Coincidence Theorems for Certain Classes of Hybrid Contractions
© S. L. Singh and S. N. Mishra. 2010
Received: 27 August 2009
Accepted: 9 October 2009
Published: 13 October 2009
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.
Nadler's multivalued contraction theorem  (see also Covitz and Nadler, Jr. ) was subsequently generalized among others by Reich  and Ćirić . For a fundamental development of fixed point theory for multivalued maps, one may refer to Rus . 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  (see also Naimpally et al.  and Singh and Mishra ).
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  and Dhompongsa and Yingtaweesittikul , 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. . 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 , Zamfirescu , Moţ and Petruşel , 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 , Bellman and Lee , Bhakta and Mitra , Baskaran and Subrahmanyam , and Pathak et al. .
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:
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
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
As usual, we write (resp., for (resp., ) when
In all that follows is a strictly decreasing function from onto defined by
Let be a complete metric space and Assume that there exists such that
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.
Maps and are said to be Suzuki-Zamfirescu hybrid contraction if and only if there exists such that
A map satisfying
for all where , is called Ćirić-generalized contraction. Indeed, Ćirić  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  has shown that the conditions (CG) and (Z) are, respectively, the conditions ( ) and ( ) when is a single-valued map, where
(Z) for all .
The following example indicates the importance of the condition (S-Z).
Then does not satisfy the condition (KSC). Indeed, for
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 ).
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
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.
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.  when In Definition 2.6, if and is the identity map on the -orbital completeness will be denoted simply by -orbitally complete.
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
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 .
If then and we are done as is a coincidence point of and So we take . In an analogous manner, choose such that
If then and we are done So we take and continue the process. Inductively, we construct sequences and such that and
Now we see that
Therefore by the condition (S-Z),
Therefore the sequence is Cauchy in Since is -orbitally complete, it has a limit in Call it Let Then and
Now as in , we show that
Therefore by the condition (S-Z),
This yields (3.7);
Next we show that
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),
This implies (3.12), and so
So since is closed.
Further, if and are IT-commuting at that is, then , and this proves that is a fixed point of
for all If there exists a such that is -orbitally complete, then has a fixed point.
It comes from Theorem 3.1 when and is the identity map on
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.
This yields that is a common fixed point of and The uniqueness of the common fixed point follows easily.
for all Then has a unique fixed point.
It comes from Corollary 3.2 that has a fixed point. The uniqueness of the fixed point follows easily.
for all Then there exists such that
Since So (3.22) gives
This means that Corollary 3.3 applies as
Hence and have a coincidence at Clearly implies
Now we have the following:
for all Then has a unique fixed point.
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.
for all ?
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:
In the multistage process, some functional equations arise in a natural way (cf. Bellman  and Bellman and Lee ); 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.
where and are defined as follows:
(DP-3)For any there exists such that
(DP-4)There exists such that
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.
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
Therefore, the first inequality in (DP-2) becomes
Similarly, (4.8), (4.9), and (4.11) imply
So, from (4.12) and (4.13), we have
Since the above inequality is true for any and is arbitrary, we find from (4.17) that
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).
Suppose that the following conditions hold.
(i) and are bounded.
where is defined by (*). Then the functional equation (4.1) possesses a unique bounded solution in
It comes from Theorem 4.1 when and as the conditions (DP-3) and (DP-4) become redundant in the present context.
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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