Coincidence Theorems for Certain Classes of Hybrid Contractions

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:

(2.1)

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

(2.2)

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

(2.3)

As usual, we write (resp., for (resp., ) when

In all that follows is a strictly decreasing function from onto defined by

(2.4)

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

(2.5)

Then does not satisfy the condition (KSC). Indeed, for

(2.6)

and this does not imply

(2.7)

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

(2.8)

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

(3.1)

If then and we are done as is a coincidence point of and So we take . In an analogous manner, choose such that

(3.2)

If then and we are done So we take and continue the process. Inductively, we construct sequences and such that and

(3.3)

Now we see that

(3.4)

Therefore by the condition (S-Z),

(3.5)

This yields

(3.6)

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

(3.7)

for any Since there exists a positive integer such that

(3.8)

Therefore for

(3.9)

Therefore by the condition (S-Z),

(3.10)

Making

(3.11)

This yields (3.7);

Next we show that

(3.12)

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),

(3.13)

Hence

(3.14)

This implies (3.12), and so

(3.15)

Making

(3.16)

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

(3.17)

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

(3.18)

implies

(3.19)

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

(3.20)

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

(3.21)

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

(3.22)

implies

(3.23)

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

(3.24)

Since So (3.22) gives

(3.25)

and this implies (3.23). Therefore

(3.26)

This means that Corollary 3.3 applies as

(3.27)

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

(3.28)

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

(3.29)

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:

(3.30)

implies

(3.31)

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:

(4.1)

(4.2)

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

(4.3)

implies

(4.4)

where and are defined as follows:

(x2a)

(4.5)

(DP-3)For any there exists such that

(4.6)

(DP-4)There exists such that

(4.7)

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

(4.8)

where

Further,

(4.9)

(4.10)

Therefore, the first inequality in (DP-2) becomes

(4.11)

and this together with (4.8) and (4.10) implies

(4.12)

Similarly, (4.8), (4.9), and (4.11) imply

(4.13)

So, from (4.12) and (4.13), we have

(4.14)

Since the above inequality is true for any and is arbitrary, we find from (4.17) that

(4.15)

implies

(4.16)

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

(4.17)

implies

(4.18)

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.