Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2013, Article ID 604105, 7 pages
http://dx.doi.org/10.1155/2013/604105
Research Article
The Solutions of Mixed Monotone Fredholm-Type Integral
Equations in Banach Spaces
Hua Su
School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, Shandong 250014, China
Correspondence should be addressed to Hua Su; jnsuhua@163.com
Received 15 January 2013; Accepted 23 February 2013
Academic Editor: Yanbin Sang
Copyright © 2013 Hua Su. 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.
By introducing new definitions of 𝜙 convex and −𝜑 concave quasioperator and V0 quasilower and 𝑢0 quasiupper, by means of
the monotone iterative techniques without any compactness conditions, we obtain the iterative unique solution of nonlinear mixed
monotone Fredholm-type integral equations in Banach spaces. Our results are even new to 𝜙 convex and −𝜑 concave quasi operator,
and then we apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations
in the ordered Banach spaces.
1. Introduction
In this paper, we will consider the following nonlinear Fredholm integral equation:
𝑢 (𝑡) = ∫ 𝐻 (𝑡, 𝑠, 𝑢 (𝑠)) 𝑑𝑠,
𝐼
𝑡 ∈ 𝐼,
(1)
where 𝐼 = [𝑎, 𝑏] and 𝐻 ∈ 𝐶[𝐼 × 𝐼 × 𝐸, 𝐸], 𝐸 is a real Banach
space with the norm ‖ ⋅ ‖, and there exists a function 𝐺 ∈
𝐶[𝐼 × 𝐼 × 𝐸 × 𝐸, 𝐸] such that for any (𝑡, 𝑠, 𝑥) ∈ 𝐼 × 𝐼 × 𝐸
𝐻 (𝑡, 𝑠, 𝑥) = 𝐺 (𝑡, 𝑠, 𝑥, 𝑥) .
(2)
Guo and Lakshmikantham [1] introduced the definition
of mixed monotone operator and coupled fixed point; there
are many good results (see [1–13]). In the special case where
𝐻(𝑡, 𝑠, 𝑥) is nondecreasing in 𝑥 for fixed 𝑡, 𝑠 ∈ 𝐼, Guo [2]
established an existence theorem of the maximal and minimal
solutions for (1) in the ordered Banach spaces by means of
monotone iterative techniques. Recently, Jingxian and Lishan
[3] and Lishan [4] obtained iterative sequences that converge
uniformly to solutions and coupled minimal and maximal
quasisolutions of the nonlinear Fredholm integral equations
in ordered Banach spaces by using the Möuch fixed point
theorem and establishing new comparison results. But these
all required the compactness conditions and the monotone
conditions in the above papers, and furthermore they did not
obtain the unique solutions. In addition, extensive studies
have also been carried out to study the global or iterative
solutions of initial value problems [8–13].
In this paper, by introducing new definitions of 𝜙 convex
and −𝜑 concave quasioperator and V0 quasilower and 𝑢0
quasiupper, by means of the monotone iterative techniques
without any compactness conditions which are of the essence
in [2–4, 7, 8, 14], we obtain the iterative unique solution
of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces and then apply these results to the twopoint boundary value problem of second-order nonlinear
ordinary differential equations.
2. Preliminaries and Definitions
Let 𝑃 be a cone in 𝐸, that is, a closed convex subset such that
𝜆𝑃 ⊂ 𝑃 for any 𝜆 ≥ 0 and 𝑃 ∩ {−𝑃} = {𝜃}. By means of 𝑃, a
partial order ≤ is defined as 𝑥 ≤ 𝑦 if and only if 𝑦 − 𝑥 ∈ 𝑃.
A cone 𝑃 is said to be normal if there exists a constant 𝑁 >
0 such that 𝑥, 𝑦 ∈ 𝐸, 𝜃 ≤ 𝑥 ≤ 𝑦 implies ‖ 𝑥 ‖≤ 𝑁 ‖ 𝑦 ‖,
where 𝜃 denotes the zero element of 𝐸 (see [2, 14]), and we call
the smallest number 𝑁 the normal constant of 𝑃 and denote
𝑁𝑃 . The cone 𝑃 is normal if and only if every ordered interval
[𝑥, 𝑦] = {𝑧 ∈ 𝐸 : 𝑥 ≤ 𝑧 ≤ 𝑦} is bounded.
Let 𝑃𝐼 = {𝑢 ∈ 𝐶[𝐼, 𝐸] : 𝑢(𝑡) ≥ 𝜃 for all 𝑡 ∈ 𝐼}, where
𝐶[𝐼, 𝐸] denotes the Banach space of all the continuous
2
Discrete Dynamics in Nature and Society
mapping 𝑢 : 𝐼 → 𝐸 with the norm ‖ 𝑢‖𝐶 = max𝑡∈𝐼 |𝑢(𝑡)|.
It is clear that 𝑃𝐼 is a cone of space 𝐶[𝐼, 𝐸], and so it defines
a partial ordering in 𝐶[𝐼, 𝐸]. Obviously, the normality of 𝑃
implies the normality of 𝑃𝐼 and the normal constants of 𝑃𝐼 ,
and 𝑃 are the same.
Let 𝑢0 , V0 ∈ 𝐶[𝐼, 𝐸]. Then, 𝑢0 , V0 are said to be coupled
lower and upper quasi-solutions of (1) if
𝑢0 (𝑡) ≤ ∫ 𝐺 (𝑡, 𝑠, 𝑢0 (𝑠) , V0 (𝑠)) 𝑑𝑠,
𝐼
(3)
V0 (𝑡) ≥ ∫ 𝐺 (𝑡, 𝑠, V0 (𝑠) , 𝑢0 (𝑠)) 𝑑𝑠,
𝐼
𝑡 ∈ 𝐼,
𝑡 ∈ 𝐼.
If the equality in (3) holds, then 𝑢0 , V0 are said to be coupled
quasi-solutions of (1).
We will always assume in this paper that 𝑃 is a normal
cone of 𝐸. For any 𝑢0 , V0 ∈ 𝐶[𝐼, 𝐸] such that V0 ≤ 𝑤0 , we
define the ordered interval 𝐷 = [𝑢0 , V0 ] = {𝑢 ∈ 𝐶[𝐼, 𝐸] : 𝑢0 ≤
𝑢 ≤ V0 }.
Next, we will give the new definition of 𝜙 convex and
−𝜑 concave quasi operator and V0 quasi-lower and 𝑢0 quasiupper.
Definition 1. Suppose that, 𝐺 ∈ 𝐶[𝐼 × 𝐼 × 𝐸 × 𝐸, 𝐸]. Then
𝐺 is called 𝜙 convex and −𝜑 concave quasi operator, if there
exist functions
𝜙 : (0, ∞) × (0, ∞) 󳨀→ (0, ∞) ,
𝜑 : (0, ∞) × (0, ∞) 󳨀→ (0, ∞) ,
3. The Main Result
The main results of this paper are the following three
theorems.
Theorem 4. Let 𝑃 be a normal cone of 𝐸, let 𝑢0 , V0 ∈ 𝑃𝐼 be
coupled lower and upper quasi-solutions of (1). Assume that
conditions (𝐻1 ), (𝐻2 ), and (𝐻3 ) hold and
(𝐻4 ) There exists 𝑤0 ∈ 𝑃𝐼 such that 𝑢0 ≤ 𝑤0 ≤ V0 , and for
𝛼0 , 𝛽0 ∈ (0, ∞) of (𝐻3 ) such that 𝑢0 ≥ 𝛼0 𝑤0 , 𝛽0 𝑤0 ≥
V0 .
Then, (1) has a unique solution 𝑥∗ (𝑡) ∈ 𝐷 = [𝑢0 , V0 ], and for
any initial 𝑥0 , 𝑦0 ∈ [𝑢0 , V0 ], one has
𝑥𝑛 (𝑡) 󳨀→ 𝑥∗ (𝑡) ,
𝑦𝑛 (𝑡) 󳨀→ 𝑥∗ (𝑡) ,
uniformly on 𝑡 ∈ 𝐼
as 𝑛 󳨀→ ∞,
where {𝑥𝑛 (𝑡)}, {𝑦𝑛 (𝑡)} are defined as
𝑥𝑛 (𝑡) = ∫ 𝐺 (𝑡, 𝑠, 𝑥𝑛−1 (𝑠) , 𝑦𝑛−1 (𝑠)) 𝑑𝑠,
𝐼
(7)
𝑦𝑛 (𝑡) = ∫ 𝐺 (𝑡, 𝑠, 𝑦𝑛−1 (𝑠) , 𝑥𝑛−1 (𝑠)) 𝑑𝑠,
𝐼
𝑡 ∈ 𝐼.
Proof. We first define the operator 𝐴 : [𝑢0 , V0 ] × [𝑢0 , V0 ] →
𝐶[𝐼, 𝐸] by the formula
(4)
𝐴 (𝑢, V) = ∫ 𝐺 (𝑡, 𝑠, 𝑢 (𝑠) , V (𝑠)) 𝑑𝑠.
𝐼
such that
(1) 𝐺(𝑡, 𝑠, 𝛼𝑢, 𝛽V) ≥ 𝜙(𝛼, 𝛽)𝐺(𝑡, 𝑠, 𝑢, V), 𝛼 < 𝛽, 𝛼, 𝛽 ∈
(0, ∞), for all 𝑢, V ∈ 𝐸,
(2) 𝐺(𝑡, 𝑠, 𝛼𝑢, 𝛽V) ≤ 𝜑(𝛼, 𝛽)𝐺(𝑡, 𝑠, 𝑢, V), 𝛼 ≥ 𝛽, 𝛼, 𝛽 ∈
(0, ∞), for all 𝑢, V ∈ 𝐸.
Definition 2. Suppose that 𝐺 ∈ 𝐶[𝐼 × 𝐼 × 𝐸 × 𝐸, 𝐸], 𝑢0 ∈ 𝑃.
Then, 𝐺 is called 𝑢0 quasi-upper, if for any 𝑢, V ∈ 𝐸, 𝑢, V < 𝑢0
such that ∫𝐼 𝐺(𝑡, 𝑠, 𝑢, V)𝑑𝑠 < 𝑢0 .
Definition 3. Suppose that 𝐺 ∈ 𝐶[𝐼 × 𝐼 × 𝐸 × 𝐸, 𝐸], V0 ∈ 𝐸.
Then, 𝐺 is called V0 quasi-lower, if for any 𝑢, V ∈ 𝐸, 𝑢, V > V0
such that ∫𝐼 𝐺(𝑡, 𝑠, 𝑢, V)𝑑𝑠 > V0 .
Let us list the following assumption for convenience.
(𝐻1 ) 𝐺 is uniformly continuous on 𝐼 × 𝐼 × 𝐸 × 𝐸, and 𝐺 is
𝜙 convex and −𝜑 concave quasi operator.
(𝐻2 ) 𝐺(𝑡, 𝑠, 𝑥, 𝑦) is nondecreasing in 𝑥 ∈ 𝐸 for fixed
(𝑡, 𝑠, 𝑦) ∈ 𝐼 × 𝐼 × 𝐸. 𝐺(𝑡, 𝑠, 𝑥, 𝑦) is nonincreasing in
𝑦 ∈ 𝐸 for fixed (𝑡, 𝑠, 𝑥) ∈ 𝐼 × 𝐼 × 𝐸.
(𝐻3 ) 𝜙(𝛼, 𝛽), 𝜑(𝛼, 𝛽) are all increasing in 𝛼, decreasing in
𝛽, and 𝜙(𝛼0 , 𝛽0 ) ≥ 𝛼0 , 𝜑(𝛽0 , 𝛼0 ) ≤ 𝛽0 and for 𝛼, 𝛽 ∈
[𝛼0 , 𝛽0 ], 𝛼 < 𝛽,
𝜑 (𝛽, 𝛼) − 𝜙 (𝛼, 𝛽) ≤ 𝑙 (𝛽 − 𝛼) ,
0 < 𝑙 < 1.
(6)
(5)
(8)
It follows from the assumption (𝐻2 ) that 𝐴 is a mixed
monotone operator, that is, 𝐴(𝑢, V) is nondecreasing in 𝑢 ∈
[𝑢0 , V0 ] and nonincreasing in V ∈ [𝑢0 , V0 ], and 𝑢0 ≤
𝐴(𝑢0 , V0 ), 𝐴(V0 , 𝑢0 ) ≤ V0 .
By (7), we have 𝑥𝑛 (𝑡) = 𝐴(𝑥𝑛−1 (𝑡), 𝑦𝑛−1 (𝑡)), 𝑦𝑛 (𝑡) =
𝐴(𝑦𝑛−1 (𝑡), 𝑥𝑛−1 (𝑡)) and set 𝑤𝑛 (𝑡) = 𝐴(𝑤𝑛−1 (𝑡), 𝑤𝑛−1 (𝑡)) for
initial 𝑤0 in (𝐻4 ), and we also define that
𝑢𝑛 (𝑡) = 𝐴 (𝑢𝑛−1 (𝑡) , V𝑛−1 (𝑡)) ,
V𝑛 (𝑡) = 𝐴 (V𝑛−1 (𝑡) , 𝑢𝑛−1 (𝑡)) .
(9)
Since 𝐴 is a mixed monotone operator, it is easy to see that
𝑢0 ≤ 𝑢1 ≤ ⋅ ⋅ ⋅ ≤ 𝑢𝑛 ≤ ⋅ ⋅ ⋅ ≤ V𝑛 ≤ ⋅ ⋅ ⋅ ≤ V1 ≤ V0 ,
𝑢𝑛 ≤ 𝑤𝑛 ≤ V𝑛 .
(10)
Obviously, by induction, it is easy to see that
𝑢𝑛 ≥ 𝛼𝑛 𝑤𝑛 ,
V𝑛 ≤ 𝛽𝑛 𝑤𝑛 ,
𝑛 = 0, 1, . . . ,
(11)
𝛼0 ≤ 𝛼1 ≤ ⋅ ⋅ ⋅ ≤ 𝛼𝑛 ≤ ⋅ ⋅ ⋅ ≤ 1 ≤ ⋅ ⋅ ⋅ ≤ 𝛽𝑛 ≤ ⋅ ⋅ ⋅ ≤ 𝛽1 ≤ 𝛽0 ,
(12)
where 𝛼𝑛 = 𝜙(𝛼𝑛−1 , 𝛽𝑛−1 ), 𝛽𝑛 = 𝜑(𝛽𝑛−1 , 𝛼𝑛−1 ), 𝑛 = 1, 2, . . ..
In fact, by the assumption (𝐻4 ), we have that inequality
(11) holds as 𝑛 = 0. Suppose that inequality (11) holds as 𝑛 = 𝑘,
Discrete Dynamics in Nature and Society
3
that is, 𝑢𝑘 ≥ 𝛼𝑘 𝑤𝑘 , V𝑘 ≤ 𝛽𝑘 𝑤𝑘 . Then, as 𝑛 = 𝑘 + 1, by the
assumption (𝐻3 ), we have
𝑢𝑘+1 = 𝐴 (𝑢𝑘 , V𝑘 ) = ∫ 𝐺 (𝑡, 𝑠, 𝑢𝑘 (𝑠) , V𝑘 (𝑠)) 𝑑𝑠
𝐼
It is easy to know by (10) and (11) that
𝜃 ≤ V𝑛 − 𝑢𝑛 ≤ 𝛽𝑛 𝑤𝑛 − 𝛼𝑛 𝑤𝑛 ≤ (𝛽𝑛 − 𝛼𝑛 ) 𝑢0 ≤ 𝑙𝑛 (𝛽0 − 𝛼0 ) 𝑢0 ,
(18)
so by the normality of 𝑃𝐼 , we have
󵄩󵄩
󵄩
󵄩 󵄩
𝑛
󵄩󵄩V𝑛 − 𝑢𝑛 󵄩󵄩󵄩𝐶 ≤ 𝑁𝑃 𝑙 (𝛽0 − 𝛼0 ) 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐶,
≥ ∫ 𝐺 (𝑡, 𝑠, 𝛼𝑘 𝑤𝑘 , 𝛽𝑘 𝑤𝑘 ) 𝑑𝑠
𝐼
≥ 𝜙 (𝛼𝑘 , 𝛽𝑘 ) ∫ 𝐺 (𝑡, 𝑠, 𝑤𝑘 (𝑠) , 𝑤𝑘 (𝑠)) 𝑑𝑠 = 𝛼𝑘+1 𝑤𝑘+1 ,
𝐼
V𝑘+1 = 𝐴 (V𝑘 , 𝑢𝑘 ) = ∫ 𝐺 (𝑡, 𝑠, V𝑘 (𝑠) , 𝑢𝑘 (𝑠)) 𝑑𝑠
𝐼
and taking limits in the above inequality as 𝑛 → ∞, we have
𝑥∗ = 𝑢∗ = V∗ ∈ [𝑢0 , V0 ], and for any natural number 𝑛, we
also have 𝑢𝑛 ≤ 𝑥∗ ≤ V𝑛 , 𝑡 ∈ 𝐼.
Then, by the mixed monotone quality of 𝐴 we have
𝑢𝑛+1 = 𝐴 (𝑢𝑛 , V𝑛 ) ≤ 𝐴 (𝑥∗ , 𝑥∗ ) ≤ 𝐴 (V𝑛 , 𝑢𝑛 ) = V𝑛+1 ,
≤ ∫ 𝐺 (𝑡, 𝑠, 𝛽𝑘 𝑤𝑘 , 𝛼𝑘 𝑤𝑘 ) 𝑑𝑠
≤ 𝜑 (𝛽𝑘 , 𝛼𝑘 ) ∫ 𝐺 (𝑡, 𝑠, 𝑤𝑘 (𝑠) , 𝑤𝑘 (𝑠)) 𝑑𝑠 = 𝛽𝑘+1 𝑤𝑘+1 .
(13)
Then, it is easy to show by induction that inequality (11) holds.
For inequality (12), by 𝑢𝑘+1 ≤ 𝑤𝑘+1 ≤ V𝑘+1 and the above
discussion, we have 0 < 𝛼𝑘+1 ≤ 1 ≤ 𝛽𝑘+1 . Obviously, it follows
from the assumption (𝐻4 ) that 𝛼0 ≤ 𝛼1 , 𝛽1 ≤ 𝛽0 . Suppose that
𝛼𝑘−1 ≤ 𝛼𝑘 , 𝛽𝑘 ≤ 𝛽𝑘−1 , so it is easy to show by (𝐻3 ) that
𝜙 (𝛼𝑘−1 , 𝛽𝑘−1 ) ≤ 𝜙 (𝛼𝑘 , 𝛽𝑘 ) ,
𝜑 (𝛽𝑘 , 𝛼𝑘 ) ≤ 𝜑 (𝛽𝑘−1 , 𝛼𝑘−1 ) ,
(14)
that is, 𝛼𝑘 ≤ 𝛼𝑘+1 , 𝛽𝑘+1 ≤ 𝛽𝑘 . Then, it is easy to show by
induction that inequality (12) holds.
Then, it follows from the inequality (12) that there exist
limits of the sequences {𝛼𝑛 }, {𝛽𝑛 }. Suppose that there exist
𝛼, 𝛽 such that 𝛼𝑛 → 𝛼, 𝛽𝑛 → 𝛽, and 𝑛 → ∞, and by
(𝐻3 ), we also have
0 ≤ 𝛽𝑛 − 𝛼𝑛 = 𝜑 (𝛽𝑛−1 , 𝛼𝑛−1 ) − 𝜙 (𝛼𝑛−1 , 𝛽𝑛−1 )
≤ 𝑙 (𝛽𝑛−1 − 𝛼𝑛−1 ) ≤ ⋅ ⋅ ⋅ ≤ 𝑙𝑛 (𝛽0 − 𝛼0 ) ,
(15)
they 0 < 𝑙 < 1, and taking limits in the above inequality as
𝑛 → ∞, we have 𝛼 = 𝛽.
Next, we will show that the sequences {𝑢𝑛 }, {V𝑛 } are all
Cauchy sequences on 𝐷.
In fact, by (10) and (11), for any natural number 𝑝, we
know that
𝜃 ≤ 𝑢𝑛+𝑝 − 𝑢𝑛 ≤ V𝑛 − 𝑢𝑛 ≤ (𝛽𝑛 − 𝛼𝑛 ) 𝑢0 ,
𝜃 ≤ V𝑛 − V𝑛+𝑝 ≤ V𝑛 − 𝑢𝑛 ≤ (𝛽𝑛 − 𝛼𝑛 ) 𝑢0 .
(16)
𝑥∗ = 𝐴 (𝑥∗ , 𝑥∗ ) ,
(21)
that is, 𝑥∗ ∈ [𝑢0 , V0 ] is the fixed point of 𝐴; thus, 𝑥∗ is the
solution of (1) on 𝐷 = [𝑢0 , V0 ].
Furthermore, we will show that the solution is unique.
Suppose that 𝑦∗ ∈ [𝑢0 , V0 ] satisfy 𝑦∗ = 𝐴(𝑦∗ , 𝑦∗ ). Then,
by the mixed monotone quality of 𝐴 and induction, for any
natural number 𝑛, it is easy to have that 𝑢𝑛 ≤ 𝑦∗ ≤ V𝑛 .
Then, by the normality of 𝑃𝐼 and taking limits in the above
inequality as 𝑛 → ∞ and the above discussion, we have
𝑦∗ = 𝑥∗ .
For any initial 𝑥0 , 𝑦0 ∈ [𝑢0 , V0 ], by (7) and (8), the mixed
monotone quality of 𝐴 and induction, for any natural number
𝑛, we have 𝑢𝑛 (𝑡) ≤ 𝑥𝑛 (𝑡) ≤ V𝑛 (𝑡), 𝑢𝑛 (𝑡) ≤ 𝑦𝑛 (𝑡) ≤ V𝑛 (𝑡), 𝑡 ∈ 𝐼.
Then, the normality of 𝑃𝐼 and (19) imply that
󵄩
󵄩󵄩
󵄩 󵄩
𝑛
󵄩󵄩𝑥𝑛 − 𝑢𝑛 󵄩󵄩󵄩𝐶 ≤ 𝑁𝑃 𝑙 (𝛽0 − 𝛼0 ) 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐶,
󵄩󵄩
󵄩
󵄩 󵄩
𝑛
󵄩󵄩𝑦𝑛 − 𝑢𝑛 󵄩󵄩󵄩𝐶 ≤ 𝑁𝑃 𝑁𝑃 𝑙 (𝛽0 − 𝛼0 ) 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐶.
Theorem 5. Let 𝑃 be a normal cone of 𝐸, let 𝑢0 , V0 ∈ 𝑃𝐼 be
coupled lower and upper quasi-solutions of (1). Assume that
conditions (𝐻1 ), (𝐻2 ), and (𝐻3 ) hold.
(𝐻4󸀠 ) 𝐺 is 𝑢0 quasi-upper, and there exists 𝑤0 ∈ 𝑃𝐼 such that
𝑤0 < 𝑢0 < V0 , and there exist 𝛼0 = sup{𝛼 > 0 : 𝑢0 ≥
𝛼𝑤0 }, 𝛽0 = inf{𝛽 > 0 : V0 ≤ 𝛽𝑤0 }.
Then, (1) has a unique solution 𝑥∗ (𝑡) ∈ 𝐷 = [𝑢0 , V0 ], and for
any initial 𝑥0 , 𝑦0 ∈ [𝑢0 , V0 ], one has
𝑦𝑛 (𝑡) 󳨀→ 𝑥∗ (𝑡) ,
uniformly on 𝑡 ∈ 𝐼 as 𝑛 󳨀→ ∞,
(17)
where 𝑁𝑃 is a normal constant. So {𝑢𝑛 }, {V𝑛 } are all Cauchy
sequences on 𝐷, and then there exists 𝑢∗ , V∗ ∈ [𝑢0 , V0 ] such
that lim𝑛 → ∞ 𝑢𝑛 = 𝑢∗ , lim𝑛 → ∞ V𝑛 = V∗ .
(22)
Thus, the sequence {𝑥𝑛 (𝑡)}, {𝑦𝑛 (𝑡)} all converges uniformly to
𝑥∗ (𝑡) on 𝑡 ∈ 𝐼. This completes the proof of Theorem 4.
𝑥𝑛 (𝑡) 󳨀→ 𝑥∗ (𝑡) ,
By the normality of 𝑃𝐼 and (15), we have
󵄩
󵄩󵄩
󵄩󵄩𝑢𝑛+𝑝 − 𝑢𝑛 󵄩󵄩󵄩 ≤ 𝑁𝑃 𝑙𝑛 (𝛽0 − 𝛼0 ) 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐶,
󵄩𝐶
󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩V𝑛 − V𝑛+𝑝 󵄩󵄩 ≤ 𝑁𝑃 𝑙𝑛 (𝛽0 − 𝛼0 ) 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐶,
󵄩𝐶
󵄩
(20)
and taking limits in above inequality as 𝑛 → ∞, we know
that
𝐼
𝐼
(19)
(23)
where {𝑥𝑛 (𝑡)}, {𝑦𝑛 (𝑡)} are defined as
𝑥𝑛 (𝑡) = ∫ 𝐺 (𝑡, 𝑠, 𝑥𝑛−1 (𝑠) , 𝑦𝑛−1 (𝑠)) 𝑑𝑠,
𝐼
𝑦𝑛 (𝑡) = ∫ 𝐺 (𝑡, 𝑠, 𝑦𝑛−1 (𝑠) , 𝑥𝑛−1 (𝑠)) 𝑑𝑠,
𝐼
(24)
𝑡 ∈ 𝐼.
4
Discrete Dynamics in Nature and Society
Proof. We first define the operator 𝐴 : [𝑢0 , V0 ] × [𝑢0 , V0 ] →
𝐶[𝐼, 𝐸] by the formula
𝐴 (𝑢, V) = ∫ 𝐺 (𝑡, 𝑠, 𝑢 (𝑠) , V (𝑠)) 𝑑𝑠.
𝐼
(8󸀠 )
It follows from the assumption (𝐻2 ) that 𝐴 is a mixed monotone operator, that is, 𝐴(𝑢, V) is nondecreasing in 𝑢 ∈
[𝑢0 , V0 ] and nonincreasing in V ∈ [𝑢0 , V0 ] and 𝑢0 ≤
𝐴(𝑢0 , V0 ), 𝐴(V0 , 𝑢0 ) ≤ V0 . By (7), we have 𝑥𝑛 (𝑡) =
𝐴(𝑥𝑛−1 (𝑡), 𝑦𝑛−1 (𝑡)), 𝑦𝑛 (𝑡) = 𝐴(𝑦𝑛−1 (𝑡), 𝑥𝑛−1 (𝑡)) and set
𝑤𝑛 (𝑡) = 𝐴(𝑤𝑛−1 (𝑡), 𝑤𝑛−1 (𝑡)), and we also define
𝑢𝑛 (𝑡) = 𝐴 (𝑢𝑛−1 (𝑡) , V𝑛−1 (𝑡)) ,
V𝑛 (𝑡) = 𝐴 (V𝑛−1 (𝑡) , 𝑢𝑛−1 (𝑡)) .
(25)
Since 𝐴 is a mixed monotone operator, it is easy to see that
𝑢0 ≤ 𝑢1 ≤ ⋅ ⋅ ⋅ ≤ 𝑢𝑛 ≤ ⋅ ⋅ ⋅ ≤ V𝑛 ≤ ⋅ ⋅ ⋅ ≤ V1 ≤ V0 .
(10󸀠 )
Suppose that for 𝑘 − 1 we have 𝑢𝑘−1 ≥ 𝛼𝑘−1 𝑤𝑘−1 , V𝑘−1 ≤
𝛽𝑘−1 𝑤𝑘−1 , and 𝛼𝑘−2 ≤ 𝛼𝑘−1 ≤ 𝛽𝑘−1 ≤ 𝛽𝑘−2 . Then, for 𝑘 + 1,
by the assumption (𝐻3 ), we have
𝑢𝑘 = 𝐴 (𝑢𝑘−1 , V𝑘−1 ) = ∫ 𝐺 (𝑡, 𝑠, 𝑢𝑘−1 (𝑠) , V𝑘−1 (𝑠)) 𝑑𝑠
𝐼
≥ ∫ 𝐺 (𝑡, 𝑠, 𝛼𝑘−1 𝑤𝑘−1 , 𝛽𝑘−1 𝑤𝑘−1 ) 𝑑𝑠
𝐼
≥ 𝜙 (𝛼𝑘−1 , 𝛽𝑘−1 ) ∫ 𝐺 (𝑡, 𝑠, 𝑤𝑘−1 (𝑠) , 𝑤𝑘−1 (𝑠)) 𝑑𝑠 = 𝛼𝑘 𝑤𝑘 ,
𝐼
V𝑘 = 𝐴 (V𝑘−1 , 𝑢𝑘−1 ) = ∫ 𝐺 (𝑡, 𝑠, V𝑘−1 (𝑠) , 𝑢𝑘−1 (𝑠)) 𝑑𝑠
𝐼
≤ ∫ 𝐺 (𝑡, 𝑠, 𝛽𝑘−1 𝑤𝑘−1 , 𝛼𝑘−1 𝑤𝑘−1 ) 𝑑𝑠
𝐼
≤ 𝜑 (𝛽𝑘−1 , 𝛼𝑘−1 ) ∫ 𝐺 (𝑡, 𝑠, 𝑤𝑘−1 (𝑠) , 𝑤𝑘−1 (𝑠)) 𝑑𝑠 = 𝛽𝑘 𝑤𝑘 .
𝐼
(30)
Because 𝐺 is 𝑢0 quasi-upper and 𝑤0 < 𝑢0 , we have
By the above two inequalities and assumption (𝐻3 ), we have
𝑤1 (𝑡) = 𝐴 (𝑤0 (𝑡) , 𝑤0 (𝑡))
= ∫ 𝐺 (𝑡, 𝑠, 𝑤0 (𝑠) , 𝑤0 (𝑠)) 𝑑𝑠 < 𝑢0 .
(26)
𝐼
= 𝜑 (𝛽𝑘−1 , 𝛼𝑘−1 ) ≤ 𝜑 (𝛽𝑘−2 , 𝛼𝑘−2 ) = 𝛽𝑘−1 .
So for any natural number 𝑛, by induction, we know that
𝑤𝑛 (𝑡) = 𝐴(𝑤𝑛−1 (𝑡), 𝑤𝑛−1 (𝑡)) < 𝑢0 .
It is easy to see by induction that
𝑢𝑘 ≥ 𝛼𝑘 𝑤𝑘 ,
V𝑘 ≤ 𝛽𝑘 𝑤𝑘 ,
𝑎0 ≤ 𝑎1 ≤ ⋅ ⋅ ⋅ ≤ 𝑎𝑘 ≤ ⋅ ⋅ ⋅ ≤ 𝑏𝑘 ≤ ⋅ ⋅ ⋅ ≤ 𝑏1 ≤ 𝑏0 ,
(11󸀠 )
(12󸀠 )
where 𝛼𝑘 = 𝜙(𝛼𝑘−1 , 𝛽𝑘−1 ), 𝛽𝑘 = 𝜑(𝛽𝑘−1 , 𝛼𝑘−1 ), 𝑘 = 1, 2, . . ..
In fact, by the assumptions (𝐻1 ) and (𝐻3 ) and the above
discussion, as 𝑛 = 0, we have
𝑢1 = 𝐴 (𝑢0 , V0 ) = ∫ 𝐺 (𝑡, 𝑠, 𝑢0 (𝑠) , V0 (𝑠)) 𝑑𝑠
𝐼
≥ ∫ 𝐺 (𝑡, 𝑠, 𝛼0 𝑤0 , 𝛽0 𝑤0 ) 𝑑𝑠
𝐼
(27)
𝐼
Then, it is easy to show by induction that inequalities (11󸀠 )
and (12󸀠 ) hold.
The following proof is similar to that of Theorem 4. This
completes the proof of Theorem 5.
By a similar argument to that of Theorem 5, we obtain the
following results.
Theorem 6. Let 𝑃 be a normal cone of 𝐸, and let 𝑢0 , V0 ∈ 𝑃𝐼
be coupled lower and upper quasi-solutions of (1). Assume that
condition (𝐻1 ), (𝐻2 ), and (𝐻3 ) hold.
(𝐻4󸀠󸀠 ) 𝐺 is V0 quasi-lower, and there exists 𝑤0 ∈ 𝑃𝐼 such that
𝑢0 < V0 < 𝑤0 , and there exist 𝛼0 = sup{𝛼 > 0 : 𝑢0 ≥
𝛼𝑤0 }, 𝛽0 = inf{𝛽 > 0 : V0 ≤ 𝛽𝑤0 }.
𝑥𝑛 (𝑡) 󳨀→ 𝑥∗ (𝑡) ,
V1 = 𝐴 (V0 , 𝑢0 ) = ∫ 𝐺 (𝑡, 𝑠, V0 (𝑠) , 𝑢0 (𝑠)) 𝑑𝑠
𝐼
𝐼
(31)
Then, (1) has a unique solution 𝑥∗ (𝑡) ∈ 𝐷 = [𝑢0 , V0 ], and for
any initial 𝑥0 , 𝑦0 ∈ [𝑢0 , V0 ], one has
≥ 𝜙 (𝛼0 , 𝛽0 ) ∫ 𝐺 (𝑡, 𝑠, 𝑤0 (𝑠) , 𝑤0 (𝑠)) 𝑑𝑠 = 𝛼1 𝑤1 ,
≤ ∫ 𝐺 (𝑡, 𝑠, 𝛽0 𝑤0 , 𝛼0 𝑤0 ) 𝑑𝑠
𝛼𝑘−1 = 𝜙 (𝛼𝑘−2 , 𝛽𝑘−2 ) ≤ 𝜙 (𝛼𝑘−1 , 𝛽𝑘−1 ) = 𝛼𝑘 ≤ 𝛽𝑘
(28)
𝑦𝑛 (𝑡) 󳨀→ 𝑥∗ (𝑡),
uniformly on 𝑡 ∈ 𝐼 as 𝑛 󳨀→ ∞,
(32)
where {𝑥𝑛 (𝑡)}, {𝑦𝑛 (𝑡)} are defined as
≤ 𝜑 (𝛽0 , 𝛼0 ) ∫ 𝐺 (𝑡, 𝑠, 𝑤0 (𝑠) , 𝑤0 (𝑠)) 𝑑𝑠 = 𝛽1 𝑤1 .
𝐼
By the above two inequalities and assumption (𝐻3 ), we have
𝑎0 ≤ 𝛼1 = 𝜙 (𝛼0 , 𝛽0 ) ≤ 𝜑 (𝛽0 , 𝛼0 ) = 𝛽1 ≤ 𝑏0 .
(29)
𝑥𝑛 (𝑡) = ∫ 𝐺 (𝑡, 𝑠, 𝑥𝑛−1 (𝑠) , 𝑦𝑛−1 (𝑠)) 𝑑𝑠,
𝐼
𝑦𝑛 (𝑡) = ∫ 𝐺 (𝑡, 𝑠, 𝑦𝑛−1 (𝑠) , 𝑥𝑛−1 (𝑠)) 𝑑𝑠,
𝐼
(33)
𝑡 ∈ 𝐼.
Discrete Dynamics in Nature and Society
5
4. Applications
Consider the following two-point BVP in the Banach space:
−𝑢󸀠󸀠 = 𝑓 (𝑡, 𝑢) ,
𝑡 ∈ 𝐽 = [0, 1] ,
𝑢 (0) = 𝑢 (1) = 0,
(34)
Obviously, by assumption (𝐶3 ), we can get that ‖ 𝐿 1 ‖≤
max𝑡∈𝐼 (𝑡(1 − 𝑡)/2) ∫𝐽 𝑎(𝑠)𝑑𝑠 = (1/8) ∫𝐽 𝑎(𝑠)𝑑𝑠 < 1, then the
equation (𝐼 − 𝐿 1 )𝑢 = 𝑥0 has a unique solution
∞
−1
𝑢0 (𝑡) = (𝐼 − 𝐿 1 ) 𝑥0 = ∑ 𝐿𝑛1 𝑥0 ∈ 𝑃𝐼 .
(40)
𝑛=0
where 𝑓 ∈ 𝐶[𝐽 × 𝑃, 𝑃], 𝑃 is a cone in a real Banach space
𝐸. Suppose that there exists a mapping 𝑔 ∈ 𝐶[𝐽 × 𝑃 × 𝑃, 𝑃]
such that 𝑓(𝑡, 𝑥) = 𝑔(𝑡, 𝑥, 𝑥), and that 𝑔 satisfies the following
conditions:
(𝐶1 ) 𝑔 is uniformly continuous on 𝐽 × 𝑃 × 𝑃, and 𝐺 is 𝜙
convex and −𝜑 concave quasi operator,
(𝐶2 ) 𝑔(𝑡, 𝑥, 𝑦) is nondecreasing in 𝑥 ∈ 𝑃 for fixed (𝑡, 𝑦) ∈
𝐽×𝑃, and 𝑔(𝑡, 𝑥, 𝑦) is nonincreasing in 𝑦 ∈ 𝑃, for fixed
(𝑡, 𝑥) ∈ 𝐽 × 𝑃,
Similarly, the equation (𝐼 − 𝐿 2 )V = 𝑦0 has a unique
solution
𝑡 ∈ 𝐽, 𝑥, 𝑦 ∈ 𝑃.
(35)
(41)
𝑛=0
Thus, by assumption (𝐶3 ), for any 𝑡 ∈ 𝐽, we have
∫ ℎ (𝑡, 𝑠) 𝑔 (𝑠, 𝑢0 (𝑠) , V0 (𝑠)) 𝑑𝑠
𝐽
(𝐶3 ) there exist the bounded nonnegative Lebesgue integrable functions 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), and 𝑑(𝑡) satisfying
∫𝐽 𝑎(𝑠)𝑑𝑠 < 8, ∫𝐽 𝑐(𝑠)𝑑𝑠 < 8 such that
𝑎 (𝑡) 𝑥 + 𝑏 (𝑡) ≤ 𝑔 (𝑡, 𝑥, 𝑦) ≤ 𝑐 (𝑡) 𝑥 + 𝑑 (𝑡) ,
∞
−1
V0 (𝑡) = (𝐼 − 𝐿 2 ) 𝑦0 = ∑ 𝐿𝑛2 𝑦0 ∈ 𝑃𝐼 .
≥ ∫ ℎ (𝑡, 𝑠) (𝑎 (𝑠) 𝑥 + 𝑏 (𝑠)) 𝑑𝑠
𝐽
= 𝐿 1 𝑢0 (𝑡) + 𝑥0 (𝑡) = 𝑢0 (𝑡) ,
(42)
∫ ℎ (𝑡, 𝑠) 𝑔 (𝑠, V0 (s) , 𝑢0 (𝑠)) 𝑑𝑠
𝐽
It is well known that 𝑢 ∈ 𝐶2 [𝐽, 𝑃] is a solution of BVP(34)
in 𝐶2 [𝐽, 𝑃] if and only if 𝑢 ∈ 𝐶[𝐽, 𝑃] is a solution of the
following integral equation:
𝑢 (𝑡) = ∫ ℎ (𝑡, 𝑠) 𝑔 (𝑠, 𝑢 (𝑠) , 𝑢 (𝑠)) 𝑑𝑠,
𝐽
𝑡 ∈ 𝐽,
(36)
𝑡 (1 − 𝑠) , 𝑡 ≤ 𝑠,
𝑠 (1 − 𝑡) , 𝑡 > 𝑠.
(37)
Lemma 7. If assumption (𝐶3 ) holds, then there exists 𝑢0 , V0 ∈
𝐶[𝐽, 𝑃] such that
𝑢0 (𝑡) ≤ ∫ ℎ (𝑡, 𝑠) 𝑔 (𝑠, 𝑢0 (𝑠) , V0 (𝑠)) 𝑑𝑠,
𝐽
𝑡 ∈ 𝐽,
(38)
V0 (𝑡) ≥ ∫ ℎ (𝑡, 𝑠) 𝑔 (𝑠, V0 (𝑠) , 𝑢0 (𝑠)) 𝑑𝑠,
𝐽
𝐽
= 𝐿 2 V0 (𝑡) + 𝑦0 (𝑡) = V0 (𝑡) ,
that is, (38) holds.
Theorem 8. Let 𝑃 be a normal cone of 𝐸. Assume that (𝐶1 )
and (𝐶3 ) hold,
where
ℎ (𝑡, 𝑠) = {
≤ ∫ ℎ (𝑡, 𝑠) (𝑐 (𝑠) V0 (𝑠) + 𝑏 (𝑠)) 𝑑𝑠
𝑡 ∈ 𝐽.
(𝐶4 ) there exists 𝑤0 ∈ 𝑃𝐼 and 𝑢0 , V0 in (38) of Lemma 7 such
that 𝑢0 < 𝑤0 < V0 , and also there exists 𝛼0 , 𝛽0 ∈ (0, ∞)
such that 𝑢0 ≥ 𝛼0 𝑤0 , 𝛽0 𝑤0 ≥ V0 ,
(𝐶5 ) 𝜙(𝛼, 𝛽), 𝜑(𝛼, 𝛽) are all increasing in 𝛼, decreasing in
𝛽 and 𝜙(𝛼0 , 𝛽0 ) ≥ 𝛼0 , 𝜑(𝛽0 , 𝛼0 ) ≤ 𝛽0 , for 𝛼, 𝛽 ∈
[𝛼0 , 𝛽0 ], 𝛼 < 𝛽,
𝜑 (𝛽, 𝛼) − 𝜙 (𝛼, 𝛽) ≤ 𝑙 (𝛽 − 𝛼) ,
0 < 𝑙 < 1.
(43)
Then, (34) has a unique solution 𝑥∗ (𝑡) ∈ 𝐷 = [𝑢0 , V0 ], and for
any initial 𝑥0 , 𝑦0 ∈ [𝑢0 , V0 ], one has
Proof. In fact, let
𝑥𝑛 (𝑡) 󳨀→ 𝑥∗ (𝑡) ,
𝐿 1 𝑢 (𝑡) = ∫ ℎ (𝑡, 𝑠) 𝑎 (𝑠) 𝑢 (𝑠) 𝑑𝑠,
𝐽
𝑥0 (𝑡) = ∫ ℎ (𝑡, 𝑠) 𝑏 (𝑠) 𝑑𝑠,
𝐽
uniformly on 𝑡 ∈ 𝐼
𝑡 ∈ 𝐽,
(39)
𝐿 2 V (𝑡) = ∫ ℎ (𝑡, 𝑠) 𝑐 (𝑠) V (𝑠) 𝑑𝑠,
𝐽
𝑦0 (𝑡) = ∫ ℎ (𝑡, 𝑠) 𝑑 (𝑠) 𝑑𝑠,
𝐽
𝑡 ∈ 𝐽.
𝑦𝑛 (𝑡) 󳨀→ 𝑥∗ (𝑡) ,
as 𝑛 󳨀→ ∞,
(44)
where {𝑥𝑛 (𝑡)}, {𝑦𝑛 (𝑡)} are defined as
𝑥𝑛 (𝑡) = ∫ ℎ (𝑡, 𝑠) 𝑔 (𝑠, 𝑥𝑛−1 (𝑠) , 𝑦𝑛−1 (𝑠)) 𝑑𝑠,
𝐽
𝑦𝑛 (𝑡) = ∫ ℎ (𝑡, 𝑠) 𝑔 (𝑠, 𝑦𝑛−1 (𝑠) , 𝑥𝑛−1 (𝑠)) 𝑑𝑠,
𝐽
(45)
𝑡 ∈ 𝐽.
6
Discrete Dynamics in Nature and Society
Proof. It is easy to see by conditions (𝐶1 ) and (𝐶2 ) that
𝐺(𝑡, 𝑠, 𝑥, 𝑦) = ℎ(𝑡, 𝑠)𝑔(𝑠, 𝑥, 𝑦) satisfy the conditions (𝐻1 ) and
(𝐻2 ) of Theorem 4. By (𝐶3 ) and (38), we have 𝑢0 , V0 ∈ 𝐶[𝐼, 𝑃]
as coupled lower and upper quasi-solutions of (34).
Thus, the assumption (𝐻1 )–(𝐻4 ) of Theorem 4 is satisfied
from the assumption (𝐶1 )–(𝐶5 ) of Theorem 8. The conclusion
of Theorem 8 follows from Theorem 4.
Example 9. In fact, we can construct the function 𝑓(𝑡, 𝑥) in
Theorem 8.
𝜙 (𝛼, 𝛽) = sin 𝛼 +
𝑡 ∈ [0, 1] ,
1
,
2𝛽
(46)
then
(47)
(48)
Thus, 𝐺 is 𝜙 convex and −𝜑 concave quasi operator and thus
satisfies (𝐶1 ).
It is easy to check that 𝑔(𝑡, 𝑥, 𝑦) is nondecreasing in 𝑥 for
fixed (𝑡, 𝑦) and is nonincreasing in 𝑦 for fixed (𝑡, 𝑥) and thus
satisfies (𝐶2 ).
There exist 𝑎(𝑡) = 𝑡/2, 𝑏(𝑡) = 𝑡/100, 𝑐(𝑡) = 2𝑡, and
𝑑(𝑡) = 1000𝑡 satisfying
1
0
1 1
1
∫ 𝑡 𝑑𝑡 = < 8,
2 0
4
1
1
0
0
(49)
∫ 𝑐 (𝑠) 𝑑𝑠 = 2 ∫ 𝑠 𝑑𝑠 = 1 < 8,
such that
𝑎 (𝑡) 𝑥 + 𝑏 (𝑡) ≤ 𝑔 (𝑡, 𝑥, 𝑦) ≤ 𝑐 (𝑡) 𝑥 + 𝑑 (𝑡) .
Thus, (𝐶3 ) holds.
(52)
2
2 4
2
1
≥ = 𝛼0 ,
𝜙 ( , ) = sin +
3 3
3 2 × (4/3) 3
(53)
1
99
≤
(𝛽 − 𝛼) .
2𝛽 100
(54)
Thus, (𝐶5 ) also holds.
The author was supported financially by Natural Science
Foundation of China (11071141) and the Project of Shandong
Province Higher Educational Science and Technology Program (J13LI12, J10LA62).
= 𝜑 (𝛼, 𝛽) 𝐺 (𝑡, 𝑠, 𝑥, 𝑦) .
∫ 𝑎 (𝑠) 𝑑𝑠 =
3
𝛽0 𝑢0 = 2𝑢0 = V0 .
2
Acknowledgment
𝐺 (𝑡, 𝑠, 𝛼𝑥, 𝛽𝑦) = ℎ (𝑡, 𝑠) 𝑔 (𝑠, 𝛼𝑥, 𝛽𝑦)
1
≤ ℎ (𝑡, 𝑠) (3𝛼 − 5𝛽) (𝑥 + )
𝑦
(51)
Choose 𝑤0 = (3/2)𝑢0 such that 𝑢0 < 𝑤0 < V0 , and also there
exist 𝛼0 = 2/3, 𝛽0 = 4/3 such that
𝜑 (𝛽, 𝛼) − 𝜙 (𝛼, 𝛽) = 3𝛽 − 5𝛼 − sin 𝛼 −
= 𝜙 (𝛼, 𝛽) 𝐺 (𝑡, 𝑠, 𝑥, 𝑦) ,
1
)
𝛽𝑦
1
] 𝑑𝑠.
V0 = 2𝑢0 = ∫ ℎ (𝑡, 𝑠) [V0 (𝑠) +
𝑢0 (𝑠)
0
and for 𝛼, 𝛽 ∈ [2/3, 4/3], 𝛼 < 𝛽, we have
𝐺 (𝑡, 𝑠, 𝛼𝑥, 𝛽𝑦) = ℎ (𝑡, 𝑠) 𝑔 (𝑠, 𝛼𝑥, 𝛽𝑦)
= ℎ (𝑡, 𝑠) (𝛼𝑥 +
1
4 2
4
2 2 4
𝜑 ( , ) = 3 × − 5 × = ≤ = 𝛽0 ,
3 3
3
3 3 3
𝜑 (𝛼, 𝛽) = 3𝛼 − 5𝛽,
1
1
≥ ℎ (𝑡, 𝑠) (sin 𝛼 +
) (𝑥 + )
2𝛽
𝑦
0
1
] 𝑑𝑠,
V0 (𝑠)
Thus, (𝐶4 ) is satisfied.
𝜙(𝛼, 𝛽), 𝜑(𝛼, 𝛽) are all increasing in 𝛼 and nondecreasing
in 𝛽,
𝜋
𝛼 ∈ [0, ] ,
2
1
= ℎ (𝑡, 𝑠) (𝛼𝑥 +
)
𝛽𝑦
1
𝑢0 = ∫ ℎ (𝑡, 𝑠) [𝑢0 (𝑠) +
3
𝑢0 = 𝛼0 𝑢0 = 𝛼0 𝑤0 ,
2
Let
1
𝑓 (𝑡, 𝑥) = 𝑔 (𝑡, 𝑥, 𝑦) = 𝑥 + ,
𝑦
There exist
(50)
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