On Multistep Iterative Scheme for Approximating the Common Fixed Points of Contractive Like Operators

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Volume 2010, Article ID 530964,11pages doi:10.1155/2010/530964

Research Article

On Multistep Iterative Scheme for

Approximating the Common Fixed Points of

Contractive-Like Operators

J. O. Olaleru and H. Akewe

Mathematics Department, University of Lagos, Lagos, Nigeria

Correspondence should be addressed to J. O. Olaleru,[email protected]

Received 11 November 2009; Revised 15 January 2010; Accepted 22 January 2010

Academic Editor: Manfred H. Moller

Copyrightq2010 J. O. Olaleru and H. Akewe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the Jungck-multistep iteration and show that it converges strongly to the unique common fixed point of a pair of weakly compatible generalized contractive-like operators defined on a Banach space. As corollaries, the results show that the Jungck-Mann, Jungck-Ishikawa, and Jungck-Noor iterations can also be used to approximate the common fixed points of such maps. The results are improvements, generalizations, and extensions of the work of Olatinwo and Imoru

2008, Olatinwo2008. Consequently, several results in literature are generalized.

1. Introduction

The convergence of Picard, Mann, Ishikawa, Noor and multistep iterations have been commonly used to approximate the fixed points of several classes of single quasicontractive operators, for example, see1–6.

LetXbe a Banach space,K, a nonempty convex subset ofXandT :K → Ka self-map ofK.

Definition 1.1. Letz0∈K. The Picard iteration scheme{zn}∞n0is defined by

zn1Tzn, n≥0. 1.1

Definition 1.2. For any givenu0∈K, the Mann iteration scheme7{un}∞n0is defined by

un1 1−αnunαnTun, 1.2

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Definition 1.3. Letx0∈K. The Ishikawa iteration scheme8{xn}∞_n0is defined by

xn1 1−αnxnαnTyn,

yn

1−βn

xnβnTxn,

1.3

where{αn}∞_n0,{βn}_n0∞ are real sequences in0,1such that∞_n0αn∞.

Observe that ifβn 0 for eachn, then the Ishikawa iteration process1.3reduces to

the Mann iteration scheme1.2.

Definition 1.4. Letx0∈K. The Noor iterationor three-stepscheme9{xn}∞n0is defined by

xn1 1−αnxnαnTyn,

yn

1−βn

xnβnTzn,

zn

1−γn

xnγnTxn,

1.4

where{αn}∞n0,{βn}∞_n0,{γn}∞_n0are real sequences in0,1such that _∞

n0αn ∞.

For motivation and the advantage of using Noor’s iteration, see5,9,10.

Observe that ifγn 0 for eachn, then the Noor iteration process1.4reduces to the

Ishikawa iteration scheme1.3.

Definition 1.5. Letx0∈K. The multistep iteration scheme11{xn}∞_n0is defined by

xn1 1−αnxnαnTyn1,

yin

1−βin

xnβinTyni1, i1,2, . . . , k−2,

ykn−1

1−βkn−1

xnβkn−1Txn, k≥2,

1.5

where{αn}∞_n0,{βin}, i1,2, . . . , k−1, are real sequences in0,1such that _∞

n0αn∞.

Observe that the multistep iteration is a generalization of the Noor, Ishikawa, and the Mann iterations. In fact, ifk1 in1.5, we have the Mann iteration1.2, ifk2 in1.5, we have the Ishikawa iteration1.3, and ifk3, we have the Noor iterations1.4.

We note that while many authors have worked on the existence of fixed points for a pair of quasicontractive maps, for example, see 1, 12–15, little is known about the approximations of those common fixed points using the convergence of iteration techniques. Jungck was the first to introduce an iteration scheme, which is now called Jungck iteration scheme 13 to approximate the common fixed points of what is now called Jungck contraction maps. Singh et al.15of recent introduced the Jungck-Mann iteration procedure and discussed its stability for a pair of contractive maps. Olatinwo and Imoru16, Olatinwo

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that one of each of the pairs of maps is injective. However, a coincidence point for a pair of quasicontractive maps needs not to be a common fixed point. We introduce the Jungck-multistep iteration and show that its convergence can be used to approximate the common fixed points of those pairs of quasicontractive maps without assuming the injectivity of any of the operators. Hence the iterative sequence used is a generalization of that used in16–18. The fact that the injectivity of any of the maps is not assumed in our results and the common fixed points of those maps are approximated and not just the coincidence points make the corollary of our results an improvement of the results of Olaleru19, Olatinwo and Imoru

16. Consequently, a lot of results dealing with convergence of Picard, Mann, Ishikawa, and multistep iterations for single quasicontractive operators on Banach spaces are generalized.

2. Preliminaries

LetXbe a Banach space,Y an arbitrary set, andS, T:Y → Xsuch thatTY⊆SY. Then we have the following definitions.

Definition 2.1see13. For anyxo∈Y, there exists a sequence{xn}∞n0∈Y such thatSxn1

Txn. The Jungck iteration is defined as the sequence{Sxn}∞n1such that

Sxn1Txn, n≥0. 2.1

This procedure becomes Picard iteration whenY XandSId, whereIdis the identity map

onX.

Similarly, the Jungck contraction maps are the mapsS, Tsatisfying

dTx, Ty≤kdSx, Sy, 0≤k <1 ∀x, y∈Y. 2.2

IfY XandSId, then maps satisfying2.2become the well-known contraction maps.

Definition 2.2see15. For any givenuo∈Y, the Jungck-Mann iteration scheme{Sun}∞_n1is

defined by

Sun1 1−αnSunαnTun, 2.3

where{αn}∞n0are real sequences in0,1such that∞n0αn∞.

Definition 2.3see18. Letxo∈Y. The Jungck-Ishikawa iteration scheme{Sxn}∞n1is defined

by

Sxn1 1−αnSxnαnTyn,

Syn

1−βn

SxnβnTxn,

2.4

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Definition 2.4 see 18. Let xo ∈ Y. The Jungck-Noor iteration or three-step scheme {Sxn}∞n1is defined by

Sxn1 1−αnSxnαnTyn,

Syn

1−βn

SxnβnTzn,

Szn

1−γn

SxnγnTxn,

2.5

where{αn}∞n0,{βn}∞_n0, and{γn}∞_n0are real sequences in0,1such that _∞

n0αn ∞.

Definition 2.5. Letxo∈Y. The Jungck-multistep iteration scheme{Sxn}∞_n1is defined by

Sxn1 1−αnSxnαnTyn1,

Syin

1−βin

SxnβniTyi1n , i1,2, . . . k−2,

Sykn−1

1−βkn−1

Sxnβkn−1Txn, k≥2,

2.6

where{αn}∞_n0,{βin}, i1,2, . . . , k−1, are real sequences in0,1such that _∞

n0αn∞.

Observe that the Jungck-multistep iteration is a generalization of the Jungck-Noor, Jungck-Ishikawa and the Jungck-Mann iterations. In fact, ifk1 in2.6, we have the Jungck-Mann iteration2.3, ifk2 in2.6, we have the Jungck-Ishikawa iteration2.4and ifk3, we have the Jungck-Noor iterations2.5.

Observe that if X Y and S Id, then the Jungck-multistep 2.6, Jungck-Noor

2.5, Jungck-Ishikawa2.4, and the Jungck-Mann2.3iterations, respectively, become the multistep1.5, Noor1.4, Ishikawa1.3, and the Mann1.2iterative procedures.

One of the most general contractive-like operators which has been studied by several authors is the Zamfirescu operators.

Suppose thatXis a Banach space. The mapT :X → Xis called a Zamfirescu operator if

_Tx₋_Ty_≤_hmax

_x₋_y_, x−Tx y−Ty

2 ,

_x₋_Ty_d_y₋_Tx

2 , 2.7

where 0≤h <1 see6.

It is known that the operators satisfying2.7are generalizations of Kannan maps4

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subject of research as to conditions under which the Mann iteration will converge faster than the Ishikawa or vice-versa when dealing with the Zamfirescu operators.

We now consider the following conditions.Xis a Banach space andYa nonempty set such thatTY⊆SYandS, T :Y → X. Forx, y∈Y andh∈0,1:

_Sx₋_Sy_, Sx−Tx Sy−Ty

2 ,

_Sx₋_Ty_Sy₋_Tx

2 , 2.8

2 ,Sx−Ty,Sy−Tx , 2.9

_Tx₋_Ty_≤_δ_Sx₋_Sy_L _Sx₋_Tx _, _{L >}0, 0< δ <1, 2.10 _Tx₋_Ty_≤ δSx−Syϕ Sx−Tx

1M Sx−Tx , 0≤δ <1, M≥0, 2.11

_Tx₋_Ty_≤_δ_Sx₋_Sy_ϕ _Sx₋_Tx _, 0≤δ <1. 2.12

whereϕ:R → Ris a monotone increasing sequence withϕ0 0.

Remark 2.6. Observe that ifX Y andS Id,2.8is the same as the Zamfirescu operator

2.7already studied by several authors;2.9becomes the operator studied by Rhoades22; while2.10becomes the operator introduced by Osilike23. Operators satisfying2.11and

2.12were introduced by Olatinwo16.

A comparison of the four maps show the following.

Proposition 2.7. 2.8⇒2.9⇒2.10⇒2.11⇒2.12but the converses are not true.

Proof. 2.8⇒2.9: This follows immediately since _Sx₋_Ty_Sy₋_Tx

2 ≤maxSx−Ty,Sy−Tx. 2.13

2.9⇒2.10: We consider each of the possibilities.

Case 1. Suppose Tx−Ty ≤ h Sx−Ty ≤ h Sx−Tx h Tx−Ty and consequently, Tx−Ty ≤h/1−hSx−Tx. SettingLh/1−hcompletes the proof.

Case 2. Suppose

_Tx₋_Ty_≤_h Sx−Tx Sy−Ty 2

≤h Sx−Tx Sy−SxSx−TxTx−Ty 2

≤h Sx−Tx h

2Sy−Sx h

2Tx−Ty.

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After computing we have Tx−Ty ≤ h/2−h Sy−Sx 2h/2−h Sx−Tx . Setting δh/2−handL2h/2−hcompletes the proof.

Case 3. Tx−Ty ≤h Sy−Tx ≤h Sy−Sx h Sx−Tx .

2.10⇒2.11: SupposeM0 andϕt Ltin2.11, we have2.10.

2.11⇒2.12: This follows from the fact that

_Tx₋_Ty_≤ δSx−SyϕSy−Tx

1MSy−Tx ≤δSx−SyϕSy−Tx. 2.15

We need the following definition.

Definition 2.8see1. A pointx∈Xis called a coincident point of a pair of self-mapsS, Tif there exists a pointwcalled a point of coincidenceinXsuch thatw SxTx. Self-maps SandTare said to be weakly compatible if they commute at their coincidence points, that is, ifSxTxfor somex∈X, thenSTxTSx.

Olatinwo and Imoru16proved that the Jungck-Mann and Jungck-Ishikawa converge to thecoincident pointofS, Tdefined by2.8whenSis an injective operator. It was shown in19that the Jungck-Ishikawa iteration converges to the coincidence point ofS, Tdefined by2.12whenSis an injective operator while the same convergence result was proved for Jungck-Noor whenS, T are defined by2.11 18.We note that the maps satisfying2.9

and of course 2.10–2.12 need not have a coincidence point 15. We rather prove the convergence of multistep iteration to the unique common fixed point of S, T defined by

2.12, without assuming thatSis injective, provided the coincident point exist forS, T.

3. Main Results

The following lemma is well known.

Lemma 3.1. Let{an}be a sequence of nonnegative numbers such thatan1 ≤1−λnanfor anyn,

whereλn∈0,1and _∞

n0λn∞. Then{an}converges to zero.

Theorem 3.2. LetX be a Banach space andS, T :Y → Xfor an arbitrary setY such that2.12

holds andTY⊆SY. Assume thatSandThave a coincidence pointzsuch thatTzSzp. For anyxo∈Y, the Jungck-multistep iteration2.6{Sxn}∞n1converges top.

Further, ifY XandS, T commute atpi.e.,SandT are weakly compatible, thenp is the unique common fixed point ofS, T.

Proof. In view of2.6and2.12coupled with the fact thatTzSzp, we have

_Sx_n1₋_p_≤1−αnSxn−pαnTz−Tyn1

≤1−αnSxn−pαn

δSz−Sy1

nϕ Sz−Tz

1−αnSxn−pδαnp−Syn1.

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An application of2.6and2.12gives

Sy1 n−p≤

1−β1

n

Sxn−pβ1nTz−Ty2n

≤1−β1n

Sxn−pβ1n

δSz−Syn2ϕ Sz−Tz

.

3.2

Substituting3.2in3.1, we have

_Sx_n1₋_p_≤₁₋_α_n_Sx_n₋_p_δα_n₁₋_β1 n

Sxn−pδ2αnβ1nSy2n−p

1−1−δαn−δαnβ1n

Sxn−pδ2αnβn1Syn2−p.

3.3

Similarly, an application of2.6and2.12give

Syn2−p ≤

1−βn2

Sxn−p δβ2n Syn3−p . 3.4

Substituting3.4in3.3we have

_Sx_n1₋_p_≤₁₋₁₋_δ_α_n₋_δα_n_β1 n

Sxn−p

δ2αnβ1n

1−βn2

Sxn−pδ3αnβn1β2nSy3n−p

1−1−δαn−1−δδαnβ1n−δ2αnβ1nβ2n

Sxn−p

δ3αnβ1nβn2Syn3−p.

3.5

Similarly, an application of2.6and2.12gives

Syn3−p ≤

1−βn3

Sxn−p δβ3n Syn4−p . 3.6

Substituting3.6in3.5we have

_Sx_n1₋_p_≤1−1−δαn−1−δδαnβ1n−δ2αnβ1nβ2n

Sxn−p

δ3αnβn1βn2

1−β3n

Sxn−pδ4αnβ1nβ2nβ3nSy4n−p

1−1−δαn−1−δδαnβ1n−1−δδ2αnβ1nβ2n−δ3αnβn1βn2β3n

×Sxn−pδ4αnβ1nβ2nβ3nSy4n−p

≤1−1−δαn−δ3αnβ1nβ2nβ3n

Sxn−pδ4αnβ1nβn2βn3Syn4−p.

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Continuing the above process we have

_Sx_n1₋_p_≤₁₋₁₋_δ_α_n₋_δk−2_α

nβ1nβn2βn3. . . βnk−2

Sxn−p

δk−1αnβ1nβ2nβ3n. . . βkn−2Synk−1−p

≤1−1−δαn−δk−2αnβ1nβn2βn3. . . βnk−2

Sxn−p

δk−1αnβ1nβ2nβ3n. . . βkn−2

1−βkn−1

Sxn−pβkn−1 Tz−Txn

Sxn−p

δk−1αnβ1nβ2nβ3n. . . βkn−2

1−βkn−1

Sxn−pδβnk−1Sxn−p

δk−1αnβ1nβ2nβ3n. . . βkn−2Sxn−p ≤1−1−δαnSxn−p.

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Hence byLemma 3.1Sxn → p.

Next we show thatpis unique. Suppose there exists another point of coincidencep∗. Then there is anz∗∈Xsuch thatTz∗Sz∗p∗. Hence, from2.12we have

_p₋_p∗ _Tz₋_Tz∗_≤_δ _Sz₋_Sz∗ _ϕ _Sz₋_Tz _δ_p₋_p∗_. _3.9

Sinceδ <1, thenpp∗and sopis unique.

SinceS, T are weakly compatible, then TSz STz and soTp Sp. Hencep is a coincidence point ofS, T and since the coincidence point is unique, thenp p∗ and hence Sp Tp p and therefore p is the unique common fixed point of S, T and the proof is complete.

Remark 3.3. Weaker versions ofTheorem 3.2are the results in16,18whereSis assumed injective and the convergence is not to the common fixed point but to the coincidence point ofS, T. Furthermore, the Jungck-multistep iteration used inTheorem 3.2is more general than the Jungck-Ishikawa and the Jungck-Noor iteration used in17,18.

It is already shown in1,20that ifSYorTYis a complete subspace ofX, then maps satisfying the generalized Zamfirescu operators2.8have a unique coincidence point. Hence we have the following results.

Theorem 3.4. LetXbe a Banach space andS, T :X → Xsuch that

2 ,

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andTX⊆SX. Assume thatSandTare weakly compatible. For anyxo∈X, the Jungck-multistep

iteration2.6{Sxn}∞n1converges to the unique common fixed point ofS, T.

Since the Jungck-Noor, Jungck-Ishikawa and Jungck-Mann iterations are special cases of Jungck-multistep iteration, then we have the following consequences.

Corollary 3.5. LetXbe a Banach space andS, T:X → Xsuch that

_Tx₋_Ty_≤_h_max_Sx₋_Sy_, Sx−Tx Sy−Ty

2 ,

2 3.11

andTX⊆SX. AssumeSandTare weakly compatible. For anyxo∈X, the Jungck-Noor iteration

2.5{Sxn}∞n1converges to the unique common fixed point ofS, T.

2 ,

2 3.12

andTX⊆SX. Assume thatSandTare weakly compatible. For anyxo∈X, the Jungck-Ishikawa

Remark 3.7. i A weaker version of Corollary 3.6 is the main result of 16 where the convergence is to the coincidence point ofS, TandSis assumed injective.

iiIfSIdinCorollary 3.5, then we have the main result of2.

2 ,

2 , 3.13

andTX⊆SX. Assume thatSandT are weakly compatible. For anyxo∈X, the Jungck-Mann

Remark 3.9. IfSId,Corollary 3.8gives the result of20.

It is already shown in1,2that ifSYorTYis a complete subspace ofX, then maps satisfying the operators2.9has a unique coincidence point. Hence we have the following results.

Theorem 3.10. LetXbe a Banach space space andS, T:X → Xsuch that

2 ,

2 3.14

andTX⊆SX. Assume thatSandTare weakly compatible. For anyxo∈X, the Jungck-multistep

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Since the Jungck-Noor, Jungck-Ishikawa, and Jungck-Mann iterations are special cases of Jungck-multistep iteration, then we have the following consequences.

Corollary 3.11. LetXbe a Banach space andS, T :X → Xsuch that

2 ,Sx−TySy−Tx , 3.15

andTX⊆ SX. Assume thatSandT are weakly compatible. For anyxo ∈ X, the Jungck-Noor

iteration2.5{Sxn}∞_n1converges to the unique common fixed point ofS, T.

2 ,Sx−TySy−Tx , 3.16

andTX⊆SX. Assume thatSandTare weakly compatible. For anyxo∈X, the Jungck-Ishikawa

2 ,Sx−TySy−Tx , 3.17

andTX ⊆SX. Assume thatSandT are weakly compatible. For anyxo ∈X, the Jungck-Mann

Acknowledgments

The research is supported by the African Mathematics Millennium Science Initiative

AMMSI. The first author is grateful to the Hampton University, Virginia, USA, for hospitality.

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