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A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems

Applied Mathematics Program, Faculty of Science and Technology, Valaya Alongkorn Rajabhat University under the Royal Patronage (VRU), 1 Moo 20 Phaholyothin Road, Klong Neung, Klong Luang, Pathumthani 13180, Thailand

KMUTTFixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, SCL 802 Fixed Point Laboratory, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

Institute of Research and Development of Processes IIDP, University of the Basque Country, 48940 Leioa, Spain

Author to whom correspondence should be addressed.

Axioms 2020, 9(3), 101; https://doi.org/10.3390/axioms9030101

Submission received: 30 July 2020 / Revised: 21 August 2020 / Accepted: 28 August 2020 / Published: 31 August 2020

(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)

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Abstract

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones.

Keywords:

convex optimization; pseudomonotone bifunction; equilibrium problems; variational inequality problems; weak convergence; fixed point problems

1. Introduction

For a nonempty, closed and convex subset

K

of a real Hilbert space

E

and

f : E \times E \to R

is a bifunction with

f (p_{1}, p_{1}) = 0

, for each

p_{1} \in K .

A equilibrium problem [1,2] for f on the set

K

is defined in the following way:

Find ℘^{*} \in K such that f (℘^{*}, p_{1}) \geq 0, \forall p_{1} \in K .

(1)

The problem (1) is very general, it includes many problems, such as fixed point problems, variational inequalities problems, the optimization problems, the Nash equilibrium of non-cooperative games, the complementarity problems, the saddle point problems, and the vector optimization problem (for further details see [1,3,4]). The equilibrium problem is also considered as the famous Ky Fan inequality [2]. This above-defined particular format of an equilibrium problem (1) is initiated by Muu and Oettli [5] in 1992 and further investigation on its theoretical properties studied by Blum and Oettli [1]. The construction of new optimization-based methods and the modification and extension of existing methods, as well as the examination of their convergence analysis, is an important research direction in equilibrium problem theory. Many methods have been developed over the last few years to numerically solve the equilibrium problems in both finite and infinite dimensional Hilbert spaces, i.e., the extragradient algorithms [6,7,8,9,10,11,12,13,14] subgradient algorithms [15,16,17,18,19,20,21] inertial methods [22,23,24,25], and others in [26,27,28,29,30,31,32,33,34].

In particular, a proximal method [35] is an efficient way to solve equilibrium problems that are equivalent to solving minimization problems on each step. This approach is also considered as the two-step extragradient-like method in [6], because of the early contribution of the Korpelevich [36] extragradient method to solve the saddle point problems. More precisely, Tran et al. introduced a method in [6], in which an iterative sequence

{u_{n + 1}}

was generated in the following manner:

\{\begin{matrix} u_{n} \in K, \\ v_{n} = \underset{}{arg min} {ξ f (u_{n}, y) + \frac{1}{2} ∥ u_{n} - y ∥^{2} : y \in K}, \\ u_{n + 1} = \underset{}{arg min} {ξ f (v_{n}, y) + \frac{1}{2} ∥ u_{n} - y ∥^{2} : y \in K}, \end{matrix}

where

0 < ξ < min \{\frac{1}{2 k_{1}}, \frac{1}{2 k_{2}}\}

and

k_{1}, k_{2}

are Lipschitz constants. Moreover,

\underset{y \in K}{arg min} f (x)

is the value of x in set

K

for which

f (x)

attains it’s minimum. The iterative sequence generated from the above-described method provides a weak convergent iterative sequence and in order to operate it, previous knowledge of the Lipschitz-like constants are required. These Lipschitz-type constants are normally unknown or hard to evaluate. In order to overcome this situation, Hieu et al. [12] introduced an extension of the method in [37] to solve the problems of equilibrium in the following manner: let

{[t]}_{+} : = max {t, 0}

and choose

u_{0} \in K,

μ \in (0, 1)

with

ξ_{0} > 0

, such that

\{\begin{matrix} v_{n} = \underset{}{arg min} {ξ_{n} f (u_{n}, y) + \frac{1}{2} ∥ u_{n} - y ∥^{2} : y \in K}, \\ u_{n + 1} = \underset{}{arg min} {ξ_{n} f (v_{n}, y) + \frac{1}{2} ∥ u_{n} - y ∥^{2} : y \in K}, \end{matrix}

where the stepsize sequence

{ξ_{n}}

is updated in the following way:

ξ_{n + 1} = min \{ξ_{n}, \frac{μ (∥ u_{n} - v_{n} ∥^{2} + ∥ u_{n + 1} - v_{n} ∥^{2})}{2 {[f (u_{n}, u_{n + 1}) - f (u_{n}, v_{n}) - f (v_{n}, u_{n + 1})]}_{+}}\} .

Recently, Vinh and Muu proposed an inertial iterative algorithm in [38] to solve a pseudomonotone equilibrium problem. The key contribution is an inertial factor in the method that used to enhance the convergence speed of the iterative sequence. The iterative sequence

{u_{n}}

was defined in the following manner:

(i): Choose $u_{- 1}, u_{0} \in K,$ $θ \in [0, 1),$ $0 < ξ < min {\frac{1}{2 k_{1}}, \frac{1}{2 k_{2}}}$ where a sequence ${ρ_{n}} \subset [0, + \infty)$ is satisfies the following conditions:

$\sum_{n = 0}^{+ \infty} ρ_{n} < + \infty .$

(2)
(ii): Choose $θ_{n}$ satisfying $0 \leq θ_{n} \leq \bar{θ_{n}}$ and

$\bar{θ_{n}} = \{\begin{matrix} min \{θ, \frac{ρ_{n}}{∥ u_{n} - u_{n - 1} ∥}\} & i f u_{n} \neq u_{n - 1}, \\ θ & e l s e . \end{matrix}$

(3)
(iii): Compute

$\{\begin{matrix} ϱ_{n} = u_{n} + θ_{n} (u_{n} - u_{n - 1}), \\ v_{n} = \underset{}{arg min} {ξ f (ϱ_{n}, y) + \frac{1}{2} ∥ ϱ_{n} - y ∥^{2} : y \in K}, \\ u_{n + 1} = \underset{}{arg min} {ξ f (v_{n}, y) + \frac{1}{2} ∥ ϱ_{n} - y ∥^{2} : y \in K} . \end{matrix}$

Recently, another efficient inertial algorithm proposed by Hieu et al. in [39] as follows: let

u_{n - 1}, u_{n}, v_{n} \in K,

θ \in [0, 1),

0 < ξ \leq \frac{1}{2 k_{2} + 8 k_{1}}

and the sequence

{u_{n}}

was defined in the following manner:

\{\begin{matrix} ϱ_{n} = u_{n} + θ (u_{n} - u_{n - 1}), \\ u_{n + 1} = \underset{}{arg min} {ξ f (v_{n}, y) + \frac{1}{2} ∥ ϱ_{n} - y ∥^{2} : y \in K}, \\ ϱ_{n + 1} = u_{n + 1} + θ (u_{n + 1} - u_{n}), \\ v_{n + 1} = \underset{}{arg min} {ξ f (v_{n}, y) + \frac{1}{2} ∥ ϱ_{n + 1} - y ∥^{2} : y \in K} . \end{matrix}

In this article, we concentrates on projection methods that are normally well-established and easy to execute due to their efficient numerical computation. Motivated by the works of [12,38], we formulate an inertial explicit subgradient extragradient method to solve the pseudomonotone equilibrium problem. These results can be seen as the modification of the methods appeared in [6,12,38,39]. Under certain mild conditions, a weak convergence theorem is proved regarding the iterative sequence of the algorithm. Moreover, experimental studies have documented that the designed method tends to be more efficient when compared to the existing methods that are presented in [38,39].

The remainder of the paper has been arranged, as follows: Section 2 contains the elementary results used in this paper. Section 3 contains our main algorithm and proves their convergence. Section 4 and Section 5 incorporate the applications of our main results. Section 6 carries out the numerical results that prove the computational effectiveness of our suggested method.

2. Preliminaries

Assume that

h : K \to R

be a convex function on a nonempty, closed and convex subset

K

of a real Hilbert space

E

and subdifferential of a function h at

p_{1} \in K

is defined by

\partial h (p_{1}) = {p_{3} \in E : h (p_{2}) - h (p_{1}) \geq 〈 p_{3}, p_{2} - p_{1} 〉, \forall p_{2} \in K} .

Assume that

K

be a nonempty, closed and convex subset of a real Hilbert space

E

and Normal cone of

K

p_{1} \in K

is defined by

N_{K} (p_{1}) = {p_{3} \in E : 〈 p_{3}, p_{2} - p_{1} 〉 \leq 0, \forall p_{2} \in K} .

A metric projection

P_{K} (p_{1})

for

p_{1} \in E

onto a closed and convex subset

K

E

is defined by

P_{K} (p_{1}) = \underset{}{arg min} {∥ p_{2} - p_{1} ∥ : p_{2} \in K} .

Now, consider the following definitions of monotonicity a bifunction (see for details [1,40]). Assume that

f : E \times E \to R

K

for

γ > 0

is said to be

(1): γ-strongly monotone if

$f (p_{1}, p_{2}) + f (p_{2}, p_{1}) \leq - γ {∥ p_{1} - p_{2} ∥}^{2}, \forall p_{1}, p_{2} \in K;$
(2): monotone if

$f (p_{1}, p_{2}) + f (p_{2}, p_{1}) \leq 0, \forall p_{1}, p_{2} \in K;$
(3): γ-strongly pseudomonotone if

$f (p_{1}, p_{2}) \geq 0 ⟹ f (p_{2}, p_{1}) \leq - γ {∥ p_{1} - p_{2} ∥}^{2}, \forall p_{1}, p_{2} \in K;$
(4): pseudomonotone if

$f (p_{1}, p_{2}) \geq 0 ⟹ f (p_{2}, p_{1}) \leq 0, \forall p_{1}, p_{2} \in K .$

We have the following implications from the above definitions:

(1) ⟹ (2) ⟹ (4) and (1) ⟹ (3) ⟹ (4) .

In general, the converses are not true. Suppose that

f : E \times E \to R

satisfy the Lipschitz-type condition [41] on a set

K

if there exist two constants

k_{1}, k_{2} > 0

, such that

f (p_{1}, p_{2}) + f (p_{2}, p_{3}) + k_{1} ∥ p_{1} - p_{2} ∥^{2} + k_{2} {∥ p_{2} - p_{3} ∥}^{2} \geq f (p_{1}, p_{3}), \forall p_{1}, p_{2}, p_{3} \in K .

Lemma 1

([42]). Suppose

K

be a nonempty, closed and convex subset of

E

and

P_{K} : E \to K

is metric projection from

E

onto

K

(i): Let $p_{1} \in K$ and $p_{2} \in E,$ we have

$∥ p_{1} - P_{K} (p_{2}) ∥^{2} + ∥ P_{K} (p_{2}) - p_{2} ∥^{2} \leq {∥ p_{1} - p_{2} ∥}^{2} .$
(ii): $p_{3} = P_{K} (p_{1})$ if and only if

$〈 p_{1} - p_{3}, p_{2} - p_{3} 〉 \leq 0, \forall p_{2} \in K .$
(iii): For any $p_{2} \in K$ and $p_{1} \in E$

$∥ p_{1} - P_{K} (p_{1}) ∥ \leq ∥ p_{1} - p_{2} ∥ .$

Lemma 2

([43,44]). Assume that

h : K \to R

be a convex, lower semicontinuous and subdifferentiable function on

K,

where

K

is a nonempty, convex and closed subset of a Hilbert space

E .

Subsequently,

p_{1} \in K

is minimizer of a function h if and only if

0 \in \partial h (p_{1}) + N_{K} (p_{1}),

where

\partial h (p_{1})

and

N_{K} (p_{1})

denotes the subdifferential of h at

p_{1} \in K

and the normal cone of

K

p_{1}

, respectively.

Lemma 3

([45]). Let

{u_{n}}

be a sequence in

E

and

K \subset E

, such that the following conditions are satisfied:

(i): for every $u \in K,$ the ${lim}_{n \to \infty} ∥ u_{n} - u ∥$ exists;
(ii): each sequentially weak cluster limit point of the sequence ${u_{n}}$ belongs to $K$ .

Then,

{u_{n}}

weakly converge to some element in

K .

Lemma 4

([46]). Let

{q_{n}}

and

{p_{n}}

be sequences of non-negative real numbers satisfying

q_{n + 1} \leq q_{n} + p_{n},

for each

n \in N .

\sum p_{n} < \infty,

then

{lim}_{n \to \infty} q_{n}

exists.

Lemma 5

([47]). For every

p_{1}, p_{2} \in E

and

ζ \in R,

then

∥ ζ p_{1} + (1 - ζ) p_{2} ∥^{2} = ζ ∥ p_{1} ∥^{2} + (1 - ζ) ∥ p_{2} ∥^{2} - ζ (1 - ζ) {∥ p_{1} - p_{2} ∥}^{2} .

Suppose that bifunction f satisfies the following conditions:

(f1): f is pseudomonotone on $K$ and $f (p_{2}, p_{2}) = 0,$ for every $p_{2} \in K$ ;
(f2): f satisfies the Lipschitz-type condition on $E$ with constants $k_{1} > 0$ and $k_{2} > 0;$
(f3): $\underset{n \to \infty}{lim sup} f (p_{n}, v) \leq f (p^{*}, v)$ for every $v \in K$ and ${p_{n}} \subset K$ satisfying $p_{n} ⇀ p^{*}$ ;
(f4): $f (p_{1}, .)$ needs to be convex and subdifferentiable on $E$ for all $p_{1} \in E .$

3. The Modified Extragradient Algorithm for the Problem (1) and Its Convergence Analysis

We provide a method consisting of two strongly convex minimization problems with an inertial term and an explicit stepsize formula that are being used to enhance the convergence rate of the iterative sequence and to make the algorithm independent of the Lipschitz constants. For the sake of simplicity in the presentation, we will use the notation

{[t]}_{+} = max {0, t}

and follow the conventions

\frac{0}{0} = + \infty

and

\frac{a}{0} = + \infty

(a \neq 0) .

The detailed method is provided below (Algorithm 1):

Algorithm 1 (Modified Extragradient Algorithm for the Problem (1))

Initialization: Choose $u_{- 1}, u_{0} \in K,$ $μ \in (0, 1),$ $β_{n} \in (0, 1],$ $θ \in [0, 1)$ and ${ρ_{n}} \subset [0, + \infty)$ satisfying

$\sum_{n = 0}^{+ \infty} ρ_{n} < + \infty .$

(4)
Iterative steps: Choose $θ_{n}$ satisfying $0 \leq θ_{n} \leq \bar{θ_{n}}$ and

$\bar{θ_{n}} = \{\begin{cases} min \{θ, \frac{ρ_{n}}{∥ u_{n} - u_{n - 1} ∥}\} & i f u_{n} \neq u_{n - 1}, \\ θ & e l s e . \end{cases}$

(5)
Step 1: Compute

$v_{n} = \underset{y \in K}{arg min} {ξ_{n} f (ϱ_{n}, y) + \frac{1}{2} ∥ ϱ_{n} - y ∥^{2}},$

where $ϱ_{n} = u_{n} + θ_{n} (u_{n} - u_{n - 1}) .$ If $ϱ_{n} = v_{n}$ ; STOP. Else, go to next step.
Step 2: Compute $u_{n + 1} = (1 - β_{n}) ϱ_{n} + β_{n} z_{n},$ where

$z_{n} = \underset{y \in K}{arg min} {ξ_{n} f (v_{n}, y) + \frac{1}{2} ∥ ϱ_{n} - y ∥^{2}} .$
Step 3: Update the stepsize in the following manner:

$ξ_{n + 1} = min \{ξ_{n}, \frac{μ ∥ ϱ_{n} - v_{n} ∥^{2} + μ {∥ z_{n} - v_{n} ∥}^{2}}{2 {[f (ϱ_{n}, z_{n}) - f (ϱ_{n}, v_{n}) - f (v_{n}, z_{n})]}_{+}}\} .$

Put $n : = n + 1$ and return to Iterative steps.

Lemma 6.

The sequence

{ξ_{n}}

is monotonically decreasing with a lower bound

min \{\frac{μ}{2 max {k_{1}, k_{2}}}, ξ_{0}\}

and it converges to

ξ > 0 .

Proof.

From the definition of sequence

{ξ_{n}}

implies that sequence

{ξ_{n}}

decreasing monotonically. It is given that f satisfy the Lipschitz-type condition with

k_{1}

and

k_{2}

. Let

f (ϱ_{n}, z_{n}) - f (ϱ_{n}, v_{n}) - f (v_{n}, z_{n}) > 0

, such that

\begin{matrix} \frac{μ (∥ ϱ_{n} - v_{n} ∥^{2} + ∥ z_{n} - v_{n} ∥^{2})}{2 [f (ϱ_{n}, z_{n}) - f (ϱ_{n}, v_{n}) - f (v_{n}, z_{n})]} & \geq \frac{μ (∥ ϱ_{n} - v_{n} ∥^{2} + ∥ z_{n} - v_{n} ∥^{2})}{2 [k_{1} ∥ ϱ_{n} - v_{n} ∥^{2} + k_{2} ∥ z_{n} - v_{n} ∥^{2}]} \\ \geq \frac{μ}{2 max {k_{1}, k_{2}}} . \end{matrix}

(6)

The above implies that

{ξ_{n}}

has a lower bound

min \{\frac{μ}{2 max {k_{1}, k_{2}}}, ξ_{0}\} .

Moreover, there exists a fixed real number

ξ > 0

, such that

{lim}_{n \to \infty} ξ_{n} = ξ .

□

Remark 1.

Because of the summability of

\sum_{n = 0}^{+ \infty} ρ_{n}

and the expression (5) implies that

\sum_{n = 1}^{\infty} θ_{n} ∥ u_{n} - u_{n - 1} ∥ \leq \sum_{n = 1}^{\infty} \bar{θ_{n}} ∥ u_{n} - u_{n - 1} ∥ \leq \sum_{n = 1}^{\infty} θ ∥ u_{n} - u_{n - 1} ∥ < \infty,

(7)

that implies

lim_{n \to \infty} θ ∥ u_{n} - u_{n - 1} ∥ = 0 .

(8)

Lemma 7.

Suppose that

f : E \times E \to R

be a bifunction satisfies the conditions(f1)–(f4). For each

℘^{*} \in E P (f, K) \neq \emptyset

, we have

∥ z_{n} - ℘^{*} ∥^{2} \leq ∥ ϱ_{n} - ℘^{*} ∥^{2} - (1 - \frac{μ ξ_{n}}{ξ_{n + 1}}) ∥ ϱ_{n} - v_{n} ∥^{2} - (1 - \frac{μ ξ_{n}}{ξ_{n + 1}}) {∥ z_{n} - v_{n} ∥}^{2} .

Proof.

From the value of

z_{n}

, we have

0 \in \partial_{2} \{ξ_{n} f (v_{n}, y) + \frac{1}{2} {∥ ϱ_{n} - y ∥}^{2}\} (z_{n}) + N_{K} (z_{n}) .

For some

ω \in \partial f (v_{n}, z_{n})

, there exists

\bar{ω} \in N_{K} (z_{n})

, such that

ξ_{n} ω + z_{n} - ϱ_{n} + \bar{ω} = 0 .

The above expression implies that

〈 ϱ_{n} - z_{n}, y - z_{n} 〉 = ξ_{n} 〈 ω, y - z_{n} 〉 + 〈 \bar{ω}, y - z_{n} 〉, \forall y \in K .

For given

\bar{ω} \in N_{K} (z_{n})

, imply that

〈 \bar{ω}, y - z_{n} 〉 \leq 0,

∀

y \in K .

It provides that

〈 ϱ_{n} - z_{n}, y - z_{n} 〉 \leq ξ_{n} 〈 ω, y - z_{n} 〉, \forall y \in K .

(9)

From

ω \in \partial f (v_{n}, z_{n})

, we have

f (v_{n}, y) - f (v_{n}, z_{n}) \geq 〈 ω, y - z_{n} 〉, \forall y \in E .

(10)

Combining expressions (9) and (10) we obtain

ξ_{n} f (v_{n}, y) - ξ_{n} f (v_{n}, z_{n}) \geq 〈 ϱ_{n} - z_{n}, y - z_{n} 〉, \forall y \in K .

(11)

By substituting

y = ℘^{*}

in (11), gives that

ξ_{n} f (v_{n}, ℘^{*}) - ξ_{n} f (v_{n}, z_{n}) \geq 〈 ϱ_{n} - z_{n}, ℘^{*} - z_{n} 〉 .

(12)

Because

f (℘^{*}, v_{n}) \geq 0

, then

f (v_{n}, ℘^{*}) \leq 0,

provides that

〈 ϱ_{n} - z_{n}, z_{n} - ℘^{*} 〉 \geq ξ_{n} f (v_{n}, z_{n}) .

(13)

From the formula of

ξ_{n + 1},

we obtain

f (ϱ_{n}, z_{n}) - f (ϱ_{n}, v_{n}) - f (v_{n}, z_{n}) \leq \frac{μ ∥ ϱ_{n} - v_{n} ∥^{2} + μ {∥ z_{n} - v_{n} ∥}^{2}}{2 ξ_{n + 1}}

(14)

From the expressions (13) and (14), we have

\begin{matrix} 〈 ϱ_{n} - z_{n}, z_{n} - ℘^{*} 〉 & \geq ξ_{n} {f (ϱ_{n}, z_{n}) - f (ϱ_{n}, v_{n})} \\ - \frac{μ ξ_{n}}{2 ξ_{n + 1}} ∥ ϱ_{n} - v_{n} ∥^{2} - \frac{μ ξ_{n}}{2 ξ_{n + 1}} {∥ z_{n} - v_{n} ∥}^{2} . \end{matrix}

(15)

Similar to expression (11), the value of

v_{n}

gives that

ξ_{n} f (ϱ_{n}, y) - ξ_{n} f (ϱ_{n}, v_{n}) \geq 〈 ϱ_{n} - v_{n}, y - v_{n} 〉, \forall y \in K .

(16)

By substituting

y = z_{n}

in the above expression, we have

ξ_{n} \{f (ϱ_{n}, z_{n}) - f (ϱ_{n}, v_{n})\} \geq 〈 ϱ_{n} - v_{n}, z_{n} - v_{n} 〉 .

(17)

Combining the expressions (15) and (17), we obtain

\begin{matrix} 〈 ϱ_{n} - z_{n}, z_{n} - ℘^{*} 〉 & \geq 〈 ϱ_{n} - v_{n}, z_{n} - v_{n} 〉 \\ - \frac{μ ξ_{n}}{2 ξ_{n + 1}} ∥ ϱ_{n} - v_{n} ∥^{2} - \frac{μ ξ_{n}}{2 ξ_{n + 1}} {∥ z_{n} - v_{n} ∥}^{2} . \end{matrix}

(18)

We have the given formulas:

- 2 〈 ϱ_{n} - z_{n}, z_{n} - ℘^{*} 〉 = - ∥ ϱ_{n} - ℘^{*} ∥^{2} + ∥ z_{n} - ϱ_{n} ∥^{2} + {∥ z_{n} - ℘^{*} ∥}^{2} .

2 〈 v_{n} - ϱ_{n}, v_{n} - z_{n} 〉 = ∥ ϱ_{n} - v_{n} ∥^{2} + ∥ z_{n} - v_{n} ∥^{2} - {∥ ϱ_{n} - z_{n} ∥}^{2} .

The above expressions with (18), we have

∥ z_{n} - ℘^{*} ∥^{2} \leq ∥ ϱ_{n} - ℘^{*} ∥^{2} - (1 - \frac{μ ξ_{n}}{ξ_{n + 1}}) ∥ ϱ_{n} - v_{n} ∥^{2} - (1 - \frac{μ ξ_{n}}{ξ_{n + 1}}) {∥ z_{n} - v_{n} ∥}^{2} .

□

Theorem 1.

Assume that

f : E \times E \to R

be a bifunction satisfies the conditions(f1)–(f4) and

℘^{*}

belongs to solution set

E P (f, K) .

Subsequently, the sequences

{ϱ_{n}},

{v_{n}},

{z_{n}}

and

{u_{n}}

generated through Algorithm 1 weakly converges to

℘^{*} .

In addition,

{lim}_{n \to \infty} P_{E P (f, K)} (u_{n}) = ℘^{*} .

Proof.

By value of

u_{n + 1}

through Lemma 5, we obtain

\begin{matrix} ∥ u_{n + 1} - ℘^{*} ∥^{2} & = ∥ (1 - β_{n}) ϱ_{n} + β_{n} z_{n} - ℘^{*} ∥^{2} \\ = ∥ (1 - β_{n}) (ϱ_{n} - ℘^{*}) + β_{n} (z_{n} - ℘^{*}) ∥^{2} \\ = (1 - β_{n}) ∥ ϱ_{n} - ℘^{*} ∥^{2} + β_{n} ∥ z_{n} - ℘^{*} ∥^{2} - β_{n} (1 - β_{n}) {∥ ϱ_{n} - z_{n} ∥}^{2} \\ \leq (1 - β_{n}) ∥ ϱ_{n} - ℘^{*} ∥^{2} + β_{n} {∥ z_{n} - ℘^{*} ∥}^{2} . \end{matrix}

(19)

By Lemma 7 and expression (19), we obtain

\begin{matrix} ∥ u_{n + 1} - ℘^{*} ∥^{2} & \leq ∥ ϱ_{n} - ℘^{*} ∥^{2} \\ - β_{n} (1 - \frac{μ ξ_{n}}{ξ_{n + 1}}) ∥ ϱ_{n} - v_{n} ∥^{2} - β_{n} (1 - \frac{μ ξ_{n}}{ξ_{n + 1}}) {∥ z_{n} - v_{n} ∥}^{2} . \end{matrix}

(20)

Because

ξ_{n} \to ξ,

then there exists a fixed number

ϵ \in (0, 1 - μ)

, such that

lim_{n \to \infty} (1 - \frac{μ ξ_{n}}{ξ_{n + 1}}) = 1 - μ > ϵ > 0 .

Subsequently, there exist a fixed real number

N_{1} \in N

such that

(1 - \frac{μ ξ_{n}}{ξ_{n + 1}}) > ϵ > 0, \forall n \geq N_{1} .

(21)

Combining the expressions (20) and (21), we obtain

∥ u_{n + 1} - ℘^{*} ∥^{2} \leq {∥ ϱ_{n} - ℘^{*} ∥}^{2}, \forall n \geq N_{1} .

(22)

By definition of the

ϱ_{n}

, we have

\begin{matrix} ∥ ϱ_{n} - ℘^{*} ∥ = ∥ u_{n} + θ_{n} (u_{n} - u_{n - 1}) - ℘^{*} ∥ \leq ∥ u_{n} - ℘^{*} ∥ + θ_{n} ∥ u_{n} - u_{n - 1} ∥ . \end{matrix}

(23)

From the definition of

ϱ_{n}

in Algorithm 1, we obtain

\begin{matrix} ∥ ϱ_{n} - ℘^{*} ∥^{2} & = ∥ u_{n} + θ_{n} (u_{n} - u_{n - 1}) - ℘^{*} ∥^{2} \\ = ∥ (1 + θ_{n}) (u_{n} - ℘^{*}) - θ_{n} (u_{n - 1} - ℘^{*}) ∥^{2} \\ = (1 + θ_{n}) ∥ u_{n} - ℘^{*} ∥^{2} - θ_{n} ∥ u_{n - 1} - ℘^{*} ∥^{2} + θ_{n} (1 + θ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} \end{matrix}

(24)

\begin{matrix} \leq (1 + θ_{n}) ∥ u_{n} - ℘^{*} ∥^{2} - θ_{n} ∥ u_{n - 1} - ℘^{*} ∥^{2} + 2 θ {∥ u_{n} - u_{n - 1} ∥}^{2} . \end{matrix}

(25)

The expression (22) can also be written as

\begin{matrix} ∥ u_{n + 1} - ℘^{*} ∥ & \leq ∥ u_{n} - ℘^{*} ∥ + θ ∥ u_{n} - u_{n - 1} ∥, \forall n \geq N_{1} . \end{matrix}

(26)

By using Lemma 4 with expressions (7) and (26), we have

lim_{n \to \infty} ∥ u_{n} - ℘^{*} ∥ = l, for some finite l \geq 0 .

(27)

The equality (8) implies that

\begin{matrix} lim_{n \to \infty} ∥ u_{n} - u_{n - 1} ∥ = 0 . \end{matrix}

(28)

By letting

n \to \infty

in (24) implies that

lim_{n \to \infty} ∥ ϱ_{n} - ℘^{*} ∥ = l .

(29)

From the expression (20) and (25), we have

\begin{matrix} ∥ u_{n + 1} - ℘^{*} ∥^{2} \\ \leq (1 + θ_{n}) ∥ u_{n} - ℘^{*} ∥^{2} - θ_{n} ∥ u_{n - 1} - ℘^{*} ∥^{2} + 2 θ {∥ u_{n} - u_{n - 1} ∥}^{2} \\ - β_{n} (1 - \frac{μ ξ_{n}}{ξ_{n + 1}}) ∥ ϱ_{n} - v_{n} ∥^{2} - β_{n} (1 - \frac{μ ξ_{n}}{ξ_{n + 1}}) {∥ z_{n} - v_{n} ∥}^{2}, \end{matrix}

(30)

which further implies that (for

n \geq N_{1}

)

\begin{matrix} ϵ β ∥ ϱ_{n} - v_{n} ∥^{2} + ϵ β {∥ v_{n} - z_{n} ∥}^{2} \\ \leq ∥ u_{n} - ℘^{*} ∥^{2} - ∥ u_{n + 1} - ℘^{*} ∥^{2} + θ_{n} (∥ u_{n} - ℘^{*} ∥^{2} - {∥ u_{n - 1} - ℘^{*} ∥}^{2}) + 2 θ {∥ u_{n} - u_{n - 1} ∥}^{2} . \end{matrix}

(31)

By letting

n \to \infty

in (31), we obtain

lim_{n \to \infty} ∥ ϱ_{n} - v_{n} ∥ = lim_{n \to \infty} ∥ v_{n} - z_{n} ∥ = 0 .

(32)

By using the Cauchy inequality and expression (32), we obtain

lim_{n \to \infty} ∥ ϱ_{n} - z_{n} ∥ \leq lim_{n \to \infty} ∥ ϱ_{n} - v_{n} ∥ + lim_{n \to \infty} ∥ z_{n} - v_{n} ∥ = 0 .

(33)

The expressions (29) and (32) imply that

lim_{n \to \infty} ∥ v_{n} - ℘^{*} ∥ = lim_{n \to \infty} ∥ z_{n} - ℘^{*} ∥ = l .

(34)

It follows from the expressions (27), (29) and (34) that the sequences

{ϱ_{n}},

{u_{n}},

{v_{n}}

and

{z_{n}}

are bounded. Now, we need to use Lemma 3, for this it is compulsory to show that any weak sequential limit points of

{u_{n}}

lies in the set

E P (f, K) .

Consider z to be a weak limit point of

{u_{n}}

i.e., there is a

{u_{n_{k}}}

{u_{n}}

that is weakly converges to

z .

Because

∥ u_{n} - v_{n} ∥ \to 0

, then

{v_{n_{k}}}

also weakly converge to z and so

z \in K .

Now, it is renaming to show that

z \in E P (f, K) .

From relation (11), due to

ξ_{n + 1}

and (17), we have

\begin{matrix} ξ_{n_{k}} f (v_{n_{k}}, y) & \geq ξ_{n_{k}} f (v_{n_{k}}, z_{n_{k}}) + 〈 ϱ_{n_{k}} - z_{n_{k}}, y - z_{n_{k}} 〉 \\ \geq ξ_{n_{k}} f (ϱ_{n_{k}}, z_{n_{k}}) - ξ_{n_{k}} f (ϱ_{n_{k}}, v_{n_{k}}) - \frac{μ ξ_{n_{k}}}{2 ξ_{n_{k} + 1}} {∥ ϱ_{n_{k}} - v_{n_{k}} ∥}^{2} \\ - \frac{μ ξ_{n_{k}}}{2 ξ_{n_{k} + 1}} {∥ v_{n_{k}} - z_{n_{k}} ∥}^{2} + 〈 ϱ_{n_{k}} - z_{n_{k}}, y - z_{n_{k}} 〉 \\ \geq 〈 ϱ_{n_{k}} - v_{n_{k}}, z_{n_{k}} - v_{n_{k}} 〉 - \frac{μ ξ_{n_{k}}}{2 ξ_{n_{k} + 1}} {∥ ϱ_{n_{k}} - v_{n_{k}} ∥}^{2} \\ - \frac{μ ξ_{n_{k}}}{2 ξ_{n_{k} + 1}} {∥ v_{n_{k}} - z_{n_{k}} ∥}^{2} + 〈 ϱ_{n_{k}} - z_{n_{k}}, y - z_{n_{k}} 〉, \end{matrix}

(35)

where

y \in K .

It follows from (28), (32), (33) and the boundedness of

{u_{n}}

right hand side tend to zero. Due to

ξ_{n_{k}} > 0,

condition (f3) and

v_{n_{k}} ⇀ z,

implies

0 \leq \underset{k \to \infty}{lim sup} f (v_{n_{k}}, y) \leq f (z, y), \forall y \in K .

(36)

Because

z \in K

imply that

f (z, y) \geq 0, \forall y \in K .

It is prove that

z \in E P (f, K) .

By Lemma 3, provides that

{ϱ_{n}},

{v_{n}},

{z_{n}}

and

{u_{n}}

weakly converges to

℘^{*}

n \to \infty .

Finally, to prove that

{lim}_{n \to \infty} P_{E P (f, K)} (u_{n}) = ℘^{*} .

Let

q_{n} : = P_{E P (f, K)} (u_{n}),

\forall n \in N .

For any

℘^{*} \in E P (f, K)

, we have

∥ q_{n} ∥ \leq ∥ q_{n} - u_{n} ∥ + ∥ u_{n} ∥ \leq ∥ ℘^{*} - u_{n} ∥ + ∥ u_{n} ∥ .

(37)

Clearly, the above implies that sequence

{q_{n}}

is bounded. Next, we need to show that

{q_{n}}

is a Cauchy sequence. By using Lemma 1(iii) and (23), we have

∥ u_{n + 1} - q_{n + 1} ∥ \leq ∥ u_{n + 1} - q_{n} ∥ \leq ∥ u_{n} - q_{n} ∥ + θ ∥ u_{n} - u_{n - 1} ∥, \forall n \geq N_{1} .

(38)

Thus, Lemma 4 provides the existence of

{lim}_{n \to \infty} ∥ u_{n} - q_{n} ∥ .

Next, take (23) ∀

m > n \geq N_{1},

we have

\begin{matrix} ∥ q_{n} - u_{m} ∥ & \leq ∥ q_{n} - u_{m - 1} ∥ + θ ∥ u_{n} - u_{n - 1} ∥ \\ \leq \dots \leq ∥ q_{n} - u_{n} ∥ + θ \sum_{k = n}^{m - 1} ∥ u_{n} - u_{n - 1} ∥ . \end{matrix}

(39)

Suppose that

q_{m}, q_{n} \in E P (f, K)

for

m > n \geq N_{1},

through Lemma 1(i) and (39), we have

\begin{matrix} ∥ q_{n} - q_{m} ∥^{2} \\ \leq ∥ q_{n} - u_{m} ∥^{2} - {∥ q_{m} - u_{m} ∥}^{2} \\ \leq ∥ q_{n} - u_{n} ∥^{2} + {(θ \sum_{k = n}^{m - 1} ∥ u_{n} - u_{n - 1} ∥)}^{2} + 2 θ ∥ q_{n} - u_{n} ∥ \sum_{k = n}^{m - 1} ∥ u_{n} - u_{n - 1} ∥ - ∥ q_{m} - u_{m} ∥^{2} . \end{matrix}

(40)

The existence of

{lim}_{n \to \infty} ∥ u_{n} - q_{n} ∥

and the summability of the series

\sum_{n} ∥ u_{n} - u_{n - 1} ∥ < + \infty,

imply

{lim}_{n \to \infty} ∥ q_{n} - q_{m} ∥ = 0,

∀

m > n .

As a result,

{q_{n}}

is a Cauchy sequence and due the closeness of the set

E P (f, K)

the sequence

{q_{n}}

strongly converges to

q^{*} \in E P (f, K) .

Next, remaining to show that

q^{*} = ℘^{*} .

From Lemma 1(ii) and

℘^{*}, q^{*} \in E P (f, K)

, we have

〈 u_{n} - q_{n}, ℘^{*} - q_{n} 〉 \leq 0 .

(41)

Because of

q_{n} \to q^{*}

and

u_{n} ⇀ ℘^{*}

, we obtain

〈 ℘^{*} - q^{*}, ℘^{*} - q^{*} 〉 \leq 0,

implies that

℘^{*} = q^{*} = {lim}_{n \to \infty} P_{E P (f, K)} (u_{n}) .

□

4. Applications to Solve Fixed Point Problems

Now, consider the applications of our results that are discussed in Section 3 to solve fixed-point problems involving

κ

-strict pseudo-contraction. Let

T : K \to K

be a mapping and the fixed point problem is formulated in the following manner:

Find ℘^{*} \in K such as T (℘^{*}) = ℘^{*} .

Let a mapping

T : K \to K

is said to be

(i): sequentially weakly continuous on $K$ if

$T (p_{n}) ⇀ T (p) for every sequence in K satisfying p_{n} ⇀ p (weakly converges);$
(ii): κ-strict pseudo-contraction [48] on $K$ if

$∥ T p_{1} - T p_{2} ∥^{2} \leq ∥ p_{1} - p_{2} ∥^{2} + κ {∥ (p_{1} - T p_{1}) - (p_{2} - T p_{2}) ∥}^{2}, \forall p_{1}, p_{2} \in K;$

(42)

that is equivalent to

$〈T p_{1} - T p_{2}, p_{1} - p_{2}〉 \leq ∥ p_{1} - p_{2} ∥^{2} - \frac{1 - κ}{2} {∥ (p_{1} - T p_{1}) - (p_{2} - T p_{2}) ∥}^{2}, \forall p_{1}, p_{2} \in K .$

(43)

Note: if we define

f (p_{1}, p_{2}) = 〈 p_{1} - T p_{1}, p_{2} - p_{1} 〉, \forall p_{1}, p_{2} \in K .

Then, the problem (1) convert into the fixed point problem with

2 k_{1} = 2 k_{2} = \frac{3 - 2 κ}{1 - κ} .

The value of

v_{n}

in Algorithm 1 convert into followings:

\begin{matrix} v_{n} & = \underset{y \in K}{arg min} {ξ_{n} f (ϱ_{n}, y) + \frac{1}{2} ∥ ϱ_{n} - y ∥^{2}} \\ = \underset{y \in K}{arg min} {ξ_{n} 〈 ϱ_{n} - T (ϱ_{n}), y - ϱ_{n} 〉 + \frac{1}{2} ∥ ϱ_{n} - y ∥^{2}} \\ = \underset{y \in K}{arg min} {ξ_{n} 〈 ϱ_{n} - T (ϱ_{n}), y - ϱ_{n} 〉 + \frac{1}{2} ∥ ϱ_{n} {- y ∥}^{2} + \frac{ξ_{n}^{2}}{2} ∥ ϱ_{n} - T (ϱ_{n}) ∥^{2} - \frac{ξ_{n}^{2}}{2} ∥ ϱ_{n} - T (ϱ_{n}) ∥^{2}} \\ = \underset{y \in K}{arg min} {\frac{1}{2} ∥ y - ϱ_{n} + ξ_{n} (ϱ_{n} - T (ϱ_{n})) ∥^{2}} \\ = P_{K} [ϱ_{n} - ξ_{n} (ϱ_{n} - T (ϱ_{n}))] = P_{K} [(1 - ξ_{n}) ϱ_{n} + ξ_{n} T (ϱ_{n})] . \end{matrix}

(44)

In the similar way to the expression (44), we obtain

z_{n} = P_{K} [ϱ_{n} - ξ_{n} (v_{n} - T (v_{n}))] .

(45)

As a consequence of the results in Section 3, we have the following fixed point theorem:

Corollary 1.

Assume that

T : K \to K

to be a weakly continuous and κ-strict pseudocontraction with

F i x (T) \neq \emptyset .

The sequences

ϱ_{n},

v_{n},

z_{n}

and

u_{n}

be generated in the following way:

(i): Choose $u_{- 1}, u_{0} \in K,$ $μ \in (0, 1),$ $β_{n} \in (0, 1],$ $θ \in [0, 1)$ and ${ρ_{n}} \subset [0, + \infty)$ satisfies the following condition:

$\sum_{n = 0}^{+ \infty} ρ_{n} < + \infty .$

(46)
(ii): Choose $θ_{n}$ satisfies $0 \leq θ_{n} \leq \bar{θ_{n}}$ , such that

$\bar{θ_{n}} = \{\begin{matrix} min \{θ, \frac{ρ_{n}}{∥ u_{n} - u_{n - 1} ∥}\} & i f u_{n} \neq u_{n - 1}, \\ θ & e l s e . \end{matrix}$

(47)
(iii): Compute $u_{n + 1} = (1 - β_{n}) ϱ_{n} + β_{n} z_{n}$ , where

$\{\begin{matrix} ϱ_{n} = u_{n} + θ_{n} (u_{n} - u_{n - 1}), \\ v_{n} = P_{K} [ϱ_{n} - ξ_{n} (ϱ_{n} - T (ϱ_{n}))], \\ z_{n} = P_{K} [ϱ_{n} - ξ_{n} (v_{n} - T (v_{n}))] . \end{matrix}$

(48)
(iv): Revised the stepsize $ξ_{n + 1}$ in the following way:

$ξ_{n + 1} = min \{ξ_{n}, \frac{μ ∥ ϱ_{n} - v_{n} ∥^{2} + μ {∥ z_{n} - v_{n} ∥}^{2}}{2 {[〈(ϱ_{n} - v_{n}) - (T (ϱ_{n}) - T (v_{n})), z_{n} - v_{n}〉]}_{+}}\}$

Subsequently,

{ϱ_{n}},

{v_{n}},

{z_{n}}

and

{u_{n}}

be the sequences converges weakly to

℘^{*} \in F i x (T) .

5. Application to Solve Variational Inequality Problems

Now, consider the applications of our results that are discussed in Section 3 in order to solve variational inequality problems involving pseudomonotone and Lipschitz-type continuous operator. Let a operator

L : K \to K

and the variational inequality problem is formulated as follows:

Find ℘^{*} \in K such that 〈L (℘^{*}), y - ℘^{*}〉 \geq 0, \forall y \in K .

A mapping

L : E \to E

is said to be

(i): L-Lipschitz continuous on $K$ if

$∥ L (p_{1}) - L (p_{2}) ∥ \leq L ∥ p_{1} - p_{2} ∥, \forall p_{1}, p_{2} \in K;$
(ii): monotone on $K$ if

$〈 L (p_{1}) - L (p_{2}), p_{1} - p_{2} 〉 \geq 0, \forall p_{1}, p_{2} \in K;$
(iii): pseudomonotone on $K$ if

$〈L (p_{1}), p_{2} - p_{1}〉 \geq 0 ⟹ 〈L (p_{2}), p_{1} - p_{2}〉 \leq 0, \forall p_{1}, p_{2} \in K .$

Note: let

f (p_{1}, p_{2}) : = 〈L (p_{1}), p_{2} - p_{1}〉,

\forall p_{1}, p_{2} \in K .

Thus, problem (1) translates into the problem (VIP) with

L = 2 k_{1} = 2 k_{2} .

From the value of

v_{n},

we have

\begin{matrix} v_{n} & = \underset{y \in K}{arg min} \{ξ_{n} f (ϱ_{n}, y) + \frac{1}{2} {∥ ϱ_{n} - y ∥}^{2}\} \\ = \underset{y \in K}{arg min} \{ξ_{n} 〈 L (ϱ_{n}), y - ϱ_{n} 〉 + \frac{1}{2} ∥ ϱ_{n} {- y ∥}^{2} + \frac{ξ_{n}^{2}}{2} ∥ L (ϱ_{n}) ∥^{2} - \frac{ξ_{n}^{2}}{2} {∥ L (ϱ_{n}) ∥}^{2}\} \\ = \underset{y \in K}{arg min} \{\frac{1}{2} {∥ y - (ϱ_{n} - ξ_{n} L (ϱ_{n})) ∥}^{2}\} \\ = P_{K} [ϱ_{n} - ξ_{n} L (ϱ_{n})] . \end{matrix}

(49)

In similar way to the expression (49), we obtain

z_{n} = P_{K} [ϱ_{n} - ξ_{n} L (v_{n})] .

Suppose that a mapping L satisfies the following conditions:

(L1): L is monotone on $K$ with $V I (L, K) \neq \emptyset$ ;
(L2): L is L-Lipschitz continuous on $K$ with $L > 0$ ;
(L3): L is pseudomonotone on $K$ with $V I (L, K) \neq \emptyset$ ; and,
(L4): $\underset{n \to \infty}{lim sup} 〈 L (p_{n}), p - p_{n} 〉 \leq 〈 L (p), y - p 〉,$ $\forall y \in K$ and ${p_{n}} \subset K$ satisfying $p_{n} ⇀ p .$

Next, let L to be monotone and (L4) can be removed. The condition (L4) is used to defined

f (u, v) = 〈 L (u), v - u 〉

and satisfy the conditions (L4). The condition (f3) is required to show

z \in E P (f, K)

see (36). The condition (L4) is required to show

z \in V I (L, K) .

Further, to show that

z \in V I (L, K) .

By letting the monotonicity of operator L, we have

〈 L (y), y - v_{n} 〉 \geq 〈 L (v_{n}), y - v_{n} 〉, \forall y \in K .

(50)

By letting

f (u, v) = 〈 L (u), v - u 〉

with expression (35), implies that

\underset{k \to \infty}{lim sup} 〈 L (v_{n_{k}}), y - v_{n_{k}} 〉 \geq 0, \forall y \in K .

(51)

Combining (50) with (51), we deduce that

\underset{k \to \infty}{lim sup} 〈 L (y), y - v_{n_{k}} 〉 \geq 0, \forall y \in K .

(52)

Therefore,

v_{n_{k}} ⇀ z \in K,

provides

〈 L (y), y - z 〉 \geq 0,

∀

y \in K .

Let

v_{t} = (1 - t) z + t y,

∀

t \in [0, 1] .

Since

v_{t} \in K

for

t \in (0, 1)

, we have

0 \leq 〈 L (v_{t}), v_{t} - z 〉 = t 〈 L (v_{t}), y - z 〉 .

(53)

That is

〈 L (v_{t}), y - z 〉 \geq 0

every

t \in (0, 1) .

Due to

v_{t} \to z

, while

t \to 0

, we have

〈 L (z), y - z 〉 \geq 0,

for all

y \in K,

consequently

z \in V I (L, K) .

Corollary 2.

Let

L : K \to E

be a mapping and satisfying the conditions(L1)–(L2). Assume that the sequences

{ϱ_{n}},

{v_{n}},

{z_{n}}

and

{u_{n}}

generated in the following manner:

(i): Choose $u_{- 1}, u_{0} \in K,$ $μ \in (0, 1),$ $β_{n} \in (0, 1],$ $θ \in [0, 1)$ and ${ρ_{n}} \subset [0, + \infty)$ , such that

$\sum_{n = 0}^{+ \infty} ρ_{n} < + \infty .$

(54)
(ii): Let $θ_{n}$ satisfies $0 \leq θ_{n} \leq \bar{θ_{n}}$ and

$\bar{θ_{n}} = \{\begin{matrix} min \{θ, \frac{ρ_{n}}{∥ u_{n} - u_{n - 1} ∥}\} & i f u_{n} \neq u_{n - 1}, \\ θ & o t h e r w i s e . \end{matrix}$

(55)
(iii): Compute $u_{n + 1} = (1 - β_{n}) ϱ_{n} + β_{n} z_{n},$ where

$\{\begin{matrix} ϱ_{n} = u_{n} + θ_{n} (u_{n} - u_{n - 1}), \\ v_{n} = P_{K} [ϱ_{n} - ξ_{n} L (ϱ_{n})], \\ z_{n} = P_{K} [ϱ_{n} - ξ_{n} L (v_{n})] . \end{matrix}$

(56)
(iv): Stepsize $ξ_{n + 1}$ is revised in the following way:

$ξ_{n + 1} = min \{ξ_{n}, \frac{μ ∥ ϱ_{n} - v_{n} ∥^{2} + μ {∥ z_{n} - v_{n} ∥}^{2}}{2 {[〈L (ϱ_{n}) - L (v_{n}), z_{n} - v_{n}〉]}_{+}}\}$

Subsequently, the sequences

{ϱ_{n}},

{v_{n}},

{z_{n}}

and

{z_{n}}

converge weakly to

℘^{*} \in V I (L, K) .

Corollary 3.

Let

L : K \to E

be a mapping and satisfying the conditions(L2)–(L4). Assume that the sequences

{ϱ_{n}},

{v_{n}},

{z_{n}}

and

{u_{n}}

generated in the following manner:

(i): Choose $u_{- 1}, u_{0} \in K,$ $μ \in (0, 1),$ $β_{n} \in (0, 1],$ $θ \in [0, 1)$ and ${ρ_{n}} \subset [0, + \infty)$ , such that

$\sum_{n = 0}^{+ \infty} ρ_{n} < + \infty .$

(57)
(ii): Choose $θ_{n}$ satisfying $0 \leq θ_{n} \leq \bar{θ_{n}}$ , such that

$\bar{θ_{n}} = \{\begin{matrix} min \{θ, \frac{ρ_{n}}{∥ u_{n} - u_{n - 1} ∥}\} & i f u_{n} \neq u_{n - 1}, \\ θ & e l s e . \end{matrix}$

(58)
(iii): Compute $u_{n + 1} = (1 - β_{n}) ϱ_{n} + β_{n} z_{n},$ where

$\{\begin{matrix} ϱ_{n} = u_{n} + θ_{n} (u_{n} - u_{n - 1}), \\ v_{n} = P_{K} [ϱ_{n} - ξ_{n} L (ϱ_{n})], \\ z_{n} = P_{K} [ϱ_{n} - ξ_{n} L (v_{n})] . \end{matrix}$

(59)
(iv): The stepsize $ξ_{n + 1}$ is updated in the following way:

$ξ_{n + 1} = min \{ξ_{n}, \frac{μ ∥ ϱ_{n} - v_{n} ∥^{2} + μ {∥ z_{n} - v_{n} ∥}^{2}}{2 {[〈L (ϱ_{n}) - L (v_{n}), z_{n} - v_{n}〉]}_{+}}\}$

Subsequently, the sequences

{ϱ_{n}},

{v_{n}},

{z_{n}}

and

{z_{n}}

converge weakly to

℘^{*} \in V I (L, K) .

6. Numerical Experiments

The computational results present this section to prove the effectiveness of Algorithm 1 when compared to Algorithm 3.1 in [39] and Algorithm 1 in [38].

(i): For Algorithm 3.1 (Alg3.1) in [39]:

$ξ = \frac{1}{10 max {k_{1}, k_{2}}}, θ = \frac{1}{2}, Error term (D_{n}) = max {∥ u_{n + 1} - v_{n} ∥^{2}, ∥ u_{n + 1} - ϱ_{n} ∥^{2}} .$
(ii): For Algorithm 1 (Alg1) in [38]:

$ξ = \frac{1}{4 max {k_{1}, k_{2}}}, θ = \frac{1}{2}, ρ_{n} = \frac{1}{n^{2}}, Error term (D_{n}) = {∥ ϱ_{n} - v_{n} ∥}^{2} .$
(iii): For Algorithm 1 (mAlg1):

$ξ = \frac{1}{2}, θ = \frac{1}{2}, μ = \frac{1}{3}, ρ_{n} = \frac{1}{n^{2}}, β_{n} = \frac{8}{10}, Error term (D_{n}) = {∥ ϱ_{n} - v_{n} ∥}^{2} .$

Example 1.

Let take the Nash–Cournot Equilibrium Model that found in the paper [6]. A bifunction f consider into the following form:

f (p_{1}, p_{2}) = 〈 P p_{1} + Q p_{2} + q, p_{2} - p_{1} 〉,

where

q \in R^{m}

with matrices P, Q of order m and Lipschitz constants are

k_{1} = k_{2} = \frac{1}{2} ∥ P - Q ∥

(see for more details [6]). In our case,

P, Q

are taken at random (choose diagonal matrices

A_{1}

and

A_{2}

randomly entries from

[0, 2]

and

[- 2, 0]

, respectively. Two random orthogonal matrices

B_{1}

and

B_{2}

provide positive semidefinite matrix

M_{1} = B_{1} A_{1} B_{1}^{T}

and negative semidefinite matrix

M_{2} = B_{2} A_{2} B_{2}^{T} .

Finally, set

Q = M_{1} + M_{1}^{T},

S = M_{2} + M_{2}^{T}

and

P = Q - S .

) and elements of q are taken arbitrary form

[- 1, 1] .

A set

K \subset R^{m}

is taken as

K : = {u \in R^{m} : - 10 \leq u_{i} \leq 10} .

Table 1 and Table 2 and Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 presented the numerical results by taking

u_{- 1} = u_{0} = v_{0} = (1, \dots, 1)

and

D_{n} \leq 10^{- 9} .

Example 2.

Suppose that

f : K \times K \to R

be a bifunction defined in the following way

f (p, q) = \sum_{i = 2}^{5} (q_{i} - p_{i}) ∥ p ∥, \forall p, q \in R^{5},

where

K = \{(p_{1}, \dots, p_{5}) : p_{1} \geq - 1, p_{i} \geq 1, i = 2, \dots, 5\} .

A bifunction f is Lipschitz-type continuous with constants

k_{1} = k_{2} = 2

and satisfy the conditions(f1)–(f4). In order to evaluate the best possible value of the control parameters, a numerical test is performed taking the variation of the inertial factor

θ .

The numerical comparison results are shown in the Table 3 by using

u_{- 1} = u_{0} = v_{0} = (2, 3, 2, 5, 5)

and

D_{n} \leq 10^{- 6} .

Example 3.

Let

E = L^{2} ([0, 1])

be a Hilbert space with an inner product

〈 p, q 〉 = \int_{0}^{1} p (r) q (r) d r,

and the induced norm

∥ p ∥ = \sqrt{\int_{0}^{1} p^{2} (r) d r},

\forall p, q \in E .

The set

K : = {p \in L^{2} ([0, 1]) : \int_{0}^{1} r p (r) d r = 2} .

Suppose that

f : E \times E \to R

is defined by

f (p, q) = 〈 L (p), q - p 〉,

where

L (p (r)) = \int_{0}^{r} p (s) d s,

for every

p \in L^{2} ([0, 1])

and

r \in [0, 1] .

The projection on set

K

is computed in the following way:

P_{K} (p) (r) : = p (r) - \frac{\int_{0}^{1} r p (r) d r - 2}{\int_{0}^{1} r^{2} d r} r, r \in [0, 1] .

Table 4 reports the numerical results by using stopping criterion

D_{n} \leq 10^{- 6}

and letting

u_{- 1} = u_{0} = v_{0} .

Example 4.

Assume that a bifunction f is defined by

f (p, q) = 〈 L (p), q - p 〉 and L (p) = G (p) + H (p),

where

G (p) = (g_{1} (p), g_{2} (p), \dots, g_{m} (p)), H (p) = E p + c, c = (- 1, - 1, \dots, - 1),

and

g_{i} (p) = p_{i - 1}^{2} + p_{i}^{2} + p_{i - 1} p_{i} + p_{i} p_{i + 1}, i = 1, 2, \dots, m, p_{0} = p_{m + 1} = 0 .

Let the matrix E of order m are consider in the following way:

e_{i, j} = \{\begin{matrix} 4 & j = i \\ 1 & i - j = 1 \\ - 2 & i - j = - 1 \\ 0 & o t h e r w i s e, \end{matrix}

where

K = \{(u_{1}, \dots, u_{m}) \in R^{m} : u_{i} \geq 1, i = 2, \dots, m\} .

Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 and Table 5 report the numerical results by taking

u_{- 1} = u_{0} = v_{0} = (1, \dots, 1)

and

D_{n} \leq 10^{- 6} .

Remark 2.

(i): It is also significant that the value of $ξ_{0}$ is crucial and performs best when it is nearer to $1 .$
(ii): It is observed that the selection of the value ϑ is often significant and roughly the value $ϑ \in (3, 6)$ performs better than most other values.

7. Conclusions

In this paper, we consider the convergence result for pseudomonotone equilibrium problems that involve Lipschitz-type continuous bifunction but the Lipschitz-type constants are unknown. We modify the extragradient methods with an inertial term and new step size formula. Weak convergence theorem is proved for sequences generated by the algorithm. Several numerical experiments confirm the effectiveness of the proposed algorithms.

Author Contributions

Conceptualization, H.u.R., N.P. and M.D.l.S.; Writing-Original Draft Preparation, N.W., N.P. and H.u.R.; Writing-Review & Editing, N.W., N.P., H.u.R. and M.D.l.S.; Methodology, N.P. and H.u.R.; Visualization, N.W. and N.P.; Software, H.u.R.; Funding Acquisition, M.D.l.S.; Supervision, M.D.l.S. and H.u.R.; Project Administration; M.D.l.S.; Resources; M.D.l.S. and H.u.R. All authors have read and agreed to the published version of this manuscript.

Funding

This research work was financially supported by Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-19.

Acknowledgments

We are very grateful to the Editor and the anonymous referees for their valuable and useful comments, which helped improve the quality of this work. Nopparat Wairojjana was partially supported by Valaya Alongkorn Rajabhat University under the Royal Patronage, Thailand. The corresponding author are grateful to the Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-19.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Example 1: numerical behaviour of Algorithm 1 by letting different options for

ξ_{0}

, while m = 10.

Figure 1. Example 1: numerical behaviour of Algorithm 1 by letting different options for

ξ_{0}

, while m = 10.

Figure 2. Example 1: numerical behaviour of Algorithm 1 by letting different options for

ξ_{0}

, while m = 20.

Figure 2. Example 1: numerical behaviour of Algorithm 1 by letting different options for

ξ_{0}

, while m = 20.

Figure 3. Example 1: numerical behaviour of Algorithm 1 by letting different options for

ξ_{0}

while m = 50.

Figure 3. Example 1: numerical behaviour of Algorithm 1 by letting different options for

ξ_{0}

while m = 50.

Figure 4. Example 1: numerical behaviour of Algorithm 1 by letting different options for

ξ_{0}

while m = 100.

Figure 4. Example 1: numerical behaviour of Algorithm 1 by letting different options for

ξ_{0}

while m = 100.

Figure 5. Example 1: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38] while m = 60.

Figure 6. Example 1: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38] while m = 120.

Figure 7. Example 1: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38] while m = 200.

Figure 8. Example 1: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38] while m = 300.

Figure 9. Example 4: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38] while m = 20.

Figure 10. Example 4: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38] while m = 50.

Figure 11. Example 4: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38] while m = 100.

Figure 12. Example 4: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38] while m = 200.

Figure 13. Example 4: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38] while m = 300.

Table 1. Example 1: Algorithm 1 numerical behaviour by letting different options for

ξ_{0}

and m.

Table 1. Example 1: Algorithm 1 numerical behaviour by letting different options for

ξ_{0}

and m.

	m = 10		m = 20		m = 50		m = 100
$ξ_{0}$	iter.	time	iter.	time	iter.	time	iter.	time
1.00	20	0.1701	25	0.2153	29	0.2726	40	0.5570
0.80	23	0.1945	27	0.2326	31	0.2788	47	0.5469
0.60	25	0.1995	30	0.2634	35	0.3285	52	0.6228
0.40	29	0.1467	33	0.2979	39	0.3549	55	0.6542
0.20	30	0.2632	35	0.2868	42	0.3849	57	0.6662

Table 2. Example 1: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38].

	Number of Iterations			Execution Time in Seconds
$m$	Alg3.1	Alg1	mAlg1	Alg3.1	Alg1	mAlg1
60	50	38	28	0.4362	0.3352	0.2705
120	57	49	33	0.6888	0.6000	0.4047
200	66	57	39	1.4708	1.0881	0.6794
300	62	55	40	1.6213	1.4251	1.0303

Table 3. Example 2: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38].

	Number of Iterations			Execution Time in Seconds
$θ$	Alg3.1	Alg1	mAlg1	Alg3.1	Alg1	mAlg1
0.90	67	56	47	2.8674	2.5324	1.6734
0.70	63	53	45	2.7813	2.6423	1.5026
0.50	57	47	41	2.0912	2.4212	1.4991
0.30	61	48	44	2.4115	2.3567	1.5092
0.10	69	60	47	2.9229	2.2881	1.5098

Table 4. Example 3: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38].

	Number of Iterations			Execution time in Seconds
$u_{0}$	Alg3.1	Alg1	mAlg1	Alg3.1	Alg1	mAlg1
$3 t$	33	28	19	4.7654	3.9782	2.9342
$3 t^{2}$	38	31	20	5.2598	4.1458	3.0987
$3 s i n (t)$	41	33	22	5.9876	5.3976	4.4298
$3 c o s (t)$	47	39	22	6.9921	5.4765	4.4611
$3 exp {(t)}^{2}$	58	43	31	8.4691	5.8329	5.0321

Table 5. Example 4: Algorithm 1 (mAlg1) numerical comparison with Algorithm 3.1 (Alg3.1) in [39] and Algorithm 1 (Alg1) in [38].

	Number of Iterations			Execution Time in Seconds
$m$	Alg3.1	Alg1	mAlg1	Alg3.1	Alg1	mAlg1
20	90	64	50	1.0089	0.6923	0.5541
50	98	70	52	1.6089	1.9092	0.8464
100	104	74	58	2.9231	2.1456	1.6970
200	109	79	61	22.5299	17.6267	13.6542
300	112	81	63	52.6776	39.0018	36.6305

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Wairojjana, N.; Rehman, H.u.; De la Sen, M.; Pakkaranang, N. A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems. Axioms 2020, 9, 101. https://doi.org/10.3390/axioms9030101

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Wairojjana N, Rehman Hu, De la Sen M, Pakkaranang N. A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems. Axioms. 2020; 9(3):101. https://doi.org/10.3390/axioms9030101

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Wairojjana, Nopparat, Habib ur Rehman, Manuel De la Sen, and Nuttapol Pakkaranang. 2020. "A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems" Axioms 9, no. 3: 101. https://doi.org/10.3390/axioms9030101

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Wairojjana, N., Rehman, H. u., De la Sen, M., & Pakkaranang, N. (2020). A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems. Axioms, 9(3), 101. https://doi.org/10.3390/axioms9030101

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A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems

Abstract

1. Introduction

2. Preliminaries

3. The Modified Extragradient Algorithm for the Problem (1) and Its Convergence Analysis

4. Applications to Solve Fixed Point Problems

5. Application to Solve Variational Inequality Problems

6. Numerical Experiments

7. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

References

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