#### Abstract

Some common fixed point theorems for -operator pairs are proved. As an application, the existence and uniqueness of the common solution for systems of functional equations arising in dynamic programming are discussed. Also, an example to validate all the conditions of the main result is presented.

#### 1. Introduction and Preliminaries

Jungck [1] introduced compatible mappings as a generalization of weakly commuting mappings. Jungck and Pathak [2] defined the concept of the biased mappings in order to generalize the concept of compatible mappings. Also, several authors [3–6] studied various classes of compatible mappings and proved common fixed point theorems for these classes. Recently, Hussain et al. [7] introduced -operator pairs as a new class of noncommuting self-mappings that contains the occasionally weakly compatible, and Sintunavarat and Kumam [8] introduced generalized -operator pairs that contain -operator pairs. On the other hand, fixed point theory has various applications in other fields, for instance, obtaining a solution of several classes of functional equations (or a system of functional equations) arising in dynamic programming (see [9–12]). Bellman and Lee [13], Zhang [14], and Chang and Ma [15] point out that the basic form of the functional equations of dynamic programming and the system of functional equations of dynamic programming are as follows: In this presented work, -operator pairs are compared with the various type of compatible mappings and it is shown that the -operator pairs reduce to symmetric Banach operator pairs under relaxed conditions. We omit the completeness condition of the space. Then some common fixed point theorems are proved for -operator pairs. Eventually, the results are used to show the existence and uniqueness of common solution for systems of functional equations without completeness of the space.

The set of fixed points of is denoted by . A point is a coincidence point (common fixed point) of and if . Let , denote the sets of all coincidence points and points of coincidence, respectively, of the pair . The pair is called a Banach operator pair if the set is -invariant, namely, . If is a Banach operator pair, then need not be a Banach operator pair. Let be a metric space and self-mappings on ; the pair is called as follows:(0)symmetric Banach operator if both and are Banach operator pairs [16];(1)compatible if , whenever is a sequence in such that and [1];(2)-operator pair if , for some [17];(3)-operator pair if there exists a point in such that see [7];(4)compatible of type if whenever is a sequence in such that and [6];(5)weakly -biased of type if implies that see [18];(6)compatible of type if whenever is a sequence in such that [3];(7)compatible of type if whenever is a sequence in such that [4];(8)compatible of type if whenever is a sequence in such that [5].

#### 2. -Operator Pair

Proposition 1. *Let and be self-mappings of metric space , and . If and are compatible, or compatible of type , or compatible of type , or compatible of type , or compatible of type , then is a -operator pair.*

*Proof. *If and are one of the assumptions listed, then and are weakly compatible and, hence, they are occasionally weakly compatible; then is a -operator pair.

*Notation 1. *The following example shows that the converse of Proposition 1 is not true, in general.

*Example 2. *Suppose that is a metric space with and are defined by

Then, , . On the other hand, for we have and Thus, is a -operator pair.

Now, suppose that is a sequence in defined by . Then, = , , and ,. Since so is not compatible.

,. Since thus is not compatible of type .

Since then is not compatible of type .

Since thus is not compatible of type .

Since therefore, is not compatible of type .

Proposition 3. *Let and be self-mappings of metric space . If is a -operator pair such that is singleton, then is symmetric Banach operator pair.*

*Proof. *By hypothesis, there is a point such that . Thus, and is a unique point of . Also, by Proposition [19] is weakly compatible and hence, by Lemma [19], is a unique common fixed point of and . Now, since the sets and are singleton, then , and ; that is, is symmetric Banach operator pair.

*Example 4. *Suppose that is a metric space with and are defined by

Then . Clearly is -operator pair and symmetric Banach operator pair.

Proposition 5. *Let and be self-mappings of metric space . If is a -operator pair and for all we have
**
where is a nondecreasing function satisfying the condition for , then is symmetric Banach operator pair.*

*Proof. *Since is a -operator pair, there is a point in such that . Now, if there is another point in and , then, by (16),
therefore, which is a contradiction. Then , that is, is singleton and, hence, by Proposition 3 is symmetric Banach operator pair.

Proposition 6. *Let and be self-mappings of metric space . If is a -operator pair such that is singleton, then is symmetric Banach operator pair.*

Corollary 7. *Let be an occasionally weakly compatible pair of self-mappings on that is singleton; then is symmetric Banach operator pair.*

*Proof. *Clearly, occasionally weakly compatible mappings are -operators; then by Proposition 6 the result is obtained.

#### 3. Common Fixed Point

*Definition 8 (see [20]). *A function is called an altering distance function if(i) is monotone increasing and continuous;(ii) if and only if .

Theorem 9. *Suppose that and are self-mappings of metric space . The pair is a -operator pair and, for all ,
**
where is an altering distance function and is a continuous function with if and only if . Then and have a unique common fixed point. Moreover, any fixed point of is a fixed point of and conversely.*

*Proof. *By hypothesis, there exists a point such that and
Suppose that there exists another point and , for which . Then, from (18), we get
accordingly, , which is a contradiction with definition of . Therefore, is singleton so *.* By using (19), ; thus, ; that is, is a unique common fixed point of and .

Now, suppose that is a fixed point of but , from (18),
thus, , which is a contradiction with definition of . Hence, . By using a similar argument, the conclusion will be obtained.

*Example 10. *Suppose that and is given by
Then is a metric space.

Let be defined as
Suppose that is defined as
Then is an altering distance function and is a continuous function with if and only if . Let be defined as
Now, we have the following cases for .

*Case 1. *.

(i) If and , then Since, , then relation (18) is established.

(ii) If and , then Since, , then relation (18) is established.

(iii) ; then Since, , then relation (18) is established.

(iv) ; then Since, , then relation (18) is established.

*Case 2. *.

In this case, it is easy to see that the relation (18) is hold.

Therefore, for all ,
Accordingly, the conditions of Theorem 9 are satisfied and is the unique common fixed point of and .

Suppose that is the collection of mappings which are upper semicontinuous, nondecreasing in each coordinate variable and for all [21].

Lemma 11 (see [21]). *If and where is a finite index set, then there exists some such that for all .*

Let , and be self-mappings of a metric space such that for all , where , ,.

Theorem 12. *Let , and be self-mappings of a metric space satisfying (31). If and are each -operator pairs, then , and have a unique common fixed point.*

*Proof. *By hypothesis there exist points such that and . If , then, from (31), we get
which implies that , a contradiction. Thus, . Suppose that there exists another point such that . Then condition (31) implies that . Hence, . That is, is singleton. Since , so and is a unique common fixed point of and . Similarly, is a unique common fixed point of and . Therefore, is a unique common fixed point of , and .

Corollary 13. *Let , and be self-mappings of a metric space satisfying , for all where . If and are each -operator pairs, then , and have a unique common fixed point.*

*Proof. *It is sufficient to set and take in Theorem 12.

Corollary 14. *Let be self-mappings of a metric space satisfying the following condition:
**
If is -operator pair, then and have a unique common fixed point.*

*Proof. *Considering that and in Theorem 12, the result is obtained.

Theorem 15. *Let be self-mappings of a metric space satisfying (33). Suppose that is nontrivial Banach operator pair on , then and have a unique common fixed point.*

*Proof. *By hypothesis and . From (33), for any
By Corollary 14 (with as identity map on ), has a unique fixed point on and hence and have a unique common fixed point.

Corollary 16. *Let be self-mappings of a metric space satisfying , for all where . If is a nontrivial Banach operator pair, then and have a unique common fixed point.*

*Proof. *It is sufficient to set and take in Theorem 15.

*Example 17. *Let be a metric space with the usual metric for all . Define , , and for all . Obviously, and for all and . Also, , ,, and . So, clearly and are each -operator pairs. Thus, all the conditions of Corollary 13 are satisfied and is the unique common fixed point of , and .

#### 4. Applications

In this section, we utilize the common fixed point theorems and their results to deduce the existence and uniqueness of the common solution for the system of functional equations in dynamic programming.

*Remark 18. *Many authors (e.g., see [9, 11–15, 22], or [3–5, 8–12, 17, 22] in [22]) used the fixed point theory to solve functional equations arising in dynamic programming on complete metric spaces such as Banach spaces. But, in the final section, we omit the completeness of the space and we state the result in the normed vector spaces and metric spaces setting.

Let be normed vector spaces, the state space, and the decision space. Denote by the set of all bounded real-valued functions on and . It is clear that is a metric space: where opt stands for sup or inf, , , and for . Suppose that the mappings and are defined:

Theorem 19. *Suppose that the following conditions are satisfied:**(i) for given , there exist such that
**(ii)**
for all , , , where , ,;**(iii) for , ;**(iv) there exist , such that
**
for some and for all , .**Then the system of functional equations (35) possesses a unique common solution in .*

*Proof. *Assume that . By condition (i) and (36), and are self-mappings of . Using (i) and (36), one can deduce that there exist such that
Note that
By virtue of (41) and (42),
From (40) and (43), we conclude that
It follows from (44) and (45) that
Equation (46) and (ii) lead to
which yields that
Let in (48); then
Now, we shall show that and are -operator pairs. By (iii) there exists ; thus, and by (iv) for all , , we have
for some . Therefore, for all
So
and, hence, . That is, is -operator pair. Similarly, is also -operator pair. Clearly, all the above process also holds for . Then all of the conditions of Theorem 15 are satisfied and is a unique common fixed point of , and ; that is, is a unique common solution of functional equations (35).

Corollary 20. *Suppose that the conditions , , and of Theorem 19 are satisfied. Moreover, if the following condition also holds:
**
for all , , , where , then the system of functional equations (35) possesses a unique common solution in .*

*Proof. *It is sufficient to set and take in Theorem 19.

*Example 21. *Let be normed vector spaces endowed with the usual norm defined by for all . Let be the state space and the decision space. Define and by
Define by
Now, for all ; , we define mappings and by
for which are defined as follows:
So,
for all ,. Also, and , ; then easily we have the following:

(i) for given , there exist such that if choose , and , for we have

(ii)