Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 541079, 8 pages
http://dx.doi.org/10.1155/2013/541079
Research Article
Strong Convergence Theorems for Solutions of
Equations of Hammerstein Type
Chih-Sheng Chuang
Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
Correspondence should be addressed to Chih-Sheng Chuang; cschuang1977@gmail.com
Received 11 March 2013; Accepted 18 April 2013
Academic Editor: Erdal Karapınar
Copyright © 2013 Chih-Sheng Chuang. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider an auxiliary operator, defined in a real Hilbert space in terms of 𝐾 and 𝐹, that is, monotone and Lipschitz mappings
(resp., monotone and bounded mappings). We use an explicit iterative process that converges strongly to a solution of equation of
Hammerstein type. Furthermore, our results improve related results in the literature.
1. Introduction
Let 𝐻 be a real Hilbert space. A mapping 𝐴 : 𝐷(𝐴) ⊆ 𝐻 →
𝐻 is said to be monotone if ⟨𝐴𝑥 − 𝐴𝑦, 𝑥 − 𝑦⟩ ≥ 0 for
every 𝑥, 𝑦 ∈ 𝐷(𝐴). 𝐴 is called maximal monotone if it is
monotone and the 𝑅(𝐼 + 𝑟𝐴) = 𝐻, the range of (𝐼 + 𝑟𝐴),
for each 𝑟 > 0, where 𝐼 is the identity mapping on 𝐻. 𝐴 is
said to satisfy the range condition if cl(𝐷(𝐴)) ⊆ 𝑅(𝐼 + 𝑟𝐴)
for each 𝑟 > 0. For monotone mappings, there are many
related equations of evolution. Several problems that arise in
differential equations, for instance, elliptic boundary value
problems whose linear parts possess Green’s function, can be
put in operator form as
𝑢 + 𝐾𝐹𝑢 = 0,
(1)
where 𝐾 and 𝐹 are monotone mappings. In fact, (1) comes
from the following integral equation of Hammerstein type [1]:
𝑢 (𝑥) + ∫ 𝑘 (𝑥, 𝑦) 𝑓 (𝑦, 𝑢 (𝑦)) 𝑑𝑦 = ℎ (𝑥) ,
Ω
(2)
where 𝑑𝑦 is a 𝜎-finite measure on the measure space Ω; the
real kernel is defined by Ω × Ω, 𝑓 is a real-valued function
defined on Ω×R and is, in general, nonlinear, and ℎ is a given
function on Ω. If we now define an operator 𝐾 by
𝐾V (𝑥) = ∫ 𝑘 (𝑥, 𝑦) V (𝑦) 𝑑𝑦,
Ω
𝑥 ∈ Ω,
(3)
and the so-called superposition or Nemytskii operator by
𝐹𝑢(𝑦) := 𝑓(𝑦, 𝑢(𝑦)), then (2) can be put in (1) (without loss
of generality, we may assume that ℎ ≡ 0).
Note that equations of Hammerstein type play a crucial role in the theory of optimal control systems and in
automation and network theory, and several existence and
uniqueness theorems have been proved for equations of the
Hammerstein type. For details, one can refer to [2–7].
In 2005, Chidume and Zegeye [8] constructed an iterative
process as follows:
𝑢𝑛+1 = 𝑢𝑛 − 𝜆 𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜆 𝑛 𝜃𝑛 (𝑢𝑛 − 𝑤) ,
V𝑛+1 = V𝑛 − 𝜆 𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜆 𝑛 𝜃𝑛 (V𝑛 − 𝑤) ,
𝑛 ∈ N,
(4)
where 𝐻 is a real Hilbert space, 𝐹 and 𝐾: 𝐻 → 𝐻 are bounded
monotone mappings satisfying the range condition, 𝑤 ∈ 𝐻,
and {𝜆 𝑛 }𝑛∈N and {𝜃𝑛 }𝑛∈N are sequences in (0, 1). Chidume and
Zegeye [8] show that this sequence converges strongly to the
solution of (1) under suitable conditions.
In 2011, Chidume and Ofoedu [9] introduced a coupled
explicit iterative process as follows:
𝑢𝑛+1 = 𝑢𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜆 𝑛 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜆 𝑛 𝜃𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N,
(5)
where 𝐸 is a uniformly smooth real Banach space, 𝐹 and 𝐾:
𝐸 → 𝐸 are bounded and monotone mappings, and {𝜆 𝑛 }𝑛∈N ,
2
Journal of Applied Mathematics
{𝜃𝑛 }𝑛∈N , and {𝛼𝑛 }𝑛∈N are sequences in (0, 1). Chidume and
Ofoedu [9] gave a strong convergence theorem for approximation of the solution of (1) under suitable conditions.
In 2012, Chidume and Djitté [10] consider the following
iterative process:
𝑢𝑛+1 = 𝑢𝑛 − 𝜆 𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜆 𝑛 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝜆 𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜆 𝑛 𝜃𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N,
(6)
where 𝐻 is a real Hilbert space, 𝐾 and 𝐹 are bounded and
maximal monotone mappings, {𝜆 𝑛 }𝑛∈N and {𝜃𝑛 }𝑛∈N are sequences in (0, 1) and Chidume and Djitté [10] show that
this iterative process converges to an approximate solution
of nonlinear equations of Hammerstein type under suitable
conditions.
Motivated by the previous works, in this paper, we consider an auxiliary operator, defined in a real Hilbert space in
terms of 𝐾 and 𝐹, that is monotone and Lipschitz mapping,
or monotone and bounded mappings. We use an explicit
iterative process that converges strongly to a solution of
equation of Hammerstein type. Furthermore, our results
improve related results in the literature.
and let 𝜇 be a Banach limit on ℓ∞ . Let 𝑔 : 𝐶 → R be defined
by 𝑔(𝑧) = 𝜇𝑛 ||𝑥𝑛 − 𝑧||2 for each 𝑧 ∈ 𝐶. Then there exists a
unique 𝑧0 ∈ 𝐶 such that 𝑔(𝑧0 ) = min{𝑔(𝑧) : 𝑧 ∈ 𝐶}.
Lemma 5 (see [15]). Let 𝐻 be a Hilbert space, let {𝑥𝑛 }𝑛∈N be a
bounded sequence in 𝐻, and let 𝜇 be a mean on ℓ∞ . Then, there
exists a unique point 𝑧0 ∈ 𝐻 such that 𝜇𝑛 ⟨𝑥𝑛 , 𝑦⟩ = ⟨𝑧0 , 𝑦⟩ for
each 𝑦 ∈ 𝐻. Indeed, 𝑧0 ∈ co{𝑥𝑛 : 𝑛 ∈ N}.
Let 𝐻 be a real Hilbert space. Let 𝑊 := 𝐻 × 𝐻 with norm
‖𝑧‖ := (‖𝑢‖2 + ‖V‖2 )
1/2
,
where 𝑧 = (𝑢, V) ∈ 𝑊.
(8)
Hence, 𝑊 is a real Hilbert space with inner product ⟨𝑤1 ,
𝑤2 ⟩ = ⟨𝑢1 , 𝑢2 ⟩ + ⟨V1 , V2 ⟩ for all 𝑤1 = (𝑢𝑙 , V1 ), 𝑤2 = (𝑢2 , V2 ) ∈
𝑊 [8].
Lemma 6. Let 𝐻 be a real Hilbert space, and let 𝑊 := 𝐻 × 𝐻.
Let 𝐹, 𝐾 : 𝐻 → 𝐻 be two mappings, and let 𝐴 : 𝑊 → 𝑊 be
defined by
𝐴𝑤 := (𝐹𝑢 − V, 𝐾V + 𝑢)
for each 𝑤 = (𝑢, V) ∈ 𝑊.
(9)
2. Preliminaries
Throughout this paper, let N be the set of positive integers and
let R be the set of real numbers. Let 𝐻 be a (real) Hilbert space
with inner product ⟨⋅, ⋅⟩ and norm || ⋅ ||, respectively.
Lemma 1. Let 𝐻 be a real Hilbert space. One has ||𝑥 + 𝑦||2 ≤
||𝑥||2 + 2⟨𝑦, 𝑥 + 𝑦⟩ for all 𝑥, 𝑦 ∈ 𝐻.
Lemma 2 (see [11]). Let {𝑎𝑛 }𝑛∈N be a sequence of nonnegative
real numbers, {𝛼𝑛 }𝑛∈N a sequence of real numbers in [0, 1] with
∑∞
𝑛=1 𝛼𝑛 = ∞, and {𝑢𝑛 }𝑛∈N a sequence of nonnegative real
numbers with ∑∞
𝑛=1 𝑢𝑛 < ∞, {𝑡𝑛 }𝑛∈N a sequence of real numbers
with lim sup 𝑡𝑛 ≤ 0. Suppose that 𝑎𝑛+1 ≤ (1 − 𝛼𝑛 )𝑎𝑛 + 𝛼𝑛 𝑡𝑛 + 𝑢𝑛
for each 𝑛 ∈ N. Then, lim𝑛 → ∞ 𝑎𝑛 = 0.
Let ℓ∞ be the Banach space of bounded sequences with
the supremum norm. A linear functional 𝜇 on ℓ∞ is called
a mean if 𝜇(𝑒) = ||𝜇|| = 1, where 𝑒 = (1, 1, 1, . . .). For 𝑥 = (𝑥1 ,
𝑥2 , 𝑥3 , . . .), the value 𝜇(𝑥) is also denoted by 𝜇𝑛 (𝑥𝑛 ). A mean 𝜇
on ℓ∞ is called a Banach limit if it satisfies 𝜇𝑛 (𝑥𝑛 ) = 𝜇𝑛 (𝑥𝑛+1 ).
If 𝜇 is a Banach limit on ℓ∞ , then for 𝑥 = (𝑥1 , 𝑥2 , 𝑥3 , . . .) ∈ ℓ∞ ,
lim inf 𝑥𝑛 ≤ 𝜇𝑛 (𝑥𝑛 ) ≤ lim sup𝑥𝑛 .
𝑛→∞
𝑛→∞
(7)
In particular, if 𝑥 = (𝑥1 , 𝑥2 , 𝑥3 , . . .) ∈ ℓ∞ and lim𝑛 → ∞ 𝑥𝑛 =
𝑎 ∈ R, then we have 𝜇(𝑥) = 𝜇𝑛 (𝑥𝑛 ) = 𝑎. For details, we can
refer to [12].
Lemma 3 (see [13]). Let 𝛼 be a real number and (𝑥0 , 𝑥1 , . . .) ∈
ℓ∞ such that 𝜇𝑛 𝑥𝑛 ≤ 𝛼 for all Banach limit 𝜇 on ℓ∞ . If
lim sup𝑛 → ∞ (𝑥𝑛+1 − 𝑥𝑛 ) ≤ 0, then, lim sup𝑛 → ∞ 𝑥𝑛 ≤ 𝛼.
Lemma 4 (see [14]). Let 𝐶 be a nonempty closed convex subset
of a Hilbert space 𝐻, let {𝑥𝑛 }𝑛∈N be a bounded sequence in 𝐻,
(i) If 𝐹 and 𝐾 are monotone mappings, then 𝐴 is a
monotone mapping [8, Lemma 3.1].
(ii) If 𝐹 and 𝐾 are bounded mappings, then 𝐴 is a bounded
mapping [8, Lemma 3.1].
(iii) If 𝐹 and 𝐾 are Lipschitz mappings with Lipschitz constants 𝐿 1 and 𝐿 2 , respectively, then 𝐴 is a Lipschitz
mapping. Indeed, the Lipschitz constant of 𝐴 is 2(𝐿+1),
where 𝐿 := max{𝐿 1 , 𝐿 2 } [16, Remark 13.6].
3. Main Results (I)
Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻 be two
mappings, and let 𝐴 : 𝐻 × 𝐻 → 𝐻 × 𝐻 be defined by 𝐴𝑤 =
(𝐹𝑢 − V, 𝐾V + 𝑢) for each 𝑤 = (𝑢, V) ∈ 𝐻 × 𝐻. Then, we
observe that 𝑢 ∈ 𝐻 is a solution of 𝑢 + 𝐾𝐹𝑢 = 0 if and only if
𝑤 = (𝑢, V) is a solution of 𝐴𝑤 = 0 in 𝐻 × 𝐻 for V = 𝐹𝑢.
Theorem 7. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be Lipschitz and monotone mappings. Suppose that 𝑢 + 𝐾𝐹𝑢 =
0 has a solution in 𝐻. Let {𝜆 𝑛 }𝑛∈N be a sequence in (0, 1). Let
{𝛼𝑛 }𝑛∈N and {𝜃𝑛 }𝑛∈N be sequences in (0, 1]. Let {𝑢𝑛 }𝑛∈N and
{V𝑛 }𝑛∈N be sequences in 𝐻 defined iteratively from arbitrary 𝑢1 ,
V1 ∈ 𝐻 by
𝑢𝑛+1 = 𝑢𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜆 𝑛 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜆 𝑛 𝜃𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
(10)
Journal of Applied Mathematics
3
For each 𝑛 ∈ N, since 𝐴 is monotone and 𝐴𝑤 = 0, we know
that
Suppose that one of the following conditions holds:
(
(i) 𝑛lim
→∞
𝛼𝑛2
) = lim 𝜆 𝑛 𝛼𝑛 = 0;
𝑛→∞
𝜃𝑛
(
(ii) 𝑛lim
→∞
𝜆𝑛
) = 0;
𝜃𝑛
(
(iii) 𝑛lim
→∞
⟨𝐴𝑤𝑛 , 𝑤𝑛+1 − 𝑤⟩
(11)
𝛼𝑛
) = 0.
𝜃𝑛
(16)
Proof. Since 𝐹 and 𝐾 are Lipschitz mappings, we may assume
that the Lipschitz constants of 𝐹 and 𝐾 are 𝐿 1 and 𝐿 2 , respectively. Let
𝑟0 :=
= ⟨𝐴𝑤𝑛 , 𝑤𝑛 − 𝑤⟩+⟨𝐴𝑤𝑛 , −𝜆 𝑛 𝛼𝑛 𝐴𝑤𝑛 −𝜆 𝑛 𝜃𝑛 (𝑤𝑛 − 𝑤1 )⟩
≥ ⟨𝐴𝑤𝑛 , −𝜆 𝑛 𝛼𝑛 𝐴𝑤𝑛 − 𝜆 𝑛 𝜃𝑛 (𝑤𝑛 − 𝑤1 )⟩ .
Then, the sequences {𝑢𝑛 }𝑛∈N , {V𝑛 }𝑛∈N , {𝐹𝑢𝑛 }𝑛∈N , and {𝐾V𝑛 }𝑛∈N
are bounded.
𝐿 = max {𝐿 1 , 𝐿 2 } ,
= ⟨𝐴𝑤𝑛 , 𝑤𝑛 − 𝜆 𝑛 𝛼𝑛 𝐴𝑤𝑛 − 𝜆 𝑛 𝜃𝑛 (𝑤𝑛 − 𝑤1 ) − 𝑤⟩
1
.
32 (𝐿 + 1)2
(12)
Let 𝑊 := 𝐻 × 𝐻 with the norm ||𝑤|| := (||𝑢||2 + ||V||2 )1/2
for each 𝑤 = (𝑢, V) ∈ 𝐻 × 𝐻. Take any 𝑢 ∈ 𝐻 such that 𝑢 is
solution of 𝑢 + 𝐾𝐹𝑢 = 0, and let 𝑢 be fixed. Let V = 𝐹𝑢 and
𝑤 = (𝑢, V). We observe that 𝑢 = −𝐾V. For each 𝑛 ∈ N, let
𝑤𝑛 := (𝑢𝑛 , V𝑛 ).
For each 𝑛 ∈ N, it follows from Lemma 1 that
󵄩󵄩
󵄩2
󵄩󵄩𝑢𝑛+1 − 𝑢󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩(1 − 𝜆 𝑛 𝜃𝑛 ) (𝑢𝑛 − 𝑢)+𝜆 𝑛 (𝜃𝑛 𝑢1 −𝛼𝑛 𝐹𝑢𝑛 +𝛼𝑛 V𝑛 −𝜃𝑛 𝑢)󵄩󵄩󵄩
2󵄩
󵄩2
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑢𝑛 − 𝑢󵄩󵄩󵄩
󵄩󵄩
󵄩2
󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
󵄩
󵄩2
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 2𝜆 𝑛 𝜃𝑛 ⟨𝑤1 − 𝑤, 𝑤𝑛+1 − 𝑤⟩
− 2𝜆 𝑛 𝛼𝑛 ⟨𝐴𝑤𝑛 , 𝑤𝑛+1 − 𝑤⟩
󵄩
󵄩2
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 2𝜆 𝑛 𝜃𝑛 ⟨𝑤1 − 𝑤, 𝑤𝑛+1 − 𝑤⟩
+ 2𝜆 𝑛 𝛼𝑛 ⟨𝐴𝑤𝑛 , 𝜆 𝑛 𝛼𝑛 𝐴𝑤𝑛 + 𝜆 𝑛 𝜃𝑛 𝑤𝑛 − 𝜆 𝑛 𝜃𝑛 𝑤1 ⟩
󵄩
󵄩2
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 2𝜆 𝑛 𝜃𝑛 ⟨𝑤1 − 𝑤, 𝑤𝑛+1 − 𝑤⟩
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝐴𝑤𝑛 󵄩󵄩󵄩 ⋅ (𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝐴𝑤𝑛 󵄩󵄩󵄩 + 𝜆 𝑛 𝜃𝑛 󵄩󵄩󵄩𝑤𝑛 − 𝑤1 󵄩󵄩󵄩)
󵄩
󵄩2
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 2𝜆 𝑛 𝜃𝑛 ⟨𝑤1 − 𝑤, 𝑤𝑛+1 − 𝑤⟩
󵄩
󵄩
+ 4𝜆 𝑛 𝛼𝑛 (𝐿 + 1) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
⋅ (2𝜆 𝑛 𝛼𝑛 (𝐿 + 1) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 𝜆 𝑛 𝜃𝑛 󵄩󵄩󵄩𝑤𝑛 − 𝑤1 󵄩󵄩󵄩)
󵄩
󵄩2
= (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 2𝜆 𝑛 𝜃𝑛 ⟨𝑤1 − 𝑤, 𝑤𝑛+1 − 𝑤⟩
+ 2𝜆 𝑛 ⟨𝜃𝑛 𝑢1 − 𝛼𝑛 𝐹𝑢𝑛 + 𝛼𝑛 V𝑛 − 𝜃𝑛 𝑢, 𝑢𝑛+1 − 𝑢⟩
󵄩
󵄩2
+ 8𝜆2𝑛 𝛼𝑛2 (𝐿 + 1)2 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩
󵄩󵄩2
󵄩
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑢𝑛 − 𝑢󵄩󵄩
󵄩
󵄩
󵄩 󵄩
+ 4𝜆2𝑛 𝛼𝑛 𝜃𝑛 (𝐿 + 1) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩𝑤𝑛 − 𝑤1 󵄩󵄩󵄩 .
+ 2𝜆 𝑛 ⟨𝜃𝑛 𝑢1 − 𝛼𝑛 𝐹𝑢𝑛 + 𝛼𝑛 V𝑛 − 𝜃𝑛 𝑢, 𝑢𝑛+1 − 𝑢⟩ .
(13)
Similarly, we have
󵄩󵄩
󵄩2
󵄩󵄩V𝑛+1 − V󵄩󵄩󵄩
󵄩
󵄩2
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩V𝑛 − V󵄩󵄩󵄩
Hence, for each 𝑛 ∈ N, it follows from (15) and (16) that
(14)
+ 2𝜆 𝑛 ⟨𝜃𝑛 V1 − 𝛼𝑛 𝐾V𝑛 − 𝛼𝑛 𝑢𝑛 − 𝜃𝑛 𝑢, V𝑛+1 − V⟩ .
For conditions (i)–(iii), we only need to consider one
case since the proof is similar. Now, we assume that
lim𝑛 → ∞ (𝜆 𝑛 /𝜃𝑛 ) = 0. Then, there exists 𝑛0 ∈ N such that 𝜆 𝑛 /
𝜃𝑛 < 𝑟0 for each 𝑛 ≥ 𝑛0 . Choose 𝑟 > 0 such that 𝑤1 ∈ 𝐵(𝑤, 𝑟/4)
and 𝑤𝑛0 ∈ 𝐵(𝑤, 𝑟/4). Let 𝐵 := 𝐵(𝑤, 𝑟).
Now, we want to show that 𝑤𝑛 ∈ 𝐵 for each 𝑛 ≥ 𝑛0 . Clearly,
𝑤𝑛0 ∈ 𝐵(𝑤, 𝑟). Suppose that 𝑤𝑛 ∈ 𝐵 for some 𝑛 ≥ 𝑛0 . Then,
𝑤𝑛+1 ∈ 𝐵. Indeed, if not, then we have
󵄩
󵄩󵄩
󵄩
󵄩
󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 ≤ 𝑟 < 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩 .
For each 𝑛 ∈ N, by (13) and (14), we know that
󵄩2
󵄩󵄩
󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
(17)
(18)
Hence, by (17) and (18), we get
󵄩
󵄩2
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩
+ 2𝜆 𝑛 ⟨𝜃𝑛 𝑤1 − 𝛼𝑛 𝐴𝑤𝑛 − 𝜃𝑛 𝑤, 𝑤𝑛+1 − 𝑤⟩
󵄩
󵄩2
= (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 2𝜆 𝑛 𝜃𝑛 ⟨𝑤1 − 𝑤, 𝑤𝑛+1 − 𝑤⟩
− 2𝜆 𝑛 𝛼𝑛 ⟨𝐴𝑤𝑛 , 𝑤𝑛+1 − 𝑤⟩ .
(15)
󵄩󵄩
󵄩2
󵄩
󵄩2
󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩 ≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 2𝜆 𝑛 𝜃𝑛 󵄩󵄩󵄩𝑤1 − 𝑤󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
+ 8𝜆2𝑛 𝛼𝑛2 (𝐿 + 1)2 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
󵄩
󵄩
󵄩 󵄩
+ 4𝜆2𝑛 𝛼𝑛 𝜃𝑛 (𝐿 + 1) 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩𝑤𝑛 − 𝑤1 󵄩󵄩󵄩 .
(19)
4
Journal of Applied Mathematics
Theorem 10. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be Lipschitz and monotone mappings. Suppose that 𝑢 + 𝐾𝐹𝑢 =
0 has a solution in 𝐻. Let {𝛼𝑛 }𝑛∈N and {𝜃𝑛 }𝑛∈N be sequences
in (0, 1]. Let {𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N be sequences in 𝐻 defined
iteratively from arbitrary 𝑢1 , V1 ∈ 𝐻 by
By (19), we have
󵄩
󵄩
𝜆 𝑛 𝜃𝑛 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ 2𝜆 𝑛 𝜃𝑛 󵄩󵄩󵄩𝑤1 − 𝑤󵄩󵄩󵄩 + 8𝜆2𝑛 𝛼𝑛2 (𝐿 + 1)2 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩
󵄩
󵄩
+ 4𝜆2𝑛 𝛼𝑛 𝜃𝑛 (𝐿 + 1) 󵄩󵄩󵄩𝑤𝑛 − 𝑤1 󵄩󵄩󵄩
𝑢𝑛+1 = 𝑢𝑛 − 𝛼𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
𝑟
+ 8𝜆2𝑛 𝛼𝑛2 (𝐿 + 1)2 𝑟 + 4𝜆2𝑛 𝛼𝑛 𝜃𝑛 (𝐿 + 1)
4
󵄩 󵄩
󵄩
󵄩
⋅ (󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑤1 − 𝑤󵄩󵄩󵄩)
V𝑛+1 = V𝑛 − 𝛼𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜃𝑛 (V𝑛 − V1 ) ,
≤ 2𝜆 𝑛 𝜃𝑛 ⋅
≤ 2𝜆 𝑛 𝜃𝑛 ⋅
(23)
Suppose that one of the following conditions holds:
5𝑟
𝑟
+ 8𝜆2𝑛 𝛼𝑛2 (𝐿 + 1)2 𝑟 + 4𝜆2𝑛 𝛼𝑛 𝜃𝑛 (𝐿 + 1) ⋅ .
4
4
(20)
(
(i) 𝑛lim
→∞
󵄩
󵄩
𝑟 < 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
𝛼𝑛2
) = lim 𝛼𝑛 = 0;
𝑛→∞
𝜃𝑛
(24)
𝛼
( 𝑛 ) = 0.
(ii) 𝑛lim
→∞ 𝜃
𝑛
This implies that
Then, the sequences {𝑢𝑛 }𝑛∈N , {V𝑛 }𝑛∈N , {𝐹𝑢𝑛 }𝑛∈N , and {𝐾V𝑛 }𝑛∈N
are bounded.
𝛼2
5𝜆 𝛼 𝜃
1
≤ 𝑟 + 8𝜆 𝑛 𝑛 (𝐿 + 1)2 𝑟 + 𝑛 𝑛 𝑛 (𝐿 + 1) 𝑟
2
𝜃𝑛
𝜃𝑛
8𝜆
8𝜆
1
≤ 𝑟 + 𝑛 (𝐿 + 1)2 𝑟 + 𝑛 (𝐿 + 1) 𝑟
2
𝜃𝑛
𝜃𝑛
Remark 11. Corollary 9 is also a special case of Theorem 10.
(21)
≤ 𝑟.
This leads to a contradiction. So, 𝑤𝑛+1 ∈ 𝐵. Hence, by mathematical induction, we know that {𝑤𝑛 }𝑛≥𝑛0 ⊆ 𝐵. Therefore,
{𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N are bounded sequences. Furthermore,
{𝐹𝑢𝑛 }𝑛∈N and {𝐾V𝑛 }𝑛∈N are bounded sequences since 𝐹 and 𝐾
are Lipschitz mappings. For conditions (ii) and (iii), the proof
is similar. Therefore, the proof is completed.
Remark 8. (i) Theorem 7 improves the conditions of [17,
Theorem 3.1] if the space 𝐸 in [17] is reduced to a real Hilbert
space. Indeed, [17, Theorem 3.1] assumes that lim𝑛 → ∞ 𝜆 𝑛 =
lim𝑛 → ∞ 𝛼𝑛 = lim𝑛 → ∞ (𝜆 𝑛 /𝜃𝑛 ) = lim𝑛 → ∞ (𝛼𝑛 /𝜃𝑛 ) = 0.
(ii) Furthermore, we know that it is impossible to assume
that 𝛼𝑛 = 𝜃𝑛 = 1 in [17, Theorem 3.1]. However, we can choose
𝛼𝑛 = 𝜃𝑛 = 1 in our result. Indeed, if 𝛼𝑛 = 𝜃𝑛 = 1 and
𝜆 𝑛 = 𝛽𝑛 , then we have the following result as a special case of
Theorem 7.
Corollary 9. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be Lipschitz and monotone mappings. Suppose that 𝑢 + 𝐾𝐹𝑢 =
0 has a solution in 𝐻. Let {𝛽𝑛 }𝑛∈N be a sequence in (0, 1). Let
{𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N be sequences in 𝐻 defined iteratively from
arbitrary 𝑢1 , V1 ∈ 𝐻 by
𝑢𝑛+1 = 𝑢𝑛 − 𝛽𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝛽𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝛽𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝛽𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
𝑛 ∈ N.
Theorem 12. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be Lipschitz and monotone mappings. Suppose that 𝑢 + 𝐾𝐹𝑢 =
0 has a solution in 𝐻. Let {𝜆 𝑛 }𝑛∈N be a sequence in (0, 1). Let
{𝛼𝑛 }𝑛∈N and {𝜃𝑛 }𝑛∈N be sequences in (0, 1]. Let {𝑢𝑛 }𝑛∈N and
{V𝑛 }𝑛∈N be sequences in 𝐻 defined iteratively from arbitrary 𝑢1 ,
V1 ∈ 𝐻 by
𝑢𝑛+1 = 𝑢𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜆 𝑛 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜆 𝑛 𝜃𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
(25)
Assume that
(i) ∑∞
𝑛=1 𝜆 𝑛 𝜃𝑛 = ∞; lim𝑛 → ∞ 𝜆 𝑛 𝛼𝑛 = lim𝑛 → ∞ 𝜆 𝑛 𝜃𝑛 = 0;
(ii) one of the following conditions holds:
(
(a) 𝑛lim
→∞
𝛼𝑛2
) = 0;
𝜃𝑛
(
(b) 𝑛lim
→∞
𝜆𝑛
) = 0;
𝜃𝑛
(
(c) 𝑛lim
→∞
𝛼𝑛
) = 0;
𝜃𝑛
(26)
(iii) one of the following conditions holds:
∞
(d) ∑ 𝜆2𝑛 𝛼𝑛2 < ∞;
(22)
If lim𝑛 → ∞ 𝛽𝑛 = 0, then the sequences {𝑢𝑛 }𝑛∈N , {V𝑛 }𝑛∈N ,
{𝐹𝑢𝑛 }𝑛∈N , and {𝐾V𝑛 }𝑛∈N are bounded.
In fact, following the same argument as the proof of
Theorem 7, we can get the following result.
𝑛=1
𝜆 𝛼2
( 𝑛 𝑛 ) = 0.
(e) 𝑛lim
→∞
𝜃𝑛
(27)
Then, there exists a subset 𝐾min of 𝐻 × 𝐻 such that if (𝑢, V) ∈
𝐾min with V = 𝐹𝑢, then the sequence {𝑢𝑛 } converges strongly
to 𝑢.
Journal of Applied Mathematics
5
Proof. Let 𝐵 and 𝑛0 be the same as the proof of Theorem 7.
Let 𝜇 be a Banach limit on ℓ∞ . Let {𝑥𝑛 }𝑛∈N be a sequence with
𝑥1 = 𝑤1 and 𝑥𝑛 = 𝑤𝑛0 +𝑛−2 for each 𝑛 ≥ 2. Clearly, {𝑥𝑛 }𝑛∈N ⊆ 𝐵.
By Lemma 4, there is a unique 𝑥 ∈ 𝐵 such that
󵄩
󵄩
󵄩2
󵄩2
𝜇𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 = min𝜇𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑦󵄩󵄩󵄩 .
𝑦∈𝐵
(28)
Let 𝐾min = {𝑥}, and we assume that 𝑤 = (𝑢, V) ∈ 𝐾min . Let
𝑡 ∈ [0, 1]. Then, 𝑡𝑤1 + (1 − 𝑡)𝑤 ∈ 𝐵. Hence, for each 𝑡 ∈ (0, 1),
it follows from Lemma 1 that
󵄩
󵄩2
≤ 𝜇𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑤󵄩󵄩󵄩
− 2𝑡𝜇𝑛 ⟨𝑤1 − 𝑤, 𝑥𝑛 − 𝑡𝑤1 − (1 − 𝑡) 𝑤⟩ .
(29)
By Lemma 5, there exists 𝑧0 ∈ 𝐻 such that 𝜇𝑛 ⟨𝑥𝑛 , 𝑦⟩ = ⟨𝑧0 , 𝑦⟩
for each 𝑦 ∈ 𝐻. By (29), for each 𝑡 ∈ (0, 1),
= 𝜇𝑛 ⟨𝑤1 − 𝑤, 𝑥𝑛 − 𝑡𝑤1 − (1 − 𝑡) 𝑤⟩ ≤ 0.
(30)
(31)
(34)
Assume that lim𝑛 → ∞ 𝛽𝑛 = 0 and ∑∞
𝑛=1 𝛽𝑛 = ∞. Then there
exists a subset 𝐾min of 𝐻 × 𝐻 such that if (𝑢, V) ∈ 𝐾min with
V = 𝐹𝑢, then the sequence {𝑢𝑛 } converges strongly to 𝑢.
Corollary 16. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be Lipschitz and monotone mappings. Suppose that 𝑢 + 𝐾𝐹𝑢 =
0 has a solution in 𝐻. Let {𝜆 𝑛 }𝑛∈N be a sequence in (0, 1). Let
{𝜃𝑛 }𝑛∈N be a sequence in (0, 1]. Let {𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N be
sequences in 𝐻 defined iteratively from arbitrary 𝑢1 , V1 ∈ 𝐻
by
V𝑛+1 = V𝑛 − 𝜆 𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜆 𝑛 𝜃𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
Assume that ∑∞
𝑛=1 𝜆 𝑛 𝜃𝑛 = ∞, lim𝑛 → ∞ 𝜆 𝑛 𝜃𝑛 = 0, and
lim𝑛 → ∞ (𝜆 𝑛 /𝜃𝑛 ) = 0. Then, there exists a subset 𝐾min of 𝐻 × 𝐻
such that if (𝑢, V) ∈ 𝐾min with V = 𝐹𝑢, then the sequence {𝑢𝑛 }
converges strongly to 𝑢.
In fact, following the same argument as the proof of
Theorem 12, we get the following result.
lim sup (⟨𝑤1 − 𝑤, 𝑥𝑛+1 − 𝑤⟩ − ⟨𝑤1 − 𝑤, 𝑥𝑛 − 𝑤⟩) ≤ 0.
𝑛→∞
(32)
By (31), (32), and Lemma 3, we know that lim sup𝑛 → ∞ ⟨𝑤1 −
𝑤, 𝑥𝑛 − 𝑤⟩ ≤ 0.
Hence,
𝑛→∞
𝑛 ∈ N.
(35)
Clearly,
lim sup ⟨𝑤1 − 𝑤, 𝑤𝑛 − 𝑤⟩ ≤ 0.
V𝑛+1 = V𝑛 − 𝛽𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝛽𝑛 (V𝑛 − V1 ) ,
𝑢𝑛+1 = 𝑢𝑛 − 𝜆 𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜆 𝑛 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
In (30), letting 𝑡 → 0, we get
𝜇𝑛 ⟨𝑤1 − 𝑤, 𝑥𝑛 − 𝑤⟩ = ⟨𝑤1 − 𝑤, 𝑧0 − 𝑤⟩ ≤ 0.
𝑢𝑛+1 = 𝑢𝑛 − 𝛽𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝛽𝑛 (𝑢𝑛 − 𝑢1 ) ,
In Theorem 12, if 𝛼𝑛 = 1 for each 𝑛 ∈ N, then we have the
following result.
󵄩
󵄩
󵄩2
󵄩2
𝜇𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑤󵄩󵄩󵄩 ≤ 𝜇𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑡𝑤1 − (1 − 𝑡) 𝑤󵄩󵄩󵄩
⟨𝑤1 − 𝑤, 𝑧0 − 𝑡𝑤1 − (1 − 𝑡) 𝑤⟩
in (0, 1). Let {𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N be sequences in 𝐻 defined
iteratively from arbitrary 𝑢1 , V1 ∈ 𝐻 by
(33)
By (17), (33), and Lemma 2, we know that lim𝑛 → ∞ ||𝑤𝑛 −𝑤|| =
0. Therefore, {𝑢𝑛 } converges strongly to 𝑢.
Remark 13. (i) The conclusion of Theorem 12 is still true if
2
∑∞
𝑛=1 𝜆 𝑛 𝜃𝑛 = ∞, lim𝑛 → ∞ (𝛼𝑛 /𝜃𝑛 ) = 0, and lim𝑛 → ∞ 𝜆 𝑛 𝛼𝑛 =
lim𝑛 → ∞ 𝜆 𝑛 𝜃𝑛 = 0.
(ii) The conclusion of Theorem 12 is still true if
∑∞
𝑛=1 𝜆 𝑛 𝜃𝑛 = ∞, lim𝑛 → ∞ (𝛼𝑛 /𝜃𝑛 ) = 0, and lim𝑛 → ∞ 𝜆 𝑛 = 0.
Remark 14. Following the same argument as in Remark 8,
we know that Theorem 12 improves the conditions of [17,
Theorem 3.2] if the space 𝐸 is reduced to a real Hilbert space.
In Theorem 12, if 𝛼𝑛 = 𝜃𝑛 = 1 and 𝜆 𝑛 = 𝛽𝑛 for each 𝑛 ∈ N,
then we have the following result.
Corollary 15. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 →
𝐻 be Lipschitz and monotone mappings. Suppose that 𝑢 +
𝐾𝐹𝑢 = 0 has a solution 𝑢 in 𝐻. Let {𝛽𝑛 }𝑛∈N be a sequence
Theorem 17. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be Lipschitz and monotone mappings. Suppose that 𝑢 + 𝐾𝐹𝑢 =
0 has a solution in 𝐻. Let {𝛼𝑛 }𝑛∈N and {𝜃𝑛 }𝑛∈N be sequences
in (0, 1]. Let {𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N be sequences in 𝐻 defined
iteratively from arbitrary 𝑢1 , V1 ∈ 𝐻 by
𝑢𝑛+1 = 𝑢𝑛 − 𝛼𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝛼𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜃𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
(36)
2
Assume that ∑∞
𝑛=1 𝜃𝑛 = ∞, lim𝑛 → ∞ 𝜃𝑛 = 0, and lim𝑛 → ∞ (𝛼𝑛 /
𝜃𝑛 ) = 0. Then, there exists a subset 𝐾min of 𝐻 × 𝐻 such that
if (𝑢, V) ∈ 𝐾min with V = 𝐹𝑢, then the sequence {𝑢𝑛 } converges
strongly to 𝑢.
Remark 18. Corollary 15 is also a special case of Theorem 17.
4. Main Results (II)
In this section, we consider that 𝐹 and 𝐾 are bounded
mappings.
Theorem 19. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be bounded and monotone mappings. Suppose that 𝑢+𝐾𝐹𝑢 = 0
has a solution in 𝐻. Let {𝜆 𝑛 }𝑛∈N be a sequence in (0, 1). Let
{𝛼𝑛 }𝑛∈N and {𝜃𝑛 }𝑛∈N be sequences in (0, 1]. Let {𝑢𝑛 }𝑛∈N and
6
Journal of Applied Mathematics
{V𝑛 }𝑛∈N be sequences in 𝐻 defined iteratively from arbitrary 𝑢1 ,
V1 ∈ 𝐻 by
󵄩󵄩
󵄩2
󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
𝑢𝑛+1 = 𝑢𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜆 𝑛 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜆 𝑛 𝜃𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
(37)
𝛼𝑛2
) = lim 𝜆 𝑛 𝛼𝑛 = 0;
𝑛→∞
𝜃𝑛
(
(ii) 𝑛lim
→∞
𝜆𝑛
) = 0;
𝜃𝑛
𝜆
( 𝑛 ) = lim 𝜆 𝑛 𝛼𝑛 = 0;
(iii) 𝑛lim
→∞ 𝜃
𝑛→∞
𝑛
(
(iv) 𝑛lim
→∞
10
81 2
𝑟 + 2𝜆2𝑛 𝛼𝑛 𝜃𝑛 ‖𝐴‖ ⋅ 𝑟2
64
8
𝑟 󵄩
󵄩
󵄩2
󵄩
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩 + 𝜆 𝑛 𝜃𝑛 ⋅ 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
2
(38)
𝛼𝑛
) = 0.
𝜃𝑛
󵄩2
󵄩󵄩
󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
+ 2𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝐴𝑤𝑛 󵄩󵄩󵄩 ⋅ (𝜆 𝑛 𝛼𝑛 󵄩󵄩󵄩𝐴𝑤𝑛 󵄩󵄩󵄩 + 𝜆 𝑛 𝜃𝑛 󵄩󵄩󵄩𝑤𝑛 − 𝑤1 󵄩󵄩󵄩)
2
󵄩󵄩
󵄩2 𝑟 󵄩
󵄩 81 𝜆 𝑛 𝛼𝑛
⋅
‖𝐴‖2 ⋅ 𝑟2
󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩 ≤ ⋅ 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩 +
2
32 𝜃𝑛
5 𝜆 𝛼𝜃
+ ⋅ 𝑛 𝑛 𝑛 ‖𝐴‖ ⋅ 𝑟2 .
2
𝜃𝑛
(42)
(43)
Furthermore, we get
≤
󵄩
󵄩2
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 2𝜆 𝑛 𝜃𝑛 ⟨𝑤1 − 𝑤, 𝑤𝑛+1 − 𝑤⟩
󵄩 󵄩 󵄩
󵄩
󵄩 󵄩2
⋅ 󵄩󵄩󵄩𝑤𝑛 󵄩󵄩󵄩 + 2𝜆2𝑛 𝛼𝑛 𝜃𝑛 ‖𝐴‖ ⋅ 󵄩󵄩󵄩𝑤𝑛 󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩𝑤𝑛 − 𝑤1 󵄩󵄩󵄩
(39)
for each 𝑛 ∈ N.
Choose 𝑟0 > 0 such that max{||𝑤1 ||, ||𝑤||} ≤ 𝑟0 /8. For
conditions (i)–(iv), we only need to consider one case since
the proof is similar. Now, we assume that lim𝑛 → ∞ (𝜆 𝑛 /𝜃𝑛 ) =
0. Then, there exists 𝑛0 ∈ N such that
(40)
for each 𝑛 ≥ 𝑛0 . Choose 𝑟 > 1 such that 𝑟0 < 𝑟 and ||𝑤𝑛0 || ≤
𝑟/8. Let 𝐵 := 𝐵(𝑤, 𝑟). Clearly, ||𝑤𝑛0 − 𝑤|| ≤ 𝑟/4.
Now, we want to show that 𝑤𝑛 ∈ 𝐵 for each 𝑛 ≥ 𝑛0 . Clearly,
𝑤𝑛0 ∈ 𝐵(𝑤, 𝑟). Suppose that 𝑤𝑛 ∈ 𝐵 for some 𝑛 ≥ 𝑛0 . Then,
𝑤𝑛+1 ∈ 𝐵. Indeed, if not, then we have
󵄩
󵄩󵄩
󵄩
󵄩
󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 ≤ 𝑟 < 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩 .
5
81 2
𝑟 + 𝜆2𝑛 𝛼𝑛 𝜃𝑛 ‖𝐴‖ ⋅ 𝑟2 .
32
2
2
5 𝜆 𝛼𝜃
󵄩 𝑟 81 𝜆 𝑛 𝛼𝑛
󵄩
𝑟 < 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩 ≤ +
‖𝐴‖2 ⋅ 𝑟 + ⋅ 𝑛 𝑛 𝑛 ‖𝐴‖ ⋅ 𝑟
2 32 𝜃𝑛
2
𝜃𝑛
󵄩
󵄩2
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 2𝜆 𝑛 𝜃𝑛 ⟨𝑤1 − 𝑤, 𝑤𝑛+1 − 𝑤⟩
𝜆𝑛
8
1
< max {
,
}
2
𝜃𝑛
81‖𝐴‖ 10 ‖𝐴‖
+ 𝜆2𝑛 𝛼𝑛2 ‖𝐴‖2 ⋅
This implies that
Proof. Since 𝐹 and 𝐾 are bounded, we know that 𝐴 is
bounded. Hence, ||𝐴|| < ∞. From the proof of Theorem 7,
we know that
+
󵄩 󵄩 󵄩
󵄩
󵄩 󵄩2
+ 2𝜆2𝑛 𝛼𝑛2 ‖𝐴‖2 ⋅ 󵄩󵄩󵄩𝑤𝑛 󵄩󵄩󵄩 + 2𝜆2𝑛 𝛼𝑛 𝜃𝑛 ‖𝐴‖ ⋅ 󵄩󵄩󵄩𝑤𝑛 󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩𝑤𝑛 − 𝑤1 󵄩󵄩󵄩
+ 2𝜆2𝑛 𝛼𝑛2 ‖𝐴‖2 ⋅
Then, the sequences {𝑢𝑛 }𝑛∈N , {V𝑛 }𝑛∈N , {𝐹𝑢𝑛 }𝑛∈N , and {𝐾V𝑛 }𝑛∈N
are bounded.
2𝜆2𝑛 𝛼𝑛2 ‖𝐴‖2
󵄩
󵄩
󵄩2
󵄩 󵄩
󵄩
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 󵄩󵄩󵄩𝑤𝑛 − 𝑤󵄩󵄩󵄩 + 2𝜆 𝑛 𝜃𝑛 󵄩󵄩󵄩𝑤1 − 𝑤󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
𝑟 󵄩
󵄩
≤ (1 − 𝜆 𝑛 𝜃𝑛 ) 𝑟2 + 2𝜆 𝑛 𝜃𝑛 ⋅ 󵄩󵄩󵄩𝑤𝑛+1 − 𝑤󵄩󵄩󵄩
4
Suppose that one of the following conditions holds:
(
(i) 𝑛lim
→∞
Clearly, ||𝑤𝑛 || ≤ ||𝑤𝑛 − 𝑤|| + ||𝑤|| ≤ 9𝑟/8. Hence, by (39) and
(41), we get
(41)
𝑟 81 𝜆 𝑛
5 𝜆
+
‖𝐴‖2 ⋅ 𝑟 + ⋅ 𝑛 ‖𝐴‖ ⋅ 𝑟
2 32 𝜃𝑛
2 𝜃𝑛
≤ 𝑟.
(44)
This leads to a contradiction. So, 𝑤𝑛+1 ∈ 𝐵. Hence, by mathematical induction, we know that {𝑤𝑛 }𝑛≥𝑛0 ⊆ 𝐵. Therefore,
{𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N are bounded sequences. Furthermore,
{𝐹𝑢𝑛 }𝑛∈N and {𝐾V𝑛 }𝑛∈N are bounded sequences since 𝐹 and
𝐾 are bounded mappings.
Remark 20. (i) Theorem 19 improves the conditions of [9,
Theorem 3.1] if the space 𝐸 is reduced to a real Hilbert space.
Indeed, [9, Theorem 3.1] assumes that lim𝑛 → ∞ (𝜆 𝑛 /𝜃𝑛 ) =
lim𝑛 → ∞ (𝛼𝑛 /𝜃𝑛 ) = 0.
(ii) Furthermore, we know that it is impossible to assume
that 𝛼𝑛 = 𝜃𝑛 = 1 in [9, Theorem 3.1]. However, we can choose
𝛼𝑛 = 𝜃𝑛 = 1 in our result. Indeed, if 𝛼𝑛 = 𝜃𝑛 = 1 and
𝜆 𝑛 = 𝛽𝑛 , then we have the following result as special case of
Theorem 19.
Corollary 21. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be bounded and monotone mappings. Suppose that 𝑢 + 𝐾𝐹𝑢 =
0 has a solution in 𝐻. Let {𝛽𝑛 }𝑛∈N be a sequence in (0, 1).
Journal of Applied Mathematics
7
Let {𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N be sequences in 𝐻 defined iteratively
from arbitrary 𝑢1 , V1 ∈ 𝐻 by
𝑢𝑛+1 = 𝑢𝑛 − 𝛽𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝛽𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝛽𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝛽𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
(45)
Following the same argument as the proof of Theorem 19,
we get the following result. Note that Corollary 21 is also a
special case of the following result.
Theorem 22. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be bounded and monotone mappings. Suppose that 𝑢 + 𝐾𝐹𝑢 =
0 has a solution in 𝐻. Let {𝛼𝑛 }𝑛∈N and {𝜃𝑛 }𝑛∈N be sequences
in (0, 1]. Let {𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N be sequences in 𝐻 defined
iteratively from arbitrary 𝑢1 , V1 ∈ 𝐻 by
V𝑛+1 = V𝑛 − 𝛼𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜃𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
(46)
Suppose that one of the following conditions holds:
(
(i) 𝑛lim
→∞
𝛼𝑛2
) = lim 𝛼𝑛 = 0;
𝑛→∞
𝜃𝑛
(47)
𝛼
( 𝑛 ) = 0.
(ii) 𝑛lim
→∞ 𝜃
𝑛
Then, the sequences {𝑢𝑛 }𝑛∈N , {V𝑛 }𝑛∈N , {𝐹𝑢𝑛 }𝑛∈N , and {𝐾V𝑛 }𝑛∈N
are bounded.
Following the similar argument as the proof of
Theorem 12, we get the following result.
Theorem 23. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be bounded and monotone mappings. Suppose that 𝑢+𝐾𝐹𝑢 = 0
has a solution in 𝐻. Let {𝜆 𝑛 }𝑛∈N be a sequence in (0, 1). Let
{𝛼𝑛 }𝑛∈N and {𝜃𝑛 }𝑛∈N be sequences in (0, 1]. Let {𝑢𝑛 }𝑛∈N and
{V𝑛 }𝑛∈N be sequences in 𝐻 defined iteratively from arbitrary 𝑢1 ,
V1 ∈ 𝐻 by
𝑢𝑛+1 = 𝑢𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜆 𝑛 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜆 𝑛 𝜃𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
(48)
Assume that
(i) ∑∞
𝑛=1 𝜆 𝑛 𝜃𝑛 = ∞; lim𝑛 → ∞ 𝜆 𝑛 𝛼𝑛 = lim𝑛 → ∞ 𝜆 𝑛 𝜃𝑛 = 0;
(ii) one of the following conditions holds:
𝛼2
( 𝑛 ) = 0;
(a) 𝑛lim
→∞ 𝜃
𝑛
(
(b) 𝑛lim
→∞
𝜆𝑛
) = 0;
𝜃𝑛
(
(c) 𝑛lim
→∞
𝛼𝑛
) = 0;
𝜃𝑛
∞
(d) ∑ 𝜆2𝑛 𝛼𝑛2 < ∞;
𝑛=1
∞
∑ 𝜆2𝑛 𝛼𝑛 𝜃𝑛 < ∞;
𝑛=1
∞
If lim𝑛 → ∞ 𝛽𝑛 = 0, then the sequences {𝑢𝑛 }𝑛∈N , {V𝑛 }𝑛∈N ,
{𝐹𝑢𝑛 }𝑛∈N , and {𝐾V𝑛 }𝑛∈N are bounded.
𝑢𝑛+1 = 𝑢𝑛 − 𝛼𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
(iii) one of the following conditions holds:
(e) ∑ 𝜆2𝑛 𝛼𝑛 < ∞;
(50)
𝑛=1
(
(f) 𝑛lim
→∞
𝜆 𝑛 𝛼𝑛2
) = 0.
𝜃𝑛
Then, there exists a subset 𝐾min of 𝐻 × 𝐻 such that if (𝑢, V) ∈
𝐾min with V = 𝐹𝑢, then the sequence {𝑢𝑛 } converges strongly to
𝑢.
Remark 24. (i) The conclusion of Theorem 23 is still true if
2
∑∞
𝑛=1 𝜆 𝑛 𝜃𝑛 = ∞, lim𝑛 → ∞ (𝛼𝑛 /𝜃𝑛 ) = 0, and lim𝑛 → ∞ 𝜆 𝑛 𝛼𝑛 =
lim𝑛 → ∞ 𝜆 𝑛 𝜃𝑛 = 0. Furthermore, the conclusion of Theorem 23
is still true if ∑∞
𝑛=1 𝜆 𝑛 𝜃𝑛 = ∞, lim𝑛 → ∞ (𝛼𝑛 /𝜃𝑛 ) = 0, and
lim𝑛 → ∞ 𝜆 𝑛 = 0.
(ii) Theorem 23 improves the conditions of [9, Theorem
3.2] if the space 𝐸 is reduced to a real Hilbert space.
Indeed, [9, Theorem 3.2] assumes that lim𝑛 → ∞ (𝜆 𝑛 /𝜃𝑛 ) =
lim𝑛 → ∞ (𝛼𝑛 /𝜃𝑛 ) = 0.
The following is a special case of Theorem 23.
Corollary 25. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 :
𝐻 → 𝐻 be bounded and monotone mappings. Suppose that
𝑢 + 𝐾𝐹𝑢 = 0 has a solution in 𝐻. Let {𝛽𝑛 }𝑛∈N be a sequence
in (0, 1). Let {𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N be sequences in 𝐻 defined
iteratively from arbitrary 𝑢1 , V1 ∈ 𝐻 by
𝑢𝑛+1 = 𝑢𝑛 − 𝛽𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝛽𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝛽𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝛽𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
Assume that ∑∞
𝑛=1 𝛽𝑛 = ∞ and lim𝑛 → ∞ 𝛽𝑛 = 0. Then, there
exists a subset 𝐾min of 𝐻 × 𝐻 such that if (𝑢, V) ∈ 𝐾min with
V = 𝐹𝑢, then the sequence {𝑢𝑛 } converges strongly to 𝑢.
The following is also a special case of Theorem 23.
Corollary 26. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be bounded and monotone mappings. Suppose that 𝑢+𝐾𝐹𝑢 = 0
has a solution in 𝐻. Let {𝜆 𝑛 }𝑛∈N be a sequence in (0, 1). Let
{𝜃𝑛 }𝑛∈N be a sequence in (0, 1]. Let {𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N be
sequences in 𝐻 defined iteratively from arbitrary 𝑢1 , V1 ∈ 𝐻
by
𝑢𝑛+1 = 𝑢𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜆 𝑛 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝜆 𝑛 𝛼𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜆 𝑛 𝜃𝑛 (V𝑛 − V1 ) ,
(49)
(51)
𝑛 ∈ N.
(52)
Assume that ∑∞
𝑛=1 𝜆 𝑛 𝜃𝑛 = ∞, lim𝑛 → ∞ 𝜆 𝑛 𝜃𝑛 = 0, and
lim𝑛 → ∞ (𝜆 𝑛 /𝜃𝑛 ) = 0. Then, there exists a subset 𝐾min of 𝐻 × 𝐻
such that if (𝑢, V) ∈ 𝐾min with V = 𝐹𝑢, then the sequence {𝑢𝑛 }
converges strongly to 𝑢.
8
Journal of Applied Mathematics
Furthermore, we get the following result. Note that
Corollary 25 is also a special case of the following result.
Theorem 27. Let 𝐻 be a real Hilbert space. Let 𝐹, 𝐾 : 𝐻 → 𝐻
be bounded and monotone mappings. Suppose that 𝑢 + 𝐾𝐹𝑢 =
0 has a solution in 𝐻. Let {𝛼𝑛 }𝑛∈N and {𝜃𝑛 }𝑛∈N be sequences
in (0, 1]. Let {𝑢𝑛 }𝑛∈N and {V𝑛 }𝑛∈N be sequences in 𝐻 defined
iteratively from arbitrary 𝑢1 , V1 ∈ 𝐻 by
𝑢𝑛+1 = 𝑢𝑛 − 𝛼𝑛 (𝐹𝑢𝑛 − V𝑛 ) − 𝜃𝑛 (𝑢𝑛 − 𝑢1 ) ,
V𝑛+1 = V𝑛 − 𝛼𝑛 (𝐾V𝑛 + 𝑢𝑛 ) − 𝜃𝑛 (V𝑛 − V1 ) ,
𝑛 ∈ N.
(53)
2
Assume that ∑∞
𝑛=1 𝜃𝑛 = ∞, lim𝑛 → ∞ 𝜃𝑛 = 0, and lim𝑛 → ∞ (𝛼𝑛 /
𝜃𝑛 ) = 0. Then, there exists a subset 𝐾min of 𝐻 × 𝐻 such that
if (𝑢, V) ∈ 𝐾min with V = 𝐹𝑢, then the sequence {𝑢𝑛 } converges
strongly to 𝑢.
References
[1] A. Hammerstein, “Nichtlineare integralgleichungen nebst anwendungen,” Acta Mathematica, vol. 54, no. 1, pp. 117–176, 1930.
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[3] H. Brezis and F. E. Browder, “Existence theorems for nonlinear
integral equations of Hammerstein type,” Bulletin of the American Mathematical Society, vol. 81, pp. 73–78, 1975.
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