 Research Article
 Open Access
 Published:
Solvability and Algorithms for Functional Equations Originating from Dynamic Programming
Fixed Point Theory and Applications volume 2011, Article number: 701519 (2011)
Abstract
The main purpose of this paper is to study the functional equation arising in dynamic programming of multistage decision processes ,. A few iterative algorithms for solving the functional equation are suggested. The existence, uniqueness and iterative approximations of solutions for the functional equation are discussed in the Banach spaces and and the complete metric space , respectively. The properties of solutions, nonnegative solutions, and nonpositive solutions and the convergence of iterative algorithms for the functional equation and other functional equations, which are special cases of the above functional equation, are investigated in the complete metric space , respectively. Eight nontrivial examples which dwell upon the importance of the results in this paper are also given.
1. Introduction
The existence, uniqueness, and iterative approximations of solutions for several classes of functional equations arising in dynamic programming were studied by a lot of researchers; see [1–23] and the references therein. Bellman [3], Bhakta and Choudhury [7], Liu [12], Liu and Kang [15], and Liu et al. [18] investigated the following functional equations:
and gave some existence and uniqueness results and iterative approximations of solutions for the functional equations in . Liu and Kang [14] and Liu and Ume [17] generalized the results in [3, 7, 12, 15, 18] and studied the properties of solutions, nonpositive solutions and nonnegative solutions and the convergence of iterative approximations for the following functional equations, respectively
in .
The purpose of this paper is to introduce and study the following functional equations arising in dynamic programming of multistage decision processes:
where denotes or , and stand for the state and decision vectors, respectively, , and represent the transformations of the processes, and represents the optimal return function with initial .
This paper is divided into four sections. In Section 2, we recall a few basic concepts and notations, establish several lemmas that will be needed further on, and suggest ten iterative algorithms for solving the functional equations (1.3)–(1.9). In Section 3, by applying the Banach fixedpoint theorem, we establish the existence, uniqueness, and iterative approximations of solutions for the functional equation (1.3) in the Banach spaces and , respectively. By means of two iterative algorithms defined in Section 2, we obtain the existence, uniqueness, and iterative approximations of solutions for the functional equation (1.3) in the complete metric space . Under certain conditions, we investigate the behaviors of solutions, nonpositive solutions, and nonnegative solutions and the convergence of iterative algorithms for the functional equations (1.3)–(1.7), respectively, in . In Section 4, we construct eight nontrivial examples to explain our results, which extend and improve substantially the results due to Bellman [3], Bhakta and Choudhury [7], Liu [12], Liu and Kang [14, 15], Liu and Ume [17], Liu et al. [18], and others.
2. Lemmas and Algorithms
Throughout this paper, we assume that , , , denotes the set of positive integers, and, for each , denotes the largest integer not exceeding . Let and be real Banach spaces, the state space, and the decision space. Define
Clearly and are Banach spaces with norm .
For any and , let
where and . It is easy to see that is a countable family of pseudometrics on . A sequence in is said to be converge to a point if, for any as and to be a Cauchy sequence if, for any , as . It is clear that is a complete metric space.
Lemma 2.1.
Let . Then
(a),
(b) for , ,
(c) for ,
(d) for ,
(e).
Proof.
Clearly (a)–(d) are true. Now we show (e). Note that (e) holds for . Suppose that (e) is true for some . It follows from (a) and Lemma 2.1 in [17] that
Hence (e) holds for any . This completes the proof.
Lemma 2.2.
Let and . Then
(a),
(b).
Proof.
It is clear that (a) is true for . Suppose that (a) is also true for some . Using Lemma 2.3 in [19] and Lemma 2.1, we infer that
That is, (a) is true for . Therefore (a) holds for any . Similarly we can prove (b). This completes the proof.
Lemma 2.3.
Let , be convergent sequences in . Then
Proof.
Put for . In view of Lemma 2.1 we deduce that
which yields that
This completes the proof.
Lemma 2.4.

(a)
Assume that is a mapping such that is bounded for some . Then
(28)

(b)
Assume that are mappings such that and are bounded for some . Then
(29)
Proof.
Now we show (a). If , (a) holds clearly. Suppose that . Note that
It follows that
which implies that
Next we show (b). If , (b) is true. Suppose that . Note that
which yields that
It follows that
which gives that
This completes the proof.
Algorithm 1.
For any , compute by
where
Algorithm 2.
For any , compute by (2.17) and (2.18).
Algorithm 3.
For any , compute by (2.17) and (2.18).
Algorithm 4.
For any , compute by
Algorithm 5.
For any , compute by
Algorithm 6.
For any , compute by
Algorithm 7.
For any , compute by
Algorithm 8.
For any , compute by
Algorithm 9.
For any , compute by
Algorithm 10.
For any , compute by
3. The Properties of Solutions and Convergence of Algorithms
Now we prove the existence, uniqueness, and iterative approximations of solutions for the functional equation (1.3) in and , respectively, by using the Banach fixedpoint theorem.
Theorem 3.1.
Let be compact. Let and satisfy the following conditions:
(C1) is bounded in ;
(C2) for some constant ;
(C3)for each fixed ,
uniformly for , respectively.
Then the functional equation (1.3) possesses a unique solution , and the sequence generated by Algorithm 1 converges to and has the error estimate
Proof.
Define a mapping by
Let and and . It follows from (C1), (C3), and the compactness of that there exist constants , , and satisfying
Using (3.3)–(3.5), (C2), and Lemmas 2.1 and 2.4, we get that
In light of (C2), (3.3), (3.5)–(3.9), and Lemmas 2.1 and 2.4, we deduce that for all with
Thus (3.10), (3.11), and (2.17) ensure that the mapping and Algorithm 1 are well defined.
Next we assert that the mapping is a contraction. Let , , and . Suppose that . Choose such that
Lemma 2.1 and (3.12) lead to
which implies that
Letting in the above inequality, we know that
Similarly we conclude that (3.15) holds for . The Banach fixedpoint theorem yields that the contraction mapping has a unique fixed point . It is easy to verify that is also a unique solution of the functional equation (1.3) in . By means of (2.17), (2.18), (3.15), and
we infer that
which yields that
and the sequence converges to by (2.18). This completes the proof.
Dropping the compactness of and (C3) in Theorem 3.1, we conclude immediately the following result.
Theorem 3.2.
Let and satisfy conditions (C1) and (C2). Then the functional equation (1.3) possesses a unique solution and the sequence generated by Algorithm 2 converges to and satisfies (3.2).
Next we prove the existence, uniqueness, and iterative approximations of solution for the functional equation (1.3) in .
Theorem 3.3.
Let and satisfy condition (C2) and the following two conditions:
(C4) is bounded on for each ;
(C5) for all .
Then the functional equation (1.3) possesses a unique solution , and the sequences and generated by Algorithms 3 and 4, respectively, converge to and have the error estimates
Proof.
Define a mapping by (3.3). Let and . Thus (C4) and (C5) guarantee that there exist and satisfying
Using (3.3), (3.20), (C2), (C5), and Lemmas 2.1 and 2.4, we infer that
which means that is a selfmapping in . Consequently, Algorithms 3 and 4 are well defined.
Now we claim that
Let , , , and . Suppose that . Select such that (3.12) holds. Thus (3.3), (3.12), (C2), (C5), and Lemma 2.1 ensure that
which implies that
Similarly we conclude that (3.24) holds for . As in (3.24), we get that (3.22) holds.
Let . It follows from Algorithm 4 that
and (3.22) leads to
which yields that is a Cauchy sequence in the complete metric space , and hence converges to some . In light of (3.22) and (C2), we know that
That is, the mapping is nonexpansive. It follows from (3.27) and Algorithm 4 that
that is, . Suppose that there exists with . Consequently there exists some satisfying . It follows from (3.22) and (C2) that
which is a contradiction. Hence the mapping has a unique fixed point , which is a unique solution of the functional equation (1.3) in . Letting in (3.26), we infer that
It follows from Algorithm 3, (2.18), and (3.22) that
which gives that as . This completes the proof.
Next we investigate the behaviors of solutions for the functional equations (1.3)–(1.5) and discuss the convergence of Algorithms 4–6 in , respectively.
Theorem 3.4.
Let , and satisfy the following conditions:
(C6) for all ;
(C7) for all ;
(C8).
Then the functional equation (1.3) possesses a solution satisfying conditions (C9)–(C12) below:
(C9)the sequence generated by Algorithm 4 converges to , where with for all ;
(C10) for all ;
(C11) for any , and , , for all ;
(C12) is unique relative to condition (C11).
Proof.
First of all we assert that
Suppose that there exists some with . It follows from that
That is,
which is impossible. That is, (3.32) holds. Let the mapping be defined by (3.3) in . Note that (C6) and (C7) imply (C4) and (C5) by (3.32) and , respectively. As in the proof of Theorem 3.3, by (C8) we conclude that the mapping maps into and satisfies
That is, the mapping is nonexpansive.
Let the sequence be generated by Algorithm 4 and with for all . We now claim that for each
Clearly (3.37) holds for . Assume that (3.37) is true for some . It follows from (C6)–(C8), (3.32), Algorithm 4, and Lemmas 2.1 and 2.4 that
That is, (3.37) is true for . Hence (3.37) holds for each .
Next we claim that is a Cauchy sequence in . Let , , and . Suppose that . Choose with
It follows from (3.39), (C8), and Lemmas 2.2 and 2.3 that
Therefore there exist and satisfying
In a similar method, we can derive that (3.41) holds also for . Proceeding in this way, we choose and for such that
On account of , (C7), (3.37), (3.41), and (3.42), we gain that
which yields that
Letting in the above inequality, we infer that
It follows from and (3.45) that is a Cauchy sequence in and it converges to some . Algorithm 4 and (3.36) lead to
which yields that . That is, the functional equation (1.3) possesses a solution .
Now we show (C10). Let . Put . It follows from (3.37), (C7), and that
that is, (C10) holds.
Next we prove (C11). Given , , and , , for . Put . Note that (C7) implies that
In view of (3.32), (3.37), (3.48), and , we know that
which means that .
Finally we prove (C12). Assume that the functional equation (1.3) has another solution that satisfies (C11). Let and . Suppose that . Select with
On account of (3.50), (C8), and Lemma 2.1, we conclude that there exist and , satisfying
that is,
Similarly we can prove that (3.52) holds for . Proceeding in this way, we select and for and such that
It follows from (3.52) and (3.53) that
Since is arbitrary, we conclude immediately that . This completes the proof.
Theorem 3.5.
Let , and satisfy conditions (C6)–(C8). Then the functional equation (1.4) possesses a solution satisfying conditions (C10)–(C12) and the following two conditions:
(C13) the sequence generated by Algorithm 5 converges to , where with for all ;
(C14) if , and are nonnegative and there exists a constant such that
then is nonnegative.
Proof.
It follows from Theorem 3.4 that the functional equation (1.4) has a solution that satisfies (C10)–(C13). Now we show (C14). Given , and . It follows from Lemma 2.2, (3.55), and (1.4) that there exist and such that
That is,
Proceeding in this way, we choose and for and such that
It follows from (3.57) and (3.58) that
In terms of (C8), (C11), and (3.55), we see that as . Letting in (3.59), we get that . Since is arbitrary, we infer immediately that . This completes the proof.
Theorem 3.6.
Let , and satisfy conditions (C6), (C7), and the following condition:
(C15), and are nonnegative and .
Then the functional equation (1.6) possesses a solution satisfying for any , where the sequence is generated by Algorithm 7 with , and for all .
Proof.
We are going to prove that, for any ,
Using and Algorithm 7, we gain that
that is, (3.60) holds for . Assume that (3.60) holds for some . Lemma 2.1 and (C15) lead to
which implies that
and hence (3.60) holds for . That is, (3.60) holds for any .
Now we claim that, for any ,
In fact, (C6) ensures that
that is, (3.64) is true for . Assume that (3.64) is true for some . In view of Lemmas 2.1 and 2.4, Algorithm 7, (C6), (C7), and C(15), we gain that
which yields that (3.64) is true for . Therefore (3.64) holds for each . Given , note that exists. It follows that there exist constants and satisfying for any . Thus (3.64) leads to
On account of (3.60), (3.67), and Algorithm 7, we deduce that is convergent for each and . Put
Obviously (3.67) ensures that . Notice that
Letting in the above inequality, by Lemmas 2.1 and 2.3 and the convergence of we infer that
which yields that
It follows from (3.60), (C15), and Lemma 2.1 that
which implies that
Letting , we gain that
It follows from (3.71) and (3.74) that is a solution of the functional equation (1.6). This completes the proof.
Following similar arguments as in the proof of Theorems 3.5 and 3.6, we have the following results.
Theorem 3.7.
Let , and satisfy conditions (C6)–(C8). Then the functional equation (1.5) possesses a solution satisfying conditions (C10)–(C12) and the two following conditions:
(C16)the sequence generated by Algorithm 6 converges to , where with for all ;
(C17)if , and are nonnegative and there exists a constant such that
then is nonpositive.
Theorem 3.8.
Let , and satisfy conditions (C6), (C7), and (C15). Then the functional equation (1.7) possesses a solution satisfying for any , where the sequence is generated by Algorithm 8 with , and for all .
4. Applications
In this section we use these results in Section 3 to establish the existence of solutions, nonnegative solutions, and nonpositive solutions and iterative approximations for several functional equations, respectively.
Example 4.1.
Let , , , and . It follows from Theorem 3.1 that the functional equation
possesses a unique solution and the sequence generated by Algorithm 1 converges to and satisfies (3.2).
Example 4.2.
Let , , , and . It is clear that Theorem 3.2 ensures that the functional equation
possesses a unique solution and the sequence generated by Algorithm 2 converges to and satisfies (3.2).
Remark 4.3.
If , for all , then Theorem 3.3 reduces to a result which generalizes the result in [3, page 149] and Theorem 3.4 in [7]. The following example demonstrates that Theorem 3.3 generalizes properly the corresponding results in [3, 7].
Example 4.4.
Let , , and . It is easy to verify that Theorem 3.3 guarantees that the functional equation
has a unique solution in . However, the results in [3, page 149] and Theorem 3.4 in [7] are valid for the functional equation (4.3).
Remark 4.5.

(1)
If , for all , then Theorems 3.4, 3.5, and 3.7 reduce to three results which generalize and unify the result in [3, page 149], Theorem 3.5 in [7], Theorem 3.5 in [12], Corollaries 2.2 and 2.3 in [14], Corollaries 3.3 and 3.4 in [17], and Theorems 2.3 and 2.4 in [18], respectively.

(2)
The results in [3, page 149], Theorem 3.5 in [7], Theorem 3.5 in [12], and Theorem 3.4 in [15] are special cases of Theorem 3.5 with , for all .
The examples below show that Theorems 3.4, 3.5, and 3.7 are indeed generalizations of the corresponding results in [3, 7, 12, 14, 15, 17, 18].
Example 4.6.
Let , . Define two functions by , for all . It is easy to see that Theorem 3.4 guarantees that the functional equation
possesses a solution that satisfies (C9)–(C12). However, the corresponding results in [3, 7, 12, 14, 17, 18] are not applicable for the functional equation (4.4).
Example 4.7.
Let . Put , , and for all . It is easy to verify that Theorem 3.5 guarantees that the functional equation
has a solution satisfying (C10)–(C14). But the corresponding results in [3, 7, 12, 14, 15, 17, 18] are not valid for the functional equation (4.5).
Example 4.8.
Let , . Put and for all . It is easy to verify that Theorem 3.6 guarantees that the functional equation
has a solution and the sequence generated by Algorithm 7 satisfies that for each , where with
Example 4.9.
Let . Put , and for all . It is easy to verify that Theorem 3.7 guarantees that the functional equation
has a solution satisfying (C10)–(C12), (C16), and (C17). But the corresponding results in [3, 7, 12, 14, 17, 18] are not valid for the functional equation (4.8).
Example 4.10.
Let , . Put and for all . It is easy to verify that Theorem 3.8 guarantees that the functional equation
possesses a solution and the sequence generated by Algorithm 8 satisfies that for each , where with
References
 1.
Belbas SA: Dynamic programming and maximum principle for discrete Goursat systems. Journal of Mathematical Analysis and Applications 1991,161(1):57–77. 10.1016/0022247X(91)903624
 2.
Bellman R: Some functional equations in the theory of dynamic programming. I. Functions of points and point transformations. Transactions of the American Mathematical Society 1955, 80: 51–71. 10.1090/S00029947195500746920
 3.
Bellman R: Dynamic Programming. Princeton University Press, Princeton, NJ, USA; 1957:xxv+342.
 4.
Bellman R: Methods of Nonlinear Analysis, Vol. II. Academic Press, New York, NY, USA; 1973:xvii+261.
 5.
Bellman R, Lee ES: Functional equations in dynamic programming. Aequationes Mathematicae 1978,17(1):1–18. 10.1007/BF01818535
 6.
Bellman R, Roosta M: A technique for the reduction of dimensionality in dynamic programming. Journal of Mathematical Analysis and Applications 1982,88(2):543–546. 10.1016/0022247X(82)902128
 7.
Bhakta PC, Choudhury SR: Some existence theorems for functional equations arising in dynamic programming. II. Journal of Mathematical Analysis and Applications 1988,131(1):217–231. 10.1016/0022247X(88)902016
 8.
Bhakta PC, Mitra S: Some existence theorems for functional equations arising in dynamic programming. Journal of Mathematical Analysis and Applications 1984,98(2):348–346. 10.1016/0022247X(84)902543
 9.
Chang S, Ma YH: Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of functional equations arising in dynamic programming. Journal of Mathematical Analysis and Applications 1991,160(2):468–479. 10.1016/0022247X(91)90319U
 10.
Liu Z: Coincidence theorems for expansion mappings with applications to the solutions of functional equations arising in dynamic programming. Acta Scientiarum Mathematicarum 1999,65(1–2):359–369.
 11.
Liu Z: Compatible mappings and fixed points. Acta Scientiarum Mathematicarum 1999,65(1–2):371–383.
 12.
Liu Z: Existence theorems of solutions for certain classes of functional equations arising in dynamic programming. Journal of Mathematical Analysis and Applications 2001,262(2):529–553. 10.1006/jmaa.2001.7551
 13.
Liu Z, Agarwal RP, Kang SM: On solvability of functional equations and system of functional equations arising in dynamic programming. Journal of Mathematical Analysis and Applications 2004,297(1):111–130. 10.1016/j.jmaa.2004.04.049
 14.
Liu Z, Kang SM: Properties of solutions for certain functional equations arising in dynamic programming. Journal of Global Optimization 2006,34(2):273–292. 10.1007/s1089800526056
 15.
Liu Z, Kang SM: Existence and uniqueness of solutions for two classes of functional equations arising in dynamic programming. Acta Mathematicae Applicatae Sinica. English Series 2007,23(2):195–208. 10.1007/s1025500703636
 16.
Liu Z, Kim JK: A common fixed point theorem with applications in dynamic programming. Nonlinear Functional Analysis and Applications 2006,11(1):11–19.
 17.
Liu Z, Ume JS: On properties of solutions for a class of functional equations arising in dynamic programming. Journal of Optimization Theory and Applications 2003,117(3):533–551. 10.1023/A:1023945621360
 18.
Liu Z, Ume JS, Kang SM: Some existence theorems for functional equations arising in dynamic programming. Journal of the Korean Mathematical Society 2006,43(1):11–28.
 19.
Liu Z, Xu Y, Ume JS, Kang SM: Solutions to two functional equations arising in dynamic programming. Journal of Computational and Applied Mathematics 2006,192(2):251–269. 10.1016/j.cam.2005.04.033
 20.
Zhang SS: Some existence theorems of common and coincidence solutions for a class of systems of functional equations arising in dynamic programming. Applied Mathematics and Mechanics 1991,12(1):31–37.
 21.
Wang CL: The principle and models of dynamic programming. II. Journal of Mathematical Analysis and Applications 1988,135(1):268–283. 10.1016/0022247X(88)901539
 22.
Wang CL: The principle and models of dynamic programming. III. Journal of Mathematical Analysis and Applications 1988,135(1):284–296. 10.1016/0022247X(88)901540
 23.
Wang CL: The principle and models of dynamic programming. V. Journal of Mathematical Analysis and Applications 1989,137(1):161–167. 10.1016/0022247X(89)902795
Acknowledgments
The authors wish to thank the referees for pointing out some printing errors. This study was supported by research funds from DongA University.
Author information
Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jiang, G., Kang, S. & Kwun, Y. Solvability and Algorithms for Functional Equations Originating from Dynamic Programming. Fixed Point Theory Appl 2011, 701519 (2011). https://doi.org/10.1155/2011/701519
Received:
Accepted:
Published:
Keywords
 Banach Space
 Unique Solution
 Functional Equation
 Dynamic Programming
 Iterative Algorithm