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
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
For the sake of brevity and proper reference, the assumption (KSC) will be called Kikkawa-Suzuki multivalued contraction.
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
The following example indicates the importance of the condition (S-Z).
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 ).
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.
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.
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
Now we see that
Therefore by the condition (S-Z),
Now as in , we show that
Therefore by the condition (S-Z),
Next we show that
Therefore using (3.7),
This implies (3.12), and so
This means that Corollary 3.3 applies as
Now we have the following:
Now we close this section with the following.
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.
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.
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
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.
where is defined by (*). Then the functional equation (4.1) possesses a unique bounded solution in
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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