On Properties of Solutions for Two Functional Equations Arising in Dynamic Programming

Volume 2010, Article ID 905858,19pages doi:10.1155/2010/905858

Research Article

On Properties of Solutions for Two Functional

Equations Arising in Dynamic Programming

Zeqing Liu,

_{Jeong Sheok Ume,}

_{and Shin Min Kang}

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,[email protected]

Received 12 July 2010; Accepted 26 October 2010

Academic Editor: Manuel De la Sen

Copyrightq2010 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:

fx max

y∈D

px, yqx, yfax, y, ∀x∈S,

fx max

y∈D

px, yfax, y, ∀x∈S,

fx min

y∈Dmax

fx min

y∈Dmax

px, y, qx, yfax, y, ∀x∈S,

fx sup

y∈D

px, y m

i1

qix, yfaix, 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 in3–7and8–11 by using various fixed point theorems and monotone iterative technique, where1.2are as follows:

fx inf

y∈DH

x, y, f, ∀x∈S,

fx opt

y∈D

px, y m

i1

qi

x, yoptvi

x, y, fai

x, y

, ∀x∈S

fx opt

y∈D

t ux, yfax, y 1−toptvx, y, fax, 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:

fx opt

y∈D

px, yHx, y, f, ∀x∈S, 1.3

fx opt

y∈D

rx, y m

i1

optpix, yqix, yfaix, y,

uix, yvix, yfbix, y

, ∀x∈S,

1.4

whereXandY are real Banach spaces,S⊆Xis 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,

Throughout this paper, we assume thatR −∞,∞,R 0,∞, andR− −∞,0. For anyt∈R,tdenotes the largest integer not exceedingt. Define

Φ1

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

Φ2

ϕ:ϕ:R−→Rand ϕt< t for t >0,

Φ3

ϕ:ϕ:R−→R is nondecreasing,

Φ4

ϕ, ψ:ϕ, ψ∈Φ3, ψt>0, ∞

n0

ψϕnt<∞fort >0

.

1.5

2. A Fixed Point Theorem

Let{dk}_k≥1be a countable family of pseudometrics on a nonvoid setXsuch that for any two different pointsx, y∈X,dkx, y>0 for somek≥1. For anyx, y∈X, let

dx, y ∞

k1 1 2k ·

dk

x, y

1dk

x, y, 2.1

then d is a metric onX. A sequence {xn}_n≥1 inX is said to converge to a point x ∈ X if

dkxn, x → 0 asn → ∞for anyk ≥1 and to be a Cauchy sequence ifdkxn, xm → 0 as n, m → ∞for anyk≥1.

Theorem 2.1. LetX, dbe a complete metric space, and letdbe defined by2.1. Iff : X → X

satisfies the following inequality:

dkfx, fy≤ϕdkx, y, ∀x, y∈X, k≥1, 2.2

whereϕis some element inΦ1∩Φ2, then

ifhas a unique fixed pointw∈Xandlimn→ ∞fnxwfor anyx∈X,

iiif, in addition,ϕ∈Φ3, then

dk

fn_{x, w≤ϕ}n_d

kx, w, ∀x∈X, n≥1, k≥1. 2.3

Proof. Givenx∈X andk ≥1, definecn dkfnx, fn−1xfor eachn≥1. In view of2.2, we

know that

cn1dk

fn1x, fnx≤ϕdk

fnx, fn−1xϕcn, ∀n≥1. 2.4

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

{cn}_n≥1 has a limitc ≥ 0. We claim thatc 0. Otherwise,c > 0. On account of2.4and

ϕ∈Φ1∩Φ2, we deduce that

c≤lim sup

which is impossible. That is,c0. We now show that{fn_x}

n≥1is a Cauchy sequence. Suppose that{fn_x}

n≥1 is not a Cauchy sequence, then there existε > 0,k ≥ 1, and two sequences of positive integers{mi}_i≥1and{ni}_i≥1withmi> niand

aidkfmix, fnix≥ε, dkfmi−1x, fnix< ε, ∀i≥1, 2.6

which yields that

ε≤ai≤dk

fmix, fmi−1xdk

fmi−1x, fnix≤cmiε, ∀i≥1. 2.7

Asi → ∞in2.7, we derive that limi→ ∞aiε. Note that2.2and2.7mean that

ai≤dkfmix, fmi1xdkfmi1x, fni1xdkfni1x, fnix

≤cmi1ϕai cni1,

2.8

for anyi≥1. Lettingi → ∞in2.8, 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→ ∞fnxw. Using2.1,2.2, andϕ∈Φ1∩Φ2, we obtain that for eachx, y∈X

dfx, fy ∞

k1 1 2k ·

dkfx, fy

1dkfx, fy ≤ ∞

k1 1 2k ·

ϕdkx, y

1ϕdkx, y

≤∞ k1 1 2k ·

dk

x, y

1dk

x, y d

x, y,

2.10

which yields that

dw, fw≤dw, fnxdfnx, fw≤dw, fnxdfn−1x, w−→0, asn−→ ∞,

2.11

that is,wis a fixed point off. If f has a fixed pointvdifferent fromw, then there existsk≥1 such thatdkw, v>0. By2.2, we have

dkw, v dkfw, fv≤ϕdkw, v< dkw, v, 2.12

Suppose thatϕ∈Φ3. By2.2, we get that for anyx∈X,n≥1, andk≥1

dk

fnx, wdk

fnx, fnw≤ϕdk

fn−1x, fn−1w≤ · · · ≤ϕndkx, w. 2.13

This completes the proof.

Remark 2.2. Theorem 2.1extends Theorem 2.1 of Bhakta and Choudhury6and Theorem 1

of Boyd and Wong12.

Lemma 2.3see11. Leta,b,c, anddbe inR, then

opt{a, b} −opt{c, d}≤max{|a−c|,|b−d|}. 2.14

3. Properties of Solutions

In this section, we assume thatX, · andY, · are real Banach spaces,S⊆Xis the state space, andD⊆Y is the decision space. Define

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

For any positive integerkandf, g∈BBS, let

dkf, gsupfx−gx:x∈B0, k,

df, g ∞

k1 1 2k ·

dkf, g

1dkf, g,

3.2

whereB0, k {x:x∈Sandx ≤k}, then{dk}_k≥1is a countable family of pseudometrics onBBS. It is clear thatBBS, dis a complete metric space.

Theorem 3.1. Letp : S×D → RandH : S×D×BBS → Rbe mappings, and letϕ be in

Φ1∩Φ2, such that

C1for anyk≥1andx, y, u, v∈B0, k×D×BBS×BBS,

Hx, y, u−Hx, y, v≤ϕdku, v, 3.3

C2for anyk≥1andu∈BBS, there existsαk, u>0satisfying

_p

then the functional equation1.3possesses a unique solutionw ∈BBS, and{Gn_g}

n≥1converges

towfor eachg∈BBS, whereGis defined by

Ggx opt

y∈D

px, yHx, y, g, ∀x, g∈S×BBS. 3.5

In addition, ifϕis inΦ3, then

dkGng, w≤ϕndkg, w, ∀g ∈BBS, n≥1, k≥1. 3.6

Proof. It follows fromC2and3.4thatGmapsBBSinto itself. Givenε > 0,k ≥ 1,x ∈

B0, k, andh, g∈BBS, suppose that opt_y∈Dsup_y∈D, then there existy, z∈Dsuch that

Ghx< px, yHx, y, hε, Ggx< px, z Hx, z, gε,

Ghx≥px, z Hx, z, h, Ggx≥px, yHx, y, g. 3.7

In view of3.3,3.5, and3.7, we deduce that

Ghx−Ggx<maxHx, y, h−Hx, y, g,Hx, z, h−Hx, z, gε ≤ϕdkh, gε,

3.8

which implies that

dkGh, GgsupGhx−Ggx:x∈B0, k≤ϕdkh, gε. 3.9

Similarly, we can show that3.9holds for opt_y∈Dinfy∈D. Asε → 0in3.9, we get that

dk

Gh, Gg≤ϕdk

h, g. 3.10

Notice that the functional equation 1.3possesses a unique solution w if and only if the mappingGhas a unique fixed pointw. 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. Letr, pi, qi, ui, vi:S×D → Randai, bi:S×D → Sbe mappings fori1,2, . . . , m.

Assume that the following conditions are satisfied:

C3for eachk≥1, there existsAk>0such that

_r

x, y m

i1

C4max{aix, y,bix, y:i∈ {1,2, . . . , m}} ≤ x, for allx, y∈S×D,

C5there exists a constantβ∈0,1such that

m

i1

maxqi

x, y,vi

x, y≤β, ∀x, y∈S×D, 3.12

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

wfor eachw0∈BBS, where{wn}n≥1is defined by

wnx opt

y∈D

rx, y m

i1

optpix, yqix, ywn−1

aix, y,

uix, yvix, ywn−1

bix, y

, ∀x∈S, n≥1.

3.13

Moreover,

dkwn, w≤βn

1−β−1dkw0, w, ∀n≥1, k≥1. 3.14

Proof. Set

Hx, y, hrx, y m

i1

optpix, yqix, yhaix, y, uix, yvix, yhbix, y

∀x, y, h∈S×D×BBS,

3.15

Ghx opt

y∈D

Hx, y, h, ∀x, h∈S×BBS. 3.16

It follows fromC3–C5and3.15that

_H

x, y, h≤rx, y m

i1

maxpix, yqix, yhaix, y,

ui

x, yvi

x, yhbi

x, y

≤rx, y m

i1

maxpi

x, y,ui

x, y m

i1

maxqi

x, y,vi

x, y

×maxhai

x, y,hbi

x, y

≤Ak

_m

i1

maxqix, y,vix, y

sup|ht|:t∈B0, k

≤Ak βsup|ht|:t∈B0, k,

for anyk≥1 andx, y, h∈B0, k×D×BBS. Consequently,Gis a self mapping onBBS. ByLemma 2.3,C4, andC5, we obtain that for anyk ≥ 1 andx, y, g, h ∈ B0, k×D× BBS×BBS,

_H

x, y, g−Hx, y, h

m

i1

optpix, yqix, ygaix, y, uix, yvix, ygbix, y

−m i1

optpi

x, yqi

x, yhai

x, y, ui

x, yvi

x, yhbi

x, y

≤m i1

maxqi

x, ygai

x, y−hai

x, y,vi

x, ygbi

x, y−hbi

x, y

≤m i1

maxqi

x, y,vi

x, y

×maxgaix, y−haix, y,gbix, y−hbix, y ≤ϕdk

g, h,

3.18

whereϕt βtfort∈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 Roosta5, page

545, Theorem 3.3 of Bhakta and Choudhury 6, and Theorems 3.3 and 3.4 of Liu 8are special cases ofTheorem 3.3. The example below shows thatTheorem 3.3extends properly the results in1,5,6,8.

Example 3.5. LetX Y S RandD R−. Putm 2,β 2/3, andAk 3k3_{for any}

k≥1. It follows fromTheorem 3.3that the functional equation

fx opt

y∈D

⎧ ⎨ ⎩x2sin

xyx−y1

opt

x3

1 x

2_y2 1x2_y22

sin2

x−yx2 3x2_y2 f

xcosx2y2,

x2ln

1 xy

1xy

cos

xy−2x−1 3x2_y−1 f

x

1|x|y2_x−_y2

opt

x3_y 1|x|y

cos2_xy−x2 3x2_y2 f

xsin1−xyx3y2,

x2_cosx2_−y2 1|x|y2

xy

4x2_y2f

x

12x2_y

, ∀x∈S

Theorem 3.6. Letr, pi, qi, ui, vi:S×D → Randai, bi:S×D → Sbe mappings fori1,2, . . . , m, and,ϕ, ψbe inΦ4satisfying

C6|rx, y|m_i1max{|pix, y|,|uix, y|} ≤ψx, for allx, y∈S×D,

C7max{aix, y,bix, y:i∈ {1,2, . . . , m}} ≤ϕx, for allx, y∈S×D,

C8sup_x,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:

C9the sequence{wn}_n≥1defined by

w0x opt

y∈D

rx, y m

i1 optpi

x, y, ui

x, y

,

wnx opt y∈D

rx, y m

i1 optpi

x, yqi

x, ywn−1

ai

x, y,

ui

x, yvi

x, ywn−1

bi

x, y

∀x∈S, n≥1,

3.20

converges tow,

C10limn→ ∞wxn 0for anyx0∈S,{yn}n≥1⊂Dandxn∈ {aixn−1, yn, bixn−1, yn:i∈ {1,2, . . . , m}}, n≥1,

C11wis unique with respect to condition (C10).

Proof. LetHandGbe defined by3.15and3.16, respectively. We now claim that

ϕt< t, ∀t >0. 3.21

If not, then there exists somet >0 such thatϕt≥t. On account ofϕ, ψ∈Φ4, we know that for anyn≥1,

ψϕnt≥ψϕn−1t≥ · · · ≥ψt>0, 3.22

whence

lim

n→ ∞ψ

ϕnt≥ψt>0, 3.23

Next, we assert that the mappingG is nonexpansive on BBS. Let k ≥ 1 and h ∈ BBS. It is easy to see that

maxaix, y,bix, y:i∈ {1,2, . . . , m}≤ϕx< k, ∀x, y∈B0, k×D, 3.24

byC7and3.21. Consequently, there exists a constantCk, h>0 satisfying

maxhai

x, y,hbi

x, y:i∈ {1,2, . . . , m}≤Ck, h, ∀x, y∈B0, k×D.

3.25

In view ofC6,3.16, and3.25, we derive that for anyx∈B0, k,

|Ghx|opt

y∈D

Hx, y, h≤sup

y∈D

Hx, y, h

≤sup

y∈D

rx, y m

i1

maxpi

x, yqi

x, yhai

x, y,

ui

x, yvi

x, yhbi

x, y

≤sup

y∈D

rx, y m

i1

maxpi

x, y,ui

x, y

m

i1

maxqi

x, y,vi

x, ymaxhai

x, y,hbi

x, y

≤ψk Ck, h,

3.26

which yields thatGmapsBBSinto itself. Givenε >0,k≥1,x∈B0, k, andh, g∈BBS, suppose that opt_y∈Dsup_y∈D, then there existy, z∈Dsuch that

Ghx< Hx, y, hε, Ggx< Hx, z, gε,

Ghx≥Hx, z, h, Ggx≥Hx, y, g.

UsingC6–C8,3.15and3.27, andLemma 2.3, we deduce that

Ghx−Ggx<maxHx, y, h−Hx, y, g,Hx, z, h−Hx, z, gε

≤max

_m

i1

maxqi

x, yhai

x, y−gai

x, y,

vi

x, yhbi

x, y−gbi

x, y, m

i1

maxqix, zhaix, z−gaix, z,

|vix, z|hbix, z−gbix, z

ε

≤max

_m

i1

maxqix, y,vix, y,

m

i1

maxqix, z,|vix, z|

dkh, gε

≤dk

h, gε,

3.28

which means that

dk

Gh, Gg≤dk

h, gε. 3.29

Similarly, we can conclude that the above inequality holds for opt_y∈D infy∈D. Lettingε →

0, we get that

dk

Gh, Gg≤dk

h, g, 3.30

which implies that

dGh, Gg ∞

k1 1 2k ·

dk

Gh, Gg

1dk

Gh, Gg ≤ ∞

k1 1 2k ·

dk

h, g

1dk

h, g d

h, g. 3.31

That is,Gis nonexpansive.

We show that for eachn≥0,

|wnx| ≤ n

j0

In terms ofC6andC9, we obtain that

|w0x| ≤sup

y∈D

_r

x, y m

i1

maxpix, y,uix, y

≤ψx, ∀x∈S, 3.33

which means that3.32holds forn0. Suppose that3.32holds for somen≥0. It follows fromC6–C8and3.25that

|wn1x|

opt_y∈D

rx, y m

i1

optpix, yqix, ywnaix, y,

ui

x, yvi

x, ywn

bi

x, y

≤sup

y∈D

_r

x, y m

i1

maxpix, yqix, ywnaix, y,

ui

x, yvi

x, ywn

bi x, y ≤sup

y∈D

_r

x, y m

i1

maxpix, y,uix, y

m

i1

maxqi

x, y,vi

x, ymaxwn

ai

x, y,wn

bi

x, y

≤ψx sup

y∈D

maxwn

ai

x, y,wn

bi

x, y:i∈ {1,2, . . . , m}

≤ψx sup

y∈D

max ⎧ ⎨ ⎩ n

j0

ψϕjaix, y,

n

j0

ψϕj_b i

x, y:i∈ {1,2, . . . , m}

⎫ ⎬ ⎭

n1

j0

ψϕjx.

3.34

Next, we prove that{wn}_n≥0is a Cauchy sequence inBBS. Givenε >0,k≥1,n≥1,

j≥1, andx0∈B0, k, suppose that opt_y∈Dsup_y∈D. We select thaty, z∈Dwith

wnx0< r

x0, y

m

i1

optpix0, y

qix0, y

wn−1

aix0, y

,

uix0, y

vix0, y

wn−1

bix0, y

2−1ε,

wnjx0< rx0, z

m

i1

optpix0, z qix0, zwnj−1aix0, z,

uix0, z vix0, zwnj−1bix0, z

2−1ε,

wnx0≥rx0, z

m

i1

optpix0, z qix0, zwn−1aix0, z,

uix0, z vix0, zwn−1bix0, z},

wnjx0≥r

x0, y

m

i1

optpix0, y

qix0, y

wnj−1

aix0, y

,

uix0, y

vix0, y

wnj−1

bix0, y

.

3.35

According toC6–C8and3.35, we have

_wnjx0−wnx0

<max

m

i1

optpix0, z qix0, zwnj−1aix0, z,

uix0, z vix0, zwnj−1bix0, z

−m i1

optpix0, z qix0, zwn−1aix0, z,

uix0, z vix0, zwn−1bix0, z}

, m

i1

optpix0, y

qix0, y

wnj−1

aix0, y

,

uix0, y

vix0, y

wnj−1

bix0, y

−m i1

optpi

x0, y

qi

x0, y

wn−1

ai

x0, y

,

uix0, y

vix0, y

wn−1

bix0, y

≤max

_m

i1

maxqix0, zwnj−1aix0, z−wn−1aix0, z,

|vix0, z|wnj−1bix0, z−wn−1bix0, z,

m

i1

maxqi

x0, ywnj−1

ai

x0, y

−wn−1

ai

x0, y,

_vi

x0, ywnj−1

bix0, y

−wn−1

bix0, y

2−1ε

≤max

_m

i1

maxqix0, z,|vix0, z|

maxwnj−1aix0, z−wn−1aix0, z,

wnj−1bix0, z−wn−1bix0, z,

m

i1

maxqix0, y,vi

x0, ymaxwnj−1

aix0, y

−wn−1

aix0, y,

wnj−1

bi

x0, y

−wn−1

bi

x0, y

2−1_ε

≤maxmaxwnj−1aix0, z−wn−1aix0, z,

wnj−1bix0, z−wn−1bix0, z:i∈ {1,2, . . . , m}

,

maxwnj−1

ai

x0, y

−wn−1

ai

x0, y,

wnj−1

bi

x0, y

−wn−1

bi

x0, y:i∈ {1,2, . . . , m}

2−1ε

wnj−1x1−wn−1x12−1ε,

3.36

for somex1 ∈ {aix0, y1, bix0, y1 :i ∈ {1,2, . . . , m}}andy1 ∈ {y, z}. In a similar way, we can conclude that3.36holds for opt_y∈D infy∈D. Proceeding in this way, we selectyt ∈D

andxt∈ {aixt−1, yt, bixt−1, yt:i∈ {1,2, . . . , m}}fort∈ {2,3, . . . , n}such that

wnj−1x1−wn−1x1<wnj−2x2−wn−2x22−2ε,

wnj−2x2−wn−2x2<wnj−3x3−wn−3x32−3ε,

.. .

_{wj1xn−1−w1xn−1<wjxn−w0xn2}−n_ε.

In terms ofC7,3.21,3.32,3.36, and3.37, we know that

_{wnjx0−wnx0<wjxn−w0xn}n i1

2−iε

<wjxn|w0xn|ε

≤ j

i0

ψϕixnψxn ε

≤ ∞ in−1

ψϕikε,

3.38

which implies that

dk

wnj, wn

≤ ∞ in−1

ψϕikε. 3.39

Lettingε → 0in the above inequality, we have

dkwnj, wn≤

∞

in−1

ψϕik, 3.40

which means that{wn}_n≥0is a Cauchy sequence inBBS, dbecause∞_n0ψϕn_t<∞for

eacht >0. Let limn→ ∞wnw∈BBS. By the nonexpansivity ofG, we get that

dw, Gw≤dw, Gwn dGwn, Gw

≤dw, wn1 dwn, w−→0, asn−→ ∞,

3.41

which implies thatwGw. That is,wis a solution of the functional equation1.4.

Now, we show that C10 holds. Given ε > 0, x0 ∈ S, {yn}n≥1 ⊂ D, and xn ∈

{aixn−1, yn,bixn−1, yn:i∈ {1,2, . . . , m}}forn≥1, setk1 x0. It is easy to verify that there exists a positive integermsatisfying

dkw, wn ∞

in

Notice that

xn ≤maxai

xn−1, yn,bi

xn−1, yn:i∈ {1,2, . . . , m}

≤ϕxn−1≤ · · · ≤ϕnx0≤ϕkk< k,

3.43

for anyn≥1. Consequently, we infer immediately that, forn≥m,

|wxn| ≤ |wxn−wnxn||wnxn| ≤dkw, wn n

i0

ψϕixn

≤dkw, wn ∞

in

ψϕik< ε,

3.44

which yields that limn→ ∞wxn 0.

At last, we show thatC11holds. Suppose that the functional equation1.4possesses another solution h ∈ BBS, which satisfiesC10. Givenε > 0 andx0 ∈ S, suppose that opt_y∈Dsup_y∈D, then there arey, z∈Ssatisfying

wx0< r

x0, y

m

i1

optpix0, y

qix0, y

waix0, y

,

uix0, y

vix0, y

wbix0, y

2−1ε,

hx0< rx0, z

m

i1

optpix0, z qix0, zhaix0, z,

uix0, z vix0, zhbix0, z}2−1ε,

wx0≥rx0, z

m

i1

optpix0, z qix0, zwaix0, z,

uix0, z vix0, zwbix0, z},

hx0≥r

x0, y

m

i1 optpi

x0, y

qi

x0, y

hai

x0, y

,

uix0, y

vix0, y

hbix0, y

,

Whence there existsy1∈ {y, z}andx1∈ {ax0, y1, bx0, y1:i∈ {1,2, . . . , m}} such that

|wx0−hx0|

<max

_m

i1

optpi

x0, y

qi

x0, y

wai

x0, y

, ui

x0, y

vi

x0, y

wbi

x0, y

−optpix0, y

qix0, y

haix0, y

, uix0, y

vix0, y

hbix0, y,

m

i1

_opt

pix0, z qix0, zwaix0, z, uix0, z vix0, zwbix0, z

−optpix0, z qix0, zhaix0, z,

uix0, z vix0, zhbix0, z}|

2−1ε

≤max

_m

i1

maxqi

x0, yw

ai

x0, y

−hai

x0, y,

vi

x0, yw

bi

x0, y

−hbi

x0, y,

m

i1

maxqix0, z|waix0, z−haix0, z|,

|vix0, z||wbix0, z−hbix0, z|

2−1ε

≤max

_m

i1

maxqix0, y,vi

x0, y,

m

i1

maxqix0, z,|vix0, z|

×maxwai

x0, y

−hai

x0, y,w

bi

x0, y

−hbi

x0, y,

|waix0, z−haix0, z|,|wbix0, z−hbix0, z|:i∈ {1,2, . . . , m}}2−1ε

≤ |wx1−hx1|2−1ε

3.46

byC8. Proceeding in this way, we select yj ∈ D and xj ∈ {aixj−1, yj, bixj−1, yj : i ∈

{1,2, . . . , m}}forj∈ {2,3, . . . , n}satisfying

|wx1−hx1|<|wx2−hx2|2−2ε,

|wx2−hx2|<|wx3−hx3|2−3ε,

.. .

|wxn−1−hxn−1|<|wxn−hxn|2−nε.

It follows that

|wx0−hx0|<|wxn−hxn|ε, 3.48

which yields that

|wx0−hx0| ≤ε, 3.49

by lettingn → ∞. Similarly,3.49also holds for opt_y∈Dinfy∈D. Asε → 0, we know that wx0 hx0. This completes the proof.

Remark 3.7. Theorem 3.6generalizes Theorem 1 of Bellman 1, page 119, Theorem 3.5 of

Bhakta and Choudhury6, Theorem 2.4 of Bhakta and Mitra7, Theorem 3.5 of Liu 8 and Theorem 3.1 of Liu and Ume11. The following example reveals thatTheorem 3.6is indeed a generalization of the results in1,6–8,11.

Example 3.8. LetXY R,SDR. Defineϕ, ψ:R → Rby

ϕt 2−1t, ψt 3t4, ∀t∈R. 3.50

It is easy to verify that the following functional equation:

fx opt

y∈D

x4

2sin1x2_y2 max

x4

1xy

xsinxy

24xy f

x2 22xy

,

x4

1xy

ln1xy

4xy f

x3_y 12x2_y

max

x4

1x−y2

ycosx−y

1x4y f

x3

12x2_sinx2_y2

,

x5_y

1xy

cosxy2x−y

4xy f

x4_ysinx3_y3_xy−1 12x3_y

, ∀x∈S

3.51

satisfies conditionsC6–C8. Consequently,Theorem 3.6ensures that it has a solutionw ∈ BBS that satisfies conditionsC9–C11. However, Theorem 1 of Bellman 1, page 119, Theorem 3.5 of Bhakta and Choudhury6, Theorem 2.4 of Bhakta and Mitra7, Theorem 3.5 of Liu8, and Theorem 3.1 of Liu and Ume11are not applicable.

Acknowledgment

References

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of Mathematical Analysis and Applications, vol. 161, no. 1, pp. 57–77, 1991.

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5 R. Bellman and M. Roosta, “A technique for the reduction of dimensionality in dynamic programming,”Journal of Mathematical Analysis and Applications, vol. 88, no. 2, pp. 543–546, 1982. 6 P. C. Bhakta and S. R. Choudhury, “Some existence theorems for functional equations arising in

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7 P. C. Bhakta and S. Mitra, “Some existence theorems for functional equations arising in dynamic programming,”Journal of Mathematical Analysis and Applications, vol. 98, no. 2, pp. 348–362, 1984. 8 Z. Liu, “Existence theorems of solutions for certain classes of functional equations arising in dynamic

programming,”Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 529–553, 2001. 9 Z. Liu, “Coincidence theorems for expansion mappings with applications to the solutions of

functional equations arising in dynamic programming,”Acta Scientiarum Mathematicarum, vol. 65, no. 1-2, pp. 359–369, 1999.

10 Z. Liu, “Compatible mappings and fixed points,”Acta Scientiarum Mathematicarum, vol. 65, no. 1-2, pp. 371–383, 1999.

11 Z. Liu and J. S. Ume, “On properties of solutions for a class of functional equations arising in dynamic programming,”Journal of Optimization Theory and Applications, vol. 117, no. 3, pp. 533–551, 2003. 12 D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,”Proceedings of the American Mathematical