(µ), ν) − F(η, T(η), ν), νi ≥ 0,

∀µ, ν ∈ H.

4. pseudo monotone, if

F(µ, T(µ), ν) ≥ 0

=⇒

−F(ν, T(ν), µ ≥ 0,

∀µ, ν ∈ H.

Deﬁnition 3. An operator T : H → H is said to be:

1. Strongly monotone, if there exists a constant α > 0, such that

2

hT(µ) − T(ν), µ − νi ≥ αkµ − νk ,

∀µ, ν ∈ H.

M.A. Noor, K.I. Noor

185

2. Lipschitz continuous, if there exists a constant β > 0, such that

kT(µ) − T(ν)k ≤ βkµ − νk,

∀µ, ν ∈ H.

3. Monotone, if

hT(µ) − T(ν), µ − νi ≥ 0,

∀µ, ν ∈ H.

4. Pseudo monotone, if

hT(µ), ν − µi ≥ 0

⇒

hT(ν), ν − µi ≥ 0,

∀µ, ν ∈ H.

Remark 1. Every strongly monotone operator is a monotone operator and

monotone operator is a pseudo monotone operator, but the converse is not

true.

3 Resolvent methods

In this section, we use the ﬁxed point formulation to suggest and analyze

some new implicit methods for solving the extended general triequilibrium

inclusions. First of all, we establish the equivalence between the extended

general triequilibrium inclusions and the ﬁxed point problem applying the

resolvent operator approach.

Lemma 1. The function µ ∈ H is a solution of the extended general triequi-

librium inclusion (1), if and only if, µ ∈ H satisﬁes the relation

g(µ) = J_A[h(µ) − ρF(µ, T(µ), ν)],

∀ν ∈ H,

(8)

where J_Ais the resolvent operator and ρ > 0 is a constant.

Proof. Let µ ∈ H be a solution of (1), then, for a constant ρ and ∀ν ∈ H,

ρF(µ, T(µ), ν) + g(µ)

−

h(µ) + ρA(g(µ)) 3 0,

⇐⇒

+

−h(µ) + ρF(µ, T(µ), ν)

g(µ) + ρA(g(µ)) 3 0

⇐⇒

g(µ) = J_A[h(µ) − ρF(µ, T(µ), ν)].

the required (8).

General triequilibrium problems

186

Lemma 1 implies that the general triequilibrium inclusion (1) is equiva-

lent to the ﬁxed point problem (8). This equivalent ﬁxed point formulation

(8) plays an important role in deriving the main results.

From equation (8), we have

ꢁ

ꢂ

µ = µ − g(µ) + J_Ah(µ) − ρF(µ, T(µ), ν)) .

We deﬁne the function Φ associated with (8) as

ꢁ

ꢂ

Φ(µ) = µ − g(µ) + J_Ah(µ) − ρF(µ, T(µ), ν)) .

(9)

To prove the unique existence of the solution of the problem (1 ), it is enough

to show that the map Φ deﬁned by (9) has a ﬁxed point.

Theorem 1. Let the operator g be strongly monotone with constant σ > 0

and Lipschitz continuous with constant ζ > 0, respectively. Let the bifunction

F(., ., .) be jointly Lipschitz continuous with constant β and the operator h

be Lipschitz continuous with constant ζ₁. If there exists a parameter ρ > 0,

such that

1 − k

ρ <

,

k < 1,

ζ₁< 1,

ζ²< 2σ,

(10)

β

where

θ = ρβ + k

(11)

(12)

p

k =

1 − 2σ + ζ²+ ζ₁.

Then there exists a unique solution of the problem (1 ).

Proof. From Lemma 1, it follows that problems (8) and (1) are equivalent.

Thus, it is enough to show that the map Φ(u), deﬁned by (9) has a ﬁxed

M.A. Noor, K.I. Noor

187

point. For all η = µ ∈ H, we have

ꢃ

Φ(µ) − Φ(η) = µ − η − (g(µ) − g(η))

ꢃ

ꢄ

ꢅ

ꢄ

ꢅ

ꢃ

+J_A

h(µ) − ρF(µ, T(µ), ν)) − J_Ah(η) − ρF(η, T(η), ν)

ꢃ

≤ µ − η − (g(µ) − g(η))

ꢃ

+ h(η) − h(µ) − ρ(F(η, T(η), ν) − F(µ, T(µ), ν))

ꢃ

≤ µ − η − (g(µ) − g(η))

ꢃ

+ h(η) − h(µ) + ρ (F(η, T(η), ν) − F(µ, T(µ), ν)

ꢃ

≤ µ − η − (g(µ) − g(η) + ζ₁η − µ + ρβ η − µ .

(13)

ꢃ

Since the operator g is strongly monotone with constant σ > 0 and Lipschitz

continuous with constant ζ > 0, it follows that

ꢃ

ꢆ

ꢇ

2

ꢃ

µ − η − (g(µ) − g(η))

≤

µ − η

− 2 g(µ) − g(η), µ − η

ꢃ

2

+ζ²g(µ) − g(η)

ꢃ

2

ꢃ

≤ (1 − 2σ + ζ ) µ − η

.

(14)

From (13) and (14), we have

n

o

p

ꢃ

Φ(µ) − Φ(ν)

≤ 2

(1 − 2σ + ζ²) + ζ₁+ ρβ

µ − νk

ꢃ

= θ µ − η ,

where

θ = ρβ + k

p

k = 2 1 − 2σ + ζ²+ ζ₁.

From (10), it follows that θ < 1, which implies that the map Φ(u) deﬁned

by (9 ) has a ﬁxed point, which is the unique solution of (1).

The ﬁxed point formulation (8) is applied to propose and suggest some

iterative methods for solving the problem (1).

General triequilibrium problems

188

Algorithm 1. For a given µ₀, compute the approximate solution {µ_n+1} by

the iterative schemes

y_n= (1 − γ_n)µ_n+ γ_n{µ_n− g(µ_n) + J_A[h(µ_n) − ρF(µ_n, T(µ_n), ν)]} (15)

z_n= (1 − β_n)µ_n+ β_n{y_n− g(y_n) + J_A[h(y_n) − ρF(y_n, T(y_n), ν)]} (16)

µ_n+1= (1 − α_n)µ_n

+α_n{z_n− g(z_n) + J_A[h(z_n) − ρF(z_n, T(z_n), ν]},

(17)

which are known as modiﬁed Noor iterations.

We now study the convergence analysis of Algorithm 1 , which is the

main motivation of our next result.

Theorem 2. Let the operators g, h and the trifunction F(., ., .) satisfy all the

assumptions of Theorem 1 . If the condition (10) holds, then the approximate

solution {u_n} obtained from Algorithm 1 converges to the exact solution

µ ∈ H of the extended general triequilibrium inclusion (1) strongly in H.

Proof. From Theorem 1, we see that there exists a unique solution µ ∈ H of

the general triequilibrium inclusions (1 ). Let µ ∈ H be the unique solution

of (1). Then, using Lemma 1, we have

µ = (1 − α_n)µ + α_n{µ − g(µ) + J_A[h(µ) − ρF(µ, T(µ), ν)]}

= (1 − β_n)µ + β_n{µ − g(µ) + J_A[h(µ) − ρF(µ, T(µ), ν)]}.

= (1 − γ_n)µ + γ_n{µ − g(µ) + J_A[h(µ) − ρF(µ, T(µ), ν)]}.

(18)

(19)

(20)

From (17),(18), we have

kµ_n+1− µk = k(1 − α_n)(µ_n− µ) + α_n(z_n− µ − (g(z_n) − g(µ)))

+α_n{A[h(µ_n) − ρF(z_n, T(z_n), ν)] − J_A([h(µ) − ρF(µ, T(µ), ν)}k

≤ (1 − α_n)kµ_n− µk + α_nkz_n− µ − (g(z_n) − g(µ))k

+α_nkh(w_n) − h(µ) − ρ(F(z_n, T(z_n), ν) − F(µ, T(µ), ν)k

≤ (1 − α_n)kµ_n− µk + α_n(k + ρβ)||z_n− µk

= (1 − α_n)ku_n− µk + α_nθkz_n− µk,

(21)

where θ is deﬁned by (11).

In a similar way, from (19) and (16), we have

kz_n− µk ≤ (1 − β_n)kµ_n− µk + 2β_nθky_n− µ − (g(y_n) − g(µ))k

+ β_nkg(y_n) − g(µ) − ρ(y_n− µ)k + β_nηky_n− µk

≤ (1 − β_n)kµ_n− µk + β_n(k + ρ)ky_n− µk,

≤ (1 − β_n)kµ_n− µk + β_nθky_n− µk,

(22)

M.A. Noor, K.I. Noor

189

where θ is deﬁned by (10).

From (15) and (19), we obtain

ky_n− µk ≤ (1 − γ_n)kµ_n− µk + γ_nθkµ_n− µk

≤ (1 − (1 − θ)γ_n)kµ_n− µk ≤ kµ_n− µk.

(23)

(24)

From (22) and (23), we obtain

kz_n− µk ≤ (1 − β_n)kµ_n− µk + β_nθkµ_n− µk

= (1 − (1 − θ)β_n)kµ_n− µk ≤ kµ_n− µk.

From the above equations, we have

kµ_n+1− µk ≤ (1 − α_n)kµ_n− µk + α_nθkµ_n− µk

= [1 − (1 − θ)α_n]kµ_n− µk

n

Y

≤

[1 − (1 − θ)α_i]kµ₀− µk.

i=0

P_∞

Since

_n=0α_ndiverges and 1 − θ > 0, we have limit_i=0[1 − (1 − θ)α_i] = 0.

Consequently the sequence {u_n} convergence strongly to µ. From (23), and

(24), it follows that the sequences {y_n} and {w_n} also converge to µ strongly

in H. This completes the proof.

We suggest new perturbed iterative schemes for solving the extended

general triequilibrium inclusion (1).

Algorithm 2. For a given µ₀, compute the approximate solution {µ_n} by

the iterative schemes

y_n= (1 − γ_n)µ_n

+γ_n{µ_n− g(µ_n) + J_A[h(µ_n) − ρF(µ_n, T(µ_n), ν)]} + γ_nh_n

z_n= (1 − β_n)µ_n

+β_n{y_n− g(y_n) + J_A[h(y_n) − ρF(y_n, T(y_n), ν)]} + β_nf_n

µ_n+1= (1 − α_n)µ_n

+α_n{z_n− g(z_n) + J_A[h(z_n) − ρF(z_nT(z_n), ν))} + α_ne_n,

where {e_n}, {f_n}, and {h_n} are the sequences of the elements of H intro-

duced to take into account possible inexact computations and J_{A(µ )}is the

n

corresponding perturbed resolvent operator and the sequences {α_n}, {β_n} and

{γ_n} satisfy

∞

X

0 ≤ α_n, β_n, γ_n≤ 1;

∀n ≥ 0,

α_n= ∞.

n=0

General triequilibrium problems

190

For γ_n= 0, we obtain the perturbed Ishikawa iterative method and for

γ_n= 0 and β_n= 0, we obtain the perturbed Mann iterative schemes for

solving general equilibrium inclusion (1).

Also, we can suggest the following iterative methods for solving the ex-

tended general triequilibrium inclusions.

Algorithm 3. For a given µ₀, compute µ_n+1by the iterative scheme

µ_n+1= µ_n− g(µ_n) + J_A[h(µ_n) − ρF(µ_n, T(µ_n), ν))],

which is known as the resolvent method.

Algorithm 4. For a given µ₀, compute µ_n+1by the iterative scheme

µ_n+1= µ_n− g(µ_n) + J_A([h(µ_n) − ρF(µ_n+1, T(µ_n+1), ν))],

which is an implicit resolvent method and is equivalent to the following two-

step method.

Algorithm 5. For a given µ₀, compute µ_n+1by the iterative scheme

z_n= µ_n− g(µ_n) + J_A[h(µ_n) − ρF(µ_n, T(µ_n), ν)]

µ_n+1= µ_n− g(µ_n) + J_A[h(µ_n) − ρF(z_n, T(z_n), ν))].

Algorithm 6. For a given µ₀, compute µ_n+1by the iterative scheme

µ_n+1= µ_n− g(µ_n) + J_A[h(µ_n+1) − ρF(µ_n+1, T(µ_n+1), ν)],

which is known as the modiﬁed resolvent method and is equivalent to the

iterative method.

Algorithm 7. For a given µ₀, compute µ_n+1by the iterative scheme

z_n= µ_n− g(µ_n) + J

[g(µ_n) − ρF(µ_n, T(µ_n), ν)]

A(µ_n)

µ_n+1= µ_n− g(µ_n) + J_A[h(z_n) − ρF(z_n, T(z_n), ν)],

which is two-step predictor-corrector method for solving the problem (1).

We can rewrite the equation (8) as:

ꢄ

ꢊ

ꢋꢅ

ꢈ

ꢉ

µ + µ

µ = µ − g(µ) + J_A

h

− ρF (

), T(

), ν

.

2

This ﬁxed point formulation is used to suggest the following implicit method.

M.A. Noor, K.I. Noor

191

Algorithm 8. For a given µ₀, compute µ_n+1by the iterative scheme

µ_n+1= µ_n− g(µ_n)

ꢄ

ꢊ

ꢋꢅ

µ_n+ µ_n+1

+J_Ah(

) − ρF (

), T(

), ν

.

2

To implement the implicit method, one uses the predictor-corrector tech-

nique to obtain a new two-step method for solving the problem (1).

Algorithm 9. For a given µ₀, compute µ_n+1by the iterative scheme

z_n= µ_n− g(µ_n) + J_A[h(µ_n) − ρF(µ_n, T(µ_n), ν)]

ꢄ

ꢅ

ꢈ

ꢉ

z_n+ µ_n

µ_n+1= µ_n− g(µ_n) + J_Ah(

− ρF((

), T(

), ν) .

2

We now suggest multi-step inertial methods for solving the extended

general triequilibrium inclusions (1).

Algorithm 10. For given µ₀, µ₁, compute µ_n+1by the recurrence relation

z_n= µ_n− θ_n(µ_n− µ_n−1) ,

n = 1, 2, ... . . .

y_n= (1 − γ_n)z_n

ꢀ

ꢄ ꢊ

ꢋ

ꢊ

ꢋꢅ ꢌ

z_n+µ_n

2

z_n+µ_n

2

− ρF (^{z +µ}), T(

), ν

,

n

+γ_nz_n− g(z_n) + J_A

h

2

ꢀ

t_n= (1 − β_n)y_n+ β_ny_n− g(y_n)

ꢄ ꢊ

ꢋ

ꢊ

ꢋꢅ ꢌ

y_n+z_n+µ_n

3

y_n+z_n+µ_n

3

− ρF (^{y +z +µ}), T(

), ν

,

n

3

n

+J_A

h

ꢀ

ꢄ ꢊ

ꢋ

z_n+y_n+t_n+µ_n

4

µ_n+1= (1 − α_n)z_n+ α_nt_n− g(t_n) + J_A

h

ꢊ

ꢋꢅꢌ

y_n+z_n+t_n+µ_n

4

−ρF (^{z +t +µ}), T(

), ν

,

n

4

n

where α_n, β_n, γ_n, θ_n∈ [0, 1],

∀n ≥ 1.

For g = h, Algorithm 10 reduces to:

General triequilibrium problems

192

Algorithm 11. For given µ₀, µ₁, compute µ_n+1by the recurrence relation

z_n= µ_n− θ_n(µ_n− µ_n−1) ,

n = 1, 2, ... . . .

y_n= (1 − γ_n)z_n

ꢀ

ꢄ ꢊ

ꢋ

ꢊ

ꢋꢅ ꢌ

z_n+µ_n

2

z_n+µ_n

2

+γ_nz_n− g(z_n) + J_A

g

− ρF (^{z +µ}), T(

), ν

,

n

2

ꢀ

t_n= (1 − β_n)y_n+ β_ny_n− g(y_n)

ꢄ ꢊ

ꢋ

ꢊ

ꢋꢅ ꢌ

y_n+z_n+µ_n

3

y_n+z_n+µ_n

3

− ρF (^{y +z +µ}), T(

), ν

,

n

3

n

+J_A

g

ꢀ

µ_n+1= (1 − α_n)z_n+ α_nt_n− g(t_n)

ꢄ ꢊ

ꢋ

ꢊ

ꢋꢅ ꢌ

z_n+y_n+t_n+µ_n

4

y_n+z_n+t_n+µ_n

4

− ρF (^{z +t +µ}), T(

), ν

,

n

4

n

+J_A

g

where α_n, β_n, γ_n, θ_n∈ [0, 1],

∀n ≥ 1, for solving the general triequilibrium

inclusions (2).

Remark 2. For diﬀerent and suitable choice of the parameters ρ, η, α, opera-

tors T, g, h, the trifunction F(., ., .) and convex sets, one can recover new and

known iterative methods for solving general triequilibrium inclusions, equi-

librium complementarity problems and related optimization problems. Using

the technique and ideas of Theorem 1 and Theorem 2, one can analyze the

convergence of Algorithm 10 and its special cases.

.

4 Dynamical systems technique

In this section, we consider the dynamical system technique for solving the

extended general triequilibrium inclusions. The projected dynamical sys-

tems associated with variational inequalities were considered by Dupuis and

Nagurney [13]. It is worth mentioning that the dynamical systems are the

initial value and boundary value problems. Consequently, variational in-

equalities and nonlinear problems arising in various branches in pure and

applied sciences can now be studied via the diﬀerential equations. It has

been shown that the dynamical systems are useful in developing some ef-

ﬁcient numerical techniques for solving variational inequalities and related

optimization problems, see [13,23 ,32,44,47,50,52 –54 ,66]. We consider some

M.A. Noor, K.I. Noor

193

new iterative methods for solving the extended general triequilibrium inclu-

sions and investigate the convergence analysis of these new methods involv-

ing only the monotonicity of the operators.

We now deﬁne the residue vector R(µ) by the relation

ꢀ

ꢌ

R(µ) = J_A[h(µ) − ρF(µ, T(µ), ν)] − g(µ) ,

∀ν ∈ H.

(25)

Invoking Lemma 1, one can easily conclude that µ ∈ H is a solution of

the problem(1), if and only if, µ ∈ H is a zero of the equation

R(µ) = 0.

(26)

We now consider a dynamical system associated with the extended gen-

eral triequilibrium inclusions. Using the equivalent formulation (8 ), we sug-

gest a class of resolvent dynamical systems as

ꢀ

ꢌ

dµ

dt

= λ J_A[h(µ) − ρF(µ, T(µ), ν)] − g(µ) ,

µ(t₀) = α,

(27)

where λ is a parameter. The system of type (27) is called the resolvent

dynamical system associated with the problem (1). Here the right hand

is related to the resolvent and is discontinuous on the boundary. From

the deﬁnition, it is clear that the solution of the dynamical system always

stays in H. This implies that the qualitative results such as the existence,

uniqueness and continuous dependence of the solution of (1) can be studied.

The equilibrium point of the dynamical system (27) is deﬁned as follows.

Deﬁnition 4. An element µ ∈ H, is an equilibrium point of the dynamical

system (27), if,

dµ

= 0.

dx

Thus, it is clear that µ ∈ H is a solution of the extended general equilib-

rium inclusion (1), if and only if, µ ∈ H is an equilibrium point. This implies

that µ ∈ H is a solution of the extended general triequilibrium inclusion (1),

if and only if, µ ∈ H is an equilibrium point.

Deﬁnition 5. ( [13]) The dynamical system is said to converge to the solu-

tion set S^∗of (27), if , irrespective of the initial point, the trajectory of the

dynamical system satisﬁes

lim dist(µ(t), S^∗) = 0,

(28)

t→∞

where

dist(µ, S^∗) = inf_ν∈S_∗kµ − νk.

General triequilibrium problems

194

It is easy to see, if the set S^∗has a unique point µ^∗, then (28) implies

that

lim µ(t) = µ^∗.

t→∞

If the dynamical system is still stable at µ^∗in the Lyapunov sense, then the

dynamical system is globally asymptotically stable at µ^∗.

Deﬁnition 6. The dynamical system is said to be globally exponentially

stable with degree η at µ^∗, if, irrespective of the initial point, the trajectory

of the system satisﬁes

kµ(t) − µ^∗k ≤ u₁kµ(t₀) − µ^∗kexp(−η(t − t₀)),

∀t ≥ t₀,

where u₁and η are positive constants independent of the initial point.

It is clear that the globally exponentially stability is necessarily globally

asymptotically stable and the dynamical system converges arbitrarily fast.

Lemma 2. (Gronwall Lemma)( [18]) Let µˆ and νˆ be real-valued nonnegative

continuous functions with domain {t : t ≤ t₀} and let α(t) = α₀(|t − t₀|),

where α₀is a monotone increasing function. If, for t ≥ t₀,

Z

t

µˆ ≤ α(t) +

µˆ(s)νˆ(s)ds,

t₀

then

Z

t

µˆ(s) ≤ α(t)exp{

νˆ(s)ds}.

t₀

We now establish that the trajectory of the solution of the resolvent

dynamical system (27) converges to the unique solution of the extended

general trifunction equilibrium inclusions (1).

Theorem 3. Let the trifunction F(., ., .) be jointly Lipschitz continuous with

constantβand the operators g, h : H −→ H be Lipschitz continuous with

constants ζ > 0, ζ₁> 0 respectively. If λ(ζ + ζ₁+ ρβ) < 1, then, for each

µ₀∈ H, there exists a unique continuous solution µ(t) of the dynamical

system (27) with µ(t₀) = µ₀over [t₀, ∞).

Proof. Let

G(µ) = {J_A[h(µ) − ρF(µ, T(µ), ν)] − g(µ)},

∀µ ∈ H,

M.A. Noor, K.I. Noor

195

dµ

dt

where λ > 0 is a constant and G(µ) =

. For ∀µ, ν ∈ H, we have

kG(µ) − G(η)k ≤ λ{J_A[h(µ) − ρF(µ, T(µ), ν)] − J_A[h(η) − ρF(η, T(η), ν)]k}

+λkg(µ) − g(η)k

= λ{kg(µ) − g(η)k + kJ_A[h(µ) − ρF(µ, T(µ), ν)] − J_A[h(η) − ρF(η, T(η), ν)]k

≤ λ{kg(µ) − g(η)k + kh(µ) − h(η) − ρ(F(µ, T(µ), ν) − F(η, T(η), ν))}

≤ λ{kg(µ) − g(η)k + kh(µ) − h(η)k + ρkF(µ, T(µ), ν) − F(η, T(η), ν)k}

≤ λ{(ζ + ζ₁+ βρ)}kµ − ηk.

This implies that the operator G(µ) is a Lipschitz continuous with con-

stant

λ{(ζ + ζ₁+ ρβ)} < 1 and for each µ ∈ H, there exists a unique and con-

tinuous solution µ(t) of the dynamical system (27), deﬁned on an interval

t₀≤ t < T₁with the initial condition µ(t₀) = µ₀. Let [t₀, T₁) be its maximal

interval of existence. Then we have to show that T₁= ∞. Consider, for any

µ ∈ Ω(µ),

dµ

kG(µ)k = k _dtk = λk[h(µ) − ρF(µ, T(µ), ν)] − g(µ)k

≤ λ{kJ_A[h(µ) − ρF(µ, T(µ), ν)] − J_A[0]k + kJ_A[0] − g(µ)k}

≤ λ{δk{g(µ) − ρF(µ, T(µ), ν)k + kJ_A[h(µ)] − J_A[0]k + kJ_A[0] − g(u)k}

≤ λ{(ρβ + ζ₁+ ζ)kuk + 2kJ_A(µ)[0]k}.

Then

Z

kµ(t)k ≤ kµ₀k +

^tkµ(s)kds

t₀

Z

≤ (kµ₀k + k₁(t − t₀)) + k₂

^tkµ(s)kds,

t₀

where k₁= 2λkJ_A(µ)[0]k and k₂= δλ(ρβ + ζ₁+ ζ). Hence, by the Gronwall

Lemma 2 , we have

kµ(t)k ≤ {ku₀k + k₁(t − t₀)}e^{k (t−t )}

,

t ∈ [t₀, T₁).

2

0

This shows that the solution is bounded on [t₀, T₁). So, T₁= ∞.

Theorem 4. If the assumptions of Theorem 3 hold, then the dynamical

system (27) converges globally exponentially to the unique solution of the

extended general equilibrium inclusion (1).

General triequilibrium problems

196

Proof. Since the trifunction F(., ., .) is jointly Lipschitz continuous and the

operators h, g are Lipschitz continuous, it follows from Theorem 3 that the

dynamical system (27) has unique solution µ(t) over [t₀, T₁) for any ﬁxed

µ₀∈ H. Let µ(t) be a solution of the initial value problem (27). For a given

µ^∗∈ H satisfying (1), consider the Lyapunov function

2

L(µ) = λkµ(t) − µ^∗k ,

u(t) ∈ H.

(29)

From (27) and (29), we have

dL

dt

dµ

dt

= 2λhµ(t) − µ^∗,

i

= 2λhµ(t) − µ^∗, J_A[h(µ(t)) − ρF(µ(t), T(µ(t)), ν)] − g(µ(t))i

= 2λhµ(t) − µ^∗, J_A[h(µ(t)) − ρF(µ(t), T(µ(t)), ν)] − g(µ^∗)

+g(µ^∗) − g(µ(t))i

= −2λhµ(t) − µ^∗, g(µ(t)) − g(µ^∗)i

+2λhµ(t) − µ^∗, J_A[h(µ(t)) − ρF(µ(t), T(µ(t)), ν)] − g(µ^∗)i

≤ −2λhρ(F(µ(t), ν) − F(µ^∗(t), T(µ^∗(t)), ν)), g(µ(t)) − g(µ^∗)i

+2λhµ(t) − µ^∗(t), J_A[g(µ(t)) − ρF(µ(t), T(µ(t)), ν)]

−J_A[h(µ^∗(t)) − ρF(µ^∗(t), T(µ^∗(t)), ν)]i,

≤ −2λσkµ(t) − µ^∗k + λkg(µ(t)) − g(µ^∗)k

2

+λkJ_A[h(µ(t)) − ρF(µ(t), T(µ(t)), ν)]

(30)

(31)

−J_A[h(µ^∗(t)) − ρF(µ^∗(t), T(µ^∗(t)), ν)k .

2

Using the jointly Lipschitz continuity of the trifunction F(., ., .) and Lips-

chitz continuity of the operator h, we have

kJ_A[h(µ(t)) − ρF(µ(t), T(µ(t)), ν)] − J_A[h(µ^∗(t))

−ρF(µ^∗(t), T(µ^∗(t)), ν))]k

≤ kh(µ(t)) − h(µ^∗(t)) − ρ(F(µ(t), T(µ(t)), ν) − F(µ^∗(t), T(µ^∗(t)), ν))k

≤ (ζ₁+ ρβ)kµ(t) − µ^∗(t)k.

(32)

From (30) and (32), we have

d

kµ(t) − µ^∗(t)k ≤ 2ξλkµ(t) − µ^∗(t)k,

dt

where

ξ = ((ζ₁+ ρβ) − 2σ).

M.A. Noor, K.I. Noor

197

Thus, for λ = −λ₁, where λ₁is a positive constant, we have

kµ(t) − µ^∗k ≤ kµ(t₀) − µ^∗ke^{−ξλ (t−t )}

,

1

0

which shows that the trajectory of the solution of the dynamical system

(27) converges globally exponentially to the unique solution of the extended

general triequilibrium inclusions (1).

We use the dynamical system (27) to suggest some iterative methods for

solving the extended general triequilibrium inclusion (1). These methods

can be viewed in the sense of Noor [25 –27] involving the double resolvent

operator.

For simplicity, we take λ = 1. Thus, the dynamical system(27) becomes

ꢁ

ꢂ

dµ

dt

+ g(µ) = J_A(µ)h(µ) − ρF(µ, T(µ), ν) ,

µ(t₀) = α.

(33)

The forward diﬀerence scheme is used to construct the implicit iterative

method. Discretizing (33 ), we have

µ_n+1− µ_n

+ g(µ_n) = J_A[h(µ_n) − ρF(µ_n+1, T(µ_n+1), ν)],

(34)

h₁

where h₁is the step size.

Now, we can suggest the following implicit iterative method for solving

the problem (1).

Algorithm 12. For a given µ₀, compute µ_n+1by the iterative scheme

ꢄ

ꢅ

µ_n+1− µ_n

µ_n+1= µ_n− g(µ_n) + J_Ah(µ_n) − ρF(µ_n+1, T(µ_n+1), ν) −

.

h₁

This is an implicit method and is equivalent to the following two-step

method.

Algorithm 13. For a given µ₀, compute µ_n+1by the iterative scheme

y_n= µ_n− g(µ_n) + J_A[h(µ_n) − ρF(µ_n, T(µ_n), ν)]

ꢄ

ꢅ

y_n− µ_n

µ_n+1= µ_n− g(µ_n) + J_Ah(µ_n) − ρF(y_n, T(y_n), ν) −

.

h₁

Discretizing (33), we now suggest other implicit iterative method for

solving the extended general triequilibrium inclusion (1)

µ_n+1− µ_n

+ g(µ_n) = J_A[h(µ_n+1) − ρF(µ_n+1, T(µ_n+1), ν)],

(35)

h

where h is the step size.

This formulation enables us to suggest the two-step iterative method.

General triequilibrium problems

198

Algorithm 14. For a given µ₀, compute µ_n+1by the iterative scheme

ꢄ

ꢅ

y_n= µ_n− g(µ_n) + J_Ah(µ_n) − ρF(µ_n, T(µ_n), ν)

ꢄ

ꢅ

y_n− µ_n

µ_n+1= µ_n− g(µ_n) + J_Ah(y_n) − ρF(y_n, T(y_n), ν) −

.

h

Discretizing (33), we propose another implicit iterative method

ꢄ

ꢅ

µ_n+1− µ_n

+ g(µ_n) = J_Ah(µ_n) − ρF(µ_n+1, T(µ_n+1), ν) ,

h₁

where h₁is the step size.

For h₁= 1, we can suggest an implicit iterative method for solving the

problem (1).

Algorithm 15. For a given µ₀, compute µ_n+1by the iterative scheme

ꢄ

ꢅ

µ_n+1= µ_n− g(µ_n) + J_Ah(µ_n) − ρF(µ_n+1, T(µ_n+1), ν) .

From (33), we have

ꢄ

dµ

+ g(µ) = J_Ah((1 − α)µ + αµ)

dt

ꢅ

−ρF((1 − α)µ + αµ), T((1 − α)µ + αµ)), ν) , (36)

where α ∈ [0, 1] is a constant.

Discretization (36) and taking h₁= 1, we have

ꢄ

ꢈ

ꢉ

µ_n+1= µ_n− g(µ_n) + J_Ah (1 − α)µ_n+ αµ_n−1

,

ꢅ

−ρF((1 − α)µ_n+ αµ_n+1), T((1 − α)µ_n+ αµ_n−1), ν) ,

which is an inertial type iterative method for solving the extended general

triequilibrium inclusion (1). Using the predictor-corrector techniques, we

have

M.A. Noor, K.I. Noor

199

Algorithm 16. For a given µ₀, µ₁, compute µ_n+1by the iterative schemes

y_n= (1 − α)µ_n+ αµ_n−1

,

n = 1, 2, ... . . .

ꢄ

ꢅ

µ_n+1= µ_n− g(µ_n) + J_Ah(y_n) − ρF(y_n, T(y_n), ν) ,

which is known as the inertial two-step iterative method.

We now introduce the second order dynamical system associated with

the extended general triequilibrium inclusion (1). To be more precise, we

consider the problem of ﬁnding µ ∈ H such that

ꢀ

ꢄ

ꢅ

ꢌ

d²µ

dx²

dµ

dx

γ

+

= λ J_Ah(µ) − ρF(µ, T(µ), ν) − g(µ) ,

(37)

µ(a) = α, µ(b) = β,

where γ > 0, λ > 0 and ρ > 0 are constants. We would like to emphasize

that the problem (37) is indeed a second order boundary vale problem. In a

similar way, we can deﬁne the second order initial value problem associated

with the dynamical system.

The equilibrium point of the dynamical system (37) is deﬁned as follows.

Deﬁnition 7. An element µ ∈ H, is an equilibrium point of the dynamical

system (37), if,

d²µ

dx²

dµ

dx

γ

+

= 0.

Thus, it is clear that µ ∈ H is a solution of the extended general triequi-

librium inclusion (1 ), if and only if, µ ∈ H is an equilibrium point.

From (37), we have

ꢄ

ꢅ

g(µ) = J_Ah(µ) − ρF(µ, T(µ), ν) .

Thus, we can rewrite (37) as follows:

ꢄ

ꢅ

d²µ

dx²

dµ

dx

g(µ) = J_Ah(µ) − ρF(µ, T(µ), ν) + γ

+

.

(38)

(39)

For λ = 1, the problem (37) is equivalent to ﬁnding µ ∈ H such that

ꢄ

ꢅ

d²µ

dx²

dµ

dx

γ

+

+ g(µ) = J_Ah(µ) − ρF(µ, T(µ), ν) ,

µ(a) = α, µ(b) = β.

General triequilibrium problems

200

The problem (39 ) is called the second dynamical system, which is in fact a

second order boundary value problem. This interlink among various ﬁelds

of mathematical and engineering sciences is fruitful in developing imple-

mentable numerical methods for ﬁnding the approximate solutions of the

extended general triequilibrium inclusions. Consequently, one can explore

the ideas and techniques of the diﬀerential equations to suggest and propose

hybrid proximal point methods for solving the extended general triequilib-

rium variational inclusions and related optimization problems.

We discretize the second-order dynamical systems (39) using central ﬁ-

nite diﬀerence and backward diﬀerence schemes to have

µ_n+1− 2µ_n+ µ_n−1

µ_n− µ_n−1

+ g(µ_n)

γ

+

h²₁

h₁

ꢄ

ꢅ

= J_Ah(µ_n) − ρF(µ_n+1, T(µ_n+1), ν) ,

(40)

where h₁is the step size.

If γ = 1, h₁= 1, then, from equation( 40) we have

Algorithm 17. For a given µ₀, compute µ_n+1by the iterative scheme

ꢄ

ꢅ

µ_n+1= µ_n+ g(µ_n) + J_Ah(µ_n) − ρF(µ_n+1, T(µ_n+1), ν) ,

which is the extraresolvent method for solving the extended general triequi-

librium inclusions (1).

Algorithm 17 is an implicit method. To implement the implicit method,

we use the predictor-corrector technique to suggest the method.

Algorithm 18. For given µ₀, µ₁, compute µ_n+1by the iterative scheme

y_n= (1 − θ_n)µ_n+ θ_nµ_n−1

,

n = 1, 2, .. . . .

ꢄ

ꢅ

µ_n+1= µ_n− g(µ_n) + J_Ah(µ_n) − ρF(y_n, T(y_n), ν) ,

is called the two-step inertial iterative method, where θ_n∈ [0, 1] is a constant.

In a similar way, we have the following two-step method.

Algorithm 19. For given µ₀, µ₁, compute µ_n+1by the iterative scheme

y_n= (1 − θ_n)µ_n+ θ_nµ_n−1

,

n = 1, 2, ... . . .

ꢄ

ꢅ

µ_n+1= µ_n− g(µ_n) + J_Ah(y_n) − ρF(y_n, T(y_n), ν) ,

M.A. Noor, K.I. Noor

201

which is also called the double inertial resolvent method for solving the ex-

tended general triequilibrium inclusions (1).

We discretize the second-order dynamical systems (27) using central ﬁ-

nite diﬀerence and backward diﬀerence schemes to have

µ_n+1− 2µ_n+ µ_n−1

µ_n− µ_n−1

+ g(µ_n)

γ

+

h²₁

h₁

ꢄ

ꢅ

= J_Ah(µ_n) − ρF(µ_n+1, T(µ_n+1), ν) ,

where h₁is the step size.

Using this discrete form, we can suggest the following an iterative method

for solving the extended general triequilibrium inclusions (1).

Algorithm 20. For given µ₀, µ₁, compute µ_n+1by the iterative scheme

µ_n+1= µ_n− g(µ_n)

ꢄ

ꢅ

µ_n+1− 2µ_n+ µ_n−1

µ_n− µ_n−1

+J_Ah(µ_n+1) − ρF(µ_n+1, T(µ_n+1), ν) − γ

+

,

h²₁

h₁

n = 1, 2, ... . . .

Algorithm 20 is called the hybrid inertial proximal method for solv-

ing the extended general triequilibrium inclusions and related optimization

problems. This is a new proposed method.

Note that, for γ = 1, h₁= 1, Algorithm 20 reduces to the following

iterative method.

Algorithm 21. For given µ₀, compute µ_n+1by the iterative scheme

µ_n+1= µ_n− g(µ_n)

ꢄ

ꢅ

+J_Ah(µ_n+1) + µ_n+1− µ_n− ρF(µ_n+1, T(µ_n+1), ν) ,

which is called the resolvent method.

We now consider the third order dynamical systems associated with the

extended general triequilibrium inclusions of the type (1). To be more pre-

cise, we consider the problem of ﬁnding µ ∈ H, such that

d³µ

dt³

d²µ

dt²

dµ

γ

+ ζ

+ ξ

+ g(µ) = J_A[h(µ) − ρF(µ, T(µ), ν)],

(41)

dt

Boundary Conditions

u(a) = α, µ˙(a) = β, µ˙(b) = β₁,

General triequilibrium problems

202

where γ > 0, ζ, ξ, β, α, β₁and ρ > 0 are constants. Problem (41) is called

third order dynamical system associated with extended general triequilib-

rium inclusions (1). The equilibrium point of the dynamical system (41) is

deﬁned as follows.

Deﬁnition 8. An element µ ∈ H, is an equilibrium point of the dynamical

system (37) if

d³µ

dt³

d²µ

dt²

dµ

dt

γ

+ ζ

+ ξ

= 0.

Thus, it is clear that µ ∈ H is a solution of the general equilibrium

inclusion (1), if and only if, µ ∈ H is an equilibrium point.

Consequently, the problem (27) can be written as

ꢄ

ꢅ

d³µ

dt³

d²µ

dt²

dµ

dt

g(µ) = J_Ah(µ) − ρF(µ, T(µ), ν) + γ

+ ζ

+ ξ

.

(42)

We discretize the third-order dynamical systems (41) using central ﬁnite

diﬀerence and backward diﬀerence schemes to have

Algorithm 22. For given µ₀, µ₁, µ₂, compute µ_n+1by the iterative scheme

µ_n+2− 2µ_n+1+ 2µ_n−1− µ_n−2

2h³₁

µ_n+1− 2µ_n+ µ_n−1

γ

+ ζ

h²₁

ꢄ

ꢅ

3µ_n− 4µ_n−1+ µ_n−2

+ξ

+ g(µ_n) = J_Ah(µ_n) − ρF(µ_n+1, T(µ_n+1), ν) ,

2h₁

n = 1, 2, . . . ,

where h₁is the step size.

Similarly, discretizing dynamical systems (42 ) using central ﬁnite diﬀer-

ence and backward diﬀerence schemes, we have

ꢄꢀ

ꢌ

µ_n+1= µ_n− g(µ_n) + J_A

h(µ_n) − ρF(µ_n+1, T(µ_n+1), ν)

µ_n+2− 2u_n+1+ 2µ_n−1− µ_n−2

µ_n+1− 2µ_n+ µ_n−1

+γ

+ ζ

2h³₁

h²₁

ꢅ

3µ_n− 4µ_n−1+ µ_n−2

+ξ

.

(43)

2h₁

If γ = 1, h₁= 1, ζ = 1, ξ = 1, then, from equation( 43) after adjustment,

we have

M.A. Noor, K.I. Noor

203

Algorithm 23. For a given µ₀, compute u_n+1by the iterative scheme

ꢁ

ꢂ

µ_n+1+ 3µ_n

u_n+1= µ_n− g(µ_n) + J_Ah(µ_n) − ρF(µ_n+1, T(µ_n+1), ν) +

.

2

which is an inertial type hybrid iterative methods for solving the extended

general triequilibrium inclusions (1).

Remark 3. For appropriate and suitable choice of the operators T, g, h, the

bifunction F(., ., .), convex set, parameters and the spaces, one can suggest a

wide class of implicit, explicit and inertial type methods for solving extended

general trifunction equilibrium inclusions and related optimization problems.

5 Sensitivity analysis

In recent years variational inequalities are being used as mathematical pro-

gramming models to study a large number of equilibrium problems arising in

ﬁnance, economics, transportation, operations research and engineering sci-

ences. The behavior of such problems as a result of changes in the problem

data is always of concern, which is called sensitivity analysis. Dafermos [12 ]

considered the sensitivity analysis considered the sensitivity of the varia-

tional inequalities using essentially the projection method. These results

were extended for variational inequalities by Noor [29] and for variational

inclusions by Noor et al. [42 , 53, 54]. We like to mention that sensitivity

analysis is important for several reasons. First, estimating problem data of-

ten introduces measurement errors, sensitivity analysis helps in identifying

sensitive parameters that should be obtained with relatively high accuracy.

Second, sensitivity analysis may help to predict the future changes of the

equilibrium as a result of changes in the governing system. Third, sensitivity

analysis provides useful information for designing or planning various equi-

librium systems. Furthermore, from mathematical and engineering point of

view, sensitivity analysis can provide new insight regarding problems be-

ing studied can stimulate new ideas and techniques for problem solving the

problems due to these and other reasons. In this section, we study the

sensitivity analysis of the extended general triequilibrium inclusions , that

is, examining how solutions of such problems change when the data of the

problems are changed.

We now consider the parametric versions of the problem (1). To formu-

late the problem, let M be an open subset of H in which the parameter λ

takes values. Let g(µ, λ) be given identity operator deﬁned on H × H × M

General triequilibrium problems

204

and take value in H × H. From now onward, we denote g_λ(.) ≡ g(., λ) and

F_λ(.) ≡ F(., λ), respectively, unless otherwise speciﬁed.

The parametric extended general triequilibrium inclusions problem is to

ﬁnd µ ∈ H such that

0 ∈ ρF_λ(µ, T_λ(µ), ν) + g_λ(µ) − h_λ(µ) + ρA(g_λ(µ)),

∀ν ∈ H × M. (44)

We also assume that, for some λ ∈ M, the problem (44) has a unique

solution µ. From Lemma 1, we see that the parametric extended general

triequilibrium inclusion are equivalent to the ﬁxed point problem:

ꢄ

ꢅ

g_λ(µ) = J_Ah_λ(µ) − ρF_λ(µ, T_λ(µ), ν) ,

or equivalently

µ = µ − g_λ(µ) + J_A[h_λ(µ) − ρF_λ(µ, T_λ(µ), ν)].

We now deﬁne the mapping Φ_λassociated with the problem (44) as

Φ_λ(µ) = µ − g_λ(µ)

(45)

+J_A[h_λ(µ) − ρF_λ(µ, T_λ(µ), ν)],

∀(µ, λ) ∈ H × M.

We use this equivalence to study the sensitivity analysis of the extended

general triequilibrium inclusion. We assume that for some λ ∈ M, problem

(44) has a solution µ and X is a closure of a ball in H centered at µ. We want

to investigate those conditions under which, for each λ in a neighborhood

of λ, problem (44) has a unique solution µ(λ) near u and the function u(λ)

is (Lipschitz) continuous and diﬀerentiable.

Deﬁnition 9. Let F_λ(.) be a trifunction on X × M. Then, the trifunction

F_λ(., ., .) with respect to an arbitrary operator T_λis said to be:

(a) Locally strongly jointly monotone with constant σ > 0, if

2

hF_λ(µ, T_λ(µ), ν) − F_λ(η, T_λ(η), ν), νi ≥ σkµ − ηk ,

∀λ ∈ M, η, µ, ν ∈ X.

(b) jointly locally Lipschitz continuous with constant ζ > 0, if

kF_λ(µ, T_λ(µ), ν) − F_λ(η, T_λ(η), ν)k ≤ ζkµ − ηk,

∀λ ∈ M, η, µ, ν ∈ X.

Deﬁnition 10. An operator T_λ: H → H is said to be:

M.A. Noor, K.I. Noor

205

1. locally strongly monotone, if there exists a constant α > 0, such that

2

hT_λ(µ) − T_λ(ν), µ − νi ≥ αkµ − νk ,

∀µ, ν ∈ H.

2. locally Lipschitz continuous, if there exists a constant β > 0, such that

kT_λ(µ) − T_λ(ν)k ≤ βkµ − νk,

∀µ, ν ∈ H.

3. locally monotone, if

hT_λ(µ) − T_λ(ν), µ − νi ≥ 0,

∀µ, ν ∈ H.

We consider the case, when the solutions of the parametric extended

general triequilibrium inclusion (44) lie in the interior of X. Following the

ideas of Dafermos [13], Noor [29] and Noor et al. [42], we consider the map

Φ_λ(µ) as deﬁned by (45). We have to show that the map Φ_λ(µ) has a ﬁxed

point, which is a solution of the parametric extended general triequilibrium

inclusion (44). First of all, we prove that the map Φ_λ(µ), deﬁned by (45 ),

is a contraction map with respect to µ uniformly in λ ∈ M.

Lemma 3. Let g_λ(.) be a locally strongly monotone with constants σ > 0

and locally Lipschitz continuous with constants ζ > 0, respectively, and the

Assumption 1 hold. If the trifunction F_λ(., ., .) is locally jointly Lipschitz

continuous and operator with constant β and h_λbe locally Lipchitz continu-

ous with constant ζ₁, we have

kΦ_λ(µ₁) − Φ_λ(µ₂)k ≤ θkµ₁− µ₂k,

for

1 − k

ρ <

k < 1,

ζ₁< 1,

ζ²< 2σ,

(46)

β

where

and

n

o

p

θ =

1 − 2σ + ζ²+ ζ₁+ ρβ = {k + ρβ}

(47)

(48)

p

k =

1 − 2σ + ζ²+ ζ₁.

General triequilibrium problems

206

Proof. In order to prove the existence of a solution of (44), it is enough to

show that the mapping Φ_λ(µ), deﬁned by (45), is a contraction mapping.

For µ₁= µ₂∈ H, and using the Assumption 1, we have

kΦ_λ(µ₁) − Φ_λ(µ₂)k ≤ kµ₁− µ₂− (g_λ(µ₁) − g_λ(µ₂))k

+kJ_A[h_λ(µ₁) − ρF_λ(µ₁, T_λ(µ₁), ν)] − J_A[h_λ(µ₂) − ρF_λ(µ₂, T_λ(µ₂), ν)]k

≤ kµ₁− µ₂− (g_λ(µ₁) − g_λ(µ₂))k

+kh_λ(µ₎− h_λ(µ₂) − ρ(F_λ(µ₁, T_λ(µ₁), ν) − F_λ(µ₂, T_λ(µ₂), ν)k

≤ kµ₁− µ₂− (g_λ(µ₁) − g_λ(µ₂))k + ρkF_λ(µ₁, T_λ(µ₁), ν) − F_λ(µ₂, T_λ(µ₂), ν)k

+kh_λ(µ₁) − h_λ(µ₂)k + +ρkF_λ(µ₁, T_λ(µ₁), ν) − F_λ(µ₂, T_λ(µ₂), ν)k

≤ kµ₁− µ₂− (g_λ(µ₁) − g_λ(µ₂))k + ζ₁kµ − νk + ρβkµ₁− µ₂)k.

(49)

Since the operator g_λis a locally strongly monotone with constant σ > 0

and locally Lipschitz continuous with constant ζ > 0, it follows that

2

||µ₁− µ₂− (g_λ(µ₁) − g_λ(µ₂)|| ≤ ||u₁− u₂|| − 2hg_λ(µ₁) − g_λ(µ₂), µ₁− µ₂i

2

+||g_λ(µ₁) − g_λ(µ₂)||

≤ (1 − 2σ + ζ²)||µ₁− µ₂|| .

(50)

2

From (48), (49), (50) and using the locally Lipschitz continuity of the bi-

function F(., .) and the operator h_λ, we have

n

o

p

kΦ_λ(µ₁) − Φ_λ(µ₂)k ≤

ζ₁+

(1 − 2σ + ζ²) + ρβ kµ₁− µ₂k

= θkµ₁− µ₂k,

where

θ = k + ρβ.

From (46), it follows that θ < 1. Thus it follows that the mapping Φ_λ(µ),

deﬁned by (45), is a contraction mapping and consequently it has a ﬁxed

point, which belongs to H satisfying extended quasi general triequilibrium

inclusion (44), the required result.

Remark 4. From Lemma 3 , we see that the map Φ_λ(µ) deﬁned by (45 )

has a unique ﬁxed point µ(λ), that is, µ(λ) = Φ_λ(µ). Also, by the assump-

tion, the function µ, for λ = λ is a solution of the parametric extended

general triequilibrium inclusion (44). Again using Lemma 3, we see that µ,

for λ = λ, is a ﬁxed point of Φ_λ(µ) and it is also a ﬁxed point of Φ_λ(µ).

Consequently, we conclude that

µ(λ) = µ = Φ_λ(µ(λ)).

M.A. Noor, K.I. Noor

207

Using Lemma 3, we can prove the continuity of the solution µ(λ) of

the parametric general triequilibrium inclusion (44) using the technique of

Noor [46].

Lemma 4. Assume that the trifunction F_λand the operator h_λare locally

Lipschitz continuous with respect to the parameter λ. If the operator g_λ(.) is

Locally Lipschitz continuous and the map λ → J_Ais continuous (or Lipschitz

continuous), then the function u(λ) satisfying (45) is (Lipschitz) continuous

at λ = λ.

We now state and prove the main result of this paper and is the moti-

vation our next result.

Theorem 5. Let µ be the solution of the parametric extended general triequi-

librium inclusion (44) for λ = λ. Let the trifunction F_λ(., ., .) be jointly

locally Lipschitz continuous and the operator h_λ(µ) be the locally strongly

monotone Lipschitz continuous operator for all µ, ν ∈ X. If the map λ → J_A

µ

is ( Lipschitz) continuous at λ = λ and the operator g_λis locally strongly

monotne Lipschitz continuous, then there exists a neighborhood N ⊂ M of λ

such that for λ ∈ N, the parametric general trifunction equilibrium inclusion

(45) has a unique solution µ(λ) in the interior of X, u(λ) = u and u(λ) is

(Lipschitz) continuous at λ = λ.

Proof. Its proof follows from Lemma3, Lemma 4 and Remark 4.

6 Conclusion

Some new classes of extended general triequilibrium inclusions are intro-

duced and investigated. We have proved that the extended general triequi-

librium inclusions are equivalent to the ﬁxed point problem. The equivalence

between the general triequilibrium inclusions and ﬁxed point problems is

used to suggest some new multi step multi-step iterative methods for solving

the general trifunction equilibrium inclusions. These new methods include

extra resolvent multi step hybrid resolvent methods as special cases. Con-

vergence analysis of the proposed method is discussed for strongly monotone

and Lipschitz continuous operators. Sensitivity analysis is also investigated

for general trifunction equilibrium inclusions using the equivalent ﬁxed point

approach. Iterative mathods suggested and analyzed in this paper for solv-

ing extended general triequilibrium inclusions are the novel generalizations,

improvements, reﬁnements and modiﬁcations of Noor (three step ) itera-

tions [30 –32], which include Ishikawa (two-step) iterations, Mann (one step)

General triequilibrium problems

208

iterations and Picard method as special cases. Using the technique and

ideas of Ashish et. al. [2,3], Cho et al. [8], Cristescu et al. [10,11], Kwuni et

al. [20], Mahato [22], Natrangan et al. [24], Noor et al. [47,48,53,54], Pam-

sang et al. [56], Rattanaseeha et al. [57], Suantai et al. [62], Tomar et al. [63],

Trinh et al. [65] and Yadav et al. [67], one can explore the applications of

these multi step methods for solving the general triequilibrium inclusions

in the ﬁxed point theory, fractal geometry, chaos theory, coding, number

theory, spectral geometry, dynamical systems, complex analysis, nonlinear

programming, graphics, artiﬁcial intelligence, control engineering, manage-

ment sciences, stock exchange, regression and link prediction problems [60],

ﬁnancial mathematical [4], and computer aided design. Comparison of these

new methods with other technique is an open problem, which needs further

research eﬀorts.

Acknowledgments. The authors sincerely thank their respected pro-

fessors, teachers, students, colleagues, collaborators, referees, editors, man-

aging editors and friends, who have contributed, directly or indirectly to

this research. The authors are grateful to the referees for their valuable

comments and suggestions for the improvements of the ﬁnal version.

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