Some results in fixed point theory and application to the convergence of some iterative processes

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R E S E A R C H

Open Access

Some results in ﬁxed point theory and

application to the convergence of some

iterative processes

Najeh Redjel

1,2*

_{and Abdelkader Dehici}

1,2

*_{Correspondence:}

[email protected]

1_{Laboratory of Informatics and}

Mathematics, University of Souk-Ahras, P.O. Box 1553, Souk-Ahras, 41000, Algeria

2_{Department of Mathematics,}

University of Constantine 1, Constantine, 25000, Algeria

Abstract

In this paper, we study the existence and uniqueness of ﬁxed points for a class of self-mappings satisfying certain rational expressions on closed, bounded and convex subsets with normal structures in reﬂexive Banach spaces. We show that, in particular, this class extends that introduced by Ray and Singh (Indian J. Pure Appl. Math. 9:216-221, 1978). As an application, we give an investigation of the convergence and stability of some iterative processes associated to these mappings.

MSC: 47H09; 47H10; 54H25

Keywords: Banach space; normal structure; rational expression; ﬁxed point; Picard iteration; Mann iteration; Ishikawa iteration; Kirk iteration

1 Introduction

It is well known that the Banach contraction principle was the starting point for the de-velopment of fixed point theory allowing the advancement of the analysis and nonlinear analysis in particular. During more than fifty years, there has been a lot of production in this area and many well-known fixed theorems have been established and investigated by several authors (see, for example, [–] and the references therein).

Deﬁnition . A function: [, +∞[−→[, +∞[ is called a comparison function if it satisﬁes the following conditions:

(i) is monotone nondecreasing;

(ii) limn−→+∞n(t) = for allt> (nstands for thenth iterate of).

Remark . Every comparison function satisﬁes() =  and(t) <t,∀t> .

Fixed point theory has been and will remain an important tool in the study of the istence and uniqueness of solutions of diﬀerential and integral equations. Among the ex-tensions of Banach’s principle, we quote the ﬁxed point result of Boyd and Wong [].

Theorem .[] Let (X,d)be a complete metric space and T be a self-mapping on X satisfying that

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dT(x),T(y)≤d(x,y) (whereis a comparison function).

Then T has a unique ﬁxed point in X.

Other rational expressions generalizing the same principle have been investigated by several authors. Among the more classical ones, those that have been established by Jaggi, Dass and Gupta.

Theorem .[] Let(X,d)be a complete metric space and T:X−→X be a continuous mapping such that

dT(x),T(y)≤αd(x,T(x))d(y,T(y))

d(x,y) +βd(x,y)

for all distinct points x,y∈X,whereα,β∈[, [withα+β< .Then T has a unique ﬁxed point.

Theorem .[] Let(X,d)be a complete metric space and let T be a mapping from X into itself satisfying

dT(x),T(y)≤αdy,T(y) +d(x,T(x))

 +d(x,y) +βd(x,y)

for all distinct points x,y∈X,whereα,β∈[, [withα+β< .Then T has a unique ﬁxed pointξ∈X.Moreover,for all x∈X,the sequence{Tn₍_x_)}_{converges to}_ξ_.

Deﬁnition . Let (X, · ) be a normed space andKbe a nonempty subset ofX. A self-mappingTonKis called a nonexpansive mapping ifT(x)–T(y) ≤ x–yfor allx,y∈K.

The set of nonexpansive self-mappings onKwill be denoted byK.

We note that the existence and uniqueness of ﬁxed points for this type of mappings are not in general ensured. However, some ﬁxed point theorems for nonexpansive mappings have been derived by several authors (we quote, for example, [, ] and Chapter  of []).

Let (X, · ) be a normed space and T be a self-mapping on X. For any nonempty bounded subsetA⊂X, the diameter ofA, which will be denoted byδ(A), is deﬁned by

δ(A) =supx–y:x,y∈A.

Letx∈Xandn∈N, we deﬁne the orbit of ordernofxbyT as follows:

O(x,n) =x,T(x), . . . ,Tn(x).

The orbit ofxbyTis deﬁned by

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Deﬁnition . Let (X,·) be a normed space andKbe a nonempty bounded subset ofX. LetT:K−→Kbe a mapping,Tis said to be orbitally controlled if∀n∈N,n≥,∀x∈X

and for alli,j∈ {, . . . ,n}, there exists ≤k≤nsuch that

Ti(x) –Tj(x)≤x–Tk(x). ()

Remark . The integerkindicated in the above deﬁnition can be chosen independently ofi,j. Indeed, letn≥ andx∈K, and assume that for alli,j∈ {, . . . ,n}, there exists an integerk(depending oni,j) such that

Ti(x) –Tj(x)≤x–Tki,j₍_x₎_.

If we deﬁnel=max{x–Tki,j₍_x₎_,_i_,_j_{= , . . . ,}_n}_{, then there exist}_i

,j∈ {, . . . ,n}such that

l=x–Tki_,j_₍_x₎_{; consequently, we get}

Ti(x) –Tj(x)≤x–Tki,j₍_x₎

for alli,j∈ {, . . . ,n}(hereki,jis independent ofiandj).

IfT is orbitally controlled, then for eachx∈K andn≥, there exist ≤β_x≤nand ≤β_x≤nsuch thatδ(O(x,n)) =x–Tβx_₍_x₎_and_δ((_O₍_T₍_x_),_n_{– ))) =}_T₍_x_{) –}_Tβ_x₍_x_). We denote byrx(T) (resp.rx(T)) the smallest integerβx(resp.βx) such thatδ(O(x,n)) = x–Tβ_x₍_x₎_(resp._δ(_O₍_T₍_x_),_n_{– )) =}_T₍_x_{) –}_Tβ_x₍_x₎_{. We note that}_δ(_O₍_x_,_n₎₎_≥_δ(_O₍_T₍_x_),

n– )).

The set of orbitally controlled self-mappings onKwill be denoted byK.

2 Main results

First of all, we will show thatK contains in particular the set of all nonexpansive

self-mappingsK.

Proposition . Let(X, · )be a normed space and K be a nonempty bounded subset

of X.ThenK⊆K.

Proof LetT∈Kand letn∈N,n≥. Ifx∈Kandi,j∈ {, . . . ,n},j>i, we have

Tj(x) –Ti(x)≤Tj–(x) –Ti–(x)≤ · · · ≤x–Tj–i(x),

which implies thatTsatisﬁes () (since ≤j–i<n), thusT∈K, which gives the result.

Let us give now the deﬁnition of quasi-contraction self-mappings.

Deﬁnition . Let (X, · ) be a normed space andKbe a nonempty subset ofX. A self-mappingT onK is said to be a quasi-contraction mapping if there existsη∈[, [ such that

T(x) –T(y)≤ηmaxx–y,x–T(x),y–T(y),x–T(y),y–T(x)

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The set of quasi-contraction self-mappings onKwill be denoted byK.

Proposition . Let(X, · )be a normed space and K be a nonempty bounded subset

of X.ThenK⊆K.

Proof If T ∈K andn≥. Let x∈K and let ≤i,j≤n. ThusTi(x),Tj(x)∈O(T(x), n– )⊂O(x,n) (T₍_x_{) =}_x_{). Since}_T _{is a quasi-contraction self-mapping on}_K_{, it follows} that

Ti(x) –Tj(x)=TTi–(x)–TTj–(x)

≤ηmaxTi–(x) –Tj–(x),Ti–(x) –Ti(x),Tj–(x) –Tj(x),

Ti–(x) –Tj(x),Tj–(x) –Ti(x) ≤ηδO(x,n)<δO(x,n).

This shows that () is satisﬁed forT and, consequently,K⊆K.

Lemma . Let(X, · )be a normed space and K be a nonempty subset of X.Let n∈N, n≥,if T∈K and x∈K such thatδ(O(x,n)) =δ(O(T(x),n– )),then rx(T) >rx(T).

Proof Assume that rx

(T)≤rx(T). By hypothesis, we obtain thatδ(O(x,n)) =δ(O(T(x),

n– )) =T(x) –Trx(T)₍_x)₌_x_–_Trx(T)₍_x). The fact that_T∈_K_{implies that}_δ(_O₍_x_,_n_{)) =}

δ(O(T(x),n– )) =T(x) –Trx_(T)₍_x₎_≤_x_–_Tr_x(T)–₍_x₎_{, which is a contradiction with}

rx_(T) is the smallest integerβ_xsuch thatδ(O(x,n)) =x–Tβ_x₍_x_).

LetPbe a real polynomial given by

P(x) =λ+λx+· · ·+λnxn withλi≥,λ> , and

n

i= λi= .

Let (X, · ) be a normed space andKbe a nonempty subset ofX. LetT:K−→K be a self-mapping. ByF(T), we will designate the set of ﬁxed points ofT inK, the convex hull ofKis denoted byco(K) and the closed convex hull ofKis denoted byco(K).

Proposition . If K is a nonempty,bounded and convex subset in X and T ∈K such that rx

(T) >rx(T)for all x∈K,then F(T) =F(P(T)).

Proof The inclusionF(T)⊆F(P(T)) is trivial. Let us show the reverse ifx∈Ksuch that

P(T)(x) =λx+λT(x) +· · ·+λnTn(x) =x. Thus,x∈co{T(x), . . . ,Tn(x)}. Now, assume that δ(O(x,n)) > . Sinceλ> , thenλ= , consequently.

x= λ

( –λ)

T(x) +

 – λ ( –λ)

z,

wherez=n_i_=–σiTi+(x) withσi=_–λλi+–λfor alli= , . . . ,n– . Moreover, it is easy to

ob-serve thatn_i_=–σi=  (since

n

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thus

δO(x,n)=x–Trx(T)₍_x₎₌ λ

( –λ)

T(x) +

 – λ ( –λ)

z–Trx(T)₍_x₎

≤ λ ( –λ)

T(x) –Trx(T)₍_x₎₊

 – λ ( –λ)

z–Trx(T)₍_x₎

= λ ( –λ)

T(x) –Trx(T)₍_x₎

+

 – λ ( –λ)

n–

i=

σiTi+(x) –

_n_–

i= σi

Trx(T)₍_x₎

≤ λ ( –λ)

T(x) –Trx(T)₍_x₎

+

 – λ ( –λ)

n–

i=

σiTi+(x) –Tr

x

(T)₍_x₎

≤ λ ( –λ)

T(x) –Trx(T)₍_x₎₊

 – λ ( –λ)

n–

i= σi

δO(x,n).

Sincen_i_=–σi= , it follows that

λ ( –λ)

δO(x,n)≤ λ ( –λ)

T(x) –Trx(T)₍_x₎_.

This gives thatδ(O(x,n)) =T(x) –Trx(T)₍_x)_{and, consequently,}_δ(_O₍_T₍_x_),_n_{– )) =}_T₍_x_{) –} Trx(T)₍_x), which contradicts the fact that_rx

(T) >rx(T). Thusδ(O(x,n)) = , which gives thatT(x) =xand, consequently,x∈F(T).

Remark . We note also that the conditionλ>  is crucial as the following example shows.

Example . LetT : [, ]−→[, ] be deﬁned as follows:T(x) =  –x. It is easy to ob-serve thatTis nonexpansive andx=_ is the unique ﬁxed point ofT. Now consider the polynomialP(z) =_z₊

. We obtain that

P(T) : [, ]−→[, ],

x−→P(T)(x) =x.

In this case, we observe thatF(P(T)) = [, ] and, consequently, we haveF(T)F(P(T)).

By the same techniques as in the proof of Proposition ., we prove the following propo-sition.

Proposition . If K is a nonempty,bounded and convex subset in a normed space X and

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Proof Indeed, in this case, the last inequality in the proof of Proposition . becomes

λ ( –λ)

δO(x,n)≤ λ ( –λ)

T(x) –Trx(T)₍_x₎≤ λ

( –λ)

x–T(x),

which gives thatδ(O(x,n)) =x–T(x)and shows thatrx

(T) = . By replacing in this in-equality, we get alsoδ(O(x,n)) = , which gives the desired result.

Now, we give the following important preparatory result.

Proposition . Let K be a nonempty subset of a normed space X,and let T be a self-mapping deﬁned on K satisfying that for all(x,y)∈K×K,x=y,if y=T(x),we have nec-essarily x=T(y)together with the following condition:

T(x) –T(y)≤(x–T(x),x–T(y)) +(y–T(y),y–T(x)) (x–T(y),y–T(x))

()

for all x,y∈K,where:

(i) ,: [, +∞[×[, +∞[−→[, +∞[nondecreasing mappings relative to the ﬁrst

variable and: [, +∞[×[, +∞[−→[, +∞[such that (t,t) = ⇐⇒(t,t) = (, );

(ii) (t, ) = for allt≥;

(iii) (t,t)+(t,t)

(t,t) ≤tfor allt≥and(t,t)∈[, +∞[×[, +∞[\{(, )}. Then

Tn(x) –Tn+(x)≤Tn–(x) –Tn(x);

and for all strictly positive integers m and n,we have

_Tn₍_x_{) –}_Tm₍_x₎_≤_x_–_T₍_x₎_.

In particular T∈K.

Proof Letx∈K, then

Tn(x) –Tn+(x)=TTn–(x)–TTn(x)

≤(Tn–(x) –Tn(x),Tn–(x) –Tn+(x)) (Tn–(x) –Tn+(x),Tn(x) –Tn(x))

+ (T

n₍_x_{) –}_Tn+₍_x_),_Tn₍_x_{) –}_Tn₍_x₎₎

(Tn–(x) –Tn+(x),Tn(x) –Tn(x)) .

Applying (ii) and (iii), we get

_Tn₍_x_{) –}_Tn+₍_x₎_≤_Tn–₍_x_{) –}_Tn₍_x₎

.. .

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Now

Tn(x) –Tm(x)=TTn–(x)–TTm–(x)

≤ (Tn–(x) –Tn(x),Tn–(x) –Tm(x)) (Tn–(x) –Tm(x),Tm–(x) –Tn(x))

+(T

m–₍_x_{) –}_Tm₍_x_),_Tm–₍_x_{) –}_Tn₍_x₎₎

(Tn–(x) –Tm(x),Tm–(x) –Tn(x)) .

Using the ﬁrst assertion together with the fact thatandare nondecreasing relative to the ﬁrst variable and (iii), we obtain that

Tn(x) –Tm(x)≤ (x–T(x),T

n–₍_x_{) –}_Tm₍_x₎₎

(Tn–(x) –Tm(x),Tm–(x) –Tn(x))

+ (x–T(x),T

m–₍_x_{) –}_Tn₍_x₎₎

(Tn–(x) –Tm(x),Tm–(x) –Tn(x))

≤x–T(x).

This gives thatTsatisﬁes () (herek=  for all integersn,m≥ and for allx∈K).

Deﬁnition . A bounded convex subsetKof a Banach spaceXis said to have a normal

structure if for each convex subsetMofKwith more than one point, there is a pointx∈M

such that

supx–y:y∈M<δ(M).

Under the assumptions of Proposition ., in the following theorem we take(t,t) = ψ(t)ψ(t),(t,t) =ψ(t)ψ(t),(t,t) =ψ(t) +ψ(t) (t,t≥), whereψ,ψ andψare self-mappings deﬁned on [, +∞[.

Theorem . Let K be a nonempty,bounded,closed,convex subset of a reﬂexive Banach

space X having a normal structure,and let T be a continuous self-mapping deﬁned on K

satisfying that for all(x,y)∈K×K,x=y,if y=T(x),we have necessarily x=T(y)together

with the following condition:

T(x) –T(y)≤ψ(x–T(x))ψ(x–T(y)) +ψ(y–T(y))ψ(y–T(x)) ψ(x–T(y)) +ψ(y–T(x))

, ()

whereψis a strictly nondecreasing function andψ(t)≤t for all t≥.

Then T has a unique ﬁxed point in K.

Proof SinceX is a reﬂexive Banach space, then every descending family of nonempty

closed convex subsets ofXhas nonempty intersection. Then we can use the Zorn lemma to obtain a minimal subsetKofKwhich is closed, convex and invariant underT.

• Ifδ(K) = , the problem is solved since in this caseK={x}, and thusT(x) =x.

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SinceKhas a normal structure, then there existsy∈Ksuch that

supx–y;x∈K

≤r<δ(K).

Hence,y–Tk₍_y_{) ≤}_r_{for all} _k_≥_{ and, consequently,} _δ(_O₍_y₎₎_≤_r_{. Let}_H ₌_{_x_∈_K

 : δ(O(x))≤r}andG=co(T(H)), which is a nonempty convex closed subset ofX. Letg∈G. Then we will consider the following three cases.

First case:g=T(h) forh∈H. Then

g–T(g)=T(h) –T(h)≤h–T(h)≤r.

Hence

g∈H and T(g)∈G.

Second case:g=n_i_=λiT(hi),hi∈H,λi≥,

n i=λi= .

Then

g–T(g)=T(g) –

n

i=

λiT(hi)

≤

n

i=

λiT(g) –T(hi)

≤

n

i= λi

ψ(g–T(g))ψ(g–T(hi)) +ψ(hi–T(hi))ψ(hi–T(g)) ψ(g–T(hi)) +ψ(hi–T(g))

≤

n

i= λi

g–T(g)ψ(g–T(hi)) +rψ(hi–T(g))

ψ(g–T(hi)) +ψ(hi–T(g))

=g–T(g)

_n

i= λi

ψ(g–T(hi))

+r _n i= λi

ψ(hi–T(g))

.

We put

γ=

_n

i= λi

ψ(g–T(hi))

and

γ=

_n

i= λi

ψ(hi–T(g))

.

We observe thatγ+γ= , thus

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which gives that

g–T(g)≤r

and, consequently,g∈HandT(g)∈G.

Third case: Ifgis a limit of the formn_i_=λiT(hi),hi∈H,λi≥,

n

i=λi= , then

g–T(g)≤T(g) –T

_n

i= λiT(hi)

+g–

n

i= λiT(hi)

+T

_n

i= λiT(hi)

–

n

i= λiT(hi)

.

By using the second case, we have that for alln≥,

T

_n

i= λiT(hi)

–

n

i=

λiT(hi)≤r.

Let> , taking into account the continuity ofT, we can choosensuﬃciently large such that g– n i= λiT(hi)

+T(g) –T

_n

i= λiT(hi)

<

and, consequently, we infer that

g–T(g)≤r+, ∀> ,

which implies that

g–T(g)≤r.

Henceg∈HandT(g)∈G. ButKis minimal, thusK=G, it follows thatδ(K) =δ(G). Now, by taking into account the fact that ifAis a subset of a Banach spaceX, we have δ(A) =δ(co(A)), then

δ(K) =δ

coT(H)

=δT(H)

=supT(x) –T(y):x,y∈H

≤sup

ψ(x–T(x))ψ(x–T(y)) +ψ(y–T(y))ψ(y–T(x)) ψ(x–T(y)) +ψ(y–T(x))

:

x,y∈H

≤sup

ψ(r)ψ(x–T(y)) +ψ(r)ψ(y–T(x)) ψ(x–T(y)) +ψ(y–T(x))

:x,y∈H

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which is a contradiction, thenδ(K) = , which gives thatK={x}and shows thatThas

a unique ﬁxed point inK.

Remark . Ifψ(t) =ψ(t) =ψ(t) =t, we ﬁnd the main result of [].

3 Applications

We start this section by giving the concept ofϕ-quasi-nonexpansive mappings.

Deﬁnition . LetT be a self-mapping deﬁned on a metric space (X,d). We say that

T :X−→Xis aϕ-quasi-nonexpansive mapping ifF(T)=∅, and there exists a function ϕ: [, +∞[−→[, +∞[ such that

dT(x),z≤ϕd(x,z)

for allx∈X,z∈F(T).

It is easy to observe that every contraction mapping isϕ-quasi-nonexpansive withϕ(t) = αt, ≤α<  andt∈[, +∞[, but the converse is false in general as the following example

shows.

Example . Letϕ: [, +∞[−→[, +∞[ be a comparison function, and letX=R, then the mappingT:R−→Rdeﬁned by

T(x) =

ϕ(|x|)sin(_x) ifx= ,

 ifx= ,

isϕ-quasi-nonexpansive but not a contraction mapping. This implies that the class of ϕ-quasi-nonexpansive mappings contains strictly that of contraction mappings. In the case whereϕ=IdX, we obtain the concept of quasi-nonexpansive mappings introduced by Tri-comi [] and studied by Diaz and Metcalf [, ] (for more details on the class of ϕ-quasi-nonexpansives, we can refer to [, ]).

Deﬁnition . A function: [, +∞[−→[, +∞[ is called subadditive if for allt,t∈ [, +∞[, we have(t+t)≤(t) +(t).

Theorem . Under assumptions of Theorem.,ifψ is a subadditive function with

ψ≤min{ψ,ψ},then T is aψ-quasi-nonexpansive self-mapping on K.

Proof Letzbe the unique ﬁxed point ofT inKandx∈Kwithx=z, then

T(x) –z=T(x) –T(z)≤ψ(x–T(x))ψ(x–z) +ψ(z–z)ψ(z–T(x)) ψ(x–z) +ψ(z–T(x))

≤ ψ(x–T(x))ψ(x–z) ψ(x–z) +ψ(z–T(x))

≤ψ(x–T(x))ψ(x–z) ψ(x–T(x))

=ψ

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Several iterative processes have been deﬁned by many mathematicians. Some of them are the following.

Picard iteration: Let (X, · ) be a normed space andT:X−→Xbe a self-mapping. Let

x∈Xbe ﬁxed, we deﬁne the sequence{xn}nrecursively by

xn+=T(xn) =Tn+(x) for alln∈N. ()

Mann iteration([]): If (X, · ) is a normed space, Mann iteration is deﬁned by the

following algorithm:

xn+= ( –αn)xn+αnT(xn) for alln∈N, ()

wherex∈Xand{αn}n⊂[, ].

Ishikawa iteration([]): Another iterative process of interest is the Ishikawa scheme of

two steps given as follows: Letx∈Xbe ﬁxed, consider the sequence{xn}ndeﬁned by

yn= ( –βn)xn+βnT(xn),

xn+= ( –αn)xn+αnT(yn) for alln∈N,

()

where{αn}nand{βn}nare sequences in [, ].

In the case whereβn= , the Ishikawa iteration reduces to the Mann iteration. In

gen-eral, there is no dependence between convergence results for Picard, Mann and Ishikawa iterations. However, some partial results on the equivalence of these processes have been given by Rhoades and Soltuz (see [–]).

Kirk iteration([]): Let (X, · ) be a normed space,Kbe a closed, convex, bounded

subset ofX. LetT be a self-mapping deﬁned onK. For eachx∈K, the sequence{Sn₍_x₎_}

deﬁned byS:K−→K, where

S=λI+λT+· · ·+λnTn, λi≥,λ> ,

n

i= λi= ,

is said to be Kirk’s iterative process.

By using Theorem . together with ([], Theorem .), we obtain the following result for the convergence of the iterative schemes of Mann and Ishikawa.

Proposition . Let(X, · )be a Banach space.Let{αn}nand{βn}nbe two real sequences in[, ]such that{αnβn}nconverges to some positive real number,let x∈X.Under the

assumptions of Theorem.withψ a comparison continuous function,the Ishikawa

se-quence given by()converges to the unique ﬁxed point of T.Moreover,if{βn}nis the

con-stant sequence equal to,the Mann iteration given by()converges to the same unique

ﬁxed point of T.

Definition . LetAbe a bounded subset of a normed space (X, · ). The Kuratowski measure of noncompactness ofAdenoted byα(A) is defined as the infimum of all>  such thatAadmits a finite covering consisting of subsets with diameter less than. A continuous mappingT:X−→Xis said to be densifying if for every bounded subsetA

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We denote byBXthe closed unit ball of the normed spaceX. The functionαdeﬁned on the set of all subsets ofXdenoted byP(X) enjoys the following properties:

(a) regularity:α() = if and only ifis totally bounded; (b) nonsingularity:αis equal to zero on every one-element set; (c) monotonicity:⊆impliesα()≤α();

(d) semi-additivity:α(∪) =max{α(),α()};

(e) Lipschitzianity:|α() –α()| ≤ρ(,), whereρdenotes the Hausdorﬀ

semi-metricρ(,) =inf{>  :+BX⊃,+BX⊃};

(f ) continuity: For any∈P(X)and any, there existsδ> such that

|α() –α()|<for allsatisfyingρ(,) <δ;

(g) semi-homogeneity:α(t) =|t|α()for any numbert; (h) algebraic semi-additivity:α(+)≤α() +α();

(i) invariance under translations:α(+x) =α()for anyx∈X. The following two theorems are omitted.

Theorem .([], Theorem ..) The Kuratowski measure of noncompactness is

invari-ant under passage to the closure and to the convex hull:α() =α() =α(co()).

Let us recall the following theorem due to Diaz and Metcalf [].

Theorem . Let T be a continuous self-mapping of the metric space(X,d)such that

(i) F(T)is nonempty;

(ii) for eachy∈Xsuch thaty∈/F(T)and for eachz∈F(T),we have

dT(y),z<d(y,z).

Then one and only one of the following properties holds:

(a) for eachx∈X,the Picard sequence{Tn(x)}contains no convergent subsequences;

(b) for eachx∈X,the sequence{Tn(x)}converges to a point belonging toF(T). Theorem .([]) Let T:K−→K be a densifying mapping deﬁned on a closed,bounded,

convex subset K of a Banach space X.Then T has at least one ﬁxed point.

Let us deﬁne the geometric propertyP(f,f,f) given as follows.

Let (X,·) be a normed space, we say thatXhas the propertyP(f,f,f) if for allx,y∈X,

f(x+y) =f(x) +f(y) andx= ,y= , thenx=cy(c> ).

In the case wheref=f=f=IR+, this property is satisﬁed by strictly convex Banach

spaces.

Let (X, · ) be a Banach space andKbe a closed, bounded subset ofX. LetT:K−→K

be a self-mapping onK. We deﬁne the set_Tas follows:

T

=αx+βT(x) :α,β∈[, ],α+β= ,x∈K\F(T).

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Theorem . Let T:K−→K be a densifying mapping deﬁned on a closed,bounded,

convex subset of a Banach space X having the property P(ψ,ψ,ψ),whereψ(t)≤t for

all t≥andψis a nondecreasing,subadditive function withψ≤min{ψ,ψ}.Assume

that T satisﬁes()and F(T)∩_T=∅.Then,for each x∈K,the sequence{Sn₍_x_)}_converges

to a ﬁxed point of T.

Proof Following Theorem ., it follows thatF(T)=∅. Moreover,Sis a densifying

map-ping. Indeed, ifAis a bounded subset ofKsuch thatα(A) > , then

S(A)⊆λA+λT(A) +· · ·+λnTn(A).

Hence, by monotonicity, algebraic semi-additivity and semi-homogeneity properties, we get

αS(A)≤λα(A) +λα

T(A)+· · ·+λnα

Tn(A).

The fact thatTis densifying shows that

αT(A)<α(A),

.. .

αTn(A)≤αTn–(A)≤ · · · ≤αT(A)<α(A);

and therefore

αS(A)< (λ+· · ·+λn)α(A) =α(A).

Also, by use of Theorem .,F(S)=∅. Moreover, by taking account of Propositions . and ., it follows thatF(T) =F(S)=∅.

Forx∈K, let

A=

+∞

n=

Sn{x}.

We haveS(A) =+_n_=∞Sn{x} ⊂Aand, sinceA={x} ∪S(A), this gives that

α(A) =maxα{x},αS(A)

=max,αS(A)

=αS(A).

SinceSis densifying, we establish thatα(A) = , by the property of regularity, we establish thatAis totally bounded and, consequently,Ais compact (sinceX is a Banach space); therefore, the sequence{Sn₍_x_)}_{contains a convergent subsequence.}

Now, ifyis a ﬁxed point ofTandx∈K\F(T), thenT(y) =yandT(x)=y; consequently,

T(x) –y≤ ψ(x–T(x))ψ(x–y)

ψ(x–y) +ψ(y–T(x))

sinceψ() = 

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Moreover, the fact thatψis nondecreasing subadditive andψ≤min{ψ,ψ}gives that

ψx–T(x)≤ψ

x–y+ψy–T(x).

The above inequality is strict, indeed, if

ψx–T(x)=ψ

x–y+ψy–T(x).

Using the propertyP(ψ,ψ,ψ) satisﬁed by the Banach spaceX, we get

x–y=cy–T(x) (c> ),

which implies thatx+cT(x) = ( +c)yand, consequently,y=_+_cx+_+c_cT(x)∈T, which

is a contradiction. This gives that

T(x) –y<ψ

x–y≤ x–y,

which implies that

Tk(x) –y<x–y.

It follows that

S(x) –y=

n

k=

λkTk(x) –y

=

n

k=

λkTk(x) –

n

k= λky

≤

n

k=

λkTk(x) –y

<

n

k=

λkx–y=x–y.

Then S satisﬁes conditions of the theorem of Diaz and Metcalf, which implies that

lim_n_−→₊_∞Sn₍_x_{) exists and belongs to}_F₍_T_{) =}_F₍_S_).

As an immediate consequence of the proof of Theorem ., we have the following.

Theorem . Let(X, · )be a reﬂexive Banach space having the property P(ψ,ψ,ψ),

whereψ(t)≤t for all t≥andψ is a nondecreasing,subadditive function withψ≤

min{ψ,ψ}.Let T:K−→K be a self-mapping deﬁned on a closed,bounded,convex subset

K with normal structure in X.Assume that T satisﬁes assumptions of Theorem ..If

x∈/T(where xis the unique ﬁxed point of T in K)and S is a densifying mapping,then,

for each x∈K,the sequence{Sn₍_x_)}_{converges to x}

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Recall that for the case of numerical stability we say that a fixed point iteration process is numerically stable if small perturbation in the initial data induces a small influence of the computed value of the fixed point. For the remainder of our study, we need the following two definitions about stability of general iterative processes.

Deﬁnition . Let (X, · ) be a normed space,T:X−→Xbe a self-mapping onX. Let {xn}n⊂Xbe the sequence generated by an iteration involvingT and deﬁned by

xn+=f(T,xn) for alln∈N, ()

wherex∈Xandf is some function. Assume that{xn}nconverges to a ﬁxed pointzofT. Let{yn}n⊂X, and we deﬁne

n:=yn+–f(T,yn) for alln∈N.

Then

(i) the iteration process () is said to beT-stable iflimn−→∞n= implies

limn−→∞yn=z;

(ii) the iteration process () is said to be almostT-stable if_n_∈_Nn<∞implies

limn−→∞yn=z.

For more information and interesting comments on these notions of stability, we can see []. On the other hand, it is easy to observe that an iterative process () which isT-stable is almostT-stable but the converse is not true in general (see the counter example given in []). Furthermore, the iterative processes can converge without being stable. Indeed, the following example given in [, ] conﬁrms this.

Example . Let (X, · ) = (R,| · |) andT: [, ]−→[, ],T(x) =x. It is easy to observe thatF(T) = [, ].

The case of Picard iteration: Letz∈], ], andxn+=T(xn) =Tn+(x) is a stationary sequence which is equal tox, this implies that its limit isx. On the other hand, if we take

y=  andyn=_n forn≥, we obtain that

yn+–T(yn)= 

n(n+ )−→,

butlimn−→+∞yn= =z. This shows that the Picard iteration converges but is notT -stable.

The case of Mann iteration: Letz∈], ] ,{αn}n⊆[, ] andxn+= ( –αn)xn+αnxn= xn=xfor alln≥. Letx=yandyn=_n_+ . Thus

n=yn+– ( –αn)yn–αnT(yn)=



(n+ )(n+ )−→,

butlimn−→+∞yn= =z. This shows that the Mann iteration converges but is notT-stable.

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Corollary . Let(X, · )be a reﬂexive Banach space and K be a nonempty,closed,

convex and bounded subset of X with normal structure.Suppose that the assumptions of

Theorem.are satisﬁed withψ a continuous comparison function.If zis the unique

ﬁxed point of T and x∈K with xn+:=T(xn),n∈Nis the Picard process and{yn}n⊂K. Deﬁne{n}nby

n:=yn+–T(yn) for all n∈N.

If_n_∈_Nn<∞,thenlimn−→∞yn=z.In other words,the Picard process is almost T - stable.

Proof The result is established by combining the fact thatT isψ-quasi-nonexpansive

together with Theorem . in [].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors would like to thank the anonymous referees and the editor for their valuable comments which helped to improve this paper.

Received: 20 March 2015 Accepted: 28 August 2015

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