A new multi-step iteration for solving a fixed point problem of nonexpansive mappings

Cholamjiak Fixed Point Theory and Applications 2013, 2013:198http://www.fixedpointtheoryandapplications.com/content/2013/1/198

RESEARCH Open Access

A newmulti-step iteration for solving a fixedpoint problem of nonexpansive mappingsPrasit Cholamjiak*

*Correspondence:[email protected] of Science, University ofPhayao, Phayao, 56000, Thailand

AbstractWe introduce a new nonlinear mapping generated by a finite family of nonexpansivemappings. Weak and strong convergence theorems are also established in the settingof a Banach space.MSC: 47H09; 47H10

Keywords: nonexpansive mapping; weak convergence; Banach space; fixed point;strong convergence

1 IntroductionLet C be a nonempty, closed and convex subset of a real Banach space E. Let T : C → Cbe a nonlinear mapping. The fixed point set of T is denoted by F(T), that is, F(T) = {x ∈C : x = Tx}. Recall that a mapping T is said to be nonexpansive if ‖Tx – Ty‖ ≤ ‖x – y‖for all x, y ∈ C, and a mapping f : C → C is called a contraction if there exists a constantα ∈ (, ) such that ‖f (x) – f (y)‖ ≤ α‖x – y‖ for all x, y ∈ C. We use �C to denote a class ofcontractions with constant α.Fixed point problems are now arising in a wide range of applications such as optimiza-

tion, physics, engineering, economics and applied sciences.Many related problems can becast as the problem of finding fixed points for nonlinear mappings. The interdisciplinarynature of fixed point problems is evident through a vast literature which includes a largebody of mathematical and algorithmic developments.In the literature, several types of iterations have been constructed and proposed in order

to get convergence results for nonexpansive mappings in various settings. One classicaliteration process is defined as follows: x ∈ C and

xn+ = ( – αn)xn + αnTxn, ∀n≥ ,

where {αn} ⊂ (, ). This method was introduced in by Mann [] and is known as theMann iteration process. However, we note that it has only weak convergence in general;for instance, see [].In , Ishikawa [] proposed the following two-step iteration: x ∈ C and

yn = ( – βn)xn + βnTxn,

xn+ = ( – αn)xn + αnTyn, ∀n≥ ,

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where {αn} and {βn} are sequences in (, ). This method is often called the Ishikawa iter-ation process.Very recently, Agarwal et al. [] introduced a new iteration process as follows: x ∈ C

and

yn = ( – βn)xn + βnTxn,

xn+ = ( – αn)Txn + αnTyn, ∀n≥ ,

where {αn} and {βn} are sequences in (, ). This method is called the S-iteration process.The weak convergence was studied in [] for nonexpansivemappings. It was also shown in[] that the convergence rate of the S-iteration process is faster than the Picard andManniteration processes for contractive mappings.Firstly, motivated by Agarwal et al. [], we have the aim to introduce and study a new

mapping defined by the following definition.

Definition . Let C be a nonempty and convex subset of a real Banach space E.Let T,T, . . . ,TN be a finite family of nonexpansive mappings of C into itself, and letλ,λ, . . . ,λN be real numbers such that ≤ λi ≤ for all i = , , . . . ,N . Define the map-ping B : C → C as follows:

U = λT + ( – λ)I,

U = λTU + ( – λ)T,

U = λTU + ( – λ)T,

...

UN– = λN–TN–UN– + ( – λN–)TN–,

B =UN = λNTNUN– + ( – λN )TN–. (.)

Such a mapping B is called the B-mapping generated by T,T, . . . ,TN and λ,λ, . . . ,λN .See [–] for the corresponding concept.Secondly, using the definition above, we studyweak convergence of the followingMann-

type iteration process in a uniformly convex Banach space with a Fréchet differentiablenorm or that satisfies Opial’s condition: x ∈ C and

xn+ = ( – αn)xn + αnBnxn, ∀n≥ , (.)

where Bn is a B-mapping generated by T,T, . . . ,TN and λn,,λn,, . . . ,λn,N (see Section ).Finally, we discuss strong convergence of the iteration scheme involving the modified

viscosity approximation method [] defined as follows: x ∈ C and

xn+ = αnf (xn) + βnxn + γnBnxn, ∀n≥ , (.)

where {αn}, {βn} and {γn} are sequences in (, ), and f ∈ �C .More references on earlier works promoting the theory of fixed points and common

fixed points for nonexpansive mappings can be found in [–].



Throughout this paper, we use the notation:• ⇀ for weak convergence and → for strong convergence.• ωω(xn) = {x : xni ⇀ x} denotes the weak ω-limit set of {xn}.

2 Preliminaries and lemmasIn this section, we begin by recalling some basic facts and lemmas which will be used inthe sequel.Let E be a real Banach space and let U = {x ∈ E : ‖x‖ = } be the unit sphere of E. A Ba-

nach space E is said to be strictly convex if for any x, y ∈U ,

x = y implies∥∥∥∥x + y

∥∥∥∥ < .

It is also said to be uniformly convex if for each ε ∈ (, ], there exists δ > such that forany x, y ∈U ,

‖x – y‖ ≥ ε implies∥∥∥∥x + y

∥∥∥∥ < – δ.

It is known that a uniformly convex Banach space is reflexive and strictly convex. Definea function δ : [, ]→ [, ] called themodulus of convexity of E as follows:

δ(ε) = inf

{ –

∥∥∥∥x + y

∥∥∥∥ : x, y ∈ E,‖x‖ = ‖y‖ = ,‖x – y‖ ≥ ε

}.

Then E is uniformly convex if and only if δ(ε) > for all ε ∈ (, ]. A Banach space E is saidto be smooth if the limit

limt→

‖x + ty‖ – ‖x‖t

(.)

exists for all x, y ∈U . The norm is said to be uniformly Gâteaux differentiable if for y ∈U ,the limit is attained uniformly for x ∈ U . It is said to be Fréchet differentiable if for x ∈ U ,the limit is attained uniformly for y ∈ U . It is said to be uniformly smooth or uniformlyFréchet differentiable if the limit (.) is attained uniformly for x, y ∈ U . The normalizedduality mapping J : E → E∗ is defined by

J(x) ={x∗ ∈ E∗ :

⟨x,x∗⟩ = ‖x‖ = ∥∥x∗∥∥}

for all x ∈ E. It is known that E is smooth if and only if the duality mapping J is singlevalued, and that if E has a uniformly Gâteaux differentiable norm, J is uniformly norm-to-weak∗ continuous on each bounded subset of E. A Banach space E is said to satisfy Opial’scondition []. If x ∈ E and xn ⇀ x, then

lim supn→∞

‖xn – x‖ < lim supn→∞

‖xn – y‖, ∀y ∈ E,x = y.

Let T : C → C. Then I – T is demiclosed at if for all sequence {xn} in C, xn ⇀ q and‖xn – Txn‖ → imply q = Tq. It is known that if E is uniformly convex, C is nonemptyclosed and convex, and T is nonexpansive, then I – T is demiclosed at []. For moredetails, we refer the reader to [, ].



Lemma . [] Let E be a smooth Banach space. Then the following hold:(i) ‖x + y‖ ≥ ‖x‖ + 〈y, J(x)〉 for all x, y ∈ E;(ii) ‖x + y‖ ≤ ‖x‖ + 〈y, J(x + y)〉 for all x, y ∈ E.

Lemma . [] In a strictly convex Banach space E, if

‖x‖ = ‖y‖ = ∥∥λx + ( – λ)y∥∥

for all x, y ∈ E and λ ∈ (, ), then x = y.

Lemma . [] Let {xn} and {zn} be two sequences in a Banach space E such that

xn+ = βnxn + ( – βn)zn, n≥ ,

where {βn} satisfies the conditions < lim infn→∞ βn ≤ lim supn→∞ βn < .If lim supn→∞(‖zn+ – zn‖ – ‖xn+ – xn‖)≤ , then ‖zn – xn‖ → as n→ ∞.

Lemma . [] Let E be a uniformly convex Banach space with a Fréchet differentiablenorm. Let C be a closed and convex subset of E, and let {Sn}∞n= be a family of Ln-Lipschitzianself-mappings on C such that

∑∞n=(Ln – ) <∞ and F =

⋂∞n= F(Sn) = ∅. For arbitrary x ∈

C, define xn+ = Snxn for all n ≥ . Then, for every p,q ∈ F , limn→∞〈xn, J(p – q)〉 exists, inparticular, for all u, v ∈ ωω(xn) and p,q ∈ F , 〈u – v, J(p – q)〉 = .

Lemma . [] Let E be a reflexive and strictly convex Banach space with a uniformlyGâteaux differentiable norm, let C be a nonempty closed convex subset of E, let A : C → Cbe a continuous strongly pseudocontractive mapping with constant k ∈ [, ), and let T :C → E be a continuous pseudocontractivemapping satisfying the weakly inward condition.If T has a fixed point in C, then the path {xt} defined by

xt = tAxt + ( – t)Txt

converges strongly to a fixed point q of T as t → , which is a unique solution of the varia-tional inequality

⟨(I –A)q, J(q – p)

⟩ ≤ , ∀p ∈ F(T).

Remark . Lemma . holds if T : C → C is a nonexpansive mapping and A = f is acontraction.

The following lemma gives us a nice property of real sequences.

Lemma . [] Assume that {an} is a sequence of nonnegative real numbers such that

an+ ≤ ( – cn)an + bn, ∀n≥ ,

where {cn} is a sequence in (, ) and {bn} is a sequence such that(a)

∑∞n= cn =∞;

(b) lim supn→∞bncn ≤ or

∑∞n= |bn| <∞.

Then limn→∞ an = .



3 Weak convergence theoremIn this section, we give some properties concerning the B-mapping and then prove a weakconvergence theorem for nonexpansive mappings.

Lemma . Let C be a nonempty, closed and convex subset of a strictly convex Banachspace E. Let {Ti}Ni= be a finite family of nonexpansive mappings of C into itself suchthat

⋂Ni= F(Ti) = ∅, and let λ,λ, . . . ,λN be real numbers such that < λi < for all

i = , , . . . ,N – and < λN ≤ . Let B be the B-mapping generated by T,T, . . . ,TN andλ,λ, . . . ,λN . Then the following hold:

(i) F(B) =⋂N

i= F(Ti);(ii) B is nonexpansive.

Proof (i) Since⋂N

i= F(Ti) ⊂ F(B) is trivial, it suffices to show that F(B) ⊂ ⋂Ni= F(Ti). To

this end, let p ∈ F(B) and x∗ ∈ ⋂Ni= F(Ti). Then we have

∥∥p – x∗∥∥ =∥∥Bp – x∗∥∥ = ∥∥λN

(TNUN–p – x∗) + ( – λN )

(TN–p – x∗)∥∥

≤ λN∥∥UN–p – x∗∥∥ + ( – λN )

∥∥p – x∗∥∥= λN

∥∥λN–(TN–UN–p – x∗) + ( – λN–)

(TN–p – x∗)∥∥ + ( – λN )

∥∥p – x∗∥∥≤ λNλN–

∥∥UN–p – x∗∥∥ + ( – λNλN–)∥∥p – x∗∥∥

= λNλN–∥∥λN–

(TN–UN–p – x∗) + ( – λN–)

(TN–p – x∗)∥∥

+ ( – λNλN–)∥∥p – x∗∥∥

≤ λNλN–λN–∥∥UN–p – x∗∥∥ + ( – λNλN–λN–)

∥∥p – x∗∥∥...

= λNλN– · · ·λ∥∥λ

(TUp – x∗) + ( – λ)

(Tp – x∗)∥∥

+ ( – λNλN– · · ·λ)∥∥p – x∗∥∥

≤ λNλN– · · ·λ∥∥TUp – x∗∥∥ + ( – λNλN– · · ·λ)

∥∥p – x∗∥∥≤ λNλN– · · ·λ

∥∥Up – x∗∥∥ + ( – λNλN– · · ·λ)∥∥p – x∗∥∥

= λNλN– · · ·λ∥∥λ

(Tp – x∗) + ( – λ)

(p – x∗)∥∥

+ ( – λNλN– · · ·λ)∥∥p – x∗∥∥

≤ λNλN– · · ·λλ∥∥Tp – x∗∥∥ + ( – λNλN– · · ·λλ)

∥∥p – x∗∥∥≤ λNλN– · · ·λλ

∥∥p – x∗∥∥ + ( – λNλN– · · ·λλ)∥∥p – x∗∥∥

=∥∥p – x∗∥∥. (.)

This shows that

∥∥p– x∗∥∥ = λNλN– · · ·λ∥∥λ

(Tp– x∗)+ (–λ)

(p– x∗)∥∥+ (–λNλN– · · ·λ)

∥∥p– x∗∥∥,which turns out to be

∥∥p – x∗∥∥ = ∥∥λ(Tp – x∗) + ( – λ)

(p – x∗)∥∥.



Again by (.), we see that ‖p – x∗‖ = ‖Tp – x∗‖ and thus∥∥p – x∗∥∥ = ∥∥Tp – x∗∥∥ = ∥∥λ

(Tp – x∗) + ( – λ)

(p – x∗)∥∥.

Using Lemma ., we get that Tp = p and hence Up = p. Again by (.), we have

∥∥p – x∗∥∥ = λNλN– · · ·λ∥∥λ

(TUp – x∗) + ( – λ)

(Tp – x∗)∥∥

+ ( – λNλN– · · ·λ)∥∥p – x∗∥∥,

which implies that

∥∥p – x∗∥∥ = ∥∥λ(TUp – x∗) + ( – λ)

(Tp – x∗)∥∥.

From (.) we see that ‖Up – x∗‖ = ‖TUp – x∗‖. Since Up = p and Tp = p,

∥∥p – x∗∥∥ = ∥∥Tp – x∗∥∥ = ∥∥λ(Tp – x∗) + ( – λ)

(p – x∗)∥∥.

Using Lemma ., we get that Tp = p and hence Up = p.By continuing this process, we can show that Tip = p andUip = p for all i = , , . . . ,N –.

Finally, we obtain

‖p – TNp‖ ≤ ‖p – Bp‖ + ‖Bp – TNp‖= ‖p – Bp‖ + ( – λN )‖p – TNp‖,

which yields that p = TNp since p ∈ F(B). Hence p = Tp = Tp = · · · = TNp and thus p ∈⋂Ni= F(Ti).(ii) The proof follows directly from (i). �

Lemma . Let C be a nonempty closed convex subset of a real Banach space E. Let {Ti}Ni=be a finite family of nonexpansive mappings of C into itself such that

⋂Ni= F(Ti) = ∅, and

let B be the B-mapping generated by T,T, . . . ,TN and λ,λ, . . . ,λN . Let {λn,i}Ni= be a realsequence in (, ). For every n ∈N, let Bn be the B-mapping generated by T,T, . . . ,TN andλn,,λn,, . . . ,λn,N as follows:

Un, = λn,T + ( – λn,)I,

Un, = λn,TU + ( – λn,)T,

Un, = λn,TU + ( – λn,)T,

...

Un,N– = λn,N–TN–UN– + ( – λn,N–)TN–,

Bn =Un,N = λn,NTNUN– + ( – λn,N )TN–.

If λn,i → λi ∈ (, ) for all i = , , . . . ,N , then limn→∞ Bnx = Bx for all x ∈ C.



Proof Let x ∈ C and Uk and Un,k be generated by T,T, . . . ,Tk and λ,λ, . . . ,λk , andT,T, . . . ,Tk and λn,,λn,, . . . ,λn,k , respectively. Then

‖Un,x –Ux‖ =∥∥(λn, – λ)(Tx – x)

∥∥ ≤ |λn, – λ|‖Tx – x‖.

Let k ∈ {, , . . . ,N} andM = max{‖TkUk–x‖ + ‖Tk–x‖ : k = , , . . . ,N}. Then

‖Un,kx –Ukx‖ =∥∥λn,kTkUn,k–x + ( – λn,k)Tk–x – λkTkUk– – ( – λk)Tk–x

∥∥= ‖λn,kTkUn,k–x – λn,kTk–x – λkTkUk– + λkTk–x‖≤ λn,k‖TkUn,k–x – TkUk–x‖ + |λn,k – λk|‖TkUk–x‖

+ |λn,k – λk|‖Tk–x‖≤ ‖Un,k–x –Uk–x‖ + |λn,k – λk|M.

It follows that

‖Bnx – Bx‖ = ‖Un,Nx –UNx‖≤ ‖Un,N–x –UN–x‖ + |λn,N – λN |M≤ ‖Un,N–x –UN–x‖ + |λn,N– – λN–|M + |λn,N – λN |M...

≤ ‖Un,x –Ux‖ + |λn, – λ|M + · · · + |λn,N– – λN–|M + |λn,N – λN |M≤ |λn, – λ|‖Tx – x‖ + |λn, – λ|M + · · · + |λn,N– – λN–|M

+ |λn,N – λN |M.

Since λn,i → λi as n→ ∞ (i = , , . . . ,N ), we thus complete the proof. �

Remark . It is easily seen that for all n ∈N, Bn is nonexpansive.

Lemma . Let C be a nonempty closed convex subset of a real Banach space E. Let {Ti}Ni=be a finite family of nonexpansive mappings of C into itself such that

⋂Ni= F(Ti) = ∅. Let

{λn,i}Ni= be a real sequence in (, ). For every n ∈ N, let Bn be the B-mapping generated byT,T, . . . ,TN and λn,,λn,, . . . ,λn,N .If limn→∞ |λn+,i – λn,i| = for all i = , , . . . ,N , then

limn→∞‖Bn+zn – Bnzn‖ =

for each bounded sequence {zn} in C.

Proof Let {zn} be a bounded sequence in C. For j ∈ {, , . . . ,N – } and for some M > ,we have

‖Un+,N–jzn –Un,N–jzn‖=

∥∥λn+,N–jTN–jUn+,N–j–zn + ( – λn+,N–j)TN–j–zn



– λn,N–jTN–jUn,N–j–zn – ( – λn,N–j)TN–j–zn∥∥

≤ λn+,N–j‖TN–jUn+,N–j–zn – TN–jUn,N–j–zn‖+ |λn+,N–j – λn,N–j|‖TN–jUn,N–j–zn‖+ |λn+,N–j – λn,N–j|‖TN–j–zn‖

≤ ‖Un+,N–j–zn –Un,N–j–zn‖ + |λn+,N–j – λn,N–j|M.

Using the relation above, we can show that

‖Bn+zn – Bnzn‖ = ‖Un+,Nzn –Un,Nzn‖

≤ MN∑j=

|λn+,j – λn,j| + |λn+, – λn,|(‖zn‖ + ‖Tzn‖

).

Since limn→∞ |λn+,i – λn,i| = for all i = , , . . . ,N , we obtain the desired result. �

Using the concept of B-mapping, we study weak convergence of the sequence generatedby Mann-type iteration process (.).

Theorem . Let E be a uniformly convex Banach space having a Fréchet differentiablenorm or that satisfies Opial’s condition. Let C be a nonempty, closed and convex sub-set of E. Let {Ti}Ni= be a finite family of nonexpansive mappings of C into itself such that⋂N

i= F(Ti) = ∅. Let {λn,i}Ni= be a real sequence in (, ) such that λn,i → λi (i = , , . . . ,N ).For every n ∈N, let Bn be the B-mapping generated by T,T, . . . ,TN and λn,,λn,, . . . ,λn,N .Let {αn} be a sequence in (, ) satisfying lim infn→∞ αn( –αn) > . Let {xn} be generated byx ∈ C and

xn+ = ( – αn)xn + αnBnxn, ∀n≥ .

Then {xn} converges weakly to x∗ ∈ ⋂Ni= F(Ti).

Proof Let p ∈ ⋂Ni= F(Ti). Then p = Bnp for all n≥ and hence

‖xn+ – p‖ ≤ ( – αn)‖xn – p‖ + αn‖Bnxn – p‖ ≤ ‖xn – p‖.

It follows that {‖xn – p‖} is nonincreasing; consequently, limn→∞ ‖xn – p‖ exists. Assume‖xn – p‖ > . Since E is uniformly convex, it follows (see, for example, []) that

‖xn+ – p‖ ≤ ‖xn – p‖{ – min{αn, – αn}δE

(‖xn – Bnxn‖‖xn – p‖

)},

which implies that

αn( – αn)‖xn – p‖δE(‖xn – Bnxn‖

‖xn – p‖)

≤ min{αn, – αn}‖xn – p‖δE(‖xn – Bnxn‖

‖xn – p‖)

≤ {‖xn – p‖ – ‖xn+ – p‖}.



Since limn→∞ ‖xn – p‖ exists and lim infn→∞ αn( – αn) > , by the continuity of δE , wehave limn→∞ ‖xn–Bnxn‖ = . Since λn,i → λi (i = , , . . . ,N ), let themapping B : C → C begenerated byT,T, . . . ,TN and λ,λ, . . . ,λN . Then, by Lemma ., we have limn→∞ ‖Bnx–Bx‖ = for all x ∈ C. So we have

‖xn – Bxn‖ ≤ ‖xn – Bnxn‖ + ‖Bnxn – Bxn‖≤ ‖xn – Bnxn‖ + sup

z∈{xn}‖Bnz – Bz‖

→ .

Since B is nonexpansive and E is uniformly convex, by the demiclosedness principle,ωω(xn)⊂ F(B). Moreover, F(B) =

⋂Ni= F(Ti) by Lemma .(i).

We next show that ωω(xn) is a singleton. Indeed, suppose that x∗, y∗ ∈ ωω(xn) ⊂⋂Ni= F(Ti). Define Sn : C → C by

Snx = ( – αn)x + αnBnx, x ∈ C.

Then Sn is nonexpansive and x∗, y∗ ∈ ⋂∞n= F(Sn). Using Lemma ., we have limn→∞〈xn,

J(x∗ –y∗)〉 exists. Suppose that {xnk } and {xmk } are subsequences of {xn} such that xnk ⇀ x∗

and xmk ⇀ y∗. Then

∥∥x∗ – y∗∥∥ = ⟨x∗ – y∗, J

(x∗ – y∗)⟩ = lim

k→∞⟨xnk – xmk , J

(x∗ – y∗)⟩ = .

This shows that x∗ = y∗.Assume that E satisfies Opial’s condition. Let x∗, y∗ ∈ ωω(xn) and {xnk } and {xmk } be sub-

sequences of {xn} such that xnk ⇀ x∗ and xmk ⇀ y∗. If x∗ = y∗, then

limn→∞

∥∥xn – x∗∥∥ = limk→∞

∥∥xnk – x∗∥∥ < limk→∞

∥∥xnk – y∗∥∥ = limk→∞

∥∥xmk – y∗∥∥< lim

k→∞∥∥xmk – x∗∥∥ = lim

n→∞∥∥xn – x∗∥∥,

which is a contradiction. It follows that x∗ = y∗. Therefore xn ⇀ x∗ ∈ ⋂Ni= F(Ti) as n → ∞.

This completes the proof. �

4 Strong convergence theoremIn this section, we prove a strong convergence theorem for a finite family of nonexpansivemappings in Banach spaces.

Theorem . Let E be a strictly convex and reflexive Banach space having a uniformlyGâteaux differentiable norm. Let C be a nonempty, closed and convex subset of E. Let {Ti}Ni=be a finite family of nonexpansive mappings of C into itself such that

⋂Ni= F(Ti) = ∅. Let

{λn,i}Ni= be a real sequence in (, ) such that λn,i → λi (i = , , . . . ,N ). For every n ∈ N, letBn be the B-mapping generated by T,T, . . . ,TN and λn,,λn,, . . . ,λn,N . Let {αn}, {βn} and{γn} be sequences in (, ) which satisfy the conditions:(C) αn + βn + γn = ;(C) limn→∞ αn = and

∑∞n= αn =∞;



(C) < lim infn→∞ βn ≤ lim supn→∞ βn < .Let f ∈ �C and define the sequence {xn} by x ∈ C and

xn+ = αnf (xn) + βnxn + γnBnxn, ∀n≥ .

Then {xn} converges strongly to q ∈ ⋂Ni= F(Ti), where q is also the unique solution of the

variational inequality

⟨(I – f )(q), J(q – p)

⟩ ≤ , ∀p ∈N⋂i=

F(Ti). (.)

Proof We divide the proof into the following steps.Step . We show that {xn} is bounded. Let p ∈ ⋂N

i= F(Ti). Then p = Bnp for all n≥ andhence, by the nonexpansiveness of {Bn}∞n=, we have

‖xn+ – p‖ =∥∥αn

(f (xn) – p

)+ βn(xn – p) + γn(Bnxn – p)

∥∥≤ αn

∥∥f (xn) – p∥∥ + βn‖xn – p‖ + γn‖xn – p‖

≤ αn∥∥f (xn) – f (p)

∥∥ + αn∥∥f (p) – p

∥∥ + ( – αn)‖xn – p‖≤ αnα‖xn – p‖ + αn

∥∥f (p) – p∥∥ + ( – αn)‖xn – p‖

=( – αn( – α)

)‖xn – p‖ + αn∥∥f (p) – p

∥∥≤ max

{‖xn – p‖,

– α

∥∥f (p) – p∥∥}

.

By induction, we can conclude that {xn} is bounded. So are {f (xn)} and {Bnxn}.Step . We show that limn→∞ ‖xn+ – xn‖ = . To this end, we define zn = xn+–βnxn

–βn. From

(.) we have

‖zn+ – zn‖ =∥∥∥∥αn+f (xn+) + γn+Bn+xn+

– βn+–

αnf (xn) + γnBnxn – βn

∥∥∥∥=

∥∥∥∥ αn+

– βn+

(f (xn+) – Bnxn

)+

αn

– βn

(Bnxn – f (xn)

)

+γn+

– βn+(Bn+xn+ – Bnxn)

∥∥∥∥≤ αn+

– βn+M +

αn

– βnM + ‖Bn+xn+ – Bnxn‖

≤(

αn+

– βn++

αn

– βn

)M + ‖Bn+xn+ – Bn+xn‖

+ ‖Bn+xn – Bnxn‖

≤(

αn+

– βn++

αn

– βn

)M + ‖xn+ – xn‖ + ‖Bn+xn – Bnxn‖

for someM > . It turns out that

‖zn+ – zn‖ – ‖xn+ – xn‖ ≤(

αn+

– βn++

αn

– βn

)M + ‖Bn+xn – Bnxn‖.



From conditions (C), (C) and Lemma ., we have

lim supn→∞

(‖zn+ – zn‖ – ‖xn+ – xn‖) ≤ .

Lemma . yields that ‖zn – xn‖ → and hence

‖xn+ – xn‖ = ( – βn)‖zn – xn‖ → .

Step . We show that limn→∞ ‖Bxn – xn‖ = . Indeed, noting that

Bnxn – xn =γn

{(xn+ – xn) + αn

(xn – f (xn)

)},

we have, by (C) and (C),

limn→∞‖Bnxn – xn‖ = .

Let B : C → C be the B-mapping generated by T,T, . . . ,TN and λ,λ, . . . ,λN . So, byLemma ., we have Bnx→ Bx for all x ∈ C. It also follows that

‖Bxn – xn‖ ≤ ‖Bxn – Bnxn‖ + ‖Bnxn – xn‖≤ sup

z∈{xn}‖Bz – Bnz‖ + ‖Bnxn – xn‖

→ .

For t ∈ (, ), we define a contraction as follows:

Stx = tf (x) + ( – t)Bx.

Then there exists a unique path xt ∈ C such that

xt = tf (xt) + ( – t)Bxt .

From Lemma ., we know that xt → q as t → , where q ∈ F(B). Lemma .(i) alsoyields that q ∈ F(B) =

⋂Ni= F(Ti). Moreover, q is the unique solution of variational inequal-

ity (.).Step . We show that lim supn→∞〈f (q) – q, J(xn – q)〉 ≤ . We see that

xt – xn = ( – t)(Bxt – xn) + t(f (xt) – xn

).

It follows, by Lemma .(ii) that

‖xt – xn‖ ≤ ( – t)‖Bxt – xn‖ + t⟨f (xt) – xn, J(xt – xn)

⟩≤ (

– t + t)(‖xt – xn‖ + ‖Bxn – xn‖

)+ t

⟨f (xt) – xt , J(xt – xn)

⟩+ t‖xt – xn‖,



which gives

⟨f (xt) – xt , J(xn – xt)

⟩ ≤ ( + t)‖xn – Bxn‖t

(‖xt – xn‖ + ‖xn – Bxn‖

)+t‖xt – xn‖.

So we have

lim supn→∞


⟩ ≤ tM (.)

for someM > . Since E has a uniformly Gâteaux differentiable norm, J is norm-to-weak∗

uniformly continuous on bounded subsets of E. So we have

⟨f (q) – q, J(xn – q) – J(xn – xt)

⟩ → (.)

and

⟨f (q) – f (xt) + xt – q, J(xn – xt)

⟩ → (.)

as t → . On the other hand, we have

⟨f (q) – q, J(xn – q)

⟩=


⟩+

⟨f (q) – f (xt) + xt – q, J(xn – xt)

⟩+

⟨f (q) – q, J(xn – q) – J(xn – xt)

⟩. (.)

Since lim supn→∞ and lim supt→ are interchangeable, using (.)-(.), we obtain

lim supn→∞

⟨f (q) – q, J(xn – q)

⟩ ≤ .

Step . We show that xn → q as n→ ∞. In fact, we have

‖xn+ – q‖ = αn⟨f (xn) – q, J(xn+ – q)

⟩+ βn

⟨xn – q, J(xn+ – q)

⟩+ γn

⟨Bnxn – q, J(xn+ – q)

⟩≤ αnα‖xn – q‖‖xn+ – q‖ + αn

⟨f (q) – q, J(xn+ – q)

⟩+ βn‖xn – q‖‖xn+ – q‖ + γn‖xn – q‖‖xn+ – q‖

=( – αn( – α)

)‖xn – q‖‖xn+ – q‖ + αn⟨f (q) – q, J(xn+ – q)

⟩

≤ ( – αn( – α)

)(‖xn – q‖ + ‖xn+ – q‖) + αn⟨f (q) – q, J(xn+ – q)

⟩,

which implies that

‖xn+ – q‖ ≤ – αn( – α) + αn( – α)

‖xn – q‖ + αn

+ αn( – α)⟨f (q) – q, J(xn+ – q)

⟩

=( –

αn( – α) + αn( – α)

)‖xn – q‖

+αn

+ αn( – α)⟨f (q) – q, J(xn+ – q)

⟩.



Put cn = αn(–α)+αn(–α) and bn = αn

+αn(–α) 〈f (q) – q, J(xn+ – q)〉. So it is easy to check that {cn} isa sequence in (, ) such that

∑∞n= cn =∞ and lim supn→∞

bncn ≤ . Hence, by Lemma .,

we conclude that xn → q as n → ∞. This completes the proof. �

Competing interestsThe author declares that he has no competing interests.

AcknowledgementsThe author wishes to thank editor/referees for valuable suggestions and Professor Suthep Suantai for the guidance. Thisresearch was supported by the Thailand Research Fund, the Commission on Higher Education, and University of Phayaounder Grant MRG5580016.

Received: 16 April 2013 Accepted: 5 July 2013 Published: 22 July 2013

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doi:10.1186/1687-1812-2013-198Cite this article as: Cholamjiak: A newmulti-step iteration for solving a fixed point problem of nonexpansivemappings. Fixed Point Theory and Applications 2013 2013:198.


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A new multi-step iteration for solving a fixed point problem of nonexpansive mappings

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