Ukrainian Mathematical Journal, Vol. 62, No. 10, March, 2011 (Ukrainian original Vol. 62, No. 10, October, 2010)
COMMON FIXED POINTS AND INVARIANT APPROXIMATION OF R-SUBWEAKLY COMMUTING MAPS IN CONVEX METRIC SPACES T. D. Narang and S. Chandok
UDC 517.5
Sufficient conditions for the existence of a common fixed point of R-subweakly commuting mappings are established within the framework of a convex metric space. As applications, we obtain various results on the best approximation for this class of mappings generalizing the results known from the literature.
1. Introduction and Preliminaries Numerous interesting and useful results have been proved in the approximation theory by applying fixed-point theorems (see, e.g., [1, 2, 7, 9–11] and the references therein). The present paper deals with the common fixed points of R-subweakly commuting mappings within the framework of convex metric spaces. We also establish the results on the invariant approximation for this class of mappings. The results proved in the present paper generalize and extend some of the results established in [1, 2, 4, 7, 9, 11, 13]. To begin with, we recall some definitions and known facts used in the sequel. For a metric space .X; d /; a continuous mapping W W X X Œ0; 1 ! X is called a convex structure on X if, for all x; y 2 X and 2 Œ0; 1; we have d.u; W .x; y; // d.u; x/ C .1
/d.u; y/
for all u 2 X: A metric space .X; d / with a convex structure is called a convex metric space [12]. A subset M of a convex metric space .X; d / is called a convex set [12] if W .x; y; / 2 M for all x; y 2 M and 2 Œ0; 1: A set M is called p-starshaped [3], where p 2 M; provided that W .x; p; / 2 M for all x 2 M and 2 Œ0; 1; i.e., if the segment Œp; x D fW .x; p; /W 0 1g connecting p with x is contained in M for all x 2 M: The set M is called starshaped if it is p-starshaped for some p 2 M: Clearly, each convex set M is starshaped but the converse assertion is not true. We say that a convex metric space .X; d / satisfies Property (I) [3] if, for all x; y; q 2 X and 2 Œ0; 1; d.W .x; q; /; W .y; q; // d.x; y/: A normed linear space X and each of its convex subsets are simple examples of convex metric spaces with W given by W .x; y; / D x C .1
/y
for x; y 2 X and 0 1: There are many convex metric spaces that are not normed linear spaces (see [3, 12]). Property (I) is always satisfied in a normed linear space. Guru Nanak Dev University, Amritsar, India. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 10, pp. 1367–1376, October, 2010. Original article submitted December 24, 2008; revision submitted July 19, 2010. 0041–5995/11/6210–1585
c 2011 Springer Science+Business Media, Inc.
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For a nonempty subset K of a metric space .X; d / and x 2 X; an element y 2 K is called a best approximant to x or a best K-approximant to x if d.x; y/ D d.x; K/ inffd.x; k/W k 2 Kg: The set of all these y 2 K is denoted by PK .x/: For a convex subset K of a convex metric space .X; d /; a mapping gW K ! X is called affine if g.W .x; y; // D W .gx; gy; / for all x; y 2 K and all 2 Œ0; 1: The mapping g is called affine with respect to p 2 K if g.W .x; p; // D W .gx; gp; / for all x 2 K and 2 Œ0; 1: Suppose that .X; d / is a metric space, M is a nonempty subset of X; and S; T; f; and g are self-mappings of M: The mapping T is called an .f; g/-contraction if there exists k 2 Œ0; 1/ such that d.T x; T y/ kd.f x; gy/; [.f; g/-nonexpansive if d.T x; T y/ d.f x; gy/ ] for all x; y 2 M: If f D g; then T is called an f -contraction (f -nonexpansive). A point x 2 M is a common fixed (coincidence) point of S and T if x D S x D T x ( S x D T x ). The pair .S; T / is called (a) commuting on M if ST x D T Sx for all x 2 M I (b) R-weakly commuting [8] on M if there exists a real number R > 0 such that d.T Sx; ST x/ Rd.T x; S x/ for all x 2 M I (c) compatible [5] if lim d.T Sxn ; ST xn / D 0; whenever .xn / is a sequence such that lim T xn D lim S xn D t for some t in M I (d) weakly compatible [6] if these mappings commute at their coincidence points, i.e., if S T x D T S x; whenever Sx D T x: Suppose .X; d / is a convex metric space, M is starshaped with respect to q; where q a fixed point of S; and is both T - and S-invariant. The mappings T and S are called (e) R-subcommuting [10] on M if, for all x 2 M; there exists a real number R > 0 such that d.T S x; S T x/ .R=/ dist .Sx; W .T x; q; //;
2 .0; 1I
(f) R-subweakly commuting [9] on M if, for all x 2 M and 2 Œ0; 1; there exists a real number R > 0 such that d.T S x; S T x/ R dist .Sx; W .T x; q; //: Clearly, compatible maps are weakly compatible but the converse assertion is not necessarily true (see [6]). Commuting mappings are R-subweakly commuting but the converse assertion may be not true (see [9]). It is well known that R-subweakly commuting maps are R-weakly commuting and R-weakly commuting maps are compatible but not conversely (see [5, 9]). The R-subcommuting and R-subweakly commuting maps are weakly compatible but the converse assertions do not hold (see [9–11]). Throughout the paper, we write M for the closure of a set M; F .S/ for the set of fixed points of a mapping S; and F .S; T / .C.S; T // for the set of fixed (coincidence) points of the mappings S and T: 2. Main Results The following four lemmas are used in proving the results in the present paper:
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Lemma A [11]. Let M be a closed subset of a metric space .X; d / and let T and S be R-weakly commuting self-mappings of M such that T .M / S.M /: Suppose there exists k 2 Œ0; 1/ such that º » 1 d.T x; T y/ k max d.S x; Sy/; d.Sx; T x/; d.Sy; T y/; Œd.Sx; T y/ C d.Sy; T x/ ; 2 for all x; y 2 M: If T .M / is complete and T is continuous, then M \ F .T / \ F .S/ is a singleton. Lemma B [10]. Let M be a closed subset of a metric space .X; d / and let S and T be R-weakly commuting self mappings of M such that T .M / S.M /: Assume that T is an S -contraction. If T .M / is complete and T is continuous, then F .T / \ F .S / is a singleton. Lemma C [1]. Let .X; d / be a convex metric space, let M X; and let x0 2 X: Then PM .x0 / @M \ M: Lemma D [5]. Let A; B; S; and T be self-mappings of a complete metric space .X; d /: Suppose that S and T are continuous, that .A; S / and .B; T / are compatible pairs, and that A.X/ T .X/ and B.X / S.X /: If there exists r 2 .0; 1/ such that º
» 1 d.Ax; By/ r max d.S x; T y/; d.Ax; Sx/; d.By; T y/; Œd.Ax; T y/ C d.Sx; By/ ; 2 for all x; y 2 X; then there is a unique point z in X such that Az D Bz D Sz D T z D z: For continuous self-mappings on closed subsets of convex metric spaces, we have the following result: Theorem 1. Let M be a closed subset of a convex metric space .X; d / with Property (I) and let T and S be continuous self-mappings on M such that T .M / S.M /: Assume that S is affine, p 2 F .S/; M is starshaped with respect to p; and T .M / is compact. If T and S are R-subweakly commuting and satisfy the inequality º d.T x; T y/ max d.S x; Sy/; dist.Sx; W .T x; p; //; dist.Sy; W .T y; p; //; » 1 Œdist.S x; W .T y; p; // C dist.Sy; W .T x; p; // ; 2 for all x; y 2 M; 2 Œ0; 1/; then M \ F .T / \ F .S/ ¤ ¿: Proof. For any n; we define Tn W M ! M by Tn x D W .T x; p; n /; x 2 M where .n / is a sequence in .0; 1/ such that n ! 1: Since M is starshaped with respect to p; S is affine with respect to p; and T .M / S.M /; we have Tn .x/ D W .T x; p; n / D W .T x; Sp; n / 2 S.M / and, hence, Tn .M / S.M / for each n: Consider
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d.Tn S x; S Tn x/ D d.W .T Sx; p; n /; SW .T x; p; n // D d.W .T Sx; p; n /; W .ST x; Sp; n //; D d.W .T Sx; p; n /; W .ST x; p; n //;
S is affine p 2 F .S/
n d.T Sx; ST x/; Property (I) n R dist.Sx; W .T x; p; // for any n: Since n 2 .0; 1/; we obtain d.Tn S x; S Tn x/ n R dist .Sx; W .T x; p; n // n Rd.Sx; Tn x/ for all x 2 M: This shows that Tn and S are n R-weakly commuting for each n: We also have d.Tn x; Tn y/ D d.W .T x; p; n /; W .T y; p; n // n d.T x; T y/; Property (I) ¼ n max d.S x; Sy/; dist .Sx; W .T x; p; //; dist.Sy; W .T y; p; //;
1 Œdist.Sx; W .T y; p; // C dist.Sy; W .T x; p; // 2
½
for any n: Since n 2 .0; 1/; we obtain ¼ d.Tn x; Tn y/ n max d.S x; Sy/; dist.Sx; W .T x; p; n //; dist.Sy; W .T y; p; n //;
½ 1 Œdist.S x; W .T y; p; n // C dist.Sy; W .T x; p; n // 2 ¼
1 n max d.S x; Sy/; d.Sx; Tn x/; d.Sy; Tn y/; Œd.Sx; Tn y/ C d.Sy; Tn x/ 2
½
for all x; y 2 M: Thus, by Lemma A, there exists some xn 2 M such that F .Tn / \ F .S/ D fxn g for any n: The compactness of T .M / implies the existence of a subsequence .xni / of .xn / such that xni ! y 2 M: By the continuity of T and S; we have y 2 F .T / \ F .S/: Hence, M \ F .T / \ F .S/ ¤ ¿:
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Corollary 1 ([11], Theorem 2.2). Let M be a closed subset of a normed linear space X and let T and S be continuous self-mappings on M such that T .M / S.M /: Suppose that S is linear, p 2 F .S/; M is starshaped with respect to p; and T .M / is compact. If T and S are R-subweakly commuting and satisfy the inequality ¼ kT x
T yk max kS x
Syk; dist.Sx; ŒT x; p/; dist.Sy; ŒT y; p/;
½ 1 dist.Sx; ŒT y; p/ C dist.Sy; ŒT x; p/ 2 for all x; y 2 M; then M \ F .T / \ F .S / ¤ ¿: Corollary 2. Let M be a closed subset of a normed linear space X and let T and S be continuous selfmappings on M such that T .M / S.M /: Assume that S is linear, p 2 F .S/; M is starshaped with respect to p; and T .M / is compact. If T and S are commuting and satisfy the inequality ¼ kT x
T yk max kS x
Syk; dist.Sx; ŒT x; p/; dist.Sy; ŒT y; p/;
½ 1 dist.Sx; ŒT y; p/ C dist.Sy; ŒT x; p/ 2 for all x; y 2 M; then M \ F .T / \ F .S / ¤ ¿: By using Lemma B, we prove the following theorem: Theorem 2. Let M be a closed subset of a convex metric space .X; d / with Property (I) and let T and S be continuous self-mappings on M such that T .M / S.M /: Assume that S is affine, p 2 F .S/; M is starshaped with respect to p; and T .M / is compact. If T and S are R-subweakly commuting and T is S-nonexpansive on M; then M \ F .T / \ F .S / ¤ ¿: Proof. Proceeding as in Theorem 1, we find d.Tn S x; S Tn x/ n R dist .Sx; W .T x; p; n // n Rd.Sx; Tn x/ for all x 2 M: This shows that Tn and S are n R-weakly commuting for each n: We also get d.Tn x; Tn y/ D d.W .T x; p; n /; W .T y; p; n // n d.T x; T y/ n d.Sx; Sy/: Thus, each Tn is an S -contraction. Since T .M / is compact, by Lemma B, there exists some xn 2 M such that F .Tn / \ F .S / D fxn g for each n: Further, since .T .xn // is a sequence in T .M /; there exists a subsequence .T .xni // with T .xni / ! x0 2 T .M /: Moreover, since
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xni D Tni xni D W .T xni ; p; ni / ! x0 ; the continuity of T and S imply that x0 2 F .T; S/: Hence, we get the required result. Corollary 3. Let M be a closed subset of a normed linear space X and let T and S be continuous selfmappings on M such that T .M / S.M /: Suppose that S is linear, p 2 F .S/; M is starshaped with respect to p; and T .M / is compact. If T and S are R-subweakly commuting and T is S -nonexpansive on M; then M \ F .T / \ F .S / ¤ ¿: Remark 1. Theorems 1 and 2 and their Corollaries 1.1, 1.2, and 2.1 generalize and extend the corresponding results in [1, 4, 9, and 11]. For a real number R > 0; let R;S R;S DM .x0 / D PM .x0 / \ GM .x0 /;
where R;S GM .x0 / D fx 2 M W d.Sx; x0 / .2R C 1/ dist .x0 ; M /g:
Lemma C and Theorem 1 yield the following theorem: Theorem 3. Let T and S be self-mappings of a convex metric space .X; d / with Property (I), let x0 2 R;S F .T; S /; and let M be a subset of X such that T .@M \ M / M: Suppose that S is affine on DM .x0 /; p 2 R;S R;S R;S F .S /; DM .x0 / is closed and starshaped with respect to p; T .DM .x0 // is compact, and S.DM .x0 // D R;S R;S DM .x0 /: If T and S are R-subweakly commuting on DM .x0 / and satisfy the inequality
d.T x; T y/
8
for y D x0 ;
: Q.x; y/
R;S for y 2 DM .x0 /;
where ¼ Q.x; y/ D max d.S x; Sy/; dist.Sx; W .T x; p; //; dist.Sy; W .T y; p; //;
½ 1 dist.S x; W .T y; p; // C dist.Sy; W .T x; p; // 2 R;S for all x 2 DM .x0 / [ fx0 g and 2 Œ0; 1/; then
PM .x0 / \ F .T / \ F .S/ ¤ ¿: R;S Proof. Let x 2 DM .x0 /: Then, by Lemma C, x 2 @M \ M and, hence, T x 2 M as T .@M \ M / M: Since
d.T x; x0 / D d.T x; T x0 / d.Sx; Sx0 / D d.Sx; x0 / D dist.x0 ; M /;
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we conclude that T x 2 PM .x0 /: It follows from the R-subweak commutativity of T and S that d.S T x; x0 / D d.S T x; T x0 / d.S T x; T Sx/ C d.T Sx; T x0 / Rd.T x; Sx/ C d.S 2 x; Sx0 / RŒd.T x; T x0 / C d.T x0 ; Sx/ C d.S 2 x; Sx0 / RŒdist.x0 ; M / C dist.x0 ; M / C dist.x0 ; M / .2R C 1/dist.x0 ; M /: R;S R;S This implies that T x 2 GM .x0 /: Consequently, T x 2 DM .x0 / and, therefore, R;S R;S R;S T .DM .x0 // DM .x0 / D S.DM .x0 //:
Theorem 1 now guarantees that PM .x0 / \ F .T / \ F .S/ ¤ ¿: Corollary 4 ([11], Theorem 2.5). Let T and S be self-mappings of a normed linear space X; let x0 2 R;S F .T; S / and let M be a subset of X such that T .@M \ M / M: Assume that S is linear on DM .xı /; p 2 R;S R;S R;S F .S /; DM .x0 / is closed and starshaped with respect to p; T .DM .x0 // is compact, and S.DM .x0 // D R;S R;S DM .x0 /: If T and S are R-subweakly commuting on DM .x0 / and satisfy the inequality
kT x
T yk
8
Syk
: Q.x; y/
for y D x0 ; R;S for y 2 DM .x0 /;
where ¼ Q.x; y/ D max kS x
Syk; dist.Sx; ŒT x; p/; dist.Sy; ŒT y; p/;
½ 1 dist.Sx; ŒT y; p/ C dist.Sy; ŒT x; p/ 2 R;S for all x 2 DM .x0 / [ fx0 g; then
PM .x0 / \ F .T / \ F .S/ ¤ ¿: Corollary 5 ([7], Theorem 2.3). Let T and S be self-mappings of a convex metric space .X; d / with Property (I), let x0 2 F .T; S /; and let M be a subset of X such that T .@M \M / M and p 2 F .S/\M: Suppose
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R;S R;S that T and S are R-subweakly commuting on DM .x0 /; T is S -nonexpansive on DM .x0 / [ fx0 g; and R;S R;S R;S S is affine on DM .x0 /: If DM .x0 / is closed and starshaped with respect to p; T .DM .x0 // is compact, R;S R;S S.DM .x0 // D DM .x0 /; and T is continuous, then PM .x0 / \ F .T / \ F .S/ is nonempty.
Theorem 4. Let T and S be self-mappings of a convex metric space .X; d / with Property (I), let x0 2 F .T; S /; and let M be a subset of X such that T .@M \ M / S.M / M: Assume that S is affine on R;S R;S R;S DM .x0 /; p 2 F .S /; DM .x0 / is closed and starshaped with respect to p; T .DM .x0 // is compact, and R;S R;S R;S R;S S.GM .x0 // \ DM .x0 / D S.DM .x0 // DM .x0 /: R;S If T and S are R-subweakly commuting and continuous on DM .x0 / and satisfy the inequality
d.T x; T y/
8
for y D x0 ;
: Q.x; y/
R;S for y 2 DM .x0 /;
where ¼ Q.x; y/ D max d.S x; Sy/; dist.Sx; W .T x; p; //; dist.Sy; W .T y; p; //;
½ 1 dist.S x; W .T y; p; // C dist.Sy; W .T x; p; // 2 R;S for all x 2 DM .x0 / [ fx0 g and 2 Œ0; 1/; then
PM .x0 / \ F .T / \ F .S/ ¤ ¿: R;S R;S R;S R;S Proof. Let x 2 DM .x0 /: Then, as in Theorem 3, T x 2 DM .x0 /; i.e., T .DM .x0 // DM .x0 /: Moreover,
d.W .x; x0 ; k/; x0 / < kd.x; x0 / C .1
k/d.x0 ; x0 / D kd.x; x0 / < dist.x0 ; M /
R;S for all k 2 .0; 1/: Further, by Lemma C, x 2 @M \ M and, hence, T .DM .x0 // T .@M \ M / S.M /: Therefore, we can choose y 2 M such that T x D Sy: Since Sy D T x 2 PM .x0 /; we conclude that y 2 R;S R;S R;S GM .x0 /: Consequently, T .DM .x0 // S.GM .x0 // PM .x0 / and, thus, R;S R;S R;S R;S R;S T .DM .x0 // S.GM .x0 // \ DM .x0 / D S.DM .x0 // DM .x0 /:
Hence, Theorem 1 guarantees that PM .x0 / \ F .T / \ F .S/ ¤ ¿: Corollary 6 ([11], Theorem 2.6). Let T and S be self-mappings of a normed linear space, let x0 2 F .T; S /; and let M be a subset of X such that T .@M \ M / S.M / M: Suppose that S is linear on
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R;S R;S R;S DM .x0 /; p 2 F .S /; DM .x0 / is closed and starshaped with respect to p; T .DM .x0 // is compact, and R;S R;S R;S R;S S.GM .x0 // \ DM .x0 / D S.DM .x0 // DM .x0 /: R;S If T and S are R-subweakly commuting and continuous on DM .x0 / and satisfy the inequality
kT x
T yk
8
Syk
: Q.x; y/
for y D x0 ; R;S for y 2 DM .x0 /;
where ¼ Q.x; y/ D max kS x
Syk; dist.Sx; ŒT x; p/; dist.Sy; ŒT y; p/;
½ 1 dist.Sx; ŒT y; p/ C dist.Sy; ŒT x; p/ 2 R;S for all x 2 DM .x0 / [ fx0 g; then
PM .x0 / \ F .T / \ F .S/ ¤ ¿: Corollary 7 ([7], Theorem 2.4). Let T and S be self-mappings of a convex metric space .X; d / with Property (I), let x0 2 F .T; S /; and let M be a subset of X such that T .@M \M / S.M / M and p 2 F .S /\M: R;S R;S Suppose that T and S are R-subweakly commuting on DM .x0 /; T is S-nonexpansive on DM .x0 / [ fx0 g; R;S R;S R;S and S is affine on DM .x0 /: If DM .x0 / is closed and starshaped with respect to, T .DM .x0 // is comR;S R;S R;S pact, S.M / \ DM .x0 / S.DM .x0 // DM .x0 /; and T is continuous, then PM .x0 / \ F .T / \ F .S / is nonempty. R;S Remark 2. Theorems 3 and 4 remain valid for DM .x0 / D PM .x0 /: If S.PM .x0 // PM .x0 /; then R;S R;S PM .x0 / GM .x0 / and, therefore, DM .x0 / D PM .x0 /: This means that Theorem 3 contains the following result as a special case:
Theorem 5 ([2], Theorem 6). Let T and S be self-mappings of a convex metric space .X; d / with Property (I), let x0 2 F .T; S /; and let M be a subset of X such that T .@M / M: Suppose that T is S -nonexpansive on PM .x0 / [ fx0 g; S is affine and continuous on PM .x0 /; and ST x D T Sx for all x in PM .x0 /: If PM .x0 / is nonempty compact and starshaped with respect to p; p 2 F .S/ and, in addition, S.PM .x0 // D PM .x0 /; then PM .x0 / \ F .T / \ F .S / is nonempty. As an application of Lemma D, we obtain the following theorem: Theorem 6. Let M be a nonempty subset of a convex metric space .X; d / with Property (I) and let T; f; and g be continuous self-maps of M: Suppose that M is starshaped with respect to q; f and g are affine
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with q 2 F .f / \ F .g/; T .M / f .M / \ g.M /; and T .M / is compact. If the pairs .T; f / and .T; g/ are R-subweakly commuting and satisfy the inequality ¼ d.T x; T y/ max d.f x; gy/; dist.f x; W .T x; q; //; dist.gy; W .T y; q; //;
½ 1 dist.f x; W .T y; q; // C dist.gy; W .T x; q; // ; 2 for all x; y 2 M; and 2 Œ0; 1/; then F .T / \ F .f / \ F .g/ is nonempty. Proof. For each n; we define Tn W M ! M by Tn x D W .T x; q; n /; x 2 D; where .n / is a sequence in .0; 1/ such that n ! 1: Since M is starshaped with respect to q; f and g are affine with respect to q; and T .M / f .M / \ g.M /; we have d.Tn f x; f Tn x/ D d.W .Tf x; q; n /; f W .T x; q; n // D d.W .Tf x; q; n /; W .f T x; f q; n //; D d.W .Tf x; q; n /; W .f T x; q; n //; n d.Tf x; f T x/;
f is affine q 2 F .f /
Property (I)
n R dist.f x; W .T x; q; // for any n: Since n 2 .0; 1/; we obtain d.Tn f x; f Tn x/ n R dist .f x; W .T x; q; n // n Rd.f x; Tn x/ for all x 2 M: This shows that Tn and f are n R-weakly commuting for each n: Similarly, Tn and g are n R-weakly commuting for each n: Moreover, d.Tn x; Tn y/ D d.W .T x; q; n /; W .T y; q; n // n d.T x; T y/ ¼ n max d.f x; gy/; dist.f x; W .T x; q; //; dist.gy; W .T y; q; //;
½ 1 Œdist.f x; W .T y; q; // C dist.gy; W .T x; q; // 2 for any n: Since n 2 .0; 1/; we conclude that
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¼ d.Tn x; Tn y/ n max d.f x; gy/; dist.f x; W .T x; q; n //; dist.gy; W .T y; q; n //;
½ 1 Œdist.f x; W .T y; q; n // C dist.gy; W .T x; q; n // 2 ¼
1 n max d.f x; gy/; d.f x; Tn x/; d.gy; Tn y/; Œd.f x; Tn y/ C d.gy; Tn x/ 2
½
for all x; y 2 M: Further, by Lemma D, for any n 1; there exists some xn 2 M such that xn D f xn D gxn D Tn xn D W .T x; q; n /: The compactness of T .M / implies the existence of a subsequence .xni / of .xn / such that T xni ! y 2 T .M /: We now find xni D f xni D gxni D Tni xni D W .T xni ; q; ni / ! y and, in addition, y 2 f .M / \ g.M / by T .M / f .M / \ g.M /: It follows from the continuity of T; f; and g that T xni ! T y; f xni ! f y; and gxni ! gy respectively. Hence, we get y D T y D f y D gy and, finally, F .T / \ F .f / \ F .g/ ¤ ¿: Corollary 8 ([13], Theorem 1). Let M be a nonempty subset of a normed linear space X and let T; f; and g be continuous self-maps of M: Suppose that M is starshaped with respect to q; f and g are affine with q 2 F .f / \ F .g/; T .M / f .M / \ g.M /; and T .M / is compact. If the pairs .T; f / and .T; g/ are R-subweakly commuting and satisfy the inequality ¼ kT x
T yk max kf x
gyk; dist.f x; ŒT x; q/; dist.gy; ŒT y; q/;
½ 1 dist.f x; ŒT y; q/ C dist.gy; ŒT x; q/ ; 2 for all x; y 2 M; then F .T / \ F .f / \ F .g/ is nonempty. Theorem 7. Let M be a nonempty subset of a convex metric space .X; d / with Property (I) and let T; f; and g be self-maps of M: Suppose that M is starshaped with respect to q; f and g are affine, T .M / f .M / \ g.M /; and T .M / is compact. If the pairs .T; f / and .T; g/ are R-subweakly commuting, T is .f; g/-nonexpansive, and either T or f; or g is continuous, then F .T / \ F .f / \ F .g/ is nonempty. Proof. Proceeding as in Theorem 6, we see that, for each n; there exists xn 2 M such that xn D f xn D gxn D Tn xn D W .T x; q; n /:
T. D. N ARANG AND S. C HANDOK
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The compactness of T .M / implies the existence of a subsequence .xni / of .xn / such that T xni ! y 2 T .M /: Thus, xni D f xni D gxni D Tni xni D W .T xni ; q; ni / ! y and, in addition, y 2 f .M / \ g.M / by T .M / f .M / \ g.M /: Hence, there exists u 2 M such that y D f u D gu: Consider d.T u; T xni / d.f u; gxni / D d.y; gxni / ! 0: Therefore, T xni ! T u D y; i.e., y D T u D f u D gu: As the R-subweak commutativity of .T; f / and .T; g/ implies weak compatibility, we get f y D f T u D Tf u D T y D T gu D gT u D gy: It follows from the continuity of either T or f; or g that T xni ! T y or f xni ! f y; or gxni ! gy: Hence, y D T y D f y D gy: Corollary 9 ([13], Theorem 2). Let M be a nonempty subset of a normed linear space X and let T; f; and g be self-maps of M: Suppose that M is starshaped with respect to q; f and g are affine, T .M / f .M / \ g.M /; and T .M / is compact. If the pairs .T; f / and .T; g/ are R-subweakly commuting, T is .f; g/-nonexpansive, and either T or f; or g is continuous, then F .T / \ F .f / \ F .g/ is nonempty. Acknowledgements The authors are thankful to the anonymous referee for careful reading and valuable suggestions. REFERENCES 1. M. A. Al-Thagafi, “Best approximation and fixed points in strong M -starshaped metric spaces,” Int. J. Math. Sci., 18, 613–616 (1995). 2. I. Beg, N. Shahzad, and M. Iqbal, “Fixed point theorems and best approximation in convex metric spaces,” J. Approx. Theory, 8, No. 12, 97–105 (1992). 3. M. D. Guay, K. L. Singh, and J. H. M. Whitfield, “Fixed point theorems for nonexpansive mappings in convex metric spaces,” in: S. P. Singh and J. H. Bury (editors), Proc. Conf. on Nonlinear Analysis, 80 (1982), pp. 179–189. 4. L. Habiniak, “Fixed point theorems and invariant approximations,” J. Approx. Theory, 56, 241–244 (1989). 5. G. Jungck, “Common fixed points for commuting and compatible maps on compacta,” Proc. Amer. Math. Soc., 103, 977–983 (1988). 6. G. Jungck and B. E. Rhoades, “Fixed point for set valued functions without continuity,” Indian J. Pure Appl. Math., 29, 227–238 (1998). 7. A. Naz, “Invariant approximations, noncommuting maps, and strongly M -starshaped metric spaces,” Demonstr. Math., 36 (2003). 8. R. P. Pant, “Common fixed points of noncommuting mappings,” J. Math. Anal. Appl., 188, 436–440 (1994). 9. N. Shahzad, “Invariant approximations and R-subweakly commuting maps,” J. Math. Anal. Appl., 257, 39–45 (2001). 10. N. Shahzad, “Noncommuting maps and best approximations,” Radovi Mat., 10, 77–83 (2001). 11. N. Shahzad, “Invariant approximations, generalized I-contractions, and R-subweakly commuting maps,” Fixed Point Theory Appl., 1, 79–86 (2005). 12. W. Takahashi, “A convexity in metric space and nonexpansive mappings I,” Kodai Math. Semin. Rep., 22, 142–149 (1970). 13. D. Wang, Y. Song, and X. Ma, “Common fixed points for noncommuting generalized .f; g/-nonexpansive maps,” Appl. Math. Sci., 2, 2597–2602 (2008).
