Báo cáo hóa học: " Research Article Solvability and Algorithms for Functional Equations Originating from Dynamic Programming" pptx

The existence, uniqueness and iterative approximations of solutions for the functional equation are discussed in the Banach spaces BCS and BS and the complete metric space BBS, respectiv

Volume 2011, Article ID 701519, 30 pages

doi:10.1155/2011/701519

Research Article

Solvability and Algorithms for

Functional Equations Originating

from Dynamic Programming

Guojing Jiang,1 Shin Min Kang,2 and Young Chel Kwun3

1 Organization Department, Dalian Vocational Technical College, Dalian, Liaoning 116035, China

2 Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

3 Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Correspondence should be addressed to Young Chel Kwun,yckwun@dau.ac.kr

Received 5 January 2011; Accepted 11 February 2011

Academic Editor: Yeol J Cho

Copyrightq 2011 Guojing Jiang et al This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

The main purpose of this paper is to study the functional equation arising in dynamic

program-ming of multistage decision processes fxopt y ∈Dopt{px, y, qx, yfax, y, rx, yfbx, y,

s x, yfcx, y} for all x ∈ S 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 BCS and BS and the complete metric space BBS,

respectively The properties of solutions, nonnegative solutions, and nonpositive solutions and theconvergence 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 BB S, 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 offunctional equations arising in dynamic programming were studied by a lot of researchers;see1 23 and the references therein Bellman 3, Bhakta and Choudhury 7, Liu 12, Liuand Kang15, 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 BBS Liu and Kang 14 and Liu and Ume 17 generalizedthe results in3,7,12,15,18 and studied the properties of solutions, nonpositive solutionsand nonnegative solutions and the convergence of iterative approximations for the followingfunctional equations, respectively

where opt denotes sup or inf, x and y stand for the state and decision vectors, respectively,

a, b, and c represent the transformations of the processes, and f x represents the optimal return function with initial x.

This paper is divided into four sections InSection 2, we recall a few basic conceptsand notations, establish several lemmas that will be needed further on, and suggestten iterative algorithms for solving the functional equations 1.3– 1.9 In Section 3, byapplying the Banach fixed-point theorem, we establish the existence, uniqueness, anditerative approximations of solutions for the functional equation1.3 in the Banach spaces

BC S and BS, respectively By means of two iterative algorithms defined in Section 2,

we obtain the existence, uniqueness, and iterative approximations of solutions for thefunctional equation1.3 in the complete metric space BBS Under certain conditions, we

investigate the behaviors of solutions, nonpositive solutions, and nonnegative solutions andthe convergence of iterative algorithms for the functional equations1.3–1.7, respectively,

in BBS In Section 4, we construct eight nontrivial examples to explain our results,which extend and improve substantially the results due to Bellman 3, Bhakta andChoudhury 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 thatR  −∞, ∞, R 0, ∞, R−  −∞, 0, N denotes the set of positive integers, and, for each t ∈ R, t denotes the largest integer not exceeding t.

LetX, · and Y,  ·  be real Banach spaces, S ⊆ X the state space, and D ⊆ Y the decision

where B0, k  {x : x ∈ S and x ≤ k} It is easy to see that {d k}k∈Nis a countable family

of pseudometrics on BBS A sequence {x n}n∈Nin BBS is said to be converge to a point

x ∈ BBS if, for any k ∈ Nd k x n , x  → 0 as n → ∞ and to be a Cauchy sequence if, for any k ∈ N, d k x n , x m  → 0 as n, m → ∞ It is clear that BBS, d is a complete metric

space

Lemma 2.1 Let {a i , b i: 1≤ i ≤ n} ⊂ R Then

a opt{a i: 1≤ i ≤ n}  opt{opt{a i: 1≤ i ≤ n − 1}, a n },

b opt{a i: 1≤ i ≤ n} ≤ opt{b i: 1≤ i ≤ n} for a i ≤ b i , 1 ≤ i ≤ n,

c max{a i b i : 1≤ i ≤ n} ≤ max{a i : 1≤ i ≤ n} max{b i : 1≤ i ≤ n} for {a i , b i : 1≤ i ≤

n} ⊂ R,

d min{a i b i: 1≤ i ≤ n} ≥ min{a i: 1≤ i ≤ n} min{b i: 1≤ i ≤ n} for {a i , b i: 1≤ i ≤ n} ⊂

R,

e |opt{a i: 1≤ i ≤ n} − opt{b i: 1≤ i ≤ n}| ≤ max{|a i − b i | : 1 ≤ i ≤ n}.

Proof Clearly a–d are true Now we show e Note that e holds for n  1 Suppose that

e is true for some n ∈ N It follows from a and Lemma 2.1 in 17 that

opt{ai: 1≤ i ≤ n  1} − opt{b i: 1≤ i ≤ n  1}

 optopt{ai: 1≤ i ≤ n}, a n1

− optopt{bi: 1≤ i ≤ n}, b n1

≤ max opt{ai: 1≤ i ≤ n} − opt{b i : 1≤ i ≤ n} , |a n1− b n1|

≤ max{|a i − b i | : 1 ≤ i ≤ n  1}.

2.3

Hencee holds for any n ∈ N This completes the proof.

Lemma 2.2 Let {a i: 1≤ i ≤ n} ⊂ R and {b i: 1≤ i ≤ n} ⊂ R Then

a max{a i b i: 1≤ i ≤ n} ≥ min{a i: 1≤ i ≤ n} max{b i: 1≤ i ≤ n},

b min{a i b i: 1≤ i ≤ n} ≤ max{a i: 1≤ i ≤ n} min{b i: 1≤ i ≤ n}.

Proof It is clear that a is true for n  1 Suppose that a is also true for some n ∈ N Using

Lemma 2.3 in19 andLemma 2.1, we infer that

That is,a is true for n  1 Therefore a holds for any n ∈ N Similarly we can prove b.

This completes the proof

Lemma 2.3 Let {a 1n}n∈N, {a 2n}n∈N, , {a kn}n∈Nbe convergent sequences in R Then

lim

n→ ∞opt{ain: 1≤ i ≤ k}  optlim

n→ ∞a in: 1≤ i ≤ k. 2.5

Proof Put lim n→ ∞a in  b ifor 1≤ i ≤ k In view ofLemma 2.1we deduce that

opt{ain: 1≤ i ≤ k} − opt{b i: 1≤ i ≤ k} ≤ max{|a in − b i | : 1 ≤ i ≤ k} −→ 0 as n −→ ∞,

n→ ∞a in: 1≤ i ≤ k



This completes the proof

Lemma 2.4 a Assume that A : S × D → R is a mapping such that opt y ∈D A x0, y  is bounded

for some x0∈ S Then

opty ∈D A



x0, y ≤supy ∈D A

b Assume that A, B : S × D → R are mappings such that opt y ∈D A x1, y  and

opty ∈D B x2, y  are bounded for some x1, x2∈ S Then

opty ∈D A

x0, y ≤supy ∈D A

x1, y

− Bx2, y . 2.16This completes the proof

Algorithm 1 For any f0∈ BCS, compute {f n}n≥0by

Algorithm 2 For any f0∈ BS, compute {f n}n≥0by2.17 and 2.18

Algorithm 3 For any f0∈ BBS, compute {f n}n≥0by2.17 and 2.18

Algorithm 4 For any w0 ∈ BBS, compute {w n}n≥0by

Algorithm 8 For any w0 ∈ BBS, compute {w n}n≥0by

3 The Properties of Solutions and Convergence of Algorithms

Now we prove the existence, uniqueness, and iterative approximations of solutions for thefunctional equation 1.3 in BCS and BS, respectively, by using the Banach fixed-point

theorem

Theorem 3.1 Let S be compact Let p, q, r, s : S × D → R and a, b, c : S × D → S satisfy the

following conditions:

C1 p is bounded in S × D;

C2 supx,y∈S×Dmax{|qx, y|, |rx, y|, |sx, y|} ≤ α for some constant α ∈ 0, 1;

C3 for each fixed x0 ∈ S,

uniformly for y ∈ D, respectively.

Then the functional equation1.3 possesses a unique solution f ∈ BCS, and the sequence {f n}n≥0generated by Algorithm 1 converges to f and has the error estimate

f n1− f  ≤ e −1−αn

i0α if0− f, ∀n ≥ 0. 3.2

Proof Define a mapping H : BC S → BCS by

In light ofC2, 3.3, 3.5–3.9, and Lemmas2.1and2.4, we deduce that for allx, y ∈ S×D

Thus3.10, 3.11, and 2.17 ensure that the mapping H : BCS → BCS andAlgorithm 1

are well defined

Next we assert that the mapping H : BCS → BCS is a contraction Let ε > 0, x ∈ S, and g, h ∈ BCS Suppose that opt y ∈D infy ∈D Choose u, v ∈ D such that

Hg x > optp x, u, qx, ugax, u, rx, ugbx, u, sx, ugcx, u− ε,

Hh x > optp x, v, qx, vhax, v, rx, vhbx, v, sx, vhcx, v− ε,

Hg x ≤ optp x, v, qx, vgax, v, rx, vgbx, v, sx, vgcx, v,

Hh x ≤ optp x, u, qx, uhax, u, rx, uhbx, u, sx, uhcx, u.

3.12

Lemma 2.1and3.12 lead to

Hg x − Hhx

< max opt

p x, u, qx, ugax, u, rx, ugbx, u, sx, ugcx, u

−optp x, u, qx, uhax, u, rx, uhbx, u, sx, uhcx, u ,

opt

p x, v, qx, vgax, v, rx, vgbx, v, sx, vgcx, v

−optp x, v, qx, vhax, v, rx, vhbx, v, sx, vhcx, v   ε

≤ maxmax q x, u g ax, u − hax, u , |rx, u| g bx, u − hbx, u ,

|sx, u| g cx, u − hcx, u ,max q x, v g ax, v − hax, v , |rx, v| g bx, v − hbx, v ,

Similarly we conclude that3.15 holds for opty ∈D supy ∈D The Banach fixed-point theorem

yields that the contraction mapping H has a unique fixed point f ∈ BCS It is easy to verify that f is also a unique solution of the functional equation1.3 in BCS By means of 2.17,

which yields that

f n1− f

i0α if0− f

and the sequence{f n}n≥0converges to f by2.18 This completes the proof

Dropping the compactness of S andC3 inTheorem 3.1, we conclude immediatelythe following result

Theorem 3.2 Let p, q, r, s : S × D → R and a, b, c : S × D → S satisfy conditions (C1) and (C2).

Then the functional equation1.3 possesses a unique solution f ∈ BS and the sequence {f n}n≥0

generated by Algorithm 2 converges to f and satisfies3.2.

Next we prove the existence, uniqueness, and iterative approximations of solution forthe functional equation1.3 in BBS.

Theorem 3.3 Let p, q, r, s : S × D → R and a, b, c : S × D → S satisfy condition (C2) and the

following two conditions:

C4 p is bounded on B0, k × D for each k ∈ N;

C5 supx,y∈B0,k×D {ax, y, bx, y, cx, y} ≤ k for all k ∈ N.

Then the functional equation1.3 possesses a unique solution w ∈ BBS, and the sequences {f n}n≥0and {w n}n≥0generated by Algorithms 3 and 4 , respectively, converge to f and have the error estimates

Proof Define a mapping H : BB S → BBS by 3.3 Let k ∈ N and h ∈ BBS Thus C4

andC5 guarantee that there exist Mk > 0 and Gk, h > 0 satisfying

which means that H is a self-mapping in BBS Consequently, Algorithms3and4are welldefined.

Now we claim that

Let k ∈ N, x ∈ B0, k, g, h ∈ BBS, and ε > 0 Suppose that opt y ∈D  infy ∈D Select u, v ∈ D

such that3.12 holds Thus 3.3, 3.12, C2, C5, andLemma 2.1ensure that

Hg x − Hhx

< max opt

p x, u, qx, ugax, u, rx, ugbx, u, sx, ugcx, u

−optp x, u, qx, uhax, u, rx, uhbx, u, sx, uhcx, u ,

opt

p x, v, qx, vgax, v, rx, vgbx, v, sx, vgcx, v

−optp x, v, qx, vhax, v, rx, vhbx, v, sx, vhcx, v   ε

≤ maxmax q x, u g ax, u − hax, u , |rx, u| g bx, u − hbx, u ,

|sx, u| g cx, u − hcx, u ,max q x, v g ax, v − hax, v , |rx, v| g bx, v − hbx, v ,

which yields that{w n}n≥0is a Cauchy sequence in the complete metric spaceBBS, d, and

hence{w n}n≥0converges to some w ∈ BBS In light of 3.22 and C2, we know that

which is a contradiction Hence the mapping H : BBS → BBS has a unique fixed point

w ∈ BBS, which is a unique solution of the functional equation 1.3 in BBS Letting

Next we investigate the behaviors of solutions for the functional equations1.3–1.5and discuss the convergence of Algorithms4 6in BBS, respectively.

Theorem 3.4 Let ϕ, ψ ∈ Φ2, p, q, r, s : S × D → R and a, b, c : S × D → S satisfy the following

conditions:

C6 supy ∈D |px, y| ≤ ψx for all x ∈ S;

C7 supy ∈Dmax{ax, y, bx, y, cx, y} ≤ ϕx for all x ∈ S;

C8 supx,y∈S×Dmax{|qx, y|, |rx, y|, |sx, y|} ≤ 1

Then the functional equation1.3 possesses a solution w ∈ BBS satisfying conditions (C9)–(C12)

C12 w is unique relative to condition (C11).

Proof First of all we assert that

Suppose that there exists some t0> 0 with ϕ t0 ≥ t0 It follows fromϕ, ψ ∈ Φ2that

ψ t0 ≤ ψϕ t0≤ ψϕ2t0≤ · · · ≤ ψϕ n t0−→ 0 as n −→ ∞. 3.33That is,

which is impossible That is,3.32 holds Let the mapping H be defined by 3.3 in BBS.

Note thatC6 and C7 imply C4 and C5 by 3.32 and ϕ, ψ ∈ Φ2, respectively As inthe proof ofTheorem 3.3, byC8 we conclude that the mapping H maps BBS into BBS

Let the sequence{w n}n≥0be generated byAlgorithm 4and w0∈ BBS with |w0x| ≤

ψ x for all x, k ∈ B0, k × N We now claim that for each n ≥ 0

|w n x| ≤ ψx, ∀x, k ∈ B0, k × N. 3.37

Clearly 3.37 holds for n  0 Assume that 3.37 is true for some n ≥ 0 It follows from

C6–C8, 3.32,Algorithm 4, and Lemmas2.1and2.4that

That is,3.37 is true for n  1 Hence 3.37 holds for each n ≥ 0.

Next we claim that{w n}n≥0is a Cauchy sequence inBBS, d Let k, n, m ∈ N, x0 ∈

B 0, k, and ε > 0 Suppose that opt y ∈D infy ∈D Choose y, z ∈ D with

It follows from3.39, C8, and Lemmas2.2and2.3that

Therefore there exist y1∈ {y, z} ⊂ D and x1∈ {ax0, y1, bx0, y1, cx0, y1} satisfying

|w n m x0 − w n x0| < |w n m−1 x1 − w n−1x1|  2−1ε. 3.41

In a similar method, we can derive that3.41 holds also for opty ∈D supy ∈D Proceeding in

this way, we choose y i ∈ D and x i ∈ {ax i−1, y i , bx i−1, y i , cx i−1, y i } for i ∈ {2, 3, , n} such

which yields that Hw  w That is, the functional equation 1.3 possesses a solution w ∈

that is,C10 holds

Next we proveC11 Given x0 ∈ S, {y n}n∈N ⊂ D, and x n ∈ {ax n−1, y n , bx n−1, y n,

c x n−1, y n } for n ∈ N Put k  x0  1 Note that C7 implies that

which means that limn→ ∞w n x n  0

Finally we proveC12 Assume that the functional equation 1.3 has another solution

h ∈ BBS that satisfies C11 Let ε > 0 and x0 ∈ S Suppose that opt y ∈D  infy ∈D Select

On account of 3.50, C8, and Lemma 2.1, we conclude that there exist y1 ∈ {y, z} and

|wx0 − hx0| ≤ |wx1 − hx1|  2−1ε. 3.52Similarly we can prove that 3.52 holds for opty ∈D  supy ∈D Proceeding in this way, we

select y i ∈ D and x i ∈ {ax i−1, y i , bx i−1, y i , cx i−1, y i } for i ∈ {2, 3, , n} and n ∈ N such

It follows from3.52 and 3.53 that

|wx0 − hx0| < |wx n  − hx n |  ε −→ ε as n −→ ∞. 3.54

Since ε is arbitrary, we conclude immediately that wx0  hx0 This completes the proof

Theorem 3.5 Let ϕ, ψ ∈ Φ2, p, q, r, s : S × D → R and a, b, c : S × D → S satisfy conditions

(C6)–(C8) Then the functional equation1.4 possesses a solution w ∈ BBS satisfying conditions

(C10)–(C12) and the following two conditions:

C13 the sequence {w n}n≥0generated by Algorithm 5 converges to w, where w0 ∈ BBS with

Proof It follows fromTheorem 3.4that the functional equation1.4 has a solution w ∈ BBS

that satisfiesC10–C13 Now we show C14 Given ε > 0, x0∈ S and n ∈ N It follows from

Lemma 2.2,3.55, and 1.4 that there exist y1 ∈ D and x1 ∈ {ax0, y1, bx0, y1, cx0, y1}such that