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Volume 2010, Article ID 905858, 19 pagesdoi:10.1155/2010/905858 Research Article On Properties of Solutions for Two Functional Equations Arising in Dynamic Programming 1 Department of Ma

Volume 2010, Article ID 905858, 19 pages

doi:10.1155/2010/905858

Research Article

On Properties of Solutions for Two Functional

Equations Arising in Dynamic Programming

1 Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China

2 Department of Applied Mathematics, Changwon National University,

Changwon 641-773, Republic of Korea

3 Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Chinju 660-701, Republic of Korea

Correspondence should be addressed to Jeong Sheok Ume,jsume@changwon.ac.kr

Received 12 July 2010; Accepted 26 October 2010

Academic Editor: Manuel De la Sen

Copyrightq 2010 Zeqing Liu et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We introduce and study two new functional equations, which contain a lot of known functional equations as special cases, arising in dynamic programming of multistage decision processes By applying a new fixed point theorem, we obtain the existence, uniqueness, iterative approximation, and error estimate of solutions for these functional equations Under certain conditions, we also study properties of solutions for one of the functional equations The results presented in this paper extend, improve, and unify the results according to Bellman, Bellman and Roosta, Bhakta and Choudhury, Bhakta and Mitra, Liu, Liu and Ume, and others Two examples are given to demonstrate the advantage of our results over existing results in the literature

1 Introduction and Preliminaries

The existence, uniqueness, and successive approximations of solutions for the following functional equations arising in dynamic programming:

f x  max

y ∈D



p

x, y

 qx, y

f

a

x, y

, ∀x ∈ S,

f x  max

y ∈D



p

x, y

 fa

x, y

, ∀x ∈ S,

f x  min

y ∈Dmax

p

x, y

, f

a

x, y

, ∀x ∈ S,

f x  min

y ∈Dmax

p

x, y

, q

x, y

f

a

x, y

, ∀x ∈ S,

f x  sup

y ∈D



p

x, y

m

i1

q i



x, y

f

a i



x, y

, ∀x ∈ S,

1.1

were first introduced and discussed by Bellman1,2 Afterwards, further analyses on the properties of solutions for the functional equations 1.1 and 1.2 and others have been studied by several authors in3 7 and 8 11 by using various fixed point theorems and monotone iterative technique, where1.2 are as follows:

f x  inf

y ∈D H

x, y, f

, ∀x ∈ S,

f x  opt

y ∈D



p

x, y

m

i1

q i

x, y opt

v i

x, y

, f

a i

x, y

, ∀x ∈ S

f x  opt

y ∈D



t

u

x, y

 fa

x, y  1 − toptv

x, y

, f

a

x, y

, ∀x ∈ S.

1.2

The aim of this paper is to investigate properties of solutions for the following more general functional equations arising in dynamic programming of multistage decision processes:

f x  opt

y ∈D



p

x, y

 Hx, y, f

f x  opt

y ∈D



r

x, y

m

i1

opt

p i



x, y

 qix, y

f

a i



x, y

,

u i



x, y

 vix, y

f

b i



x, y

, ∀x ∈ S,

1.4

where X and Y are real Banach spaces, S ⊆ X is the state space, D ⊆ Y is the decision space, opt denotes the sup or inf, x and y stand for the state and decision vectors, respectively,

a1, a2, , a m, b1, b2, , b m represent the transformations of the processes, and fx denotes the optimal return function with initial state x The rest of the paper is organized as follows.

InSection 2, we state the definitions, notions, and a lemma and establish a new fixed point theorem, which will be used in the rest of the paper The main results are presented in

Section 3 By applying the new fixed point theorem, we establish the existence, uniqueness, iterative approximation, and error estimate of solutions for the functional equation 1.3 and 1.4 Under certain conditions, we also study other properties of solutions for the functional equations 1.4 The results present in this paper extend, improve, and unify the corresponding results according to Bellman 1, Bellman and Roosta 5, Bhakta and Choudhury6, Bhakta and Mitra 7, Liu 8, Liu and Ume 11, and others Two examples are given to demonstrate the advantage of our results over existing results in the literature

Throughout this paper, we assume that R  −∞, ∞, R 0, ∞, and R− −∞, 0 For any t ∈ R, t denotes the largest integer not exceeding t Define

Φ1ϕ : ϕ : R−→ R is upper semicontinuous from the right on R

,

Φ2ϕ : ϕ : R−→ Rand ϕt < t for t > 0,

Φ3ϕ : ϕ : R−→ R is nondecreasing

,

Φ4





ϕ, ψ

: ϕ, ψ∈ Φ3, ψ t > 0, ∞

n0

ψ

ϕ n t< ∞ for t > 0



.

1.5

2 A Fixed Point Theorem

Let{dk} k≥1be a countable family of pseudometrics on a nonvoid set X such that for any two

different points x, y ∈ X, dkx, y > 0 for some k ≥ 1 For any x, y ∈ X, let

d

x, y

∞

k1

1

2k · d k



x, y

1 dkx, y , 2.1

then d is a metric on X A sequence {xn} n≥1 in X is said to converge to a point x ∈ X if

d kxn , x  → 0 as n → ∞ for any k ≥ 1 and to be a Cauchy sequence if dkxn , x m → 0 as

n, m → ∞ for any k ≥ 1.

Theorem 2.1 Let X, d be a complete metric space, and let d be defined by 2.1 If f : X → X

satisfies the following inequality:

d k



fx, fy

≤ ϕd k



x, y

, ∀x, y ∈ X, k ≥ 1, 2.2

where ϕ is some element inΦ1∩ Φ2, then

i f has a unique fixed point w ∈ X and limn→ ∞f n x  w for any x ∈ X,

ii if, in addition, ϕ ∈ Φ3, then

d k

f n x, w

≤ ϕ n dkx, w, ∀x ∈ X, n ≥ 1, k ≥ 1. 2.3

Proof Given x ∈ X and k ≥ 1, define cn  dkf n x, f n−1x  for each n ≥ 1 In view of 2.2, we know that

c n1 dkf n1x, f n x

≤ ϕd k

f n x, f n−1x

 ϕcn, ∀n ≥ 1. 2.4

Since ϕ∈ Φ1∩ Φ2, by2.4 we easily conclude that {cn} n≥1is nonincreasing It follows that

{cn} n≥1 has a limit c ≥ 0 We claim that c  0 Otherwise, c > 0 On account of 2.4 and

ϕ∈ Φ1∩ Φ2, we deduce that

c≤ lim sup

n→ ∞ ϕ cn ≤ ϕc < c, 2.5

which is impossible That is, c  0 We now show that {f n x}n≥1is a Cauchy sequence Suppose that{f n x}n≥1is not a Cauchy sequence, then there exist ε > 0, k ≥ 1, and two sequences of positive integers{mi} i≥1and{ni} i≥1with mi > ni and

a i  dkf m i x, f n i x

≥ ε, dkf m i−1 x, f n i x

< ε, ∀i ≥ 1, 2.6 which yields that

ε ≤ ai ≤ dkf m i x, f m i−1 x

 dkf m i−1 x, f n i x

≤ cm i  ε, ∀i ≥ 1. 2.7

As i → ∞ in 2.7, we derive that limi→ ∞a i  ε Note that 2.2 and 2.7 mean that

a i ≤ dkf m i x, f m i1 x

 dkf m i1 x, f n i1 x

 dkf n i1 x, f n i x

≤ cm i1  ϕai  cn i1 ,

2.8

for any i ≥ 1 Letting i → ∞ in 2.8, we see that

ε ≤ ϕε < ε. 2.9

This is a contradiction By completeness of X, d, there exists a point w ∈ X, such that

limn→ ∞f n x  w Using 2.1, 2.2, and ϕ ∈ Φ1∩ Φ2, we obtain that for each x, y ∈ X

d

fx, fy

∞

k1

1

2k · d k



fx, fy

1 dkfx, fy ≤∞

k1

1

2k · ϕ



d k



x, y

1 ϕd k



x, y

≤∞

k1

1

2k · d k



x, y

1 dkx, y   dx, y

,

2.10

which yields that

d

w, fw

≤ dw, f n x

 df n x, fw

≤ dw, f n x

 df n−1x, w

−→ 0, as n −→ ∞,

2.11

that is, w is a fixed point of f If f has a fixed point v different from w, then there exists k ≥ 1 such that d kw, v > 0 By 2.2, we have

d kw, v  dkfw, fv

≤ ϕdkw, v < dkw, v, 2.12

which is a contradiction Consequently, w is a unique fixed point of f.

Suppose that ϕ∈ Φ3 By2.2, we get that for any x ∈ X, n ≥ 1, and k ≥ 1

d k

f n x, w

 dkf n x, f n w

≤ ϕd k

f n−1x, f n−1w

≤ · · · ≤ ϕ n dkx, w. 2.13 This completes the proof

Remark 2.2. Theorem 2.1extends Theorem 2.1 of Bhakta and Choudhury6 and Theorem 1

of Boyd and Wong12

Lemma 2.3 see 11 Let a, b, c, and d be in R, then

opt{a, b} − opt{c, d} 2.14

3 Properties of Solutions

In this section, we assume that  are real Banach spaces, S ⊆ X is the state space, and D ⊆ Y is the decision space Define

BB S f : f : S −→ R is bounded on bounded subsets of S. 3.1

For any positive integer k and f, g ∈ BBS, let

d k



f, g

 sup : x ∈ B0, k,

d

f, g

∞

k1

1

2k · d k



f, g

1 dkf, g , 3.2

where B k} k≥1is a countable family of pseudometrics

on BBS It is clear that BBS, d is a complete metric space.

Theorem 3.1 Let p : S × D → R and H : S × D × BBS → R be mappings, and let ϕ be in

Φ1∩ Φ2, such that

C1 for any k ≥ 1 and x, y, u, v ∈ B0, k × D × BBS × BBS,

H

x, y, u

C2 for any k ≥ 1 and u ∈ BBS, there exists αk, u > 0 satisfying

p

then the functional equation1.3 possesses a unique solution w ∈ BBS, and {G n g}n≥1converges

to w for each g ∈ BBS, where G is defined by

Gg x  opt

y ∈D



p

x, y

 Hx, y, g

, ∀x, g

∈ S × BBS. 3.5

In addition, if ϕ is inΦ3, then

d k



G n g, w

≤ ϕ n

d k



g, w

, ∀g ∈ BBS, n ≥ 1, k ≥ 1. 3.6

Proof It follows fromC2 and 3.4 that G maps BBS into itself Given ε > 0, k ≥ 1, x ∈

B 0, k, and h, g ∈ BBS, suppose that opt y ∈D supy ∈D , then there exist y, z ∈ D such that

Gh x < px, y

 Hx, y, h

 ε, Gg x < px, z  Hx, z, g

 ε,

Gh x ≥ px, z  Hx, z, h, Gg x ≥ px, y

 Hx, y, g

. 3.7

In view of3.3, 3.5, and 3.7, we deduce that

Gh x − Ggx < max H

x, y, h

− Hx, y, g , H x, z, h − Hx, z, g

≤ ϕd k



h, g

which implies that

d k



Gh, Gg

 sup : x ∈ B0, k≤ ϕd k



h, g

 ε. 3.9 Similarly, we can show that3.9 holds for opty ∈D infy∈D As ε → 0in3.9, we get that

d k

Gh, Gg

≤ ϕd k

h, g

Notice that the functional equation 1.3 possesses a unique solution w if and only if the mapping G has a unique fixed point w Thus,Theorem 3.1follows fromTheorem 2.1 This completes the proof

Remark 3.2 The conditions ofTheorem 3.1are weaker than the conditions of Theorem 3.1 of Bhakta and Choudhury6

Theorem 3.3 Let r, p i , q i , u i , v i : S×D → R and ai , b i : S×D → S be mappings for i  1, 2, , m.

Assume that the following conditions are satisfied:

C3 for each k ≥ 1, there exists Ak > 0 such that

r

x, y

m



i1

max p i



x, y , u i



i i

C5 there exists a constant β ∈ 0, 1 such that

m



i1

max q i

x, y , v i

then the functional equation1.4 possesses a unique solution w ∈ BBS, and {wn} n≥1converges to

w for each w0∈ BBS, where {wn} n≥1is defined by

w nx  opt

y ∈D



r

x, y

m

i1

opt

p i



x, y

 qix, y

w n−1

a i



x, y

,

u i



x, y

 vix, y

w n−1

b i



x, y

, ∀x ∈ S, n ≥ 1.

3.13

Moreover,

d kwn , w  ≤ β n

1− β−1d kw0, w , ∀n ≥ 1, k ≥ 1. 3.14

Proof Set

H

x, y, h

 rx, y

m

i1

opt

p i



x, y

 qix, y

h

a i



x, y

, u i



x, y

 vix, y

h

b i



x, y

∀x, y, h

∈ S × D × BBS,

3.15

Gh x  opt

y ∈D H

x, y, h

, ∀x, h ∈ S × BBS. 3.16

It follows fromC3–C5 and 3.15 that

H

x, y, h

m



i1

max p i





x, y h

a i



x, y ,

u i

x, y i

x, y h

b i

x, y

≤ r

x, y

m



i1

max p i

x, y , u i

x, y

m



i1

max q i

x, y , v i

x, y

× max h

a i

x, y , h

b i

x, y

≤ Ak 

m



i1

max q i



x, y , v i



x, y

 sup

|ht| : t ∈ B0, k

≤ Ak  β sup|ht| : t ∈ B0, k,

3.17

for any k ≥ 1 and x, y, h ∈ B0, k × D × BBS Consequently, G is a self mapping on BBS.

ByLemma 2.3,C4, and C5, we obtain that for any k ≥ 1 and x, y, g, h ∈ B0, k × D ×

BB S × BBS,

H

x, y, g

− Hx, y, h

 m

i1

opt

p i



x, y

 qix, y

g

a i



x, y

, u i



x, y

 vix, y

g

b i



x, y

−m

i1

opt

p i

x, y

 qix, y

h

a i

x, y

, u i

x, y

 vix, y

h

bi

x, y

≤m

i1

max q i

x, y g

a i

x, y

− ha i

x, y , v i

x, y g

b i

x, y

− hb i

x, y

≤m

i1

max q i

x, y , v i

x, y

× max g

a i



x, y

− ha i



x, y , g

b i



x, y

− hb i



x, y

≤ ϕd k

g, h

,

3.18

where ϕt  βt for t ∈ R Thus,Theorem 3.3follows fromTheorem 3.1 This completes the proof

Remark 3.4 Theorem 2 of Bellman1, page 121, the result of Bellman and Roosta 5, page 545, Theorem 3.3 of Bhakta and Choudhury 6, and Theorems 3.3 and 3.4 of Liu 8 are special cases ofTheorem 3.3 The example below shows thatTheorem 3.3extends properly the results in1,5,6,8

Example 3.5 Let X  Y  S  R and D  R− Put m  2, β  2/3, and Ak  3k3for any

k≥ 1 It follows fromTheorem 3.3that the functional equation

f x  opt

y ∈D

⎧

⎨

⎩x2sin



xy  x − y  1

 opt



x3



1 x2 y2

1x2 y22



sin2



x − y  x2

3 x2 y2 f

x cos

x2 y2

,

x2ln



1 xy

1 xy



cos



xy − 2x − 1

3 x2y− 1



x

1 |x|y2x − y2



 opt



x3y

1 |x|  y

cos2

xy − x2

3 x2y2 f

x sin

1− xy  x3y2

,

x2cos

x2− y2

1 |x|  y2  xy

4 x2y2f



x

1 2x2y



, ∀x ∈ S

3.19

possesses a unique solution w ∈ BBS However, the results in 1,5,6,8 are not applicable

Theorem 3.6 Let r, p i , q i , u i , v i : S×D → R and ai , b i : S×D → S be mappings for i  1, 2, , m,

and, ϕ, ψ be in Φ4satisfying

C6 |rx, y| m

i1max{|pix, y|, |ui

C8 supx,y∈S×Dm

i1max{|qix, y|, |vix, y|} ≤ 1,

then the functional equation 1.4 possesses a solution w ∈ BBS that satisfies the following

conditions:

C9 the sequence {wn} n≥1defined by

w0x  opt

y ∈D



r

x, y

m

i1

opt

p i

x, y

, u i



x, y

,

w nx  opt

y ∈D



r

x, y

m

i1

opt

p i

x, y

 qix, y

w n−1

a i

x, y

,

u i

x, y

 vix, y

w n−1

b i

x, y

∀x ∈ S, n ≥ 1,

3.20

converges to w,

C10 limn→ ∞w xn  0 for any x0∈ S, {yn} n≥1⊂ D and xn ∈ {aixn−1, y n, bixn−1, y n : i ∈ {1, 2, , m}}, n ≥ 1,

C11 w is unique with respect to condition (C10).

Proof Let H and G be defined by3.15 and 3.16, respectively We now claim that

ϕ t < t, ∀t > 0. 3.21

If not, then there exists some t > 0 such that ϕt ≥ t On account of ϕ, ψ ∈ Φ4, we know that

for any n≥ 1,

ψ

ϕ n t≥ ψϕ n−1t≥ · · · ≥ ψt > 0, 3.22 whence

lim

n→ ∞ψ

ϕ n t≥ ψt > 0, 3.23

which is a contradiction since∞

n0ψ ϕ n t < ∞.

Next, we assert that the mapping G is nonexpansive on BBS Let k ≥ 1 and h ∈

BB S It is easy to see that

maxa i

x, y,b i

x, y: i ∈ {1, 2, , m} 

x, y

∈ B0, k × D, 3.24

byC7 and 3.21 Consequently, there exists a constant Ck, h > 0 satisfying

max h

a i

x, y , h

b i

x, y : i ∈ {1, 2, , m}≤ Ck, h, ∀x, y

∈ B0, k × D.

3.25

In view ofC6, 3.16, and 3.25, we derive that for any x ∈ B0, k,

|Ghx|  opt

y ∈D H

x, y, h

sup

y ∈D H

x, y, h

≤ sup

y ∈D



r

x, y

m



i1

max p i

x, y i

x, y h

a i

x, y ,

u i

x, y i

x, y h

b i

x, y



≤ sup

y ∈D



r

x, y

m



i1

max p i

x, y , u i

x, y

m

i1

max q i

x, y , v i

x, y max h

a i

x, y , h

b i

x, y



≤ ψk  Ck, h,

3.26

which yields that G maps BBS into itself Given ε > 0, k ≥ 1, x ∈ B0, k, and h, g ∈ BBS,

suppose that opty ∈D supy ∈D , then there exist y, z ∈ D such that

Gh x < Hx, y, h

 ε, Gg x < Hx, z, g

 ε,

Gh x ≥ Hx, z, h, Gg x ≥ Hx, y, g

.

3.27

UsingC6–C8, 3.15 and 3.27, andLemma 2.3, we deduce that

Gh x − Ggx < max H

x, y, h

− Hx, y, g , H x, z, h − Hx, z, g

≤ max

m

i1

max q i

x, y h

a i

x, y

− ga i

x, y ,

v i

x, y h

b i

x, y

− gb i

x, y ,

m



i1

max q ix, z h aix, z − gaix, z ,

|vix, z| h bix, z − gbix, z



 ε

≤ max

m



i1

max q i



x, y , v i



x, y ,

m



i1

max q ix, z , |vix, z|



d k



h, g

 ε

≤ dkh, g

 ε,

3.28

which means that

d k

Gh, Gg

≤ dkh, g

Similarly, we can conclude that the above inequality holds for opty ∈D  infy∈D Letting ε →

0, we get that

d k

Gh, Gg

≤ dkh, g

which implies that

d

Gh, Gg

∞

k1

1

2k · d k



Gh, Gg

1 dkGh, Gg ≤∞

k1

1

2k · d k



h, g

1 dkh, g   dh, g

. 3.31

That is, G is nonexpansive.

We show that for each n≥ 0,

|wnx| ≤n

j0

ψ

, ∀x ∈ S. 3.32

3 Properties of Solutions< /b>

In this section, we assume that  are real Banach spaces, S ⊆ X is the state space, and D ⊆ Y is the decision space Define

BB... that the functional equation 1.3 possesses a unique solution w if and only if the mapping G has a unique fixed point w Thus,Theorem 3.1follows fromTheorem 2.1 This completes the proof

Remark... class="page_container" data-page="8">

for any k ≥ and x, y, h ∈ B0, k × D × BBS Consequently, G is a self mapping on BBS.

ByLemma 2.3,C4, and C5, we obtain that for any