Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 390634,10 pages doi:10.1155/2009/390634

Research Article

Weak Contractions, Common Fixed Points, and Invariant Approximations

Nawab Hussain1 and Yeol Je Cho2

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

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

Correspondence should be addressed to YeolJe Cho, yjcho@gsnu.ac.kr

Received 19 January 2009; Accepted 23 February 2009

Recommended by Charles E. Chidume

The existence of common fixed points is established for the mappings, where T is (f, d, 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.

Copyright © 2009 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

Let M be a subset of a normed space (X, || ■ ||). The set

Pm(u) = {x e M : ||x - u|| = dist(u, M)} (1.1)

is called the set of best approximants to u e X out of M, where

dist(u,M) = inf {||y - u|| : y e M}. (1.2)

We denote N and cl(M) (resp., wcl(M)) by the set of positive integers and the closure (resp., weak closure) of a set M in X, respectively. Let f, T : M ^ M be mappings. The set of fixed points of T is denoted by F(T). A point x e M is a coincidence point (resp., common fixed point) of f and T if fx = Tx (resp., x = fx = Tx). The set of coincidence points of f and T is denoted by C(f, T).

The pair {f, T} is said to be

(1) commuting [1] if Tfx = fTx for all x e M,

(2) compatible [2,3] if limn^^\\Tfxn - fTxn\\ = 0 whenever {xn} is a sequence such that limn ^^Txn = limn ^<xfxn = t for some t in M,

(3) weakly compatible if they commute at their coincidence points, that is, if fTx = Tfx whenever fx = Tx,

(4) a Banach operator pair if the set F(f) is T-invariant, namely, T(F(f)) c F(f).

Obviously, the commuting pair (T, f) is a Banach operator pair, but converse is not true in general (see [4, 5].) If (T,f) is a Banach operator pair, then (f,T) needs not be a Banach operator pair (see [4, Example 1]).

The set M is said to be q-starshaped with q e M if the segment [q, x] = {(1 - k)q + kx : 0 < k < 1} joining q to x is contained in M for all x e M. The mapping f defined on a q-starshaped set M is said to be affine if

f ((1 - k)q + kx) =(1 - k)fq + kfx, Vx e M. (1.3)

Suppose that the set M is q-starshaped with q e F (f) and is both T- and f -invariant. Then T and f are said to be

(5) Cq-commuting [3, 6] if fTx = Tfx for all x e Cq (f, T), where Cq (f, T) = U{C(f, Tk) : 0 < k < 1} where Tkx = (1 - k)q + kTx,

(6) pointwise R-subweakly commuting [7] if, for given x e M, there exists a real number R> 0 such that \\fTx - Tfx\\ < R dist(fx, [q, Tx]),

(7) R-subweakly commuting on M [8] if, for all x e M, there exists a real number R> 0 such that \\fTx - Tfx\\ < R dist(fx, [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 Habiniak [10], Jungck and Sessa [11], and Singh [12].

Since then, Al-Thagafi [13] extended these works and proved some results on invariant approximations for commuting mappings. Hussain and Jungck [8] , Hussain [5] , Jungck and Hussain [3], O'Regan and Hussain [7], Pathak and Hussain [14], and Pathak et al. [15] extended the work of Al-Thagafi [13] for more general noncommuting mappings.

Recently, Chen and Li [4] introduced the class of Banach operator pairs as a new class of noncommuting mappings and it has been further studied by Hussain [5], Khan and Akbar [16], and Pathak and Hussain [14].

In this paper, we extend and improve the recent common fixed point and invariant approximation results of Al-Thagafi [13], Al-Thagafi and Shahzad [17], Berinde [18], Chen and Li [4], Habiniak [10], Jungck and Sessa [11], Pathak and Hussain [14], and Singh [12] to the class of (f, 0, 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 others (see [14,15,19, 20]).

Journal of Inequalities and Applications 2. Main Results

Let (X, d) be a metric space. A mapping T : X ^ X is called a weak contraction if there exist two constants 9 e (0,1) and L > 0 such that

d(Tx, Ty) < 9d(x, y) + Ld(y, Tx), Vx, y e X. (2.1)

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

d(Tx, Ty) < 9d(x, y) + L(x, Ty), Vx, y e X, (2.2)

which is obtained from (2.1) by formally replacing d(Tx,Ty), d(x,y) by d(Ty,Tx), d(y,x), respectively, and then interchanging x and y.

Consequently, in order to check the weak contraction of T, it is necessary to check both (2.1) and (2.2). Obviously, a Banach contraction satisfies (2.1) and hence is a weak contraction. Some examples of weak contractions are given in [18, 21, 22]. The next example shows that a weak contraction needs not to be continuous.

Example 2.2 (see [18, 22]). Let [0,1] be the unit interval with the usual norm and let T : [0,1] ^ [0,1] be given by Tx = 2/3 for all x e [0,1) and T1 = 0. Then T satisfies the inequality (2.1) with 1 >9 > 2/3 and L > 9 and T has a unique fixed point x = 2/3, but T is not continuous.

Let f be a self-mapping on X. A mapping T : X ^ X is said to be f-weak contraction or (f, 9,L)-weak contraction if there exist two constants 9 e (0,1) and L > 0 such that

d(Tx, Ty) < 9d(fx, fy) + Ld(fy, Tx), Vx, y e X. (2.3)

Berinde [18] introduced the notion of a (9, L)-weak contraction and proved that a lot of the well-known contractive conditions do imply the (9, L)-weak contraction. The concept of (9, L)-weak contraction does not ask 9 + L 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 d(x,y), d(x,Tx), d(y,Ty), d(x,Ty), d(y,Tx). For more details, we refer to [18, 21] and references cited in these papers.

The following result is a consequence of the main theorem of Berinde [18].

Lemma 2.3. Let M be a nonempty subset of a metric space (X,d) and let T be a self-mapping of M. Assume that cl T(M) c M, cl T(M) is complete, and T is a (9, L)-weak contraction. Then M n F(T) is nonempty.

Theorem 2.4. Let M be a nonempty subset of a metric space (X,d) and let T, f be self-mappings of M. Assume that F(f) is nonempty, cl T(F(f)) c F(f), cl(T(M)) is complete, and T is an (f,9,L)-weak contraction. Then M n F(T) n F(f) = 0.

Proof. Since cl(T(F(f))) is a closed subset of cl(T(M)), cl(T(F(f))) is complete. Further, by the (f, d, L)-weak contraction of T, for all x,y e F(f), we have

d(Tx,Ty) < Qd(fx,fy) + L ■ d(fy,Tx) = d(x,y) + L ■ d(y,Tx). (2.4)

Hence T is a (d, L)-weak contraction on F(f) and cl T(F(f)) c F(f). Therefore, by Lemma 2.3, T has a fixed point z in F(f) and so M n F(T) n F(f) / 0. □

Corollary 2.5. Let M be a nonempty subset of a metric space (X,d) and let (T,f) be a Banach operator pair on M. Assume that cl(T(M)) is complete, T is (f, 9,L)-weak contraction, and F(f) is nonempty and closed. Then M n F(T) n F(f) = 0.

In Theorem 2.4 and Corollary 2.5, if L = 0, then we easily obtain the following result, which improves Lemma 3.1 of Chen and Li [4].

Corollary 2.6 (see [17, Theorem 2.2]). Let M be a nonempty subset of a metric space (X,d) and let T,f be self-mappings of M. Assume that F(f) is nonempty, cl(T(F(f))) c F(f), cl(T(M)) is complete, and T is an f -contraction. Then M n F (T) n F(f) is a singleton.

The following result properly contains [4, Theorems 3.2-3.3] and improves [13, Theorem 2.2], [10, Theorem 4], and [11, Theorem 6].

Theorem 2.7. Let M be a nonempty subset of a normed (resp., Banach) space X and let T, f be self-mappings of M. Suppose that F(f) is q-starshaped, cl T(F(f)) c F(f) (resp., wcl T(F(f)) c F(f)), cl(T(M)) is compact (resp., wcl(T(M)) is weakly compact, and either I - T is demiclosed at 0 or X satisfies Opial's condition, where I stands for the identity mapping), and there exists a constant L > 0 such that

||Tx - Ty\\<\fx - fy|| + L ■ distfy, [q,Tx]), Vx,y e M. (2.5)

Then M n F(T) n F(f) = 0.

Proof. For each n e N, define Tn : F(f) ^ F(f) by Tnx = (1 - kn)q + knTx for all x e F(f) and a fixed sequence {kn} of real numbers (0 < kn < 1) converging to 1. Since F(f) is q-starshaped and clT(F(f)) c F(f) (resp., wclT(F(f)) c F(f)), we have clTn(F(f)) C F(f) (resp., wcl Tn(F(f)) c F(f)) for each n e N. Also, by the inequality (2.5),

\\Tnx - Tny\\ = kn\\Tx - Ty\\

< kn\\fx - fy\\ + knL ■ distfy, [q,Tx]) (2.6)

< kn\\fx - fy\\ + Ln ■ \\fy - Tn x

for all x,y e F(f), Ln := knL, and 0 < kn < 1. Thus, for n e N, Tn is a (f,kn,Ln)-weak contraction, where Ln > 0.

Ifcl(T(M)) is compact, then, for each n e N,cl(Tn(M)) is compact and hence complete. By Theorem 2.4, for each n e N, there exists xn e F(f) such that xn = fxn = Tnxn. The compactness of cl(T(M)) implies that there exists a subsequence {Txm} of {Txn} such that

Txm ^ z e cl(T(M)) as m ^ to. Since {Txm} is a sequence in T(F(f)) and clT(F(f)) c F(f), we have z e F(f). Further, it follows that

X~m — TmXm — (1 k-m^q + km.TX-m ^ Z, ||Xm TX-m || ^ 0 (n ^ ^O)• (2.7)

Moreover, we have

11TXm - Tz11 < fXm - fz11 + L • distfz, [q,TXm])

— ||Xm - z|| + L • dist(z,[q,TXm]) (2.8)

< 11 Xm TXm.11 + 11 "TXm z11 + L • 11 z TXm. 11 •

Taking the limit as m ^ to, we get z — Tz and so M n F(T) n F(f) — 0.

Next, the weak compactness of wcl(T(M)) implies that wcl(Tn(M)) is weakly compact and hence complete due to completeness of X (see [3]). From Theorem 2.4, for each n e N, there exists Xn e F(f) such that Xn — fXn — TnXn• The weak compactness of wcl(T(M)) implies that there is a subsequence {TXm} of {TXn} converging weakly to y e wcl(T(M)) as m ^ to. Since {TXm} is a sequence in T(F(f)), we have y e wcl(T(F(f))) c F(f). Also, we have Xm - TXm ^ 0 as m ^ to. If I - T is demiclosed at 0, then y — Ty and so M n F(T) n F (f)/ 0^

If fy — Ty, then we have

liminffXm - fyH

< liminf|| fXm - TyII

m^TO 11 11

< lim inf fXm - TXm + lim inf TXm - Ty

m^TO 11 11 m^TO 11 11

< lim inf fXm - TXm + lim inf fXm - fy + lim inf L • dist fy, q, TXm

m ^to 11 11 m ^to" 11 m ^to

< liminf|| fXm - /v\\ + liminf|| fXm - TXm\\ + L • liminf||y - TXm||

m ^to m ^to m ^to

— liminffXm - fyy,

which is a contradiction. Thus Ty = fy = y and hence M n F(T) n F(f) / 0. This completes the proof. □

Obviously, f -nonexpansive mappings satisfy the inequality (2.5) and so we obtain the following.

Corollary 2.8 (see [17, Theorem 2.4]). Let M be a nonempty subset of a normed (resp., Banach) space X and let T, f be self-mappings of M. Suppose that F(f) is q-starshaped, cl T(F(f)) c F(f)

(resp., wcl T(F(f)) c F(f)), cl(T(M)) is compact, (resp., wcl(T(M)) is 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 n F(T) n F(f)/ 0.

Corollary 2.9 (see [4, Theorems 3.2-3.3]). Let M be a nonempty subset of a normed (resp., Banach) space X and let T, f be self-mappings of M. Suppose that F (f) is q-starshaped and closed (resp., weakly closed), cl(T(M)) is compact (resp., wcl(T(M)) 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 on M. Then M n F (T) n F(f) = 0.

Corollary 2.10 (see [13, Theorem 2.1]). Let M be a nonempty closed and q-starshaped subset of a normed space X and let T, f be self-mappings of M such that T(M) c f (M). Suppose that T commutes with f and q e F(f). If cl(T(M)) is compact, f is continuous, linear, and T is f-nonexpansive on M, then M n F(T) n F(f) = 0.

Let C = Pm(u) n CfM(u), where CfM(u) = {x e M : fx e Pm(u)}.

Corollary 2.11. Let X be a normed (resp., Banach) space X and let T, f be self-mappings of X. If u e X, D c C, D0 := DnF(f) is q-starshaped, cl(T(D0)) c D0 (resp., wcl(T(D0)) c D0), cl(T(D)) is compact, (resp., wcl(T (D)) is weakly compact, and I - T is demiclosed at 0). If the inequality (2.5) holds for all x,y e D, then PM(u) n F(T) n F (f) / 0.

Corollary 2.12. Let X be a normed (resp., Banach) space X and let T, f be self-mappings of X. If u e X, D c Pm(u), D0 := D n F(f) is q-starshaped, cl(T(D0)) c D0 (resp., wcl(T(D0)) c D0), cl(T(D)) is compact, (resp., wcl(T(D)) is weakly compact, and I - T is demiclosed at 0). If the inequality (2.5) holds for all x,y e D, then PM(u) n F(T) n F (f) / 0.

Corollary 2.13 (see [11, Theorem 7]). Let f, T be self-mappings of a Banach space X with u e F(f) nF(T) and M c X with T(dM) c M. Suppose that D = PM(u) is q-starshaped with q e F(f), f (D) = D, and f is affine, continuous in the weak and strong topology on D. Iff and T are commuting on D and T is f-nonexpansive on D u {u}, then PM(u) n F(T) n F(f) = 0 provided either (i) D is weakly compact and (f-T) is demiclosed or (ii) D is weakly compact and X satisfies Opial's condition.

Remark 2.14. Corollary 2.5 in [17] and Theorems 4.1-4.2 of Chen and Li [4] are special cases of Corollaries 2.11-2.12

We denote I0 by the class of closed convex subsets of X containing 0. For any M e I0, we define Mu = {x e M : ||x|| < 2||u||}. It is clear that Pm(u) c Mu e I0 (see [8,13]).

Theorem 2.15. Let f, T be self-mappings of a normed (resp., Banach) space X. If u e X and M e I0 such that T(Mu) c M, cl(T(Mu)) is compact (resp., wcl(T(Mu)) is weakly compact), and ||Tx -u|| < ||x-u|| for all x e Mu, thenPM(u) is nonempty closed and convex with T(PM(u)) c PM(u).If, in addition, D c PM(u), D0 := D n F(f) is q-starshaped, cl(T(D0)) c D0 (resp., wcl(T(D0)) c D0, and I - T is demiclosed at 0), and the inequality (2.5) holds for all x,y e D, then PM (u) n F (T) n

F(f)/ 0.

Proof. We may assume that u/M. If x e M \ Mu, then ||x|| > 2||u||. Note that

||x - u|| > ||x|| - ||u|| > ||u|| > dist(u,M).

(2.10)

Thus dist(u, Mu) = dist(u, M) < ||u||. If cl(T(Mu)) is compact, then, by the continuity of the norm, we get ||z - u|| = dist(u,cl(T(Mu))) for some z e cl(T(Mu)). If we assume that wcl(X(Mu)) is weakly compact, then, using [23, Lemma 5.5, page 192], we can show the existence of a z e wcl(X(Mu)) such that dist(u,wcl(X(Mu))) = ||z - u||. Thus, in both cases, we have

dist(u,Mu) < dist(u,clT(Mu)) < dist (u,T(Mu)) < ||Tx - u|| < ||x - u|| (2.11)

for all x e Mu. Hence ||z - u|| = dist(u, M) and so PM(u) is nonempty closed and convex with T(PM(u)) c PM(u). The compactness of cl(T(Mu)) (resp., the weak compactness of wcl(T(Mu))) implies that cl(T(D)) is compact (resp., wcl(T(D)) is weakly compact). Therefore, the result now follows from Corollary 2.12. This completes the proof. □

Corollary 2.16. Let f, T be self-mappings of a normed (resp., Banach) space X. Ifu e X and M e such that T(Mu) c M, cl(T(Mu)) is compact (resp., wcl(T(Mu)) is weakly compact), and ||Tx -u|| < ||x - u|| for all x e Mu, then PM(u) is nonempty closed and convex with T(PM(u)) c PM(u). If, in addition, D c PM(u), D0 := D n F(f) is q-starshaped and closed (resp., weakly closed and I - T is demiclosed at 0), (T,f) is a Banach operator pair on D, and the inequality (2.5) holds for all x,y e D, then Pm(u) n F(T) n F(f) / 0.

Remark 2.17. Theorem 2.15 and Corollary 2.16 extend [13, Theorems 4.1 and 4.2], [17, Theorem 2.6], and [10, Theorem 8].

Banach's Fixed Point Theorem states that if (X, d) is a complete metric space, K is a nonempty closed subset of X, and T : K ^ K is a self-mapping satisfying the following condition: there exists X e [0,1) such that

d(Tx, Ty) < Xd(x, y), Vx, y e K, (2.12)

then T has a unique fixed point, say z in K, and the Picard iterative sequence {T"x} converges to the point z for all x e K. Since then, Ciric [24] introduced and studied self-mappings on K satisfying the following condition: there exists X e [0,1) such that

d(Tx,Ty) < Xm(x,y), Vx,y e K, (2.13)

m(x,y) = max {d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}. (2.14)

Further, many investigations were developed by Berinde [19], Jungck [1, 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 Ciric [24].

Theorem 2.18 (see [3, Theorem 2.1]). Let M be a nonempty subset of a metric space (X,d) and let f, g be self-mappings of M. Assume that cl f (M) c g(M), cl f (M) is complete, and f, g satisfy the following condition: there exists h e [0,1) such that

d(fx,fy) < h max {d(gx,gy),d(fx,gx),d(fy,gy),d(fx,gy),d(fy,gx)}, (2.15)

for all x,y e M. Then C(f, g) / 0.

The following result (Theorem 2.19) properly contains [17, Theorem 3.3], [4, Theorems 3.2-3.3], [5, Theorem 2.11], and [14, 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. Let M be a nonempty subset of a normed (resp., Banach) space X and let T, f, g be self-mappings of M. Suppose that F(f) n F(g) is q-starshaped, cl T(F(f) n F(g)) c F(f) n F(g) (resp., wclT(F(f) n F(g)) c F(f) n F(g)), cl(T(M)) is compact (resp., wcl(T(M)) is weakly compact), and T is continuous on M (resp., I - T is demiclosed at 0). If the following condition holds:

llTx - Tyll < max {Hfx - gyhdist(fx [q,Tx]),dist(gy, [q, Ty]),

(2.16)

dist( gy, [q,Tx]), dist( fx, [q,Ty])}

for all x,y e M, then M n F(T) n F(f) n F(g) / 0.

Theorem 2.20. Let f, g, T be self-mappings of a Banach space X with u e F(T) n F(f) n F(g) and M e I0 such that T(Mu) c f (M) c M = g(M). Suppose that ||fx - u|| < ||x - u\\, ||gx - u|| = ||x - u||, ||Tx - u|| < ||fx - gu|| for all x e M, and cl(f (Mu)) is compact. Then one has the following:

(1) PM (u) is nonempty closed and convex,

(2) T(Pm(u)) c f (Pm(u)) c Pm(u) = g(Pm(u)),

(3) PM(u) n F(T) n F(f) n F(g) = 0 provided T is continuous, F(g) is q-starshaped, cl(f (F(g))) C F(g), and the pair (f,g) satisfies the inequality (2.5) for all x,y e Pm(u),F(f) is q-starshaped with q e F(f) n F(g) n Pm(u), clT(F(f) n F(g)) c F(f) n F(g), and the inequality (2.16) holds for all q e F(f) n F (g) and x,y e PM(u).

Proof. (1) and (2) follow from [5, 8, Theorem 2.14]. By (2), the compactness of cl(f (Mu)) implies thatcl(f (PM(u))) andcl(T(PM(u))) is compact. Theorem 2.7 implies that F(f )nF(g)n PM(u) = 0. Further, F(f) n F(g) is q-starshaped with q e F(f) n F(g) n PM(u). Therefore, the conclusion now follows from Theorem 2.19 applied to PM(u). □

Remark2.21. (1) Theorem 2.20 extends [13, Theorem 4.1], [10, Theorem 8], [5, Theorem 2.13], [8, Theorem 2.14], and [14, Theorem 2.11].

(2) Theorems 2.7-2.16 represent very strong variants of the results in [3, 8, 11, 13] in the sense that the commutativity or compatibility of the mappings T and f is replaced by the hypothesis that (T,f) is a Banach operator pair, f needs not be linear or affine, and T needs not be f-nonexpansive.

(3) The Banach operator pairs are different from those of weakly compatible, Cq-commuting and K-subweakly commuting mappings and so our results are different from those in [3, 7, 8, 17]. Consider M = R2 with the norm ||(x,y)|| = |x| + |y| for all (x,y) e M. Define two self-mappings T and f on M as follows:

•y x2 + y3 - 1 T(x, y) = ( x3 + x - 1, -

f (x,y) = (x3 + x - 1,^x2 + y3 - 1).

(2.17)

Then we have the following:

F (T ) = {(1,0)}, F (f ) = {(1,y) : y e R1}, C(T,f ) = {(x,y) : y = Vï-x2, x e R1}, (2.18)

T(F(f )) = {T(1,y) : y e R1} = { (1,|) : y e R1} ç {(1,y) : y e R1} = F(f ).

Thus (T,f ) is a Banach operator pair. It is easy to see that T is (f, d, L)-weak contraction and

T, f do not commute on the set C(T,f ), and so are not weakly compatible. Clearly, f is not

affine or linear, F (f ) is convex and (1,0) is a common fixed point of T and f.

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