An Averaged Halpern Type Algorithm for Solving Fixed Point Problems and Variational Inequality Problems

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An Averaged Halpern Type Algorithm for Solving Fixed Point Problems and Variational Inequality Problems

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Submitted:

06 October 2024

Posted:

07 October 2024

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Abstract

In this paper we propose and study in the setting of a Hilbert space an averaged Halpern type algorithm for approximating the solution of a common fixed point problem for a couple of nonexpansive and demicontractive mappings with a variational inequality constraint. The strong convergence of the sequence generated by the algorithm is established under feasible assumptions on the parameters involved. In particular, we also obtain the common solution of the fixed point problem for nonexpansive or demicontractive mappings and of a variational inequality problem. Our results extend and generalize various important related results in literature that were established for the pairs (nonexpansive, nonspreading) or (nonexpansive, strongly quasi-nonexpansive) mappings. Numerical tests to illustrate the superiority of our algorithm over the ones existing in literature are also reported.

Keywords:

Subject: Computer Science and Mathematics - Analysis

1. Introduction

Let

H

be a real Hilbert space with norm

∥ \cdot ∥

and inner product

〈 \cdot, \cdot 〉 .

Let

D \subset H

be a closed convex set, and consider the self-mapping

G : D \to D .

Throughout this paper the set of all fixed points of G in D is denoted by

F i x (G) = {u \in D : G u = u} .

The mapping G is said to be

(a): nonexpansive if

$∥ G u - G v ∥ \leq ∥ u - v ∥, for all u, v \in D;$

(1)
(b): quasi-nonexpansive if $F i x (G) \neq \emptyset$ and

$∥ G u - v ∥ \leq ∥ u - v ∥, for all u \in D and v \in F i x (G);$

(2)
(c): β-demicontractive if $F i x (G) \neq \emptyset$ and there exists a positive number $β < 1$ such that

${∥ G u - v ∥}^{2} \leq {∥ u - v ∥}^{2} + β {∥ u - G u ∥}^{2},$

(3)

for all $u \in D$ and $v \in F i x (G)$ .
(d): strongly quasi-nonexpansive if $F i x (G) \neq \emptyset,$ G is quasi-nonexpansive and $u_{p} - G u_{p} \to 0$ whenever ${u_{p}}$ is a bounded sequence such that $∥ u_{p} - u^{*} ∥ - ∥ G u_{p} - u^{*} ∥ \to 0$ for some $u^{*} \in F i x (G) .$

By the previous definitions, it is obvious that any nonexpansive mapping G with

F i x (G) \neq \emptyset

is quasi-nonexpansive, any strongly quasi-nonexpansive is quasi-nonexpansive, and that any quasi-nonexpansive mapping is demicontractive, too, but the reverses are no more true, as illustrated by the next example.

Example 1

([4]). Let

H

be the real line with the usual norm and

D = [0, 1] .

Define F on D as

F (u) = \{\begin{matrix} 7 / 8, & i f 0 \leq u < 1 \\ 1 / 4, & if u = 1 . \end{matrix}

(4)

Then F is

\frac{2}{3}

-demicontractive but F is neither nonexpansive nor quasi-nonexpasive (and hence not strongly quasi-nonexpansive).

There are several papers in literature which are devoted to the approximation of common fixed points of nonexpansive type mappings. For example, in order to approximate the common fixed points of a pair of nonexpansive self mappings

(T_{1}, T_{2})

with

F i x (T_{1}) \cap F i x (T_{2}) \neq \emptyset

, Takahashi and Tamura [21] considered the following iterative procedure:

\{\begin{matrix} x_{1} \in C, \\ x_{n + 1} = (1 - α_{n}) x_{n} + α_{n} T_{1} [(1 - β_{n}) x_{n} + β_{n} T_{2} x_{n}], n \geq 1, \end{matrix}

(5)

for which they established a weak convergence theorem.

Moudafi [15] considered a slightly different Krasnoselsij-Mann iterative procedure for the same problem, that he called "hierarchical fixed-point problem":

\{\begin{matrix} x_{1} \in C, \\ x_{n + 1} = (1 - α_{n}) x_{n} + α_{n} [(1 - β_{n}) T_{2} x_{n} + β_{n} T_{1} x_{n}], n \geq 1, \end{matrix}

(6)

where

F i x (T_{1})

and

F i x (T_{2})

are assumed to be nonempty.

Further, Iemoto and Takahashi [21] considered the problem of approximating the common fixed points of a nonexpansive mapping T and of a nonspreading mappings S in a Hilbert space, and utilized the iterative scheme

\{\begin{matrix} x_{1} \in C, \\ x_{n + 1} = (1 - α_{n}) x_{n} + α_{n} [(1 - β_{n}) T x_{n} + β_{n} S x_{n}], n \geq 1, \end{matrix}

(7)

for which they formulated and proved some weak convergence theorems.

Starting from this background, our aim in this paper is to solve the common fixed point problem in the setting of Hilbert spaces for the case of the larger class of demicontractive mappings, thus extending and unifying the main results in Cianciaruso et al. [7], Falset et al. [8], Iemoto and Takahashi [9] and many others.

Our main result (Theorem 1) provides a convergence theorem for an averaged iterative Halpern type algorithm used to approximate a solution of the common fixed point problem for a pair consisting of a nonexpansive mapping and a demicontractive mapping, which also solves a certain variational inequality problem.

2. Preliminaries

We recall some important lemmas used in the proofs of our main results. The following two lemmas are taken from Berinde [3].

Lemma 1.

[3] Let

H

be a real Hilbert space and

D \subset H

a closed and convex set. If

G : D \to D

is β-demicontractive, then the Krasnoselskij perturbation

G_{ν} = (1 - ν) I + ν G

of G is

(1 + β / ν - 1 / ν)

-demicontractive.

Lemma 2.

[3] Let

H

be a real Hilbert space and

D \subset H

a closed and convex set. If

G : D \to D

is β-demicontractive, then for any

ν \in (0, 1 - β)

G_{ν} = (1 - ν) I + ν G

is quasi-nonexpansive.

Lemma 3

(Zhou [30]). Let C be a nonempty subset of a real Hilbert space and let

T : C \to C

be a k-strictly pseudocontractive mapping. Then the averaged mapping

T_{λ} = (1 - λ) I + λ T

is nonexpansive for any

λ \in (0, 1 - k)

Lemma 4.

Let

H

be a real Hilbert space,

D \subset H

a closed and convex set and

F : D \to D

a mapping. Then, for any

ν \in (0, 1),

we have

F i x (F_{ν}) = F i x (F) .

Lemma 5.

[24] Let

{α_{n}}

be a sequence of nonnegative numbers such that

α_{n} \leq (1 - c_{n}) α_{n} + c_{n} μ_{n} + δ_{n}, n \geq 0,

where

{c_{n}}

is a sequence in

[0, 1]

and

{μ_{n}}

is a sequence in

R

such that

\sum_{n = 1}^{\infty} c_{n} = \infty, \underset{n \to \infty}{lim sup} μ_{n} \leq 0, δ_{n} \geq 0, and \sum_{n = 1}^{\infty} δ_{n} < \infty .

Then

{lim}_{n \to \infty} α_{n} = 0 .

Lemma 6.

Let

{γ_{p}}

be a sequence of real numbers that has a subsequence

{γ_{p_{k}}}

which satisfies

γ_{p_{k}} < γ_{p_{k} + 1}

for all

k \in N .

There exists an increasing sequence of integers

{τ (p)}_{p \geq p_{0}}

satisfying:

lim_{p \to \infty} τ (p) = \infty, γ_{τ (p)} \leq γ_{τ (p) + 1}, γ_{p} \leq γ_{τ (p) + 1}, \forall p \geq p_{0} .

3. Fixed Points and Variational Inequalities

In this section we state and prove our main results. To do this, we first consider the following property.

A mapping G satisfies Condition A if

u_{p} - G u_{p} \to 0

whenever

{u_{p}}

is a bounded sequence such that

∥ u_{p} - u^{*} ∥ - ∥ (1 - ν) u_{p} + ν G u_{p} - u^{*} ∥ \to 0

for some

u^{*} \in F i x (G)

and

ν \in [0, 1] .

Theorem 1.

Let

H

be a Hilbert space and C be a closed convex subset of

H .

Let

F : C \to C

be a nonexpansive mapping and

G : C \to C

be a β-demicontractive mapping satisfying Condition A such that

I - G

is demiclosed at

0 .

Assume that

F i x (F) \cap F i x (G) \neq \emptyset .

Let

{a_{p}}

and

{b_{p}}

be sequences in

[0, 1]

such that

a_{p} \to 0

and

\sum_{p = 1}^{\infty} a_{p} = \infty .

Let

{u_{p}}

be the sequence generated in the following manner:

\{\begin{matrix} x, u_{1} \in C, \\ u_{p + 1} = a_{p} x + (1 - a_{p}) [b_{p} F u_{p} + (1 - b_{p}) ((1 - α) u_{p} + α G u_{p})], p \geq 1 . \end{matrix}

(8)

Then, the following assertions hold.

(I): If $\sum_{p = 1}^{\infty} (1 - b_{p}) < \infty$ and $\sum_{p = 1}^{\infty} | a_{p} = a_{p + 1} | < \infty,$ then ${u_{p}}$ strongly converges to $u^{*} \in F i x (F)$ which is the unique point in $F i x (F)$ that solves the variational inequality

$〈 u^{*} - x, u - u^{*} 〉 \geq 0, \forall u \in F i x (F),$

(9)

i.e. $u^{*} = P_{F i x (F)} x .$
(II): If $\sum_{p = 1}^{\infty} (1 - b_{p}) < \infty$ and $\frac{b_{p}}{a_{p}} \to 0,$ then ${u_{p}}$ converges strongly to $v^{*} \in F i x (G)$ which is the unique point in $F i x (G)$ that solves the variational inequality

$〈 u^{*} - x, u - v^{*} 〉 \geq 0, \forall u \in F i x (G),$

(10)

i.e. $v^{*} = P_{F i x (G)} x .$
(III): If ${lim inf}_{p \to \infty} b_{p} (1 - b_{p}) > 0,$ then ${u_{p}}$ strongly converges to $\bar{u} \in F i x (F) \cap F i x (G)$ that is the unique solution of the variational inequality

$〈 \bar{u} - x, u - \bar{u} 〉 \geq 0, \forall u \in F i x (F) \cap F i x (G),$

(11)

i.e. $\bar{u} = P_{F i x (F) \cap F i x (G)} x .$

Proof.

Since G is a

β

-demicontractive mapping, by Lemma 2 it follows that the averaged mapping

G_{α} = (1 - α) I + α G

is quasi-nonexpansive, for

α \in (0, 1 - β) .

Clearly,

G_{α} - I

is demiclosed at zero. One can also see that

G_{α}

is strongly quasi-nonexpansive from the fact that G satisfies Condition A. Now we can write

u_{p + 1} = a_{p} x + (1 - a_{p}) [b_{p} F u_{p} + (1 - b_{p}) G_{α} u_{p}], p \in N .

Let w be a common fixed point of F and

G_{α} .

Define

T_{p} = b_{p} F + (1 - b_{p}) G_{α}, \forall p \in N .

For all

p \in N,

\begin{matrix} ∥ u_{p + 1} - w ∥ & = ∥ a_{p} x + (1 - a_{p}) T_{p} u_{p} - w ∥ \\ \leq (1 - a_{p}) ∥ u_{p} - w ∥ + a_{p} ∥ x - w ∥ \\ \leq max {∥ u_{p} - w ∥, ∥ x - w ∥} \\ \leq max {∥ u_{1} - w ∥, ∥ x - w ∥}, \end{matrix}

that is,

{u_{p}}

is a bounded sequence.

Furthermore, since

a_{p} \to 0

p \to \infty,

we have

u_{p + 1} - T_{p} u_{p} = a_{p} (x - T_{p} u_{p}) \to 0, as p \to \infty .

(12)

To prove (I), for all

p \in N

compute

\begin{matrix} ∥ u_{p + 1} - u_{p} ∥ & = ∥ (1 - a_{p}) (T_{p} u_{p} - T_{p - 1} u_{p - 1}) - (a_{p} - a_{p - 1}) T_{p - 1} u_{p - 1} + (a_{p} - a_{p - 1}) x ∥ \\ = ∥ (1 - a_{p}) (T_{p} u_{p} - T_{p - 1} u_{p - 1}) + (a_{p - 1} - a_{p}) [T_{p - 1} u_{p - 1} - x] ∥ \\ = ∥ (a_{p - 1} - a_{p}) [T_{p - 1} u_{p - 1} - x] \\ + (1 - a_{p}) [b_{p} F u_{p} + (1 - b_{p}) G_{α} F u_{p} - b_{p - 1} F u_{p - 1} - (1 - b_{p - 1}) G_{α} u_{p - 1}] ∥ \\ = ∥ (a_{p - 1} - a_{p}) [T_{p - 1} u_{p - 1} - x] + (1 - a_{p}) [b_{p} (F u_{p} - F u_{p - 1}) \\ + (1 - b_{p}) (G_{α} u_{p} - G_{α} u_{p - 1}) + (b_{p} - b_{p - 1} (F u_{p - 1} - G_{α} u_{p - 1})]] ∥ \\ \leq | a_{p - 1} - a_{p} | ∥ T_{p - 1} u_{p - 1} - x ∥ + (1 - a_{p}) [b_{p} ∥ u_{p} - u_{p - 1} ∥ \\ + (1 - b_{p}) ∥ G_{α} u_{p} - G_{α} u_{p - 1} ∥ + | b_{p} - b_{p - 1} | ∥ F u_{p - 1} - G_{α} u_{p - 1}] ∥ \\ = (1 - c_{p}) ∥ u_{p} - u_{p - 1} ∥ + μ_{p}, \end{matrix}

where

c_{p} = 1 - b_{p} + a_{p} b_{p}

and

μ_{p} = | a_{p - 1} - a_{p} | ∥ T_{p - 1} u_{p - 1} - x ∥ + (1 - a_{p}) [(1 - b_{p}) + | b_{p} - b_{p - 1} | ∥ F u_{p - 1} - G_{α} u_{p - 1}] ∥ .

We see that

c_{p} \to 0,

\sum_{p = 1}^{\infty} c_{p} = \infty

and

\sum_{p = 1}^{\infty} μ_{p} < \infty .

Hence, by Lemma 5 we conclude that

u_{p + 1} - u_{p} \to 0 .

This and (12) imply that

u_{p} - T_{p} u_{p} \to 0 .

(13)

Moreover,

\begin{matrix} ∥ u_{p} - T_{p} u_{p} ∥ & = ∥ u_{p} - b_{p} F u_{p} - (1 - b_{p}) ∥ \\ \geq ∥ u_{p} - b_{p} F u_{p} ∥ - (1 - b_{p}) ∥ G_{α} u_{p} ∥, \\ ∥ u_{p} - b_{p} F u_{p} ∥ & \leq ∥ u_{p} - T_{p} u_{p} ∥ + (1 - b_{p}) ∥ G_{α} u_{p} ∥, \end{matrix}

and thus, from the hypothesis that

\sum_{p = 1}^{\infty} (1 - b_{p}) < \infty,

we also have

u_{p} - b_{p} F u_{p} \to 0 .

We can conclude that

u_{p} - F u_{p} \to 0 .

Hence, any weak limit of

{u_{p}}

is in

F i x (F) .

Let

{u_{p_{k}}}

be a subsequence of

{u_{p}}

such that

\underset{p \to \infty}{lim sup} 〈 u_{p} - u^{*}, x - u^{*} 〉 = lim_{k \to \infty} 〈 u_{p_{k}} - u^{*}, x - u^{*} 〉

(14)

and

u_{p_{k}} \to y .

Thus,

y \in F i x (F)

and

\underset{p \to \infty}{lim sup} 〈 u_{p} - u^{*}, x - u^{*} 〉 = 〈 y - u^{*}, x - u^{*} 〉,

which is nonpositive by the definition of

u^{*} .

We obtain

\begin{matrix} \underset{p \to \infty}{lim sup} 〈 T_{p} u_{p} - u^{*}, x - u^{*} 〉 & = \underset{p \to \infty}{lim sup} [〈 u_{p} - u^{*}, x - u^{*} 〉 + 〈 T_{p} u_{p} - u_{p}, x - u^{*} 〉] \\ = \underset{p \to \infty}{lim sup} 〈 u_{p} - u^{*}, x - u^{*} 〉 \leq 0 . \end{matrix}

Finally,

\begin{matrix} ∥ u_{p + 1} - u^{*} ∥^{2} & = ∥ (1 - a_{p}) (T_{p} u_{p} - u^{*}) + a_{p} (x - u^{*}) ∥^{2} \\ = {(1 - a_{p})}^{2} ∥ T_{p} u_{p} - u^{*} ∥^{2} + a_{p}^{2} {∥ x - u^{*} ∥}^{2} \\ + 2 a_{p} 〈 (1 - a_{p}) (T_{p} u_{p} - u^{*}), x - u^{*} 〉 \\ = {(1 - a_{p})}^{2} ∥ b_{p} (F u_{p} - u^{*}) + (1 - b_{n}) (G_{α} u_{p} - u^{*}) ∥^{2} + a_{p}^{2} {∥ x - u^{*} ∥}^{2} \\ + 2 a_{p} 〈 T_{p} u_{p} - u^{*}, x - u^{*} 〉 - 2 a_{p}^{2} 〈 T_{p} u_{p} - u^{*}, x - u^{*} 〉 \\ \leq {(1 - a_{p})}^{2} [b_{p} ∥ u_{p} - u^{*} ∥ + (1 - b_{p}) {∥ G_{α} u_{p} - u^{*} ∥}^{2}] + a_{p}^{2} {∥ x - u^{*} ∥}^{2} \\ + 2 a_{p} 〈 T_{p} u_{p} - u^{*}, x - u^{*} 〉 \\ \leq {(1 - a_{p})}^{2} ∥ u_{p} - u^{*} ∥^{2} + (1 - b_{p}) ∥ G_{α} u_{p} - u^{*} {) ∥}^{2} + a_{p} {∥ x - u^{*} ∥}^{2} \\ + 2 a_{p} 〈 T_{p} u_{p} - u^{*}, x - u^{*} 〉 \\ = (1 - t_{p}) {∥ u_{p} - u^{*} ∥}^{2} + t_{p} r_{p} + s_{p}, \end{matrix}

where

\begin{matrix} t_{p} = 2 a_{p} - a_{p}^{2}, r_{p} = a_{p} [∥ x - u^{*} ∥^{2} + 2 〈 T_{p} u_{p} - u^{*}, x - u^{*} 〉], \\ s_{p} = (1 - b_{p}) {∥ G_{α} u_{p} - u^{*} ∥}^{2} . \end{matrix}

Now, Lemma 5 implies that

u_{p} \to u^{*} .

To prove (II), let

v^{*}

be the unique solution of the variational inequality (10) and compute

\begin{matrix} ∥ u_{p + 1} - v^{*} ∥^{2} & = ∥ a_{n} x + (1 - a_{n}) T_{p} u_{p} - v^{*} + a_{p} v^{*} - a_{p} v^{*} ∥^{2} \\ = ∥ a_{p} (x - v^{*}) + (1 - a_{n}) (T_{p} u_{p} - v^{*}) ∥^{2} \\ \leq {(1 - a_{n})}^{2} {∥ (T_{p} u_{p} - v^{*}) ∥}^{2} + 2 a_{p} 〈 x - v^{*}, u_{p + 1} - v^{*} 〉 \\ = {(1 - a_{n})}^{2} {∥ b_{p} (F u_{p} - v^{*}) + (1 - b_{p}) (G_{α} u_{p} - v^{*}) ∥}^{2} \\ + 2 a_{p} 〈 x - v^{*}, u_{p + 1} - v^{*} 〉 \\ \leq {(1 - a_{p})}^{2} ∥ u_{p} - v^{*} ∥^{2} + b_{p} {∥ F u_{p} - v^{*} ∥}^{2} \\ + 2 a_{p} 〈 x - v^{*}, u_{p + 1} - v^{*} 〉 \end{matrix}

(15)

We have two cases, namely, the sequence

{∥ u_{p} - v^{*} ∥}

is eventually not increasing or not eventually not increasing.

Case (II) 1. There exists

p_{0} \in N

such that

∥ u_{p + 1} - v^{*} ∥ \leq ∥ u_{p} - v^{*} ∥

for all

p \geq p_{0} .

Put

\begin{matrix} φ_{p} & = 2 〈 u_{p + 1} - v^{*}, x - v^{*} 〉, and \\ μ_{p} & = (1 - b_{p}) {∥ (G_{α} u_{p} - v^{*}) ∥}^{2} . \end{matrix}

Since

{(1 - a_{p})}^{2} \leq (1 - a_{p}),

we have

∥ u_{p + 1} - v^{*} ∥^{2} \leq (1 - a_{p}) {∥ u_{p} - v^{*} ∥}^{2} + a_{p} φ_{p} + μ_{p} .

Since

{∥ u_{p} - v^{*} ∥}

is not eventually increasing,

{lim}_{p \to \infty} ∥ u_{p} - v^{*} ∥

exists. Thus,

\begin{matrix} 0 & = lim_{p \to \infty} (∥ u_{p + 1} - v^{*} ∥ - ∥ u_{p} - v^{*} ∥) \\ \leq \underset{p \to \infty}{lim inf} (a_{p} ∥ x - v^{*} ∥ + (1 - a_{n}) ∥ T_{p} u_{p} - v^{*} ∥ - ∥ u_{p} - v^{*} ∥) \\ = \underset{p \to \infty}{lim inf} (∥ T_{p} u_{p} - v^{*} ∥ - ∥ u_{p} - v^{*} ∥) \\ = \underset{p \to \infty}{lim inf} (∥ b_{p} (F u_{p} - v^{*}) + (1 - b_{p}) (G_{α} u_{p} - v^{*}) ∥ - ∥ u_{p} - v^{*} ∥) \\ = \underset{p \to \infty}{lim inf} (∥ G_{α} u_{p} - v^{*} ∥ - ∥ u_{p} - v^{*} ∥) \\ = \underset{p \to \infty}{lim sup} (∥ u_{p} - v^{*} ∥ - ∥ u_{p} - v^{*} ∥) = 0 \end{matrix}

Hence,

lim_{p \to \infty} (∥ G_{α} u_{p} - v^{*} ∥ - ∥ u_{p} - v^{*} ∥) = 0 .

From the strong quasi-nonexpansiveness of

G_{α}

, we conclude that

G_{α} u_{p} - u_{p} \to 0 .

(16)

The rest of the proof is similar to the proof of (I).

Case (II) 2. The sequence

{∥ u_{p} - v^{*} ∥}

is not eventually not increasing. There exists a subsequence

{∥ u_{p_{k}} - v^{*} ∥}

such that

∥ u_{p_{k}} - v^{*} ∥ < ∥ u_{p_{k} + 1} - v^{*} ∥

for all

k \in N .

By Lemma 6, there exists an increasing sequence of integers

{τ (p)}

satisfying

\begin{matrix} lim_{p \to \infty} τ (p) = \infty, ∥ u_{τ (p)} - v^{*} ∥ \leq ∥ u_{τ (p) + 1} - v^{*} ∥ \\ ∥ u_{p} - v^{*} ∥ \leq ∥ u_{τ (p) + 1} - v^{*} ∥, \forall p \geq p_{0} . \end{matrix}

(17)

Thus,

0 \leq \underset{p \to \infty}{lim inf} (∥ u_{τ (p) + 1} - v^{*} ∥ - ∥ u_{τ (p)} - v^{*} ∥) .

Using (16) with

τ (p)

instead of

p,

we obtain

lim_{p \to \infty} ∥ G_{α} u_{τ (p)} - v^{*} ∥ - ∥ u_{τ (p)} - v^{*} ∥ = 0 .

The strong quasi-nonexpasiveness of

G_{α}

implies

G_{α} u_{τ (p)} - u_{τ (p)} \to 0,

(18)

Since

I - G_{α}

is demiclosed at

0,

we conclude that

\underset{p \to \infty}{lim sup} 〈 u_{τ (p) + 1} - v^{*}, x - v^{*} 〉 \leq 0 .

(19)

One may observe that

\begin{matrix} ∥ u_{τ (p) + 1} - u_{τ (p)} ∥ & = a_{τ (p)} ∥ x - u_{τ (p)} ∥ + (1 - a_{τ (p)}) b_{τ (p)} O (1) \\ + (1 - a_{τ (p)}) ∥ G_{α} u_{τ (p)} -_{τ (p)} ∥, \end{matrix}

where

O (1)

represents a bounded sequence. Thus, from (18) it follows that

u_{τ (p) + 1} - u_{τ (p)} \to 0 .

Replacing p by

τ (p)

in (15) yields

\begin{matrix} ∥ u_{τ (p) + 1} - v^{*} ∥ & \leq {(1 - a_{τ (p)})}^{2} {∥ u_{τ (p)} - v^{*} ∥}^{2} \\ + 2 a_{τ (p)} 〈 u_{τ (p) + 1} - v^{*}, x - v^{*} 〉 + b_{τ (p)} {∥ F u_{τ (p)} - v^{*} ∥}^{2} \\ \leq {(1 - a_{τ (p)})}^{2} {∥ u_{τ (p) + 1} - v^{*} ∥}^{2} \\ + 2 a_{τ (p)} 〈 u_{τ (p) + 1} - v^{*}, x - v^{*} 〉 + b_{τ (p)} {∥ F u_{τ (p)} - v^{*} ∥}^{2} . \end{matrix}

As a consequent, we have

\begin{matrix} 2 a_{τ (p)} {∥ u_{τ (p) + 1} - v^{*} ∥}^{2} & \leq {(a_{τ (p)})}^{2} {∥ u_{τ (p) + 1} - v^{*} ∥}^{2} \\ + 2 a_{τ (p)} 〈 u_{τ (p) + 1} - v^{*}, x - v^{*} 〉 + b_{τ (p)} {∥ F u_{τ (p)} - v^{*} ∥}^{2} . \end{matrix}

Dividing by

a_{τ (p)}

gives

\begin{matrix} 0 \leq 2 ∥ u_{τ (p) + 1} - v^{*} ∥^{2} & \leq a_{τ (p)} {∥ u_{τ (p) + 1} - v^{*} ∥}^{2} \\ + 2 a_{τ (p)} 〈 u_{τ (p) + 1} - v^{*}, x - v^{*} 〉 + \frac{b_{τ (p)}}{a_{τ (p)}} {∥ F u_{τ (p)} - v^{*} ∥}^{2} . \end{matrix}

Since

b_{p} / a_{p} \to 0,

by (19) we obtain

lim_{p \to \infty} ∥ u_{τ (p)} - v^{*} ∥ \leq lim_{p \to \infty} ∥ u_{τ (p) + 1} - v^{*} ∥ = 0 .

From (17), we obtain that

u_{p} \to v^{*}

To prove (III), let

\bar{u}

be the unique point in

F i x (F) \cap F i x (G)

that satisfies the variational inequality (11). We have

\begin{matrix} ∥ T_{p} u_{p} - \bar{u} ∥^{2} & = ∥ b_{p} (F u_{p} - \bar{u}) + (1 - b_{p}) {(G u_{p} - \bar{u} ∥}^{2} \\ = b_{p} ∥ F u_{p} - \bar{u} ∥^{2} + (1 - b_{p}) ∥ G_{α} u_{p} - \bar{u} ∥^{2} - b_{p} (1 - b_{p}) {∥ F u_{p} - G_{α} u_{p} ∥}^{2} \\ \leq ∥ u_{p} - \bar{u} ∥^{2} - b_{p} (1 - b_{p}) {∥ F u_{p} - G_{α} u_{p} ∥}^{2}, \end{matrix}

\begin{matrix} ∥ u_{p + 1} - \bar{u} ∥^{2} & = ∥ a_{p} (x - \bar{u}) + (1 - a_{p}) {(T_{p} u_{p} - \bar{u} ∥}^{2} \\ \leq {(1 - a_{p})}^{2} ∥ T_{p} u_{p} - \bar{u} ∥^{2} + a_{p}^{2} ∥ x - \bar{u} ∥^{2} + a_{p} ∥ (x - \bar{u}) (T_{p} u_{p} - \bar{u}) ∥ \\ \leq ∥ u_{p} - \bar{u} ∥^{2} - b_{p} (1 - b_{p}) {∥ F u_{p} - G_{α} u_{p} ∥}^{2} \\ + a_{p}^{2} ∥ x - \bar{u} ∥^{2} + a_{p} ∥ (x - \bar{u}) (T_{p} u_{p} - \bar{u}) ∥ . \end{matrix}

(20)

Similar to (II), we have two cases.

Case (III) 1.

{∥ u_{p} - \bar{u} ∥}

is eventually not increasing. There exists

p_{0} \in N

such that

∥ u_{p + 1} - \bar{u} ∥ \leq ∥ u_{p} - \bar{u} ∥,

for all

n \geq n_{0} .

Thus,

{lim}_{p \to \infty} ∥ u_{p} - \bar{u} ∥

exists. We have

\begin{matrix} b_{p} (1 - b_{p}) ∥ F u_{p} - G u_{p} ∥ \leq & ∥ u_{p} - \bar{u} ∥^{2} - ∥ u_{p + 1} - \bar{u} ∥^{2} + a_{p}^{2} {∥ x - \bar{u} ∥}^{2} \\ + a_{p} ∥ (x - \bar{u}) (T_{p} u_{p} - \bar{u}) ∥ . \end{matrix}

Since

{lim inf}_{p \to \infty} b_{p} (1 - b_{p}) > 0,

we have

F u_{p} - G_{α} u_{p} \to 0 .

(21)

Moreover,

\begin{matrix} 0 & = lim_{p \to \infty} (∥ u_{p + 1} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥) \\ \leq \underset{p \to \infty}{lim inf} (a_{p} ∥ x - \bar{u} ∥ + (1 - a_{p}) ∥ T_{p} u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥) \\ = \underset{p \to \infty}{lim inf} (∥ T_{p} u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥) \\ = \underset{p \to \infty}{lim inf} (∥ b_{p} (F u_{p} - \bar{u} + (1 - b_{p}) (G_{α} u_{p} - \bar{u}) ∥ - ∥ u_{p} - \bar{u} ∥) \\ \leq \underset{p \to \infty}{lim inf} (b_{p} (∥ F u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥) + (1 - b_{p}) (∥ G_{α} u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥)) \\ \leq \underset{p \to \infty}{lim sup} (b_{p} (∥ F u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥) + (1 - b_{p}) (∥ G_{α} u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥)) \\ \leq 0 \end{matrix}

Therefore

lim_{p \to \infty} (b_{p} (∥ F u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥) + (1 - b_{p}) (∥ G_{α} u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥)) = 0 .

It follows that

lim_{p \to \infty} (b_{p} (∥ F u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥) = lim_{p \to \infty} (1 - b_{p}) (∥ G_{α} u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥)) = 0 .

(22)

Thus,

lim_{p \to \infty} ((∥ F u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥) = lim_{p \to \infty} (∥ G_{α} u_{p} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥)) = 0 .

(23)

From strong quasi-nonexpansiveness of

G_{α}

it follows

S u_{p} - u_{p} \to 0 .

(24)

Since

u_{p} - F u_{p} = u_{p} - G_{α} u_{p} + G_{α} u_{p} - F u_{p},

we obtain

u_{p} - F u_{p} \to 0 .

(25)

Choose a subsequence

u_{p_{k}} \to z

such that

\underset{p \to \infty}{lim sup} 〈 u_{p} - \bar{u}, x - \bar{u} 〉 = lim_{p \to \infty} 〈 u_{p_{k}} - \bar{u}, x - \bar{u} 〉 = 〈 z - \bar{u}, x - \bar{u} 〉 .

Since both F and

G_{α}

are demiclosed at 0 and by (24), (25), one can conclude that

z \in F i x (F) \cap F i x (G_{α}) .

Hence, by the definition of

\bar{u},

we obtain (26). Furthermore, from (24) and (25) we have

T_{p} u_{p} - u_{p} \to 0 .

(26)

Finally

\begin{matrix} ∥ u_{p + 1} - \bar{u} ∥^{2} & = ∥ (1 - a_{p}) (T_{p} u_{p} - \bar{u}) + a_{p} (x - \bar{u}) ∥ \\ = {(1 - a_{p})}^{2} ∥ T_{p} u_{p} - \bar{u} ∥^{2} + a_{p}^{2} {∥ x - \bar{u} ∥}^{2} + 2 a_{p} 〈 (1 - a_{p}) (T_{p} u_{p} - \bar{u}), x - \bar{u} 〉 \\ \leq {(1 - a_{p})}^{2} ∥ u_{p} - \bar{u} ∥^{2} + a_{p}^{2} [∥ x - \bar{u} ∥^{2} - 2 〈 T_{p} u_{p} - \bar{u}, x - \bar{u} 〉] \\ + 2 a_{p} 〈 T_{p} u_{p} - u_{p}, x - \bar{u} 〉 + 2 a_{p} 〈 u_{p} - \bar{u}, x - \bar{u} 〉 \\ \leq (1 - a_{p}) ∥ u_{p} - \bar{u} ∥^{2} + a_{p} [∥ x - \bar{u} ∥^{2} - 2 〈 T_{p} u_{p} - \bar{u}, x - \bar{u} 〉] \\ + 2 a_{p} 〈 T_{p} u_{p} - u_{p}, x - \bar{u} 〉 + 2 a_{p} 〈 u_{p} - \bar{u}, x - \bar{u} 〉 . \end{matrix}

Putting

\begin{matrix} s_{p} & = 1 - {(1 - a_{p})}^{2}, \\ σ_{p} & = 2 〈 T_{p} u_{p} - u_{p}, x - p_{0} 〉 + 2 〈 u_{p} - \bar{u}, x - \bar{u} 〉 + a_{p} [∥ x - \bar{u} ∥^{2} - 2 〈 T_{p} u_{p} - \bar{u}, x - \bar{u} 〉], \end{matrix}

the conclusion follows from Lemma 5.

Case (III) 2. The sequence

{∥ u_{p} - \bar{u} ∥}

is not eventually not increasing, i.e., there exists a subsequence

{∥ u_{p_{k}} - \bar{u} ∥}

such that

∥ u_{p_{k}} - \bar{u} ∥ < ∥ u_{p_{k + 1}} - \bar{u} ∥, \forall j \in N .

Lemma 6 implies that there exists an increasing sequence of integers

{τ (p)}_{p \in N}

satisfying (17). Therefore

\begin{matrix} o & \leq \underset{p \to \infty}{lim inf} (∥ u_{τ (p) + 1} - \bar{u} ∥ - ∥ u_{τ (p)} - \bar{u} ∥) \\ \leq \underset{p \to \infty}{lim sup} (∥ u_{τ (p) + 1} - \bar{u} ∥ - ∥ u_{τ (p)} - \bar{u} ∥) \\ \leq \underset{p \to \infty}{lim sup} (∥ u_{p + 1} - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥) \\ \leq \underset{p \to \infty}{lim sup} (∥ (1 - a_{p}) (T_{p} u_{p} - \bar{u}) + a_{p} (x - \bar{u}) ∥ - ∥ u_{p} - \bar{u} ∥) \\ \leq \underset{p \to \infty}{lim sup} ((1 - a_{p}) ∥ u_{p} - \bar{u} ∥ + a_{p} ∥ x - \bar{u} ∥ - ∥ u_{p} - \bar{u} ∥) \\ = 0 \end{matrix}

We obtain

lim_{p \to \infty} (∥ u_{τ (p) + 1} - \bar{u} ∥ - ∥ u_{τ (p)} - \bar{u} ∥) = 0

(27)

Now, using (24) with

τ (p)

instead of

p,

we have

G_{α} u_{τ (p)} - u_{τ (p)} \to 0,

(28)

and (20) can be written as

\begin{matrix} 0 & \leq b_{τ (p)} (1 - b_{τ (p)}) {∥ F u_{τ (p)} - G_{α} u_{τ (p)} ∥}^{2} \\ \leq ∥ u_{τ (p)} - \bar{u} ∥^{2} - ∥ u_{τ (p) + 1} - \bar{u} ∥^{2} + a_{p} ∥ x - \bar{u} ∥ (1 + ∥ (T_{p} u_{p} - \bar{u}) ∥) . \end{matrix}

From (27) and since

{lim inf}_{p \to \infty} b_{p} (1 - b_{p}) > 0,

F u_{τ (p)} - G_{α} u_{τ (p)} \to 0 .

(29)

We also obtain that

u_{τ (p)} - F u_{τ (p)} = u_{τ (p)} - G_{α} u_{τ (p)} + G_{α} u_{τ (p)} - F u_{τ (p)} .

By (28) and (29), we have that

u_{τ (p)} - F u_{τ (p)} \to 0 and u_{τ (p)} - T_{τ (p)} u_{τ (p)} \to 0 .

(30)

Similar to (26) changing

τ (p)

with

p,

we have

\underset{p \to \infty}{lim sup} 〈 u_{τ (p)} - \bar{u}, x - \bar{u} 〉 \leq 0 .

Following the same proof of (26), replacing p with

τ (p),

we obtain

\underset{p \to \infty}{lim sup} 〈 u_{τ (p)} - \bar{u}, x - \bar{u} 〉 \leq 0 .

(31)

Now we compute,

\begin{matrix} ∥ u_{τ (p) + 1} - \bar{u} ∥^{2} & \leq {(1 - a_{τ (p)})}^{2} {∥ u_{τ (p)} - \bar{u} ∥}^{2} \\ + a_{τ (p)}^{2} [∥ x - \bar{u} ∥^{2} - 2 〈 T_{τ (p)} u_{τ (p)} - \bar{u}, x - \bar{u} 〉] \\ + 2 a_{τ (p)} 〈 T_{τ (p)} u_{τ (p)} - u_{τ (p)}, x - \bar{u} 〉 + 2 a_{τ (p)} 〈 u_{τ (p)} - \bar{u}, x - \bar{u} 〉 \\ \leq {(1 - a_{τ (p)})}^{2} {∥ u_{τ (p) + 1} - \bar{u} ∥}^{2} \\ + a_{τ (p)}^{2} [∥ x - \bar{u} ∥^{2} - 2 〈 T_{τ (p)} u_{τ (p)} - \bar{u}, x - \bar{u} 〉] \\ + 2 a_{τ (p)} 〈 T_{τ (p)} u_{τ (p)} - u_{τ (p)}, x - \bar{u} 〉 + 2 a_{τ (p)} 〈 u_{τ (p)} - \bar{u}, x - \bar{u} 〉 \end{matrix}

Consequently,

\begin{matrix} 2 a_{τ (p)} {∥ u_{τ (p) + 1} - \bar{u} ∥}^{2} & \leq a_{τ (p)}^{2} [∥ x - \bar{u} ∥^{2} - 2 〈 T_{τ (p)} u_{τ (p)} - \bar{u}, x - \bar{u} 〉] \\ + 2 a_{τ (p)} 〈 u_{τ (p)} - \bar{u}, x - \bar{u} 〉 \\ + 2 a_{τ (p)} 〈 T_{τ (p)} u_{τ (p)} - u_{τ (p)}, x - \bar{u} 〉 \end{matrix}

dividing by

a_{τ (p)},

we get

\begin{matrix} 0 & \leq 2 ∥ u_{τ (p) + 1} - \bar{u} ∥^{2} \\ \leq a_{τ (p)} [∥ x - \bar{u} ∥^{2} - 2 〈 T_{τ (p)} u_{τ (p)} - \bar{u}, x - \bar{u} 〉] + 2 〈 u_{τ (p)} - \bar{u}, x - \bar{u} 〉 \\ + 2 〈 T_{τ (p)} u_{τ (p)} - u_{τ (p)}, x - \bar{u} 〉 \end{matrix}

Taking the limsup and recalling the hypothesis (30) and (31), we obtain

lim_{p \to \infty} ∥ u_{τ (p)} - \bar{u} ∥ \leq lim_{p \to \infty} ∥ u_{τ (p) + 1} - \bar{u} ∥ = 0

Now by (17), we conclude that

u_{p} \to \bar{u} .

□

A more general result could be proved similarly to the proof of Theorem 1.

Theorem 2.

Let

H

be a Hilbert space and C be a closed convex subset of

H .

Let

F : C \to C

be a k-strictly pseudocontractive mapping and

G : C \to C

be a β-demicontractive mapping satisfying Condition A such that

I - G

is demiclosed at

0 .

Assume that

F i x (F) \cap F i x (G) \neq \emptyset .

Let

{a_{p}}

and

{b_{p}}

be sequences in

[0, 1]

such that

a_{p} \to 0

and

\sum_{p = 1}^{\infty} a_{p} = \infty .

Let

{u_{p}}

be a sequence generated in the following manner:

\{\begin{matrix} x, u_{1} \in C, \\ u_{p + 1} = a_{p} x + (1 - a_{p}) [b_{p} F_{λ} u_{p} + (1 - b_{p}) ((1 - α) u_{p} + α G u_{p})], p \geq 1 . \end{matrix}

(32)

where

F_{λ} u = (1 - λ) u + λ F u

, with

λ \in (0, 1 - k)

Then, the following assertions hold.

(I): If $\sum_{p = 1}^{\infty} (1 - b_{p}) < \infty, \sum_{p = 1}^{\infty} | a_{p} = a_{p + 1} | < \infty,$ then ${u_{p}}$ strongly converges to $u^{*} \in F i x (F)$ that is the unique point in $F i x (F)$ that solves the variational inequality

$〈 u^{*} - x, u - u^{*} 〉 \geq 0, \forall u \in F i x (F),$

i.e. $u^{*} = P_{F i x (F)} x .$
(II): If $\sum_{p = 1}^{\infty} (1 - b_{p}) < \infty, \frac{b_{p}}{a_{p}} \to 0,$ then ${u_{p}}$ converges strongly to $v^{*} \in F i x (G)$ that is the unique point in $F i x (G)$ that solves the variational inequality

$〈 v^{*} - x, u - v^{*} 〉 \geq 0, \forall u \in F i x (G),$

(33)

i.e. $v^{*} = P_{F i x (G)} x$ .
(III): If ${lim inf}_{p \to \infty} b_{p} (1 - b_{p}) > 0,$ then ${u_{p}}$ strongly converges to $\bar{u} \in F i x (F) \cap F i x (G)$ that is the unique solution of the variational inequality

$〈 \bar{u} - x, u - \bar{u} 〉 \geq 0, \forall u \in F i x (F) \cap F i x (G),$

i.e. $\bar{u} = P_{F i x (F) \cap F i x (G)} x .$

Proof.

As F is k-strictly pseudocontractive, by Lemma 3 we have that the averaged mapping

F_{λ} u = (1 - λ) u + λ F u

is nonexpansive, for any

λ \in (0, 1 - k)

and that

F i x (F) = F i x (F_{λ})

. We apply Theorem 1 for

F_{λ}

and G and get the conclusion. □

Remark 1.

Most of the results obtained in Takahashi and Tamura [21], Moudafi [15], Cianciaruso et al. [7], Falset et al. [8], Iemoto and Takahashi [9] could be obtained as corollaries of our main results or could be slightly improved by considering our averaged Halpern type algorithm (8).

We illustrate this fact in the following for four different instances.

If F is nonexpansive and G is nonspreading, then by Theorem 1 we obtain an improvement of Theorem 4.1 in Iemoto and Takahashi [9], in the sense that for our averaged Halpern type algorithm (8) we have strong convergence, while for the Krasnoselsij-Mann iterative procedure (5) only weak convergence was obtained by Iemoto and Takahashi [9];
If F is nonexpansive and G is nonspreading, then by Theorem 1 we obtain the main result (i.e., Theorem 14) in Cianciaruso et al. [7];
If F and G are both nonexpansive, then by Theorem 1 we obtain an improvement of the main result in Takahashi and Tamura [21], in the sense that for our averaged Halpern type algorithm (8) we get strong convergence, while for the Krasnoselsij-Mann iterative procedure (5) only weak convergence is obtained by Takahashi and Tamura [21];
If F is nonexpansive and G is strongly quasi-nonexpansive, then by Theorem 1 we obtain the main result (i.e., Theorem 3) in Falset et al. [8];
...

4. Numerical Illustrations

In this section, we consider some numerical examples to illustrate the numerical behaviour of Algorithm (8), for approximating a common fixed point for a nonexpansive mapping and a

β

-demicontractive mapping.

Example 2.

Let

H

be the real line with the usual norm and

D = [0, 1] .

Define F and G on D as follows as

F (u) = 5 / 3 - u, u \in D

(34)

and

G (u) = \{\begin{matrix} 5 / 6, & i f 0 \leq u < 1 \\ 1 / 3, & if u = 1 . \end{matrix}

(35)

Note that F is nonexpansive, and G is

1 / 2

-demicontractive. It is easy to check that

F i x (F) \cap F i x (G) = {5 / 6} .

The mapping G is neither quasi-nonexpasive nor nonexpansive (and hence it is neither strongly quasi-nonexpansive nor nonspeading).

Therefore, we cannot apply any of the results in Takahashi and Tamura [21], Moudafi [15], Cianciaruso et al. [7], Falset et al. [8], Iemoto and Takahashi [9] etc. to solve the common fixed point problem for F and G.

If we put

a_{p} = \frac{1}{r p}, b_{p} = \frac{2 p}{1 + 3 p}, p \in N, r \in R and r \geq 1,

then all assumptions in Theorem 1 part (iii) are satisfied. This implies that the sequence

{u_{p}}

generated by the algorithm (8) converges to

5 / 6

, the unique common fixed point of F and G.

Several numerical experiments were conducted in MATLAB using the algorithm(8)with different values of the parameters.

The numerical results for three initial values with

r = 1000

are presented in Table 1.

Table 2 shows numerical results for three initial values with

r = 5000

and

x = 0.7 .

One can see that for x near the common fixed point and r large, the iterations converge faster.

5. Conclusions

We have introduced an averaged iterative Halpern type algorithm intended to find a common fixed point for a pair consisting of a nonexpansive mapping and a demicontractive mapping which also solves a certain variational inequality problem;
We established a strong convergence theorem (Theorem 1) for the sequence generated by our algorithm;
We extended Theorem 1 to the more general case of a pair of mappings consisting of a k-strictly pseudocontractive mapping F and a $β$ -demicontractive mapping G (Theorem 2), by considering the double averaged Halpern type algorithm (32).
We validated the effectiveness of our general theoretical results by some appropriate numerical experiments, corresponding to part (iii) of Theorem 1, which are reported in Section 4. These results clearly illustrate the progress of our convergence results over existing literature.
For other related results we refer the reader to Agwu et al. [1], Araveeporn et al. [2], Ceng and Yao [5], Ceng and Yuan [6], Jaipranop and Saejung [10], Kraikaew and Saejung [12], Mebawondu et al. [14], Nakajo et al. [17], Petruşel and Yao [18], Rizvi [19], Sahu et al. [20], Thuy [22], Uba et al. [23], Xu [25], Yao et al. [26,27], Yotkaew et al. [28],...

Acknowledgments

The first draft of this paper was carried out during the first author’s short visit (December 2023) at the Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. He is grateful to Professor Monther Alfuraidan, the Chairman of Department of Mathematics, for the invitation and for providing excellent facilities during the visit.

Conflicts of Interest

The authors declare that they have no conflict of interest.

Funding

Not applicable.

Data Availability Statement

Not applicable.

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Table 1. Numerical results for

x = 0.2, r = 1000

with initial values

u_{0} = 1, u_{0} = 0.7

and

u_{0} = 0.1 .

Table 1. Numerical results for

x = 0.2, r = 1000

with initial values

u_{0} = 1, u_{0} = 0.7

and

u_{0} = 0.1 .

Iteration (p)	$u_{p}$	$u_{p}$	$u_{p}$
0	1.000000	0.700000	0.100000
1	0.786549	0.822542	0.774552
2	0.837948	0.834349	0.839148
3	0.832452	0.833106	0.832234
4	0.833503	0.833354	0.833553
5	0.833264	0.833303	0.833251
6	0.833332	0.833321	0.833335
7	0.833316	0.833319	0.833315
8	0.833323	0.833322	0.833323
9	0.833323	0.833323	0.833322
10	0.833324	0.833324	0.833324
⋯	⋯	⋯	⋯
15	0.833327	0.833327	0.833327
⋯	⋯	⋯	⋯
20	0.833329	0.833329	0.833329
⋯	⋯	⋯	⋯
50	0.833332	0.833332	0.833332
⋯	⋯	⋯	⋯
51	0.833332	0.833332	0.833332
⋯	⋯	⋯	⋯
111	0.833333	0.833333	0.833333
112	0.833333	0.833333	0.833333

Table 2. Numerical results for

x = 0.7, r = 5000

with initial values

u_{0} = 1, u_{0} = 0.7

and

u_{0} = 0.1 .

Table 2. Numerical results for

x = 0.7, r = 5000

with initial values

u_{0} = 1, u_{0} = 0.7

and

u_{0} = 0.1 .

Iteration (p)	$u_{p}$	$u_{p}$	$u_{p}$
0	1.000000	0.700000	0.100000
1	0.786649	0.822642	0.774652
2	0.837988	0.834389	0.839188
3	0.832478	0.833133	0.832260
4	0.833522	0.833373	0.833572
5	0.833279	0.833318	0.833266
6	0.833344	0.833330	0.833348
7	0.833326	0.833330	0.833325
8	0.833332	0.833331	0.833332
9	0.833332	0.833331	0.833332
10	0.833332	0.833331	0.833332
⋯	⋯	⋯	⋯
15	0.833332	0.833332	0.833332
⋯	⋯	⋯	⋯
20	0.833332	0.833332	0.833332
⋯	⋯	⋯	⋯
26	0.833333	0.833333	0.833333
27	0.833333	0.833333	0.833333

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An Averaged Halpern Type Algorithm for Solving Fixed Point Problems and Variational Inequality Problems

Abstract

1. Introduction

2. Preliminaries

3. Fixed Points and Variational Inequalities

4. Numerical Illustrations

5. Conclusions

Acknowledgments

Conflicts of Interest

Funding

Data Availability Statement

References

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