A Method of approximation for a zero of the sum of maximally monotone mappings in Hilbert spaces

Wega, Getahun Bekele; Zegeye, Habtu

doi:10.1016/j.ajmsc.2019.05.004

Our purpose of this study is to construct an algorithm for finding a zero of the sum of two maximally monotone mappings in Hilbert spaces and discus its convergence. The assumption that one of the mappings is $α$ -inverse strongly monotone is dispensed with. In addition, we give some applications to the minimization problem. Our method of proof is of independent interest. Finally, a numerical example which supports our main result is presented. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

1. Introduction

Let $H$ be a real Hilbert space with inner product $〈 ., . 〉$ and induced norm $‖ . ‖$ ⁠. Let $A : H \to 2^{H}$ be a nonlinear mapping. The domain, range, zero, and graph of $A$ are respectively the sets $D o m (A) = {x \in H : A x \neq Ø}, R (A) = {A x : x \in D o m (A)}, Z e r o (A) = {x \in H : 0 \in A x}$ and $G p h (A) = {(x, y) \in H \times H : y \in A x}$ ⁠. A mapping $A : H \to 2^{H}$ is called monotone if for any $x, y \in H$ and $u \in A x, v \in A y$ we have

〈 u - v, x - y 〉 \geq 0.

(1.1)

A monotone mapping $A : H \to 2^{H}$ is called maximally monotone if $G p h (A)$ is not properly contained in the graph of any other monotone operator. The resolvent of $A$ is given by $J_{λ A} = {(I + λ A)}^{- 1}$ ⁠, where $I$ is the identity mapping on $H$ and $λ > 0$ ⁠.

Let $A, B : H \to 2^{H}$ be maximally monotone mappings. Consider the problem of finding $z \in H$ such that

0 \in A z + B z .

(1.2)

We denote the solution set of (1.2) by ${(A + B)}^{- 1} (0)$ ⁠. This problem includes, as special cases, convex programming, variational inequalities, split feasibility problem and minimization problem. For solving problem (1.2), we remark that several authors have studied different iterative schemes (see, for example, [3,4,10,11,14,21] and the references therein).

In 1979, Passty [11] introduced Forward–backward splitting method which defines a sequence ${x_{n}}$ by

x_{n + 1} = {(I + r_{n} B)}^{- 1} (I - r_{n} A) x_{n}, n \geq 1,

(1.3)

where ${r_{n}}$ is a sequence of positive numbers, $A$ and $B$ are maximal monotone mappings with $D o m (A) \subset D o m (B)$ ⁠, and $A$ is single valued. This method is essentially a generalization of the classical gradient method for constrained convex optimization and monotone variational inequalities, and inherit restrictions similar to those methods such as $A$ is single valued. In general, this method provides weak convergence even with the restriction that $A$ is single-valued. It fails to provide weak convergence results to the zero of the sum of the general maximal monotone mappings.

In 1979, Lions and Mercier [7] introduced Peaceman–Rachford splitting method whose iteration method is given by

\begin{array}{l} y_{n} = (2 {(I + λ B)}^{- 1} - I) x_{n}, \\ z_{n} = (2 {(I + λ A)}^{- 1} - I) y_{n}, \\ x_{n + 1} = (1 - ρ_{n}) x_{n} + ρ_{n} z_{n}, n \geq 1, \end{array}

(1.4)

where $λ > 0$ is a fixed scalar, and ${ρ_{n}} \subset (0, 1]$ is a sequence of relaxation parameters. They proved weak convergence of this sequence to the solution of problem (1.2) under certain conditions.

In 2008, Eckstein and Svaiter [5] constructed new approach splitting algorithms which starts by reformulating (1.2) as the problem of locating a point in a certain extended solution set $S_{e} (A, B) \subset H \times H$ and proved weak convergence results provided that $H$ has finite dimension or $A + B$ is maximal monotone mapping. The extended solution set for the problem (1.2), which is the subset of $H \times H$ is defined by:

S_{e} (A, B) = {(z, w) \in H \times H : w \in B (z), - w \in A (z)} .

(1.5)

We treat $H \times H$ as a Hilbert space by endowing it with the canonical inner product

〈 (x, x^{'}), (y, y^{'}) 〉 = 〈 x, y 〉 + 〈 x^{'}, y^{'} 〉 .

More recently, in order to solve problem (1.2), Svaiter [17] studied the following Algorithm: for any $λ > 0, x_{0}, b_{0} \in H$ ⁠,

Step 1. $Compute, y_{n}, a_{n} such that a_{n} \in A (y_{n}), λ a_{n} + y_{n} = x_{n - 1} - λ b_{n - 1} .$
Step 2. $Compute, x_{n}, b_{n} such that b_{n} \in B (x_{n}), λ b_{n} + x_{n} = y_{n} + λ b_{n - 1}, n \geq 1.$

They proved that ${(x_{n}, b_{n})}$ and ${(y_{n}, - a_{n})}$ converge weakly to a point $(z, w)$ in $S_{e} (A, B)$ ⁠, where $z \in {(A + B)}^{- 1} (0)$ ⁠. We remark that the convergence is still weak convergence.

With regard to a strong convergence, several authors have studied different iterative schemes (see for example, [12,18–20,22] and the references therein) for a zero of the sum of monotone mappings $A$ and $B$ ⁠.

In 2012, Takahashi et al. [20] proved some strong convergence theorems of the Halpern-type iteration in a Hilbert space $H$ ⁠, which is defined by the following manner: for any $x_{1} \in H$ ⁠,

x_{n + 1} = β_{n} x_{n} + (1 - β_{n}) (α_{n} u + (1 - α_{n}) J_{r_{n} B} (x_{n} - r_{n} A x_{n})), \forall n \geq 1,

(1.6)

where $u \in H$ is a fixed point and A is an $α$ -inverse strongly monotone mapping on $H$ and $B$ is a maximal monotone mapping on $H$ ⁠. Under suitable conditions, they proved that the sequence ${x_{n}}$ generated by (1.6) converges strongly to a zero point of $A + B$ ⁠. A monotone mapping $A : H \to H$ is said to be $α$ -inverse strongly monotone if there exists a positive real number $α$ such that

〈 A x - A y, x - y 〉 \geq α {‖ A x - A y ‖}^{2}, \forall x, y \in H .

(1.7)

Several researchers have also studied and obtained similar results in Hilbert and Banach spaces more general than Hilbert spaces (see, e.g, [8,13,16]). However, we observe that the strong convergence results available for the zero of the sum of monotone mappings $A$ and $B$ ⁠, are either one of the mappings is $α$ -inverse strongly monotone or single-valued.

Question 1.

A natural question arises whether we can obtain strong convergence result for approximating a zero of the sum of two maximally monotone mappings?

Motivated and inspired by the above results, our purpose in this paper is to construct an algorithm for finding a zero of the sum of maximally monotone mappings via the extended solution set $S_{e} (A, B) \subset H \times H$ and discus its strong convergence. The assumption that one of the mappings is $α$ -inverse strongly monotone is dispensed with. Our results provide an affirmative answer to our concern. Our method of proof is of independent interest. Our results improve, extend, and generalize many results in the literature.

2. Preliminaries

Let $H$ be a real Hilbert space, $C$ be a nonempty closed convex subset of $H$ ⁠. The set of fixed point of the mapping $T : C$ → $C$ is denoted by $F (T)$ ⁠, that is, $F (T) = {x \in C : T x = x}$ ⁠. The following lemmas shall be used in the sequel.

Definition 2.1.

Let $H$ be a real Hilbert space, $C$ be a nonempty closed convex subset of $H$ ⁠. A mapping $T : C \to C$ is said to be Lipschitz if there exists $L \geq 0$ such that

‖ T x - T y ‖ \leq L ‖ x - y ‖, \forall x, y \in C .

(2.1)

T

is said to be nonexpansive mapping if

L = 1

and it is called contraction if

L \in [0, 1)

⁠.

We remark that every contraction mapping is nonexpansive and every nonexpansive mapping is Lipschitz mapping.

Definition 2.2.

Let $H$ be a real Hilbert space, $C$ be a nonempty closed convex subset of $H$ ⁠. A mapping $T : C$ → $C$ is said to be firmly nonexpansive if

{‖ T x - T y ‖}^{2} \leq 〈 T x - T y, x - y 〉, \forall x, y \in C .

(2.2)

Lemma 2.3

([5]). Finding a point in $S_{e} (A, B)$ is equivalent to solving (1.2) in the sense that

z \in {(A + B)}^{- 1} (0) \Leftrightarrow \exists w \in H : (z, w) \in S_{e} (A, B) .

(2.3)

Lemma 2.4

([5]). Given any two maximal monotone mappings $A, B : H \to 2^{H}$ , the corresponding extended solution set $S_{e} (A, B)$ is closed and convex in $H \times H$ .

Lemma 2.5

([5]). Given any two points $(x, b) \in G p h (B)$ and $(y, a) \in G p h (A)$ , define the function $ϕ : H \times H \to ℝ v i a$

ϕ (z, w) = 〈 z - x, b - w 〉 + 〈 z - y, a + w 〉 .

Then, for any

(z, w) \in S_{e} (A, B)

we have

ϕ (z, w) \leq 0

, that is

S_{e} (A, B) \subseteq {(z, w) \in H \times H | ϕ (z, w) \leq 0} .

Additionally, $ϕ$ is both continuous and affine, $\nabla ϕ = (a + b, x - y)$ , and

\nabla ϕ = 0 \Leftrightarrow (x, b) \in S_{e} (A, B), x = y, a = - b .

Furthermore, $\nabla ϕ = 0$ implies $ϕ (z, w) = 0$ for all $z, w \in H$ ⁠.

The function $ϕ$ in Lemma 2.5 is called decomposable separators.

Lemma 2.6

([23]). Let ${a_{n}}$ be a sequence of nonnegative real numbers satisfying the following relation:

a_{n + 1} \leq (1 - α_{n}) a_{n} + α_{n} δ_{n}, n \geq n_{0},

where

{α_{n}} \subset (0, 1)

and

{δ_{n}} \subset R

satisfying the following conditions:

\sum_{n = 1}^{\infty} α_{n} = \infty

, and

\lim \sup_{n \to \infty} δ_{n} \leq 0

or

\sum_{n = 1}^{\infty} | α_{n} δ_{n} | < \infty

. Then,

\lim_{n \to \infty} a_{n} = 0

.

Lemma 2.7

([2]). Let $x, y \in H$ . If $H$ is a real Hilbert space, then the following inequality holds:

{‖ x + y ‖}^{2} \leq {‖ x ‖}^{2} + 2 〈 y, x + y 〉 .

Lemma 2.8

([6] Demiclosedness Principle). Let $H$ be a Hilbert space, $C$ be a closed and convex subset of $H$ , and $T : C \to C$ be a nonexpansive operator with $F (T) \neq Ø .$ If ${x_{n}}$ is a sequence in $C$ weakly converge to $x \in C$ and ${(I - T) x_{n}}$ converge strongly to $y \in C$ , then $(I - T) x = y$ . In particular, if $y = 0$ , then $x \in F (T) .$

Lemma 2.9

([24]). Let $C$ be a closed and convex subset of a real Hilbert space $H$ , given $x \in H$ and $y \in C$ . Then $y = P_{C} x$ if and only if the following inequality holds:

〈 x - y, z - y 〉 \leq 0

for all

z \in C

.

3. Main results

In this section, we introduce our Algorithm for finding a point which is a solution of the sum of maximally monotone mappings as a problem of locating a point in the extended solution set $S_{e} (A, B)$ in Hilbert spaces and discus its convergence.

Let $H$ be a real Hilbert space. Hereafter, let $A, B : H \to 2^{H}$ be maximal monotone mappings satisfying $Ω : = {x \in H : 0 \in A x + B x} \neq Ø$ ⁠. Let ${α_{n}} \subset (0, 1)$ such that $\lim_{n \to \infty} α_{n} = 0, \sum_{n = 1}^{\infty} α_{n} = \infty$ and $\sum_{n = 1}^{\infty} | α_{n} - α_{n - 1} | < \infty$ ⁠, and let ${β_{n}}$ be a decreasing sequence in $(0,1]$ such that $β_{0} = 1$ ⁠, and $\sum_{n = 1}^{\infty} | β_{n + 1} - β_{n} | < \infty .$

We now propose the following algorithm which uses basically Algorithm 2 of [5].

Algorithm 3.1.

Step 0: Select initial guess $p_{0} = (z_{0}, w_{0}) \in H \times H .$
Step 1: Given $(z_{n}, w_{n}) \in H \times H$ ⁠, for $n \geq 0$ ⁠, compute:

x_{n} = {(I + λ_{n} B)}^{- 1} (z_{n} + λ_{n} w_{n}),

y_{n} = {(I + μ_{n} A)}^{- 1} ((1 - θ_{n}) z_{n} + θ_{n} x_{n} - μ_{n} w_{n}),

where

λ_{n}, μ_{n} \in [\underline{λ}, \bar{λ}]

for some

\bar{λ} > \underline{λ} > 0

and

θ_{n} \in R

⁠.

Step 2: Compute $a_{n} = \frac{1}{μ_{n}} ((1 - θ_{n}) z_{n} + θ_{n} x_{n} - y_{n}) - w_{n}$ and $b_{n} = w_{n} + \frac{1}{λ_{n}} (z_{n} - x_{n}) .$
Step 3: Define $ϕ_{i} : H \times H \to ℝ$ to be

ϕ_{i} (z, w) : = 〈 z - x_{i}, b_{i} - w 〉 + 〈 z - y_{i}, a_{i} + w 〉,

and compute

T_{i} p_{n} = {\begin{cases} (z_{n}, w_{n}) if ϕ_{i} (z_{n}, w_{n}) \leq 0, \\ (z_{n}, w_{n}) - (\frac{ϕ_{i} (z_{n}, w_{n})}{{‖ \nabla ϕ_{i} ‖}^{2}}) \nabla ϕ_{i} if ϕ_{i} (z_{n}, w_{n}) > 0, \end{cases}

(3.1)

where

T_{i} p_{n}

is the metric projection of

p_{n} = (z_{n}, w_{n})

onto the set

H_{i} = {(z, w) \in H \times H : ϕ_{i} (z, w) \leq 0}, for each i = 1, 2, \dots, n .

Step 4: Compute

{\begin{array}{l} q_{n} = β_{n} p_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) T_{i} p_{n}, \\ p_{n + 1} = α_{n} f (p_{n}) + (1 - α_{n}) q_{n}, \forall n \geq 0, \end{array}

(3.2)

where

f : H \times H \to H \times H

is a contraction mapping with constant

α

⁠. Set

n : = n + 1

and go to Step 1.

Remark 3.2.

By maximality of $B$ the resolvent mapping ${(I + λ_{n} B)}^{- 1}$ is single-valued and well-defined (see, e.g, [9]). Hence

(x_{n}, b_{n}) = ({(I + λ_{n} B)}^{- 1} (z_{n} + λ_{n} w_{n}), w_{n} + \frac{1}{λ_{n}} (z_{n} - x_{n})),

exists and is unique. Similarly by maximality of

A

the resolvent mapping

{(I + μ_{n} A)}^{- 1}

is single-valued and well-defined, thus

(y_{n}, a_{n}) = ({(I + μ_{n} A)}^{- 1} ((1 - θ_{n}) z_{n} + θ_{n} x_{n} - μ_{n} w_{n}), \frac{1}{μ_{n}} ((1 - θ_{n}) z_{n} + θ_{n} x_{n} - y_{n}) - w_{n}),

exists and is unique. By rearranging equations in Step 1, one has the following equations:

{\begin{cases} x_{n} + λ_{n} b_{n} = z_{n} + λ_{n} w_{n}; \\ y_{n} + μ_{n} a_{n} = (1 - θ_{n}) z_{n} + θ_{n} x_{n} - μ_{n} w_{n}; \end{cases}

(3.3)

where

a_{n} \in A (y_{n})

and

b_{n} \in B (x_{n})

⁠. Hence decomposable separators function

ϕ

in Lemma2.5 is well defined.

We proceed with the following lemmas which will be used to prove our main theorem.

Lemma 3.3.

The sequence ${p_{n}}$ generated by Algorithm 3.1 is bounded.

Proof.

Let $p^{*} = (z^{*}, w^{*}) \in S_{e} (A, B)$ ⁠, then by Lemma 2.5 we have $p^{*} \in H_{i}$ for all $i \geq 0$ and hence $S_{e} (A, B) \subset \cap_{i = 1}^{\infty} F (T_{i}) = ℱ$ ⁠. Now, from (3.2) and the fact that $T_{i}$ is firmly nonexpansive we get

\begin{array}{l} ‖ q_{n} - p^{*} ‖ = ‖ β_{n} p_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) T_{i} p_{n} - p^{*} ‖ \\ = ‖ β_{n} p_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) T_{i} p_{n} - \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) p^{*} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) p^{*} - p^{*} ‖ \\ = ‖ β_{n} (p_{n} - p^{*}) + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) (T_{i} p_{n} - p^{*}) ‖ \\ \leq β_{n} ‖ p_{n} - p^{*} ‖ + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) ‖ p_{n} - p^{*} ‖ = ‖ p_{n} - p^{*} ‖, \end{array}

(3.4)

and

\begin{array}{l} ‖ p_{n + 1} - p^{*} ‖ = ‖ α_{n} f (p_{n}) + (1 - α_{n}) q_{n} - p^{*} ‖ \\ = ‖ α_{n} (f (p_{n}) - p^{*}) + (1 - α_{n}) (q_{n} - p^{*}) ‖ \\ \leq α_{n} ‖ f (p_{n}) - p^{*} ‖ + (1 - α_{n}) ‖ q_{n} - p^{*} ‖ \\ \leq α_{n} ‖ f (p_{n}) - f (p^{*}) ‖ + α_{n} ‖ f (p^{*}) - p^{*} ‖ + (1 - α_{n}) ‖ q_{n} - p^{*} ‖ \\ \leq α_{n} ‖ f (p_{n}) - f (p^{*}) ‖ + α_{n} ‖ f (p^{*}) - p^{*} ‖ + (1 - α_{n}) ‖ p_{n} - p^{*} ‖ \\ \leq α α_{n} ‖ p_{n} - p^{*} ‖ + α_{n} ‖ f (p^{*}) - p^{*} ‖ + (1 - α_{n}) ‖ p_{n} - p^{*} ‖ \\ = (1 - α_{n} (1 - α)) ‖ p_{n} - p^{*} ‖ + α_{n} ‖ f (p^{*}) - p^{*} ‖ \\ \leq \max {‖ p_{n} - p^{*} ‖, \frac{‖ f (p^{*}) - p^{*} ‖}{1 - α}}, \end{array}

and hence by induction we have

‖ p_{n} - p^{*} ‖ \leq \max {‖ p_{0} - p^{*} ‖, \frac{‖ f (p^{*}) - p^{*} ‖}{1 - α}},

which implies that

{p_{n}}

is bounded, so are

{T_{i} p_{n}}, {q_{n}}

and

{f (p_{n})}

⁠.

□

Lemma 3.4.

The sequence ${p_{n}}$ generated by Algorithm 3.1 converges strongly to $p^{*} = (z^{*}, w^{*})$ in $ℱ = \cap_{i = 1}^{\infty} F {(T)}_{i}$ .

Proof.

We divide the proof into the following steps.

Step 1. We prove that $\lim_{n \to \infty} ‖ p_{n + 1} - p_{n} ‖ = 0$ ⁠.

From (3.2), we have

\begin{array}{l} ‖ q_{n} - q_{n - 1} ‖ = ‖ β_{n} p_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) T_{i} p_{n} - β_{n - 1} p_{n - 1} \\ - \sum_{i = 1}^{n - 1} (β_{i - 1} - β_{i}) T_{i} p_{n - 1} ‖ \\ \leq β_{n} ‖ p_{n} - p_{n - 1} ‖ + | β_{n} - β_{n - 1} | ‖ p_{n - 1} ‖ \\ + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) ‖ T_{i} p_{n} - T_{i} p_{n - 1} ‖ + | β_{n} - β_{n - 1} | ‖ T_{i} p_{n - 1} ‖ \\ \leq β_{n} ‖ p_{n} - p_{n - 1} ‖ + | β_{n} - β_{n - 1} | ‖ p_{n - 1} ‖ \\ + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) ‖ p_{n} - p_{n - 1} ‖ + | β_{n} - β_{n - 1} | ‖ T_{i} p_{n - 1} ‖ \\ = ‖ p_{n} - p_{n - 1} ‖ + | β_{n} - β_{n - 1} | (‖ p_{n - 1} ‖ + ‖ T_{i} p_{n - 1} ‖), \end{array}

(3.5)

hence Eqs. and yield

\begin{array}{l} ‖ p_{n + 1} - p_{n} ‖ = ‖ α_{n} f (p_{n}) + (1 - α_{n}) q_{n} - α_{n - 1} f (p_{n - 1}) - (1 - α_{n - 1}) q_{n - 1} ‖ \\ \leq α_{n} ‖ f (p_{n}) - f (p_{n - 1}) ‖ + | α_{n} - α_{n - 1} | ‖ f (p_{n - 1}) ‖ \\ + (1 - α_{n}) ‖ q_{n} - q_{n - 1} ‖ + | α_{n} - α_{n - 1} | ‖ q_{n - 1} ‖ \\ \leq α α_{n} ‖ p_{n} - p_{n - 1} ‖ + (1 - α_{n}) ‖ q_{n} - q_{n - 1} ‖ \\ + | α_{n} - α_{n - 1} | (‖ f (p_{n - 1}) ‖ + ‖ q_{n - 1} ‖) \\ \leq α α_{n} ‖ p_{n} - p_{n - 1} ‖ + | α_{n} - α_{n - 1} | (‖ f (p_{n - 1}) ‖ + ‖ q_{n - 1} ‖) \\ + (1 - α_{n}) (‖ p_{n} - p_{n - 1} ‖ + | β_{n} - β_{n - 1} | (‖ p_{n - 1} ‖ + ‖ T_{i} p_{n - 1} ‖)) \\ = (1 - α_{n} (1 - α)) ‖ p_{n} - p_{n - 1} ‖ + | α_{n} - α_{n - 1} | (‖ f (p_{n - 1}) ‖ + ‖ q_{n - 1} ‖) \\ + (1 - α_{n}) | β_{n} - β_{n - 1} | (‖ p_{n - 1} ‖ + ‖ T_{i} p_{n - 1} ‖) . \end{array}

Thus, from Lemma 2.6 and conditions of ${α_{n}}$ and ${β_{n}}$ ⁠, we obtain that

‖ p_{n + 1} - p_{n} ‖ \to 0 a s n \to \infty .

(3.6)

Step 2. We prove that $‖ T_{i} p_{n} - p_{n} ‖ \to 0$ as $n \to \infty$ ⁠. Take $p^{*} = P_{ℱ} (f (p^{*}))$ ⁠. Then, we have

\begin{array}{l} {‖ p_{n} - p^{*} ‖}^{2} \geq {‖ T_{i} p_{n} - T_{i} p^{*} ‖}^{2} \\ = {‖ T_{i} p_{n} - p_{n} + p_{n} - T_{i} p^{*} ‖}^{2} \\ = {‖ T_{i} p_{n} - p_{n} ‖}^{2} + {‖ p_{n} - p^{*} ‖}^{2} + 2 〈 T_{i} p_{n} - p_{n}, p_{n} - p^{*} 〉, \end{array}

(3.7)

which implies that

\frac{1}{2} {‖ T_{i} p_{n} - p_{n} ‖}^{2} \leq 〈 p_{n} - T_{i} p_{n}, p_{n} - p^{*} 〉 .

(3.8)

Thus, from (3.2) and (3.8), we obtain

\begin{array}{l} \frac{1}{2} \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) {‖ T_{i} p_{n} - p_{n} ‖}^{2} \leq \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) 〈 p_{n} - T_{i} p_{n}, p_{n} - p^{*} 〉 \\ = 〈 (1 - β_{n}) p_{n} - \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) T_{i} p_{n}, p_{n} - p^{*} 〉 \\ = 〈 (1 - β_{n}) p_{n} - q_{n} + β_{n} p_{n}, p_{n} - p^{*} 〉 \\ = 〈 p_{n} - q_{n}, p_{n} - p^{*} 〉 \\ = 〈 p_{n} - p_{n + 1}, p_{n} - p^{*} 〉 \\ + 〈 p_{n + 1} - q_{n}, p_{n} - p^{*} 〉 . \end{array}

(3.9)

From (3.2) and (3.9), we also have

\begin{array}{l} \frac{1}{2} \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) {‖ T_{i} p_{n} - p_{n} ‖}^{2} \leq 〈 p_{n} - p_{n + 1}, p_{n} - p^{*} 〉 \\ + 〈 α_{n} f (p_{n}) + (1 - α_{n}) q_{n} - q_{n}, p_{n} - p^{⋆} 〉 \\ \leq ‖ p_{n} - p_{n + 1} ‖ ‖ p_{n} - p^{*} ‖ \\ + 〈 α_{n} f (p_{n}) - α_{n} q_{n}, p_{n} - p^{⋆} 〉 \\ \leq ‖ p_{n} - p_{n + 1} ‖ ‖ p_{n} - p^{*} ‖ \\ + α_{n} ‖ f (p_{n}) - q_{n} ‖ ‖ p_{n} - p^{⋆} ‖ . \end{array}

(3.10)

Now, from Lemma 3.3, (3.6), (3.10) and the condition of ${α_{n}}$ ⁠, we obtain that

\lim_{n \to \infty} \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) {‖ T_{i} p_{n} - p_{n} ‖}^{2} = 0.

(3.11)

Since ${β_{n}}$ is strictly decreasing, for every $i \in N$ ⁠, equality (3.11) yields

\lim_{n \to \infty} ‖ T_{i} p_{n} - p_{n} ‖ = 0,

(3.12)

and

\lim_{n \to \infty} ‖ T_{n} p_{n} - p_{n} ‖ = 0 .

(3.13)

Step 3. We prove that $\lim \sup_{n \to \infty} 〈 f (p^{*}) - p^{*}, p_{n} - p^{*} 〉 \leq 0$ ⁠.

Since ${p_{n}}$ is bounded, there exist $u \in H \times H$ and a subsequence ${p_{n_{k}}}$ of $p_{n}$ such that

\underset{n \to \infty}{\lim \sup} 〈 f (p^{*}) - p^{*}, p_{n} - p^{*} 〉 = \lim_{k \to \infty} 〈 f (p^{*}) - p^{*}, p_{n_{k}} - p^{*} 〉,

and

p_{n_{k}} ⇀ u

⁠. Now, using and Lemma 2.8, we obtain that

u \in F (T_{i})

for each

i \geq 1

and hence

u \in ℱ

⁠. Since

p^{*} = P_{ℱ} (f (p^{*}))

⁠, by Lemma 2.9, we obtain

\begin{array}{l} \underset{n \to \infty}{\lim \sup} 〈 f (p^{*}) - p^{*}, p_{n} - p^{*} 〉 = \lim_{k \to \infty} 〈 f (p^{*}) - p^{*}, p_{n_{k}} - p^{*} 〉 \\ = 〈 f (p^{*}) - p^{*}, u - p^{*} 〉 \leq 0. \end{array}

(3.14)

Step 4. Finally, we prove that $p_{n} \to p^{*}$ as $n \to \infty$ ⁠. From Lemma 2.7, (3.2) and (3.4), we have

\begin{array}{l} {‖ p_{n + 1} - p^{*} ‖}^{2} = {‖ α_{n} f (p_{n}) + (1 - α_{n}) q_{n} - p^{*} ‖}^{2} \\ \leq {‖ α_{n} (f (p_{n}) - f (p^{*})) + (1 - α_{n}) (q_{n} - p^{*}) ‖}^{2} \\ + 2 α_{n} 〈 f (p^{*}) - p^{*}, p_{n + 1} - p^{*} 〉 \\ \leq {(α α_{n} ‖ p_{n} - p^{*} ‖ + (1 - α_{n}) ‖ p_{n} - p^{*} ‖)}^{2} \\ + 2 α_{n} 〈 f (p^{*}) - p^{*}, p_{n + 1} - p^{*} 〉 \\ = {(1 - (1 - α) α_{n})}^{2} {‖ p_{n} - p^{*} ‖}^{2} + 2 α_{n} 〈 f (p^{*}) - p^{*}, p_{n + 1} - p^{*} 〉 \\ \leq (1 - (1 - α) α_{n}) {‖ p_{n} - p^{*} ‖}^{2} + 2 α_{n} 〈 f (p^{*}) - p^{*}, p_{n + 1} - p^{*} 〉 \\ \leq (1 - (1 - α) α_{n}) {‖ p_{n} - p^{*} ‖}^{2} + 2 α_{n} 〈 f (p^{*}) - p^{*}, p_{n} - p^{*} 〉 \\ + 2 α_{n} ‖ f (p^{*}) - p^{*} ‖ ‖ p_{n + 1} - p_{n} ‖ . \end{array}

(3.15)

Hence, from (3.14), (3.15) and Lemma 2.6, we get that $p_{n} \to p^{*}$ as $n \to \infty$ ⁠. □

Following the method of proof of [5] we obtain the following lemma.

Lemma 3.5.

If in Algorithm 3.1 , the sequences ${λ_{n}}, {μ_{n}} \subset [\underline{λ}, \bar{λ}]$ for some $\bar{λ} > \underline{λ} > 0$ and ${θ_{n}} \subset ℝ$ satisfy the condition

\inf_{n \geq 0} {\frac{μ_{n}}{λ_{n}} - {(\frac{θ_{n}}{2})}^{2}} > 0,

(3.16)

then, there exists some scalar

η > 0

such that

ϕ_{n} (p_{n}) \geq η {‖ \nabla ϕ_{n} ‖}^{2}

for all

n \geq 0

.

Next, we state and prove our main theorem.

Theorem 3.6.

Let $H$ be a real Hilbert space and let $A, B : H \to 2^{H}$ be maximal monotone mappings satisfying $Ω : = {x \in H : 0 \in A x + B x} \neq Ø$ . If the sequences ${λ_{n}}, {μ_{n}} \subset [\underline{λ}, \bar{λ}]$ for some $\bar{λ} > \underline{λ} > 0$ and ${θ_{n}} \subset R$ satisfy the condition

\inf_{n \geq 0} {\frac{μ_{n}}{λ_{n}} - {(\frac{θ_{n}}{2})}^{2}} > 0,

then, the sequence

{p_{n}}

generated byAlgorithm 3.1converges strongly to an element

p^{*} = (z^{*}, w^{*})

of

S_{e} (A, B)

, where

z^{*} \in Ω

.

Proof.

By Lemma 3.5, there exists $η > 0$ such that

ϕ_{n} (p_{n}) \geq η {‖ \nabla ϕ_{n} ‖}^{2} .

(3.17)

This implies that $ϕ_{n} (p_{n})$ is always nonnegative, so from (3.1), we have

‖ T_{n} p_{n} - p_{n} ‖ = ‖ - (\frac{ϕ_{n} (z_{n}, w_{n})}{{‖ \nabla ϕ_{n} ‖}^{2}}) \nabla ϕ_{n} ‖ = \frac{ϕ_{n} (z_{n}, w_{n})}{‖ \nabla ϕ_{n} ‖},

(3.18)

for all

n

for which

\nabla ϕ_{n} \neq 0

⁠. Thus, dividing both sides of the inequality by

‖ \nabla ϕ_{n} ‖

⁠, we obtain

‖ T_{n} p_{n} - p_{n} ‖ \geq η ‖ \nabla ϕ_{n} ‖,

(3.19)

which is also true for

n

with

\nabla ϕ_{n} = 0

⁠. Now, and imply that

\nabla ϕ_{n} \to 0

as

n \to \infty

⁠. Now, from the expression for

\nabla ϕ_{n} = (a_{n} + b_{n}, x_{n} - y_{n})

⁠, we have

x_{n} - y_{n} → 0 and a_{n} + b_{n} \to 0 as n \to \infty .

(3.20)

Moreover, subtracting $y_{n}$ from both sides of the first equation and adding $μ_{n} b_{n} - θ_{n} y_{n}$ to both sides of the second equation in (3.3) and rearranging we obtain

{\begin{cases} (z_{n} - y_{n}) + λ_{n} (w_{n} - b_{n}) = (x_{n} - y_{n}), \\ (θ_{n} - 1) (z_{n} - y_{n}) + μ_{n} (w_{n} - b_{n}) = θ_{n} (x_{n} - y_{n}) - μ_{n} (a_{n} + b_{n}), n \geq 0. \end{cases}

Thus, solving for $(w_{n} - b_{n})$ from the above equations we obtain that

(μ_{n} + (1 - θ_{n}) λ_{n}) (w_{n} - b_{n}) = (x_{n} - y_{n}) - μ_{n} (a_{n} + b_{n}) .

(3.21)

Therefore, since $(μ_{n} + (1 - θ_{n}) λ_{n}) > 0$ ⁠, from (3.20) and (3.21) we have $w_{n} - b_{n} \to 0$ and hence this with (3.20) yields $z_{n} - y_{n} \to 0$ ⁠, as $n \to \infty$ ⁠. As a consequence, we obtain

z_{n} - x_{n} → 0, z_{n} - y_{n} \to 0, w_{n} - b_{n} \to 0, w_{n} + a_{n} \to 0 a s n \to \infty .

(3.22)

In addition, from Lemma 3.4 we know that the sequence ${p_{n}} = {(z_{n}, w_{n})}$ converges strongly to a point $p^{*} = (z^{*}, w^{*}) \in ℱ = ⋂_{i = 1}^{\infty} F (T_{i})$ ⁠. Now, we show that $p^{*} \in S_{e} (A, B)$ ⁠. But $z_{n} - x_{n} \to 0$ implies that $x_{n} \to z^{*}$ ⁠. Furthermore, since $w_{n} - b_{n} \to 0$ ⁠, we have $b_{n} \to w^{*}$ ⁠. As a consequence, since the $G p h (B)$ is closed and $(x_{n}, b_{n}) \in G p h (B)$ for all $n$ ⁠, we conclude that $(z^{*}, w^{*}) \in G p h (B)$ ⁠. Similarly, $(z^{*}, - w^{*}) \in G p h (A)$ ⁠, so $(z^{*}, w^{*}) \in S_{e} (A, B)$ ⁠. Therefore, by Lemma 2.3 we obtain $z^{*} \in {(A + B)}^{- 1} (0)$ ⁠. The proof is complete. □

If, in Algorithm 3.1, we replace the contraction mapping $f$ by a constant $p = (z, w) \in H \times H$ ⁠, then we get the following corollary for the approximation of a zero of the sum of maximally monotone mappings.

Corollary 3.7.

Let $H$ be a real Hilbert space. Let $A, B : H \to 2^{H}$ be maximal monotone mappings satisfying $Ω : = {x \in H : 0 \in A x + B x} \neq Ø$ . Let $λ_{n}, μ_{n} \in [\underline{λ}, \bar{λ}]$ for some $\bar{λ} > \underline{λ} > 0$ and ${θ_{n}} \subset ℝ$ satisfy the following:

\inf_{n \geq 0} {\frac{μ_{n}}{λ_{n}} - {(\frac{θ_{n}}{2})}^{2}} > 0.

For arbitrary

(z_{0}, w_{0}) \in H \times H

define an iterative algorithm by

{\begin{matrix} q_{n} = β_{n} p_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) T_{i} p_{n}, \\ p_{n + 1} = α_{n} p + (1 - α_{n}) q_{n}, \forall n \geq 0, \end{matrix}

(3.23)

where

p = (z, w) \in H \times H

and

T_{i} p_{n}

is as in. Then

{p_{n}}

converges strongly to an element

p^{*} = (z^{*}, w^{*})

of

S_{e} (A, B)

, where

z^{*} \in Ω

.

Remark 3.8.

We observe that Algorithm 3.1 is equivalent to the following scheme:

{\begin{array}{l} p_{0} = (z_{0}, w_{0}) \in H \times H c h o s e n a r b i t r a r l y, \\ x_{n} = {(I + λ_{n} B)}^{- 1} (z_{n} + λ_{n} w_{n}), \\ y_{n} = {(I + μ_{n} A)}^{- 1} ((1 - θ_{n}) z_{n} + θ_{n} x_{n} - μ_{n} w_{n}), \\ (z_{n + 1}, w_{n + 1}) = α_{n} f (z_{n}, w_{n}) + (1 - α_{n}) (c_{n}, d_{n}), \end{array}

(3.24)

where

c_{n} = {\begin{cases} β_{n} z_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) z_{n} if δ_{i} \leq 0, \\ β_{n} z_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) (z_{n} - δ_{i} (a_{i} + b_{i})) if δ_{i} > 0, \end{cases}

(3.25)

d_{n} = {\begin{cases} β_{n} w_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) w_{n} if δ_{i} \leq 0, \\ β_{n} w_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) (w_{n} - δ_{i} (x_{i} - y_{i})) if δ_{i} > 0, \end{cases}

(3.26)

and

λ_{n}, μ_{n} \in [\underline{λ}, \bar{λ}]

for some

\bar{λ} > \underline{λ} > 0

⁠,

θ_{n} \in ℝ

⁠, with

δ_{i} = \frac{〈 z_{n} - x_{i}, b_{i} - w_{n} 〉 + 〈 z_{n} - y_{i}, a_{i} + w_{n} 〉}{{‖ a_{i} + b_{i} ‖}^{2} + {‖ x_{i} - y_{i} ‖}^{2}},

for

b_{i} \in B (x_{i})

and

a_{i} \in A (y_{i})

⁠.

If, in (3.24) we assume $θ_{n} = 0$ for all $n \geq 0$ then we get the following corollary for approximating a zero of the sum of maximally monotone mappings.

Corollary 3.9.

Let $H$ be a real Hilbert space. Let $A, B : H \to 2^{H}$ be maximal monotone mappings satisfying $Ω : = {x \in H : 0 \in A x + B x} \neq Ø$ ⁠. Let $λ_{n}, μ_{n} \in [\underline{λ}, \bar{λ}]$ for some $\bar{λ} > \underline{λ} > 0$ ⁠. Let $f : H \times H \to H \times H$ be a contraction mapping with constant $α$ ⁠. For arbitrary $(z_{0}, w_{0}) \in H \times H$ define an iterative algorithm by

{\begin{cases} x_{n} = {(I + λ_{n} B)}^{- 1} (z_{n} + λ_{n} w_{n}), \\ y_{n} = {(I + μ_{n} A)}^{- 1} (z_{n} - μ_{n} w_{n}), \\ (z_{n + 1}, w_{n + 1}) = α_{n} f (z_{n}, w_{n}) + (1 - α_{n}) (c_{n}, d_{n}), \end{cases}

(3.27)

where

{c_{n}}

and

{d_{n}}

are as inand, respectively. Then

{p_{n}}

converges strongly to an element

p^{*} = (z^{*}, w^{*})

of

S_{e} (A, B)

, where

z^{*} \in Ω

.

If, in (3.24) we assume $f = p = (z, w) \in H \times H$ and $θ_{n} = 0$ for all $n \geq 0$ ⁠, then we get the following corollary for approximating a zero of the sum of maximally monotone mappings.

Corollary 3.10.

Let $H$ be a real Hilbert space. Let $A, B : H \to 2^{H}$ be maximal monotone mappings satisfying $Ω:= {x \in H : 0 \in A x + B x} \neq Ø$ . Let $λ_{n}, μ_{n} \in [\underline{λ}, \bar{λ}]$ for some $\bar{λ} > \underline{λ} > 0 .$ For arbitrary $(z_{o}, w_{0}) \in H \times H$ define an iterative algorithm by

{\begin{cases} x_{n} = {(I + λ_{n} B)}^{- 1} (z_{n} + λ_{n} w_{n}), \\ y_{n} = {(I + μ_{n} A)}^{- 1} (z_{n} - μ_{n} w_{n}), \\ (z_{n + 1}, w_{n + 1}) = α_{n} (z, w) + (1 - α_{n}) (c_{n}, d_{n}), \end{cases}

(3.28)

where

{c_{n}}

and

{d_{n}}

are as inand, respectively. Then

{p_{n}}

converges strongly to an element

p^{*} = (z^{*}, w^{*})

of

S_{e} (A, B)

, where

z^{*} \in Ω

.

We note that if in Corollary 3.10 we assume that $p = (0, 0)$ we get the following theorem for approximating the minimum-norm point of the extended solution set of the sum of maximally monotone mappings.

Theorem 3.11.

Let $H$ be a real Hilbert space. Let $A, B : H \to 2^{H}$ be maximal monotone mappings satisfying $Ω : = {x \in H : 0 \in A x + B x} \neq Ø$ . Let $λ_{n}, μ_{n} \in [\underline{λ}, \bar{λ}]$ for some $\bar{λ} > \underline{λ} > 0$ . For arbitrary $(z_{0}, w_{0}) \in H \times H$ define an iterative algorithm by

{\begin{array}{l} x_{n} = {(I + λ_{n} B)}^{- 1} (z_{n} + λ_{n} w_{n}), \\ y_{n} = {(I + μ_{n} A)}^{- 1} (z_{n} - μ_{n} w_{n}), \\ (z_{n + 1}, w_{n + 1}) = (1 - α_{n}) (c_{n}, d_{n}), \end{array}

(3.29)

where

{c_{n}}

and

{d_{n}}

are as inand, respectively. Then

{p_{n}}

converges strongly to the minimum-norm point

p^{*} = (z^{*}, w^{*})

of

S_{e} (A, B)

.

Proof.

We note that since $p^{*} = (z^{*}, w^{*}) = P_{ℱ} (0)$ ⁠, where $p^{*} \in S_{e} (A, B) \subset ℱ$ we obtain that $p^{*}$ is the minimum-norm point of $S_{e} (A, B)$ ⁠. □

4. Application

4.1 Application to convex minimization problem

In this section, we apply Theorem 3.6 to study the convex minimization problem. Let $h : H \to ℝ$ be a convex smooth function and $g : H \to ℝ$ be a convex, lower semicontinuous functions. We consider the problem of finding $x^{*} \in H$ such that

h (x^{*}) + g (x^{*}) = \min_{x \in H} {h (x) + g (x)},

(4.1)

This problem is equivalent, by Fermats rule, to the problem of finding $x^{*} \in H$ such that

0 \in \nabla h (x^{*}) + \partial g (x^{*}),

(4.2)

where $\nabla h$ is a gradient of $h$ and $\partial g$ is a subdifferential of $g$ ⁠. Note that $\nabla h$ and $\partial g$ are maximally monotone mappings (see, for example, [1,15]). Thus, the following result can be obtained from Theorem 3.6.

Theorem 4.1.

Let $H$ be a real Hilbert space. Let $h : H \to ℝ$ be a convex smooth function and $g : H \to ℝ$ be a convex, lower semicontinuous functions such that $Ω = \min_{x \in H} {h (x) + g (x)} \neq Ø$ . Let ${λ_{n}}$ and ${μ_{n}}$ be real sequences such that $λ_{n}, μ_{n} \in [\underline{λ}, \bar{λ}]$ for some $\bar{λ} > \underline{λ} > 0$ , and ${θ_{n}} \subset ℝ$ , satisfy the condition

\inf_{n \geq 0} {\frac{μ_{n}}{λ_{n}} - {(\frac{θ_{n}}{2})}^{2}} > 0.

Let $f : H \times H \to H \times H$ be a contraction mapping with constant $α$ ⁠. For arbitrary $(z_{0}, w_{0}) \in H \times H$ define an iterative algorithm by

{\begin{array}{l} p_{0} = (z_{0}, w_{0}) \in H \times H chosen arbitrarly, \\ x_{n} = {(I + λ_{n} \partial g)}^{- 1} (z_{n} + λ_{n} w_{n}), \\ y_{n} = {(I + μ_{n} \nabla h)}^{- 1} ((1 - θ_{n}) z_{n} + θ_{n} x_{n} - μ_{n} w_{n}), \\ (z_{n + 1}, w_{n + 1}) = α_{n} f (z_{n}, w_{n}) + (1 - α_{n}) (c_{n}, d_{n}), \end{array}

(4.3)

where

c_{n} = {\begin{array}{l} β_{n} z_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) z_{n} i f δ_{i} \leq 0, \\ β_{n} z_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) (z_{n} - δ_{i} (a_{i} + b_{i})) i f δ_{i} > 0, \end{array}

(4.4)

d_{n} = {\begin{array}{l} β_{n} w_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) w_{n} i f δ_{i} \leq 0, \\ β_{n} w_{n} + \sum_{i = 1}^{n} (β_{i - 1} - β_{i}) (w_{n} - δ_{i} (x_{i} - y_{i})) i f δ_{i} > 0, \end{array}

(4.5)

with

δ_{i} = \frac{〈 z_{n} - x_{i}, b_{i} - w_{n} 〉 + 〈 z_{n} - y_{i}, a_{i} + w_{n} 〉}{{‖ a_{i} + b_{i} ‖}^{2} + {‖ x_{i} - y_{i} ‖}^{2}},

for

b_{i} \in \partial g (x_{i})

and

a_{i} \in \nabla h (y_{i})

. Then

{p_{n}}

converges strongly to an element

p^{*} = (z^{*}, w^{*})

of

S_{e} (\nabla h, \partial g)

, where

z^{*} \in Ω

Proof.

Let $A = \nabla h$ and $B = \partial g$ in Theorem 4.1. Thus, Theorem 3.6 provides the conclusion of the theorem. □

5. Numerical example

In this section, we present some numerical experiment result to explain the conclusion of our result. The following numerical example verifies the conclusion of Corollary 3.9.

Example 5.1.

Let $H = l_{2}$ ⁠, where $l_{2}$ is the space of sequences. Let $A, B : l_{2} \to l_{2}$ be defined by $A x = 2 x + (1, 2, 3, 0, 0, \dots)$ and $B x = x + (1, 1, 1, 0, 0, \dots)$ ⁠, where $x = (x_{1}, x_{2}, \dots) \in l_{2}$ ⁠. We see that $A$ and $B$ are maximally monotone with $R (I + γ A) = l_{2} = R (I + γ B)$ for all $γ > 0$ ⁠. Now by direct calculation we get that

\begin{array}{l} x_{n} = \frac{z_{n} + λ_{n} w_{n} - λ_{n} (1, 1, 1, 0, 0, \dots)}{1 + λ_{n}}, y_{n} = \frac{z_{n} - μ_{n} w_{n} - μ_{n} (1, 2, 3, 0, 0, \dots)}{1 + 2 μ_{n}}, \\ a_{n} = \frac{2 z_{n} - 2 μ_{n} w_{n} + (1, 2, 3, 0, 0, \dots)}{1 + 2 μ_{n}} a n d b_{n} = \frac{z_{n} + λ_{n} w_{n} + (1, 1, 1, 0, 0, \dots)}{1 + λ_{n}} . \end{array}

Thus, if we assume $λ_{n} = μ_{n} = 1$ ⁠, $α_{n} = \frac{1}{1000 (n + 10)}$ , $β_{n} = \frac{1}{n}$ , for all $n \geq 1$ ⁠, and $f (x) = \frac{x}{1000}$ with initial point $z_{1} = (1, 4, 5, 0, 0, \dots)$ and $w_{1} = (0, - 1, - 2, 0, 0, \dots)$ then the iteration Scheme (3.27) provides the following numerical experiment result using MATLAB (see, Table 1). From this we obtain that the Algorithm $z_{n}$ converges strongly to $z^{*} = (- \frac{2}{3}, - 1, - \frac{4}{3}, 0, 0, \dots) \in {(A + B)}^{- 1} (0)$ ⁠.

Table 1

Convergence of the sequence $z_{n}$ ⁠.

$n$	$z_{n}$	${\| \| z_{n + 1} - z_{n} \| \|}_{l_{2}}$
1	$(1.0000, 4.0000, 5.0000, 0, 0, \dots)$
2	$(0.5000, 2.4998, 3.0831, 0, 0, \dots)$	2.4850
3	$(0.1319, 1.3956, 1.6734, 0, 0, \dots)$	1.8269
4	$(- 0.0320, 0.9039, 1.0465, 0, 0, \dots)$	0.8134
5	$(- 0.1089, 0.6730, 0.7528, 0, 0, \dots)$	0.3814
10	$(- 0.3328, 0.0014, - 0.0986, 0, 0, \dots)$	0.1263
100	$(- 0.6237, - 0.8714, - 1.1904, 0, 0, \dots)$	0.0022
200	$(- 0.6478, - 0.9436, - 1.2774, 0, 0, \dots)$	0.0010
300	$(- 0.6563, - 0.9694, - 1.3077, 0, 0, \dots)$	2.8284e−04
400	$(- 0.6607, - 0.9825, - 1.3228, 0, 0, \dots)$	1.4142e−04
500	$(- 0.6633, - 0.9904, - 1.3318, 0, 0, \dots)$	1.0000e−04

	$⋮$	$⋮$
	$↓$	$↓$
	$(- \frac{2}{3}, - 1, - \frac{4}{3}, 0, 0, \dots)$	0

6. Conclusion

In this paper, we have constructed and studied splitting algorithms which starts by reformulating (1.2) as the problem of locating a point in a certain extended solution set $S_{e} (A, B) \subset H \times H$ which converges strongly to a zero of the sum of two maximally monotone mappings in Hilbert spaces. The assumption that one of the mappings is single-valued, $α$ -inverse strongly monotone or $α$ -strongly monotone is dispensed with. In addition, we applied our main results to study the convex minimization problem. Finally, we provided a numerical example to support our results. Our results extend the results of [5] in the sense that our theorems provide strong convergence in arbitrary Hilbert spaces. In particular, Theorem 3.6 extends Proposition 3 of Eckstein [5] from weak to strong convergence. Moreover, our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

The authors thank the anonymous referees for useful suggestions which improved the contents of this paper. Funding: The first author gratefully acknowledges the funding received from Simons Foundation based at Botswana International University of Science and Technology (BIUST), Palapye, Botswana.Declaration of Competing Interest: None.The publisher wishes to inform readers that the article “A Method of approximation for a zero of the sum of maximally monotone mappings in Hilbert spaces” was originally published by the previous publisher of the Arab Journal of Mathematical Sciences and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was necessary for the journal to transition from the previous publisher to the new one. The publisher sincerely apologises for any inconvenience caused. To access and cite this article, please use Wega, G. B., Zegeye, H. (2019), “A Method of approximation for a zero of the sum of maximally monotone mappings in Hilbert spaces”, Arab Journal of Mathematical Sciences, Vol. 27 No. 1, pp. 26-40. The original publication date for this paper was 22/05/2019.

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Getahun Bekele Wega and Habtu Zegeye

Published in the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

A Method of approximation for a zero of the sum of maximally monotone mappings in Hilbert spaces

1. Introduction

2. Preliminaries

3. Main results

4. Application

4.1 Application to convex minimization problem

5. Numerical example

6. Conclusion

References

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A Method of approximation for a zero of the sum of maximally monotone mappings in Hilbert spaces

1. Introduction

2. Preliminaries

3. Main results

4. Application

4.1 Application to convex minimization problem

5. Numerical example

6. Conclusion

References

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