Volume 2011, Article ID 701519,30pages doi:10.1155/2011/701519
Research Article
Solvability and Algorithms for
Functional Equations Originating
from Dynamic Programming
Guojing Jiang,
1Shin Min Kang,
2and Young Chel Kwun
3 1Organization Department, Dalian Vocational Technical College, Dalian, Liaoning 116035, China2Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea 3Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea
Correspondence should be addressed to Young Chel Kwun,[email protected]
Received 5 January 2011; Accepted 11 February 2011
Academic Editor: Yeol J. Cho
Copyrightq2011 Guojing Jiang 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.
The main purpose of this paper is to study the functional equation arising in dynamic program-ming of multistage decision processesfxopty∈Dopt{px, y, qx, yfax, y, rx, yfbx, y,
sx, yfcx, y}for allx∈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 spacesBCSandBSand the complete metric spaceBBS, 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 spaceBBS, 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; see1–23and the references therein. Bellman3, Bhakta and Choudhury7, Liu12, Liu and Kang15, and Liu et al.18investigated the following functional equations:
fx inf
y∈Dmax
px, y, fax, y, ∀x∈S,
fx sup
y∈D
maxpx, y, fax, y, ∀x∈S,
fx inf
y∈Dmax
px, y, qx, yfax, y, ∀x∈S
and gave some existence and uniqueness results and iterative approximations of solutions for the functional equations inBBS. Liu and Kang14and Liu and Ume17generalized the results in3,7,12,15,18and studied the properties of solutions, nonpositive solutions and nonnegative solutions and the convergence of iterative approximations for the following functional equations, respectively
fx opt
y∈D
maxpx, y, fax, y, ∀x∈S,
fx opt
y∈D
minpx, y, fax, y, ∀x∈S,
fx opt
y∈D
maxpx, y, qx, yfax, y, ∀x∈S,
fx opt
y∈D
minpx, y, qx, yfax, y, ∀x∈S
1.2
inBBS.
The purpose of this paper is to introduce and study the following functional equations arising in dynamic programming of multistage decision processes:
fx opt
y∈D
optpx, y, qx, yfax, y, rx, yfbx, y, sx, yfcx, y, ∀x∈S,
1.3
fx opt
y∈D
maxpx, y, qx, yfax, y, rx, yfbx, y, sx, yfcx, y, ∀x∈S,
1.4
fx opt
y∈D
minpx, y, qx, yfax, y, rx, yfbx, y, sx, yfcx, y, ∀x∈S,
1.5
fx sup
y∈D
maxpx, y, qx, yfax, y, rx, yfbx, y, sx, yfcx, y, ∀x∈S,
1.6
fx inf
y∈Dmin
px, y, qx, yfax, y, rx, yfbx, y, sx, yfcx, y, ∀x∈S,
1.7
fx sup
y∈D
minpx, y, qx, yfax, y, rx, yfbx, y, sx, yfcx, y, ∀x∈S,
1.8
fx inf
y∈Dmax
px, y, qx, yfax, y, rx, yfbx, y, sx, yfcx, y, ∀x∈S,
1.9
where opt denotes sup or inf,xandystand for the state and decision vectors, respectively,
This paper is divided into four sections. InSection 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 fixed-point theorem, we establish the existence, uniqueness, and iterative approximations of solutions for the functional equation1.3in the Banach spaces
BCS and BS, respectively. By means of two iterative algorithms defined in Section 2, we obtain the existence, uniqueness, and iterative approximations of solutions for the functional equation1.3in the complete metric spaceBBS. Under certain conditions, we investigate the behaviors of solutions, nonpositive solutions, and nonnegative solutions and the 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 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 thatR −∞,∞,R 0,∞,R− −∞,0,Ndenotes the set of positive integers, and, for eacht∈R,tdenotes the largest integer not exceedingt. LetX,·andY, · be real Banach spaces,S⊆Xthe state space, andD⊆Ythe decision space. Define
Φ1
ϕ:ϕ:R−→R is nondecreasing,
Φ2ϕ, ψ
:ϕ, ψ∈Φ1, ψt>0, lim n→ ∞ψ
ϕnt0 fort >0
,
Φ3
ϕ, ψ:ϕ, ψ∈Φ1, lim n→ ∞ψ
ϕnt exists fort >0
,
BS f:f:S−→Ris bounded,
BCS f :f∈BSis continuous,
BBS f :f :S−→Ris bounded on bounded subsets ofS.
2.1
ClearlyBS, · 1andBCS, · 1are Banach spaces with normf1supx∈S|fx|. For anyk∈Nandf, g∈BBS, let
dk
f, gsup fx−gx :x∈B0, k,
df, g
∞
k1
1 2k ·
dk
f, g
1dk
f, g,
2.2
whereB0, k {x: x∈Sandx ≤k}. It is easy to see that{dk}k∈Nis a countable family
x ∈ BBSif, for anyk ∈ Ndkxn, x → 0 as n → ∞ and to be a Cauchy sequenceif, for
any k ∈ N, dkxn, xm → 0 asn, m → ∞. It is clear that BBS, d is a complete metric
space.
Lemma 2.1. Let{ai, bi: 1≤i≤n} ⊂R. Then
aopt{ai: 1≤i≤n}opt{opt{ai: 1≤i≤n−1}, an},
bopt{ai: 1≤i≤n} ≤opt{bi: 1≤i≤n}forai≤bi,1≤i≤n,
cmax{aibi : 1≤i≤ n} ≤max{ai : 1≤i≤n}max{bi : 1≤i≤ n}for{ai, bi : 1≤ i≤
n} ⊂R,
dmin{aibi: 1≤i≤n} ≥min{ai: 1≤i≤n}min{bi: 1≤i≤n}for{ai, bi: 1≤i≤n} ⊂ R,
e|opt{ai: 1≤i≤n} −opt{bi: 1≤i≤n}| ≤max{|ai−bi|: 1≤i≤n}.
Proof. Clearlya–dare true. Now we showe. Note thateholds forn1. Suppose that
eis true for somen∈N. It follows fromaand Lemma 2.1 in17that
opt{ai: 1≤i≤n1} −opt{bi: 1≤i≤n1}
optopt{ai: 1≤i≤n}, an1
−optopt{bi: 1≤i≤n}, bn1
≤max opt{ai: 1≤i≤n} −opt{bi : 1≤i≤n} ,|an1−bn1|
≤max{|ai−bi|: 1≤i≤n1}.
2.3
Henceeholds for anyn∈N. This completes the proof.
Lemma 2.2. Let{ai: 1≤i≤n} ⊂Rand{bi: 1≤i≤n} ⊂R. Then
amax{aibi: 1≤i≤n} ≥min{ai: 1≤i≤n}max{bi: 1≤i≤n},
bmin{aibi: 1≤i≤n} ≤max{ai: 1≤i≤n}min{bi: 1≤i≤n}.
Proof. It is clear thatais true forn1. Suppose thatais also true for somen∈N. Using Lemma 2.3 in19andLemma 2.1, we infer that
max{aibi: 1≤i≤n1}
max{max{aibi: 1≤i≤n}, an1bn1}
≥max{min{ai : 1≤i≤n}max{bi: 1≤i≤n}, an1bn1} ≥min{ai: 1≤i≤n1}max{bi: 1≤i≤n1}.
That is,ais true forn1. Thereforeaholds for anyn ∈ N. Similarly we can proveb. This completes the proof.
Lemma 2.3. Let{a1n}n∈N,{a2n}n∈N, . . . ,{akn}n∈Nbe convergent sequences inR. Then
lim
n→ ∞opt{ain: 1≤i≤k}opt
lim
n→ ∞ain: 1≤i≤k
. 2.5
Proof. Put limn→ ∞ainbifor 1≤i≤k. In view ofLemma 2.1we deduce that
opt{ain: 1≤i≤k} −opt{bi: 1≤i≤k} ≤max{|ain−bi|: 1≤i≤k} −→0 asn−→ ∞,
2.6
which yields that
lim
n→ ∞opt{ain: 1≤i≤k}opt
lim
n→ ∞ain: 1≤i≤k
. 2.7
This completes the proof.
Lemma 2.4. aAssume thatA:S×D → Ris a mapping such thatopty∈DAx0, yis bounded for somex0∈S. Then
opt
y∈D
Ax0, y
≤sup
y∈D
Ax0, y . 2.8
b Assume that A, B : S × D → R are mappings such that opty∈DAx1, y and
opty∈DBx2, yare bounded for somex1, x2∈S. Then
opt
y∈D
Ax1, y
−opt
y∈D
Bx2, y
≤sup
y∈D
A
x1, y
−Bx2, y . 2.9
Proof. Now we show a. If supy∈D|Ax0, y| ∞, a holds clearly. Suppose that
supy∈D|Ax0, y|<∞. Note that
− Ax0, y ≤A
x0, y
It follows that
−sup
y∈D
Ax0, y inf
y∈D
− Ax0, y ≤ inf y∈DA
x0, y
≤opt
y∈D
Ax0, y
≤sup
y∈D
Ax0, y
≤sup
y∈D
Ax0, y ,
2.11
which implies that
opt
y∈D
Ax0, y
≤sup
y∈D
Ax0, y . 2.12
Next we show b. If supy∈D|Ax1, y −Bx2, y| ∞, b is true. Suppose that
supy∈D|Ax1, y−Bx2, y|<∞. Note that
Ax1, y
−Bx2, y ≤sup y∈D
Ax1, y
−Bx2, y <∞, ∀y∈D, 2.13
which yields that
Bx2, y
−sup
y∈D
A
x1, y
−Bx2, y
≤Ax1, y
≤Bx2, y
sup
y∈D
A
x1, y
−Bx2, y , ∀y∈D.
2.14
It follows that
opt
y∈D
Bx2, y
−sup
y∈D
A
x1, y
−Bx2, y
opt
y∈D
Bx2, y
−sup
y∈D
A
x1, y
−Bx2, y
≤opt
y∈D
Ax1, y
≤opt
y∈D
Bx2, y
sup
y∈D
A
x1, y
−Bx2, y
opt
y∈D
Bx2, y
sup
y∈D
A
x1, y
−Bx2, y ,
2.15
which gives that
opt
y∈D
Ax1, y
−opt
y∈D
Bx2, y
≤sup
y∈D
A
x1, y
−Bx2, y . 2.16
Algorithm 1. For anyf0∈BCS, compute{fn}n≥0by
fn1x 1−αnfnx αnopt
y∈D
optpx, y, qx, yfn
ax, y,
rx, yfn
bx, y, sx, yfn
cx, y, ∀x∈S, n≥0,
2.17
where
{αn}n≥0 is any sequence in0,1,
∞
n0
αn ∞. 2.18
Algorithm 2. For anyf0∈BS, compute{fn}n≥0by2.17and2.18.
Algorithm 3. For anyf0∈BBS, compute{fn}n≥0by2.17and2.18.
Algorithm 4. For anyw0 ∈BBS, compute{wn}n≥0by
wn1x opt
y∈D
optpx, y, qx, ywn
ax, y, rx, ywn
bx, y,
sx, ywn
cx, y, ∀x∈S, n≥0.
2.19
Algorithm 5. For anyw0 ∈BBS, compute{wn}n≥0by
wn1x opt
y∈D
maxpx, y, qx, ywn
ax, y, rx, ywn
bx, y,
sx, ywn
cx, y, ∀x∈S, n≥0.
2.20
Algorithm 6. For anyw0 ∈BBS, compute{wn}n≥0by
wn1x opt
y∈D
minpx, y, qx, ywn
ax, y, rx, ywn
bx, y,
sx, ywn
cx, y, ∀x∈S, n≥0.
2.21
Algorithm 7. For anyw0 ∈BBS, compute{wn}n≥0by
wn1x sup
y∈D
maxpx, y, qx, ywn
ax, y, rx, ywn
bx, y,
sx, ywn
cx, y, ∀x∈S, n≥0.
Algorithm 8. For anyw0 ∈BBS, compute{wn}n≥0by
wn1x inf
y∈Dmin
px, y, qx, ywn
ax, y, rx, ywn
bx, y,
sx, ywn
cx, y, ∀x∈S, n≥0.
2.23
Algorithm 9. For anyw0 ∈BBS, compute{wn}n≥0by
wn1x sup
y∈D
minpx, y, qx, ywn
ax, y, rx, ywn
bx, y,
sx, ywn
cx, y, ∀x∈S, n≥0.
2.24
Algorithm 10. For anyw0∈BBS, compute{wn}n≥0by
wn1x inf
y∈Dmax
px, y, qx, ywn
ax, y, rx, ywn
bx, y,
sx, ywn
cx, y, ∀x∈S, n≥0.
2.25
3. The Properties of Solutions and Convergence of Algorithms
Now we prove the existence, uniqueness, and iterative approximations of solutions for the functional equation 1.3inBCSandBS, respectively, by using the Banach fixed-point theorem.
Theorem 3.1. LetSbe compact. Letp, q, r, s :S×D → Randa, b, c : S×D → S satisfy the following conditions:
C1pis bounded inS×D;
C2supx,y∈S×Dmax{|qx, y|,|rx, y|,|sx, y|} ≤αfor some constantα∈0,1;
C3for each fixedx0 ∈S,
lim
x→x0
px, ypx0, y
, lim
x→x0
qx, yqx0, y
,
lim
x→x0
rx, yrx0, y
, lim
x→x0
sx, ysx0, y
,
lim
x→x0
ax, yax0, y
, lim
x→x0
bx, ybx0, y
,
lim
x→x0
cx, ycx0, y
3.1
uniformly fory∈D, respectively.
Then the functional equation1.3possesses a unique solutionf ∈BCS, and the sequence {fn}n≥0generated byAlgorithm 1converges tofand has the error estimate
fn1−f≤e−1−αn
Proof. Define a mappingH:BCS → BCSby
Hhx opt
y∈D
optpx, y, qx, yhax, y, rx, yhbx, y
sx, yhcx, y, ∀x, h∈S×BCS.
3.3
Leth∈BCSandx0∈Sandε >0. It follows fromC1,C3, and the compactness ofSthat
there exist constantsM >0,δ >0, andδ1>0 satisfying
sup
x,y∈S×D
px, y ≤M, 3.4
sup
x,y∈S×D
max|hx|, hax, y , hbx, y , hcx, y ≤M, 3.5
px, y−px0, y < ε
3, ∀
x, y∈S×D withx−x0< δ, 3.6
max qx, y−qx0, y , r
x, y−rx0, y , s
x, y−sx0, y <
ε
6M,
∀x, y∈S×D withx−x0< δ,
3.7
|hx1−hx2|< ε
6, ∀x1, x2∈Swithx1−x2< δ1, 3.8
maxax, y−ax0, y,b
x, y−bx0, y,c
x, y−cx0, y< δ1,
∀x, y∈S×D withx−x0< δ.
3.9
Using3.3–3.5,C2, and Lemmas2.1and2.4, we get that
|Hhx| ≤sup
y∈D
optpx, y, qx, yhax, y, rx, yhbx, y, sx, yhcx, y
≤sup
y∈D
max px, y , qx, y hax, y ,
rx, y hbx, y , sx, y hcx, y
≤sup
y∈D
max px, y ,max qx, y , rx, y , sx, y
×max hax, y , hbx, y , hcx, y
≤max{M, αM}
M, ∀x∈S.
In light ofC2,3.3,3.5–3.9, and Lemmas2.1and2.4, we deduce that for allx, y∈S×D
withx−x0< δ
|Hhx−Hhx0|
opt
y∈D
optpx, y, qx, yhax, y, rx, yhbx, y, sx, yhcx, y
−opt
y∈D
optpx0, y
, qx0, y
hax0, y
, rx0, y
hbx0, y
, sx0, y
hcx0, y
≤sup
y∈D
max px, y−px0, y , q
x, yhax, y−qx0, y
hax0, y ,
rx, yhbx, y−rx0, y
hbx0, y ,
s
x, yhcx, y−sx0, y
hcx0, y
≤sup
y∈D
max px, y−px0, y , q
x, y−qx0, y h
ax, y
qx0, y h
ax, y−hax0, y ,
rx, y−rx0, y h
bx, y rx0, y h
bx, y−hbx0, y ,
sx, y−sx0, y h
cx, y sx0, y h
cx, y−hcx0, y ≤sup
y∈D
max px, y−px0, y ,
max qx, y−qx0, y , r
x, y−rx0, y , s
x, y−sx0, y
×max hax, y , hbx, y , hcx, y
max qx0, y , r
x0, y , s
x0, y ×max hax, y−hax0, y , h
bx, y−hbx0, y ,
hcx, y−hcx0, y
≤max
ε
3, M·
ε
6Mα· ε
6
< ε.
3.11
Thus3.10,3.11, and2.17ensure that the mappingH:BCS → BCSandAlgorithm 1
are well defined.
Next we assert that the mappingH:BCS → BCSis a contraction. Letε >0,x∈S, andg, h∈BCS. Suppose that opty∈Dinfy∈D. Chooseu, v∈Dsuch that
Hgx>optpx, u, qx, ugax, u, rx, ugbx, u, sx, ugcx, u−ε,
Hhx>optpx, v, qx, vhax, v, rx, vhbx, v, sx, vhcx, v−ε,
Hgx≤optpx, v, qx, vgax, v, rx, vgbx, v, sx, vgcx, v,
Hhx≤optpx, u, qx, uhax, u, rx, uhbx, u, sx, uhcx, u.
Lemma 2.1and3.12lead to
Hgx−Hhx
<max optpx, u, qx, ugax, u, rx, ugbx, u, sx, ugcx, u
−optpx, u, qx, uhax, u, rx, uhbx, u, sx, uhcx, u ,
optpx, v, qx, vgax, v, rx, vgbx, v, sx, vgcx, v
−optpx, v, qx, vhax, v, rx, vhbx, v, sx, vhcx, v ε
≤maxmax qx, u gax, u−hax, u ,|rx, u| gbx, u−hbx, u ,
|sx, u| gcx, u−hcx, u ,
max qx, v gax, v−hax, v ,|rx, v| gbx, v−hbx, v ,
|sx, v| gcx, v−hcx, v ε
≤max qx, u ,|rx, u|,|sx, u|, qx, v ,|rx, v|,|sx, v|g−h1ε
≤αg−h1ε,
3.13
which implies that
Hg−Hh1≤αg−h1ε, ∀g, h∈BCS. 3.14
Lettingε → 0in the above inequality, we know that
Hg−Hh1≤αg−h1, ∀g, h∈BCS. 3.15
Similarly we conclude that3.15holds for opty∈Dsupy∈D. The Banach fixed-point theorem yields that the contraction mappingHhas a unique fixed pointf∈BCS. It is easy to verify thatfis also a unique solution of the functional equation1.3inBCS. By means of2.17,
2.18,3.15, and
fx 1−αnfx αnopt
y∈D
optpx, y, qx, yfax, y,
rx, yfbx, y, sx, yfcx, y, ∀x∈S, n≥0,
3.16
we infer that
fn1x−fx ≤1−αn fnx−fx αn Hfnx−Hfx
≤1−1−ααn fnx−fx
≤e−1−αin0αif0−f
1, ∀x∈S, n≥0,
which yields that
fn1−f1≤e−1−α
n i0αif
0−f1, ∀n≥0, 3.18
and the sequence{fn}n≥0converges tofby2.18. This completes the proof.
Dropping the compactness ofS andC3inTheorem 3.1, we conclude immediately the following result.
Theorem 3.2. Letp, q, r, s:S×D → Randa, b, c:S×D → Ssatisfy conditions (C1) and (C2). Then the functional equation1.3possesses a unique solutionf ∈ BSand the sequence{fn}n≥0 generated byAlgorithm 2converges tofand satisfies3.2.
Next we prove the existence, uniqueness, and iterative approximations of solution for the functional equation1.3inBBS.
Theorem 3.3. Letp, q, r, s :S×D → Randa, b, c: S×D → Ssatisfy condition (C2) and the following two conditions:
C4pis bounded onB0, k×Dfor eachk∈N;
C5supx,y∈B0,k×D{ax, y,bx, y,cx, y} ≤kfor allk∈N.
Then the functional equation1.3possesses a unique solutionw∈BBS, and the sequences {fn}n≥0and{wn}n≥0generated by Algorithms3and4, respectively, converge tofand have the error estimates
dk
fn1, w
≤e−1−αni0αidkf0, w, ∀n≥0, k∈N,
dkwn1, w≤ α n1
1−αdkw1, w0, ∀n≥0, k∈N.
3.19
Proof. Define a mappingH :BBS → BBSby3.3. Letk ∈Nandh∈BBS. ThusC4 andC5guarantee that there existMk>0 andGk, h>0 satisfying
sup
x,y∈B0,k×D
px, y ≤Mk,
sup
x,y∈B0,k×D
h
ax, y , hbx, y , hcx, y ≤Gk, h.
3.20
Using3.3,3.20,C2,C5, and Lemmas2.1and2.4, we infer that
|Hhx| ≤sup
y∈D
max px, y , qx, y hax, y ,
rx, y hbx, y , sx, y hcx, y
≤sup
y∈D
max px, y ,max qx, y , rx, y , sx, y
×max hax, y , hbx, y , hcx, y
≤max{Mk, αGk, h}, ∀x∈B0, k,
which means thatHis a self-mapping inBBS. Consequently, Algorithms3and4are well defined.
Now we claim that
dk
Hg, Hh≤αdk
g, h, ∀g, h∈BBS, k∈N. 3.22
Letk ∈N,x∈B0, k,g, h∈BBS, andε >0. Suppose that opty∈Dinfy∈D. Selectu, v∈D
such that3.12holds. Thus3.3,3.12,C2,C5, andLemma 2.1ensure that
Hgx−Hhx
<max optpx, u, qx, ugax, u, rx, ugbx, u, sx, ugcx, u
−optpx, u, qx, uhax, u, rx, uhbx, u, sx, uhcx, u ,
opt
px, v, qx, vgax, v, rx, vgbx, v, sx, vgcx, v
−optpx, v, qx, vhax, v, rx, vhbx, v, sx, vhcx, v ε
≤maxmax qx, u gax, u−hax, u ,|rx, u| gbx, u−hbx, u ,
|sx, u| gcx, u−hcx, u ,
max qx, v gax, v−hax, v ,|rx, v| gbx, v−hbx, v ,
|sx, v| gcx, v−hcx, v ε
≤max qx, u ,|rx, u|,|sx, u|, qx, v ,|rx, v|,|sx, v|dk
g, hε
≤αdk
g, hε,
3.23
which implies that
dk
Hg, Hh≤αdk
g, hε, ∀g, h∈BBS. 3.24
Similarly we conclude that3.24holds for opty∈Dsupy∈D. Asε → 0in3.24, we get that
3.22holds.
Letw0∈BBS. It follows fromAlgorithm 4that
and3.22leads to
dkwn1, wn1m≤ nm
in1
dkwi, wi1 nm
in1
dkHwi−1, Hwi
≤ nm
in1
αdkwi−1, wi≤ nm
in1
αidkw0, w1
≤ αn1
1−αdkw0, w1, ∀n≥0, k, m∈N,
3.26
which yields that{wn}n≥0is a Cauchy sequence in the complete metric spaceBBS, d, and
hence{wn}n≥0converges to somew∈BBS. In light of3.22andC2, we know that
dHg, Hh
∞
k1
1 2k ·
dk
Hg, Hh
1dk
Hg, Hh ≤
∞
k1
1 2k ·
αdk
g, h
1αdk
g, h
≤∞
k1
1 2k ·
αdk
g, h
ααdk
g, h d
g, h, ∀g, h∈BBS.
3.27
That is, the mappingHis nonexpansive. It follows from3.27andAlgorithm 4that
dHw, w lim
n→ ∞dHwn, w nlim→ ∞dwn1, w 0, 3.28
that is,wHw. Suppose that there existsu∈BBS\ {w}withuHu. Consequently there exists somek0∈Nsatisfyingdk0w, u>0. It follows from3.22andC2that
0< dk0w, u dk0Hw, Hu≤αdk0w, u< dk0w, u, 3.29
which is a contradiction. Hence the mappingH :BBS → BBShas a unique fixed point
w ∈ BBS, which is a unique solution of the functional equation 1.3 in BBS. Letting
m → ∞in3.26, we infer that
dkwn1, w≤ α n1
1−αdkw0, w1, ∀n≥0, k∈N. 3.30
It follows fromAlgorithm 3,2.18, and3.22that
dk
fn1, w
sup
x∈B0,k
1−αn
fnx−wx
αn
Hfnx−Hwx
≤1−αn sup x∈B0,k
fnx−wx αn sup
x∈B0,k
Hfnx−Hwx
≤1−αndk
fn, w
αndk
Hfn, Hw
≤1−1−ααndk
fn, w
≤e−1−αni0αidkf0, w, ∀n≥0, k∈N,
3.31
Next we investigate the behaviors of solutions for the functional equations1.3–1.5
and discuss the convergence of Algorithms4–6inBBS, respectively.
Theorem 3.4. Letϕ, ψ∈Φ2,p, q, r, s:S×D → Randa, b, c:S×D → Ssatisfy the following conditions:
C6supy∈D|px, y| ≤ψxfor allx∈S;
C7supy∈Dmax{ax, y,bx, y,cx, y} ≤ϕxfor allx∈S;
C8supx,y∈S×Dmax{|qx, y|,|rx, y|,|sx, y|} ≤1.
Then the functional equation1.3possesses a solutionw∈BBSsatisfying conditions (C9)–(C12) below:
C9the sequence{wn}n≥0generated byAlgorithm 4converges tow, wherew0 ∈BBSwith |w0x| ≤ψxfor allx, k∈B0, k×N;
C10|wx| ≤ψxfor allx∈S;
C11limn→ ∞wxn 0for any x0 ∈ S,{yn}n∈N ⊂ D and xn ∈ {axn−1, yn,bxn−1, yn,
cxn−1, yn}for alln∈N;
C12wis unique relative to condition (C11).
Proof. First of all we assert that
ϕt< t, ∀t >0. 3.32
Suppose that there exists somet0>0 withϕt0≥t0. It follows fromϕ, ψ∈Φ2that
ψt0≤ψ
ϕt0
≤ψϕ2t0
≤ · · · ≤ψϕnt0
−→0 asn−→ ∞. 3.33
That is,
ψt0≤0< ψt0, 3.34
which is impossible. That is,3.32holds. Let the mappingHbe defined by3.3inBBS. Note thatC6andC7implyC4andC5by3.32andϕ, ψ ∈ Φ2, respectively. As in
the proof ofTheorem 3.3, byC8we conclude that the mappingHmapsBBSintoBBS
and satisfies
dk
Hg, Hh≤dk
g, h, ∀g, h∈BBS, k∈N, 3.35
dHg, Hh
∞
k1
1 2k ·
dk
Hg, Hh
1dk
Hg, Hh ≤
∞
k1
1 2k ·
dk
g, h
1dk
g, h
dg, h, ∀g, h∈BBS.
3.36
Let the sequence{wn}n≥0be generated byAlgorithm 4andw0∈BBSwith|w0x| ≤
ψxfor allx, k∈B0, k×N. We now claim that for eachn≥0
|wnx| ≤ψx, ∀x, k∈B0, k×N. 3.37
Clearly 3.37holds for n 0. Assume that3.37is true for somen ≥ 0. It follows from
C6–C8,3.32,Algorithm 4, and Lemmas2.1and2.4that
|wn1x|
opt
y∈D
optpx, y, qx, ywn
ax, y, rx, ywn
bx, y, sx, ywn
cx, y
≤sup
y∈D
max px, y , qx, y wn
ax, y ,
r
x, y wn
bx, y , sx, y wn
cx, y
≤sup
y∈D
max px, y ,max qx, y , rx, y , sx, y
×max wn
ax, y , wn
bx, y , wn
cx, y
≤sup
y∈D
maxψx,maxψax, y, ψbx, y, ψcx, y
≤maxψx, ψϕx
ψx.
3.38
That is,3.37is true forn1. Hence3.37holds for eachn≥0.
Next we claim that{wn}n≥0 is a Cauchy sequence inBBS, d. Letk, n, m∈N,x0 ∈
B0, k, andε >0. Suppose that opty∈Dinfy∈D. Choosey, z∈Dwith
wnx0>opt
px0, y
, qx0, y
wn−1
ax0, y
,
rx0, y
wn−1
bx0, y
, sx0, y
wn−1
cx0, y
−2−1ε,
wnmx0>opt
px0, z, qx0, zwnm−1ax0, z,
rx0, zwnm−1bx0, z, sx0, zwnm−1cx0, z} −2−1ε,
wnx0≤opt
px0, z, qx0, zwn−1ax0, z,
rx0, zwn−1bx0, z, sx0, zwn−1cx0, z},
wnmx0≤opt
px0, y
, qx0, y
wnm−1
ax0, y
,
rx0, y
wnm−1
bx0, y
, sx0, y
wnm−1
cx0, y
.
It follows from3.39,C8, and Lemmas2.2and2.3that
|wnmx0−wnx0|
<max optpx0, y
, qx0, y
wnm−1
ax0, y
,
rx0, y
wnm−1
bx0, y
, sx0, y
wnm−1
cx0, y
−optpx0, y
, qx0, y
wn−1
ax0, y
,
rx0, y
wn−1
bx0, y
, sx0, y
wn−1
cx0, y ,
optpx0, z, qx0, zwnm−1ax0, z,
rx0, zwnm−1bx0, z, sx0, zwnm−1cx0, z}
−optpx0, z, qx0, zwn−1ax0, z,
rx0, zwn−1bx0, z, sx0, zwn−1cx0, z} 2−1ε
≤maxmax qx0, y wnm−1
ax0, y
−wn−1
ax0, y ,
rx0, y wnm−1
bx0, y
−wn−1
bx0, y ,
sx0, y wnm−1
cx0, y
−wn−1
cx0, y ,
max qx0, z |wnm−1ax0, z−wn−1ax0, z|,
|rx0, z||wnm−1bx0, z−wn−1bx0, z|,
|sx0, z||wnm−1cx0, z−wn−1cx0, z|}}2−1ε
≤maxmax qx0, y , r
x0, y , s
x0, y
×max wnm−1
ax0, y
−wn−1
ax0, y , wnm−1
bx0, y
−wn−1
bx0, y ,
wnm−1 cx0, y
−wn−1
cx0, y ,max qx0, z ,|rx0, z|,|sx0, z|
×max{|wnm−1ax0, z−wn−1ax0, z|,|wnm−1bx0, z−wn−1bx0, z|,
|wnm−1cx0, z−wn−1cx0, z|}}2−1ε
≤max wnm−1
ax0, y
−wn−1
ax0, y ,
wnm−1
bx0, y
−wn−1
bx0, y , wnm−1
cx0, y
−wn−1
cx0, y ,
|wnm−1ax0, z−wn−1ax0, z|,|wnm−1bx0, z−wn−1bx0, z|,
|wnm−1cx0, z−wn−1cx0, z|}2−1ε.
Therefore there existy1∈ {y, z} ⊂Dandx1∈ {ax0, y1, bx0, y1, cx0, y1}satisfying
|wnmx0−wnx0|<|wnm−1x1−wn−1x1|2−1ε. 3.41
In a similar method, we can derive that3.41holds also for opty∈Dsupy∈D. Proceeding in this way, we chooseyi∈Dandxi∈ {axi−1, yi, bxi−1, yi, cxi−1, yi}fori∈ {2,3, . . . , n}such
that
|wnm−1x1−wn−1x1|<|wnm−2x2−wn−2x2|2−2ε,
|wnm−2x2−wn−2x2|<|wnm−3x3−wn−3x3|2−3ε,
.. .
|wm1xn−1−w1xn−1|<|wmxn−w0xn|2−nε.
3.42
On account ofϕ, ψ∈Φ2,C7,3.37,3.41, and3.42, we gain that
|wnmx0−wnx0|<|wmxn−w0xn| n
i1
2−iε,
<|wmxn||w0xn|ε
≤2ψxn ε
≤2ψϕnx0
ε,
3.43
which yields that
dkwnm, wn≤2ψ
ϕnkε. 3.44
Lettingε → 0in the above inequality, we infer that
dkwnm, wn≤2ψ
ϕnk. 3.45
It follows fromϕ, ψ∈Φ2and3.45that{wn}n≥0is a Cauchy sequence inBBS, dand it
converges to somew∈BBS.Algorithm 4and3.36lead to
dHw, w≤dHw, Hwn dwn1, w
≤dw, wn dwn1, w−→0 asn−→ ∞,
which yields thatHw w. That is, the functional equation1.3possesses a solutionw ∈ BBS.
Now we showC10. Letx ∈ S. Putk 1 x. It follows from3.37,C7, and
ϕ, ψ∈Φ2that
|wx| ≤ |wx−wnx||wnx|
≤dkw, wn ψx−→ψx asn−→ ∞,
3.47
that is,C10holds.
Next we proveC11. Givenx0 ∈ S,{yn}n∈N ⊂ D, andxn ∈ {axn−1, yn,bxn−1, yn,
cxn−1, yn}forn∈N. Putk x0 1. Note thatC7implies that
xn ≤maxa
xn−1, yn,b
xn−1, yn,c
xn−1, yn
≤ϕxn−1≤ · · · ≤ϕnx0≤ϕnk, ∀n∈N.
3.48
In view of3.32,3.37,3.48, andϕ, ψ∈Φ2, we know that
|wxn| ≤ |wxn−wnxn||wnxn|
≤dkw, wn ψxn
≤dkw, wn ψ
ϕnk
−→0 asn−→ ∞,
3.49
which means that limn→ ∞wnxn 0.
Finally we proveC12. Assume that the functional equation1.3has another solution
h ∈ BBSthat satisfiesC11. Letε > 0 andx0 ∈ S. Suppose that opty∈D infy∈D. Select
y, z∈Dwith
wx0>opt
px0, y
, qx0, y
wax0, y
, rx0, y
wbx0, y
, sx0, y
wcx0, y
−2−1ε,
hx0>opt
px0, z, qx0, zhax0, z, rx0, zhbx0, z, sx0, zhcx0, z
−2−1ε,
wx0≤opt
px0, z, qx0, zwax0, z, qx0, zwbx0, z, rx0, zwcx0, z
,
hx0≤opt
px0, y
, qx0, y
hax0, y
, rx0, y
hbx0, y
, sx0, y
hcx0, y
.
On account of 3.50,C8, and Lemma 2.1, we conclude that there exist y1 ∈ {y, z} and
x1∈ {ax0, y1,bx0, y1, cx0, y1}satisfying
|wx0−hx0|
<max optpx0, y
, qx0, y
wax0, y
, rx0, y
wbx0, y
, sx0, y
wcx0, y
−optpx0, y
, qx0, y
hax0, y
, rx0, y
hbx0, y
, sx0, y
hcx0, y ,
optpx0, z, qx0, zwax0, z, rx0, zwbx0, z, sx0, zwcx0, z
−optpx0, z, qx0, zhax0, z, rx0, zhbx0, z, sx0, zhcx0, z 2−1ε ≤maxmax qx0, y w
ax0, y
−hax0, y , r
x0, y w
bx0, y
−hbx0, y ,
sx0, y w
cx0, y
−hcx0, y ,
max qx0, z |wax0, z−hax0, z|,|rx0, z||wbx0, z−hbx0, z|, |sx0, z||wcx0, z−hcx0, z|}}2−1ε
≤maxmax qx0, y , r
x0, y , s
x0, y
max wax0, y
−hax0, y ,
wbx0, y
−hbx0, y , w
cx0, y
−wcx0, y ,
max qx0, z ,|rx0, z|,|sx0, z|
max{|wax0, z−hax0, z|,
|wbx0, z−hbx0, z|,|wcx0, z−hcx0, z|}}2−1ε ≤max wax0, y
−hax0, y , w
bx0, y
−hbx0, y ,
wcx0, y
−hcx0, y ,|wax0, z−hax0, z|,
|wbx0, z−hbx0, z|,|wcx0, z−hcx0, z|}2−1ε
|wx1−hx1|2−1ε,
3.51
that is,
|wx0−hx0| ≤ |wx1−hx1|2−1ε. 3.52
Similarly we can prove that 3.52holds for opty∈D supy∈D. Proceeding in this way, we selectyi ∈D andxi ∈ {axi−1, yi, bxi−1, yi, cxi−1, yi}fori ∈ {2,3, . . . , n}andn ∈ Nsuch
that
|wx1−hx1|<|wx2−hx2|2−2ε,
|wx2−hx2|<|wx3−hx3|2−3ε,
.. .
|wxn−1−hxn−1|<|wxn−hxn|2−nε.
It follows from3.52and3.53that
|wx0−hx0|<|wxn−hxn|ε−→ε asn−→ ∞. 3.54
Sinceεis arbitrary, we conclude immediately thatwx0 hx0. This completes the proof.
Theorem 3.5. Letϕ, ψ∈Φ2,p, q, r, s:S×D → Randa, b, c:S×D → Ssatisfy conditions (C6)–(C8). Then the functional equation1.4possesses a solutionw ∈BBSsatisfying conditions (C10)–(C12) and the following two conditions:
C13the sequence{wn}n≥0generated byAlgorithm 5converges tow, wherew0 ∈BBSwith |w0x| ≤ψxfor allx, k∈B0, k×N;
C14ifq, r, andsare nonnegative and there exists a constantβ∈0,1such that
maxqx, y, rx, y, sx, y≡β, ∀x, y∈S×D, 3.55
thenwis nonnegative.
Proof. It follows fromTheorem 3.4that the functional equation1.4has a solutionw∈BBS
that satisfiesC10–C13. Now we showC14. Givenε >0,x0∈Sandn∈N. It follows from Lemma 2.2,3.55, and1.4that there existy1 ∈ D andx1 ∈ {ax0, y1, bx0, y1, cx0, y1}
such that
wx0>max
px0, y1
, qx0, y1
wax0, y1
, rx0, y1
wbx0, y1
,
sx0, y1
wcx0, y1
−2−1ε
≥maxpx0, y1
,maxqx0, y1
, rx0, y1
, sx0, y1
×minwax0, y1
, wbx0, y1
, wcx0, y1
−2−1ε
≥maxpx0, y1
, βwx1
−2−1ε
≥βwx1−2−1ε.
3.56
That is,
wx0> βwx1−2−1ε. 3.57
Proceeding in this way, we chooseyi ∈ Dandxi ∈ {axi−1, yi, bxi−1, yi, cxi−1, yi}fori∈
{2,3, . . . , n}andn∈Nsuch that
wx1> βwx2−2−2β−1ε,
wx2> βwx3−2−3β−2ε,
.. .
wxn−1> βwxn−2−nβ−n1ε.
It follows from3.57and3.58that
wx0> βnwxn− n
i1
2−iε≥βnwxn−ε, ∀n∈N. 3.59
In terms ofC8,C11, and3.55, we see that|βnwx
n| → 0 asn → ∞. Lettingn → ∞in 3.59, we get thatwx0≥ −ε. Sinceε >0 is arbitrary, we infer immediately thatwx0≥0.
This completes the proof.
Theorem 3.6. Letϕ, ψ∈Φ3,p, q, r, s:S×D → Randa, b, c:S×D → Ssatisfy conditions (C6), (C7), and the following condition:
C15q, r, andsare nonnegative andsupx,y∈S×Dmax{qx, y, rx, y, sx, y} ≤1.
Then the functional equation 1.6 possesses a solution w ∈ BBS satisfying
limn→ ∞wnx wxfor any x ∈ S, where the sequence{wn}n≥0 is generated byAlgorithm 7 withw0∈BBS, w0x≤supy∈Dpx, y, and|w0x| ≤supy∈D|px, y| for allx∈S.
Proof. We are going to prove that, for anyn∈N,
w0x≤w1x≤ · · · ≤wnx, ∀x∈S. 3.60
Usingϕ, ψ∈Φ3andAlgorithm 7, we gain that
w0x≤sup y∈D
px, y
≤sup
y∈D
maxpx, y, qx, yw0
ax, y, rx, yw0
bx, y, sx, yw0
cx, y
w1x, ∀x∈S,
3.61
that is,3.60holds forn1. Assume that3.60holds for somen∈N.Lemma 2.1andC15 lead to
maxpx, y, qx, ywn−1
ax, y, rx, ywn−1
bx, y, sx, ywn−1
cx, y
≤maxpx, y, qx, ywn
ax, y, rx, ywn
bx, y, sx, ywn
cx, y,
∀x, y∈S×D,
which implies that
wnx sup
y∈D
maxpx, y, qx, ywn−1
ax, y, rx, ywn−1
bx, y, sx, ywn−1
cx, y
≤sup
y∈D
maxpx, y, qx, ywn
ax, y, rx, ywn
bx, y, sx, ywn
cx, y
wn1x, ∀x∈S,
3.63
and hence3.60holds forn1. That is,3.60holds for anyn∈N. Now we claim that, for anyn≥0,
|wnx| ≤max
ψϕix: 0≤i≤n, ∀x∈S. 3.64
In fact,C6ensures that
|w0x| ≤sup y∈D
px, y ≤ψx, ∀x∈S, 3.65
that is,3.64is true forn0. Assume that3.64is true for somen≥0. In view of Lemmas
2.1and2.4,Algorithm 7,C6,C7, and C15, we gain that
|wn1x| ≤sup y∈D
max px, y , qx, y wn
ax, y ,
rx, y wn
bx, y , sx, y wn
cx, y
≤sup
y∈D
max px, y ,maxqx, y, rx, y, sx, y
×max wn
ax, y , wn
bx, y , wn
cx, y
≤sup
y∈D
maxψx,maxψϕiax, y: 0≤i≤n,
maxψϕibx, y: 0≤i≤n,
maxψϕicx, y: 0≤i≤n
≤maxψx,maxψϕi1x: 0≤i≤n
≤maxψϕix: 0≤i≤n1, ∀x∈S,
which yields that3.64is true forn1. Therefore3.64holds for eachn≥0. Givenk∈N, note that limn→ ∞ψϕnk exists. It follows that there exist constants M > 0 and n0 ∈ N
satisfyingψϕnk< Mfor anyn≥n
0. Thus3.64leads to
|wnx| ≤max
M,maxψϕik: 0≤i≤n0−1
, ∀n≥0, k, x∈N×B0, k. 3.67
On account of3.60,3.67, andAlgorithm 7, we deduce that{wnx}n≥0 is convergent for
eachx∈Sand{wn}n≥0∈BBS. Put
lim
n→ ∞wnx wx, ∀x∈S,
Ax sup
y∈D
maxpx, y, qx, ywax, y, rx, ywbx, y,
sx, ywcx, y, ∀x∈S.
3.68
Obviously3.67ensures thatw∈BBS. Notice that
maxpx, y, qx, ywn−1
ax, y, rx, ywn−1
bx, y,
sx, ywn−1
cx, y≤wnx, ∀
x, y, n∈S×D×N.
3.69
Letting n → ∞ in the above inequality, by Lemmas 2.1 and 2.3 and the convergence of
{wnx}n≥0we infer that
maxpx, y, qx, ywax, y, rx, ywbx, y,
sx, ywcx, y≤wx, ∀x, y∈S×D,
3.70
which yields that
Ax sup
y∈D
maxpx, y, qx, ywax, y, rx, ywbx, y, sx, ywcx, y
≤wx, ∀x∈S.
3.71
It follows from3.60,C15, andLemma 2.1that
maxpx, y, qx, ywn−1
ax, y, rx, ywn−1
bx, y, sx, ywn−1
cx, y
≤maxpx, y, qx, ywax, y, rx, ywbx, y,
sx, ywcx, y, ∀x, y, n∈S×D×N,
which implies that
wnx sup
y∈D
maxpx, y, qx, ywn−1
ax, y, rx, ywn−1
bx, y,
sx, ywn−1
cx, y
≤sup
y∈D
maxpx, y, qx, ywax, y, rx, ywbx, y, sx, ywcx, y
Ax, ∀x, n∈S×N.
3.73
Lettingn → ∞, we gain that
wx≤Ax, ∀x∈S. 3.74
It follows from 3.71and 3.74thatw is a solution of the functional equation1.6. This completes the proof.
Following similar arguments as in the proof of Theorems3.5 and 3.6, we have the following results.
Theorem 3.7. Letϕ, ψ∈Φ2,p, q, r, s:S×D → Randa, b, c:S×D → Ssatisfy conditions (C6)–(C8). Then the functional equation1.5possesses a solutionw ∈BBSsatisfying conditions (C10)–(C12) and the two following conditions:
C16the sequence{wn}n≥0generated byAlgorithm 6converges tow, wherew0 ∈BBSwith |w0x| ≤ψxfor allx, k∈B0, k×N;
C17ifq, r, andsare nonnegative and there exists a constantβ∈0,1such that
minqx, y, rx, y, sx, y≡β, ∀x, y∈S×D, 3.75
thenwis nonpositive.
Theorem 3.8. Letϕ, ψ∈Φ3,p, q, r, s:S×D → Randa, b, c:S×D → Ssatisfy conditions (C6), (C7), and (C15). Then the functional equation1.7possesses a solutionw∈BBSsatisfying
limn→ ∞wnx wxfor anyx∈S, where the sequence{wn}n≥0is generated byAlgorithm 8with
w0∈BBS,w0x≥infy∈Dpx, yand|w0x| ≤supy∈D|px, y| for allx∈S.
4. Applications
Example 4.1. LetXY R,S 1,2,DR, andα2/3. It follows fromTheorem 3.1that the functional equation
fx opt
y∈D
opt
x10y2
xy2,
sinxy2−cosx2y
3 f
xysin2y−y2
1y2
,
xy2
x123y2f
2xy2 xy2
, xy
2
2x3y2f
2x2yln1y
x2yln1y
, ∀x∈S,
4.1
possesses a unique solutionf ∈ BCSand the sequence{fn}n≥0 generated byAlgorithm 1
converges tofand satisfies3.2.
Example 4.2. LetXY R,SR,DR−, andα2/3. It is clear thatTheorem 3.2ensures that the functional equation
fx opt
y∈D
opt
sin2xycosx2−y, 2x
13xln21x−yf
−xy,
2x2y2sinx2y
13x2y2 f
x2y2,cos
xy
3x−y f
x−y2 , ∀x∈S
4.2
possesses a unique solutionf ∈ BSand the sequence{fn}n≥0 generated byAlgorithm 2
converges tofand satisfies3.2.
Remark 4.3. If qx, y rx, y sx, y, ax, y bx, y cx, y for all x, y ∈ S×D, then Theorem 3.3reduces to a result which generalizes the result in 3, page 149 and Theorem 3.4 in7. The following example demonstrates that Theorem 3.3generalizes properly the corresponding results in3,7.
Example 4.4. LetX Y R,S D R, andα 5/6. It is easy to verify thatTheorem 3.3
guarantees that the functional equation
fx opt
y∈D
opt
x4
1 x−y ,
2 sinx2−y3 cosx−y2
6ln1 x−y f
x3y2
1x2y2
,
2x2y
112x22yf
x3ysin2x22y−1 1x2y
,
4 sinx−y−cosx2−y2
6 cosx2y2 f
x2y sinx2y4
1xy
, ∀x∈S,
4.3
has a unique solution inBBS. However, the results in3, page 149and Theorem 3.4 in7
Remark 4.5. 1Ifax, y bx, y cx, y,qx, y rx, y sx, yfor allx, y ∈ S× D, then Theorems3.4,3.5, and3.7 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 in17, and Theorems 2.3 and 2.4 in18, respectively.
2 The results in 3, page 149, Theorem 3.5 in 7, Theorem 3.5 in 12, and Theorem 3.4 in15are special cases ofTheorem 3.5withqx, y 1,rx, y sx, y 0 for allx, y∈S×D.
The examples below show that Theorems3.4,3.5, and3.7are indeed generalizations of the corresponding results in3,7,12,14,15,17,18.
Example 4.6. LetX Y R,SD R. Define two functionsψ, ϕ:R → Rbyψt t2,
ϕt t/2 for allt ∈ R. It is easy to see thatTheorem 3.4guarantees that the functional equation
fx opt
y∈D
opt
x2
1 x−y ,cos
3x2y2f
x2y3
12xy3
,
sinx−y2f
x2ycos2x−yln1 x−y
1x2y2
,
1x2y3
2x2y3f
x2y2
12xy2cos2x2y2
, ∀x∈S,
4.4
possesses a solutionw∈BBSthat satisfiesC9–C12. However, the corresponding results in3,7,12,14,17,18are not applicable for the functional equation4.4.
Example 4.7. LetX Y R SD. Putβ 1,ψt t2, andϕt t/3 for allt∈R. It is
easy to verify thatTheorem 3.5guarantees that the functional equation
fx opt
y∈D
max
⎧ ⎪ ⎨ ⎪ ⎩
x3y
1 xy , f ⎛ ⎜
⎝ xsin
2x−y
3ln1
x2y2
⎞ ⎟ ⎠,
x2x−y2
1x2x−y2f
x3y2
13x2y2 x2−y2
,
sin2x2−y1
1x2y2 f
x2y4sinx2y2
13|x|y4cos2x−y
⎫⎪⎬
⎪
⎭, ∀x∈S,
4.5
has a solutionw ∈BBSsatisfyingC10–C14. But the corresponding results in3,7,12,
