- Research Article
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Coincidence Theorems for Certain Classes of Hybrid Contractions
Fixed Point Theory and Applications volume 2010, Article number: 898109 (2009)
Abstract
Coincidence and fixed point theorems for a new class of hybrid contractions consisting of a pair of single-valued and multivalued maps on an arbitrary nonempty set with values in a metric space are proved. In addition, the existence of a common solution for certain class of functional equations arising in dynamic programming, under much weaker conditions are discussed. The results obtained here in generalize many well known results.
1. Introduction
Nadler's multivalued contraction theorem [1] (see also Covitz and Nadler, Jr. [2]) was subsequently generalized among others by Reich [3] and Ćirić [4]. For a fundamental development of fixed point theory for multivalued maps, one may refer to Rus [5]. Hybrid contractive conditions, that is, contractive conditions involving single-valued and multivalued maps are the further addition to metric fixed point theory and its applications. For a comprehensive survey of fundamental development of hybrid contractions and historical remarks, refer to Singh and Mishra [6] (see also Naimpally et al. [7] and Singh and Mishra [8]).
Recently Suzuki [9, Theorem 2] obtained a forceful generalization of the classical Banach contraction theorem in a remarkable way. Its further outcomes by Kikkawa and Suzuki [10, 11], Moţ and Petruşel [12] and Dhompongsa and Yingtaweesittikul [13], are important contributions to metric fixed point theory. Indeed, [10, Theorem 2] (see Theorem 2.1 below) presents an extension of [9, Theorem 2] and a generalization of the multivalued contraction theorem due to Nadler, Jr. [1]. In this paper we obtain a coincidence theorem (Theorem 3.1) for a pair of single-valued and multivalued maps on an arbitrary nonempty set with values in a metric space and derive fixed point theorems which generalize Theorem 2.1 and certain results of Reich [3], Zamfirescu [14], Moţ and Petruşel [12], and others. Further, using a corollary of Theorem 3.1, we obtain another fixed point theorem for multivalued maps. We also deduce the existence of a common solution for Suzuki-Zamfirescu type class of functional equations under much weaker contractive conditions than those in Bellman [15], Bellman and Lee [16], Bhakta and Mitra [17], Baskaran and Subrahmanyam [18], and Pathak et al. [19].
2. Suzuki-Zamfirescu Hybrid Contraction
For the sake of brevity, we follow the following notations, wherein and
are maps to be defined specifically in a particular context while
and
are the elements of specific domains:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ1_HTML.gif)
Consistent with Nadler, Jr. [20, page 620], will denote an arbitrary nonempty set,
a metric space, and
(resp.
) the collection of nonempty closed (resp., closed and bounded) subsets of
For
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ2_HTML.gif)
The hyperspace is called the generalized Hausdorff metric space induced by the metric
on
For any subsets of
,
denotes the ordinary distance between the subsets
and
while
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ3_HTML.gif)
As usual, we write (resp.,
for
(resp.,
) when
In all that follows is a strictly decreasing function from
onto
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ4_HTML.gif)
Recently Kikkawa and Suzuki [10] obtained the following generalization of Nadler, Jr. [1].
Theorem 2.1.
Let be a complete metric space and
Assume that there exists
such that
(KSC)
for all Then
has a fixed point.
For the sake of brevity and proper reference, the assumption (KSC) will be called Kikkawa-Suzuki multivalued contraction.
Definition 2.2.
Maps and
are said to be Suzuki-Zamfirescu hybrid contraction if and only if there exists
such that
(S-Z) implies
for all
A map satisfying
(CG)
for all where
, is called Ćirić-generalized contraction. Indeed, Ćirić [4] showed that a Ćirić generalized contraction has a fixed point in a
-orbitally complete metric space
It may be mentioned that in a comprehensive comparison of 25 contractive conditions for a single-valued map in a metric space, Rhoades [21] has shown that the conditions (CG) and (Z) are, respectively, the conditions () and (
) when
is a single-valued map, where
(Z) for all
.
Obiviously, (Z) implies (CG). Further, Zamfirescu's condition [14] is equivalent to (Z) when is single-valued (see Rhoades [21, pages 259 and 266]).
The following example indicates the importance of the condition (S-Z).
Example 2.3.
Let be endowed with the usual metric and let
and
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ5_HTML.gif)
Then does not satisfy the condition (KSC). Indeed, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ6_HTML.gif)
and this does not imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ7_HTML.gif)
Further, as easily seen, does not satisfy (CG) for
. However, it can be verified that the pair
and
satisfies the assumption (S-Z). Notice that
does not satisfy the condition (S-Z) when
and
is the identity map.
We will need the following definitions as well.
Definition 2.4 (see [4]).
An orbit for at
is a sequence
A space
is called
-orbitally complete if and only if every Cauchy sequence of the form
converges in
Definition 2.5.
Let and
If for a point
there exists a sequence
in
such that
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ8_HTML.gif)
is the orbit for at
We will use
as a set and a sequence as the situation demands. Further, a space
is
-orbitally complete if and only if every Cauchy sequence of the form
converges in
As regards the existence of a sequence in the metric space
, the sufficient condition is that
However, in the absence of this requirement, for some
a sequence
may be constructed some times. For instance, in the above example, the range of
is not contained in the range of
but we have the sequence
for
So we have the following definition.
Definition 2.6.
If for a point there exists a sequence
in
such that the sequence
converges in
then
is called
-orbitally complete with respect to
or simply
-orbitally complete.
We remark that Definitions 2.5 and 2.6 are essentially due to Rhoades et al. [22] when In Definition 2.6, if
and
is the identity map on
the
-orbital completeness will be denoted simply by
-orbitally complete.
Definition 2.7 ([23], see also [8]).
Maps and
are IT-commuting at
if
We remark that IT-commuting maps are more general than commuting maps, weakly commuting maps and weakly compatible maps at a point. Notice that if is also single-valued, then their IT-commutativity and commutativity are the same.
3. Coincidence and Fixed Point Theorems
Theorem 3.1.
Assume that the pair of maps and
is a Suzuki-Zamfirescu hybrid contraction such that
If there exists an
such that
is
-orbitally complete, then
and
have a coincidence point; that is, there exists
such that
Further, if then
and
have a common fixed point provided that
and
are IT-commuting at
and
is a fixed point of
.
Proof.
Without any loss of generality, we may take and
a nonconstant map. Let
Pick
We construct two sequences
and
in the following manner. Since
we take an element
such that
Similarly, we choose
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ9_HTML.gif)
If then
and we are done as
is a coincidence point of
and
So we take
. In an analogous manner, choose
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ10_HTML.gif)
If then
and we are done
So we take
and continue the process. Inductively, we construct sequences
and
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ11_HTML.gif)
Now we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ12_HTML.gif)
Therefore by the condition (S-Z),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ13_HTML.gif)
This yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ14_HTML.gif)
where
Therefore the sequence is Cauchy in
Since
is
-orbitally complete, it has a limit in
Call it
Let
Then
and
Now as in [10], we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ15_HTML.gif)
for any Since
there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ16_HTML.gif)
Therefore for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ17_HTML.gif)
Therefore by the condition (S-Z),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ18_HTML.gif)
Making
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ19_HTML.gif)
This yields (3.7);
Next we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ20_HTML.gif)
for any If
then it holds trivially. So we suppose
such that
Such a choice is permissible as
is not a constant map.
Therefore using (3.7),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ21_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ22_HTML.gif)
This implies (3.12), and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ23_HTML.gif)
Making
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ24_HTML.gif)
So since
is closed.
Further, if and
are IT-commuting at
that is,
then
, and this proves that
is a fixed point of
We remark that, in general, a pair of continuous commuting maps at their coincidences need not have a common fixed point unless has a fixed point (see, e.g., [6–8]).
Corollary 3.2.
Let Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ25_HTML.gif)
for all If there exists a
such that
is
-orbitally complete, then
has a fixed point.
Proof.
It comes from Theorem 3.1 when and
is the identity map on
The following two results are the extensions of Suzuki [9, Theorem 2]. Corollary 3.3 also generalizes the results of Kikkawa and Suzuki [10, Theorem 3] and Jungck [24].
Corollary 3.3.
Let be such that
and
is an
-orbitally complete subspace of
Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ26_HTML.gif)
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ27_HTML.gif)
for all Then
and
have a coincidence point; that is, there exists
such that
Further, if and
and
commute at
then
and
have a unique common fixed point.
Proof.
Set for every
Then it comes from Theorem 3.1 that there exists
such that
Further, if
and
and
commute at
then
Also,
and this implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ28_HTML.gif)
This yields that is a common fixed point of
and
The uniqueness of the common fixed point follows easily.
Corollary 3.4.
Let be such that
is
-orbitally complete. Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ29_HTML.gif)
for all Then
has a unique fixed point.
Proof.
It comes from Corollary 3.2 that has a fixed point. The uniqueness of the fixed point follows easily.
Theorem 3.5.
Let and
be such that
and let
be
-orbitally complete. Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ30_HTML.gif)
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ31_HTML.gif)
for all Then there exists
such that
Proof.
Choose Define a single-valued map
as follows. For each
let
be a point of
which satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ32_HTML.gif)
Since So (3.22) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ33_HTML.gif)
and this implies (3.23). Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ34_HTML.gif)
This means that Corollary 3.3 applies as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ35_HTML.gif)
Hence and
have a coincidence at
Clearly
implies
Now we have the following:
Theorem 3.6.
Let and let
be
-orbitally complete. Assume that there exists
such that
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ36_HTML.gif)
for all Then
has a unique fixed point.
Proof.
For , define a single-valued map
as follows. For each
let
be a point of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ37_HTML.gif)
Now following the proof technique of Theorem 3.5 and using Corollary 3.4, we conclude that has a unique fixed point
Clearly
implies that
Now we close this section with the following.
Question 1.
Can we replace Assumption (3.17) in Corollary 3.2 by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ38_HTML.gif)
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ39_HTML.gif)
for all ?
4. Applications
Throughout this section, we assume that and
are Banach spaces,
and
Let
denote the field of reals,
and
Viewing
and
as the state and decision spaces respectively, the problem of dynamic programming reduces to the problem of solving the functional equations:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ40_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ41_HTML.gif)
In the multistage process, some functional equations arise in a natural way (cf. Bellman [15] and Bellman and Lee [16]); see also [17–19, 25]). In this section, we study the existence of the common solution of the functional equations (4.1), (4.2) arising in dynamic programming.
Let denote the set of all bounded real-valued functions on
For an arbitrary
, define
Then
is a Banach space. Suppose that the following conditions hold:
(DP-1) and
are bounded.
(DP-2)Let be defined as in the previous section. There exists
such that for every
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ42_HTML.gif)
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ43_HTML.gif)
where and
are defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ44_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ45_HTML.gif)
(DP-3)For any there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ46_HTML.gif)
(DP-4)There exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ47_HTML.gif)
Theorem 4.1.
Assume that the conditions (DP-1)–(DP-4) are satisfied. If is a closed convex subspace of
then the functional equations (4.1) and (4.2) have a unique common bounded solution.
Proof.
Notice that is a complete metric space, where
is the metric induced by the supremum norm on
By (DP-1)
and
are self-maps of
The condition (DP-3) implies that
It follows from (DP-4) that
and
commute at their coincidence points.
Let be an arbitrary positive number and
Pick
and choose
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ48_HTML.gif)
where
Further,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ49_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ50_HTML.gif)
Therefore, the first inequality in (DP-2) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ51_HTML.gif)
and this together with (4.8) and (4.10) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ52_HTML.gif)
Similarly, (4.8), (4.9), and (4.11) imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ53_HTML.gif)
So, from (4.12) and (4.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ54_HTML.gif)
Since the above inequality is true for any and
is arbitrary, we find from (4.17) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ55_HTML.gif)
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ56_HTML.gif)
Therefore Corollary 3.3 applies, wherein and
correspond, respectively, to the maps
and
Therefore,
and
have a unique common fixed point
that is,
is the unique bounded common solution of the functional equations (4.1) and (4.2).
Corollary 4.2.
Suppose that the following conditions hold.
(i) and
are bounded.
(ii)For defined earlier (cf. (DP-2) above), there exists
such that for every
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ57_HTML.gif)
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F898109/MediaObjects/13663_2009_Article_1360_Equ58_HTML.gif)
where is defined by (*). Then the functional equation (4.1) possesses a unique bounded solution in
Proof.
It comes from Theorem 4.1 when and
as the conditions (DP-3) and (DP-4) become redundant in the present context.
References
Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.
Covitz H, Nadler SB Jr.: Multi-valued contraction mappings in generalized metric spaces. Israel Journal of Mathematics 1970, 8: 5–11. 10.1007/BF02771543
Reich S: Fixed points of contractive functions. Bollettino della Unione Matematica Italiana 1972, 5: 26–42.
Ćirić LB: Fixed points for generalized multi-valued contractions. Matematički Vesnik 1972, 9(24): 265–272.
Rus IA: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca, Romania; 2001:198.
Singh SL, Mishra SN: Nonlinear hybrid contractions. Journal of Natural & Physical Sciences 1994, 5–8: 191–206.
Naimpally SA, Singh SL, Whitfield JHM: Coincidence theorems for hybrid contractions. Mathematische Nachrichten 1986, 127: 177–180. 10.1002/mana.19861270112
Singh SL, Mishra SN: Coincidences and fixed points of nonself hybrid contractions. Journal of Mathematical Analysis and Applications 2001,256(2):486–497. 10.1006/jmaa.2000.7301
Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society 2008,136(5):1861–1869.
Kikkawa M, Suzuki T: Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(9):2942–2949. 10.1016/j.na.2007.08.064
Kikkawa M, Suzuki T: Some similarity between contractions and Kannan mappings. Fixed Point Theory and Applications 2008, 2008:-8.
Moţ G, Petruşel A: Fixed point theory for a new type of contractive multivalued operators. Nonlinear Analysis: Theory, Methods & Applications 2009,70(9):3371–3377. 10.1016/j.na.2008.05.005
Dhompongsa S, Yingtaweesittikul H: Fixed points for multivalued mappings and the metric completeness. Fixed Point Theory and Applications 2009, 2009:-15.
Zamfirescu T: Fix point theorems in metric spaces. Archiv der Mathematik 1972, 23: 292–298. 10.1007/BF01304884
Bellman R: Methods of Nonliner Analysis. Vol. II, Mathematics in Science and Engineering. Volume 61. Academic Press, New York, NY, USA; 1973:xvii+261.
Bellman R, Lee ES: Functional equations in dynamic programming. Aequationes Mathematicae 1978,17(1):1–18. 10.1007/BF01818535
Bhakta PC, Mitra S: Some existence theorems for functional equations arising in dynamic programming. Journal of Mathematical Analysis and Applications 1984,98(2):348–362. 10.1016/0022-247X(84)90254-3
Baskaran R, Subrahmanyam PV: A note on the solution of a class of functional equations. Applicable Analysis 1986,22(3–4):235–241. 10.1080/00036818608839621
Pathak HK, Cho YJ, Kang SM, Lee BS: Fixed point theorems for compatible mappings of type (P) and applications to dynamic programming. Le Matematiche 1995,50(1):15–33.
Nadler SB Jr.: Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Mathematics. Volume 4. Marcel Dekke, New York, NY, USA; 1978:xvi+707.
Rhoades BE: A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society 1977, 226: 257–290.
Rhoades BE, Singh SL, Kulshrestha C: Coincidence theorems for some multivalued mappings. International Journal of Mathematics and Mathematical Sciences 1984,7(3):429–434. 10.1155/S0161171284000466
Itoh S, Takahashi W: Single-valued mappings, multivalued mappings and fixed-point theorems. Journal of Mathematical Analysis and Applications 1977,59(3):514–521. 10.1016/0022-247X(77)90078-6
Jungck G: Commuting mappings and fixed points. The American Mathematical Monthly 1976,83(4):261–263. 10.2307/2318216
Singh SL, Mishra SN: On a Ljubomir Ćirić fixed point theorem for nonexpansive type maps with applications. Indian Journal of Pure and Applied Mathematics 2002,33(4):531–542.
Acknowledgments
The authors thank the referees and Professor M. A. Khamsi for their appreciation and suggestions regarding this work. This research is supported by the Directorate of Research Development, Walter Sisulu University.
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Singh, S., Mishra, S. Coincidence Theorems for Certain Classes of Hybrid Contractions. Fixed Point Theory Appl 2010, 898109 (2009). https://doi.org/10.1155/2010/898109
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DOI: https://doi.org/10.1155/2010/898109