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Journal of Inequalities and Applications

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Open Access 01.12.2014 | Research

Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces

verfasst von: Yonghong Yao, Giuseppe Marino, Hong-Kun Xu, Yeong-Cheng Liou

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2014

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Abstract

An iterative algorithm is introduced for the construction of the minimum-norm fixed point of a pseudocontraction on a Hilbert space. The algorithm is proved to be strongly convergent.

MSC:47H05, 47H10, 47H17.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

1 Introduction

Construction of fixed points of nonlinear mappings is a classical and active area of nonlinear functional analysis due to the fact that many nonlinear problems can be reformulated as fixed point equations of nonlinear mappings. The research of this area dates back to Picard’s and Banach’s time. As a matter of fact, the well-known Banach contraction principle states that the Picard iterates

{T^{n} x}

converge to the unique fixed point of T whenever T is a contraction of a complete metric space. However, if T is not a contraction (nonexpansive, say), then the Picard iterates

{T^{n} x}

fail, in general, to converge; hence, other iterative methods are needed. In 1953, Mann [1] introduced the now called Mann’s iterative method which generates a sequence

{x_{n}}

via the averaged algorithm

x_{n + 1} = (1 - α_{n}) x_{n} + α_{n} T x_{n}, n \geq 0,

(1.1)

where

{α_{n}}

is a sequence in the unit interval

[0, 1]

, T is a self-mapping of a closed convex subset C of a Hilbert space H, and the initial guess

x_{0}

is an arbitrary (but fixed) point of C.

Mann’s algorithm (1.1) has extensively been studied [2‐7], and in particular, it is known that if T is nonexpansive (i.e.,

∥ T x - T y ∥ \leq ∥ x - y ∥

for all

x, y \in C

) and if T has a fixed point, then the sequence

{x_{n}}

generated by Mann’s algorithm (1.1) converges weakly to a fixed point of T provided the sequence

{α_{n}}

satisfies the condition

\sum_{n = 1}^{\infty} α_{n} (1 - α_{n}) = \infty .

(1.2)

This algorithm, however, does not converge in the strong topology in general (see [[8], Corollary 5.2]).

Browder and Petryshyn [9] studied weak convergence of Mann’s algorithm (1.1) for the class of strict pseudocontractions (in the case of constant stepsizes

α_{n} = α

for all n; see [10] for the general case of variable stepsizes). However, Mann’s algorithm fails to converge for Lipschitzian pseudocontractions (see the counterexample of Chidume and Mutangadura [11]). It is therefore an interesting question of inventing iterative algorithms which generate a sequence converging in the norm topology to a fixed point of a Lipschitzian pseudocontraction (if any). The interest of pseudocontractions lies in their connection with monotone operators; namely, T is a pseudocontraction if and only if the complement

I - T

is a monotone operator.

We also notice that it is quite usual to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subset C of a Hilbert space

H_{1}

and a bounded linear operator

A : H_{1} \to H_{2}

, where

H_{2}

is another Hilbert space. The C-constrained pseudoinverse of A,

A_{C}^{†}

is then defined as the minimum-norm solution of the constrained minimization problem

A_{C}^{†} (b) : = arg min_{x \in C} ∥ A x - b ∥

(1.3)

which is equivalent to the fixed point problem

x = P_{C} (x - λ A^{*} (A x - b)),

(1.4)

where

P_{C}

is the metric projection from

H_{1}

onto C,

A^{*}

is the adjoint of A,

λ > 0

is a constant, and

b \in H_{2}

is such that

P_{\bar{A (C)}} (b) \in A (C)

It is therefore an interesting problem to invent iterative algorithms that can generate sequences which converge strongly to the minimum-norm solution of a given fixed point problem. The purpose of this paper is to solve such a problem for pseudocontractions. More precisely, we shall introduce an iterative algorithm for the construction of fixed points of Lipschitzian pseudocontractions and prove that our algorithm (see (3.1) in Section 3) converges in the strong topology to the minimum-norm fixed point of the mapping.

For the existing literature on iterative methods for pseudocontractions, the reader can consult [10, 12‐26]; for finding minimum-norm solutions of nonlinear fixed point and variational inequality problems, see [27‐29]; and for related iterative methods for nonexpansive mappings, see [2, 3, 30, 31] and the references therein.

2 Preliminaries

Let H be a real Hilbert space with the inner product

〈 \cdot, \cdot 〉

and the norm

∥ \cdot ∥

, respectively. Let C be a nonempty closed convex subset of H. The class of nonlinear mappings which we will study is the class of pseudocontractions. Recall that a mapping

T : C \to C

is a pseudocontraction if it satisfies the property

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

(2.1)

It is not hard to find that T is a pseudocontraction if and only if T satisfies one of the following two equivalent properties:

(a)

{∥ T x - T y ∥}^{2} \leq {∥ x - y ∥}^{2} + {∥ (I - T) x - (I - T) y ∥}^{2}

for all

x, y \in C

; or

(b)

I - T

is monotone on C:

〈 x - y, (I - T) x - (I - T) y 〉 \geq 0

for all

x, y \in C

Recall that a mapping

T : C \to C

is nonexpansive if

∥ T x - T y ∥ \leq ∥ x - y ∥, \forall x, y \in C .

It is immediately clear that nonexpansive mappings are pseudocontractions.

Recall also that the nearest point (or metric) projection from H onto C is defined as follows: For each point

x \in H

P_{C} x

is the unique point in C with the property

∥ x - P_{C} x ∥ \leq ∥ x - y ∥, y \in C .

Note that

P_{C}

is characterized by the inequality

P_{C} x \in C, 〈 x - P_{C} x, y - P_{C} x 〉 \leq 0, y \in C .

(2.2)

Consequently,

P_{C}

is nonexpansive.

In the sequel we shall use the following notations:

$Fix (S)$ stands for the set of fixed points of S;
$x_{n} ⇀ x$ stands for the weak convergence of $(x_{n})$ to x;
$x_{n} \to x$ stands for the strong convergence of $(x_{n})$ to x.

Below is the so-called demiclosedness principle for nonexpansive mappings.

Lemma 2.1 (cf. [32])

Let C be a nonempty closed convex subset of a real Hilbert space H, and let

S : C \to C

be a nonexpansive mapping with fixed points. If

{x_{n}}

is a sequence in C such that

x_{n} ⇀ x^{*}

and

(I - S) x_{n} \to y

, then

(I - S) x^{*} = y

We also need the following lemma whose proof can be found in literature (cf. [33]).

Lemma 2.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that a mapping

F : C \to H

is monotone and weakly continuous along segments (i.e.,

F (x + t y) \to F (x)

weakly as

t \to 0

, whenever

x + t y \in C

for

x, y \in C

). Then the variational inequality

x^{*} \in C, 〈 F x^{*}, x - x^{*} 〉 \geq 0, \forall x \in C

(2.3)

is equivalent to the dual variational inequality

x^{*} \in C, 〈 F x, x - x^{*} 〉 \geq 0, \forall x \in C .

(2.4)

Finally, we state the following elementary result on convergence of real sequences.

Lemma 2.3 ([30])

Let

{a_{n}}

be a sequence of nonnegative real numbers satisfying

a_{n + 1} \leq (1 - γ_{n}) a_{n} + γ_{n} σ_{n}, n \geq 0,

where

{γ_{n}} \subset (0, 1)

and

{σ_{n}}

satisfy

(i)

\sum_{n = 0}^{\infty} γ_{n} = \infty

;

(ii)

either

{lim sup}_{n \to \infty} σ_{n} \leq 0

\sum_{n = 0}^{\infty} | γ_{n} σ_{n} | < \infty

Then

{a_{n}}

converges to 0.

3 An iterative algorithm and its convergence

Throughout this section we assume that C is a nonempty closed subset of a real Hilbert space H and

T : C \to C

is a pseudocontraction with a nonempty fixed point set

Fix (T)

. The aim of this section is to introduce an iterative method for finding the minimum-norm fixed point of T. Towards this, we select two sequences of real numbers,

{α_{n}}

and

{β_{n}}

in the interval

(0, 1)

such that

α_{n} + β_{n} < 1

(3.1)

for all n. We also take an arbitrary initial guess

x_{0} \in C

. We then define an iterative algorithm which generates a sequence

{x_{n}}

via the following recursion:

x_{n + 1} = P_{C} [(1 - α_{n} - β_{n}) x_{n} + β_{n} T x_{n}], n \geq 0 .

(3.2)

We shall prove that this sequence strongly converges to the minimum-norm fixed point of T provided

{α_{n}}

and

{β_{n}}

satisfy certain conditions. To this end, we need the following lemma.

Lemma 3.1 Let

f : C \to H

be a contraction with coefficient

ρ \in (0, 1)

. Let

S : C \to C

be a nonexpansive mapping with

Fix (S) \neq \emptyset

. For each

t \in (0, 1)

, let

x_{t}

be defined as the unique solution of the fixed point equation

x_{t} = S P_{C} [t f (x_{t}) + (1 - t) x_{t}] .

(3.3)

Then, as

t \to 0^{+}

, the net

{x_{t}}

converges strongly to a point

x^{*} \in Fix (S)

which solves the following variational inequality:

x^{*} \in Fix (S), 〈 (I - f) x^{*}, x - x^{*} 〉 \geq 0, x \in Fix (S) .

In particular, if we take

f = 0

, then the net

{x_{t}}

defined via the fixed point equation

x_{t} = S P_{C} [(1 - t) x_{t}],

(3.4)

converges in norm, as

t \to 0^{+}

, to the minimum-norm fixed point of S.

Proof First observe that, for each

t \in (0, 1)

x_{t}

is well defined. Indeed, if we define a mapping

S_{t} : C \to C

S_{t} x = S P_{C} [t f (x) + (1 - t) x], x \in C .

For

x, y \in C

, we have

\begin{array}{rcl} ∥ S_{t} x - S_{t} y ∥ & = & ∥ S P_{C} [t f (x) + (1 - t) x] - S P_{C} [t f (y) + (1 - t) y] ∥ \\ \leq & t ∥ f (x) - f (y) ∥ + (1 - t) ∥ x - y ∥ \\ \leq & [1 - (1 - ρ) t] ∥ x - y ∥, \end{array}

which implies that

S_{t}

is a self-contraction of C. Hence

S_{t}

has a unique fixed point

x_{t} \in C

which is the unique solution of fixed point equation (3.3).

Next we prove that

{x_{t}}

is bounded. Take

u \in Fix (S)

. From (3.3) we have

\begin{array}{rcl} ∥ x_{t} - u ∥ & = & ∥ S P_{C} [t f (x_{t}) + (1 - t) x_{t}] - S P_{C} u ∥ \\ \leq & t ∥ f (x_{t}) - f (u) ∥ + t ∥ f (u) - u ∥ + (1 - t) ∥ x_{t} - u ∥ \\ \leq & [1 - (1 - ρ) t] ∥ x_{t} - u ∥ + t ∥ f (u) - u ∥, \end{array}

that is,

∥ x_{t} - u ∥ \leq \frac{∥ f (u) - u ∥}{1 - ρ} .

Hence,

{x_{t}}

is bounded and so is

{f (x_{t})}

From (3.3) we have

\begin{array}{rcl} ∥ x_{t} - S x_{t} ∥ & = & ∥ S P_{C} [t f (x_{t}) + (1 - t) x_{t}] - S P_{C} x_{t} ∥ \\ \leq & t ∥ f (x_{t}) - x_{t} ∥ \to 0 as t \to 0^{+} . \end{array}

(3.5)

Next we show that

{x_{t}}

is relatively norm-compact as

t \to 0^{+}

, i.e., we show that from any sequence in

{x_{t}}

, a convergent subsequence can be extracted. Let

{t_{n}} \subset (0, 1)

be a sequence such that

t_{n} \to 0^{+}

n \to \infty

. Put

x_{n} : = x_{t_{n}}

. From (3.5) we have

∥ x_{n} - S x_{n} ∥ \to 0 .

(3.6)

Again from (3.3) we get

\begin{array}{rcl} {∥ x_{t} - u ∥}^{2} & = & {∥ S P_{C} [t f (x_{t}) + (1 - t) x_{t}] - S P_{C} u ∥}^{2} \\ \leq & {∥ x_{t} - u + t (f (x_{t}) - x_{t}) ∥}^{2} \\ = & {∥ x_{t} - u ∥}^{2} + 2 t 〈 f (x_{t}) - x_{t}, x_{t} - u 〉 + t^{2} {∥ f (x_{t}) - x_{t} ∥}^{2} \\ = & {∥ x_{t} - u ∥}^{2} + 2 t 〈 f (x_{t}) - f (u), x_{t} - u 〉 + 2 t 〈 f (u) - u, x_{t} - u 〉 \\ + 2 t 〈 u - x_{t}, x_{t} - u 〉 + t^{2} {∥ f (x_{t}) - x_{t} ∥}^{2} \\ \leq & [1 - 2 (1 - ρ) t] {∥ x_{t} - u ∥}^{2} + 2 t 〈 f (u) - u, x_{t} - u 〉 + t^{2} {∥ f (x_{t}) - x_{t} ∥}^{2} . \end{array}

It turns out that

{∥ x_{t} - u ∥}^{2} \leq \frac{1}{1 - ρ} 〈 f (u) - u, x_{t} - u 〉 + t M,

(3.7)

where

M > 0

is a constant such that

M > \frac{1}{2 (1 - ρ)} sup {{∥ f (x_{t}) - x_{t} ∥}^{2} : t \in (0, 1)} .

In particular, we get from (3.7)

{∥ x_{n} - u ∥}^{2} \leq \frac{1}{1 - ρ} 〈 f (u) - u, x_{n} - u 〉 + t_{n} M, u \in Fix (S) .

(3.8)

Since

{x_{n}}

is bounded, without loss of generality, we may assume that

{x_{n}}

converges weakly to a point

x^{*} \in C

. Noticing (3.6) we can use Lemma 2.1 to get

x^{*} \in Fix (S)

. Therefore we can substitute

x^{*}

for u in (3.8) to get

{∥ x_{n} - x^{*} ∥}^{2} \leq \frac{1}{1 - ρ} 〈 f (x^{*}) - x^{*}, x_{n} - x^{*} 〉 + t_{n} M .

(3.9)

However,

x_{n} ⇀ x^{*}

. This together with (3.9) guarantees that

x_{n} \to x^{*}

. The net

{x_{t}}

is therefore relatively compact, as

t \to 0^{+}

, in the norm topology.

Now we return to (3.8) and take the limit as

n \to \infty

to get

{∥ x^{*} - u ∥}^{2} \leq \frac{1}{1 - ρ} 〈 f (u) - u, x^{*} - u 〉, u \in Fix (S) .

In particular,

x^{*}

solves the following variational inequality:

x^{*} \in Fix (S), 〈 (I - f) u, u - x^{*} 〉 \geq 0, u \in Fix (S) .

By Lemma 2.2, we see that

x^{*}

solves the variational inequality

x^{*} \in Fix (S), 〈 (I - f) x^{*}, u - x^{*} 〉 \geq 0, u \in Fix (S) .

(3.10)

Therefore,

x^{*} = (P_{Fix (S)} f) x^{*}

. That is,

x^{*}

is the unique fixed point in

Fix (S)

of the contraction

P_{Fix (S)} f

. Clearly this is sufficient to conclude that the entire net

{x_{t}}

converges in norm to

x^{*}

t \to 0^{+}

Finally, if we take

f = 0

, then variational inequality (3.10) is reduced to

0 \leq 〈 x^{*}, u - x^{*} 〉, u \in Fix (S) .

Equivalently,

{∥ x^{*} ∥}^{2} \leq 〈 x^{*}, u 〉, u \in Fix (S) .

This clearly implies that

∥ x^{*} ∥ \leq ∥ u ∥, u \in Fix (S) .

Therefore,

x^{*}

is the minimum-norm fixed point of S. This completes the proof. □

We are now in a position to prove the strong convergence of algorithm (3.2).

Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, and let

T : C \to C

be L-Lipschitzian and pseudocontractive with

Fix (T) \neq \emptyset

. Suppose that the following conditions are satisfied:

(i)

{lim}_{n \to \infty} α_{n} = 0

and

\sum_{n = 0}^{\infty} α_{n} = \infty

;

(ii)

{lim}_{n \to \infty} \frac{α_{n}}{β_{n}} = {lim}_{n \to \infty} \frac{β_{n}^{2}}{α_{n}} = 0

;

(iii)

{lim}_{n \to \infty} \frac{α_{n} β_{n - 1} - α_{n - 1} β_{n}}{α_{n}^{2} β_{n - 1}} = 0

Then the sequence

{x_{n}}

generated by algorithm (3.2) converges strongly to the minimum-norm fixed point of T.

Proof First we prove that the sequence

{x_{n}}

is bounded. We will show this fact by induction. According to conditions (i) and (ii), there exists a sufficiently large positive integer m such that

1 - 2 (L + 1) (L + 2) (α_{n} + 2 β_{n} + \frac{β_{n}^{2}}{α_{n}}) > 0, n \geq m .

(3.11)

Fix

p \in Fix (T)

and take a constant

M_{1} > 0

such that

max {∥ x_{0} - p ∥, ∥ x_{1} - p ∥, \dots, ∥ x_{m} - p ∥, 2 ∥ p ∥} \leq M_{1} .

(3.12)

Next, we show that

∥ x_{m + 1} - p ∥ \leq M_{1}

Set

y_{m} = (1 - α_{m} - β_{m}) x_{m} + β_{m} T x_{m}; thus x_{m + 1} = P_{C} [y_{m}] .

Then, by using property (2.2) of the metric projection, we have

〈 x_{m + 1} - y_{m}, x_{m + 1} - p 〉 \leq 0 .

(3.13)

By the fact that

I - T

is monotone, we have

〈 (I - T) x_{m + 1} - (I - T) p, x_{m + 1} - p 〉 \geq 0 .

(3.14)

From (3.2), (3.13) and (3.14), we obtain

\begin{array}{rcl} {∥ x_{m + 1} - p ∥}^{2} & = & 〈 x_{m + 1} - p, x_{m + 1} - p 〉 \\ = & 〈 x_{m + 1} - y_{m}, x_{m + 1} - p 〉 + 〈 y_{m} - p, x_{m + 1} - p 〉 \\ \leq & 〈 y_{m} - p, x_{m + 1} - p 〉 \\ = & 〈 x_{m} - p, x_{m + 1} - p 〉 - α_{m} 〈 x_{m}, x_{m + 1} - p 〉 + β_{m} 〈 T x_{m} - x_{m}, x_{m + 1} - p 〉 \\ = & 〈 x_{m} - p, x_{m + 1} - p 〉 + α_{m} 〈 x_{m + 1} - x_{m}, x_{m + 1} - p 〉 - α_{m} 〈 p, x_{m + 1} - p 〉 \\ - α_{m} 〈 x_{m + 1} - p, x_{m + 1} - p 〉 + β_{m} 〈 T x_{m} - T x_{m + 1}, x_{m + 1} - p 〉 \\ + β_{m} 〈 x_{m + 1} - x_{m}, x_{m + 1} - p 〉 - β_{m} 〈 x_{m + 1} - T x_{m + 1}, x_{m + 1} - p 〉 \\ \leq & ∥ x_{m} - p ∥ ∥ x_{m + 1} - p ∥ + α_{m} ∥ x_{m + 1} - x_{m} ∥ ∥ x_{m + 1} - p ∥ \\ + α_{m} ∥ p ∥ ∥ x_{m + 1} - p ∥ - α_{m} {∥ x_{m + 1} - p ∥}^{2} \\ + β_{m} (∥ T x_{m} - T x_{m + 1} ∥ + ∥ x_{m + 1} - x_{m} ∥) ∥ x_{m + 1} - p ∥ \\ \leq & ∥ x_{m} - p ∥ ∥ x_{m + 1} - p ∥ + α_{m} ∥ p ∥ ∥ x_{m + 1} - p ∥ - α_{m} {∥ x_{m + 1} - p ∥}^{2} \\ + (L + 1) (α_{m} + β_{m}) ∥ x_{m + 1} - x_{m} ∥ ∥ x_{m + 1} - p ∥ . \end{array}

It follows that

(1 + α_{m}) ∥ x_{m + 1} - p ∥ \leq ∥ x_{m} - p ∥ + α_{m} ∥ p ∥ + (L + 1) (α_{m} + β_{m}) ∥ x_{m + 1} - x_{m} ∥ .

(3.15)

By (3.2), we have

\begin{array}{rcl} ∥ x_{m + 1} - x_{m} ∥ & = & ∥ P_{C} [(1 - α_{m} - β_{m}) x_{m} + β_{m} T x_{m}] - P_{C} [x_{m}] ∥ \\ \leq & ∥ (1 - α_{m} - β_{m}) x_{m} + β_{m} T x_{m} - x_{m} ∥ \\ \leq & α_{m} (∥ p ∥ + ∥ x_{m} - p ∥) + β_{m} (∥ T x_{m} - p ∥ + ∥ x_{m} - p ∥) \\ \leq & α_{m} (∥ p ∥ + ∥ x_{m} - p ∥) + (L + 1) β_{m} ∥ x_{m} - p ∥ \\ \leq & (L + 1) (α_{m} + β_{m}) ∥ x_{m} - p ∥ + α_{m} ∥ p ∥ \\ \leq & (L + 2) (α_{m} + β_{m}) M_{1} . \end{array}

(3.16)

Substitute (3.16) into (3.15) to obtain

\begin{array}{rcl} (1 + α_{m}) ∥ x_{m + 1} - p ∥ & \leq & ∥ x_{m} - p ∥ + α_{m} ∥ p ∥ + (L + 1) (L + 2) {(α_{m} + β_{m})}^{2} M_{1} \\ \leq & (1 + \frac{1}{2} α_{m}) M_{1} + (L + 1) (L + 2) {(α_{m} + β_{m})}^{2} M_{1}, \end{array}

that is,

\begin{array}{rcl} ∥ x_{m + 1} - p ∥ & \leq & [1 - \frac{(α_{m} / 2) - (L + 1) (L + 2) {(α_{m} + β_{m})}^{2}}{1 + α_{m}}] M_{1} \\ = & {1 - \frac{(α_{m} / 2) [1 - 2 (L + 1) (L + 2) (α_{m} + 2 β_{m} + (β_{m}^{2} / α_{m}))]}{1 + α_{m}}} M_{1} \\ \leq & M_{1} . \end{array}

By induction, we get

∥ x_{n} - p ∥ \leq M_{1}, \forall n \geq 0,

(3.17)

which implies that

{x_{n}}

is bounded and so is

{T x_{n}}

. Now we take a constant

M_{2} > 0

such that

M_{2} = sup_{n} {∥ x_{n} ∥ \lor ∥ T x_{n} - x_{n} ∥} .

[Here

a \lor b = max {a, b}

for

a, b \in R

Set

S = {(2 I - T)}^{- 1}

(i.e., S is a resolvent of the monotone operator

I - T

). We then have that S is a nonexpansive self-mapping of C and

Fix (S) = Fix (T)

(cf. Theorem 6 of [34]).

By Lemma 3.1, we know that whenever

{γ_{n}} \subset (0, 1)

and

γ_{n} \to 0^{+}

, the sequence

{z_{n}}

defined by

z_{n} = S P_{C} [(1 - γ_{n}) z_{n}]

(3.18)

converges strongly to the minimum-norm fixed point

x^{*}

of S (and of T as

Fix (S) = Fix (T)

). Without loss of generality, we may assume that

∥ z_{n} ∥ \leq M_{2}

for all n.

It suffices to prove that

∥ x_{n + 1} - z_{n} ∥ \to 0

n \to \infty

(for some

γ_{n} \to 0^{+}

). To this end, we rewrite (3.18) as

(2 I - T) z_{n} = P_{C} [(1 - γ_{n}) z_{n}], n \geq 0 .

By using the property of metric projection (2.2), we have

\begin{aligned} 〈 (1 - γ_{n}) z_{n} - (2 z_{n} - T z_{n}), x_{n + 1} - (2 z_{n} - T z_{n}) 〉 \leq 0 \\ \Rightarrow 〈 - γ_{n} z_{n}, x_{n + 1} - z_{n} - (z_{n} - T z_{n}) 〉 + 〈 T z_{n} - z_{n}, x_{n + 1} - z_{n} - (z_{n} - T z_{n}) 〉 \leq 0 \\ \Rightarrow 〈 - γ_{n} z_{n} + T z_{n} - z_{n}, x_{n + 1} - z_{n} 〉 + {∥ z_{n} - T z_{n} ∥}^{2} \leq 〈 γ_{n} z_{n}, T z_{n} - z_{n} 〉 \\ \Rightarrow 〈 - γ_{n} z_{n} + T z_{n} - z_{n}, x_{n + 1} - z_{n} 〉 \leq γ_{n} ∥ z_{n} ∥ ∥ T z_{n} - z_{n} ∥ \\ \Rightarrow 〈 - z_{n} + \frac{T z_{n} - z_{n}}{γ_{n}}, x_{n + 1} - z_{n} 〉 \leq ∥ z_{n} ∥ ∥ T z_{n} - z_{n} ∥ . \end{aligned}

Note that

\begin{array}{rcl} ∥ z_{n} - T z_{n} ∥ & = & ∥ P_{C} [(1 - γ_{n}) z_{n}] - z_{n} ∥ \\ \leq & ∥ (1 - γ_{n}) z_{n} - z_{n} ∥ \\ = & γ_{n} ∥ z_{n} ∥ . \end{array}

Hence, we get

〈 - z_{n} + \frac{T z_{n} - z_{n}}{γ_{n}}, x_{n + 1} - z_{n} 〉 \leq γ_{n} {∥ z_{n} ∥}^{2} .

(3.19)

From (3.18) we have

\begin{array}{rcl} ∥ z_{n + 1} - z_{n} ∥ & = & ∥ S P_{C} [(1 - γ_{n + 1}) z_{n + 1}] - S P_{C} [(1 - γ_{n}) z_{n}] ∥ \\ \leq & ∥ (1 - γ_{n + 1}) z_{n + 1} - (1 - γ_{n}) z_{n} ∥ \\ = & ∥ (1 - γ_{n + 1}) (z_{n + 1} - z_{n}) + (γ_{n} - γ_{n + 1}) z_{n} ∥ \\ \leq & (1 - γ_{n + 1}) ∥ z_{n + 1} - z_{n} ∥ + | γ_{n + 1} - γ_{n} | ∥ z_{n} ∥ . \end{array}

It follows that

∥ z_{n + 1} - z_{n} ∥ \leq \frac{| γ_{n + 1} - γ_{n} |}{γ_{n + 1}} ∥ z_{n} ∥ .

(3.20)

Set

γ_{n} : = \frac{α_{n}}{β_{n}} .

By condition (ii),

γ_{n} \to 0^{+}

and

γ_{n} \in (0, 1)

for n large enough. Hence, by (3.19) and (3.20) we have

〈 - z_{n} + \frac{β_{n} (T z_{n} - z_{n})}{α_{n}}, x_{n + 1} - z_{n} 〉 \leq \frac{α_{n}}{β_{n}} {∥ z_{n} ∥}^{2} \leq \frac{α_{n}}{β_{n}} M_{2}^{2}

(3.21)

and

∥ z_{n} - z_{n - 1} ∥ \leq \frac{α_{n} β_{n - 1} - α_{n - 1} β_{n}}{α_{n} β_{n - 1}} M_{2} .

(3.22)

By (3.2) we have

\begin{array}{rcl} ∥ x_{n + 1} - x_{n} ∥ & = & ∥ P_{C} [(1 - α_{n} - β_{n}) x_{n} + β_{n} T x_{n}] - P_{C} x_{n} ∥ \\ \leq & α_{n} ∥ x_{n} ∥ + β_{n} ∥ T x_{n} - x_{n} ∥ \\ \leq & (α_{n} + β_{n}) M_{2} . \end{array}

(3.23)

Next, we estimate

∥ x_{n + 1} - z_{n + 1} ∥

. Since

x_{n + 1} = P_{C} [y_{n}]

〈 x_{n + 1} - y_{n}, x_{n + 1} - z_{n} 〉 \leq 0

. Using (3.21) and by the fact that T is L-Lipschitzian and pseudocontractive, we infer that

\begin{array}{rcl} {∥ x_{n + 1} - z_{n} ∥}^{2} & = & 〈 x_{n + 1} - z_{n}, x_{n + 1} - z_{n} 〉 \\ = & 〈 x_{n + 1} - y_{n}, x_{n + 1} - z_{n} 〉 + 〈 y_{n} - z_{n}, x_{n + 1} - z_{n} 〉 \\ \leq & 〈 y_{n} - z_{n}, x_{n + 1} - z_{n} 〉 \\ = & 〈 [(1 - α_{n} - β_{n}) x_{n} + β_{n} T x_{n}] - z_{n}, x_{n + 1} - z_{n} 〉 \\ = & (1 - α_{n} - β_{n}) 〈 x_{n} - z_{n}, x_{n + 1} - z_{n} 〉 + β_{n} 〈 T x_{n} - T x_{n + 1}, x_{n + 1} - z_{n} 〉 \\ + β_{n} 〈 T x_{n + 1} - T z_{n}, x_{n + 1} - z_{n} 〉 + 〈 - α_{n} z_{n} + β_{n} (T z_{n} - z_{n}), x_{n + 1} - z_{n} 〉, \end{array}

which leads to

\begin{array}{rcl} {∥ x_{n + 1} - z_{n} ∥}^{2} & \leq & (1 - α_{n} - β_{n}) ∥ x_{n} - z_{n} ∥ ∥ x_{n + 1} - z_{n} ∥ + β_{n} L ∥ x_{n} - x_{n + 1} ∥ ∥ x_{n + 1} - z_{n} ∥ \\ + β_{n} {∥ x_{n + 1} - z_{n} ∥}^{2} + α_{n} 〈 - z_{n} + \frac{β_{n}}{α_{n}} (T z_{n} - z_{n}), x_{n + 1} - z_{n} 〉 \\ \leq & \frac{1 - α_{n} - β_{n}}{2} ({∥ x_{n} - z_{n} ∥}^{2} + {∥ x_{n + 1} - z_{n} ∥}^{2}) + \frac{β_{n}^{2}}{2} {∥ x_{n + 1} - z_{n} ∥}^{2} \\ + \frac{L^{2}}{2} {∥ x_{n} - x_{n + 1} ∥}^{2} + β_{n} {∥ x_{n + 1} - z_{n} ∥}^{2} + \frac{α_{n}^{2}}{β_{n}} {∥ z_{n} ∥}^{2} . \end{array}

It follows that, using (3.21), (3.22) and (3.23), we get

\begin{array}{rcl} {∥ x_{n + 1} - z_{n} ∥}^{2} & \leq & \frac{1 - α_{n} - β_{n}}{1 + α_{n} - β_{n}} {∥ x_{n} - z_{n} ∥}^{2} + \frac{L^{2}}{1 + α_{n} - β_{n}} {∥ x_{n + 1} - x_{n} ∥}^{2} \\ + \frac{2 α_{n}^{2}}{(1 + α_{n} - β_{n}) β_{n}} {∥ z_{n} ∥}^{2} + \frac{β_{n}^{2}}{1 + α_{n} - β_{n}} {∥ x_{n + 1} - z_{n} ∥}^{2} \\ \leq & (1 - \frac{2 α_{n}}{1 + α_{n} - β_{n}}) {∥ x_{n} - z_{n} ∥}^{2} + \frac{{(α_{n} + β_{n})}^{2}}{1 + α_{n} - β_{n}} L^{2} M_{2}^{2} \\ + \frac{2 α_{n}^{2}}{(1 + α_{n} - β_{n}) β_{n}} M_{2}^{2} + \frac{β_{n}^{2}}{1 + α_{n} - β_{n}} 4 M_{2}^{2} \\ \leq & (1 - \frac{2 α_{n}}{1 + α_{n} - β_{n}}) {(∥ x_{n} - z_{n - 1} ∥ + ∥ z_{n} - z_{n - 1} ∥)}^{2} \\ + {\frac{{(α_{n} + β_{n})}^{2}}{1 + α_{n} - β_{n}} + \frac{2 α_{n}^{2}}{(1 + α_{n} - β_{n}) β_{n}} + \frac{β_{n}^{2}}{1 + α_{n} - β_{n}}} M \\ \leq & (1 - \frac{2 α_{n}}{1 + α_{n} - β_{n}}) {∥ x_{n} - z_{n - 1} ∥}^{2} \\ + \frac{1}{1 + α_{n} - β_{n}} ∥ z_{n} - z_{n - 1} ∥ (2 ∥ x_{n} - z_{n - 1} ∥ + ∥ z_{n} - z_{n - 1} ∥) \\ + {\frac{{(α_{n} + β_{n})}^{2}}{1 + α_{n} - β_{n}} + \frac{2 α_{n}^{2}}{(1 + α_{n} - β_{n}) β_{n}} + \frac{β_{n}^{2}}{1 + α_{n} - β_{n}}} M \\ \leq & (1 - \frac{2 α_{n}}{1 + α_{n} - β_{n}}) {∥ x_{n} - z_{n - 1} ∥}^{2} + \frac{1}{1 + α_{n} - β_{n}} \frac{α_{n} β_{n - 1} - α_{n - 1} β_{n}}{α_{n} β_{n - 1}} M \\ + {\frac{{(α_{n} + β_{n})}^{2}}{1 + α_{n} - β_{n}} + \frac{2 α_{n}^{2}}{(1 + α_{n} - β_{n}) β_{n}} + \frac{β_{n}^{2}}{1 + α_{n} - β_{n}}} M, \end{array}

(3.24)

where the finite constant

M > 0

is given by

M : = max {L^{2} M_{2}^{2}, 4 M_{2}^{2}, M_{2} sup_{n} (2 ∥ x_{n} - z_{n - 1} ∥ + ∥ z_{n} - z_{n - 1} ∥)} .

Let

δ_{n} = \frac{2 α_{n}}{1 + α_{n} - β_{n}} \approx 2 α_{n} (as n \to \infty)

and note that by (3.1) it follows that

{δ_{n}} \subset (0, 1)

. Moreover, set

θ_{n} = {\frac{α_{n} β_{n - 1} - α_{n - 1} β_{n}}{2 α_{n}^{2} β_{n - 1}} + \frac{1}{2} (α_{n} + 2 β_{n} + \frac{β_{n}^{2}}{α_{n}}) + \frac{α_{n}}{β_{n}} + \frac{β_{n}^{2}}{2 α_{n}}} M .

Then relation (3.24) is rewritten as

{∥ x_{n + 1} - z_{n} ∥}^{2} \leq (1 - δ_{n}) {∥ x_{n} - z_{n - 1} ∥}^{2} + δ_{n} θ_{n} .

(3.25)

By conditions (i), (ii) and (iii), it is easily found that

lim_{n \to \infty} δ_{n} = 0, \sum_{n = 1}^{\infty} δ_{n} = \infty, lim_{n \to \infty} θ_{n} = 0 .

We can therefore apply Lemma 2.3 to (3.25) and conclude that

{∥ x_{n + 1} - z_{n} ∥}^{2} \to 0

n \to \infty

. This completes the proof. □

Remark 3.3 Choose the sequences

(α_{n})

and

(β_{n})

such that

α_{n} = \frac{1}{{(n + 1)}^{a}} and β_{n} = \frac{1}{{(n + 1)}^{b}}, n \geq 0,

where

0 < b < a < 2 b < 1

. It is clear that conditions (i) and (ii) of Theorem 3.2 are satisfied. To verify condition (iii), we compute

\begin{array}{rcl} | \frac{α_{n} β_{n - 1} - α_{n - 1} β_{n}}{α_{n}^{2} β_{n - 1}} | & = & \frac{1}{α_{n}} | 1 - \frac{α_{n - 1} β_{n}}{α_{n} β_{n - 1}} | \\ = & {(n + 1)}^{a} | 1 - \frac{{(n + 1)}^{a - b}}{n^{a - b}} | \\ = & {(n + 1)}^{a} [{(1 + \frac{1}{n})}^{a - b} - 1] \\ \approx & \frac{a - b}{n} {(n + 1)}^{a} \to 0 . \end{array}

Therefore,

{α_{n}}

and

{β_{n}}

satisfy all three conditions (i)-(iii) in Theorem 3.2.

4 Application

To show an application of our results, we deal with the following problem.

Problem 4.1 Let

0 < x_{0} < 1

and define the sequence

{x_{n}}

by the recursion

x_{n + 1} = (1 - n^{- 1 / 2} - n^{- 1 / 3}) x_{n} + n^{- 1 / 3} \frac{x_{n}^{2}}{1 + x_{n}} .

(4.1)

At which value does

{x_{n}}

approach as n goes to infinity?

We claim that

{lim}_{n \to \infty} x_{n} = 0

and it can be easily derived by applying Theorem 3.2.

Proof In order to apply our result, let

H = R

C = [0, 1]

and define

T : C \to C

T x : = \frac{x^{2}}{1 + x} .

Observe that T is Lipschitzian, pseudocontractive and that

Fix (T) = {0}

. Moreover, if we set

α_{n} = n^{- 1 / 2}

and

β_{n} = n^{- 1 / 3}

, then

(i)

{lim}_{n \to \infty} α_{n} = 0

and

\sum_{n = 0}^{\infty} α_{n} = \infty

;

(ii)

{lim}_{n \to \infty} \frac{α_{n}}{β_{n}} = {lim}_{n \to \infty} \frac{β_{n}^{2}}{α_{n}} = 0

;

(iii)

{lim}_{n \to \infty} \frac{α_{n} β_{n - 1} - α_{n - 1} β_{n}}{α_{n}^{2} β_{n - 1}} = 0

Then Theorem 3.2 ensures that

lim_{n \to \infty} x_{n} = 0 .

□

Acknowledgements

Yonghong Yao was supported in part by NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Titel: Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces
verfasst von: Yonghong Yao
Giuseppe Marino
Hong-Kun Xu
Yeong-Cheng Liou
Publikationsdatum: 01.12.2014
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2014
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2014-206

Springer Professional

Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Preliminaries

3 An iterative algorithm and its convergence

4 Application

Acknowledgements

Competing interests

Authors’ contributions

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Springer Professional

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Preliminaries

3 An iterative algorithm and its convergence

4 Application

Acknowledgements

Competing interests

Authors’ contributions

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