R E S E A R C H
Open Access
Fixed points by certain iterative schemes with
applications
Ahmed El-Sayed Ahmed
1,2*and Sayed Attia Ahmed
3,4*Correspondence:
[email protected] 1Department of Mathematics,
Faculty of Science, Sohag University, Sohag, 82524, Egypt
2Mathematics Department, Faculty of Science, Taif University, El-Hawiyah, P.O. Box 888, El-Taif, 5700, Saudi Arabia
Full list of author information is available at the end of the article
Abstract
The main aim of this paper is to present the concept of general Mann and general Ishikawa type double-sequences iterations with errors to approximate fixed points. We prove that the general Mann type double-sequence iteration process with errors converges strongly to a coincidence point of two continuous pseudo-contractive mappings, each of which maps a bounded closed convex nonempty subset of a real Hilbert space into itself. Moreover, we discuss equivalence from theS,T-stabilities point of view under certain restrictions between the general Mann type
double-sequence iteration process with errors and the general Ishikawa iterations with errors. An application is also given to support our idea using compatible-type mappings.
MSC: 47H10; 54H25
Keywords: fixed points; Hilbert spaces; Mann iteration; pseudo-contractive maps; weakly compatible maps
1 Introduction
In the last few decades investigations of fixed points by some iterative schemes have at-tracted many mathematicians. With the recent rapid developments in fixed point theory, there has been a renewed interest in iterative schemes. The properties of iterations be-tween the type of sequences and kind of operators have not been completely studied and are now under discussion. The theory of operators has occupied a central place in modern research using iterative schemes because of its promise of enormous utility in fixed point theory and its applications. There are a number of papers that have studied fixed points by some iterative schemes (see []). It is rather interesting to note that the type of operators play a crucial role in investigations of fixed points.
The Mann iterative scheme was invented in (see [–]), and it is used to obtain con-vergence to a fixed point for many classes of mappings (see [–] and others). The idea of considering fixed point iteration procedures with errors comes from practical numeri-cal computations. This topic of research plays an important role in the stability problem of fixed point iterations. In , Liu [] initiated a study of fixed point iterations with errors. Several authors have proved some fixed point theorems for Mann-type iterations with errors using several classes of mappings (see [–] and others).
Suppose thatHis a real Hilbert space andAis a nonlinear mapping ofHinto itself. The mapAis said to be accretive if∀x,y∈D(A), we have that
Ax–Ay,x–y ≥, ()
and it is said to be strongly accretive ifA–kIis accretive, wherek∈(, ) is a constant and Idenotes the identity operator onH.
The mapAis said to beφ-strongly accretive if∀x,y∈E, exists a strictly increasing func-tionφ: [,∞)→[,∞) withφ() = such that
Ax–Ay,x–y ≥φ x–y x–y ,
and it is called uniformly accretive if there exists a strictly increasing functionψ: [,∞)→ [,∞) withψ() = such thatAx–Ay,x–y ≥ψ( x–y ).
LetN(A) ={x∗∈H:Ax∗= }denote the null space (set of zero) ofA. IfN(A)=φand () holds for all x∈D(A) and y∈N(A), then A is said to be quasi-accretive. The no-tions of strongly,φ-strongly, uniformly quasi-accretive are similarly defined.Ais said to bem-accretive if∀r> the operator (I+rA) is surjective. Closely related to the class of accretive maps is the class of pseudo-contractive maps.
A mapT:H→His said to be pseudo-contractive if∀x,y∈D(T) we have that
(I–T)x– (I–T)y,x–y≥, ()
observe thatTis pseudo-contractive if and only ifA= (I–T) is accretive.
A mappingT:H→His called Lipschitzian (orL-Lipschitzian) if there existsL> such that
Tx–Ty ≤L x–y , ∀x,y∈H.
In the sequel we useL> .
Definition .(seee.g.[]) LetNdenote the set of all natural numbers, and letEbe a normed linear space. By a double sequence inEwe mean a functionf :N×N→Edefined byf(n,m) =xn,m∈E.
The double sequence{xn,m}is said to converge strongly tox∗ if for a given> , there
exist integersN,M> such that∀n≥N,m≥M, we have that
xn,m–x∗<.
If∀n,r≥N,m,t≥M, we have that
xn,r–xm,t <,
then the double sequence is said to be Cauchy. Furthermore, if for each fixedn,xn,m→x∗n
asm→ ∞and thenx∗n→x∗asn→ ∞, soxn,m→x∗asn,m→ ∞.
In , Moore [] introduced the following theorem.
Theorem A Let C be a bounded closed convex nonempty subset of a(real)Hilbert space H,and let T:C→C be a continuous pseudo-contractive map.Let{αn}n≥,{ak}k≥⊂(, )
be real sequences satisfying the following conditions:
(ii) limk,r→∞a–k–aakr = ,∀ <r≤k;
(iii) limn→∞αn= ;
(iv) n≥αn=∞.
For an arbitrary but fixed w∈C,and for each k≥,define Tk:C→C by
Tkx:= ( –ak)w+akTx, ∀x∈C.
Then the double sequence{xk,n}k≥,n≥generated from an arbitrary x,∈C by
xk,n+= ( –αn)xk,n+αnTkxk,n, k,n≥, ()
converges strongly to a fixed point x∗∞of T in C.
The two most popular iteration procedures for obtaining fixed points ofT, when the Banach principle fails, are doubly Mann iterations with errors [] defined by
uk,n+= ( –αn)uk,n+αnTuk,n+αnun,
and doubly Ishikawa iterations with errors defined by
xk,n+= ( –αn)xk,n+αnTzk,n+αnvn,
zk,n= ( –βn)xk,n+βnTxk,n+βnwn.
The sequences{αn} ⊂(, ),{βn} ⊂[, ) satisfy
lim
n→∞αn=nlim→∞βn= ,
∞
n=
αn=∞.
A reasonable conjecture is that the doubly Ishikawa iteration with error and the corre-sponding doubly Mann iteration with error are equivalent for all maps for which either method provides convergence to a fixed point.
In the present paper, we define the following iteration which will be called the general Mann iteration process with errors:
Suk,n+= ( –αn)Suk,n+αnTuk,n+αnun. ()
Using this general Mann iteration process, we give a strong convergence theorem in the double-sequence setting.
It should be remarked that in (), if we putS=I, whereIdenotes the identity mapping, then we obtain the Mann iteration process with errors (see []).
The general doubly Ishikawa iteration with error is defined by
Sxk,n+= ( –αn)Sxk,n+αnTzk,n+αnvn, ()
Szk,n= ( –βn)Sxk,n+βnTxk,n+βnwn. ()
The sequences{αn} ⊂(, ),{βn} ⊂[, ) satisfy
lim
n→∞αn=nlim→∞βn= , ∞
n=
It should be remarked that in () and (), if we putS=I, where Idenotes the identity mapping, then we obtain the Ishikawa iteration process with errors (see [, ]).
2 A strong convergence theorem
In this section, it is proved that a general Mann-type double-sequence iteration process with error converges strongly to a coincidence point of the continuous pseudo-contractive
mappingsS andT both of them mapC into C (whereC is a bounded closed convex
nonempty subset of a (real) Hilbert space). Now, we give the following theorem.
Theorem . Let C be a bounded closed convex nonempty subset of a(real)Hilbert space H, and let S,T :C→C be continuous pseudo-contractive maps. Let{αn}n≥,{ak}k≥⊂
(, )be real sequences satisfying the following conditions:
(i) limk→∞ak= (monotonically);
(ii) limk,r→∞ak–ar
–ak = ,∀ <r≤k;
(iii) limn→∞αn= ;
(iv) n≥αn=∞.
For an arbitrary but fixed w∈C,and for each k≥,define Tk:C→C by
Tkx= ( –ak)w+akTx+ ( –ak)uk, ∀x∈C.
Then the double sequence{xk,n}k≥,n≥generated from an arbitrary x,∈C by
Sxk,n+= ( –αn)Sxk,n+αnTkxk,n+ ( –αn)uk,n, k,n≥, ()
converges strongly to a coincidence point x∗∞of S and T∈C.
Proof Clearly,CF(T)=∅andCF(S)=∅(seee.g.[]), where the set of coincidence points ofTis denoted byCF(T) and the set of coincidence points ofSis denoted byCF(S).
Now, we have
Tkx–Tky,Sx–Sy=akTx–Ty,Sx–Sy ≤ak Sx–Sy
so that for allk≥,Tkis continuous and strongly pseudo-contractive. Also,Cis invariant
underTkfor allkby convexity. Hence,Tkhas a unique fixed pointx∗k∈C,∀k≥. It thus
suffices to prove the following:
() for each fixedk≥,Sxk,n→Sx∗k∈Casn→ ∞;
() Sx∗k→Sx∗∞∈Cask→ ∞; () x∗∞∈CF(S)∩CF(T).
The first is known, but for completeness we give the details. Now, letd=diamCandbk= –ak∈(, ),∀k. Then
Sxk,n+–Sx∗k
=( –αn)Sxk,n+αnTkxk,n+ ( –αn)uk,n–Sx∗n
=Sxk,n–Sx∗k–αn(Sxk,n–Tkxk,n) + ( –αn)uk,n
=Sxk,n–Sx∗k
– αn
Sxk,n–Tkxk,n,Sxk,n–Sx∗k
– ( –αn)
uk,n,Sxk,n–Sx∗k
+αn Sxk,n–Tkxk,n + αn( –αn)xk,n–Tkxk,n,un+ ( –αn) uk,n
≤Sxk,n–Sx∗k
– αnakSxk,n–Sx∗k
– ( –αn) uk,n Sxk,n–Sx∗kαnd
+ αn( –αn) Sxk,n–Tkxk,n uk,n + ( –αn) uk,n . ()
If we set
θk,n=Sxk,n–Sx∗k, δk,n= akαn,
then from () we obtain
θk,n+≤( –δk,n)θk,n+dαn+
αn( –αn)d– ( –αn)θk,n
uk,n
+ ( –αn) uk,n
= ( –δk,n)θk,n+dαn+ ( –αn)
αnd+ ( –αn) uk,n – θk,n
uk,n .
Observing that
dαn=O(δk,n), lim
n→∞δk,n= and
n≥
dαn=∞,
we obtainθk,n→ asn→ ∞. So the first part is proved. Now, we have
Sx∗k–Tx∗k=Sx∗k–a–k Sx∗k–a–k ( –ak)w–a–k ( –ak)uk
=
– ak
Sx∗k–
–ak
ak
(w+uk)
=–
–ak
ak
Sx∗k–
–ak
ak
(w+uk)
=
–ak
ak
–Sx∗k+w+uk
≤
–ak
ak
Sx∗k+ w + uk
≤
–ak
ak
d+ uk
,
which implies that
lim k→∞Sx
∗
k–Tx∗k≤.
Then
lim k→∞Sx
∗
k–Tx∗k= ,
hence{x∗k}is a coincidence point sequence forSandT. Also, assuming thatx∗∞is a coin-cidence point ofSandT, then
Sx∗∞–Tx∗∞≤ lim k→∞
d+ uk
–ak
ak
Now, for all <r≤k, we have
Sx∗k–Sx∗r
=Sx∗k–Sx∗r,Sx∗k–Sx∗r=Tkx∗k–Trx∗r,Sx∗k–Sx∗r
=w+akTxk–arTxr+ ( –ak)uk– ( –ar)ur,Sx∗k–Sx∗r
= (ar–ak)
w,Sx∗k–Sx∗r+akTxk∗–arTx∗r,Sx∗k–Sx∗r
+ ( –ak)
uk,Sx∗k–Sx∗r
– ( –ar)
ur,Sx∗k–Sx∗r
= (ar–ak)
w,Sx∗k–Sx∗r+ (ak–ar)
Tx∗r,Sx∗k–Sx∗r+ak
Tx∗k–Tx∗r,Sx∗k–Sx∗r
+ ( –ak)
uk,Sx∗k–Sx∗r
– ( –ar)
ur,Sx∗k–Sxr∗Sx∗k–Sx∗r
≤(ak–ar) w Sx∗k–Sx∗r+ (ak–ar)Tx∗rSx∗k–Sx∗r+akSx∗k–Sx∗r
+ ( –ak) uk Sx∗k–Sx∗r– ( –ar) ur Sx∗k–Sx∗r
≤(ak–ar)Sx∗k–Sx∗r w +Tx∗r+akSx∗k–Sx∗r
+( –ak) uk – ( –ar) ur Sx∗k–Sx∗r.
Then we obtain
( –ak)Sx∗k–Sx∗r
≤
(ak–ar)
w +Tx∗rSx∗k–Sx∗r
+( –ak) uk – ( –ar) ur Sx∗k–Sx∗r.
Then
Sx∗k–Sx∗r≤ak–ar –ak
w +Tx∗rSx∗k–Sx∗r
+
–ak
–ak
uk –
–ar
–ak
ur
Sx∗k–Sx∗r
≤ak–ar
–ak
(d)Sx∗k–Sx∗r+
uk –
–ar
–ak
ur
Sx∗k–Sx∗r,
which implies that
Sx∗
k–Sx∗r≤
ak–ar
–ak
(d) + uk –
–ar
–ak
ur .
Hence,
lim k,r→∞
Sx∗k–Sx∗r≤d lim k,r→∞
ak–ar
–ak
+ lim
k→∞ uk –k,limr→∞
–ar
–ak ·
ur
= .
Thus{Sx∗k}is a Cauchy sequence, and hence there exists{Sx∞∗ } ∈Csuch thatSx∗k→Sx∗∞ ask→ ∞. Therefore, the second part is proved. By continuity,Tx∗k→Tx∗∞ask→ ∞. But Sx∗k–Tx∗k→ ask→ ∞. Hence,x∗∞∈CF(S)∩CF(T). This completes the proof.
( –ak)Suk.Then Tkmaps C into itself and{xk,n}converges strongly to a coincidence point
of S and T.
Proof The proof follows from Theorem . by settingw= ∈C.
Corollary . In Theorem.,let S,T be two nonexpansive self-mappings.Then the same conclusion is obtained.
Proof The proof of this corollary can be followed directly by observing that every
nonex-pansive mapping is a continuous pseudo-contraction.
Remark . If we putuk= in Theorem ., we obtain the result of Moore in []. 3 The equivalence betweenS,T-stabilities
In this section, we give the concept ofS,T-stabilities, then we show thatS,T-stabilities of general doubly Mann and general doubly Ishikawa iterations are equivalent.
Let{Sxk,n}be the doubly general Ishikawa iteration with errors and{Suk,n}be the general
doubly Mann iteration with errors. Let{qk,n},{pk,n} ⊂Ebe such thatq,=p,, and let
(αn)n⊂(, ), (βn)n⊂[, );n∈Nsatisfy () and
Syk,n= ( –βn)Sqk,n+βnTqk,n. ()
We consider the following nonnegative sequences for alln∈N:
k,n:=Sqk,n+– ( –αn)Sqk,n–αnTyk,n+αnvn ()
and
δk,n:=Spk,n+– ( –αn)Spk,n–αnTpk,n+αnvn. ()
LetEbe a normed space andTbe a self-map ofE. Letx,be a point ofE, and assume that xk,n+=f(T,xk,n) is an iteration procedure, involvingT, which yields a sequence{xk,n}of
points fromE. Suppose thatxk,nconverges to a fixed pointx∗ofT. Letξk,nbe an arbitrary
sequence inE, and set
n=ξk,n+–f(T,ξk,n), ∀n∈N.
Definition . Iflimn→∞= ⇒limn→∞ξk,n=p, then the iteration procedurexk,n+= f(T,xk,n) is said to beT-stable with respect toT.
Remark . In practice, such a sequence{ξk,n}could arise in the following way. Letx,be
a point inE. Setxk,n+=f(T,xk,n). Letξ,=x,. Nowx,=f(T,x,). Because of round-ing in the functionT, a new valueξ,approximately equal tox,might be computed to yieldξ,, an approximation off(T,ξ,). This computation is continued to obtain{ξk,n}an
approximate sequence of{xk,n}.
(i) Iflimk,n→∞k,n= implies thatlimk,n→∞Sqk,n=Sx∗, then the general Ishikawa
iteration as defined in () and () is said to beS,T-stable.
(ii) Iflimk,n→∞δk,n= implies thatlimk,n→∞Spk,n=Sx∗, then the general Mann
iteration process as defined in () is said to beS,T-stable.
Remark . LetEbe a normed space andS,T:E→E. The following are equivalent:
(a) for all{αn} ⊂(, ),{βn} ⊂[, )satisfying (), the Ishikawa iteration isS,T-stable,
(b) for all{αn} ⊂(, ),{βn} ⊂[, )satisfying (),∀{qk,n} ⊂E,
lim
k,n→∞k,n=k,limn→∞
Sqk,n+– ( –αn)Sqk,n–αnTyk,n+αnvn=
⇒ lim
k,n→∞Sqk,n=Sx
∗. ()
Remark . LetEbe a normed space andS,T:E→E. Then the following are equivalent:
(a) for all{αn} ⊂(, )satisfying (), the general Mann iteration isS,T-stable,
(a) for all{αn} ⊂(, )satisfying (),∀{pk,n} ⊂E,
lim
k,n→∞δk,n=k,limn→∞Spk,n+– ( –αn)Spk,n–αnTpk,n+αnvn=
⇒ lim
k,n→∞Spk,n=Sx
∗. ()
The next result states that these two methods of iterations with errors are equivalent from theS,T-stability point of view under certain restrictions.
Theorem . Let E be a normed space and S,T:E→E.Then the following are equiva-lent:
(I) For all{αn} ⊂(, ),{βn} ⊂[, )satisfying(),the general Ishikawa iteration process as defined by()and()isS,T-stable.
(II) For all{αn} ⊂(, ),satisfying(),the general Mann iteration process as defined in
()isS,T-stable.
Proof Let
M:=max
sup k,n∈N
T(yk,n),sup k,n∈N
T(qk,n), sup k,n∈N
T(pk,n),sup n∈N
un
.
Since the general Mann and general Ishikawa iterations converge andM<∞, Remarks . and . assure that (I)⇔(II) is equivalent to (b)⇔(a). We shall prove that (b)⇒(a).
In (b) and () setSqk,n:=Spk,n, we obtain
Spk,n+– ( –αn)Spk,n–αnTpk,n+αnun
≤Spk,n+– ( –αn)Spk,n–αnTyk,n+ αnTyk,n–αnTpn+αnun ≤Spk,n+– ( –αn)Spk,n–αnTyk,n+αn
Tyk,n + Tpk,n + un
≤Spk,n+– ( –αn)Spk,n–αnTyk,n+ αnM→ asn→ ∞. ()
Condition (b) assures that
lim
Thus, for{Spk,n}satisfying
lim k,n→∞
Spk,n+– ( –αn)Spk,n–αnTyk,n+αnun= ,
we have shown that
lim
k,n→∞Spk,n=Sx ∗.
Conversely, we prove (a)⇒(b). In (a) and () setSpk,n=Sqk,nto obtain
Sqk,n+– ( –αn)Sqk,n–αnTyk,n+αnun
≤Sqk,n+– ( –αn)Sqk,n–αnTsk,n+ αnTyk,n–αnTsn+αnun
≤Sqk,n+– ( –αn)Sqk,n–αnTSqk,n+ αnM→ asn→ ∞. ()
Condition (a) assures that
lim k,n→∞
Sqk,n+– ( –αn)Sqk,n–αnTSqk,n+αnun= ⇒ lim
k,n→∞Sqk,n=Sx
∗.
Thus, for{Sqk,n}satisfying
lim
k,n→∞Sqk,n+– ( –αn)Sqk,n–αnTyk,n+αnun= ,
we have shown that
lim
k,n→∞Sqk,n=Sx ∗.
This completes the proof of the theorem.
Corollary . Let E be a normed space and S,T:E→E.Then the following are equiva-lent:
(i) For all{αn} ⊂(, ),{βn} ⊂[, )satisfying(),the Ishikawa iteration process defined by
xk,n+= ( –αn)xk,n+αnTzk,n+αnvn,
zk,n= ( –βn)xk,n+βnTxk,n+βnwn
isT-stable.
(ii) For all{αn} ⊂(, ),satisfying(),the Mann iteration process defined by
uk,n+= ( –αn)uk,n+αnTuk,n+αnun ()
isT-stable.
Proof The proof of this result can be obtained directly by settingS=I in Theorem .,
4 Application
In this section, we investigate the solvability of certain nonlinear functional equations in a Banach spaceXby the help of compatible mappings of type (B) in the double-sequence setting.
The concept of compatible mappings of type (B) was introduced by Pathak and Khan (see []).
Definition .(see [] and []) LetSandT be mappings from a normed spaceEinto itself. The mappingsSandTare said to be compatible mappings of type (B) if
lim
n→∞ STxn–TTxn ≤
lim
n→∞ STxn–St +nlim→∞ St–SSxn
and
lim
n→∞ TSxn–SSxn ≤
lim
n→∞ TSxn–Tt +nlim→∞ Tt–TTxn
whenever{xn}is a sequence inEsuch thatlimn→∞Sxn=limn→∞Txn=tfor somet∈E.
Now, we extend the above definition to double-sequence setting as follows.
Definition . LetSandTbe mappings from a normed spaceEinto itself. The mappings SandTare said to be compatible mappings of type (B) if
lim
n,m→∞ STxn,m–TTxn,m ≤
lim
n,m→∞ STxn,m–St +n,mlim→∞ St–SSxn,m
and
lim
n,m→∞ TSxn,m–SSxn,m ≤
lim
n,m→∞ TSxn,m–Tt +n,mlim→∞ Tt–TTxn,m
whenever{xn,m}is a sequence inEsuch thatlimn,m→∞Sxn,m=limn,m→∞Txn,m=tfor some
t∈E.
Now, we state and prove the following result.
Theorem . Let {fn,m}, {gn,m}, {tn,m} and{rn,m} be sequences of elements in a Banach
space X.Let{νn,m}be the unique solution of the system of equations
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
Fν–ABν=fn,m,
Fν–BBν=gn,m,
Fν–STν=tn,m,
Fν–TTν=rn,m,
where F,A,B,S,T:X→X satisfy the following conditions:
(d) The pairs{A,S}and{B,T}are compatible of type(B),
(d)
Ax–By ≤qmax
Sx–Ty , Sx–Ax , Sx–Ax × Ty–By ,
Ty–Ax × Sx–By ,
Ty–Ax + Sx–By
for allx,y∈X,whereq∈(, ).IfFν=νand
lim
n,m→∞ fn,m =n,mlim→∞ gn,m =n,mlim→∞ tn,m =n,mlim→∞ rn,m = ,
then the sequence{νn,m}converges to the solution of the equation
ν=Fν=Aν=Bν=Sν=Tν.
Proof We will show that{νn,m}is a Cauchy sequence. Since
νn,m–νn,m
= νn,m–STνn,m + STνn,m–TTνn,m + TTνn,m–ABνn,m
+ ABνn,m–BBνn,m + BBνn,m–νn,m
≤ νn,m–STνn,m + STνn,m–TTνn,m + TTνn,m–ABνn,m
+ BBνn,m–νn,m
+ νn,m–STνn,m + STνn,m–TTνn,m
+ TTνn,m–ABνn,m + BBνn,m–νn,m
ABνn,m–νn,m
+ νn,m–νn,m + νn,m–BBνn,m
+ ABνn,m–BBνn,m
≤ νn,m–STνn,m + STνn,m–TTνn,m + TTνn,m–ABνn,m
+ BBνn,m–νn,m
+ νn,m–STνn,m + STνn,m–TTνn,m
+ TTνn,m–ABνn,m + BBνn,m–νn,m
ABνn,m–νn,m
+ νn,m–νn,m + νn,m–BBνn,m
+qmax
SBνn,m–TBνn,m
,
SBνn,m–ABνn,m , SBνn,m–ABνn,m × TBνn,m–BBνn,m ,
TBνn,m–ABνn,m × SBνn,m–BBνn,m ,
TBνn,m–ABνn,m
+ SBνn,m–BBνn,m
.
Lettingn,n→ ∞withm>nandm>n, we deduce
lim n,n→∞
νn,m–νn,m
≤q lim
n,n→∞
νn,m–νn,m
,
which implies that
lim n,n→∞
νn,m–νn,m
Thus{νn,m}is a Cauchy sequence and converges to a pointνinX. Further,
ν–ABν ≤ ν–νn,m + νn–BBνn,m + BBνn,m–ABν
≤ ν–νn,m + νn,m–BBνn,m +qmax
SBνn,m–TBν ,
SBν–ABν , SBν–ABν × TBνn,m–BBνn,m , TBνn,m–ABν
× SBν–BBνn,m ,
TBνn–ABν
+ SBν–BBν
n
≤ ν–νn,m + νn,m–BBνn,m +qmax
SBνn,m–νn,m + νn,m–ν
× SBν–ν + ν–ABν , SBν–ABν × TBνn–BBνn,m ,
TBνn,m–νn,m + νn,m–ABν
× SBν–ν + ν–BBνn,m
,
TBνn,m–ABν + SBν–BBνn,m
.
Lettingn→ ∞, we getν=ABν, which from (d) implies thatAν=Tν. Similarly,Tν=Sν. From (d), we now have
ABν=BAν=ν=SBν=BSν=TBν=BTν.
Using (i) and (d), we have
v–Bv =Av–Bv
≤qmax
SAv–Tv ,SAv–Av,SAv–Av× Tv–Bv ,
Tv–Av× SAv–Bv ,
Tv–A
v
+ SAv–Bv
≤qmax
v–Tv , , , Tv–v × v–Bv , Tv–v
+ v–Bv
≤qmax v–Tv , , , v–Tv , Tv–v ,
which implies thatν=Tν. It follows that
Tν=TSν=STν=ν=ABν=BAν=BTν=TBν,
completing the proof of the theorem.
As a consequence of Theorem ., we have the following corollary.
Corollary . Let{fn,m}, {gn,m},{tn,m}and{rn,m} be sequences of elements in a Banach
space X.Let{νn,m}be the unique solution of the system of equations
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
ν–ABν=fn,m,
ν–BBν=gn,m,
ν–STν=tn,m,
where A,B,S,T:X→X satisfy the following conditions:
(d) The pairs{A,S}and{B,T}are compatible of type(B),
(d) A=B=S=T=I,whereIdenotes the identity mapping,and
(d)
Ax–By ≤qmax
Sx–Ty , Sx–Ax , Sx–Ax × Ty–By ,
Ty–Ax × Sx–By ,
Ty–Ax + Sx–By
for allx,y∈X,whereq∈(, ).If
lim
n,m→∞ fn,m =n,mlim→∞ gn,m =n,mlim→∞ tn,m =n,mlim→∞ rn,m = ,
then the sequence{νn,m}converges to the solution of the equation
ν=Aν=Bν=Sν=Tν.
Proof The proof can be obtained by puttingF=Iin Theorem ., whereIdenotes the
identity mapping.
Open problem It is still an open problem to extend some defined iterative schemes in the sense of double-sequence setting. For some recent studies on various iterative schemes, we refer to [, –] and others.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Faculty of Science, Sohag University, Sohag, 82524, Egypt.2Mathematics Department,
Faculty of Science, Taif University, El-Hawiyah, P.O. Box 888, El-Taif, 5700, Saudi Arabia.3Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt.4Department of Mathematics, University College, Umm Al-Qura University, Mecca, Saudi Arabia.
Acknowledgements
The authors would like to thank Scientific Research Deanship at Umm Al-Qura University (Project ID 43305020) for the financial support.
Received: 9 February 2014 Accepted: 15 April 2014 Published:16 May 2014
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10.1186/1687-1812-2014-121
