Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 487545, 6 pages
http://dx.doi.org/10.1155/2013/487545
Research Article
Multivalued Variational Inequalities with 𝐷𝐽-Pseudomonotone
Mappings in Reflexive Banach Spaces
A. M. Saddeek1 and S. A. Ahmed1,2
1
2
Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
Department of Mathematics, University College, Umm Al-Qura University, Saudi Arabia
Correspondence should be addressed to A. M. Saddeek; a m saddeek@yahoo.com
Received 28 August 2012; Accepted 31 January 2013
Academic Editor: Feyzi Başar
Copyright © 2013 A. M. Saddeek and S. A. Ahmed. 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.
This paper is concerned with the study of a class of variational inequalities with multivalued D𝐽 -pseudomonotone mappings in
reflexive Banach spaces by using the D𝐽 -antiresolvent technique. An application to the multivalued nonlinear D𝐽 -complementarity
problem is also presented. The results coincide with the corresponding results announced by many others for the gradient state.
1. Introduction and Preliminaries
Variational inequalities give a convenient mathematical
framework for discussing a large variety of interesting problems appearing in pure and applied sciences. It is well
known that the theory of pseudomonotone mappings plays
an important part in the study of the above-mentioned
variational inequalities.
In recent years, pseudomonotone theory has become an
attractive field for many mathematicians (see [1–8]).
In a very recent paper [9], by using the 𝐷𝐽 -antiresolvent
technique (where 𝐽 is the duality mapping) devised by the first
author, the author introduced a new concept of monotonicity,
which is called the 𝐷𝐽 -pseudomonotone type.
In the present paper, the concept of multivalued 𝐷𝐽 pseudomonotone mappings in reflexive Banach spaces is
used to study a wide class of variational inequalities, called the
multivalued 𝐷𝐽 -pseudomonotone variational inequalities.
Moreover, the results obtained in this paper can be
applied to the multivalued nonlinear 𝐷𝐽 -complementarity
problem. This problem contains, in particular, a mathematical model arising in the study of the postcritical equilibrium
state of a thin plate resting, without friction on a flat
rigid support (see [10–12]). The results coincide with the
corresponding results (see [2, 13–15]) in the case of gradient
mappings.
Unless otherwise stated, 𝑉 stands for a real reflexive
Banach space with norm ‖⋅‖𝑉 and 𝑉⋆ stands for the uniformly
convex dual of 𝑉 with the dual norm ‖ ⋅ ‖𝑉⋆ . The duality
pairing between 𝑉 and 𝑉⋆ is denoted by ⟨⋅, ⋅⟩. The set of
all nonnegative integers is denoted by N. The field of real
(resp., positive real) numbers is denoted by R (resp., R+ ).
Notation “ → ” stands for strong convergence and “⇀” for
weak convergence.
A mapping 𝐽 : 𝑉 → 𝑉⋆ is said to be a duality mapping
(see, e.g., [16]) with gauge function Φ (i.e., Φ is continuous
strictly increasing real-valued function satisfying Φ(0) = 0
and lim𝑡 → +∞ Φ(𝑡) = +∞) if for every 𝑢 ∈ 𝑉, ⟨𝐽𝑢, 𝑢⟩ =
‖𝐽𝑢‖𝑉⋆ ‖𝑢‖𝑉 = Φ(‖𝑢‖𝑉)(‖𝑢‖𝑉). If 𝑉 = 𝐻 is a Hilbert space,
then 𝐽 ≡ 𝐼, the identity mapping.
Assume that 𝑉⋆ has a weakly sequentially continuous
duality mapping 𝐽 (i.e., if {𝑢𝑛 }𝑛∈N is a sequence in 𝑉 which
weakly convergent to a point 𝑢, then the sequence {𝐽(𝑢𝑛 )}𝑛∈N
converges to 𝐽(𝑢) (see, e.g., [17])).
Let 𝑔 : 𝑉 → R ∪ {+∞} be a function. The domain of
𝑔 is dom 𝑔 = {𝑢 ∈ 𝑉 : 𝑔(𝑢) < +∞}. When dom 𝑔 ≠ 𝜙, 𝑔 is
called proper (see, e.g., [18]). The interior of the domain of 𝑔
is denoted by int dom 𝑔. The function 𝑔 is said to be Ĝateaux
differentiable at 𝑢 ∈ int dom 𝑔 (see, e.g., [18]), if
𝑔󸀠 (𝑢, 𝜂) = lim
𝑡→0
exists for all 𝜂 ∈ 𝑉.
𝑔 (𝑢 + 𝑡𝜂) − 𝑔 (𝑢)
𝑡
(1)
2
Abstract and Applied Analysis
Let 𝑔 be proper, convex, lower semicontinuous, and
Ĝateaux differentiable at 𝑢 ∈ int dom 𝑔; then the gradient of
𝑔 at 𝑢 is the function ∇𝑔(𝑢) which is defined by ⟨∇𝑔(𝑢), 𝜂⟩ =
𝑔󸀠 (𝑢, 𝜂) for any 𝜂 ∈ 𝑉. It is known (see, e.g., [19]) that the
conjugate 𝑔⋆ : 𝑉⋆ → R ∪ {+∞} is also proper, convex, and
lower semicontinuous.
The convex function 𝑔 is said to be of Legendre type (see,
e.g., [20]) if the following conditions hold:
(𝐿 1 ) int dom(𝑔) ≠ 𝜙, 𝑔 is Ĝateaux differentiable on
int dom(𝑔) and dom ∇𝑔 = int dom 𝑔;
⋆
⋆
(𝐿 2 ) int dom(𝑔 ) ≠ 𝜙, 𝑔 is Ĝateaux differentiable on
int dom(𝑔⋆ ) and dom ∇𝑔⋆ = int dom(𝑔⋆ ).
It is well known (see, e.g., [21]) that if 𝑔 is a proper, convex,
lower semicontinuous, and Legendre type, then ∇𝑔⋆ =
(∇𝑔)−1 and range ∇𝑔⋆ = dom ∇𝑔.
Throughout this paper, the function 𝑔 : 𝑉 → R+ ∪ {+∞}
is proper, convex, and lower semicontinuous which is also
Legendre on int dom(𝑔).
The Bregman distance (see, e.g., [22]) is the function 𝐷𝑔 :
𝑉 × int dom(𝑔) → R ∪ {+∞} defined by
𝐷𝑔 (], 𝑢) = 𝑔 (]) − 𝑔 (𝑢) − ⟨∇𝑔 (𝑢) , ] − 𝑢⟩ ,
(2)
dom 𝐷𝑔 = (dom 𝑔) × int dom 𝑔.
(3)
with
It should be pointed out that if 𝑉 = 𝐻 is a Hilbert space and
𝑔 = (1/2)‖ ⋅ ‖2𝐻, then ∇𝑔 = 𝐼 (the identity mapping) and
𝐷𝐽 (], 𝑢) = (1/2)‖] − 𝑢‖2𝐻.
⋆
For a multivalued mapping 𝑇 : 𝑉 → 2𝑉 , the associated
𝐷𝐽 -antiresolvent (where 𝐽 : 𝑉 → 𝑉⋆ is the duality mapping)
of 𝐽 − 𝑇 (see [9]) is the mapping 𝑇𝐽 : 𝑉 → 2𝑉, defined by
𝑇𝐽 = ∇𝑔⋆ (𝐽 − 𝑇) .
(4)
Such a mapping is known as (see [23]) a 𝐷𝑔 -antiresolvent
mapping of 𝑇 when 𝐽 = ∇𝑔 (in this case, the mapping 𝑇𝐽
is denoted by 𝑇𝑔 ).
In light of the above-mentioned discussion, we note that
if 𝐽 − 𝑇 = ∇𝑔, then 𝑇𝐽 is the identity mapping 𝐼.
⋆
Following [9], the mapping 𝐽 − 𝑇 : 𝑉 → 2𝑉 is said to be
𝐷𝐽 -pseudomonotone, if for every 𝑢, 𝜂 ∈ dom 𝐽 ∩ dom 𝑇, ] ∈
(𝐽 − 𝑇)(𝑢), 𝜔 ∈ (𝐽 − 𝑇)(𝜂), and every sequence {𝑢𝑛 }𝑛∈N ⊂
dom 𝐽 ∩ dom 𝑇 and ]𝑛 ∈ (𝐽 − 𝑇)(𝑢𝑛 ) the conditions
∇𝑔⋆ (]𝑛 ) ⇀ ∇𝑔⋆ (]) ,
lim sup ⟨]𝑛 , ∇𝑔⋆ (]𝑛 ) − ∇𝑔⋆ (])⟩ ≤ 0
(5)
𝑛→∞
𝐷𝑔 -pseudomonotonicity in the sense of Bregman distance
𝐷𝑔 ) of the mapping ∇𝑔, if 𝐽 − 𝑇 = ∇𝑔.
The multivalued variational inequality defined by the 𝐷𝐽 mapping (or multivalued 𝐷𝐽 -variational inequality) 𝐽 − 𝑇 :
⋆
𝑉 → 2𝑉 and the set 𝐾 ⊂ 𝑉 is to find 𝑢 ∈ 𝐾 such that
∃] ∈ (𝐽 − 𝑇) (𝑢) , 𝜔
∈ (𝐽 − 𝑇) (𝜂) :
⟨] − 𝑓, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (])⟩ ≥ 0,
(7)
∀𝜂 ∈ 𝐾,
⋆
where 𝑓 ∈ 2𝑉 .
The multivalued nonlinear complementarity problem
defined by the 𝐷𝐽 -mapping (or multivalued nonlinear 𝐷𝐽 ⋆
complementarity problem) 𝐽 − 𝑇 : 𝑉 → 2𝑉 and the set
𝐾 is to find 𝑢 ∈ 𝐾 such that
⟨] − 𝑓, ∇𝑔⋆ (])⟩ = 0,
⟨] − 𝑓, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (])⟩ ≥ 0,
∀𝜂 ∈ 𝐾,
(8)
⋆
where 𝑓 ∈ 2𝑉 , ] ∈ (𝐽 − 𝑇)(𝑢), and 𝜔 ∈ (𝐽 − 𝑇)(𝜂).
The multivalued 𝐷𝐽 -variational inequality and multivalued nonlinear 𝐷𝐽 -complementarity problem are very general
in the sense that they include, as special cases, multivalued
variational inequality and multivalued nonlinear complementarity problem.
The following definition and results will be used in the
sequel.
Definition 1 (see, e.g., [15, p. 84]). The mapping 𝐴 : 𝑉 →
⋆
2𝑉 is continuous on finite dimensional subspaces if for any
finite dimensional subspace 𝑉0 ⊂ 𝑉, the restriction of 𝐴 to
𝑉0 ∩ dom(𝐴) is weakly continuous.
Corollary 2 (see [24]). Let 𝑗 : 𝑉0 ⊂ 𝑉 → 𝑉 be the injection
mapping. Let 𝑗⋆ : 𝑉⋆ → 𝑉0⋆ be its dual mapping. Then, 𝑗⋆ 𝐴𝑗 :
⋆
dom(𝐴) ∩ 𝑗𝑉0 → 2𝑉0 is continuous.
Corollary 3 (see [25]). Let 𝐾 be a nonempty compact convex
set of R𝑛 and let 𝑆 : 𝐾 → 2𝐾 be continuous. Then 𝑆 admits a
fixed point.
2. Main Results
Theorem 4. Let 𝐾 be a closed convex set in 𝐻 and let 𝑇 :
𝐾 → 2𝐻 be a multivalued mapping. Then the following are
equivalent:
(1) ∇𝑔⋆ (]󸀠 ) ∈ 𝑃𝑟𝐾 (𝑥)
= arg min {𝐷𝐼 (∇𝑔⋆ (𝜔󸀠 ) , 𝑥)
imply that
lim inf ⟨]𝑛 , ∇𝑔⋆ (]𝑛 ) − ∇𝑔⋆ (𝜔)⟩ ≥ ⟨𝜔, ∇𝑔⋆ (]) − ∇𝑔⋆ (𝜔)⟩ .
𝑛→∞
(6)
As remarked in [9], the 𝐷𝐽 -pseudomonotonicity of the
mapping 𝐽−𝑇 coincides with the pseudomonotonicity (or the
1󵄩
󵄩2
= 󵄩󵄩󵄩󵄩∇𝑔⋆ (𝜔󸀠 ) − 𝑥󵄩󵄩󵄩󵄩𝐻 : ∇𝑔⋆ (𝜔󸀠 )
2
∈ 𝐾, 𝜔󸀠 ∈ (𝐼 − 𝑇) (𝜂) , 𝜂 ∈ 𝐾} ,
(9)
Abstract and Applied Analysis
3
the multivalued projection for 𝐾;
Then there exists a solution to the multivalued 𝐷𝐽 -variational
inequality (7).
(2) ∇𝑔⋆ (]󸀠 ) ∈ 𝐾 : ⟨∇𝑔⋆ (]󸀠 ) − 𝑥, ∇𝑔⋆ (𝜔󸀠 )
−∇𝑔⋆ (]󸀠 )⟩ ≥ 0
∀∇𝑔⋆ (𝜔󸀠 ) ∈ 𝐾,
(10)
where 𝜔󸀠 ∈ (𝐼 − 𝑇) (𝜂) ,
]󸀠 ∈ (𝐼 − 𝑇) (𝑦) , 𝑦, 𝜂 ∈ 𝐾.
Proof. Assume that (1) holds. Let 𝑥 ∈ 2𝐻 and ∇𝑔⋆ (]󸀠 ) ∈
𝑃𝑟𝐾 (𝑥) ⊂ 𝐾. For every ∇𝑔⋆ (𝜔󸀠 ) ∈ 𝐾 and 𝑡 ∈ (0, 1], we have
𝐷𝐼 (𝑥, ∇𝑔⋆ (]󸀠 ))
⋆
󸀠
⋆
⋆
⟨𝑠𝑚 − 𝑗𝑚
𝑓, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (]𝑚 )⟩ ≥ 0.
󸀠
≤ 𝐷𝐼 (𝑥, (1 − 𝑡) ∇𝑔 (] ) + 𝑡∇𝑔 (𝜔 ))
= 𝐷𝐼 (𝑥, ∇𝑔⋆ (]󸀠 ))
(11)
− 𝑡 ⟨𝑥 − ∇𝑔⋆ (]󸀠 ) , ∇𝑔⋆ (𝜔󸀠 ) − ∇𝑔⋆ (]󸀠 )⟩
2
⋆
󸀠
⋆
+ 𝑡 𝐷𝐼 (∇𝑔 (𝜔 ) , ∇𝑔 (] )) .
⋆
𝑓 − 𝑠𝑚 , ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (]𝑚 )⟩ .
≥ ⟨∇𝑔⋆ (]𝑚 ) + 𝑗𝑚
This implies
⟨𝑥 − ∇𝑔⋆ (]󸀠 ) , ∇𝑔⋆ (𝜔󸀠 ) − ∇𝑔⋆ (]󸀠 )⟩
(12)
Hence, 𝑡 → 0+ implies (2).
On the other hand, assume that (2) holds. For every
∇𝑔⋆ (𝜔󸀠 ) ∈ 𝐾, we have
𝐷𝐼 (∇𝑔⋆ (𝜔󸀠 ) , 𝑥)
⋆
󸀠
⋆
󸀠
= 𝐷𝐼 (∇𝑔 (𝜔 ) , ∇𝑔 (] ))
+ ⟨∇𝑔⋆ (𝜔󸀠 ) − 𝑥, ∇𝑔⋆ (𝜔󸀠 ) − ∇𝑔⋆ (]󸀠 )⟩
(13)
+ 𝐷𝐼 (∇𝑔⋆ (]󸀠 ) , 𝑥) ≥ 𝐷𝐼 (∇𝑔⋆ (]󸀠 ) , 𝑥) .
This implies (1).
Remark 5. In the particular situation when 𝐼 − 𝑇 = ∇𝑔
Theorem 4 coincides (in gradient setting) with Theorem 2.3
in [15] and also with Proposition 2.1 (1) and (2) in [2].
Theorem 6. In addition to conditions on 𝑉, 𝑉⋆ , 𝑔⋆ , and 𝐽, one
assumes that 𝑉 is separable, 𝐾 ⊂ 𝑉 is nonempty closed and
⋆
convex, 𝑇 : 𝐾 → 2𝑉 , ∇𝑔⋆ are weakly continuous mappings,
and either 𝑇 or ∇𝑔⋆ is continuous. Moreover, assume that the
⋆
mapping 𝐽 − 𝑇 : 𝐾 → 2𝑉 is a bounded 𝐷𝐽 -pseudomonotone
mapping and that, for each 𝜂 ∈ 𝐾, there exist 𝜂0 ∈ 𝐾, 𝜔0 ∈
(𝐽 − 𝑇)(𝜂0 ), 𝜔 ∈ (𝐽 − 𝑇)(𝜂), and 𝑟 > 0 such that
⟨𝜔, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (𝜔0 )⟩ > ⟨𝑓, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (𝜔0 )⟩ ,
⋆
𝑓 ∈ 2𝑉 ,
󵄩
󵄩󵄩 ⋆
󵄩󵄩∇𝑔 (𝜔)󵄩󵄩󵄩𝑉 ≥ 𝑟.
(15)
The equivalent form of problem (15) is to find 𝑢𝑚 ∈ 𝐾𝑚 such
⋆
(𝐽 − 𝑇)𝑗𝑚 (𝑢𝑚 ), ]𝑚 ∈
that for each 𝜂 ∈ 𝐾𝑚 , there exist 𝑠𝑚 ∈ 𝑗𝑚
(𝐽 − 𝑇)(𝑢𝑚 ), and 𝜔 ∈ (𝐽 − 𝑇)(𝜂) such that
⟨∇𝑔⋆ (]𝑚 ) , ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (]𝑚 )⟩
󸀠
≤ 𝑡𝐷𝐼 (∇𝑔⋆ (𝜔󸀠 ) , ∇𝑔⋆ (]󸀠 )) .
Proof. Suppose that {𝑥1 , 𝑥2 , 𝑥3 , . . .} is an infinite dense set in
𝐾 and 𝑉𝑚 , 𝑚 ∈ N, is the linear span of {𝑥1 , 𝑥2 , 𝑥3 , . . . , 𝑥𝑚 }.
Let 𝐾𝑚 = con{𝑥1 , 𝑥2 , 𝑥3 , . . . , 𝑥𝑚 } = {∑𝑚
𝑖=1 𝜆 𝑖 𝑥𝑖 , 𝜆 𝑖 ≥
0, ∑𝑚
𝑖=1 𝜆 𝑖 = 1}. Let 𝑗𝑚 : 𝑉𝑚 → 𝑉 be the injection mapping
⋆
: 𝑉⋆ → 𝑉𝑚⋆ be its restriction dual. Observe that
and let 𝑗𝑚
∪𝐾𝑚 is dense in 𝐾, for 𝑚 ∈ N.
Now, fix an integer 𝑚 ≥ 1 and consider the finite
dimensional problem.
Find 𝑢𝑚 ∈ 𝐾𝑚 such that for each 𝜂 ∈ 𝐾𝑚 , there exist
⋆
(𝐽 − 𝑇)𝑗𝑚 (𝑢𝑚 ), ]𝑚 ∈ (𝐽 − 𝑇)(𝑢𝑚 ), and 𝜔 ∈ (𝐽 − 𝑇)(𝜂)
𝑠𝑚 ∈ 𝑗𝑚
such that
(16)
Using the identification of 𝑉𝑚 with R𝑚 and 𝑉𝑚⋆ and Theorem 4 (with R𝑚 = 𝐻 and 𝐽 = 𝐼), we see that (16) is equivalent
⋆
𝑓 − 𝑠𝑚 ).
to ∇𝑔⋆ (]𝑚 ) ∈ 𝑃𝑟𝐾𝑚 (∇𝑔⋆ (]𝑚 ) + 𝑗𝑚
Let 𝐵𝑟 (0) be any closed ball containing 𝐾𝑚 . It is well
known (see, e.g., [26, p. 54, 224]) that 𝐵𝑟 (0) is compact and
convex in 𝐾𝑚 ; thus it is weakly closed.
From Corollary 2, 𝑗⋆ (𝐽 − 𝑇)𝑗 : 𝐵𝑟 (0) → 2𝐵𝑟 (0) is
continuous; hence the function ∇𝑔⋆ (]) 󳨃→ 𝑃𝑟𝐵𝑟 (0) (∇𝑔⋆ (]) +
⋆
𝑗𝑚
𝑓 − 𝑠𝑚 ) is continuous from 𝐵𝑟 (0) into 2𝐵𝑟 (0) .
Hence, by Corollary 3, this equation admits a solution. If
the closed convex set 𝐾 is assumed to be bounded, then by
the reflexivity of 𝑉 it is weakly compact (by employing the
Banach -Alaoglu theorem (see, e.g., [16, p. 3]).
Then we have a subsequence denoted by {𝑢𝑚 }𝑚∈N such
that 𝑢𝑚 ⇀ 𝑢 ∈ 𝐾. Since 𝐽 − 𝑇 is bounded, we have ‖]𝑚 ‖𝑉⋆ ≤
𝑀 for all 𝑚 ∈ N. Since 𝐽 − 𝑇 is weakly continuous and since
either 𝐽 − 𝑇 or ∇𝑔⋆ is continuous by hypothesis, it follows
that 𝑇𝐽 is weakly continuous by [27, Lemma 1]. So, we have
∇𝑔⋆ (]𝑚 ) ⇀ ∇𝑔⋆ (]).
Now, we prove that
lim sup ⟨]𝑚 , ∇𝑔⋆ (]𝑚 ) − ∇𝑔⋆ (])⟩ ≤ 0.
𝑚→∞
(17)
̃ ∈ 𝐾𝑁 such that
For any 𝜖 > 0, choose 𝑁 so large and 𝑢
󵄩
󵄩󵄩 ⋆
⋆
𝑢) .
󵄩󵄩∇𝑔 (]) − ∇𝑔 (̃])󵄩󵄩󵄩𝑉 < 𝜖 for ̃] ∈ (𝐽 − 𝑇) (̃
(18)
Therefore, we have
(14)
⟨]𝑚 − 𝑓, ∇𝑔⋆ (]𝑚 ) − ∇𝑔⋆ (̃])⟩ ≤ 0 for 𝑚 ≥ 𝑁.
(19)
4
Abstract and Applied Analysis
Since 𝐾𝑁 ⊂ 𝐾𝑚 , we have
⋆
Theorem 8. Let all assumptions of Theorem 6 hold, except for
condition (14) let it be replaced by the 𝐷𝐽 -coercive condition:
for 𝜔 ∈ (𝐽 − 𝑇)(𝜂),
⋆
lim sup ⟨]𝑚 , ∇𝑔 (]𝑚 ) − ∇𝑔 (])⟩
𝑚→∞
= lim sup [⟨]𝑚 , ∇𝑔⋆ (]𝑚 ) − ∇𝑔⋆ (̃])⟩
𝑚→∞
lim
⋆
(20)
‖∇𝑔 (𝜔)‖𝑉 → ∞
+ ⟨]𝑚 , ∇𝑔⋆ (̃]) − ∇𝑔⋆ (])⟩]
Since 𝜖 is arbitrary, this shows the desired inequality.
By the 𝐷𝐽 -pseudomonotonicity of 𝐽 − 𝑇, it follows that
lim inf ⟨]𝑚 , ∇𝑔⋆ (]𝑚 ) − ∇𝑔⋆ (𝜔)⟩ ≥ ⟨], ∇𝑔⋆ (]) − 𝑔⋆ (𝜔)⟩
𝑚→∞
(21)
for all 𝜂 ∈ dom 𝐽 ∩ dom 𝑇 and 𝜔 ∈ (𝐽 − 𝑇)(𝜂).
If 𝜂 ∈ 𝐾𝑛 , 𝑚 ≥ 𝑛, we have
Hence
(23)
for every 𝜂 in 𝐾𝑛 , 𝑛 ∈ N, 𝜔 ∈ (𝐽 − 𝑇)(𝜂).
Since ∪𝑛 𝐾𝑛 is dense in 𝐾𝑛 , so we have that 𝑢 is a solution
to (7).
Now, to complete the proof, we consider the case when 𝐾
is unbounded.
In this case we consider the set 𝐾𝜌 = {𝜂 ∈ 𝐾 :
‖∇𝑔⋆ (𝜔)‖𝑉 ≤ 𝜌}, where 𝜌 = max{‖∇𝑔⋆ (𝜔0 )‖𝑉, 𝑟}.
Since 𝐾𝜌 is bounded, there exists at least one 𝑢𝜌 ∈ 𝐾𝜌 :
⋆
⟨]𝜌 − 𝑓, ∇𝑔 (𝜔) − ∇𝑔 (]𝜌 )⟩ ≥ 0
𝜂, 𝜂0 ∈ 𝐾.
Suppose further that 𝐾 has the following property (W):
𝛼∇𝑔⋆ (𝜔) ∈ 𝐾 for all ∇𝑔⋆ (𝜔) ∈ 𝐾 and 𝛼 ≥ 0.
⋆
Then for every 𝑓 ∈ 2𝑉 there exist 𝑢 ∈ 𝐾, ] ∈ (𝐽 − 𝑇)(𝑢)
such that
⟨] − 𝑓, ∇𝑔⋆ (])⟩ = 0,
⟨] − 𝑓, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (])⟩ ≥ 0
(28)
⋆
Proof. Let 𝑓 ∈ 2𝑉 satisfy ‖𝑓‖𝑉⋆ < 𝑀 and
󵄩 󵄩
2𝑀 − 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉⋆
󵄩󵄩 ⋆
󵄩󵄩
∇𝑔
(𝜔
)
<
𝜌.
󵄩𝑉
󵄩󵄩 󵄩󵄩
0 󵄩
󵄩󵄩
󵄩󵄩𝑓󵄩󵄩𝑉⋆
(29)
The 𝐷𝐽 -coercivity of 𝐽 − 𝑇 implies that there exists 𝜌 > 0 such
that
󵄩
󵄩
⟨𝜔, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (𝜔0 )⟩ ≥ 2𝑀󵄩󵄩󵄩∇𝑔⋆ (𝜔)󵄩󵄩󵄩𝑉
(30)
for 𝜔 ∈ (𝐽 − 𝑇)(𝜂), 𝜂 ∈ 𝐾 with ‖∇𝑔⋆ (𝜔)‖𝑉 ≥ 𝜌.
So we conclude
⟨𝜔 − 𝑓, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (𝜔0 )⟩
󵄩 󵄩 󵄩 󵄩
󵄩 󵄩 󵄩 󵄩
󵄩
󵄩
≥ 2𝑀󵄩󵄩󵄩∇𝑔⋆ (𝜔)󵄩󵄩󵄩𝑉 − 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉⋆ 󵄩󵄩󵄩∇𝑔⋆ (𝜔)󵄩󵄩󵄩𝑉 − 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉⋆ 󵄩󵄩󵄩∇𝑔⋆ (𝜔0 )󵄩󵄩󵄩𝑉
󵄩 󵄩
󵄩 󵄩 󵄩
󵄩
≥ (2𝑀 − 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉⋆ ) 𝜌 − 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉⋆ 󵄩󵄩󵄩∇𝑔⋆ (𝜔0 )󵄩󵄩󵄩𝑉
(24)
for ]𝜌 ∈ (𝐽 − 𝑇)(𝑢𝜌 ) and 𝜂 ∈ 𝐾𝜌 .
Since 𝜂0 ∈ 𝐾𝜌 , we have
⋆
(27)
for all 𝜂 ∈ 𝐾, 𝜔 ∈ (𝐽 − 𝑇)(𝜂).
⟨]𝑚 , ∇𝑔⋆ (]𝑚 ) − ∇𝑔⋆ (𝜔)⟩ ≤ ⟨𝑓, ∇𝑔⋆ (]𝑚 ) − ∇𝑔⋆ (𝜔)⟩ .
(22)
⋆
⟨𝜔, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (𝜔0 )⟩
] = +∞,
󵄩
󵄩󵄩 ⋆
󵄩󵄩∇𝑔 (𝜔)󵄩󵄩󵄩𝑉
𝜔0 ∈ (𝐽 − 𝑇) (𝜂0 ) ,
󵄩 󵄩
≤ (󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉⋆ + 𝑀) 𝜖.
⟨], ∇𝑔⋆ (]) − ∇𝑔⋆ (𝜔)⟩ ≤ ⟨𝑓, ∇𝑔⋆ (]) − ∇𝑔⋆ (𝜔)⟩
[
󵄩
󵄩
> 0 for 𝜂 ∈ 𝐾, 𝜔 ∈ (𝐽 − 𝑇) (𝜂) 𝑤𝑖𝑡ℎ 󵄩󵄩󵄩∇𝑔⋆ (𝜔)󵄩󵄩󵄩𝑉 ≥ 𝜌.
(31)
⋆
⟨]𝜌 − 𝑓, ∇𝑔 (]𝜌 ) − ∇𝑔 (𝜔0 )⟩ ≤ 0 for 𝜔0 ∈ (𝐽 − 𝑇) (𝜂0 ) .
(25)
This, together with (14), implies that ‖∇𝑔⋆ (]𝜌 )‖𝑉 < 𝜌.
To clarify that 𝑢𝜌 is also a solution to original problem on
𝐾, for any 𝜂 ∈ 𝐾, set ∇𝑔⋆ (𝜔𝑡 ) = (1 − 𝑡)∇𝑔⋆ (]𝜌 ) + 𝑡∇𝑔⋆ (𝜔) for
𝑡 > 0 is sufficiently small, where 𝜔𝑡 ∈ (𝐽 − 𝑇)(𝜂𝑡 ) and 𝜂𝑡 ∈ 𝐾𝜌 .
Consequently
𝑢𝜌 ∈ 𝐾𝜌 ⊂ 𝐾 : 0 ≤ ⟨]𝜌 − 𝑓, ∇𝑔⋆ (𝜔𝑡 ) − ∇𝑔⋆ (]𝜌 )⟩
= 𝑡 ⟨]𝜌 − 𝑓, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (]𝜌 )⟩
(26)
for 𝜂 ∈ 𝐾.
The second part of (28) thus follows from Theorem 6.
To prove the first part of (28), observe that we can choose
a point 𝜂 in 𝐾 and 𝜔 ∈ (𝐽−𝑇)(𝜂) and assume that ∇𝑔⋆ (𝜔) = 0.
Therefore, from Theorem 6, we have
⟨] − 𝑓, ∇𝑔⋆ (])⟩ ≤ 0
for all ] ∈ (𝐽 − 𝑇)(𝑢), 𝑢 ∈ 𝐾.
On the other hand, setting ∇𝑔⋆ (𝜔) = 𝛼∇𝑔⋆ (]), where 𝛼 >
1, we get
0 ≤ ⟨] − 𝑓, ∇𝑔⋆ (𝜔) − ∇𝑔⋆ (])⟩ = (𝛼 − 1) ⟨] − 𝑓, ∇𝑔⋆ (])⟩ .
(33)
This implies
This completes the proof.
Remark 7. In the particular situation when 𝐽 − 𝑇 = ∇𝑔,
Theorem 6 coincides with the Brezis Theorem (see, e.g., [13,
14]) for the case of gradient mapping.
We are now in a position to state and prove the following
theorem.
(32)
⟨] − 𝑓, ∇𝑔⋆ (])⟩ ≥ 0.
(34)
⟨] − 𝑓, ∇𝑔⋆ (])⟩ = 0.
(35)
So,
This completes the proof of (28).
Abstract and Applied Analysis
5
The following proposition gives a characterization of the
sum of two 𝐷𝐽 -Pseudomonotone mappings.
3. Application to Multivalued Nonlinear 𝐷𝐽 Complementarity Problem
Proposition 9. Let 𝑉, 𝑉⋆ , and 𝐽 be as above and let 𝑇𝑖 : 𝑉 →
⋆
2𝑉 , 𝑖 = 1, 2, and ∇𝑔⋆ be weakly continuous mappings. If 𝐽 −
⋆
𝑇𝑖 : 𝑉 → 2𝑉 , 𝑖 = 1, 2, are 𝐷𝐽 -pseudomonotone mappings
such that dom 𝐽 ∩ dom 𝑇𝑖 ≠ 𝜙, 𝑖 = 1, 2, then ∑2𝑖=1 (𝐽 − 𝑇𝑖 ) is
𝐷𝐽 -pseudomonotone.
As applications of Theorem 8 we consider the multivalued
nonlinear 𝐷𝐽 -complementarity problem (8) with 𝑇 = 𝜆𝑇1 +
𝑇2 − 𝜆𝐽, where 𝑇𝑖 , 𝑖 = 1, 2 are two nonlinear multivalued
⋆
mappings from 𝐾 to 2𝑉 , and 𝜆 ∈ (0, ∞).
̃ 𝑛 ∈ ∑2𝑖=1 (𝐽 − 𝑇𝑖 )(𝜂𝑛 ), 𝑦
̃ ∈ ∑2𝑖=1 (𝐽 − 𝑇𝑖 )(𝜂), 𝜂𝑛 , 𝜂 ∈
Proof. Let 𝑦
𝑦𝑛 ) ⇀ ∇𝑔⋆ (̃
𝑦) and
dom 𝐽 ∩ dom 𝑇𝑖 , 𝑖 = 1, 2, with ∇𝑔⋆ (̃
𝑦𝑛 , ∇𝑔⋆ (̃
𝑦𝑛 ) − ∇𝑔⋆ (̃
𝑦)⟩ ≤ 0.
lim sup ⟨̃
𝑛→∞
(36)
Now, we prove for 𝑦𝑛(𝑖) ∈ (𝐽 − 𝑇𝑖 )(𝜂𝑛 ), 𝑦(𝑖) ∈ (𝐽 − 𝑇𝑖 )(𝜂), 𝑖 =
1, 2, that
lim sup ⟨𝑦𝑛(1) , ∇𝑔⋆ (𝑦𝑛(1) ) − ∇𝑔⋆ (𝑦(1) )⟩ ≤ 0,
𝑛→∞
lim sup ⟨𝑦𝑛(2) , ∇𝑔⋆ (𝑦𝑛(2) ) − ∇𝑔⋆ (𝑦(2) )⟩ ≤ 0.
Theorem 10. Let 𝑉, 𝑉⋆ , 𝑔⋆ , 𝐽, and 𝐾 be the same as in
Theorem 6, and suppose that 𝐾 has the property (W). Let the
mappings ∇𝑔⋆ and 𝑇𝑖 , 𝑖 = 1, 2 be weakly continuous and let
either ∇𝑔⋆ or both 𝑇𝑖 , 𝑖 = 1, 2 be continuous. Let 𝐽 − 𝑇𝑖 : 𝐾 →
⋆
2𝑉 , 𝑖 = 1, 2 be two bounded 𝐷𝐽 -pseudomonotone mappings.
Let
1
=
𝜌1
(37)
inf
𝑛∈𝐾
‖∇𝑔⋆ (𝜆𝑤1 +𝑤2 )‖𝑉 ≠ 0
( ⟨𝑤1 , ∇𝑔⋆ (𝜆𝑤1 + 𝑤2 )
−∇𝑔⋆ (𝜆𝑤0(1) + 𝑤0(2) )⟩
𝑛→∞
If
lim sup ⟨𝑦𝑛(2) , ∇𝑔⋆ (𝑦𝑛(2) ) − ∇𝑔⋆ (𝑦(2) )⟩ = 𝜖 > 0,
𝑛→∞
(38)
(note that otherwise, by symmetry), then there exists a
subsequence {𝑦𝑛(2)
}
⊂ {𝑦𝑛(2) }𝑛∈N such that
𝑘 𝑘∈N
lim sup ⟨𝑦𝑛(2)
, ∇𝑔⋆ (𝑦𝑛(2)
) − ∇𝑔⋆ (𝑦(2) )⟩ = 𝜖.
𝑘
𝑘
𝑘→∞
󵄩2 −1
󵄩
×(󵄩󵄩󵄩∇𝑔⋆ (𝜆𝑤1 + 𝑤2 )󵄩󵄩󵄩𝑉) ) ,
𝑎=
lim inf
( ⟨𝑤2 , ∇𝑔⋆ (𝜆𝑤1 + 𝑤2 )
𝑛∈𝐾
‖∇𝑔⋆ (𝜆𝑤1 +𝑤2 )‖𝑉 → ∞
−∇𝑔⋆ (𝜆𝑤0(1) + 𝑤0(2) )⟩
(39)
󵄩2 −1
󵄩
×(󵄩󵄩󵄩∇𝑔⋆ (𝜆𝑤1 + 𝑤2 )󵄩󵄩󵄩𝑉) ) ,
This implies that
lim sup ⟨𝑦𝑛(1)
, ∇𝑔⋆
𝑘
𝑘→∞
(𝑦𝑛(1)
)
𝑘
⋆
(1)
− ∇𝑔 (𝑦 )⟩
⋆
𝑏 = lim inf
𝑡→∞
⋆
= lim sup [ ⟨̃
𝑦𝑛𝑘 , ∇𝑔 (̃
𝑦𝑛𝑘 ) − ∇𝑔 (̃
𝑦)⟩
𝑘→∞
− ⟨𝑦𝑛(2)
, ∇𝑔⋆
𝑘
(𝑦𝑛(2)
)
𝑘
⋆
(40)
(2)
− ∇𝑔 (𝑦 )⟩]
≤ 0 − 𝜖.
From the 𝐷𝐽 -pseudomonotonicity of 𝐽−𝑇1 , we get for all 𝑦󸀠 ∈
(𝐽 − 𝑇1 )(𝜂󸀠 ), 𝜂󸀠 ∈ dom 𝐽 ∩ dom 𝑇1
(1)
⋆
(1)
⋆
󸀠
⟨𝑦 , ∇𝑔 (𝑦 ) − ∇𝑔 (𝑦 )⟩
, ∇𝑔⋆ (𝑦𝑛(1)
) − ∇𝑔⋆ (𝑦󸀠 )⟩ .
≤ lim inf ⟨𝑦𝑛(1)
𝑘
𝑘
(41)
𝑘→∞
Letting 𝑦 = 𝑦(1) , we obtain
0 ≤ lim inf
𝑘→∞
(𝑦𝑛(1)
)
𝑘
⋆
(1)
− ∇𝑔 (𝑦 )⟩ ≤ 0 − 𝜖,
Proof. By Proposition 9, 𝐽 − 𝑇 = 𝜆(𝐽 − 𝑇1 ) + (𝐽 − 𝑇2 ) is 𝐷𝐽 pseudomonotone for every 𝜆 ≥ 0. Set 𝜆 > 𝜌(𝑏 − 𝑎). Then
lim inf
󵄩2 −1
󵄩
×(󵄩󵄩󵄩∇𝑔⋆ (𝜆𝑤1 + 𝑤2 )󵄩󵄩󵄩𝑉) )
a contradiction.
Hence,
𝑘→∞
lim sup ⟨𝑦𝑛(2)
, ∇𝑔⋆ (𝑦𝑛(2)
) − ∇𝑔⋆ (𝑦(2) )⟩ ≤ 0.
𝑘
𝑘
≥
(43)
𝑘→∞
This holds for any subsequence, so (37) holds and the proof
follows immediately by the superadditivity of the lim inf.
( ⟨𝜆𝑤1 + 𝑤2 , ∇𝑔⋆ (𝜆𝑤1 + 𝑤2 )
−∇𝑔⋆ (𝜆𝑤0(1) + 𝑤0(2) )⟩
(42)
, ∇𝑔⋆ (𝑦𝑛(1)
) − ∇𝑔⋆ (𝑦(1) )⟩ ≤ 0,
lim sup ⟨𝑦𝑛(1)
𝑘
𝑘
Φ (𝑡)
𝑡
Be such that 𝑎 < 𝑏, where 𝑤𝑖 ∈ (𝐽 − 𝑇𝑖 )(𝜂), 𝑤0(𝑖) ∈ (𝐽 −
𝑇𝑖 )(𝜂0 ), 𝑖 = 1, 2, 𝜂, 𝜂0 ∈ 𝐾, 𝜌1 > 0 and Φ is the gauge
function. Then for every 𝜆 > 𝜌1 (𝑏 − 𝑎) problem (8) with
𝑇 = 𝜆𝑇1 + 𝑇2 − 𝜆𝐽 has a solution in 𝐾.
𝜂∈𝐾
‖∇𝑔⋆ (𝜆𝑤1 +𝑤2 )‖𝑉 → ∞
󸀠
⟨𝑦𝑛(1)
, ∇𝑔⋆
𝑘
(44)
(45)
𝜆
+ 𝑎 > 𝑏 > 0.
𝜌1
This implies that the mapping 𝐽 − 𝑇 = 𝜆(𝐽 − 𝑇1 ) + (𝐽 − 𝑇2 ) is
𝐷𝐽 -coercive.
The conclusion follows from Theorem 8.
6
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