Common fixed points and invariant approximation of R-subweakly commuting maps in convex metric spaces
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 lit...
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Інститут математики НАН України
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irk-123456789-1662722020-02-19T01:27:39Z Common fixed points and invariant approximation of R-subweakly commuting maps in convex metric spaces Narang, T.D. Chandok, S. Статті 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. Встановлено достатні умови існування спільної нерухомої точки R-субслабко комутуючих відображень у рамках опуклого метричного простору. Як застосування, одержано різні результати щодо найкращих наближень для згаданого класу відображень, які узагальнюють інші відомі з літератури результати. 2010 Article Common fixed points and invariant approximation of R-subweakly commuting maps in convex metric spaces / T.D. Narang, S. Chandok // Український математичний журнал. — 2010. — Т. 62, № 10. — С. 1367–1376. — Бібліогр.: 13 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166272 517.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Narang, T.D. Chandok, S. Common fixed points and invariant approximation of R-subweakly commuting maps in convex metric spaces Український математичний журнал |
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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. |
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Narang, T.D. Chandok, S. |
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Narang, T.D. Chandok, S. |
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Narang, T.D. |
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Common fixed points and invariant approximation of R-subweakly commuting maps in convex metric spaces |
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Common fixed points and invariant approximation of R-subweakly commuting maps in convex metric spaces |
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Common fixed points and invariant approximation of R-subweakly commuting maps in convex metric spaces |
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Common fixed points and invariant approximation of R-subweakly commuting maps in convex metric spaces |
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Common fixed points and invariant approximation of R-subweakly commuting maps in convex metric spaces |
| title_sort |
common fixed points and invariant approximation of r-subweakly commuting maps in convex metric spaces |
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Інститут математики НАН України |
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2010 |
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| citation_txt |
Common fixed points and invariant approximation of R-subweakly commuting maps in convex metric spaces / T.D. Narang, S. Chandok // Український математичний журнал. — 2010. — Т. 62, № 10. — С. 1367–1376. — Бібліогр.: 13 назв. — англ. |
| series |
Український математичний журнал |
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AT narangtd commonfixedpointsandinvariantapproximationofrsubweaklycommutingmapsinconvexmetricspaces AT chandoks commonfixedpointsandinvariantapproximationofrsubweaklycommutingmapsinconvexmetricspaces |
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2025-07-14T21:05:46Z |
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2025-07-14T21:05:46Z |
| _version_ |
1837657905854152704 |
| fulltext |
UDC 517.5
T. D. Narang, S. Chandok (Guru Nanak Dev Univ., Amritsar, India)
COMMON FIXED POINTS AND INVARIANT
APPROXIMATION OF R-SUBWEAKLY COMMUTING MAPS
IN CONVEX METRIC SPACES
СПIЛЬНI НЕРУХОМI ТОЧКИ ТА IНВАРIАНТНE
НАБЛИЖЕННЯ R-СУБСЛАБКО КОМУТУЮЧИХ
ВIДОБРАЖЕНЬ В ОПУКЛИХ МЕТРИЧНИХ ПРОСТОРАХ
Sufficient conditions for the existence of a common fixed point of R-subweakly commuting mappings in the
framework of a convex metric space are derived. As applications, various best approximation results for this
class of mappings are obtained that generalize various results known in the literature.
Встановлено достатнi умови iснування спiльної нерухомої точки R-субслабко комутуючих вiдображень
у рамках опуклого метричного простору. Як застосування, одержано рiзнi результати щодо найкращих
наближень для згаданого класу вiдображень, якi узагальнюють iншi вiдомi з лiтератури результати.
1. Introduction and preliminaries. Applying fixed point theorems, many interesting
and useful results have been proved in approximation theory (see, e.g., [1, 2, 7, 9 –
11] and the references cited therein). This paper deals with the common fixed points
of R-subweakly commuting mappings in the framework of convex metric spaces. We
also establish results on invariant approximation for this class of mappings. The results
proved in the paper generalize and extend some of the results of [1, 2, 4, 7, 9, 11, 13].
To begin with, we recall some definitions and known facts to be used in the sequel.
For a metric space (X, d), a continuous mapping W : X ×X × [0, 1] → X is said
to be a convex structure on X if for all x, y ∈ X and λ ∈ [0, 1], we have
d(u,W (x, y, λ)) ≤ λd(u, x) + (1− λ)d(u, y)
for all u ∈ 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 said to be a convex set [12] if
W (x, y, λ) ∈ M for all x, y ∈ M and λ ∈ [0, 1]. A set M is said to be p-starshaped
[3] where p ∈ M , provided W (x, p, λ) ∈ M for all x ∈ M and λ ∈ [0, 1], i. e., if the
segment [p, x] = {W (x, p, λ) : 0 ≤ λ ≤ 1} joining p to x is contained in M for all
x ∈M . M is said to be starshaped if it is p-starshaped for some p ∈M .
Clearly, each convex set M is starshaped but converse is not true.
A convex metric space (X, d) is said to satisfy Property (I) [3] if for all x, y, q ∈ X
and λ ∈ [0, 1],
d(W (x, q, λ),W (y, q, λ)) ≤ λd(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, λ) = λx + (1 − λ)y for x, y ∈ X and
0 ≤ λ ≤ 1. There are many convex metric spaces which are not normed linear spaces
(see [3, 12]). Property (I) is always satisfied in a normed linear space.
c© T. D. NARANG, S. CHANDOK, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10 1367
1368 T. D. NARANG, S. CHANDOK
For a non-empty subset K of a metric space (X, d) and x ∈ X , an element y ∈ K
is said to be a best approximant to x or a best K-approximant to x if d(x, y) =
= d(x,K) ≡ inf{d(x, k) : k ∈ K}. The set of all such y ∈ K is denoted by PK(x).
For a convex subset K of a convex metric space (X, d), a mapping g : K → X is
said to be affine if for all x, y ∈ K, g(W (x, y, λ)) = W (gx, gy, λ) for all λ ∈ [0, 1].
g is said to be affine with respect to p ∈ K if g(W (x, p, λ)) = W (gx, gp, λ) for
all x ∈ K and λ ∈ [0, 1].
Suppose (X, d) is a metric space, M a nonempty subset of X , and S, T, f, g are
self mappings of M . T is said to be an (f, g)-contraction if there exists k ∈ [0, 1)
such that d(Tx, Ty) ≤ kd(fx, gy), ((f, g)-nonexpansive if d(Tx, Ty) ≤ d(fx, gy))
for all x, y ∈ M . If f = g then T is said to be an f -contraction (f -nonexpansive).
A point x ∈ M is a common fixed (coincidence) point of S and T if x = Sx = Tx
(Sx = Tx). The pair (S, T ) is said to be (a) commuting on M if STx = TSx
for all x ∈ M (b) R-weakly commuting [8] on M if there exists a real number
R > 0 such that d(TSx, STx) ≤ Rd(Tx, Sx) for all x ∈ M (c) compatible [5] if
lim d(TSxn, STxn) = 0 whenever (xn) is a sequence such that lim Txn = limSxn =
= t for some t in M (d) weakly compatible [6] if they commute at their coincidence
points, i.e., if STx = TSx whenever Sx = Tx.
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. T and S are called (e) R-
subcommuting [10] on M if for all x ∈M , there exists a real number R > 0 such that
d(TSx, STx) ≤ (R/λ) dist (Sx,W (Tx, q, λ)), λ ∈ (0, 1] (f) R-subweakly commut-
ing [9] on M if for all x ∈ M and λ ∈ [0, 1], there exists a real number R > 0 such
that d(TSx, STx) ≤ R dist (Sx,W (Tx, q, λ)).
Clearly Compatible maps are weakly compatible but the converse need not be true
(see [6]). Commuting mappings are R-subweakly commuting, but the converse may not
be true (see [9]). It is well known that R-subweakly commuting maps are R-weakly
commuting and R-weakly commuting are compatible but not conversely (see [5, 9]).
R-subcommuting and R-subweakly commuting maps are weakly compatible but their
converses do not hold (see [9 – 11]).
Throughout, we shall write M for closure of the set M , F (S) for the set of fixed
points of a mapping S, F (S, T ) (C(S, T )) for the set of fixed points (coincidence points)
of mappings S and T .
2. Main results. The following four lemmas will be used in proving results of this
paper.
Lemma A [11]. Let M be a closed subset of a metric space (X, d), and let T, S
be R-weakly commuting self mappings of M such that T (M) ⊆ S(M). Suppose there
exists k ∈ [0, 1) such that
d(Tx, Ty) ≤ kmax
{
d(Sx, Sy), d(Sx, Tx), d(Sy, Ty),
1
2
[d(Sx, Ty) + d(Sy, Tx)]
}
,
for all x, y ∈ 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, T
be R-weakly commuting self mappings of M such that T (M) ⊆ S(M). Suppose T is
an S-contraction. If T (M) is complete and T is continuous, then F (T ) ∩ F (S) is a
singleton.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
COMMON FIXED POINTS AND INVARIANT APPROXIMATION . . . 1369
Lemma C [1]. Let (X, d) be a convex metric space, M ⊂ X and x0 ∈ 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, T are continuous, the pairs (A,S) and (B, T ) are compatible
pairs, and that A(X) ⊂ T (X) and B(X) ⊂ S(X). If there exists r ∈ (0, 1) such that
d(Ax,By) ≤ rmax
{
d(Sx, Ty), d(Ax, Sx), d(By, Ty),
1
2
[d(Ax, Ty) + d(Sx,By)]
}
,
for all x, y ∈ X, then there is a unique point z in X such that Az = Bz = Sz = Tz =
= 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, S be continuous self mappings on M such that T (M) ⊆ S(M).
Suppose S is affine, p ∈ F (S), M is starshaped with respect to p, and T (M) is compact.
If T and S are R-subweakly commuting and satisfy
d(Tx, Ty) ≤ max
{
d(Sx, Sy),dist(Sx,W (Tx, p, λ)),dist(Sy,W (Ty, p, λ)),
1
2
[dist(Sx,W (Ty, p, λ)) + dist(Sy,W (Tx, p, λ))]
}
,
for all x, y ∈M, λ ∈ [0, 1), then M ∩ F (T ) ∩ F (S) 6= Ø.
Proof. For each n, define Tn : M → M by Tnx = W (Tx, p, λn), x ∈ M where
(λn) is a sequence in (0, 1) such that λn → 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) = W (Tx, p, λn) = W (Tx, Sp, λn) ∈ S(M)
and so Tn(M) ⊆ S(M) for each n. Consider
d(TnSx, STnx) = d(W (TSx, p, λn), SW (Tx, p, λn))
= d(W (TSx, p, λn),W (STx, Sp, λn)), S is affine
= d(W (TSx, p, λn),W (STx, p, λn)), p ∈ F (S)
≤ λnd(TSx, STx), Property (I)
≤ λnR dist(Sx,W (Tx, p, λ)),
for each n since λn ∈ (0, 1), we obtain
d(TnSx, STnx) ≤ λnR dist (Sx,W (Tx, p, λn)) ≤ λnRd(Sx, Tnx)
for all x ∈M . This shows that Tn and S are λnR-weakly commuting for each n. Also
d(Tnx, Tny) = d(W (Tx, p, λn),W (Ty, p, λn))
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1370 T. D. NARANG, S. CHANDOK
≤ λnd(Tx, Ty), Property (I)
≤ λn max
{
d(Sx, Sy),dist(Sx,W (Tx, p, λ)),dist(Sy,W (Ty, p, λ)),
1
2
[dist(Sx,W (Ty, p, λ)) + dist(Sy,W (Tx, p, λ))]
}
,
for each n since λn ∈ (0, 1), we obtain
d(Tnx, Tny) ≤ λn max
{
d(Sx, Sy),dist(Sx,W (Tx, p, λn)),dist(Sy,W (Ty, p, λn)),
1
2
[dist(Sx,W (Ty, p, λn)) + dist(Sy,W (Tx, p, λn))]
}
≤ λn max
{
d(Sx, Sy), d(Sx, Tnx), d(Sy, Tny),
1
2
[d(Sx, Tny) + d(Sy, Tnx)]
}
for all x, y ∈ M . Now by Lemma A, there exists some xn ∈ M such that F (Tn) ∩
∩ F (S) = {xn} for each n. The compactness of T (M) implies the existence of a
subsequence (xni) of (xn) such that xni → y ∈ M . By the continuity of T and S, we
have y ∈ F (T ) ∩ F (S). Hence M ∩ F (T ) ∩ F (S) 6= Ø.
Corollary 1.1 ([11], Theorem 2.2). Let M be a closed subset of a normed linear
space X, and let T, S be continuous self mappings on M such that T (M) ⊆ S(M).
Suppose S is linear, p ∈ F (S), M is starshaped with respect to p, and T (M) is compact.
If T and S are R-subweakly commuting and satisfy
‖Tx− Ty‖ ≤ max
{
‖Sx− Sy‖ ,dist(Sx, [Tx, p]),dist(Sy, [Ty, p]),
1
2
[
dist(Sx, [Ty, p]) + dist(Sy, [Tx, p])
]}
,
for all x, y ∈M , then M ∩ F (T ) ∩ F (S) 6= Ø.
Corollary 1.2. Let M be a closed subset of a normed linear space X and let T, S
be continuous self mappings on M such that T (M) ⊆ S(M). Suppose S is linear,
p ∈ F (S), M is starshaped with respect to p, and T (M) is compact. If T and S are
commuting and satisfy
‖Tx− Ty‖ ≤ max
{
‖Sx− Sy‖ ,dist(Sx, [Tx, p]),dist(Sy, [Ty, p]),
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
COMMON FIXED POINTS AND INVARIANT APPROXIMATION . . . 1371
1
2
[
dist(Sx, [Ty, p]) + dist(Sy, [Tx, p])
]}
,
for all x, y ∈M, then M ∩ F (T ) ∩ F (S) 6= Ø.
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, S be continuous self mappings on M such that T (M) ⊆ S(M).
Suppose S is affine, p ∈ F (S), M is starshaped with respect to p, and T (M) is
compact. If T, S are R-subweakly commuting and T is S-nonexpansive on M, then
M ∩ F (T ) ∩ F (S) 6= Ø.
Proof. Proceeding as in Theorem 1, we have
d(TnSx, STnx) ≤ λnR dist (Sx,W (Tx, p, λn)) ≤ λnRd(Sx, Tnx)
for all x ∈M . This shows that Tn and S are λnR-weakly commuting for each n. Also
d(Tnx, Tny) = d(W (Tx, p, λn),W (Ty, p, λn)) ≤ λnd(Tx, Ty) ≤ λnd(Sx, Sy).
Thus each Tn is S-contraction. Since T (M) is compact, by Lemma B, there exists some
xn ∈ M such that F (Tn) ∩ F (S) = {xn} for each n. Since (T (xn)) is a sequence in
T (M), there exists a subsequence (T (xni
)) with T (xni
) → x0 ∈ T (M). Since xni
=
= Tnixni = W (Txni , p, λni) → x0, the continuity of T and S imply x0 ∈ F (T, S).
Hence the result.
Corollary 2.1. Let M be a closed subset of a normed linear space X, and let T, S
be continuous self mappings on M such that T (M) ⊆ S(M). Suppose S is linear,
p ∈ F (S), M is starshaped with respect 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) 6= Ø.
Remark 1. Theorems 1, 2 and their Corollaries 1.1, 1.2, and 2.1 generalize and
extend the corresponding results of [1, 4, 9, and 11].
For a real number R > 0, let DR,S
M (x0) = PM (x0)∩GR,S
M (x0), where GR,S
M (x0) =
= {x ∈M : d(Sx, x0) ≤ (2R+ 1)dist(x0,M)}.
Applying Lemma C and Theorem 1, we prove the following theorem.
Theorem 3. Let T and S be self mappings of a convex metric space (X, d) with
Property (I), x0 ∈ F (T, S) and M be a subset of X such that T (∂M ∩M) ⊂ M.
Suppose S is affine on DR,S
M (x0), p ∈ F (S), DR,S
M (x0) is closed and starshaped with
respect to p, T (DR,S
M (x0)) is compact, and S(DR,S
M (x0)) = DR,S
M (x0). If T, S are
R-subweakly commuting on DR,S
M (x0) and satisfy
d(Tx, Ty) ≤
d(Sx, Sx0), if y = x0,
Q(x, y), if y ∈ DR,S
M (x0),
where
Q(x, y) = max
{
d(Sx, Sy),dist(Sx,W (Tx, p, λ)),dist(Sy,W (Ty, p, λ)),
1
2
[
dist(Sx,W (Ty, p, λ)) + dist(Sy,W (Tx, p, λ))
]}
,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1372 T. D. NARANG, S. CHANDOK
for all x ∈ DR,S
M (x0) ∪ {x0}, and λ ∈ [0, 1), then PM (x0) ∩ F (T ) ∩ F (S) 6= Ø.
Proof. Let x ∈ DR,S
M (x0). Then by Lemma C, x ∈ ∂M ∩ M and so Tx ∈ M
as T (∂M ∩M) ⊂ M . Since d(Tx, x0) = d(Tx, Tx0) ≤ d(Sx, Sx0) = d(Sx, x0) =
= dist(x0,M), we obtain Tx ∈ PM (x0). From the R-subweak commutativity of T and
S it follows that
d(STx, x0) = d(STx, Tx0)
≤ d(STx, TSx) + d(TSx, Tx0)
≤ Rd(Tx, Sx) + d(S2x, Sx0)
≤ R[d(Tx, Tx0) + d(Tx0, Sx)] + d(S2x, Sx0)
≤ R[dist(x0,M) + dist(x0,M)] + dist(x0,M)
≤ (2R+ 1)dist(x0,M).
This implies that Tx ∈ GR,S
M (x0). Consequently, Tx ∈ DR,S
M (x0) and so
T (DR,S
M (x0)) ⊆ DR,S
M (x0) = S(DR,S
M (x0)). Now, Theorem 1 guarantees that PM (x0)∩
∩ F (T ) ∩ F (S) 6= Ø.
Corollary 3.1 ([11], Theorem 2.5). Let T and S be self mappings of a normed linear
space X, x0 ∈ F (T, S) and M be a subset of X such that T (∂M∩M) ⊂M. Suppose S
is linear on DR,S
M (x◦), p ∈ F (S), DR,S
M (x0) is closed and starshaped with respect to p,
T (DR,S
M (x0)) is compact, and S(DR,S
M (x0)) = DR,S
M (x0). If T and S are R-subweakly
commuting on DR,S
M (x0) and satisfy
‖Tx− Ty‖ ≤
‖Sx− Sy‖ , if y = x0,
Q(x, y), if y ∈ DR,S
M (x0),
where
Q(x, y) = max
{
‖Sx− Sy‖ ,dist(Sx, [Tx, p]),dist(Sy, [Ty, p]),
1
2
[
dist(Sx, [Ty, p]) + dist(Sy, [Tx, p])
]}
,
for all x ∈ DR,S
M (x0) ∪ {x0}, then PM (x0) ∩ F (T ) ∩ F (S) 6= Ø.
Corollary 3.2 ([7], Theorem 2.3). Let T and S be self mappings of a convex met-
ric space (X, d) with Property (I), x0 ∈ F (T, S) and M be a subset of X such that
T (∂M ∩ M) ⊂ M and p ∈ F (S) ∩ M. Suppose T and S are R-subweakly com-
muting on DR,S
M (x0), T is S-nonexpansive on DR,S
M (x0) ∪ {x0} and S is affine on
DR,S
M (x0). If DR,S
M (x0) is closed and starshaped with respect to p, T (DR,S
M (x0)) is
compact, S(DR,S
M (x0)) = DR,S
M (x0), and T is continuous, then PM (x0)∩F (T )∩F (S)
is nonempty.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
COMMON FIXED POINTS AND INVARIANT APPROXIMATION . . . 1373
Theorem 4. Let T and S be self mappings of a convex metric space (X, d)
with Property (I), x0 ∈ F (T, S) and M be a subset of X such that T (∂M ∩M) ⊂
⊂ S(M) ⊂ M. Suppose S is affine on DR,S
M (x0), p ∈ F (S), DR,S
M (x0) is closed and
starshaped with respect to p, T (DR,S
M (x0)) is compact, and S(GR,S
M (x0))∩DR,S
M (x0) =
= S(DR,S
M (x0)) ⊂ DR,S
M (x0). If T and S are R-subweakly commuting and continuous
on DR,S
M (x0) and satisfy
d(Tx, Ty) ≤
d(Sx, Sx0), if y = x0,
Q(x, y), if y ∈ DR,S
M (x0),
where
Q(x, y) = max
{
d(Sx, Sy),dist(Sx,W (Tx, p, λ)),dist(Sy,W (Ty, p, λ)),
1
2
[
dist(Sx,W (Ty, p, λ)) + dist(Sy,W (Tx, p, λ))
]}
,
for all x ∈ DR,S
M (x0) ∪ {x0}, and λ ∈ [0, 1), then PM (x0) ∩ F (T ) ∩ F (S) 6= Ø.
Proof. Let x ∈ DR,S
M (x0). Then as in Theorem 3, Tx ∈ DR,S
M (x0), i.e.,
T (DR,S
M (x0)) ⊆ DR,S
M (x0). Also, d(W (x, x0, k), x0) < kd(x, x0)+(1−k)d(x0, x0) =
= kd(x, x0) < dist(x0,M) for all k ∈ (0, 1). Then by Lemma C, x ∈ ∂M ∩
∩M and so T (DR,S
M (x0)) ⊆ T (∂M ∩M) ⊂ S(M) and so we can choose y ∈ M
such that Tx = Sy. Since Sy = Tx ∈ PM (x0), it follows that y ∈ GR,S
M (x0).
Consequently, T (DR,S
M (x0)) ⊆ S(GR,S
M (x0)) ⊂ PM (x0). Therefore, T (DR,S
M (x0)) ⊆
⊆ S(GR,S
M (x0)) ∩DR,S
M (x0) = S(DR,S
M (x0)) ⊂ DR,S
M (x0). So, Theorem 1 guarantees
that PM (x0) ∩ F (T ) ∩ F (S) 6= Ø.
Corollary 4.1 ([11], Theorem 2.6). Let T and S be self mappings of a normed linear
space, x0 ∈ F (T, S) and M be a subset of X such that T (∂M ∩M) ⊂ S(M) ⊂ M.
Suppose S is linear on DR,S
M (x0), p ∈ F (S), DR,S
M (x0) is closed and starshaped with
respect to p, T (DR,S
M (x0)) is compact, and S(GR,S
M (x0))∩DR,S
M (x0) = S(DR,S
M (x0)) ⊂
⊂ DR,S
M (x0). If T and S are R-subweakly commuting and continuous on DR,S
M (x0) and
satisfy
‖Tx− Ty‖ ≤
‖Sx− Sy‖ , if y = x0,
Q(x, y), if y ∈ DR,S
M (x0),
where
Q(x, y) = max
{
‖Sx− Sy‖ ,dist(Sx, [Tx, p]),dist(Sy, [Ty, p]),
1
2
[
dist(Sx, [Ty, p]) + dist(Sy, [Tx, p])
]}
,
for all x ∈ DR,S
M (x0) ∪ {x0}, then PM (x0) ∩ F (T ) ∩ F (S) 6= Ø.
Corollary 4.2 ([7], Theorem 2.4). Let T and S be self mappings of a convex metric
space (X, d) with Property (I), x0 ∈ F (T, S) and M be a subset of X such that
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1374 T. D. NARANG, S. CHANDOK
T (∂M ∩M) ⊂ S(M) ⊂ M and p ∈ F (S) ∩M. Suppose T and S are R-subweakly
commuting on DR,S
M (x0), T is S-nonexpansive on DR,S
M (x0) ∪ {x0} and S is affine
on DR,S
M (x0). If DR,S
M (x0) is closed and starshaped with respect to, T (DR,S
M (x0)) is
compact, S(M) ∩ DR,S
M (x0) ⊂ S(DR,S
M (x0)) ⊂ DR,S
M (x0), and T is continuous, then
PM (x0) ∩ F (T ) ∩ F (S) is nonempty.
Remark 2. Theorems 3 and 4 remain valid when DR,S
M (x0) = PM (x0). If
S(PM (x0)) ⊂ PM (x0), then PM (x0) ⊂ GR,S
M (x0) and so DR,S
M (x0) = PM (x0). Con-
sequently, 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), x0 ∈ F (T, S) and M be a subset of X such that
T (∂M) ⊆ M. Suppose T is S-nonexpansive on PM (x0) ∪ {x0}, S is affine and con-
tinuous on PM (x0) and STx = TSx for all x in PM (x0). If PM (x0) is nonempty,
compact and starshaped with respect to p, p ∈ F (S), and if S(PM (x0)) = 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 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 ∈ 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
d(Tx, Ty) ≤ max
{
d(fx, gy),dist(fx,W (Tx, q, λ)),dist(gy,W (Ty, q, λ)),
1
2
[
dist(fx,W (Ty, q, λ)) + dist(gy,W (Tx, q, λ))
]}
,
for all x, y ∈M, and λ ∈ [0, 1), then F (T ) ∩ F (f) ∩ F (g) is nonempty.
Proof. For each n, define Tn : M → M by Tnx = W (Tx, q, λn), x ∈ D where
(λn) is a sequence in (0, 1) such that λn → 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(Tnfx, fTnx) = d(W (Tfx, q, λn), fW (Tx, q, λn))
= d(W (Tfx, q, λn),W (fTx, fq, λn)), f is affine
= d(W (Tfx, q, λn),W (fTx, q, λn)), q ∈ F (f)
≤ λnd(Tfx, fTx), Property (I)
≤ λnR dist(fx,W (Tx, q, λ))
for each n since λn ∈ (0, 1), we obtain
d(Tnfx, fTnx) ≤ λnR dist (fx,W (Tx, q, λn)) ≤ λnRd(fx, Tnx)
for all x ∈ M . This shows that Tn and f are λnR-weakly commuting for each n,
similarly Tn and g are λnR-weakly commuting for each n. Also
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COMMON FIXED POINTS AND INVARIANT APPROXIMATION . . . 1375
d(Tnx, Tny) = d(W (Tx, q, λn),W (Ty, q, λn))
≤ λnd(Tx, Ty)
≤ λn max
{
d(fx, gy),dist(fx,W (Tx, q, λ)),dist(gy,W (Ty, q, λ)),
1
2
[dist(fx,W (Ty, q, λ)) + dist(gy,W (Tx, q, λ))]
}
for each n since λn ∈ (0, 1), we obtain
d(Tnx, Tny) ≤ λn max
{
d(fx, gy),dist(fx,W (Tx, q, λn)),dist(gy,W (Ty, q, λn)),
1
2
[dist(fx,W (Ty, q, λn)) + dist(gy,W (Tx, q, λn))]
}
≤ λn max
{
d(fx, gy), d(fx, Tnx), d(gy, Tny),
1
2
[d(fx, Tny) + d(gy, Tnx)]
}
for all x, y ∈ M . Now by Lemma D, for each n ≥ 1, there exists some xn ∈ M such
that xn = fxn = gxn = Tnxn = W (Tx, q, λn). The compactness of T (M) implies
the existence of a subsequence (xni
) of (xn) such that Txni
→ y ∈ T (M). Now
xni = fxni = gxni = Tnixni = W (Txni , q, λni)→ y and also y ∈ f(M) ∩ g(M) by
T (M) ⊂ f(M) ∩ g(M). It follows from the continuity of T , f and g that Txni
→ Ty,
fxni
→ fy and gxni
→ gy respectively. So, we get y = Ty = fy = gy. Hence
F (T ) ∩ F (f) ∩ F (g) 6= Ø.
Corollary 6.1 ([13], Theorem 1). Let M be a nonempty subset of a normed linear
space X and 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 ∈ 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
‖Tx− Ty‖ ≤ max
{
‖fx− gy‖ ,dist(fx, [Tx, q]),dist(gy, [Ty, q]),
1
2
[
dist(fx, [Ty, q]) + dist(gy, [Tx, q])
]}
,
for all x, y ∈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 T, f and g be self maps of M. Suppose that M is starshaped with
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1376 T. D. NARANG, S. CHANDOK
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 ∈M such
that xn = fxn = gxn = Tnxn = W (Tx, q, λn). The compactness of T (M) implies
the existence of a subsequence (xni
) of (xn) such that Txni
→ y ∈ T (M). Now
xni = fxni = gxni = Tnixni = W (Txni , q, λni)→ y and also y ∈ f(M) ∩ g(M) by
T (M) ⊂ f(M) ∩ g(M). Hence there exists u ∈M such that y = fu = gu. Consider
d(Tu, Txni
) ≤ d(fu, gxni
) = d(y, gxni
)→ 0,
therefore Txni
→ Tu = y i.e. y = Tu = fu = gu. As R-subweak commutativity of
(T, f) and (T, g) imply weak compatibility, fy = fTu = Tfu = Ty = Tgu = gTu =
= gy. It follows from the continuity of either T or f or g that Txni
→ Ty or fxni
→ fy
or gxni
→ gy. Hence y = Ty = fy = gy.
Corollary 7.1 ([13], Theorem 2). Let M be a nonempty subset of a normed linear
space X and 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.
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Received 24.12.08,
after revision — 19.07.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
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