# Existence Theorems for Solvability of a Functional Equation Arising in Dynamic Programming

Authors: Deepmala
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

---

## Abstract

The paper deals with the existence, uniqueness, and iterative approximations of solutions for the functional equations arising in dynamic programming of multistage decision making processes in Banach spacesBC(S) and B(S) and complete metric space BB(S), respectively. Our main results extend, improve, and generalize the results due to several authors. Some examples are also given to demonstrate the advantage of our results over existing one in the literature.


---

## Body

## 1. Introduction

In this paper, we introduce and study the existence and uniqueness of solutions for the following functional equation arising in dynamic programming of multistage decision processes:(1)
f
(
x
)
=
opt
y
∈
D
⁡
{
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
aaaaaaaaaaaaaaaaaa
aa
×
f
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
}
,
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaai
i
∀
x
∈
S
,
where “opt” denotes the “sup” or “inf,” x and y stand for the state and decision vectors, respectively, a
i represents the transformation of the processes, and f
(
x
) denotes the optimal return function with initial state x.It is clear that (1) includes many functional equations and system of functional equations as special case, respectively. Bellman [1] was the first to investigate the existence and uniqueness of solutions for the following functional equation:
(2)
f
(
x
)
=
inf
⁡
y
∈
D
max
⁡
{
r
(
x
,
y
)
,
s
(
x
,
y
)
,
f
(
b
(
x
,
y
)
)
}
,
x
∈
S
,
f
(
x
)
=
inf
⁡
y
∈
D
max
⁡
{
r
(
x
,
y
)
,
f
(
b
(
x
,
y
)
)
}
,
x
∈
S
in a complete metric space B
B
(
S
).Bhakta and Mitra [2] obtained the existence and uniqueness of solutions for the functional equations
(3)
f
(
x
)
=
sup
⁡
y
∈
D
{
p
(
x
,
y
)
+
A
(
x
,
y
,
f
(
a
(
x
,
y
)
)
)
}
,
x
∈
S
in a Banach space B
(
S
) and
(4)
f
(
x
)
=
sup
⁡
y
∈
D
{
p
(
x
,
y
)
+
f
(
a
(
x
,
y
)
)
}
,
x
∈
S
in B
B
(
S
), respectively. Bhakta and Choudhury [3] established the existence of solutions for the functional equations (2) in B
B
(
S
).In 2003, Liu and Ume [4] pointed out that the form of the functional equations of dynamic programming is as follows:
(5)
f
(
x
)
=
opt
y
∈
D
⁡
{
a
[
u
(
x
,
y
)
+
f
(
T
(
x
,
y
)
)
]
aaaaaa
+
b
opt
[
v
(
x
,
y
)
,
f
(
T
(
x
,
y
)
)
]
}
,
aaaaaaaaaaaaaaaaaai
x
∈
S
,
a
+
b
=
1
.
In 2004, Liu et al. [5] obtained an existence, uniqueness, and iterative approximation of solutions for the functional equation
(6)
f
(
x
)
=
opt
y
∈
D
⁡
{
u
(
x
,
y
)
+
opt
{
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
aaaaassssssssssssssaaaaaaai
×
f
i
(
a
i
(
x
,
y
)
)
:
i
=
1,2
}
}
,
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaiaaaaaaa
i
x
∈
S
.
In 2006, Liu et al. [6] provided the sufficient conditions which ensure the existence and uniqueness and iterative approximation of solution for the functional equation
(7)
f
(
x
)
=
opt
y
∈
D
⁡
{
p
(
x
,
y
)
+
q
(
x
,
y
)
f
(
a
(
x
,
y
)
)
aaaaiaa
+
opt
{
r
(
x
,
y
)
,
s
(
x
,
y
)
f
(
b
(
x
,
y
)
)
,
a
a
a
a
a
a
a
i
i
a
a
a
t
(
x
,
y
)
f
(
c
(
x
,
y
)
)
}
}
,
aaaaaaaaaaaaaaaaaaaaaaaaaaaa
x
∈
S
.
In 2007, Liu and Kang [7] studied the following functional equation:
(8)
f
(
x
)
=
inf
⁡
y
∈
D
max
⁡
{
p
(
x
,
y
)
,
f
(
a
(
x
,
y
)
)
,
q
(
x
,
y
)
a
a
a
a
a
a
a
a
a
+
f
(
b
(
x
,
y
)
)
}
,
∀
x
∈
S
and gave an existence and uniqueness result of solution for the functional equation.In 2011, Jiang et al. [8] investigated the following functional equation:
(9)
f
(
x
)
=
opt
y
∈
D
⁡
opt
{
p
(
x
,
y
)
,
q
(
x
,
y
)
f
(
a
(
x
,
y
)
)
,
r
(
x
,
y
)
a
a
a
i
a
a
a
a
a
×
f
(
b
(
x
,
y
)
)
,
s
(
x
,
y
)
f
(
c
(
x
,
y
)
)
}
,
aaaaaaaaaaaaaaaaaaaaiaaaaaaaaaaaaaaa
∀
x
∈
S
and gave some existence and uniqueness results and iterative approximations of solutions for the functional equation in B
B
(
S
).In Section2, we recall some basic concepts, notations, and lemmas. In Section 3, we utilize the fixed point theorem due to Boyd and Wong [9] to establish the existence, uniqueness, and iterative approximation of solution for the functional equation (1) in Banach spaces and complete metric spaces. Also, we construct some nontrivial examples to explain our results. The results presented here generalize, improve, and extend the results of Bellman [1], Bhakta and Mitra [2], Bhakta and Choudhury [3], Liu and Ume [4], Liu et al. [5], Liu et al. [6], Liu and Kang [7], Jiang et al. [8], Pathak and Deepmala [10], and Liu [11]. The existence problems of solutions of various functional equations arising in dynamic programming are both theoretical and practical interest. Fixed point theorems are applied in various fields (see [12–16]). Here, we use a fixed point theorem to show the solvability of the functional equation arising in dynamic programming.


## 2. Preliminaries

Throughout this paper, we assume thatℝ
=
(
-
∞
,
+
∞
), R
+
=
[
0
,
+
∞
), and R
-
=
(
-
∞
,
0
]. For any t
∈
ℝ, [
t
] denotes the largest integer not exceeding t and (
X
,
∥
·
∥
) and (
Y
,
∥
·
∥
′
) are real Banach spaces. Let S
⊆
X be the state space and let D
⊆
Y be decision space. DefineΦ
1
=
{
φ
:
φ
:
R
+
→
R
+
is nondecreasing
},
Φ
2
=
{
(
φ
,
ψ
)
:
φ
,
ψ
∈
Φ
1
,
ψ
(
t
)
>
0
and
∑
n
=
0
∞
‍
ψ
(
φ
n
(
t
)
)
<
∞
for
t
>
0
},
Φ
3
=
{
(
φ
,
ψ
)
:
φ
,
ψ
∈
Φ
1
,
ψ
(
t
)
>
0
and
lim
⁡
n
→
∞
ψ
(
φ
n
(
t
)
)
=
0
for
t
>
0
},
Φ
4
=
{
φ
:
φ
∈
Φ
1
and
∑
n
=
0
∞
‍
φ
n
(
t
)
<
∞
for
t
>
0
},
B
(
S
)
=
{
f
:
f
:
S
→
ℝ
is
bounded
},
B
C
(
S
)
=
{
f
:
f
∈
B
(
S
)
is
continuous
},
B
B
(
S
)
=
{
f
:
f
:
S
→
ℝ
is bounded on bounded subsets of
S
}.Here, (
B
(
S
)
,
∥
·
∥
1
) and (
B
C
(
S
)
,
∥
·
∥
1
) are Banach spaces with norm
⁡
∥
f
∥
1
=
sup
⁡
x
∈
S
|
f
(
x
)
|. For any positive integer k and f, g
∈
B
B
(
S
), let
(10)
d
k
(
f
,
g
)
=
sup
⁡
{
|
f
(
x
)
-
g
(
x
)
|
:
x
∈
B
¯
(
0
,
k
)
}
,
d
(
f
,
g
)
=
∑
k
=
1
∞
‍
1
2
k
·
d
k
(
f
,
g
)
1
+
d
k
(
f
,
g
)
,
where B
¯
(
0
,
k
)
=
{
x
:
x
∈
S
and
∥
x
∥
≤
k
}. Then, {
d
k
}
k
≥
1 is a countable family of pseudometrices on B
B
(
S
). A sequence {
x
n
}
n
≥
1 in B
B
(
S
) is said to converge to a point x
∈
B
B
(
S
) if for any k
≥
1, d
k
(
x
n
,
x
m
)
→
0 as n
,
m
→
∞. It is clear that (
B
B
(
S
)
,
d
) is a complete metric space. A metric space (
M
,
ρ
) is said to metrically convex if for each x
,
y
∈
M, there is a z
≠
x
,
y for which ρ
(
x
,
y
)
=
ρ
(
x
,
z
)
+
ρ
(
z
,
y
). Clearly, any Banach space is metrically convex.Theorem 1 (see [9]).
Suppose that(
M
,
ρ
) is a complete metrically convex metric space and that f
:
M
→
M satisfies
(11)
ρ
(
f
(
x
)
,
f
(
y
)
)
≤
φ
(
ρ
(
x
,
y
)
)
f
o
r
x
,
y
∈
M
,
where φ
:
P
¯
→
R
+ satisfies φ
(
t
)
<
t for t
∈
P
¯
-
{
0
}, where P
=
{
ρ
(
x
,
y
)
:
x
,
y
∈
M
} and P
¯ denotes the closure of P. Then, f has a fixed point u
∈
M and lim
⁡
n
→
∞
⁡
f
n
(
x
)
=
u, for each x
∈
M.Lemma 2 (see [8]).
{
a
i
,
b
i
:
1
≤
i
≤
n
}
⊆
ℝ. Then,
(12)
|
 opt
⁡
{
a
i
:
1
≤
i
≤
n
}
-
opt
⁡

{
b
i
:
1
≤
i
≤
n
}
|
≤
max
⁡
{
|
a
i
-
b
i
|
:
1
≤
i
≤
n
}
.Lemma 3 (see [8]).
(i) Assume thatA
:
S
×
D
→
ℝ is a mapping such that opt
y
∈
D
A
(
x
0
,
y
) is bounded for some x
0
∈
S. Then,
(13)
|
opt
y
∈
D
⁡
A
(
x
0
,
y
)
|
≤
sup
⁡
y
∈
D
⁡
|
A
(
x
0
,
y
)
|
.
(ii) Assume thatA
,
B
:
S
×
D
→
ℝ is a mapping such that opt
y
∈
D
⁡
A
(
x
1
,
y
) and opt
y
∈
D
⁡
B
(
x
2
,
y
) is bounded for some x
1
,
x
2
∈
S. Then,
(14)
|
opt
y
∈
D
⁡
A
(
x
1
,
y
)
-

opt
y
∈
D
⁡
B
(
x
2
,
y
)
|
≤
sup
⁡
y
∈
D
⁡
|
A
(
x
1
,
y
)
-
B
(
x
2
,
y
)
|
.


## 3. Main Results

In this section, we discuss existence and uniqueness of solutions inB
C
(
S
) and B
(
S
).Theorem 4.
LetS be compact. Let u
,
v
,
p
i
,
q
i
:
S
×
D
→
ℝ and a
i
:
S
×
D
→
S for i
=
1,2
,
3 and satisfy the following conditions: (C
1)
u, v and p
i are bounded for i
=
1,2
,
3.
(C
2)
for eachx
0
∈
S, u
(
x
,
y
)
→
u
(
x
0
,
y
), v
(
x
,
y
)
→
v
(
x
0
,
y
), p
i
(
x
,
y
)
→
p
i
(
x
0
,
y
)
,
q
i
(
x
,
y
)
→
q
i
(
x
0
,
y
)
,
a
i
(
x
,
y
)
→
a
i
(
x
0
,
y
) as x
→
x
0 uniformly for y
∈
D and i
=
1,2
,
3.
(C
3)
max
⁡
{
|
q
i
(
x
,
y
)
|
:
i
=
1,2
,
3
}
≤
α, for some α
∈
(
0,1
) and (
x
,
y
)
∈
S
×
D. Then, the functional equation (1) possesses a unique solution w
∈
B
C
(
S
) and {
H
n
h
}
n
≥
1 converges to w for each h
∈
B
C
(
S
), where H is defined by
(15)
H
h
(
x
)
=
opt
y
∈
D
⁡
{
u
(
x
,
y
)
+

opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
×
h
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
}
,
∀
x
∈
S
.Proof.
Leth
∈
B
C
(
S
) and x
0
∈
S and ϵ
>
0; it follows from (
C
1
), (
C
2
) and compactness of S that there exists a constant M
>
0, δ
>
0, and δ
1
>
0 such that
(16)
max
⁡
{
|
u
(
x
,
y
)
|
,
|
v
(
x
,
y
)
|
,
|
p
i
(
x
,
y
)
|
:
i
=
1,2
,
3
}
≤
M
,
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
∀
(
x
,
y
)
∈
S
×
D
,
(17)
max
⁡
{
|
h
(
x
)
|
,
|
h
(
a
i
(
x
,
y
)
)
|
:
i
=
1,2
,
3
}
≤
M
,
aaaaaaaaaaaaaaaaaaaaaaaaa
∀
(
x
,
y
)
∈
S
×
D
,
sup
⁡
y
∈
D
{
|
u
(
x
,
y
)
-
u
(
x
0
,
y
)
|
}
≤
ϵ
4
,
with
∥
x
-
x
0
∥
<
δ
,
max
⁡
{
|
p
i
(
x
,
y
)
-
p
i
(
x
0
,
y
)
|
:
i
=
1,2
,
3
}
≤
ϵ
4
,
aaaaaaaaaaaaaaaaaaaaaaaa
with
∥
x
-
x
0
∥
<
δ
,
max
⁡
{
|
q
i
(
x
,
y
)
-
q
i
(
x
0
,
y
)
|
:
i
=
1,2
,
3
}
≤
ϵ
4
M
,
aaaaaaaaaaaaaaaaaaaaaaaaa
with
∥
x
-
x
0
∥
<
δ
,
|
h
(
x
1
)
-
h
(
x
2
)
|
<
ϵ
4
,
∀
x
1
,
x
2
∈
S
,
aaaaaaaaaaaaaaa
with
∥
x
1
-
x
2
∥
<
δ
1
,
max
⁡
{
∥
a
i
(
x
,
y
)
-
a
i
(
x
0
,
y
)
∥
:
i
=
1,2
,
3
}
<
δ
1
,
aaaaaaa
∀
(
x
,
y
)
∈
S
×
D
with
∥
x
-
x
0
∥
<
δ
,
(18)
|
H
h
(
x
)
|
≤
sup
⁡
y
∈
D
⁡
|
{
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
+
q
i
(
x
,
y
)
h
(
a
i
(
x
,
y
)
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
:
i
=
1,2
,
3
h
(
a
i
(
x
,
y
)
)
{
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
}
}
|
≤
sup
⁡
y
∈
D
⁡
{
|
u
(
x
,
y
)
|
aaaaaaaaa
+
max
⁡
{
|
v
(
x
,
y
)
|
,
|
p
i
(
x
,
y
)
|
+
|
q
i
(
x
,
y
)
|
aaaaaaaaaaaiaaaii
×
|
h
(
a
i
(
x
,
y
)
)
|
:
i
=
1,2
,
3
|
v
(
x
,
y
)
|
,
|
p
i
(
x
,
y
)
|
+
|
q
i
(
x
,
y
)
|
}
}
≤
M
+
max
⁡
{
M
,
M
+
α
M
}
≤
2
M
+
α
M
.
Thus, H is bounded.
In light of(
C
3
), (15), (17), and Lemmas 2 and 3, we deduce that for all (
x
,
y
)
∈
S
×
D with ∥
x
-
x
0
∥
<
δ
(19)
|
H
h
(
x
)
-
H
h
(
x
0
)
|
=
|
opt
y
∈
D
⁡
{
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
i
a
a
×
h
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
}
aaaaaa
-
opt
y
∈
D
⁡
{
u
(
x
0
,
y
)
aaaaaaaaaaaa
+
opt
{
v
(
x
0
,
y
)
,
p
i
(
x
0
,
y
)
+
q
i
(
x
0
,
y
)
aaaaaaaaaaaaaaiaaa
×
h
(
a
i
(
x
0
,
y
)
)
:
i
=
1,2
,
3
}
}
opt
y
∈
D
⁡
{
u
(
x
,
y
)
|
≤
sup
⁡
y
∈
D
⁡
{
|
u
(
x
,
y
)
-
u
(
x
0
,
y
)
|
aaaaaaaaa
+
max
⁡
⁡
{
|
v
(
x
,
y
)
-
v
(
x
0
,
y
)
|
,
a
a
a
a
a
a
a
a
a
a
a
a
a
a
i
|
p
i
(
x
,
y
)
-
p
i
(
x
0
,
y
)
|
aaaaaaaaaaaaaaaai
+
|
q
i
(
x
,
y
)
h
(
a
i
(
x
,
y
)
)
-
q
i
(
x
0
,
y
)
aaaaaaaaaaaaaaaiaaa
×
h
(
a
i
(
x
0
,
y
)
)
|
:
i
=
1,2
,
3
}
}
≤
sup
⁡
y
∈
D
{
|
u
(
x
,
y
)
-
u
(
x
0
,
y
)
|
aaaaaaaaa
+
max
⁡
⁡
{
|
v
(
x
,
y
)
-
v
(
x
0
,
y
)
|

,
|
p
i
(
x
,
y
)
-
p
i
(
x
0
,
y
)
|
aaaaaaaaaaaaaaaa
+
|
q
i
(
x
,
y
)
-
q
i
(
x
0
,
y
)
|
|
h
(
a
i
(
x
,
y
)
)
|
aaaaaaaaaaaaaaaa
+
|
q
i
(
x
0
,
y
)
|
|
h
(
a
i
(
x
,
y
)
)
-
h
(
a
i
(
x
0
,
y
)
)
|
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
:
i
=
1,2
,
3
}
}
≤
ϵ
4
+
max
⁡
{
ϵ
4
,
ϵ
4
+
ϵ
4
M
M
+
α
ϵ
4
}
<
ϵ
,
which implies that H
h is continuous at x
0. Thus, H is a self-mapping on B
C
(
S
).
Givenϵ
>
0, x
∈
S and h
,
g
∈
B
C
(
S
). Suppose that op
t
y
∈
D
=
sup
⁡
y
∈
D. Then, ∃
y
,
z
∈
D such that
(20)
H
h
(
x
)
<
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
h
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
+
ϵ
,
H
g
(
x
)
<
u
(
x
,
z
)
+
opt
{
v
(
x
,
z
)
,
p
i
(
x
,
z
)
+
q
i
(
x
,
z
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
g
(
a
i
(
x
,
z
)
)
:
i
=
1,2
,
3
}
+
ϵ
,
H
h
(
x
)
≥
u
(
x
,
z
)
+
opt
{
v
(
x
,
z
)
,
p
i
(
x
,
z
)
+
q
i
(
x
,
z
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
h
(
a
i
(
x
,
z
)
)
:
i
=
1,2
,
3
}
,
H
g
(
x
)
≥
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
g
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
.
Using (20), we arrive at
(21)
|
H
h
(
x
)
-
H
g
(
x
)
|
<
max
⁡
{
max
⁡
{
|
q
i
(
x
,
y
)
|
a
a
a
a
a
a
a
a
a
a
a
a
a
×
|
h
(
a
i
(
x
,
y
)
)
-
g
(
a
i
(
x
,
y
)
)
|
:
i
=
1,2
,
3
}
,
aaaaaaaaaa
max
⁡
{
|
q
i
(
x
,
z
)
|
a
a
a
a
a
a
a
a
a
a
a
a
a
×
|
h
(
a
i
(
x
,
z
)
)
-
g
(
a
i
(
x
,
z
)
)
|
:
i
=
1,2
,
3
}
}
+
ϵ
≤
max
⁡
{
|
q
i
(
x
,
y
)
|
,
|
q
i
(
x
,
z
)
|
}
∥
h
-
g
∥
1
+
ϵ
≤
α
∥
h
-
g
∥
1
+
ϵ
which implies that
(22)
∥
H
h
-
H
g
∥
1
≤
φ
(
∥
h
-
g
∥
1
)
+
ϵ
,
∀
h
,
g
∈
B
C
(
S
)
,
where φ
(
λ
)
=
α
λ
,
λ
∈
R
+. Letting ϵ
→
0
+, we get
(23)
∥
H
h
-
H
g
∥
1
≤
φ
(
∥
h
-
g
∥
1
)
,
∀
h
,
g
∈
B
C
(
S
)
.
Similarly, we conclude that (23) holds for op
t
y
∈
D
=
inf
⁡
y
∈
D.
Theorem1 ensures that H has a unique fixed point w
∈
B
C
(
S
) and {
H
n
h
}
n
≥
1 converges to w for each h
∈
B
C
(
S
). It is obvious that w is also a unique solution of the functional equation (1) in B
C
(
S
). This completes the proof.Dropping the compactness ofS and (
C
2
) in the proof of Theorem 4, we get the following result.Theorem 5.
Letu
,
v
,
p
i
,
q
i
:
S
×
D
→
ℝ and a
i
:
S
×
D
→
S for i
=
1,2
,
3 and satisfy conditions (
C
1
) and (
C
3
). Then, the functional equation (1) possesses a unique solution w
∈
B
(
S
) and {
H
n
h
}
n
≥
1 converges to w for each h
∈
B
(
S
), where H is defined by (15).Remark 6.
Ifu
(
x
,
y
)
=
p
i
(
x
,
y
)
=
0, for i
=
1,2
,
3 and for
all
(
x
,
y
)
∈
S
×
D, then Theorems 4 and 5 reduce to the results of Jiang et al. [8]. The example below shows that Theorems 4 and 5 extend substantially the results in [8].Example 7.
LetX
=
Y
=
ℝ, S
=
[
1,3
], D
=
R
+; then, Theorem 5 ensures that the following functional equation:
(24)
f
(
x
)
=
opt
y
∈
D
⁡
{
(
1
+
x
x
+
y
2
)
aaaaaaaaa
+
opt
{
x
2
x
+
y
+
7
,
log
⁡
(
x
+
y
2
+
1
)
aaaaaaaaaaaaaaaa
+
sin
(
x
+
2
y
)
f
(
3
x
2
+
y
3
x
2
+
y
3
)
,
aaaaaaaaaaaaaaaaii
sin
(
x
+
y
2
+
3
)
aaaaaaaaaaaaiiaaa
+
sin
2
(
x
y
)
f
(
5
x
+
y
2
x
+
y
2
)
,
x
3
x
+
y
2
+
1
aaaaaaaaaaaaaiiaa
+
x
+
log
⁡
(
2
x
y
2
)
3
x
+
log
⁡
(
2
x
y
2
)
f
(
2
x
2
+
y
x
2
+
y
)
}
}
,
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
∀
x
∈
S
possesses a unique solution in B
(
S
). However, the corresponding results in [8] are not applicable for the functional equation (24) because
(25)
(
1
+
x
x
+
y
2
)
>
0
,
∀
(
x
,
y
)
∈
S
×
D
.
We point out that the functional equation (24) possesses also a unique solution in B
C
(
S
).In our next results, we discuss properties of solutions inB
B
(
S
).Theorem 8.
Letu
,
v
,
p
i
,
q
i
:
S
×
D
→
ℝ and a
i
:
S
×
D
→
S for i
=
1,2
,
3 and satisfy the following conditions: (B
1)
u
,
v, and p
i are bounded on B
¯
(
0
,
k
)
×
D for k
≥
1 and i
=
1,2
,
3,
(
B
2
)
max
⁡
{
∥
a
i
(
x
,
y
)
∥
:
i
=
1,2
,
3
}
≤
∥
x
∥, for (
x
,
y
)
∈
S
×
D,
(
B
3
)
there exists a constantα such that sup
⁡
(
x
,
y
)
∈
S
×
D
{
|
q
i
(
x
,
y
)
|
:
i
=
1,2
,
3
}
≤
α
<
1. Then, the functional equation (1) possesses a unique solution w
∈
B
B
(
S
) and {
H
n
h
}
n
≥
1 converges to w for each h
∈
B
B
(
S
), where H is defined by (15).Proof.
It follows from(
B
1
) and (
B
2
) that for each k
≥
1 and h
∈
B
B
(
S
), there exists β
(
k
)
>
0 and η
(
k
,
h
)
>
0 such that
(26)
sup
⁡
(
x
,
y
)
∈
B
¯
(
0
,
k
)
×
D
{
|
u
(
x
,
y
)
|
,
|
v
(
x
,
y
)
|
,
|
p
i
(
x
,
y
)
|
:
i
=
1,2
,
3
}
≤
β
(
k
)
,
sup
⁡
(
x
,
y
)
∈
B
¯
(
0
,
k
)
×
D
{
|
h
(
a
i
(
x
,
y
)
)
|
:
i
=
1,2
,
3
}
≤
η
(
k
,
h
)
.
By virtue of (
B
3
), (26), we know that
(27)
|
H
h
(
x
)
|
≤
sup
⁡
y
∈
D
⁡
{
|
u
(
x
,
y
)
|
aaaaaai
+
max
⁡
{
|
v
(
x
,
y
)
|
,
|
p
i
(
x
,
y
)
|
+
|
q
i
(
x
,
y
)
|
a
a
a
a
a
a
a
a
a
a
a
a
×
|
h
(
a
i
(
x
,
y
)
)
|
:
i
=
1,2
,
3
}
}
≤
β
(
k
)
+
max
⁡
{
β
(
k
)
,
β
(
k
)
+
η
(
k
,
h
)
}
≤
2
β
(
k
)
+
η
(
k
,
h
)
.
Thus, H
h is bounded; that is, H is self-mapping on B
B
(
S
).
Letk
∈
N, x
∈
B
¯
(
0
,
k
)
,
g
,
h
∈
B
B
(
S
) and ϵ
>
0. Suppose that op
t
y
∈
D
=
sup
⁡
y
∈
D. Then, there exists y
,
z
∈
D satisfying
(28)
H
h
(
x
)
<
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
a
a
a
a
a
a
a
a
a
a
a
a
a
×
h
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
+
ϵ
,
H
g
(
x
)
<
u
(
x
,
z
)
+
opt
{
v
(
x
,
z
)
,
p
i
(
x
,
z
)
+
q
i
(
x
,
z
)
a
a
a
a
a
a
a
a
a
a
a
a
a
×
g
(
a
i
(
x
,
z
)
)
:
i
=
1,2
,
3
}
+
ϵ
,
H
h
(
x
)
≥
u
(
x
,
z
)
+
opt
{
v
(
x
,
z
)
,
p
i
(
x
,
z
)
+
q
i
(
x
,
z
)
aaaaaaaaaaaaaaa
×
h
(
a
i
(
x
,
z
)
)
:
i
=
1,2
,
3
}
,
H
g
(
x
)
≥
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
aaaaaaaaaaaaaaa
×
g
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
.
In terms of the above, (
B
3
) and Lemma 2, we get
(29)
|
H
h
(
x
)
-
H
g
(
x
)
|
<
max
⁡
{
max
⁡
{
|
q
i
(
x
,
y
)
|
|
h
(
a
i
(
x
,
y
)
)
-
g
(
a
i
(
x
,
y
)
)
|
aaaaaaaaaaaaaaa
:
i
=
1,2
,
3
|
h
(
a
i
(
x
,
y
)
)
-
g
(
a
i
(
x
,
y
)
)
|
}
,
aaaaaaaaaa
max
⁡
{
|
q
i
(
x
,
z
)
|
|
h
(
a
i
(
x
,
z
)
)
-
g
(
a
i
(
x
,
z
)
)
|
a
a
a
a
a
a
a
a
a
a
a
a
a
i
:
i
=
1,2
,
3
|
h
(
a
i
(
x
,
y
)
)
-
g
(
a
i
(
x
,
y
)
)
|
}
}
+
ϵ
≤
α
max
⁡
{
{
|
h
(
a
i
(
x
,
y
)
)
-
g
(
a
i
(
x
,
y
)
)
|
:
i
=
1,2
,
3
}
,
aaaaaaaaaaa
{
|
h
(
a
i
(
x
,
z
)
)
-
g
(
a
i
(
x
,
z
)
)
|
:
i
=
1,2
,
3
}
}
+
ϵ
,
which implies that
(30)
|
H
h
(
x
)
-
H
g
(
x
)
|
≤
α
d
k
(
h
,
g
)
+
ϵ
,
∀
h
,
g
∈
B
B
(
S
)
.
In a similar way, we can show that (30) holds for op
t
y
∈
D
=
inf
⁡
y
∈
D. It follows that
(31)
d
k
(
H
g
,
H
h
)
≤
φ
(
d
k
(
h
,
g
)
)
+
ϵ
,
where φ
(
λ
)
=
α
λ for λ
∈
R
+. As ϵ
→
0
+ in (31), we get that
(32)
d
k
(
H
g
,
H
h
)
≤
φ
(
d
k
(
h
,
g
)
)
.
It follows from Theorem 2.2 in [3] that H has a unique fixed point w
∈
B
B
(
S
) and {
H
n
h
}
n
≥
1 converges to w for each h
∈
B
B
(
S
). Obviously, w is also a unique solution of the functional equation (1). This completes the proof.Remark 9.
(1) Ifu
=
v
=
q
1
=
q
3
=
p
1
=
p
2
=
0, op
t
y
∈
D
=
inf
⁡
y
∈
D, and opt
=
max
⁡, then Theorem 8 reduces to Theorem 3.4 of Bhakta and Choudhury [3] and a result of Bellman [1].
(2) Ifv
=
p
3
=
q
3
=
0, then Theorem 8 reduces to a result of Liu et al. [5].
(3) If we replaceu
(
x
,
y
) by p
(
x
,
y
)
+
q
(
x
,
y
)
f
(
a
(
x
,
y
)
) and p
1
=
p
2
=
p
3
=
q
3
=
0, then Theorem 8 reduces to Theorem 4.1 of Liu et al. [6], which, in turn, generalizes the results in [1, 3].
(4) Theorem 3.3 of Jiang et al. [8] is a particular case of Theorem 8.
(5) Theorem 4.1 of Pathak and Deepmala [10] is reduced to Theorem 8, whenever u
=
0.Theorem 10.
Letu
,
v
,
p
i
,
q
i
:
S
×
D
→
ℝ and a
i
:
S
×
D
→
S for i
=
1,2
,
3, and let (
φ
,
ψ
) be in (
Φ
2
) satisfying the following conditions: (
B
4
)
sup
⁡
y
∈
D
{
|
u
(
x
,
y
)
|
,
|
v
(
x
,
y
)
|
,
|
p
i
(
x
,
y
)
|
:
i
=
1,2
,
3
}
≤
ψ
(
∥
x
∥
)
,
for
all
(
x
,
y
)
∈
S
×
D.
(
B
5
)
max
⁡
{
∥
a
i
(
x
,
y
)
∥
:
i
=
1,2
,
3
}
≤
φ
(
∥
x
∥
)
,
for
all
(
x
,
y
)
∈
S
×
D.
(
B
6
)
max
⁡
{
|
q
i
(
x
,
y
)
|
:
i
=
1,2
,
3
}
≤
1
,
for
all
(
x
,
y
)
∈
S
×
D. Then, the functional equation (1) possesses a solution w
∈
B
B
(
S
) that satisfies the following conditions: (
B
7
)
the sequence{
w
n
}
n
≥
0 is defined by
w
0
(
x
)
=
opt
y
∈
D
{
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
:
i
=
1,2
,
3
}
}, for all x
∈
S;
w
n
(
x
)
=
opt
y
∈
D
{
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
w
n
-
1
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
}, for all x
∈
S
,
n
≥
1 converges to w.
(
B
8
)
lim
⁡
n
→
∞
w
(
x
n
)
=
0 for any x
0
∈
S, {
y
n
}
n
≥
1
⊂
D and x
n
∈
{
a
i
(
x
n
-
1
,
y
n
-
1
)
:
i
=
1,2
,
3
}, for all n
∈
N.
(B
9)
w is unique with respect to condition (
B
8
).Proof.
Since(
φ
,
ψ
) is in (
Φ
2
), it is easy to verify that
(33)
φ
(
t
)
<
t
for
t
<
0
.
First of all we assert that the mapping H defined by (15) is nonexpansive on B
B
(
S
) on account of (33) and (
B
5
); we obtain that
(34)
max
⁡
{
∥
a
i
(
x
,
y
)
∥
:
i
=
1,2
,
3
}
≤
φ
(
∥
x
∥
)
<
k
,
aaaaaaaaaaaaaaaai
for
(
x
,
y
)
∈
B
¯
(
0
,
k
)
×
D
,
which implies that ∃ a constant θ
(
k
,
h
)
>
0 with
(35)
max
⁡
{
|
h
(
a
i
(
x
,
y
)
)
|
:
i
=
1,2
,
3
}
≤
θ
(
k
,
h
)
,
aaaaaaaaaaaaaaai
for
(
x
,
y
)
∈
B
¯
(
0
,
k
)
×
D
.
By virtue of (
B
4
), (
B
6
), (15), (35), and Lemmas 2 and 3, we deduce that
(36)
|
H
h
(
x
)
|
=
sup
⁡
y
∈
D
{
|
u
(
x
,
y
)
|
aaaaaa
+
|
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
a
a
a
a
a
a
a
i
a
a
a
×
h
(
a
i
(
x
,
y
)
)
}
|
:
i
=
1,2
,
3
}
≤
sup
⁡
y
∈
D
{
|
u
(
x
,
y
)
|
+
max
⁡
{
|
v
(
x
,
y
)
|
,
aaaaaaaaaaaaaaaaaaaaaa
|
p
i
(
x
,
y
)
|
+
|
q
i
(
x
,
y
)
|
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
|
h
(
a
i
(
x
,
y
)
)
|
:
i
=
1,2
,
3
}
}
≤
ψ
(
k
)
+
max
⁡
{
ψ
(
k
)
,
ψ
(
k
)
+
θ
(
k
,
h
)
}
≤
2
ψ
(
k
)
+
θ
(
k
,
h
)
for
x
∈
B
¯
(
0
,
k
)
.
Thus, H is self-mapping on B
B
(
S
).
As in the proof of Theorem8, by (
B
6
) we immediately conclude that for h
,
g
∈
B
B
(
S
) and k
≥
1,
(37)
d
k
(
H
h
,
H
g
)
≤
d
k
(
h
,
g
)
,
which yields that
(38)
d
(
H
h
,
H
g
)
=
∑
k
=
1
∞
‍
1
2
k
d
k
(
H
h
,
H
g
)
1
+
d
k
(
H
h
,
H
g
)
≤
∑
k
=
1
∞
‍
1
2
k
d
k
(
h
,
g
)
1
+
d
k
(
h
,
g
)
=
d
(
h
,
g
)
for h
,
g
∈
B
B
(
S
). That is, H is nonexpansive.
Now we assert that for eachn
≥
0,
(39)
|
w
n
(
x
)
|
≤
2
∑
j
=
0
n
‍
ψ
(
φ
j
(
∥
x
∥
)
)
,
x
∈
S
.
Now by (
B
4
) we see that
(40)
|
w
0
(
x
)
|
≤
sup
⁡
y
∈
D
⁡
{
|
u
(
x
,
y
)
|
aaaaaaa
+
max
⁡
{
|
v
(
x
,
y
)
|
,
|
p
i
(
x
,
y
)
|
:
i
=
1,2
,
3
}
}
≤
2
ψ
(
∥
x
∥
)
,
x
∈
S
.
That is, (39) is true for n
=
0. Suppose that (39) holds for some n
≥
0. From (B
4)–(B
6) we know that
(41)
|
w
n
+
1
(
x
)
|
=
|
opt
y
∈
D
⁡
{
×
w
n
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
}
u
(
x
,
y
)
+
opt
{
v
(
x
,
y
)
,
p
i
(
x
,
y
)
+
q
i
(
x
,
y
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
w
n
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
}
opt
y
∈
D
|
≤
sup
⁡
y
∈
D
⁡
{
×
w
n
(
a
i
(
x
,
y
)
)
:
i
=
1,2
,
3
}
}
|
u
(
x
,
y
)
|
aaaaaaaaa
+
max
⁡
{
×
|
w
n
(
a
i
(
x
,
y
)
)
|
:
i
=
1,2
,
3
}
|
v
(
x
,
y
)
|
,
|
p
i
(
x
,
y
)
|
+
|
q
i
(
x
,
y
)
|
aaaaaaaaaaaaiaaa
×
|
w
n
(
a
i
(
x
,
y
)
)
|
:
i
=
1,2
,
3
}
}
≤
sup
⁡
y
∈
D
⁡
{
∑
j
=
0
n
ψ
(
φ
j
(
∥
a
i
(
x
,
y
)
∥
)
)
:
i
=
1,2
,
3
2
ψ
(
∥
x
∥
)
a
a
a
a
a
a
a
a
+
max
⁡
{
2
∑
j
=
0
n
ψ
(
φ
j
(
∥
a
i
(
x
,
y
)
∥
)
)
:
i
=
1,2
,
3
}
}
≤
2
∑
j
=
0
n
+
1
ψ
(
φ
j
(
∥
x
∥
)
)
.
Hence, (39) holds for n
≥
0.
Next we claim that{
w
n
}
n
≥
0 is a cauchy sequence in B
B
(
S
). Given k
≥
1 and x
0
∈
B
¯
(
0
,
k
). Let ϵ
>
0, n
,
m
∈
N. Suppose that op
t
y
∈
D
=
sup
⁡
y
∈
D. Then, we select y
,
z
∈
D such that
(42)
w
n
(
x
0
)
<
u
(
x
0
,
y
)
+
opt
{
v
(
x
0
,
y
)
,
p
i
(
x
0
,
y
)
+
q
i
(
x
0
,
y
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
w
n
-
1
(
a
i
(
x
0
,
y
)
)
:
i
=
1,2
,
3
}
+
ϵ
2
,
w
n
+
m
(
x
0
)
<
u
(
x
0
,
z
)
+
opt
{
v
(
x
0
,
z
)
,
p
i
(
x
0
,
z
)
+
q
i
(
x
0
,
z
)
aaaaaaaaa
×
w
n
+
m
-
1
(
a
i
(
x
0
,
z
)
)
:
i
=
1,2
,
3
}
+
ϵ
2
,
w
n
(
x
0
)
≥
u
(
x
0
,
z
)
+
opt
{
v
(
x
0
,
z
)
,
p
i
(
x
0
,
z
)
+
q
i
(
x
0
,
z
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
w
n
-
1
(
a
i
(
x
0
,
z
)
)
:
i
=
1,2
,
3
}
,
w
n
+
m
(
x
0
)
≥
u
(
x
0
,
y
)
+
opt
{
v
(
x
0
,
y
)
,
p
i
(
x
0
,
y
)
+
q
i
(
x
0
,
y
)
aaaaaaaaa
×
w
n
+
m
-
1
(
a
i
(
x
0
,
y
)
)
:
i
=
1,2
,
3
}
.
In view of (42), (
B
6
), and Lemma 2, we infer that
(43)
w
n
+
m
(
x
0
)
-
w
n
(
x
0
)
≤
max
⁡
{
|
q
i
(
x
0
,
z
)
|
|
w
n
+
m
-
1
(
a
i
(
x
0
,
z
)
)
-
w
n
-
1
(
a
i
(
x
0
,
z
)
)
|
aaaaaaaaaa
:
i
=
1,2
,
3
(
a
i
(
x
0
,
z
)
)
}
+
ϵ
2
≤
max
⁡
{
|
q
i
(
x
0
,
z
)
|
:
i
=
1,2
,
3
}
×
max
⁡
⁡
{
|
w
n
+
m
-
1
(
a
i
(
x
0
,
z
)
)
-
w
n
-
1
(
a
i
(
x
0
,
z
)
)
|
aaaaaaaaaaaa
:
i
=
1,2
,
3
(
a
i
(
x
0
,
z
)
)
}
+
ϵ
2
≤
max
⁡
{
|
w
n
+
m
-
1
(
a
i
(
x
0
,
z
)
)
-
w
n
-
1
(
a
i
(
x
0
,
z
)
)
|
aaaaaaaaaa
:
i
=
1,2
,
3
(
a
i
(
x
0
,
z
)
)
}
+
ϵ
2
,
w
n
+
m
(
x
0
)
-
w
n
(
x
0
)
≥
-
max
⁡
{
|
w
n
+
m
-
1
(
a
i
(
x
0
,
y
)
)
-
w
n
-
1
(
a
i
(
x
0
,
y
)
)
|
⁡
aaaaaaaaaaaa
:
i
=
1,2
,
3
(
a
i
(
x
0
,
y
)
)
}
-
ϵ
2
,
which means that
(44)
w
n
+
m
(
x
0
)
-
w
n
(
x
0
)
<
max
⁡
{
max
⁡
{
|
w
n
+
m
-
1
(
a
i
(
x
0
,
z
)
)
-
w
n
-
1
(
a
i
(
x
0
,
z
)
)
|
aaaaaaaaaiaaaaa
:
i
=
1,2
,
3
|
w
n
+
m
-
1
(
a
i
(
x
0
,
z
)
)
-
w
n
-
1
(
a
i
(
x
0
,
z
)
)
|
}
,
aaaaaaaiaa
max
⁡
{
|
w
n
+
m
-
1
(
a
i
(
x
0
,
y
)
)
-
w
n
-
1
(
a
i
(
x
0
,
y
)
)
|
aaaaaaaaaiiaaaa
:
i
=
1,2
,
3
|
w
n
+
m
-
1
(
a
i
(
x
0
,
y
)
)
-
w
n
-
1
(
a
i
(
x
0
,
y
)
)
|
}
|
w
n
+
m
-
1
(
a
i
(
x
0
,
y
)
)
-
w
n
-
1
(
a
i
(
x
0
,
y
)
)
|
}
+
ϵ
2
=
|
w
n
+
m
-
1
(
x
1
)
-
w
n
-
1
(
x
1
)
|
+
ϵ
2
,
for some y
1
∈
{
y
,
z
} and x
1
∈
{
a
i
(
x
0
,
y
1
)
:
i
=
1,2
,
3
}.
Similarly, we conclude that the above inequality (44) holds for op
t
y
∈
D
=
inf
⁡
y
∈
D. Proceeding in this way, we select y
j
∈
D and x
j
∈
{
a
i
(
x
j
-
1
,
y
j
)
:
i
=
1,2
,
3
} for j
=
2,3
,
…
,
n such that
(45)
|
w
n
+
m
-
1
(
x
1
)
-
w
n
-
1
(
x
1
)
|

<

|
w
n
+
m
-
2
(
x
2
)
-
w
n
-
2
(
x
2
)
|
+
2
-
2
ϵ
,
|
w
n
+
m
-
2
(
x
2
)
-
w
n
-
2
(
x
2
)
|

<

|
w
n
+
m
-
3
(
x
3
)
-
w
n
-
3
(
x
3
)
|
+
2
-
3
ϵ
,
⋮
a
|
w
m
+
1
(
x
n
-
1
)
-
w
1
(
x
n
-
1
)
|
<
|
w
m
(
x
n
)
-
w
0
(
x
n
)
|
+
2
-
n
ϵ
.
a
a
a
a
It follows from (
B
5
), (33), (39), (44), and (45) that
(46)
|
w
n
+
m
(
x
0
)
-
w
n
(
x
0
)
|
<
|
w
m
(
x
n
)
-
w
0
(
x
n
)
|
+
∑
i
=
1
n
2
-
i
ϵ
<
|
w
m
(
x
n
)
|
+
|
w
0
(
x
n
)
|
+
ϵ
≤
2
∑
i
=
0
m
ψ
(
φ
i
(
∥
x
n
∥
)
)
+
2
ψ
(
∥
x
n
∥
)
+
ϵ
≤
2
∑
i
=
0
m
ψ
(
φ
i
+
n
(
∥
x
0
∥
)
)
+
2
ψ
(
φ
n
(
∥
x
0
∥
)
)
+
ϵ
≤
2
∑
j
=
n
-
1
∞
ψ
(
φ
j
(
k
)
)
+
ϵ
,
which implies that
(47)
d
k
(
w
n
+
m
,
w
n
)
≤
2
∑
j
=
n
-
1
∞
‍
ψ
(
φ
j
(
k
)
)
+
ϵ
.
As ϵ
→
0
+ in the above inequality, we deduce that
(48)
d
k
(
w
n
+
m
,
w
n
)
≤
2
∑
j
=
n
-
1
∞
ψ
(
φ
j
(
k
)
)
,
which yields that {
w
n
}
n
≥
0 is a cauchy sequence in (
B
B
(
S
)
,
d
) since ∑
i
=
0
∞
‍
ψ
(
φ
n
(
t
)
)
<
∞, for each t
>
0. Suppose that {
w
n
}
n
≥
0 converges to some w
∈
B
B
(
S
). Notice that H is nonexpansive. It follows that
(49)
d
(
w
,
H
w
)
≤
d
(
w
,
H
w
n
)
+
d
(
H
w
n
,
H
w
)
≤
d
(
w
,
w
n
+
1
)
+
d
(
w
n
,
w
)
⟶
0
as
n
⟶
∞
.
That is, H
w
=
w. Thus, the functional equation (1) possess a solution w.
Now we show that(
B
8
) holds. Let ϵ
>
0
,
x
0
∈
S, {
y
n
}
n
≥
1
⊂
D, and x
n
∈
{
a
i
(
x
n
-
1
,
y
n
)
,
i
=
1,2
,
3
}, for n
≥
1. Put k
=
[
∥
x
0
∥
]
+
1. Then, there exists a positive integer m satisfying
(50)
d
k
(
w
,
w
n
)
+
2
∑
j
=
n
∞
‍
ψ
(
φ
j
(
k
)
)
<
ϵ
,
for
n
>
m
.
By (39), (
B
5
), and (50), we infer that, for n
>
m,
(51)
|
w
(
x
n
)
|
≤
|
w
(
x
n
)
-
w
n
(
x
n
)
|
+
|
w
n
(
x
n
)
|
≤
d
k
(
w
,
w
n
)
+
2
∑
j
=
0
∞
ψ
(
φ
j
(
∥
x
n
∥
)
)
≤
d
k
(
w
,
w
n
)
+
2
∑
j
=
n
∞
ψ
(
φ
j
(
k
)
)
≤
ϵ
,
which means that lim
⁡
n
→
∞
w
(
x
n
)
=
0.
Finally, we show that(
B
9
) holds. Suppose that the functional equation (1) possesses another solution h
∈
B
B
(
S
), which satisfies condition (
B
8
). Let ϵ
>
0 and x
0
∈
S. If op
t
y
∈
D
=
sup
⁡
y
∈
D, then there exists y
,
z
∈
S such that
(52)
w
(
x
0
)
<
u
(
x
0
,
y
)
+
opt
{
v
(
x
0
,
y
)
,
p
i
(
x
0
,
y
)
+
q
i
(
x
0
,
y
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
w
(
a
i
(
x
0
,
y
)
)
:
i
=
1,2
,
3
}
+
ϵ
2
,
h
(
x
0
)
<
u
(
x
0
,
z
)
+
opt
{
v
(
x
0
,
z
)
,
p
i
(
x
0
,
z
)
+
q
i
(
x
0
,
z
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
h
(
a
i
(
x
0
,
z
)
)
:
i
=
1,2
,
3
}
+
ϵ
2
,
w
(
x
0
)
≥
u
(
x
0
,
z
)
+
opt
{
v
(
x
0
,
z
)
,
p
i
(
x
0
,
z
)
+
q
i
(
x
0
,
z
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
w
(
a
i
(
x
0
,
z
)
)
:
i
=
1,2
,
3
}
,
h
(
x
0
)
≥
u
(
x
0
,
y
)
+
opt
{
v
(
x
0
,
y
)
,
p
i
(
x
0
,
y
)
+
q
i
(
x
0
,
y
)
a
a
a
a
a
a
a
a
a
a
a
a
a
a
i
×
h
(
a
i
(
x
0
,
y
)
)
:
i
=
1,2
,
3
}
.
Using Lemma 2, (
B
6
), and (52), we get
(53)
w
(
x
0
)
-
h
(
x
0
)
<
max
⁡
{
|
q
i
(
x
0
,
y
)
|
|
w
(
a
i
(
x
0
,
y
)
)
-
h
(
a
i
(
x
0
,
y
)
)
|
aaaaaaaaaa
:
i
=
1,2
,
3
|
q
i
(
x
0
,
y
)
|
|
w
(
a
i
(
x
0
,
y
)
)
-
h
(
a
i
(
x
0
,
y
)
)
|
}
+
ϵ
2
≤
max
⁡
{
|
q
i
(
x
0
,
y
)
|
:
i
=
1,2
,
3
}
×
max
⁡
⁡
{
|
w
(
a
i
(
x
0
,
y
)
)
-
h
(
a
i
(
x
0
,
y
)
)
|
:
i
=
1,2
,
3
}
+
ϵ
2
≤
max
⁡
{
|
w
(
a
i
(
x
0
,
y
)
)
-
h
(
a
i
(
x
0
,
y
)
)
|
:
i
=
1,2
,
3
}
+
ϵ
2
,
w
(
x
0
)
-
h
(
x
0
)
≥
-
max
⁡
{
|
w
(
a
i
(
x
0
,
z
)
)
-
h
(
a
i
(
x
0
,
z
)
)
|
:
i
=
1,2
,
3
}
-
ϵ
2
which implies that
(54)
|
w
(
x
0
)
-
h
(
x
0
)
|
<
max
⁡
{
max
⁡
{
|
w
(
a
i
(
x
0
,
y
)
)
-
h
(
a
i
(
x
0
,
y
)
)
|
:
i
=
1,2
,
3
}

,
aaaaaaaaaa
max
⁡
{
|
w
(
a
i
(
x
0
,
z
)
)
-
h
(
a
i
(
x
0
,
z
)
)
|
:
i
=
1,2
,
3
}
}
+
ϵ
2
=
|
w
(
x
1
)
-
h
(
x
1
)
|
+
ϵ
2
for some y
1
∈
{
y
,
z
} and x
1
∈
{
a
i
(
x
0
,
y
i
)
:
i
=
1,2
,
3
}. Similarly, we conclude that (54) holds for opt
y
∈
D
=
inf
⁡
y
∈
D. Proceeding in this way, we select y
j
∈
D and x
j
∈
{
a
i
(
x
j
-
1
,
y
j
)
} for j
=
2,3
,
…
,
n satisfying
(55)
|
w
(
x
1
)
-
h
(
x
1
)
|
<
|
w
(
x
2
)
-
h
(
x
2
)
|
+
2
-
2
ϵ
,
|
w
(
x
2
)
-
h
(
x
2
)
|
<
|
w
(
x
3
)
-
h
(
x
3
)
|
+
2
-
3
ϵ
,
⋮
|
w
(
x
n
-
1
)
-
h
(
x
n
-
1
)
|
<
|
w
(
x
n
)
-
h
(
x
n
)
|
+
2
-
n
ϵ
.
a
a
i
Combining (54) and (55), we obtain that
(56)
|
w
(
x
0
)
-
h
(
x
0
)
|
<
|
w
(
x
n
)
-
h
(
x
n
)
|
+
∑
j
=
1
n
2
-
j
ϵ
<
|
w
(
x
n
)
-
h
(
x
n
)
|
+
ϵ
.
Letting n
→
∞ in the above inequalities, by (
B
8
) we get that
(57)
|
w
(
x
0
)
-
h
(
x
0
)
|
≤
ϵ
.
As ϵ
→
0
+ in the above inequality, we know that w
(
x
0
)
=
h
(
x
0
). This completes the proof.Remark 11.
(1) Theorem10 extends, improves, and unifies the results in Liu et al. [5, 6].
(2) In caseu
=
p
1
=
p
2
=
p
3
=
0 and (
φ
,
ψ
)
∈
Φ
3, then Theorem 10 reduces to Theorem 3.4 of Jiang et al. [8], which, in turn, generalizes and unifies the results of Bellman [1], Bhakta and Choudhury [3], Liu [11], and Liu and Ume [4].
(3) If we replacep
1
=
p
2
=
p
3
=
0, q
1
=
q
2
=
q
3
=
(
1
-
λ
), a
1
=
a
2
=
a
3
=
a
(
x
,
y
), v
(
x
,
y
)
=
(
1
-
λ
)
v
1
(
x
,
y
), u
(
x
,
y
)
=
λ
[
u
(
x
,
y
)
+
f
(
a
(
x
,
y
)
)
], ψ
(
t
)
=
M
t, and φ
∈
Φ
4, for (
x
,
y
)
∈
S
×
D, t
∈
R
+, where λ is a constant in [
0,1
] and M is a positive constant, then Theorem 10 reduces to Theorem 3.1 of Liu and Ume [4], which, in turn, generalizes Theorem 3.5 in [3], Theorem 2.4 in [2], and a result in [1].
(4) Dropping condition(
B
6
) and replacing op
t
y
∈
D
=
inf
⁡
y
∈
D, opt
=
max
⁡, q
1
=
q
2
=
1, u
=
p
1
=
p
3
=
q
3
=
0, then Theorem 10 reduces to Theorem 3.4 of Liu and Kang [7], which, in turn, generalizes the results in [1, 3, 7].
The example below demonstrates that Theorem10 generalizes, extends, and unifies the results in [1–8, 11].Example 12.
LetX
=
Y
=
R, S
=
D
=
[
1
,
∞
), ψ
(
t
)
=
4
t
4, φ
(
t
)
=
t
/
2, for all t
∈
R
+. Consider the following functional equation:
(58)
f
(
x
)
=
opt
y
∈
D
⁡
{
x
4
(
1
+
1
x
+
y
2
)
aaaaaaaaa
+
opt
{
x
3
(
1
+
|
sin
2
(
x
+
2
y
+
3
)
|
)
,
2
x
3
1
+
x
+
y
aaaaaaaaaaaaiaaa
+
sin
2
(
x
+
2
y
)
f
(
x
2
1
+
x
2
y
2
)
,
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
x
8
x
5
+
cos
⁡
2
(
x
+
2
y
-
1
)
aaaaaaaaaaaiaaaa
+
1
1
+
x
y
f
(
x
3
1
+
x
2
+
y
2
)
,
x
4
1
+
x
y
aaaaaaaaaaaaaiaa
+
cos
⁡
2
(
x
+
y
3
)
aaaaaaaaaaaiaaaa
×
f
(
x
2
+
x
y
2
sin
2
(
x
+
2
y
)
)
}
}
,
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
∀
x
∈
S
.
It follows from Theorem 10 that the functional equation (58) possesses a solution w
∈
B
B
(
S
). However, the corresponding results in [1–8, 11] are not valid for the functional equation (58) because
(59)
|
x
4
(
1
+
1
x
+
y
2
)
|
≤
M
|
x
|
does not hold for (
x
,
y
)
=
(
1
+
M
,
1
)
∈
S
×
D, where M is a positive constant; also
(60)
x
4
(
1
+
1
x
+
y
2
)
>
0
,
x
3
(
1
+
|
sin
2
(
x
+
2
y
+
3
)
|
)
>
0
.



---

*Source: 706585-2014-04-08.xml*