#### A modified two-layer iteration via a boundary point approach to generalized multivalued pseudomonotone mixed variational inequalities

Saddeek Journal of Inequalities and Applications
A modified two-layer iteration via a boundary point approach to generalized multivalued pseudomonotone mixed variational inequalities
Ali Mohamed Saddeek
Most mathematical models arising in stationary filtration processes as well as in the theory of soft shells can be described by single-valued or generalized multivalued pseudomonotone mixed variational inequalities with proper convex nondifferentiable functionals. Therefore, for finding the minimum norm solution of such inequalities, the current paper attempts to introduce a modified two-layer iteration via a boundary point approach and to prove its strong convergence. The results here improve and extend the corresponding recent results announced by Badriev, Zadvornov and Saddeek (Differ. Equ. 37:934-942, 2001).
modified two-layer iteration; multivalued pseudomonotone mapping; generalized mixed variational inequalities; strong convergence; uniformly convex spaces
1 Introduction
Let V be a real Banach space, V ∗ be its dual space, · V∗ be the dual norm of the given
norm
· V , and ·, · be the duality pairing between V ∗ and V . Let M be a nonempty
closed convex subset of V . Let C(V ∗) be the family of nonempty compact subsets of V ∗.
Let H be a real Hilbert space with the inner product (·, ·) and the norm · H , respectively.
We denote by → and
strong and weak convergence, respectively. Let A : V → V ∗
be a nonlinear single-valued mapping.
Definition . (see [–]) For all u, η ∈ V , the mapping A : V → V ∗ is said to be as
follows:
(i) pseudomonotone, if it is bounded and for every sequence {un} ⊂ V such that
un
u ∈ V and lim sup Aun, un – u ≤
n→∞
imply
lim inf Aun, un – η ≥ Aη, u – η ;
n→∞
(ii) coercive, if there exists a function ρ : R+ → R+ with limξ→∞ ρ(ξ ) = +∞ such that
Au, u ≥ ρ u V
u V ;
(iii) potential, if
(iv) bounded Lipschitz continuous, if
Au – Aη V ∗ ≤ μ(R)
u – η V ,
A t(u + η) , u + η – A(tu), u dt =
A(u + tη), η dt;
where R = max{ u V , η V }, μ is a nondecreasing function on [, +∞), and is
the gauge function (i.e., it is a strictly increasing continuous function on [, +∞)
such that () = and limξ→∞ (ξ ) = +∞);
(v) uniformly monotone, if there exists a gauge such that
Au – Aη, u – η ≥
(vi) inverse strongly monotone, if there exists a constant γ > such that
If (ξ ) = ξ and μ(R) = γ > , in (iv), the mapping A is called γ -Lipschitzian mapping, and
if there exists α > such that (ξ ) = αξ , in (v), the mapping A is called strongly
monotone mapping. It is obvious that any inverse strongly monotone mapping is γ -Lipschitzian
mapping.
The single-valued pseudomonotone mixed variational inequality problem is formulated
as finding a point u ∈ M such that
Au, η – u + F(η) – F(u) ≥ f , η – u
∀η ∈ M,
where A : V → V ∗ is a single-valued pseudomonotone mapping, F : V → R ∪ {+∞} is a
proper convex and lower semicontinuous (but, in general, nondifferentiable) functional,
and f ∈ V ∗ is a given element.
Problem (.) is equivalent to finding u ∈ V such that
∈ Au – f + ∂F(u),
where ∂F(u) is the subdifferential of F, i.e.,
∂F(u) = u∗ ∈ V ∗ : F(η) – F(u) ≥ u∗, η – u ∀η ∈ V .
The interior of the domain of F is denoted by int(D(F)).
Such problems appear in many fields of physics (e.g., in hydrodynamics, elasticity or
plasticity), more specifically, when describing or analyzing the steady state filtration (see,
(.)
(.)
for example, [, –] and the references cited therein) and the problem of finding the
equilibrium of soft shells (see, for example, [, , –] and the references cited therein).
The existence of at least one solution to problem (.) can be guaranteed by
imposing pseudomonotonicity and coercivity conditions on the mapping A (see, for example,
[, ]).
If f = and F(u) = IM(u) ∀u ∈ M, where IM is the indicator functional of M defined by
u ∈ M such that IM(u) = +,∞, uo.∈wM,, then problem (.) is equivalent to finding u ∈ M such
that
(.)
(.)
Au, η – u ≥
∀η ∈ M,
Au = f .
which is known as the classical variational inequality problem firstly introduced and
studied by Stampacchia []. Problem (.) is equivalent to the following nonlinear operator
equation: find u ∈ M such that
A mapping J : V → V ∗ is called a duality mapping with gauge function if, for every
u ∈ V , Ju, u = ( u V ) u V and Ju V ∗ = ( u V ). If V = H, then the duality mapping
with the gauge function (ξ ) = ξ can be identified with the identity mapping of H into
itself.
It is well known (see, for example, [, ]) that J() = , J is odd, single-valued,
bijective and is uniformly continuous on bounded sets if V is a reflexive Banach space and V ∗
is uniformly convex; moreover, J– is also single-valued, bijective, and JJ– = IV ∗ , J–J = IV .
Therefore, we always assume that the dual space of a reflexive Banach space is uniformly
convex.
Remark . (see, for example, []) The single-valued duality mapping J is bounded
Lipschitz continuous and uniformly monotone.
In order to find a solution of problem (.), Badriev et al. [] suggested the following
two-layer iteration method: for an arbitrary u ∈ M, define un+ ∈ M as follows:
J(un+ – un), η – un+ + τ F(η) – F(un+) ≥ τ f – Aun, η – un+
where τ > is an iteration parameter and n ≥ .
In this way the original variational inequality problem (.) is thus reduced to another
variational inequality problem involving the duality mapping J instead of the original
pseudomonotone mapping A. Such a problem can then be solved by known methods (see, for
example, [, ]).
If V = H, then the iteration generated by (.) can be written in the following form:
(un+ – un, η – un+) + τ F(η) – F(un+) ≥ τ (f – Aun, η – un+) ∀η ∈ M,
(.)
for an arbitrary u ∈ M and τ > .
J(un+ – un) = τ (f – Aun),
n ≥ ,
where u is an arbitrary point in M and τ > .
In the case when V = H, iteration (.) can be written as follows:
un+ = un – τ (Aun – f ),
n ≥ ,
for τ > and u is an arbitrary point in M.
Saddeek and Ahmed [] proved some weak convergence theorems of iterations (.)
and (.) for approximating the solution of nonlinear equation (.).
Attempts to modify the two-layer iterations (.) and (.) so that strong convergence is
guaranteed have recently been made.
In [], Saddeek introduced the following modification of (.) in a Hilbert space H
(boundary point method):
un+ = un – τ h(un)(Aun – f ),
n ≥ ,
h(u) = inf α ∈ [, ] : αu ∈ M
∀u ∈ M.
where τ > , u is an arbitrary point in M, and h : M → [, ] is a function defined by He
and Zhu [] as follows:
In [], Saddeek and Ahmed considered the following two-layer iteration method for
solving the nonlinear operator equation (.) in a Banach space V :
w, η – u + F(η) – F(u) ≥ f , η – u
∀η ∈ M,
where A : V → C(V ∗) is a multivalued pseudomonotone mapping (see definition below),
F : V → R ∪ {+∞} is a functional as above, and f ∈ V ∗ is a given element.
Clearly, problems (.) and (.) are special cases of problem (.).
The set of all u ∈ M satisfying (.) is denoted by SOL(M, F, A – f ).
In [], Badriev et al. obtained the following weak convergence theorems using the
twolayer iteration (.).
He obtained strong convergence results for finding the minimum norm solution of
nonlinear equation (.).
In [], He and Zhu have observed that, if ∈/ M, calculating h(un) implies determining
h(un)un, a boundary point of M, so iteration (.) is known as the boundary point method.
In [], Saddeek extended the results of Saddeek [] to a uniformly convex Banach space
and introduced the following modification of the two-layer iteration (.) (boundary point
method):
Jun+ = Jun – τ h(un)(Aun – f ),
n ≥ ,
where τ > , u is an arbitrary point in M, τ > , and h is defined by (.).
In [], Noor introduced and studied the following generalized multivalued
pseudomonotone mixed variational inequality problem: find u ∈ M, w ∈ A(u) such that
(.)
(.)
(.)
(.)
(.)
(.)
Theorem . (see [], Theorem ) Let V be a real reflexive Banach space with a uniformly
convex dual space V ∗, and let J : V → V ∗ be the duality mapping. Let M be a nonempty
closed convex subset of V . Let A : V → V ∗ be a pseudomonotone, coercive, potential, and
bounded Lipschitz continuous mapping. Let F : V → R ∪ {+∞} be a proper convex and
γ -Lipschitzian (i.e., | F(u) – F(η) |≤ γ u – η V ∀u, η ∈ V , γ > ) functional. Define a
functional F : V → R ∪ {+∞} by
F(u) = F(u) + F(u) – f , u ,
F(u) =
A t(u) , u dt, f ∈ V ∗.
Assume also that
where
< τ < min , μ ,
μ = μ R +
–(R + γ ) ,
(.)
(.)
R = sup u V ,
u∈S
R = sup Au – f V ∗ ,
u∈S
S = u ∈ M : F(u) ≤ F(u) .
Then the sequence {un} defined by (.) is bounded in V , and all of its weak limit points are
solutions of problem (.).
Badriev et al. [] have remarked that, due to the reflexivity of V , the mixed variational
inequality (.) is solvable by Theorem ..
In Theorem ., the assumption that V is reflexive can be dropped. Indeed, if V ∗ is
uniformly convex, then V is uniformly smooth (and hence V is reflexive).
Theorem . (see [], Theorem ) Let V = H be a real Hilbert space, and let M be a
nonempty closed convex subset of H. Let A : H → H be a pseudomonotone, coercive,
potential, and inverse strongly monotone mapping. Let Fi : H → R ∪ {+∞}, i = , , be the
same as in Theorem ..
Then the sequence {un} defined by (.) with < τ < τ = γ converges weakly in H to a
solution of problem (.).
Some attempts to prove the weak convergence of the whole sequence in the framework
of Banach spaces have been made by Saddeek and Ahmed [] and Saddeek [, ].
Although the above mentioned theorems and all their extensions are unquestionably
interesting, only weak convergence theorems are obtained unless very strong assumptions
are made.
This suggests an important question: can the two-layer iteration method (.) be
modified to prove its strong convergence to the minimum norm solution of problem (.).
In this paper, inspired by [, ], and [], a generalized multivalued pseudomonotone
mixed variational inequality is considered, and a modified two-layer iteration via a
boundary point approach to find the minimum norm solution of such inequalities is introduced,
and its strong convergence is proved in the framework of uniformly convex spaces. The
results obtained in this paper improve and generalize the corresponding recent results
announced by [].
2 Definitions and preliminary
Definition . (see [, , ]) A multivalued mapping A : V → C(V ∗) is called
(i) pseudomonotone, if it is bounded and, for every sequence {un} ⊂ V , {wn} ⊂ A(un),
the conditions
un
u ∈ V
and
lim sup wn, un – u ≤
n→∞
imply that for every η ∈ V there exists w ∈ A(u) such that
lim inf wn, un – η ≥ w, u – η ;
n→∞
w, u ≥ ρ
u V
u V
∀u ∈ V , w ∈ A(u);
(ii) coercive, if there exists a function ρ : R+ → R+ with limξ→∞ ρ(ξ ) = +∞ such that
(iii) potential, if
w, u + η – w, u dt =
w, η dt
for all u, η ∈ V , w ∈ A(t(u + η)), w ∈ A(tu), w ∈ A(u + tη), t ∈ [, ];
(iv) bounded Lipschitz continuous, if
for all u, η ∈ V , w ∈ A(u), w´ ∈ A(η), where μ(R) and (ξ ) as above;
(v) inverse strongly monotone, if there exists a constant γ > such that
w – w´ V ∗ ≤ μ(R)
w – w´ , u – η ≥ γ w – w´ V
for all u, η ∈ V , w ∈ A(u), w´ ∈ A(η).
Definition . is an extension of Definition .((i)-(iv), (vi)) of single-valued mappings to
multivalued mappings.
Let G : M × V ∗ → R ∪ {+∞} be a functional defined as follows:
G(u, Jη) = u V – Jη, u + Jη V ∗ + F(u),
(.)
where u ∈ M, η ∈ V , Jη ∈ V ∗.
Definition . (see, for example, []) The mapping FM : V → C(M) is called
generalized F-projection mapping if FM (η) = arg minu∈M G(u, Jη), ∀η ∈ V .
If V = H and F(u) = ∀u ∈ M, then (.) reduces to the following simple form:
G(u, Jη) = u – η H ,
∀u ∈ M, η ∈ H,
and the generalized F-projection reduces to the projection
The following two lemmas are also useful in the sequel.
M from H to C(M).
Lemma . (see []) The generalized F-projection FM (η) has the following properties:
(i) FM (η) is a nonempty closed convex subset of M for all η ∈ V ;
(ii) for all η ∈ V , u¯ ∈ FM (η) if and only if
Jη – Ju¯ , u¯ – v + (F(v) – F(u¯ ) ≥
∀v ∈ M;
(iii) if V is strictly convex, then FM (η) is a single-valued mapping.
Let G : V × V → R+ ∪ {} be a functional defined as follows:
G(u, η) = u V – Jη, u + η V ,
∀u, η ∈ V .
(.)
Lemma . (see []) Let V be a real Banach space with a uniformly convex dual space V ∗,
let M be a nonempty closed convex subset of V , and let η ∈ V , u¯ ∈ FM (η). Then
(i) G(u, u¯ ) + G(u¯ , η) ≤ G(u, η) ∀u ∈ M;
(ii) for u, η ∈ V , G(u, η) = iff u = η.
A Banach space V is said to have the Kadec-Klee property (see, for example, []) if, for
every sequence {un} in V with un u and un V → u V together imply that limn→∞ un –
u V = .
Every Hilbert space is uniformly convex, and every uniformly convex Banach space has
the Kadec-Klee property.
3 Main results
In this section, we propose a modification of the two-layer iteration method (.) by the
boundary point method to establish strong convergence theorems of the modified
iteration for finding the minimum norm solution of the following generalized
pseudomonotone mixed variational inequality in uniformly convex spaces: find u ∈ M, w ∈ A(u) such
that
(.)
(.)
h(w), η – u + F(η) – F(u) ≥ f , η – u
∀η ∈ M,
where A, F, f are defined as above and h is a positive constant.
3.1 The modified two-layer iteration
For an arbitrary point u ∈ M, define un+ ∈ M as follows:
where wn ∈ A(un), and τ , J , h are defined as above.
Observe that iteration (.) is a modification and generalization of iterations (.)
and (.).
Jun+ – Jun, η – un+ + τ F(η) – F(un+) ≥ τ f – h(un)wn, η – un+
∀η ∈ M, (.)
where τ > is the iteration parameter, n ≥ , J is the duality mapping, wn ∈ A(un) and h
is defined by (.).
For M = V , F(u) = ∀u ∈ M, and η = un+ ± z, z ∈ M, (.) is equivalent to
Jun+ = Jun – τ h(un)wn – f ,
∀n ≥ ,
If V = H, A is a single-valued mapping in (.) and h(un) = ∀n ≥ , we have
iteration (.).
Iteration (.) can be considered as a modified method for solving the following operator
inclusion problem: find u ∈ V such that
f ∈ Au, f ∈ V ∗.
For each u ∈ V , w ∈ A(tu), let F˜ : V → R ∪ {+∞} be a functional defined by
F˜ (u) = F˜(u) + F(u) – f , u ,
F˜(u) =
h(u)w, u dt, f ∈ V ∗.
Let us assume also that
R˜ = sup u V ,
u∈S˜
R˜ = sup h(u)w – f V ∗ ,
u∈S˜
S˜ = u ∈ M : F˜ (u) ≤ F˜ (u) ,
where w ∈ A(u).
Let μ˜ be a positive constant such that
μ˜ = μ R˜ +
–(R˜ + γ ) .
(.)
(.)
(.)
(.)
Theorem . Let V be a real uniformly convex Banach space with a uniformly convex dual
space V ∗, J : V → V ∗ be the duality mapping, and let M be a nonempty closed convex subset
of V . Let A : V → C(V ∗) be a multivalued mapping. Suppose that A is pseudomonotone,
coercive, potential, and bounded Lipschitz continuous mapping. Let F : V → R ∪ {+∞} be
a proper convex (not necessarily differentiable) and γ -Lipschitzian functional with M ⊂
int(D(F)). Let F˜ , R˜, R˜, S˜, and μ˜ be defined by (.), (.), and (.). Assume that <
τ = min{, μ˜ }. Let {h(un)} be an increasing and bounded real sequence in [, ].
Then, for an arbitrary u = u ∈ M, the sequence {un} defined by (.) converges strongly
to u˜ = SFOL(M,F,h(w)–f ) (i.e., the minimum norm element in SOL(M, F, h(w) – f )).
Proof Since F is supposed to be convex and γ -Lipschitzian, and A is coercive and
bounded, it results from [] and [] that F is weakly lower semicontinuous and F˜ is
coercive; moreover, R˜ < +∞ and R˜ < +∞. Hence μ˜ < +∞. This means that the iterative
sequence (.) is well defined.
Now we divide the proof into steps.
Step . We prove that {un} is bounded. To this end, it suffices to prove that
{un} ⊂ S˜,
un V ≤ R˜,
n ≥ .
(.)
Let us prove (.) by induction on n. For n = , we have u ∈ S˜. Suppose now that un ∈ S˜.
We will show that un+ ∈ S˜.
Setting η = un in (.) and taking into account that the functional F is γ -Lipschitzian
and J is uniformly monotone, and the inequality τ ≤ , we obtain
Now, using (.) together with the strict monotonicity of , we have
un+ – un V ≤
–(R˜ + γ ).
Furthermore, it follows from the bounded Lipschitz continuity of A that, for any t ∈ [, ],
wn ∈ Aun, wn ∈ A(un+ + t(un – un+))
wn – wn, un+ – un ≤ μ(R∗)
( – t)(un+ – un) V
where R∗ = max{ un+ + t(un – un+) V , un V }.
Since
it follows from the definition of R∗ that
R∗ ≤ R˜ +
–(R˜ + γ ).
Since μ˜ is an increasing function, we must have μ˜ (R∗) ≤ μ˜ .
Consequently, it follows from (.) and (.) that
– wn – wn, un+ – un ≥ –μ˜
Moreover, since A is potential, we have
F˜ (un) – F˜ (un+) =
h(un)wn, un – un+ dt – f , un – un+ + F(un) – F(un+)
=
h(un) wn – wn , un – un+ dt – f – h(un)wn, un – un+
≥ –
+ F(un) – F(un+)
+ F(un) – F(un+) .
h(un) wn – wn , un – un+ dt + τ – τ f – h(un)wn, un+ – un
(.)
(.)
(.)
(.)
(.)
(.)
(.)
(.)
Setting η = un in (.) and using the uniform monotonicity of J , (.), (.), it results that
F˜ (un) – F˜ (un+) ≥ –μ˜
This implies that F˜ (un+) ≤ F˜ (un) ≤ F˜ (u) and so un+ ∈ S˜. So {un} is bounded.
Step . We prove that limn→∞ un+ – un V = and limn→∞ Jun+ – Jun V ∗ = .
It follows from (.) that the sequence {F˜ (un)} is bounded and monotone, and thus we
have that limn→∞ F˜ (un) exists. This together with (.) implies that
lim λ
n→∞
lim
n→∞
un+ – un V = .
Since J is bounded Lipschitz continuous,
from (.) that
lim Jun+ – Jun V ∗ = .
n→∞
Step . We show that there exists a subsequence {unk } of {un} such that unk u¯ ∈ V ,
limk→∞ F(unk ) ≥ F(u¯ ), and lim supk→∞ h(unk ) wnk , unk – u¯ ≤ .
Since {un} is bounded and V is reflexive, we can choose a subsequence {unk } of {un} such
that unk u¯ ∈ V as k → ∞.
This together with the weak lower semicontinuity of F implies that limk→∞ F(unk ) ≥
F(u¯ ).
Since F is γ -Lipschitzian, {h(un)} ⊂ [, ], it follows from (.) that, for arbitrary η ∈ M,
h(unk ) wnk , unk – η = h(unk ) wnk , unk – unk+ + h(unk ) wnk , unk+ – η
≤ h(unk ) wnk , unk – unk+ + τ – Junk+ – Junk , η – unk+
+ F(η) – F(unk ) + F(unk ) – F(unk+ )
+ f , unk+ – unk + f , unk – η
≤
wnk V ∗ + f V ∗ + γ
unk+ – unk V + τ – Junk+ – Junk V ∗
× η – unk+ V + F(η) – F(unk ) + f , unk – η
≤ Cη Junk+ – Junk V ∗ + unk+ – unk V
+ F(η) – F(unk ) + f , unk – η ,
(.)
where Cη is a positive constant depending on η.
Setting η = u¯ in (.) and using the weak lower semicontinuity of F, (.), (.), we
have
F(unk+ ) – F(u¯ ) ≥ Junk , unk+ – u¯ , k ≥ .
lim sup h(unk ) wnk , unk – u¯ ≤ lim sup Cu¯ Junk+ – Junk V ∗ + unk+ – unk V
k→∞ k→∞
+ lim sup F(u¯ ) – F(unk ) + lim sup f , unk – u¯
k→∞ k→∞
≤ .
Step . We show that u¯ ∈ SOL(M, F, h(w) – f ).
Since {h(un)} ⊂ [, ] is bounded and monotone increasing, it follows that
lim h(un) = h > .
n→∞
By (.)-(.), the lower semicontinuity of F and by the pseudomonotonicity of A, we
have
= lim inf Cη Junk+ – Junk V ∗ + unk+ – unk V
k→∞
≥ likm→i∞nf h(unk ) wnk , unk – η + lim inf F(unk) – F(η) + lim inf f , η – unk
k→∞ k→∞
≥ h( w¯), u¯ – η + F(u¯ ) – F(η) + f , η – u¯ ,
where w¯ ∈ A u¯. This means that u¯ ∈ SOL(M, F, h(w) – f ).
Step . We prove that
lim sup –Ju¯ , unk+ – u¯ + F(u¯ ) – F(unk+ ) ≤ ,
k→∞
where u¯ = SFOL(M,F,h(w)–f ).
Indeed take a subsequence {unk+ } of {un} such that unk+ u¯ .
Note that u¯ = SFOL(M,F,h(w)–f ). Then from u¯ ∈ SOL(M, F, h(w) – f ), the weak lower
semicontinuity of F, and Lemma .(ii), the desired inequality (.) follows immediately.
Step . We show that limn→∞ un – u¯ V = .
Since unk+ u¯ , it follows from the weak lower semicontinuity of · V that
lim inf unk+ V ≥ u¯ V .
k→∞
F(u) – F(u¯ ) ≥ u∗, u – u¯ ,
and hence
From the convexity of D(F), F and from the weak lower semicontinuity of F, we obtain
that F is subdifferentiable in int(D(F)). Thus, for all u ∈ D(F), there exists an element
u∗ ∈ V ∗ such that
(.)
(.)
(.)
(.)
(.)
In view of u
n
k+
=
F
SOL(M,F ,h(w)–f )
–
J (Ju
n
k
τ
– h(u )(f – w )), we have
n n
k k
G u
n
k+
, Ju
n
k
τ
– h(u )(f – w )
n n
k k
≤
G u, Ju
¯
n
k
τ
– h(u )(f – w ) .
n n
k k
η
By using (.) with J = Ju
n
k
τ
– h(u )(f – w ), we have
n n
k k
u
n
k+ V
u
≤ ¯
V
V
– Ju
n
k
τ
– h(u )(f – w ), u
n n ¯
k k
+ Ju
n
k
τ
– h(u )(f – w )
n n
k k
V
∗
+ F (u)
¯
= u
¯
– Ju
n
k
τ
– h(u )(f – w ), u
n n
k k
n
k+
+ Ju
n
k
τ
– h(u )(f – w ), u
n n
k k
n
k+
– u
¯
+ Ju
n
k
τ
– (u )(f – w )
n n
k k
V
∗
+ F (u),
¯
– Ju
n
k
τ
– h(u )(f – w ), u
n n
k k
n
k+
+ Ju
n
k
τ
– h(u )(f – w )
n n
k k
+ F (u
n
k+
)
V
∗
u
n
k+ V
u
≤ ¯
V
+ Ju , u
n
k
n
k+
– u + F (u) – F (u
¯ ¯
n
k+
)
τ
+ h(u ) w , u
n n
k k
n
k+
τ
– u + h(u ) f , u – u
¯ n ¯
k
n
k+
.
(.)
on the both sides of (.) and using u
n
k+
u, (.)-(.),
¯
which implies that
Taking the
lim sup
and (.) yields
k
→∞
which implies that
lim sup
k
→∞
lim sup
k
→∞
u
n
k+ V
u ,
≤ ¯
V
u
n
k+
V ≤ ¯ V
u .
This shows that
k
lim
→∞
u
n
k+
V
= u .
¯ V
k
lim
→∞
u
n
k+
= u.
¯
Combining (.) and (.), we have
u
¯ V ≤
lim inf
k
→∞
u
n
k+
V ≤
lim sup
k
→∞
u
n
k+
V ≤ ¯ V
u .
Since V is a uniformly convex Banach space, then it has the Kadec-Klee property, and so
from u
n
k+
u and (.) we obtain
¯
Let us now show that the whole sequence converges strongly to u.
¯
(.)
(.)
(.)
(.)
Since {G(un+, )} is bounded and nondecreasing (indeed, by Lemma .(i), we have
G(un+, un) + G(un+, ) ≤ G(un, ) and G(un+, un) ≥ ( un+ V – un V ) ≥ ), it follows
that {G(un+, )} is convergent.
This together with (.) implies that
lim G(un+, ) = G(u¯ , ).
n→∞
(.)
Now, following to [], we suppose that there exists some subsequence {unj+ } of {un} such
that limj→∞ unj+ = uˆ , then by Lemma .(i) we obtain
≤ k,lji→m∞ G(unk+ , ) – G
≤ G(u¯ , uˆ ) = lim G(unk+ , unj+ ) = lim G unk+ , SFOL(M,F,h(w)–f )
k,j→∞ k,j→∞
F
SOL(M,F,h(w)–f ),
= lim
k,j→∞
G(unk+ , ) – G(unj+ , )
= G(u¯ , ) – G(u¯ , ) = ,
which means that G(u¯ , uˆ) = and hence, by Lemma .(ii), it results that uˆ = u¯ .
Consequently, limn→∞ un = u¯ . This completes the proof of Theorem ..
Theorem . Let V = H be a real Hilbert space, and let M be a nonempty closed convex
subset of H. Let A : H → C(H) be a multivalued mapping. Suppose that A is a
pseudomonotone, coercive, potential, and inverse strongly monotone mapping. Let {h(un)}, M,
F˜ , S˜, μ˜ and F˜, F, R˜ , R˜ be the same as in Theorem ..
Then, for arbitrary u = u ∈ M, the sequence {un} defined by
(un+ – un, η – un+) + τ F(η) – F(un+) ≥ τ f – h(un)wn, η – un+
∀η ∈ M, (.)
with < τ < τ = hγ , h > , converges strongly to u˜ =
F
SOL(M,F,h(w)–f ).
Proof Since any inverse strongly monotone mapping is γ -Lipschitzian mapping, i.e.,
bounded Lipschitz continuous with μ(ξ ) = γ and (ξ ) = ξ , then by simple modifications
of the proof of Theorem ., we can easily show that there exists a subsequence {unk+ } of
{un} such that unk+ u¯ ∈ SOL(M, F, h(w) – f ) and limk→∞ unk+ H = u¯ H .
Since every Hilbert space is uniformly convex, by virtue of the Kadec-Klee property of
H, we have limk→∞ unk+ = u¯ ∈ SOL(M, F, h(w) – f ).
Now, we prove that un u¯ and limn→∞ un H = u¯ H .
From u¯ ∈ SOL(M, F, h(w) – f ), we have
τ F(η) – F(u¯ ) ≥ τ f – h( w¯), η – u¯ ,
∀η ∈ M.
Setting η = un+ in (.) and η = u¯ in (.), we have
τ F(un+) – F(u¯ ) ≥ τ f – h(w¯ ), un+ – u¯
and
τ F(u¯ ) – F(un+) ≥ (un+ – un, un+ – u¯ ) + τ f – h(un)wn, u¯ – un+ .
(.)
(.)
(.)
Adding (.) and (.), we have
(un+ – u¯ , un+ – u¯ ) ≤ (un – u¯ , un+ – u¯ ) – τ h(un)wn – h( w¯), un+ – u¯
= un – u¯ – τ h(un)wn – h(w¯ ) , un+ – u¯ ,
which implies that
un+ – u¯ H ≤ un – u¯ – τ h(un)wn – h(w¯ )
H .
Then, by the inverse strong monotonicity of A, we obtain for all sufficiently large n
un+ – u¯ H ≤ un – u¯ H – τ h(un)wn – h(w¯ ), un – u¯ + τ h(un)wn – h(w¯ ) H
= un – u¯ H – τ h(wn – w¯ , un – u¯ ) + τ h wn – w¯ H
τ h
≤ un – u¯ H – τ h – γ
wn – w¯ H .
Since – τγh > , it follows that un+ – u¯ H ≤ un – u¯ H and so limn→∞ un – u¯ H = σu¯.
By following the same arguments as in [] and [], we can readily claim that all weak
limit points of the sequence {un} coincide, and hence un u¯ as n → ∞.
By the weak lower semicontinuity of · H , this implies that
lim inf un H > u¯ H .
n→∞
lim sup un H ≤ u¯ H .
n→∞
Analogically to the proof of step with obvious modifications, we have
(.)
(.)
This, together with (.), implies that limn→∞ un H = u¯ H .
Applying again the virtue of the Kadec-Klee property of H, we obtain limn→∞ un = u¯ .
This completes the proof of Theorem ..
Remark . Theorems . and . extend and improve the corresponding Theorems .
and ..
Example . (Axisymmetric shell problem) A quintessential example of a single-valued
mapping satisfying all the assumptions contemplated in Theorems . and . which
appears in determining the axisymmetric equilibrium position of a soft netlike rotation shell
is as follows:
The shell surface (in a strainless state) is assumed to be a cylinder of length l and radius .
Let s be a Lagrangian coordinate in the longitudinal direction such that < s < l.
Let V = [W◦ (p)(, l)] and V ∗ = [W◦ (q–)(, l)], q = pp– , p > . Set u(s) = (u(s), u(s)), η(s) =
(η(s), η(s)), M = {u ∈ V : u(s) + ≥ ∀s ∈ (, l)}, and λ = [( + ddus ) + ( ddus )] , λ = + u.
Consider the surface force is characterized by a known constant function P. Let Ti(λi),
i = , , be two functions (tightening force) satisfying conditions ()-() in Badriev and
Banderov [].
Consider the mappings A, B, C, D : V → V ∗ defined by
l
l
l
Au, η =
Bu, η =
Cu, η =
l T(λ)
λ
dη
u
ds
+
du
ds
+
du du
,
ds
ds
,
dη
ds
ds;
+ +
η +
du
ds
du dη
ds ds
uη
ds;
ds;
Du, η =
T(λ)η ds.
If A = (A + D) + P(B + C), then by Theorems and in [] it follows that the mapping
A satisfies all the assumptions postulated in Theorems . and ..
4 Conclusion
results.
A generalized multivalued pseudomonotone mixed variational inequality is considered,
and a modified two-layer iteration via a boundary point approach to find the minimum
norm solution of such inequalities is introduced, and its strong convergence is proved in
the framework of uniformly convex spaces. The results develop the corresponding recent
Acknowledgements
The author would like to extend his sincere gratitude to the two referees for their laudable comments and precious
suggestions. I am also profoundly grateful to professor doctor IB Badriev for many valuable discussions.
Competing interests
The author declares that he has no competing interests.
Authors’ contributions
I am the only author. I have read and approved the final manuscript.
Publisher’s Note
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