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# The implicit midpoint rule for nonexpansive mappings

- Maryam A Alghamdi
^{1}, - Mohammad Ali Alghamdi
^{2}, - Naseer Shahzad
^{2}and - Hong-Kun Xu
^{2, 3}Email author

**2014**:96

https://doi.org/10.1186/1687-1812-2014-96

© Alghamdi et al.; licensee Springer. 2014

**Received:**21 January 2014**Accepted:**16 March 2014**Published:**9 April 2014

## Abstract

The implicit midpoint rule (IMR) for nonexpansive mappings is established. The IMR generates a sequence by an implicit algorithm. Weak convergence of this algorithm is proved in a Hilbert space. Applications to the periodic solution of a nonlinear time-dependent evolution equation and to a Fredholm integral equation are included.

**MSC:**47J25, 47N20, 34G20, 65J15.

## Keywords

- implicit midpoint rule
- nonexpansive mapping
- periodic solution
- nonlinear evolution equation
- Fredholm integral equation
- Hilbert space

## 1 Introduction

The implicit midpoint rule (IMR) is one of the powerful numerical methods for solving ordinary differential equations (in particular, the stiff equations) [1–6] and differential-algebra equations [7].

where $h>0$ is a stepsize. It is known that if $f:{\mathbb{R}}^{k}\to {\mathbb{R}}^{k}$ is Lipschitz continuous and sufficiently smooth, then the sequence $\{{y}_{n}\}$ converges to the exact solution of (1.1) as $h\to 0$ uniformly over $t\in [0,\overline{t}]$ for any fixed $\overline{t}>0$.

*f*in the form $f(y)=y-g(y)$, then differential equation (1.1) becomes

where *T* is, in general, a nonlinear operator in a Hilbert space. We below introduce our implicit midpoint rule (IMR) for the fixed point problem (1.6) in two iterative algorithms. The first algorithm generates a sequence $\{{x}_{n}\}$ in the following manner.

**Algorithm I**Initialize ${x}_{0}\in H$ arbitrarily and iterate

where ${t}_{n}\in (0,1)$ for all *n*.

Our second IMR is an algorithm that generates a sequence $\{{x}_{n}\}$ as follows.

**Algorithm II**Initialize ${x}_{0}\in H$ arbitrarily and iterate

where ${t}_{n}\in (0,1)$ for all *n*.

Consequently, we may concentrate on Algorithm II.

*T*is a nonexpansive mapping in a general Hilbert space

*H*, that is,

The iterative methods for finding fixed points of nonexpansive mappings have received much attention due to the fact that in many practical problems, the governing operators are nonexpansive (*cf.* [8, 9]). Two iterative methods are basic and they are Mann’s method [10, 11] and Halpern’s method [12–16]. An implicit method is also proposed in [17].

## 2 Convergence analysis

*H*is a Hilbert space with the inner product $\u3008\cdot \phantom{\rule{0.2em}{0ex}},\cdot \u3009$ and the norm $\parallel \cdot \parallel $ and that $T:H\to H$ is a nonexpansive mapping with a fixed point. We use $Fix(T)$ to denote the set of fixed points of

*T*. Namely, $Fix(T)=\{x\in H:Tx=x\}$. It is not hard to find that both IMR (1.7) and (1.8) are well defined. As a matter of fact, for each fixed $u\in H$ and $t\in (0,1)$, the mapping

This is immediately clear due to the nonexpansivity of *T*.

is a contraction with coefficient $t/2$.

### 2.1 Properties of Algorithm II

We first discuss the properties of Algorithm II.

**Lemma 2.1**

*Let*$\{{x}_{n}\}$

*be the sequence generated by Algorithm*II.

*Then*

- (i)
$\parallel {x}_{n+1}-p\parallel \le \parallel {x}_{n}-p\parallel $

*for all*$n\ge 0$*and*$p\in Fix(T)$. - (ii)
${\sum}_{n=1}^{\mathrm{\infty}}{t}_{n}{\parallel {x}_{n}-{x}_{n+1}\parallel}^{2}<\mathrm{\infty}$.

- (iii)
${\sum}_{n=1}^{\mathrm{\infty}}{t}_{n}(1-{t}_{n}){\parallel {x}_{n}-T(\frac{{x}_{n}+{x}_{n+1}}{2})\parallel}^{2}<\mathrm{\infty}$.

*Proof*Let $p\in Fix(T)$. We deduce that

The proof of the lemma is complete. □

**Lemma 2.2**

*Let*$\{{x}_{n}\}$

*be the sequence generated by Algorithm*II.

*Suppose that*${t}_{n+1}^{2}\le a{t}_{n}$

*for all*$n\ge 0$

*and some*$a>0$.

*Then*

*Proof*By definition (1.8) of Algorithm II, we derive that

This in turn implies (2.8). □

### 2.2 Convergence of Algorithms I and II

As Algorithm I is a variant of Algorithm II, we focus on the convergence of Algorithm II. To this end, we need two conditions for the sequence of parameters $\{{t}_{n}\}$ as follows:

(C1) ${t}_{n+1}^{2}\le a{t}_{n}$ for all $n\ge 0$ and some $a>0$,

(C2) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{t}_{n}>0$.

satisfies (C1) and (C2).

**Lemma 2.3**

*Assume*(C1)

*and*(C2).

*Then the sequence*$\{{x}_{n}\}$

*generated by Algorithm*II

*satisfies the property*

*Proof*From (1.8) it follows that

*n*. Hence from Lemma 2.2, we immediately get

□

To prove the convergence of Algorithm II, we need the following so-called demiclosedness principle for nonexpansive mappings.

**Lemma 2.4** ([18])

*Let* *C* *be a nonempty closed convex subset of a Hilbert space* *H*, *and let* $V:C\to H$ *be a nonexpansive mapping with a fixed point*. *Assume that* $\{{x}_{n}\}$ *is a sequence in* *C* *such that* ${x}_{n}\to x$ *weakly and* $(I-V){x}_{n}\to 0$ *strongly*. *Then* $(I-T)x=0$ (*i*.*e*., $Tx=x$).

We use the notation ${\omega}_{w}({x}_{n})$ to denote the set of all weak cluster points of the sequence $\{{x}_{n}\}$.

The following result is easily proved (see [19]).

**Lemma 2.5**

*Let*

*K*

*be a nonempty closed convex subset of a Hilbert space*

*H*,

*and let*$\{{x}_{n}\}$

*be a bounded sequence in*

*H*.

*Assume that*

- (i)
${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-p\parallel $

*exists for all*$p\in K$, - (ii)
${\omega}_{w}({x}_{n})\subset K$.

*Then* $\{{x}_{n}\}$ *weakly converges to a point in* *K*.

We are now in a position to state and prove the main convergence result of this paper.

**Theorem 2.6** *Let* *H* *be a Hilbert space and* $T:H\to H$ *be a nonexpansive mapping with* $Fix(T)\ne \mathrm{\varnothing}$. *Assume that* $\{{x}_{n}\}$ *is generated by IMR* (1.8) *where the sequence* $\{{t}_{n}\}$ *of parameters satisfies conditions* (C1) *and* (C2). *Then* $\{{x}_{n}\}$ *converges weakly to a fixed point of* *T*.

*Proof* By Lemmas 2.3 and 2.4, we have ${\omega}_{w}({x}_{n})\subset Fix(T)$. Furthermore, by Lemma 2.1, ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-p\parallel $ exists for all $p\in Fix(T)$. Consequently, we can apply Lemma 2.5 with $K=Fix(T)$ to assert the weak convergence of $\{{x}_{n}\}$ to a point in $Fix(T)$. □

We then have the following convergence result for IMR (1.7).

**Theorem 2.7** *Let* *H* *be a Hilbert space and* $T:H\to H$ *be a nonexpansive mapping with* $Fix(T)\ne \mathrm{\varnothing}$. *Assume that* $\{{x}_{n}\}$ *is generated by IMR* (1.7) *where the sequence* $\{{t}_{n}\}$ *of parameters satisfies conditions* (C1) *and* (C2). *Then* $\{{x}_{n}\}$ *converges weakly to a fixed point of* *T*.

*Proof*Since $\{{x}_{n}\}$ is also generated by algorithm (1.9), it suffices to verify that the sequence $\{{s}_{n}\}$ defined in (1.10) satisfies conditions (C1) and (C2). As $0\le {t}_{n}\le 1$ and satisfies (C2), it is evident that $\{{s}_{n}\}$ satisfies (C2) as well. To see that $\{{s}_{n}\}$ also fulfils (C1), we argue as follows, using the fact that $\{{t}_{n}\}$ satisfies (C1):

□

## 3 Applications

### 3.1 Periodic solution of a nonlinear evolution equation

*H*,

where $A(t)$ is a family of closed linear operators in *H* and $f:\mathbb{R}\times H\to H$.

Browder [20] proved the following existence of periodic solutions of equation (3.1).

**Theorem 3.1** ([20])

*Suppose that*$A(t)$

*and*$f(t,u)$

*are periodic in*

*t*

*of period*$\xi >0$

*and satisfy the following assumptions*:

- (i)
*For each**t**and each pair*$u,v\in H$,$Re\u3008f(t,u)-f(t,v),u-v\u3009\le 0.$ - (ii)
*For each**t**and each*$u\in D(A(t))$, $Re\u3008A(t)u,u\u3009\ge 0$. - (iii)
*There exists a mild solution**u**of equation*(3.1)*on*${\mathbb{R}}^{+}$*for each initial value*$v\in H$.*Recall that**u**is a mild solution of*(3.1)*with the initial value*$u(0)=v$*if*,*for each*$t>0$,$u(t)=U(t,0)v+{\int}_{0}^{t}U(t,s)f(s,u(s))\phantom{\rule{0.2em}{0ex}}ds,$

*where*${\{U(t,s)\}}_{t\ge s\ge 0}$

*is the evolution system for the homogeneous linear system*

- (iv)
*There exists some*$R>0$*such that*$Re\u3008f(t,u),u\u3009<0$

*for* $\parallel u\parallel =R$ *and all* $t\in [0,\xi ]$.

*Then there exists an element* *v* *of* *H* *with* $\parallel v\parallel <R$ *such that the mild solution of equation* (3.1) *with the initial condition* $u(0)=v$ *is periodic of period* *ξ*.

We next apply our IMR for nonexpansive mappings to provide an iterative method for finding a periodic solution of (3.1).

*u*is the solution of (3.1) satisfying the initial condition $u(0)=v$. Namely, we define

*T*by

*T*is nonexpansive. Moreover, assumption (iv) forces

*T*to map the closed ball $B:=\{v\in H:\parallel v\parallel \le R\}$ into itself. Consequently,

*T*has a fixed point which we denote by

*v*, and the corresponding solution

*u*of (3.1) with the initial condition $u(0)=v$ is a desired periodic solution of (3.1) with period

*ξ*. In other words, to find a periodic solution

*u*of (3.1) is equivalent to finding a fixed point of

*T*. Our IMR is thus applicable to (3.1). It turns out that the sequence $\{{v}_{n}\}$ defined by the IMR

converges weakly to a fixed point *v* of *T*, and the mild solution of (3.1) with the initial value $u(0)=\xi $ is a periodic solution of (3.1). Note that the iteration method (3.3) is essentially to find a mild solution of (3.1) with the initial value of $({v}_{n}+{v}_{n+1})/2$.

### 3.2 Fredholm integral equation

*g*is a continuous function on $[0,1]$ and $F:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous. The existence of solutions has been investigated in the literature (see [21] and the references therein). In particular, if

*F*satisfies the Lipschitz continuity condition

*T*is nonexpansive. As a matter of fact, we have, for $x,y\in {L}^{2}[0,1]$,

*T*in the Hilbert space ${L}^{2}[0,1]$. Hence our IMR is again applicable. Initiating with any function ${x}_{0}\in {L}^{2}[0,1]$, we define a sequence of functions $\{{x}_{n}\}$ in ${L}^{2}[0,1]$ by

Then the sequence $\{{x}_{n}\}$ converges weakly in ${L}^{2}[0,1]$ to the solution of integral equation (3.6).

## Declarations

### Acknowledgements

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant No. 2-363-1433-HiCi. The authors, therefore, acknowledge technical and financial support of KAU.

## Authors’ Affiliations

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