Hindawi Publishing Corporation
Fixed Point Theory and Applications
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 Kwun
3
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, [email protected]
Received 5 January 2011; Accepted 11 February 2011
Academic Editor: Yeol J. Cho
Copyright q 2011 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 processes fxopt
y∈D
opt{px, y,qx, yfax, y,rx, yfbx, y,
sx, yfcx, y} for all x ∈ S. A few iterative algorithms for solving the functional equation are
, ∀x ∈ S,
f
x
sup
y∈D
max
p
x, y
,f
a
x, y
, ∀x ∈ S,
f
x
inf
y∈D
max
p
x, y
,f
a
x, y
, ∀x ∈ S,
f
x
opt
y∈D
min
p
x, y
,f
a
x, y
, ∀x ∈ S,
f
x, y
,q
x, y
f
a
x, y
, ∀x ∈ S
1.2
in BBS.
The purpose of this paper is to introduce and study the following functional equations
arising in dynamic programming of multistage decision processes:
f
x
opt
y∈D
opt
p
x, y
, ∀x ∈ S,
1.3
f
x
opt
y∈D
max
p
x, y
,q
x, y
f
a
x, y
,r
x, y
f
x, y
f
a
x, y
,r
x, y
f
b
x, y
,s
x, y
f
c
x, y
, ∀x ∈ S,
1.5
x, y
,s
x, y
f
c
x, y
, ∀x ∈ S,
1.6
f
x
inf
y∈D
min
p
x, y
,q
x, y
x
sup
y∈D
min
p
x, y
,q
x, y
f
a
x, y
,r
x, y
f
b
x, y
a
x, y
,r
x, y
f
b
x, y
,s
x, y
f
c
x, y
, ∀x ∈ S,
1.9
where opt denotes sup or inf, x and y stand for the state and decision vectors, respectively,
a, b,andc represent the transformations of the processes, and fx represents the optimal
return function with initial x.
Fixed Point Theory and Applications 3
−→ R
is nondecreasing
,
Φ
2
ϕ, ψ
: ϕ, ψ ∈ Φ
1
,ψ
t
> 0, lim
n →∞
ψ
ϕ
n
t
0fort>0
f : f : S −→ R is bounded
,
BC
S
f : f ∈ B
S
is continuous
,
BB
S
f : f : S −→ R is bounded on bounded subsets of S
.
2.1
Clearly BS, ·
1
and BCS, ·
1
,
d
f, g
∞
k1
1
2
k
·
d
k
f, g
1 d
k
f, g
,
2.2
where
B0,k{x : x ∈ S and x≤k}.Itiseasytoseethat{d
k
}
: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
i
:1≤ i ≤ n 1
}
− opt
{
b
i
:1≤ i ≤ n 1
}
opt
opt
{
a
i
:1≤ i ≤ n
}
,a
n1
− opt
opt
{
b
− b
n1
|
≤ max
{|
a
i
− b
i
|
:1≤ i ≤ n 1
}
.
2.3
Hence e 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
}
,a
n1
b
n1
}
≥ max
{
min
{
a
i
:1≤ i ≤ n
}
max
{
b
i
:1≤ i ≤ n
}
,a
n1
b
n1
}
≥ min
{
a
i
:1≤ i ≤ n 1
in
:1≤ i ≤ k
}
opt
lim
n →∞
a
in
:1≤ i ≤ k
. 2.5
Proof. Put lim
n →∞
a
in
b
i
for 1 ≤ i ≤ k.InviewofLemma 2.1 we deduce that
opt
{
a
in
:1≤ i ≤ k
}
− opt
{
b
in
:1≤ i ≤ k
. 2.7
This completes the proof.
Lemma 2.4. a Assume that A : S × D → R is a mapping such that opt
y∈D
Ax
0
,y is bounded
for some x
0
∈ S.Then
opt
y∈D
A
x
0
,y
opt
y∈D
A
x
1
,y
− opt
y∈D
B
x
2
,y
≤ sup
y∈D
A
,y
≤ A
x
0
,y
≤
A
x
0
,y
, ∀y ∈ D. 2.10
6 Fixed Point Theory and Applications
It follows that
−sup
y∈D
A
y∈D
A
x
0
,y
≤ sup
y∈D
A
x
0
,y
≤ sup
y∈D
A
x
0
,y
,
2.11
which implies that
. 2.12
Next we show b.Ifsup
y∈D
|Ax
1
,y − Bx
2
,y| ∞, b is true. Suppose that
sup
y∈D
|Ax
1
,y − Bx
2
,y| < ∞.Notethat
A
x
1
,y
− B
x
2
,y
A
x
1
,y
− B
x
2
,y
≤ A
x
1
,y
≤ B
x
2
,y
sup
y∈D
A
x
1
,y
− B
x
2
,y
opt
y∈D
B
x
2
,y
− sup
y∈D
A
x
y∈D
A
x
1
,y
− B
x
2
,y
opt
y∈D
B
x
2
,y
sup
y∈D
− opt
y∈D
B
x
2
,y
≤ sup
y∈D
A
x
1
,y
− B
x
2
,y
opt
p
x, y
,q
x, y
f
n
a
x, y
,
r
x, y
f
n
b
x, y
,s
0
∈ BS, compute {f
n
}
n≥0
by 2.17 and 2.18.
Algorithm 3. For any f
0
∈ BBS, compute {f
n
}
n≥0
by 2.17 and 2.18.
Algorithm 4. For any w
0
∈ BBS, compute {w
n
}
n≥0
by
w
n1
x
opt
y∈D
opt
p
w
n
c
x, y
, ∀x ∈ S, n ≥ 0.
2.19
Algorithm 5. For any w
0
∈ BBS, compute {w
n
}
n≥0
by
w
n1
x
opt
y∈D
max
p
x, y
,q
x, y
, ∀x ∈ S, n ≥ 0.
2.20
Algorithm 6. For any w
0
∈ BBS, compute {w
n
}
n≥0
by
w
n1
x
opt
y∈D
min
p
x, y
,q
x, y
w
2.21
Algorithm 7. For any w
0
∈ BBS, compute {w
n
}
n≥0
by
w
n1
x
sup
y∈D
max
p
x, y
,q
x, y
w
n
a
∈ BBS, compute {w
n
}
n≥0
by
w
n1
x
inf
y∈D
min
p
x, y
,q
x, y
w
n
a
x, y
,r
by
w
n1
x
sup
y∈D
min
p
x, y
,q
x, y
w
n
a
x, y
,r
x, y
w
x
inf
y∈D
max
p
x, y
,q
x, y
w
n
a
x, y
,r
x, y
w
n
b
x → x
0
p
x, y
p
x
0
,y
, lim
x → x
0
q
x, y
q
x
0
,y
,
lim
x → x
0
r
a
x
0
,y
, lim
x → x
0
b
x, y
b
x
0
,y
,
lim
x → x
0
c
x, y
c
x
, ∀n ≥ 0. 3.2
Fixed Point Theory and Applications 9
Proof. Define a mapping H : BCS → BCS by
Hh
x
opt
y∈D
opt
p
x, y
,q
x, y
h
a
x, y
,r
x, y
there exist constants M>0, δ>0, and δ
1
> 0 satisfying
sup
x,y∈S×D
p
x, y
≤ M,
3.4
sup
x,y∈S×D
max
|
h
x
|
,
h
p
x, y
− p
x
0
,y
<
ε
3
, ∀
x, y
∈ S × D with
x − x
0
<δ, 3.6
max
s
x, y
− s
x
0
,y
<
ε
6M
,
∀
x, y
∈ S × D with
x − x
0
<δ,
3.7
|
a
x, y
− a
x
0
,y
,
b
x, y
− b
x
0
,y
,
|
Hh
x
|
≤ sup
y∈D
opt
p
x, y
,q
x, y
h
a
x, y
,r
x, y
,
q
x, y
h
a
x, y
,
r
x, y
y∈D
max
p
x, y
, max
q
x, y
,
r
x, y
x, y
,
h
c
x, y
≤ max
{
M, αM
}
M, ∀x ∈ S.
3.10
10 Fixed Point Theory and Applications
In light of C2, 3.3, 3.5–3.9, and Lemmas 2.1 and 2.4, we deduce that for all x, y ∈ S×D
with x − x
0
<δ
|
Hh
x, y
,r
x, y
h
b
x, y
,s
x, y
h
c
x, y
−opt
y∈D
opt
p
,y
,s
x
0
,y
h
c
x
0
,y
≤ sup
y∈D
max
p
a
x
0
,y
,
r
x, y
h
b
x, y
− r
x
0
,y
h
x
0
,y
≤ sup
y∈D
max
p
x, y
− p
x
0
,y
,
h
a
x, y
− h
a
x
0
,y
,
r
x, y
− r
b
x, y
− h
b
x
0
,y
,
s
x, y
− s
x
0
,y
c
x
0
,y
≤ sup
y∈D
max
p
x, y
− p
x
0
,y
,
,
s
x, y
− s
x
0
,y
× max
h
a
x, y
,
,y
,
r
x
0
,y
,
s
x
0
,y
× max
x
0
,y
,
h
c
x, y
− h
c
x
0
,y
≤ max
ε
3
g
a
x, u
,r
x, u
g
b
x, u
,s
x, u
g
c
x, u
− ε,
Hh
x, v
h
c
x, v
− ε,
Hg
x
≤ opt
p
x, v
,q
x, v
g
a
x, v
x, u
,q
x, u
h
a
x, u
,r
x, u
h
b
x, u
,s
x, u
h
c
,q
x, u
g
a
x, u
,r
x, u
g
b
x, u
,s
x, u
g
c
x, u
h
c
x, u
,
opt
p
x, v
,q
x, v
g
a
x, v
,r
h
a
x, v
,r
x, v
h
b
x, v
,s
x, v
h
c
x, v
|
r
x, u
|
g
b
x, u
− h
b
x, u
,
|
s
x, u
|
a
x, v
− h
a
x, v
,
|
r
x, v
|
g
b
x, v
− h
ε
≤ max
q
x, u
,
|
r
x, u
|
,
|
s
x, u
|
,
q
1
ε,
3.13
which implies that
Hg − Hh
1
≤ α
g − h
1
ε, ∀g,h ∈ BC
S
. 3.14
Letting ε → 0
in the above inequality, we know that
Hg − Hh
n
f
x
α
n
opt
y∈D
opt
p
x, y
,q
x, y
f
a
x, y
,
r
x
≤
1 − α
n
f
n
x
− f
x
α
n
Hf
n
x
−1−α
n
i0
α
i
f
0
− f
1
, ∀x ∈ S, n ≥ 0,
3.17
12 Fixed Point Theory and Applications
which yields that
f
n1
− f
1
≤ e
−1−α
n
B0,k × D for each k ∈ N;
C5 sup
x,y∈B0,k×D
{ax, y, bx, y, cx, y} ≤ k for all k ∈ N.
Then the functional equation 1.3 possesses a unique solution w ∈ BBS, and the sequences
{f
n
}
n≥0
and {w
n
}
n≥0
generated by Algorithms 3 and 4, respectively, converge to f and have the error
estimates
d
k
f
n1
,w
≤ e
−1−α
n
i0
α
i
d
sup
x,y∈B0,k×D
p
x, y
≤ M
k
,
sup
x,y∈B0,k×D
h
a
x, y
,
x
|
≤ sup
y∈D
max
p
x, y
,
q
x, y
h
x, y
h
c
x, y
≤ sup
y∈D
max
p
x, y
, max
h
a
x, y
,
h
b
x, y
,
h
c
x, y
k
g,h
, ∀g,h ∈ BB
S
,k∈ N. 3.22
Let k ∈ N, x ∈
B0,k, g,h ∈ BBS,andε>0. Suppose that opt
y∈D
inf
y∈D
. Select u, v ∈ D
such that 3.12 holds. Thus 3.3, 3.12, C2, C5,andLemma 2.1 ensure that
Hg
x
− Hh
x
< max
x, u
g
c
x, u
−opt
p
x, u
,q
x, u
h
a
x, u
,r
x, u
h
g
a
x, v
,r
x, v
g
b
x, v
,s
x, v
g
c
x, v
−opt
c
x, v
ε
≤ max
max
q
x, u
g
a
x, u
− h
,
|
s
x, u
|
g
c
x, u
− h
c
x, u
,
max
q
g
b
x, v
− h
b
x, v
,
|
s
x, v
|
g
c
x, v
− h
x, u
|
,
q
x, v
,
|
r
x, v
|
,
|
s
x, v
|
d
k
.Asε → 0
in 3.24,wegetthat
3.22 holds.
Let w
0
∈ BBS. It follows from Algorithm 4 that
w
n1
x
Hw
n
x
, ∀n ≥ 0,x∈ S, 3.25
14 Fixed Point Theory and Applications
and 3.22 leads to
d
k
w
n1
,w
n1m
≤
nm
i−1
,w
i
≤
nm
in1
α
i
d
k
w
0
,w
1
≤
α
n1
1 − α
d
k
w
0
,w
1
k
Hg,Hh
≤
∞
k1
1
2
k
·
αd
k
g,h
1 αd
k
g,h
≤
∞
k1
1
2
k
·
n
,w
lim
n →∞
d
w
n1
,w
0, 3.28
that is, w Hw. Suppose that there exists u ∈ BBS \{w} with u Hu. Consequently there
exists some k
0
∈ N satisfying d
k
0
w, u > 0. It follows from 3.22 and C2 that
0 <d
k
0
w, u
d
k
0
Hw,Hu
w
0
,w
1
, ∀n ≥ 0,k∈ N. 3.30
It follows from Algorithm 3, 2.18,and3.22 that
d
k
f
n1
,w
sup
x∈B0,k
1 − α
n
f
n
x
− w
f
n
x
− w
x
α
n
sup
x∈B0,k
Hf
n
x
− Hw
x
≤
k
f
n
,w
≤ e
−1−α
n
i0
α
i
d
k
f
0
,w
, ∀n ≥ 0,k∈ N,
3.31
which gives that f
n
→ w as n →∞. This completes the proof.
Fixed Point Theory and Applications 15
Next we investigate the behaviors of solutions for the functional equations 1.3–1.5
and discuss the convergence of Algorithms 4–6 in BBS, respectively.
Theorem 3.4. Let ϕ, ψ ∈ Φ
2
∈ S, {y
n
}
n∈N
⊂ D and x
n
∈{ax
n−1
,y
n
, bx
n−1
,y
n
,
cx
n−1
,y
n
} for all n ∈ N;
C12 w is unique relative to condition (C11).
Proof. First of all we assert that
ϕ
t
<t, ∀t>0. 3.32
Suppose that there exists some t
0
> 0withϕt
ϕ
n
t
0
−→ 0asn −→ ∞ . 3.33
That is,
ψ
t
0
≤ 0 <ψ
t
0
, 3.34
which is impossible. That is, 3.32 holds. Let the mapping H be defined by 3.3 in BBS.
Note that C6 and C7 imply C4 and C5 by 3.32 and ϕ, ψ ∈ Φ
2
, respectively. As in
the proof of Theorem 3.3,byC8 we conclude that the mapping H maps BBS into BBS
and satisfies
d
k
Hg,Hh
Hg,Hh
≤
∞
k1
1
2
k
·
d
k
g,h
1 d
k
g,h
d
g,h
, ∀g,h ∈ BB
S
.
∈
B
0,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 Lemmas 2.1 and 2.4 that
|
w
n1
x
|
opt
y∈D
opt
p
x, y
x, y
≤ sup
y∈D
max
p
x, y
,
q
x, y
,
s
x, y
w
n
c
x, y
≤ sup
y∈D
max
p
x, y
× max
w
n
a
x, y
,
w
n
b
x, y
,
,ψ
b
x, y
,ψ
c
x, y
≤ max
ψ
inf
y∈D
. Choose y, z ∈ D with
w
n
x
0
> opt
p
x
0
,y
,q
x
0
,y
w
n−1
a
x
0
0
,y
− 2
−1
ε,
w
nm
x
0
> opt
p
x
0
,z
,q
x
0
,z
w
nm−1
a
c
x
0
,z
}
− 2
−1
ε,
w
n
x
0
≤ opt
p
x
0
,z
,q
x
0
,z
w
n−1
c
x
0
,z
}
,
w
nm
x
0
≤ opt
p
x
0
,y
,q
x
0
0
,y
w
nm−1
c
x
0
,y
.
3.39
Fixed Point Theory and Applications 17
It follows from 3.39, C8, and Lemmas 2.2 and 2.3 that
|
w
nm
x
0
− w
n
x
0
|
x
0
,y
w
nm−1
b
x
0
,y
,s
x
0
,y
w
nm−1
c
x
0
,y
− opt
b
x
0
,y
,s
x
0
,y
w
n−1
c
x
0
,y
,
opt
p
b
x
0
,z
,s
x
0
,z
w
nm−1
c
x
0
,z
}
− opt
p
x
0
,z
,s
x
0
,z
w
n−1
c
x
0
,z
}
2
−1
ε
≤ max
max
q
r
x
0
,y
w
nm−1
b
x
0
,y
− w
n−1
b
x
0
,y
0
,y
,
max
q
x
0
,z
|
w
nm−1
a
x
0
,z
− w
b
x
0
,z
|
,
|
s
x
0
,z
||
w
nm−1
c
x
0
,z
− w
n−1
c
,
s
x
0
,y
× max
w
nm−1
a
x
0
,y
− w
n−1
,
w
nm−1
c
x
0
,y
− w
n−1
c
x
0
,y
, max
nm−1
a
x
0
,z
− w
n−1
a
x
0
,z
|
,
|
w
nm−1
b
x
0
,z
− w
2
−1
ε
≤ max
w
nm−1
a
x
0
,y
− w
n−1
a
x
0
,y
,
,y
− w
n−1
c
x
0
,y
,
|
w
nm−1
a
x
0
,z
− w
n−1
a
x
c
x
0
,z
− w
n−1
c
x
0
,z
|}
2
−1
ε.
3.40
18 Fixed Point Theory and Applications
Therefore there exist y
1
∈{y, z}⊂D and x
1
∈{ax
0
,y
1
,bx
− w
n−1
x
1
|
2
−1
ε. 3.41
In a similar method, we can derive that 3.41 holds also for opt
y∈D
sup
y∈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
|
2
−2
ε,
|
w
nm−2
x
2
− w
n−2
x
2
|
<
|
w
nm−3
x
3
− w
n−3
n
− w
0
x
n
|
2
−n
ε.
3.42
On account of ϕ, ψ ∈ Φ
2
, C7, 3.37, 3.41,and3.42, we gain that
|
w
nm
x
0
− w
n
x
0
|
|
|
w
0
x
n
|
ε
≤ 2ψ
x
n
ε
≤ 2ψ
ϕ
n
x
0
≤ 2ψ
ϕ
n
k
. 3.45
It follows from ϕ, ψ ∈ Φ
2
and 3.45 that {w
n
}
n≥0
is a Cauchy sequence in BBS,d and it
converges to some w ∈ BBS. Algorithm 4 and 3.36 lead to
d
Hw,w
≤ d
Hw,Hw
n
d
w
|
w
x
− w
n
x
|
|
w
n
x
|
≤ d
k
w, w
n
ψ
x
n
} for n ∈ N.Putk x
0
1. Note that C7 implies that
x
n
≤ max
a
x
n−1
,y
n
,
b
x
n−1
,y
n
≤ ϕ
n
k
, ∀n ∈ N.
3.48
In view of 3.32, 3.37, 3.48,andϕ, ψ ∈ Φ
2
, we know that
|
w
x
n
|
≤
|
w
x
n
− w
n
x
n
ψ
ϕ
n
k
−→ 0asn −→ ∞ ,
3.49
which means that lim
n →∞
w
n
x
n
0.
Finally we prove C12. Assume that the functional equation 1.3 has another solution
h ∈ BBS that satisfies C11.Letε>0andx
0
∈ S. Suppose that opt
y∈D
inf
y∈D
. Select
y, z ∈ D with
w
w
b
x
0
,y
,s
x
0
,y
w
c
x
0
,y
− 2
−1
ε,
h
x
0
b
x
0
,z
,s
x
0
,z
h
c
x
0
,z
− 2
−1
ε,
w
x
0
b
x
0
,z
,r
x
0
,z
w
c
x
0
,z
,
h
x
0
≤ opt
p
0
,y
,s
x
0
,y
h
c
x
0
,y
.
3.50
20 Fixed Point Theory and Applications
On account of 3.50, C8,andLemma 2.1, we conclude that there exist y
1
∈{y, z} and
x
1
∈{ax
0
,y
1
, bx
,y
,q
x
0
,y
w
a
x
0
,y
,r
x
0
,y
w
b
x
0
,y
a
x
0
,y
,r
x
0
,y
h
b
x
0
,y
,s
x
0
,y
h
c
,r
x
0
,z
w
b
x
0
,z
,s
x
0
,z
w
c
x
0
,z
b
x
0
,z
,s
x
0
,z
h
c
x
0
,z
2
−1
ε
≤ max
max
,
r
x
0
,y
w
b
x
0
,y
− h
b
x
0
,y
,
max
q
x
0
,z
|
w
a
x
0
,z
− h
a
|
,
|
s
x
0
,z
||
w
c
x
0
,z
− h
c
x
0
,z
|}}
2
−1
ε
x
0
,y
max
w
a
x
0
,y
− h
a
x
0
,y
,
− w
c
x
0
,y
,
max
q
x
0
,z
,
|
r
x
,z
|
,
|
w
b
x
0
,z
− h
b
x
0
,z
|
,
|
w
c
x
0
x
0
,y
,
w
b
x
0
,y
− h
b
x
0
,y
,
a
x
0
,z
|
,
|
w
b
x
0
,z
− h
b
x
0
,z
|
,
|
w
|
2
−1
ε,
3.51
that is,
|
w
x
0
− h
x
0
|
≤
|
w
x
1
− h
x
1
x
1
− h
x
1
|
<
|
w
x
2
− h
x
2
|
2
−2
ε,
|
w
x
2
n−1
− h
x
n−1
|
<
|
w
x
n
− h
x
n
|
2
−n
ε.
3.53
Fixed Point Theory and Applications 21
It follows from 3.52 and 3.53 that
|
w
, p, q,r, s : S × D → R and a, b, c : S × D → S satisfy conditions
(C6)–(C8). Then the functional equation 1.4 possesses a solution w ∈ BBS satisfying conditions
(C10)–(C12) and the following two conditions:
C13 the sequence {w
n
}
n≥0
generated by Algorithm 5 converges to w,wherew
0
∈ BBS with
|w
0
x|≤ψx for all x, k ∈ B0,k × N;
C14 if q, r, and s are nonnegative and there exists a constant β ∈ 0, 1 such that
max
q
x, y
,r
x, y
,s
x, y
≡ β, ∀
> max
p
x
0
,y
1
,q
x
0
,y
1
w
a
x
0
,y
1
,r
x
0
−1
ε
≥ max
p
x
0
,y
1
, max
q
x
0
,y
1
,r
x
0
,y
1
,s
x
,y
1
− 2
−1
ε
≥ max
p
x
0
,y
1
,βw
x
1
− 2
−1
ε
≥ βw
x
1
− 2
i−1
,y
i
} for i ∈
{2, 3, ,n} and n ∈ N such that
w
x
1
>βw
x
2
− 2
−2
β
−1
ε,
w
x
2
>βw
x
3
>β
n
w
x
n
−
n
i1
2
−i
ε ≥ β
n
w
x
n
− ε, ∀n ∈ N. 3.59
In terms of C8, C11,and3.55,weseethat|β
n
wx
n
|→0asn →∞. Letting n →∞in
3.59,wegetthatwx
0
≥−ε. Since ε>0 is arbitrary, we infer immediately that wx
|px, y| for all x ∈ S.
Proof. We are going to prove that, for any n ∈ N,
w
0
x
≤ w
1
x
≤···≤w
n
x
, ∀x ∈ S. 3.60
Using ϕ, ψ ∈ Φ
3
and Algorithm 7, we gain that
w
0
x
≤ sup
y∈D
p
x, y
,s
x, y
w
0
c
x, y
w
1
x
, ∀x ∈ S,
3.61
that is, 3.60 holds for n 1. Assume that 3.60 holds for some n ∈ N. Lemma 2.1 and C15
lead to
max
p
x, y
,q
≤ max
p
x, y
,q
x, y
w
n
a
x, y
,r
x, y
w
n
b
x, y
,s
x, y
,q
x, y
w
n−1
a
x, y
,r
x, y
w
n−1
b
x, y
,s
x, y
w
n−1
b
x, y
,s
x, y
w
n
c
x, y
w
n1
x
, ∀x ∈ S,
3.63
and hence 3.60 holds for n 1. That is, 3.60 holds for any n ∈ N.
Now we claim that, for any n ≥ 0,
|
w
n
x
x, y
≤ ψ
x
, ∀x ∈ S, 3.65
that is, 3.64 is true for n 0. Assume that 3.64 is true for some n ≥ 0. In view of Lemmas
2.1 and 2.4, Algorithm 7, C6, C7,andC15, we gain that
|
w
n1
x
|
≤ sup
y∈D
max
p
x, y
x, y
,s
x, y
w
n
c
x, y
≤ sup
y∈D
max
p
x, y
w
n
b
x, y
,
w
n
c
x, y
≤ sup
y∈D
max
ψ
b
x, y
:0≤ i ≤ n
,
max
ψ
ϕ
i
c
x, y
i
x
:0≤ i ≤ n 1
, ∀x ∈ S,
3.66
24 Fixed Point Theory and Applications
which yields that 3.64 is true for n 1. Therefore 3.64 holds for each n ≥ 0. Given k ∈ N,
note that lim
n →∞
ψϕ
n
k exists. It follows that there exist constants M>0andn
0
∈ N
satisfying ψϕ
n
k <Mfor any n ≥ n
0
.Thus3.64 leads to
|
w
n
x
n≥0
is convergent for
each x ∈ S and {w
n
}
n≥0
∈ BBS.Put
lim
n →∞
w
n
x
w
x
, ∀x ∈ S,
A
x
sup
y∈D
max
p
x, y
x, y
, ∀x ∈ S.
3.68
Obviously 3.67 ensures that w ∈ BBS.Noticethat
max
p
x, y
,q
x, y
w
n−1
a
x, y
,r
x, y
w
n−1
b
n≥0
we infer that
max
p
x, y
,q
x, y
w
a
x, y
,r
x, y
w
b
x, y
,
s
x, y
,q
x, y
w
a
x, y
,r
x, y
w
b
x, y
,s
x, y
w
c
w
n−1
b
x, y
,s
x, y
w
n−1
c
x, y
≤ max
p
x, y
,q
x, y
w
3.72
Fixed Point Theory and Applications 25
which implies that
w
n
x
sup
y∈D
max
p
x, y
,q
x, y
w
n−1
a
x, y
,r
x, y
x, y
w
a
x, y
,r
x, y
w
b
x, y
,s
x, y
w
c
x, y
A
n≥0
generated by Algorithm 6 converges to w,wherew
0
∈ BBS with
|w
0
x|≤ψx for all x, k ∈ B0,k × N;
C17 if q, r, and s are nonnegative and there exists a constant β ∈ 0, 1 such that
min
q
x, y
,r
x, y
,s
x, y
≡ β, ∀
x, y
∈ S × D, 3.75
then w is nonpositive.
Theorem 3.8. Let ϕ, ψ ∈ Φ
3
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