- Research
- Open Access

# Strong convergence and stability of Picard iteration sequences for a general class of contractive-type mappings

- Charles E Chidume
^{1}Email author

**2014**:233

https://doi.org/10.1186/1687-1812-2014-233

© Chidume; licensee Springer. 2014

**Received:**18 April 2014**Accepted:**21 October 2014**Published:**17 November 2014

## Abstract

Let $(E,\parallel \cdot \parallel )$ be a normed linear space, $T:E\to E$ be a mapping of *E* into itself satisfying the following contractive condition: $\parallel {T}^{i}x-{T}^{i}y\parallel \u2a7d{a}^{i}\parallel x-y\parallel +{\phi}_{i}(\parallel x-Tx\parallel )$, for each $x,y\in E$, $0\u2a7d{a}^{i}<1$, where ${\phi}_{i}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a sub-additive monotone increasing function with ${\phi}_{i}(0)=0$ and ${\phi}_{i}(Lu)=L{\phi}_{i}(u)$, $L\u2a7e0$, $u\in {\mathbb{R}}^{+}$. It is shown that the Picard iteration process converges strongly to the unique fixed point of *T*. Furthermore, several classes of nonlinear operators studied by various authors are shown to belong to this class of mappings. Our theorem improves several recent important results. In particular, it improves a recent result of Akewe *et al.* (Fixed Point Theory Appl 2014:45, 2014), and a host of other results.

**MSC:**accretive-type mappings, pseudocontractive mappings, Picard sequence, contractive-type mappings.

## Keywords

- 47H04
- 47H06
- 47H15
- 47H17
- 47J25

## 1 Nonlinear operators of accretive-type and fixed points

*K*be a nonempty subset of a real normed space

*E*. A mapping $T:K\to E$ is called

*Lipschitz*if there exists $L\ge 0$ such that

If $L\in [0,1)$, the map *T* is called a *contraction map*, and if $L=1$, *T* is called *nonexpansive*.

*H*be a real Hilbert space; a mapping $A:H\to H$ is called

*monotone*if

*E*. A map $J:E\to {2}^{{E}^{\ast}}$ defined by

*normalized duality map on*

*E*. It is well known that if ${E}^{\ast}$ is strictly convex then

*J*is single-valued. In the sequel, single-valued normalized duality map will be denoted by

*j*. In real Hilbert spaces, the normalized duality map is the identity map. A mapping

*A*with domain $D(A)$ and range $R(A)$ in

*E*is called

*accretive*if, for all $x,y\in D(A)$, the following inequality is satisfied:

*A*is accretive if, for each $x,y\in D(A)$, there exists $j(x-y)\in J(x-y)$ such that

where $J:E\to {2}^{{E}^{\ast}}$ is the normalized duality map on *E*. It follows, again from inequality (1.3), that *A* is accretive if and only if $(I+sA)$ is *expansive* and, consequently, its inverse ${(I+sA)}^{-1}$ exists and is nonexpansive as a mapping from $R(I+sA)$ into $D(A)$, where $R(I+sA)$ denotes the range of $(I+sA)$. The range of $(I+sA)$ does not need to be all of *E*. This leads to the following definition.

**Definition 1.1** An operator *A* is said to be *m*-accretive if *A* is accretive and the range of $(I+sA)$ is all of *E* for some $s>0$.

*some*$s>0$, then it holds for

*all*$s>0$. The operator −Δ, where Δ denotes the Laplacian, is an

*m*-accretive operator. Let $f:E\to \mathbb{R}$ be a convex functional on a real normed space

*E*. The

*subdifferential of*

*f*, denoted by

*∂f*, is a map $\partial f:E\to {2}^{{E}^{\ast}}$ defined for each $x\in E$ by

*subdifferential of*

*f*is a

*maximal monotone operator*. Furthermore, it follows from (1.5) that if zero is in the subdifferential of

*f*at some ${u}^{\ast}\in E$, then ${u}^{\ast}$ is a minimizer of

*f*. Thus, for a convex functional

*f*on a real Hilbert space, solving the inclusion

*f*. More generally, we have the inclusion

where *A* is a maximal monotone operator is of great interest in nonlinear operator theory.

*A*is an accretive map on an appropriate Banach space. At equilibrium, $\frac{du}{dt}=0$ and solving the equation

where *A* is an accretive operator amounts to solving for the equilibrium points of the evolution system (1.7).

Browder converted (1.8) to a fixed point problem. He introduced an operator *T* defined as follows: $T:=I-A$, where *A* is accretive and called such a *T*, *pseudocontractive*. It is clear that fixed points of *T* correspond to zeros of *A*.

is pseudocontractive but is neither nonexpansive nor continuous.

Existence of solutions of system (1.7) has been established. Browder [2] proved that the system is solvable if *A* is locally Lipschitzian and accretive on *E*, and utilizing the existence result for system (1.7), he proved that if *A* is locally Lipschitz and accretive on *E*, then *A* is *m*-accretive.

Martin [3] proved that if *A* is *continuous* and accretive on *E*, then *A* is *m*-accretive. Browder [2] further proved that if $A:E\to E$ is Lipschitz and strongly accretive (*i.e.*, there exists $k\in \mathbb{R}$ such that for each $x,y\in D(A)$, there exists $j(x-y)\in J(x-y)$ such that $\u3008Ax-Ay,j(x-y)\u3009\ge k{\parallel x-y\parallel}^{2}$) then *A* is surjective. This result was subsequently generalized by Deimling [4] to the *continuous* strongly accretive operators (see, *e.g.*, Deimling [[5], Theorem 13.1]). For details of accretive and monotone operators, the reader may consult Reich [6, 7].

## 2 Iterative methods for solutions of certain nonlinear equations

We begin with the well-known and celebrated contraction mapping principle.

**Theorem 2.1** (Contraction mapping principle)

*Let*$(X,\rho )$

*be a complete metric space and*$T:X\to X$

*be a contraction map of*

*X*

*into itself*.

*Then*

- (a)
*T**has a unique fixed point*,*say*${x}^{\ast}$*in**X*; - (b)
*the sequence*${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$*in**X**defined by*${x}_{0}\in X$,${x}_{n+1}=T{x}_{n},\phantom{\rule{1em}{0ex}}n=0,1,2,3,\dots $(2.1)

*converges to* ${x}^{\ast}$.

Theorem 2.1 is, perhaps, the most important fixed point theorem. The sequence of the recursion formula (2.1) is called the *Picard sequence*.

One important (see, *e.g.*, [[8], p.57]) class of nonlinear mappings generalizing the class of contraction mappings is the class of nonexpansive mappings. Readers interested in nonexpansive mappings may consult, for example, Goebel and Reich [9], Reich [10].

If *K* is a nonempty *compact convex* subset of ${\mathbb{R}}^{2}$ and $T:K\to K$ is *a nonexpansive map*, even with a unique fixed point, the Picard sequence defined by (2.1) may fail to converge to the fixed point. It suffices to take $K=\{x\in {\mathbb{R}}^{2}:\parallel x\parallel \le 1\}$ and let $T:K\to K$ be a rotation of *K* about the origin of coordinates through a fixed angle *θ*, $0<\theta <\frac{\pi}{2}$ (say). It is easy to check that *T* is nonexpansive, zero is the unique fixed point of *T* and that the Picard sequence (2.1) with ${x}_{0}=(1,0)$ fails to converge to zero.

Following research efforts by Mann [11], Krasnoselskii [12], Schaefer [13], Ishikawa [14], Edelstein [15–17], Reinermann [18], Edelstein and O’Brian [17], Chidume [19], and a host of other authors, the following recursion formula was developed and found to be effective for approximating fixed points of *nonexpansive mappings*.

*K*be a nonempty convex subset of a normed space

*E*and $T:K\to K$ be a nonexpansive map. Let the sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ in

*K*be defined by

*If the sequence*${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$

*is bounded*, Ishikawa [14] proved that the sequence is an

*approximate fixed point sequence*in the sense that

where *T* maps *K* into *K* and proved that *if* *K* *is bounded*, then the convergence in (2.3) is uniform.

Chidume [19] considered the recursion formula (2.2), introduced the concept of admissible sequence, and proved that *if* *K* *is bounded*, then the convergence in (2.3) is uniform *for the sequence defined by* (2.2).

**Remark 1** We note here that the recursion formula (2.2) which is certainly cumbersome when compared with Picard iteration was developed for the class of nonexpansive maps because the simpler Picard sequence will not always converge for nonexpansive maps. Furthermore, the recursion formula (2.2) can only yield the result that the sequence defined by (2.2) satisfies (2.3). In general, it does not yield convergence of the sequence to a fixed point of *T*. To obtain convergence to a fixed point of *T*, *some type of compactness condition* must be imposed either on *K* or on the map *T* (*e.g.*, *T* may be required to be *demicompact at zero*, or $(I-T)$ may be required to *map closed bounded subsets of* *E* *into closed subsets of* *E*, *etc.*; see, *e.g.*, Chidume [8]). The recursion formula (2.2) is now generally referred to as *Mann formula* in the light of Mann [11].

An important class of mappings generalizing the class of nonexpansive mappings is the class of *Lipschitz pseudocontractive maps*. It is not difficult to check that every nonexpansive map is a Lipschitz pseudocontraction. We have already given an example of a pseudocontractive map which is not even continuous. All attempts to use the Mann formula, which has been successfully employed for nonexpansive mappings, to approximate a fixed point of a Lipschitz pseudocontractive map even on a compact convex domain in a real Hilbert space, proved abortive. In 1974, Ishikawa [20] proved the following theorem.

**Theorem IS**

*Let*

*K*

*be a nonempty compact convex subset of a real Hilbert space*

*H*

*and*$T:K\to K$

*be a Lipschitz pseudocontractive map*.

*Let the sequence*${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$

*be defined by*${x}_{0}\in K$,

*where* $\{{\alpha}_{n}\}$ *and* $\{{\beta}_{n}\}$ *are real sequences satisfying the following conditions*: (i) $0\u2a7d{\alpha}_{n}\u2a7d{\beta}_{n}<1$ $\mathrm{\forall}n\u2a7e1$; (ii) $\sum {\alpha}_{n}{\beta}_{n}=\mathrm{\infty}$; (iii) ${lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0$. *Then* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *converges strongly to a fixed point of* *T*.

**Remark 2** It is clear that the recursion formulas (2.5) and (2.6) of the Ishikawa scheme are more cumbersome than the Mann formula (2.2). However, since it was not known whether or not the simpler Mann sequence would always converge to fixed points of Lipschitz pseudocontractive maps, the cumbersome Ishikawa scheme was applied for this class of maps. The question of whether or not the simpler Mann sequence had actually failed for this class of maps remained open for many years. This was resolved in 2001 by Chidume and Mutangadura [21] who produced an example of a Lipschitz pseudocontractive map defined on a compact convex subset of ${\mathbb{R}}^{2}$ with a unique fixed point for which no Mann sequence converges.

**Remark 3** (a) We first observe that if we set ${\beta}_{n}=0$ ∀*n* in the recursion formula (2.6) then condition (i) in Theorem IS shows that ${\alpha}_{n}=0$ ∀*n* and so (2.5) and (2.6) reduce to ${x}_{n+1}={x}_{n}$ ∀*n*, so that ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ converges to ${x}_{0}$, the initial approximation which may not be a fixed point of *T*.

(b) Because the Ishikawa formulas were used successfully in approximating a fixed point of *T* in Theorem IS, several authors started studying a modification of it in which condition (i) is replaced by the condition: (i)^{∗} $0\u2a7d{\alpha}_{n},{\beta}_{n}<1$, and condition (ii) is modified accordingly. In this modification, ${\alpha}_{n}$ and ${\beta}_{n}$ are independent and it is permissible to set ${\beta}_{n}=0$ for all *n*. They still called such a modified formula an Ishikawa formula. *This is wrong*. To see this, it suffices to set ${\beta}_{n}=0$ ∀*n* and see that the sequence obtained from the modified scheme will not converge to a fixed point of *T* in Theorem IS. In particular, if ${\beta}_{n}=0$ ∀*n*, *the modified formula generally reduces to the Mann formula* and then the example of Chidume and Mutangadura [21] shows that the modified formula will not converge to a fixed point of *T* in the setting of Theorem IS (see, *e.g.*, [8] for more comments on the Ishikawa iteration formula).

(c) The order of convergence of the Picard sequence is that of a geometric progression, that of the Mann sequence is of the form $O(\frac{1}{n})$, while that of the Ishikawa sequence is of the form $O(\frac{1}{\sqrt{n}})$. Furthermore, whenever Picard sequence converges, it is preferred to the Mann sequence which itself is preferred to the Ishikawa formula whenever it converges, because the preferred recursion formula is simpler (consequently requiring less computation and therefore reducing cost of computation).

*Three* other iteration methods have been introduced and have successfully been employed to approximate fixed points of Lipschitz pseudocontractive mappings even in *Banach spaces more general than Hilbert spaces*.

*K*be a nonempty closed convex and bounded subset of a Hilbert space

*H*. Suppose that $T:K\to K$ is a pseudocontractive and Lipschitzian map with constant $L\u2a7e0$. For arbitrary ${z}_{0},w\in K$, Schu [22] defined the following

*two*-

*step*iteration process:

where the real sequences ${\{{\mu}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{{\alpha}_{n}\}}_{n=1}^{\mathrm{\infty}}$ are in $(0,1)$ and satisfy appropriate conditions and are such that $(\{{\mu}_{n}\},\{{\alpha}_{n}\})$ has *property* A (see Schu [22] for a definition). Schu proved that ${\{{z}_{n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to the unique fixed point of *T* nearest to *w*.

This result was extended by Chidume [23] to real Banach spaces possessing weakly sequential continuous duality maps (*e.g.*, ${l}_{p}$ spaces, $1<p<\mathrm{\infty}$).

A second iteration scheme for approximating fixed points of Lipschitz pseudocontractive mappings was implicitly introduced by Bruck [24] who actually applied the scheme, still in Hilbert spaces, to approximate a solution of the inclusion $0\in Ax$ where *A* is an *m*-*monotone operator*.

*H*be a Hilbert space, $A:H\to H$ be an

*m*-

*monotone operator*with $0\in R(A)$, the range of

*A*. For arbitrary $z\in H$, Bruck considered the sequence $\{{x}_{n}\}$ in

*H*defined by ${x}_{0}\in H$,

and proved that if $\{{x}_{n}\}$ and $\{A{x}_{n}\}$ are bounded, then $\{{x}_{n}\}$ converges strongly to some ${x}^{\ast}$, solution of $0\in Au$, provided ${\lambda}_{n}$ and ${\theta}_{n}$ are *acceptably paired sequences* (*e.g.*, see [24] for a definition).

An example of acceptably paired sequences given in [24] is ${\lambda}_{n}={n}^{-1}$, ${\theta}_{n}={(log(logn))}^{-1}$, $n(i)={i}^{i}$.

The ideas of sequences with *property* A and sequences that are *acceptably paired* are due to Halpern [25]. Reich [26] also studied the recursion formula (2.9) for Lipschitz accretive operators on real uniformly convex Banach spaces with *a duality mapping that is weakly sequentially continuous at zero*.

*et al.*[27], Chidume [28], Reich [26, 29, 30], Shahzad and Al-Dubiban [31], Chidume and Zegeye [32] studied the following perturbation of the Mann recurrence relation to approximate fixed points of Lipschitz pseudocontractive mappings in real Banach spaces much more general than Hilbert spaces. Let

*E*be a real normed space,

*K*be a nonempty convex subset of

*E*, $T:K\to K$ be a Lipschitz pseudocontractive map. For arbitrary ${x}_{1}\in K$, let the sequence $\{{x}_{n}\}$ be defined iteratively by

where ${\lambda}_{n}$ and ${\theta}_{n}$ are real sequences in $(0,1)$ satisfying appropriate conditions. They proved the following theorem.

**Theorem CZ** ([32])

*Let*

*K*

*be a nonempty closed convex subset of a real Banach space*

*E*.

*Let*$T:K\to K$

*be a Lipschitz pseudocontractive map with constant*$L>0$

*and*$F(T):=\{x\in K:Tx=x\}\ne \mathrm{\varnothing}$.

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

*be generated from arbitrary*${x}_{1}\in K$

*by*

*for all positive integers* *n*, *where* ${\lambda}_{n}$ *and* ${\theta}_{n}$ *are real sequences in* $(0,1)$ *satisfying appropriate conditions*.

*Then* $\parallel {x}_{n}-T{x}_{n}\parallel \to 0$ *as* $n\to \mathrm{\infty}$.

**Remark 4**Real sequences applicable in Theorem CZ are

**Remark 5** We have shown in this section that a cumbersome recurrence relation is desirable and introduced *only when a simpler recurrence formula is not available for the class of mappings under consideration*. It is obvious that whenever a *k*-step method works for any class of maps, it is trivial to construct an *n*-step method that will work for the same class of maps, $n>k$, $n\in \mathbb{N}$. Such an *n*-step method will, in general, require more computation time and therefore will be less efficient than the *k*-step method. In general, the rate of convergence of such an *n*-step method is at best the same as that of the *k*-step method. Consequently, such *n*-step methods serve no useful purpose and are therefore not desirable.

## 3 A strong convergence theorem

Akewe *et al.* [33] stated the following theorem.

**Theorem AOO** ([[33], Theorem 2.2, p.7])

*Let*$(E,\parallel \cdot \parallel )$

*be a normed linear space*, $T:E\to E$

*be a self*-

*map of*

*E*

*satisfying the following contractive condition*:

*for each*$x,y\in E$, $0\u2a7d{a}^{i}<1$,

*where*$\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$

*is a sub*-

*additive monotone increasing function with*$\phi (0)=0$

*and*$\phi (Lu)=L\phi (u)$, $L\u2a7e0$, $u\in {\mathbb{R}}^{+}$.

*For*${x}_{0}\in E$,

*let*${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$

*be the Kirk*-

*multistep iterative scheme defined by*

*where* ${k}_{1}\ge {k}_{2}\ge {k}_{3}\ge \cdots \ge {k}_{q}$, *for each* $j,{\alpha}_{n,i}\ge 0$, ${\alpha}_{n,0}\ne 0$, ${\beta}_{n,j}^{j}\ge 0$, ${\beta}_{n,0}^{j}\ge 0$.

*Then*

- (i)
*T**defined by*(3.1)*has a unique fixed point**p*; - (ii)
*the Kirk*-*multistep iterative scheme*(*defined by*(3.2))*converges strongly to the fixed point**p**of**T*.

**Remark 6** The authors of Theorem AOO *did not prove* (i) as claimed. The *existence* of a fixed point of *T* was not proved. What the authors showed is that *if* *T* *has a fixed point*, then the fixed point is unique.

**Remark 7**Kirk [34] introduced the following

*one*-

*step*iterative method for approximating a fixed point of a nonexpansive map

*T*: ${x}_{0}\in E$:

He never introduced the *multistep* method defined in (3.2).

**Remark 8**In Theorem AOO, the summation ${\sum}_{j=0}^{i}\left(\genfrac{}{}{0ex}{}{i}{j}\right){a}^{i-j}$ is simply ${(1+a)}^{i}$. Therefore, the contractive condition (3.1) reduces to the following:

which is basically a compact form of (3.1).

Now, assuming the existence of a fixed point for a mapping *T* satisfying contractive condition (3.1) or (3.5), we prove that a Picard sequence for ${T}^{i}$ converges strongly to the unique fixed point of *T*.

**Theorem 3.1**

*Let*$(E,\parallel \cdot \parallel )$

*be a real normed space and*$T:E\to E$

*be a map satisfying the contractive condition*(3.1)

*or*(3.5),

*with constant*${a}^{i}=a$.

*Assume that*

*T*

*has a fixed point*$p\in E$.

*For arbitrary*${x}_{1}\in E$,

*let*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*be a sequence defined by*

*Then* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *converges strongly to* *p*.

*Proof*Since $Tp=p$, put $x=p$ and ${T}^{i}=T$ in the contractive condition (3.1) or (3.5), to obtain

Hence, ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ converges strongly to *p*. This completes the proof. □

*T*has a unique fixed point. Several authors proved fixed point theorems for mappings satisfying special cases of the contractive condition of Hardy and Rogers. We observe that if

*p*denotes the unique fixed point of

*T*in the theorem of Hardy and Rogers, then the following inequality holds:

where $a:=\frac{{a}_{1}+{a}_{2}+{a}_{3}}{1-{a}_{2}-{a}_{5}}\in (0,1)$.

**Remark 9** If an operator satisfies inequality (3.7) where *p* is a fixed point of *T*, then *p* is necessarily unique. For assume that there exists $q\ne p$ such that $Tq=q$. Then $\parallel p-q\parallel =\parallel Tp-Tq\parallel \le {a}^{i}\parallel p-q\parallel $ so that $(1-{a}^{i})\parallel p-q\parallel \le 0$, which yields $p=q$.

**Remark 10** The *T*-stability of the Picard iterative scheme, whenever it converges, is well known (see, *e.g.*, Ostrowski [36], Berinde [37, 38], Bruck [24], Rhoades [39], Harder and Hicks [40], Shahzad and Zegeye [41]).

**Remark 11** In the light of Remark 5, our theorem is a significant improvement on the results of Akewe *et al.* [33] in the sense that the Picard sequence defined by (3.6) is much simpler than the multistep methods (3.2) considered in [33]. Furthermore, the Picard sequence converges as fast as a geometric progression whereas convergence with the multistep methods considered in [33] is either of order $O(\frac{1}{n})$ or of order $O(\frac{1}{\sqrt{n}})$.

## Declarations

## Authors’ Affiliations

## References

- Kato T: Nonlinear semigroups and evolution equations.
*J. Math. Soc. Jpn.*1967, 19: 508–520. 10.2969/jmsj/01940508View ArticleGoogle Scholar - Browder FE: Nonlinear mappings of nonexpansive and accretive type in Banach spaces.
*Bull. Am. Math. Soc.*1969, 73: 875–882.View ArticleMathSciNetGoogle Scholar - Martin RH: A global existence theorem for autonomous differential equation in Banach spaces.
*Proc. Am. Math. Soc.*1970, 26: 307–314. 10.1090/S0002-9939-1970-0264195-6View ArticleGoogle Scholar - Deimling K: Zeros of accretive operators.
*Manuscr. Math.*1974, 13: 365–374. 10.1007/BF01171148View ArticleMathSciNetGoogle Scholar - Deimling K:
*Nonlinear Functional Analysis*. Springer, Berlin; 1985.View ArticleGoogle Scholar - Reich S: Extension problems for accretive sets in Banach spaces.
*J. Funct. Anal.*1977, 26: 378–395. 10.1016/0022-1236(77)90022-2View ArticleGoogle Scholar - Reich S: Constructive techniques for accretive and monotone operators. In
*Applied Nonlinear Analysis*. Academic press, New York; 1979:335–345.Google Scholar - Chidume C Lecture Notes in Mathematics 1965.
*Geometric Properties of Banach Spaces and Nonlinear Iterations*2009.Google Scholar - Goebel K, Reich S:
*Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings*. Dekker, New York; 1984.Google Scholar - Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces.
*J. Math. Anal. Appl.*1979, 67: 274–276. 10.1016/0022-247X(79)90024-6View ArticleMathSciNetGoogle Scholar - Mann WR: Mean value methods in iteration.
*Proc. Am. Math. Soc.*1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleGoogle Scholar - Krasnoselskii MA: Two observations about the method of successive approximations.
*Usp. Mat. Nauk*1957, 10(1):131–140.Google Scholar - Schaefer H: Über die Methode Sukzessiver Approximationen.
*Jahresber. Dtsch. Math.-Ver.*1957, 59(1):131–140. (German)MathSciNetGoogle Scholar - Ishikawa S: Fixed points and iteration of nonexpansive mapping in a Banach space.
*Proc. Am. Math. Soc.*1976, 73: 61–71.MathSciNetGoogle Scholar - Edelstein M: A remark on a theorem of Krasnoselskii.
*Am. Math. Mon.*1966, 13: 507–510.Google Scholar - Edelstein M: On nonexpansive mappings.
*Proc. Am. Math. Soc.*1964, 15: 689–695. 10.1090/S0002-9939-1964-0165498-3View ArticleMathSciNetGoogle Scholar - Edelstein M, O’Brian RC: Nonexpansive mappings, asymptotic regularity and successive approximation.
*J. Lond. Math. Soc.*1978, 17(3):547–554.View ArticleGoogle Scholar - Reinermann J: Über Fixpunkte kontrahierender Abbildungen und schwach konvergente Toeplitz-Verfahren.
*Arch. Math.*1969, 20: 59–64. 10.1007/BF01898992View ArticleMathSciNetGoogle Scholar - Chidume CE: On the approximation of fixed points of nonexpansive mapping.
*Houst. J. Math.*1981, 7: 345–554.MathSciNetGoogle Scholar - Ishikawa S: Fixed points by a new iteration method.
*Proc. Am. Math. Soc.*1974, 44(1):147–150. 10.1090/S0002-9939-1974-0336469-5View ArticleMathSciNetGoogle Scholar - Chidume CE, Mutangadura SA: An example of Mann iteration method for Lipschitz pseudocontractions.
*Proc. Am. Math. Soc.*2001, 129(8):2359–2363. 10.1090/S0002-9939-01-06009-9View ArticleMathSciNetGoogle Scholar - Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings.
*J. Math. Anal. Appl.*1991, 158: 407–413. 10.1016/0022-247X(91)90245-UView ArticleMathSciNetGoogle Scholar - Chidume CE: Iterative approximation of fixed points of Lipschitz pseudocontractive maps.
*Proc. Am. Math. Soc.*2001, 129(8):2245–2251. 10.1090/S0002-9939-01-06078-6View ArticleMathSciNetGoogle Scholar - Bruck RE:A strongly convergent iterative method for the solution of $0\in Ux$ for a maximal monotone operator in Hilbert space.
*J. Math. Anal. Appl.*1974, 48: 114–126. 10.1016/0022-247X(74)90219-4View ArticleMathSciNetGoogle Scholar - Halpern B: Fixed points of nonexpansive mappings.
*Bull. Am. Math. Soc.*1967, 3: 957–961.View ArticleGoogle Scholar - Reich S: Iterative methods for accretive sets. In
*Nonlinear Equations in Abstract Spaces*. Academic Press, New York; 1978:317–326.Google Scholar - Abbas M, Khan SH, Rhoades BE: Simpler is also better approximating fixed points.
*Appl. Math. Comput.*2008, 205(1):428–431. 10.1016/j.amc.2008.08.021View ArticleMathSciNetGoogle Scholar - Chidume CE: Nonexpansive mappings, generalizations and iterative algorithms. In
*Nonlinear Analysis and Applications*. Edited by: Agarwal RP, O’Reagan D. Kluwer Academic, Dordrecht; 2003:383–430.Google Scholar - Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces.
*J. Math. Anal. Appl.*1980, 75: 287–292. 10.1016/0022-247X(80)90323-6View ArticleMathSciNetGoogle Scholar - Reich S: Product of formulas, nonlinear semigroups and accretive operators.
*J. Funct. Anal.*1980, 36: 147–168. 10.1016/0022-1236(80)90097-XView ArticleMathSciNetGoogle Scholar - Shahzard N, Al-Dubiban R: Approximating common fixed points of nonexpansive mappings in Banach spaces.
*Georgian Math. J.*2006, 13(3):529–537.MathSciNetGoogle Scholar - Chidume CE, Zegeye H: Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps.
*Proc. Am. Math. Soc.*2004, 132(3):831–840. 10.1090/S0002-9939-03-07101-6View ArticleMathSciNetGoogle Scholar - Akewe H: Strong convergence and stability of Kirk-multistep-type iterative schemes for contractive-type operators.
*Fixed Point Theory Appl.*2014., 2014: Article ID 45Google Scholar - Kirk WA: On successive approximations for nonexpansive mappings in Banach spaces.
*Glasg. Math. J.*1971, 12: 6–9. 10.1017/S0017089500001063View ArticleMathSciNetGoogle Scholar - Hardy G, Rogers T: A generalization of fixed point theorem of Reich.
*Can. Math. Bull.*1973, 16(2):706–721.MathSciNetGoogle Scholar - Ostrowski AM: The round-off stability of iterations.
*Z. Angew. Math. Mech.*1967, 47: 77–81. 10.1002/zamm.19670470202View ArticleMathSciNetGoogle Scholar - Berinde V: On the stability of some fixed point procedures.
*Bul. Ştiinţ. - Univ. Baia Mare, Ser. B Fasc. Mat.-Inform.*2002, XVIII(1):7–14.MathSciNetGoogle Scholar - Berinde V: Iterative approximation of fixed points and Mann iteration for a general class of functions.
*J. Adv. Math. Stud.*2010, 3(2):1–3.MathSciNetGoogle Scholar - Rhoades BE: Fixed point theorems and stability results for fixed point iteration procedures II.
*Indian J. Pure Appl. Math.*1993, 24(11):691–703.MathSciNetGoogle Scholar - Harder AM, Hicks TL: Stability results for fixed point iteration procedures.
*Math. Jpn.*1988, 33(5):693–706.MathSciNetGoogle Scholar - Shahzad N, Zegeye H: On stability results for
*ϕ*-strongly pseudocontractive mappings.*Nonlinear Anal.*2006, 64(12):2619–2630. 10.1016/j.na.2005.09.007View ArticleMathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.