Weak Contractions, Common Fixed Points, and Invariant Approximations

Volume 2009, Article ID 390634,10pages doi:10.1155/2009/390634

Research Article

Weak Contractions, Common Fixed Points, and Invariant Approximations

Nawab Hussain

and Yeol Je Cho

1Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

2Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, South Korea

Correspondence should be addressed to Yeol Je Cho,[email protected] Received 19 January 2009; Accepted 23 February 2009

Recommended by Charles E. Chidume

The existence of common fixed points is established for the mappings, whereT isf, θ, L-weak contraction on a nonempty subset of a Banach space. As application, some results on the invariant best approximation are proved. Our results unify and substantially improve several recent results given by some authors.

Copyrightq2009 N. Hussain and Y. J. Cho. 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.

1. Introduction and Preliminaries

LetMbe a subset of a normed spaceX, · . The set PMu

x∈M:x−udistu, M

1.1

is called the set of best approximants tou∈Xout ofM,where distu, M inf

y−u:y∈M

. 1.2

We denoteNand clM resp., wclMby the set of positive integers and the closureresp., weak closureof a setMinX, respectively. Letf, T :M → Mbe mappings. The set of fixed points ofT is denoted byFT. A pointx∈ Mis a coincidence pointresp., common fixed pointoffandT iffxTxresp.,xfxTx. The set of coincidence points offandT is denoted byCf, T.

The pair{f, T}is said to be

1commuting1ifTfxfTxfor allx∈M,

2compatible2,3if lim_{n→ ∞}Tfxn−fTx_n0 whenever{xn}is a sequence such that lim_{n→ ∞}Tx_nlim_{n→ ∞}fx_ntfor sometinM,

3weakly compatible if they commute at their coincidence points, that is, iffTxTfx wheneverfxTx,

4a Banach operator pair if the setFfisT-invariant, namely,TFf⊆Ff.

Obviously, the commuting pairT, fis a Banach operator pair, but converse is not true in generalsee4,5.IfT, fis a Banach operator pair, then f, Tneeds not be a Banach operator pairsee4, Example 1.

The setMis said to beq-starshaped withq∈Mif the segmentq, x {1−kqkx: 0 ≤ k ≤ 1} joiningq tox is contained inMfor all x ∈ M.The mapping f defined on a q-starshaped setMis said to be affine if

f

1−kqkx

1−kfqkfx, ∀x∈M. 1.3

Suppose that the setMisq-starshaped withq∈Ffand is bothT- andf-invariant.

ThenT andfare said to be

5Cq-commuting3,6iffTxTfxfor allx∈Cqf, T, whereCqf, T ∪{Cf, Tk: 0≤k≤1}whereT_kx 1−kqkTx,

6pointwiseR-subweakly commuting7if, for givenx∈M,there exists a real number R >0 such thatfTx−Tfx ≤Rdistfx,q, Tx,

7R-subweakly commuting onM8if, for allx∈M,there exists a real numberR >0 such thatfTx−Tfx ≤Rdistfx,q, Tx.

In 1963, Meinardus 9 employed Schauder’s fixed point theorem to prove a result regarding invariant approximation. Further, some generalizations of the result of Meinardus were obtained by Habiniak10, Jungck and Sessa11, and Singh12.

Since then, Al-Thagafi13extended these works and proved some results on invariant approximations for commuting mappings. Hussain and Jungck8, Hussain5, Jungck and Hussain 3, O’Regan and Hussain 7, Pathak and Hussain 14, and Pathak et al. 15 extended the work of Al-Thagafi13for more general noncommuting mappings.

Recently, Chen and Li4introduced the class of Banach operator pairs as a new class of noncommuting mappings and it has been further studied by Hussain5, Khan and Akbar 16, and Pathak and Hussain14.

In this paper, we extend and improve the recent common fixed point and invariant approximation results of Al-Thagafi13, Al-Thagafi and Shahzad17, Berinde18, Chen and Li4, Habiniak10, Jungck and Sessa11, Pathak and Hussain14, and Singh12 to the class off, θ, L-weak contractions. The applications of the fixed point theorems are remarkable in diverse disciplines of mathematics, statistics, engineering, and economics in dealing with the problems arising in approximation theory, potential theory, game theory, theory of differential equations, theory of integral equations, and otherssee14,15,19,20.

2. Main Results

LetX, dbe a metric space. A mappingT :X → Xis called a weak contraction if there exist two constantsθ∈0,1andL≥0 such that

dTx, Ty≤θdx, y Ldy, Tx, ∀x, y∈X. 2.1

Remark 2.1. Due to the symmetry of the distance, the weak contraction condition 2.1 includes the following:

dTx, Ty≤θdx, y Lx, Ty, ∀x, y∈X, 2.2

which is obtained from2.1by formally replacingdTx, Ty,dx, ybydTy, Tx,dy, x, respectively, and then interchangingxandy.

Consequently, in order to check the weak contraction ofT, it is necessary to check both 2.1and2.2. Obviously, a Banach contraction satisfies2.1and hence is a weak contraction.

Some examples of weak contractions are given in18,21,22. The next example shows that a weak contraction needs not to be continuous.

Example 2.2 see18, 22. Let0,1be the unit interval with the usual norm and let T : 0,1 → 0,1 be given byTx 2/3 for all x ∈ 0,1 and T1 0. Then T satisfies the inequality2.1with 1> θ ≥2/3 andL≥θandT has a unique fixed pointx2/3, butT is not continuous.

Letf be a self-mapping onX. A mappingT :X → Xis said to bef-weak contraction orf, θ, L-weak contraction if there exist two constantsθ∈0,1andL≥0 such that

dTx, Ty≤θdfx, fy Ldfy, Tx, ∀x, y∈X. 2.3

Berinde18introduced the notion of aθ, L-weak contraction and proved that a lot of the well-known contractive conditions do imply theθ, L-weak contraction. The concept of θ, L-weak contraction does not ask θL to be less than 1 as happens in many kinds of fixed point theorems for the contractive conditions that involve one or more of the displacements dx, y, dx, Tx, dy, Ty, dx, Ty, dy, Tx. For more details, we refer to 18,21and references cited in these papers.

The following result is a consequence of the main theorem of Berinde18.

Lemma 2.3. LetMbe a nonempty subset of a metric spaceX, dand letTbe a self-mapping ofM.

Assume that clTM⊂M, clTMis complete, andTis aθ, L-weak contraction. ThenM∩FT is nonempty.

Theorem 2.4. LetMbe a nonempty subset of a metric spaceX, dand letT, fbe self-mappings of M.Assume thatFfis nonempty, clTFf⊆Ff, clTMis complete, andTis anf, θ, L- weak contraction. ThenM∩FT∩Ff/∅.

Proof. Since clTFfis a closed subset of clTM, clTFfis complete. Further, by thef, θ, L-weak contraction ofT, for allx, y∈Ff, we have

dTx, Ty≤θdfx, fy L·dfy, Tx dx, y L·dy, Tx. 2.4

HenceTis aθ, L-weak contraction onFfand clTFf⊆Ff. Therefore, byLemma 2.3, Thas a fixed pointzinFfand soM∩FT∩Ff/∅.

Corollary 2.5. Let Mbe a nonempty subset of a metric space X, d and let T, fbe a Banach operator pair onM. Assume that clTMis complete,T isf, θ, L-weak contraction, andFfis nonempty and closed. ThenM∩FT∩Ff/∅.

InTheorem 2.4andCorollary 2.5, ifL 0, then we easily obtain the following result, which improves Lemma 3.1 of Chen and Li4.

Corollary 2.6see17, Theorem 2.2. LetMbe a nonempty subset of a metric spaceX, dand letT, f be self-mappings ofM.Assume thatFfis nonempty, clTFf ⊆ Ff, clTMis complete, andT is anf-contraction. ThenM∩FT∩Ffis a singleton.

The following result properly contains 4, Theorems 3.2-3.3 and improves 13, Theorem 2.2,10, Theorem 4, and11, Theorem 6.

Theorem 2.7. LetMbe a nonempty subset of a normed (resp., Banach) spaceXand letT, fbe self- mappings ofM.Suppose thatFfisq-starshaped, clTFf⊆Ff(resp., wclTFf⊆Ff), clTMis compact (resp., wclTMis weakly compact, and eitherI−T is demiclosed at 0 orX satisfies Opial’s condition, whereIstands for the identity mapping), and there exists a constantL≥0 such that

Tx−Ty ≤ fx−fyL·dist

fy,q, Tx

, ∀x, y∈M. 2.5 ThenM∩FT∩Ff/∅.

Proof. For eachn ∈N, defineTn :Ff → FfbyTnx 1−knqknTxfor allx ∈Ff and a fixed sequence {kn} of real numbers0 < kn < 1converging to 1. Since Ffisq- starshaped and clTFf ⊆ Ff resp., wclTFf ⊆ Ff, we have clT_nFf ⊆ Ff resp., wclTnFf⊆Fffor eachn∈N. Also, by the inequality2.5,

Tnx−TnyknTx−Ty

≤k_nfx−fyk_nL·dist

fy,q, Tx

≤k_nfx−fyL_n·fy−T_nx 2.6 for all x, y ∈ Ff,Ln : knL, and 0 < kn < 1.Thus, for n ∈ N,Tn is af, kn, Ln-weak contraction, whereL_n≥0.

If clTMis compact, then, for eachn∈N, clTnMis compact and hence complete.

By Theorem 2.4, for each n ∈ N, there exists xn ∈ Ff such that xn fxn Tnxn. The compactness of clTMimplies that there exists a subsequence{Txm}of{Txn}such that

Txm → z ∈ clTMasm → ∞. Since{Txm}is a sequence inTFfand clTFf ⊆ Ff, we havez∈Ff. Further, it follows that

xmTmxm 1−km

qkmTxm−→z, xm−Txm−→0 n−→ ∞. 2.7

Moreover, we have

Txm−Tz≤fxm−fzL·dist fz,

q, Txm

x_m−zL· dist

z,

q, Tx_m

≤x_m−Tx_mTx_m−zL·z−Tx_m.

2.8

Taking the limit asm → ∞,we getzTzand soM∩FT∩Ff/∅.

Next, the weak compactness of wclTMimplies that wclTnMis weakly compact and hence complete due to completeness ofXsee3. FromTheorem 2.4, for eachn ∈N, there exists x_n ∈ Ff such that x_n fx_n T_nx_n.The weak compactness of wclTM implies that there is a subsequence{Txm}of{Txn}converging weakly toy∈wclTMas m → ∞. Since{Txm}is a sequence inTFf, we havey∈wclTFf⊆Ff. Also, we havex_m−Tx_m → 0 asm → ∞. IfI−Tis demiclosed at 0, thenyTyand soM∩FT∩ Ff/∅.

Iffy /Ty,then we have

lim inf

m→ ∞ fx_m−fy

<lim inf

m→ ∞ fx_m−Ty

≤lim inf

m→ ∞ fxm−Txmlim inf

m→ ∞Txm−Ty

≤lim inf

m→ ∞ fx_m−Tx_mlim inf

m→ ∞fx_m−fylim inf

m→ ∞ L·dist fy,

q, Tx_m

≤lim inf

m→ ∞ fx_m−fylim inf

m→ ∞ fx_m−Tx_mL·lim inf

m→ ∞y−Tx_m lim inf

m→ ∞ fxm−fy,

2.9

which is a contradiction. ThusTyfyyand henceM∩FT∩Ff/∅.This completes the proof.

Obviously,f-nonexpansive mappings satisfy the inequality2.5and so we obtain the following.

Corollary 2.8see17, Theorem 2.4. LetMbe a nonempty subset of a normed (resp., Banach) spaceX and letT,fbe self-mappings ofM.Suppose thatFfisq-starshaped, clTFf⊆Ff

(resp., wclTFf⊆Ff), clTMis compact, (resp., wclTMis weakly compact, and either I −T is demiclosed at 0 or X satisfies Opial’s condition), and T is f-nonexpansive on M. Then M∩FT∩Ff/∅.

Corollary 2.9see4, Theorems 3.2-3.3. LetMbe a nonempty subset of a normed (resp., Banach) spaceX and letT, f be self-mappings of M.Suppose that Ff isq-starshaped and closed (resp., weakly closed), clTM is compact (resp., wclTM is weakly compact, and either I − T is demiclosed at 0 or X satisfies Opial’s condition), T, f is a Banach operator pair, and T is f- nonexpansive onM. ThenM∩FT∩Ff/∅.

Corollary 2.10see13, Theorem 2.1. LetMbe a nonempty closed andq-starshaped subset of a normed space X and let T,f be self-mappings of M such that TM ⊆ fM. Suppose that T commutes with f and q ∈ Ff. If clTM is compact,f is continuous, linear, and T is f- nonexpansive onM, thenM∩FT∩Ff/∅.

LetCP_Mu ∩ C^f_Mu,whereC^f_Mu {x∈M: fx∈P_Mu}.

Corollary 2.11. LetX be a normed (resp., Banach) spaceX and letT, f be self-mappings ofX. If u∈X,D⊆C,D₀:D∩Ffisq-starshaped, clTD0⊆D₀(resp., wclTD0⊆D₀), clTD is compact, (resp., wclTDis weakly compact, andI−Tis demiclosed at 0). If the inequality2.5 holds for allx, y∈D,thenPMu∩FT∩Ff/∅.

Corollary 2.12. LetX be a normed (resp., Banach) spaceX and letT, f be self-mappings ofX. If u∈ X,D ⊆ PMu,D0 : D∩Ffisq-starshaped, clTD0 ⊆ D0(resp., wclTD0 ⊆ D0), clTD is compact, (resp., wclTD is weakly compact, and I −T is demiclosed at 0). If the inequality2.5holds for allx, y∈D,thenP_Mu∩FT∩Ff/∅.

Corollary 2.13see11, Theorem 7. Letf,T be self-mappings of a Banach space X withu ∈ Ff∩FTandM⊂XwithT∂M⊂M. Suppose thatDPMuis q-starshaped withq∈Ff, fD D, andfis affine, continuous in the weak and strong topology onD. IffandTare commuting onDandT isf-nonexpansive onD∪ {u}, thenP_Mu∩FT∩Ff/∅provided eitheriDis weakly compact andf−Tis demiclosed oriiDis weakly compact andXsatisfies Opial’s condition.

Remark 2.14. Corollary 2.5 in17and Theorems 4.1-4.2 of Chen and Li4are special cases of Corollaries2.11-2.12

We denoteI0by the class of closed convex subsets ofXcontaining 0. For anyM∈I0, we defineM_u{x∈M:x ≤2u}.It is clear thatP_Mu⊂M_u∈I0see8,13.

Theorem 2.15. Letf, T be self-mappings of a normed (resp., Banach) spaceX. Ifu∈XandM∈I0

such thatTMu ⊆ M, clTMuis compact (resp., wclTMuis weakly compact), andTx− u ≤ x−ufor allx∈M_u, thenP_Muis nonempty closed and convex withTPMu⊆P_Mu. If, in addition,D⊆PMu,D0:D∩Ffisq-starshaped, clTD0⊆D0(resp., wclTD0⊆D0, andI−T is demiclosed at 0), and the inequality2.5holds for allx, y ∈D,thenPMu∩FT∩ Ff/∅.

Proof. We may assume thatu/∈M. Ifx∈M\Mu,thenx>2u. Note that

x−u ≥ x − u>u ≥distu, M. 2.10

Thus distu, Mu distu, M ≤ u.If clTMu is compact, then, by the continuity of the norm, we getz−u distu,clTMufor somez ∈ clTMu.If we assume that wclTMuis weakly compact, then, using 23, Lemma 5.5, page 192, we can show the existence of az∈wclTMusuch that distu,wclTMu z−u. Thus, in both cases, we have

dist u, M_u

≤dist

u,clTMu

≤dist

u, TMu

≤ Tx−u ≤ x−u 2.11

for allx ∈ M_u.Hencez−u distu, Mand soP_Muis nonempty closed and convex with TPMu ⊆ P_Mu. The compactness of clTMu resp., the weak compactness of wclTMu implies that clTD is compact resp., wclTD is weakly compact.

Therefore, the result now follows fromCorollary 2.12. This completes the proof.

Corollary 2.16. Letf, Tbe self-mappings of a normed (resp., Banach) spaceX. Ifu∈XandM∈I0

such thatTMu ⊆ M, clTMuis compact (resp., wclTMuis weakly compact), andTx− u ≤ x−ufor allx∈M_u, thenP_Muis nonempty closed and convex withTPMu⊆P_Mu.

If, in addition,D ⊆ P_Mu,D₀ : D∩Ffisq-starshaped and closed (resp., weakly closed and I−T is demiclosed at 0),T, fis a Banach operator pair onD, and the inequality2.5holds for all x, y∈D,thenP_Mu∩FT∩Ff/∅.

Remark 2.17. Theorem 2.15 and Corollary 2.16 extend 13, Theorems 4.1 and 4.2, 17, Theorem 2.6, and10, Theorem 8.

Banach’s Fixed Point Theorem states that ifX, dis a complete metric space,K is a nonempty closed subset ofX, and T : K → K is a self-mapping satisfying the following condition: there existsλ∈0,1such that

dTx, Ty≤λdx, y, ∀x, y∈K, 2.12

thenThas a unique fixed point, sayzinK,and the Picard iterative sequence{Tⁿx}converges to the pointzfor allx∈K.Since then, ´Ciri´c24introduced and studied self-mappings onK satisfying the following condition: there existsλ∈0,1such that

dTx, Ty≤λmx, y, ∀x, y∈K, 2.13

where

mx, y max

dx, y, dx, Tx, dy, Ty, dx, Ty, dy, Tx

. 2.14

Further, many investigations were developed by Berinde19, Jungck1,2, Hussain and Jungck 8, Hussain and Rhoades 6, O’Regan and Hussain 7, and many other mathematicians see 14, 25 and references therein. Recently, Jungck and Hussain 3 proved the following extension of the result of ´Ciri´c24.

Theorem 2.18see3, Theorem 2.1. LetMbe a nonempty subset of a metric spaceX, dand let f, gbe self-mappings ofM. Assume that clfM⊂gM, clfMis complete, andf,gsatisfy the following condition: there existsh∈0,1such that

dfx, fy≤h max

dgx, gy, dfx, gx, dfy, gy, dfx, gy, dfy, gx

, 2.15 for allx, y∈M. ThenCf, g/∅.

The following resultTheorem 2.19properly contains17, Theorem 3.3,4, Theorems 3.2-3.3, 5, Theorem 2.11, and14, Theorem 2.2. The proof is analogous to the proof of Theorem 2.7. In fact, instead of applying Theorem 2.4, we apply Theorem 2.18 to get the conclusion.

Theorem 2.19. LetMbe a nonempty subset of a normed (resp., Banach) spaceXand letT, f, gbe self-mappings ofM.Suppose thatFf∩Fgisq-starshaped, clTFf∩Fg ⊆Ff∩Fg (resp., wclTFf∩Fg ⊆ Ff∩Fg), clTM is compact (resp., wclTMis weakly compact), andTis continuous onM(resp.,I−T is demiclosed at 0). If the following condition holds:

Tx−Ty ≤max

fx−gy,dist

fx,q, Tx ,dist

gy,q, Ty , dist

gy,q, Tx ,dist

fx,q, Ty 2.16 for allx, y∈M,thenM∩FT∩Ff∩Fg/∅.

Theorem 2.20. Letf,g,T be self-mappings of a Banach spaceX withu ∈ FT∩Ff∩Fg and M ∈ I0 such thatTMu ⊂ fM ⊂ M gM. Suppose that fx −u ≤ x−u, gx−ux−u,Tx−u ≤ fx−gufor allx∈M, and clfMuis compact. Then one has the following:

1P_Muis nonempty closed and convex,

2TPMu⊂fPMu⊂PMu gPMu,

3P_Mu ∩ FT ∩ Ff ∩ Fg/∅ provided T is continuous, Fg is q-starshaped, clfFg ⊆ Fg, and the pair f, g satisfies the inequality 2.5 for all x, y ∈ P_Mu, Ff is q-starshaped with q ∈ Ff∩ Fg∩ P_Mu, clTFf ∩Fg ⊆ Ff∩Fg, and the inequality2.16holds for allq∈Ff∩Fgandx, y∈P_Mu.

Proof. 1 and 2 follow from 5,8, Theorem 2.14. By 2, the compactness of clfMu implies that clfPMuand clTPMuis compact.Theorem 2.7implies thatFf∩Fg∩

PMu/∅. Further,Ff∩Fgisq-starshaped withq∈Ff∩Fg∩PMu. Therefore, the conclusion now follows fromTheorem 2.19applied toP_Mu.

Remark 2.21. 1Theorem 2.20extends13, Theorem 4.1,10, Theorem 8,5, Theorem 2.13, 8, Theorem 2.14, and14, Theorem 2.11.

2Theorems2.7–2.16represent very strong variants of the results in3,8,11,13in the sense that the commutativity or compatibility of the mappingsT andfis replaced by the hypothesis thatT, fis a Banach operator pair,f needs not be linear or affine, andT needs not bef-nonexpansive.

3 The Banach operator pairs are different from those of weakly compatible, Cq- commuting and R-subweakly commuting mappings and so our results are different from those in3,7,8,17. ConsiderM R² with the normx, y |x||y|for allx, y ∈M.

Define two self-mappingsT andfonMas follows:

Tx, y x³x−1,

3

x²y³−1 3

, fx, y

x³x−1, ³

x²y³−1 .

2.17

Then we have the following:

FT

1,0

, Ff

1, y:y∈R¹ , CT, f

x, y:y³

1−x², x∈R¹ , T

Ff

T1, y:y∈R¹

1,y

3

:y∈R¹

⊆

1, y:y∈R¹

Ff.

2.18

ThusT, fis a Banach operator pair. It is easy to see thatT isf, θ, L-weak contraction and T,f do not commute on the setCT, f, and so are not weakly compatible. Clearly,f is not affine or linear,Ffis convex and1,0is a common fixed point ofTandf.

References

1 G. Jungck, “Commuting mappings and fixed points,” The American Mathematical Monthly, vol. 83, no.

4, pp. 261–263, 1976.

2 G. Jungck, “Common fixed points for commuting and compatible maps on compacta,” Proceedings of the American Mathematical Society, vol. 103, no. 3, pp. 977–983, 1988.

3 G. Jungck and N. Hussain, “Compatible maps and invariant approximations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1003–1012, 2007.

4 J. Chen and Z. Li, “Common fixed-points for Banach operator pairs in best approximation,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1466–1475, 2007.

5 N. Hussain, “Common fixed points in best approximation for Banach operator pairs with ´Ciri´c type I-contractions,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1351–1363, 2008.

6 N. Hussain and B. E. Rhoades, “Cq-commuting maps and invariant approximations,” Fixed Point Theory and Applications, vol. 2006, Article ID 24543, 9 pages, 2006.

7 D. O’Regan and N. Hussain, “GeneralizedI-contractions and pointwiseR-subweakly commuting maps,” Acta Mathematica Sinica, vol. 23, no. 8, pp. 1505–1508, 2007.

8 N. Hussain and G. Jungck, “Common fixed point and invariant approximation results for noncom- muting generalizedf, g-nonexpansive maps,” Journal of Mathematical Analysis and Applications, vol.

321, no. 2, pp. 851–861, 2006.

9 G. Meinardus, “Invarianz bei linearen approximationen,” Archive for Rational Mechanics and Analysis, vol. 14, pp. 301–303, 1963.

10 L. Habiniak, “Fixed point theorems and invariant approximations,” Journal of Approximation Theory, vol. 56, no. 3, pp. 241–244, 1989.

11 G. Jungck and S. Sessa, “Fixed point theorems in best approximation theory,” Mathematica Japonica, vol. 42, no. 2, pp. 249–252, 1995.

12 S. P. Singh, “An application of a fixed-point theorem to approximation theory,” Journal of Approximation Theory, vol. 25, no. 1, pp. 89–90, 1979.

13 M. A. Al-Thagafi, “Common fixed points and best approximation,” Journal of Approximation Theory, vol. 85, no. 3, pp. 318–323, 1996.

14 H. K. Pathak and N. Hussain, “Common fixed points for Banach operator pairs with applications,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2788–2802, 2008.

15 H. K. Pathak, Y. J. Cho, and S. M. Kang, “An application of fixed point theorems in best approximation theory,” International Journal of Mathematics and Mathematical Sciences, vol. 21, no. 3, pp. 467–470, 1998.

16 A. R. Khan and F. Akbar, “Common fixed points from best simultaneous approximations,” Taiwanese Journal of Mathematics, vol. 13, 2009.

17 M. A. Al-Thagafi and N. Shahzad, “Banach operator pairs, common fixed-points, invariant approximations, and∗-nonexpansive multimaps,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2733–2739, 2008.

18 V. Berinde, “On the approximation of fixed points of weak contractive mappings,” Carpathian Journal of Mathematics, vol. 19, no. 1, pp. 7–22, 2003.

19 V. Berinde, Iterative Approximation of Fixed Points, vol. 1912 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2nd edition, 2007.

20 M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics, Wiley-Interscience, New York, NY, USA, 2001.

21 V. Berinde, “Approximating fixed points of weak contractions using the Picard iteration,” Nonlinear Analysis Forum, vol. 9, no. 1, pp. 43–53, 2004.

22 V. Berinde and M. P˘acurar, “Fixed points and continuity of almost contractions,” Fixed Point Theory, vol. 9, no. 1, pp. 23–34, 2008.

23 S. P. Singh, B. Watson, and P. Srivastava, Fixed Point Theory and Best Approximation: The KKM-Map Principle, vol. 424 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.

24 L. B. ´Ciri´c, “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 45, no. 2, pp. 267–273, 1974.

25 L. B. ´Ciri´c, “Contractive type non-self mappings on metric spaces of hyperbolic type,” Journal of Mathematical Analysis and Applications, vol. 317, no. 1, pp. 28–42, 2006.