GENERAL SYSTEM OF STRONGLY
PSEUDOMONOTONE NONLINEAR VARIATIONAL
INEQUALITIES BASED ON PROJECTION SYSTEMS
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
R. U. VERMA
Department of Mathematics, University of Toledo
Toledo, Ohio 43606, USA
EMail: verma99@msn.com
Received:
26 February, 2006
Accepted:
11 December, 2006
Communicated by:
R.N. Mohapatra
2000 AMS Sub. Class.:
49J40, 65B05, 47H20.
Key words:
Strongly pseudomonotone mappings, Approximation solvability, Projection methods,
System of nonlinear variational inequalities.
Abstract:
Let K1 and K2 , respectively, be non empty closed convex subsets of real Hilbert
spaces H1 and H2 . The Approximation − solvability of a generalized system of
nonlinear variational inequality (SN V I) problems based on the convergence of projection methods is discussed. The SNVI problem is stated as follows: find an element
(x∗ , y ∗ ) ∈ K1 × K2 such that
hρS(x∗ , y ∗ ), x − x∗ i ≥ 0, ∀x ∈ K1 and for ρ > 0,
hηT (x∗ , y ∗ ), y − y ∗ i ≥ 0, ∀y ∈ K2 and for η > 0,
where S : K1 × K2 → H1 and T : K1 × K2 → H2 are nonlinear mappings.
vol. 8, iss. 1, art. 6, 2007
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Contents
1
Introduction
3
2
General Projection Methods
7
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
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1.
Introduction
Projection-like methods in general have been one of the most fundamental techniques for establishing the convergence analysis for solutions of problems arising
from several fields, such as complementarity theory, convex quadratic programming,
and variational problems. There exists a vast literature on approximation-solvability
of several classes of variational/hemivariational inequalities in different space settings. The author [6, 7] introduced and studied a new system of nonlinear variational
inequalities in Hilbert space settings. This class encompasses several classes of nonlinear variational inequality problems. In this paper we intend to explore, based on a
general system of projection-like methods, the approximation-solvability of a system
of nonlinear strongly pseudomonotone variational inequalities in Hilbert spaces. The
obtained results extend/generalize the results in [1], [5] – [7] to the case of strongly
pseudomonotone system of nonlinear variational inequalities. Approximation solvability of this system can also be established using the resolvent operator technique
but in the more relaxed setting of Hilbert spaces. For more details, we refer the
reader to [1] – [10].
Let H1 and H2 be two real Hilbert spaces with the inner product h·, ·i and norm
k · k. Let S : K1 × K2 → H1 and T : K1 × K2 → H2 be any mappings on
K1 × K2 , where K1 and K2 are nonempty closed convex subsets of H1 and H2 ,
respectively. We consider a system of nonlinear variational inequality (abbreviated
as SNVI) problems: determine an element (x∗ , y ∗ ) ∈ K1 × K2 such that
(1.1)
hρS(x∗ , y ∗ ), x − x∗ i ≥ 0 ∀x ∈ K1
(1.2)
hηT (x∗ , y ∗ ), y − y ∗ i ≥ 0 ∀y ∈ K2 ,
where ρ, η > 0.
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
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The SNVI (1.1)−(1.2) problem is equivalent to the following projection formulas
x∗ = Pk [x∗ − ρS(x∗ , y ∗ )]
for ρ > 0
y ∗ = Qk [y ∗ − ηT (x∗ , y ∗ )]
for η > 0,
where Pk is the projection of H1 onto K1 and QK is the projection of H2 onto K2 .
We note that the SNVI (1.1)−(1.2) problem extends the NVI problem: determine
an element x∗ ∈ K1 such that
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
hS(x∗ ), x − x∗ i ≥ 0, ∀x ∈ K1 .
(1.3)
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
Also, we note that the SNVI (1.1) − (1.2) problem is equivalent to a system of
nonlinear complementarities (abbreviated as SNC): find an element (x∗ , y ∗ ) ∈ K1 ×
K2 such that S(x∗ , y ∗ ) ∈ K1∗ , T (x∗ , y ∗ ) ∈ K2∗ , and
hρS(x∗ , y ∗ ), x∗ i = 0 for
(1.4)
∗
∗
∗
hηT (x , y ), y i = 0 for
(1.5)
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ρ > 0,
η > 0,
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K1∗ = {f ∈ H1 : hf, xi ≥ 0, ∀x ∈ K1 }.
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K2∗ = {g ∈ H2 : hg, yi ≥ 0, ∀g ∈ K2 }.
Close
Now, we recall some auxiliary results and notions crucial to the problem on hand.
Lemma 1.1. For an element z ∈ H, we have
and
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where K1∗ and K2∗ , respectively, are polar cones to K1 and K2 defined by
x∈K
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hx − z, y − xi ≥ 0,
∀y ∈ K
if and only if
x = Pk (z).
Lemma 1.2 ([3]). Let {αk }, {β k }, and {γ k } be three nonnegative sequences such
that
αk+1 ≤ (1 − tk )αk + β k + γ k for k = 0, 1, 2, ...,
P
P∞ k
k
k
k
k
where tk ∈ [0, 1], ∞
k=0 t = ∞, β = o(t ), and
k=0 γ < ∞. Then α → 0 as
k → ∞.
A mapping T : H → H from a Hilbert space H into H is called monotone if
hT (x)−T (y), x−yi ≥ 0 for all x, y ∈ H. The mapping T is (r)−strongly monotone
if for each x, y ∈ H, we have
hT (x) − T (y), x − yi ≥ r||x − y||2
for a constant r > 0.
This implies that kT (x) − T (y)k ≥ rkx − yk, that is, T is (r)-expansive, and
when r = 1, it is expansive. The mapping T is called (s)-Lipschitz continuous (or Lipschitzian) if there exists a constant s ≥ 0 such that kT (x) − T (y)k ≤
skx − yk, ∀ x, y ∈ H. T is called (µ)-cocoercive if for each x, y ∈ H, we have
hT (x) − T (y), x − yi ≥ µ||T (x) − T (y)||2
for a constant µ > 0.
Clearly, every (µ)-cocoercive mapping T is ( µ1 )-Lipschitz continuous. We can easily see that the following implications on monotonicity, strong monotonicity and
expansiveness hold:
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
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strong monotonicity ⇒ expansiveness
⇓
monotonicity
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T is called relaxed (γ)-cocoercive if there exists a constant γ > 0 such that
hT (x) − T (y), x − yi ≥ (−γ)kT (x) − T (y)k2 ,
∀x, y ∈ H.
T is said to be (r)-strongly pseudomonotone if there exists a positive constant r such
that
hT (y), x − yi ≥ 0 ⇒ hT (x), x − yi ≥ r kx − yk2 , ∀x, y ∈ H.
T is said to be relaxed (γ, r)-cocoercive if there exist constants γ,r > 0 such that
hT (x) − T (y), x − yi ≥ (−γ) kT (x) − T (y)k2 + r kx − yk2 .
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
Clearly, it implies that
hT (x) − T (y), x − yi ≥ (−γ) kT (x) − T (y)k2 ,
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
that is, T is relaxed (γ)-cocoercive.
T is said to be relaxed (γ, r)-pseudococoercive if there exist positive constants γ
and r such that
hT (y), x − yi ≥ 0 ⇒ hT (x), x − yi ≥ (−γ) kT (x) − T (y)k2 +r kx − yk2 ,
Thus, we have following implications:
⇒ strong (r)-pseudomonotonicity
(r)-strong monotonicity
⇓
relaxed (γ, r)-cocoercivity
⇓
relaxed (γ, r)-pseudococoercivity
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∀x, y ∈ H.
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2.
General Projection Methods
This section deals with the convergence of projection methods in the context of the
approximation-solvability of the SNVI (1.1) − (1.2) problem.
Algorithm 1. For an arbitrarily chosen initial point (x0 , y 0 ) ∈ K1 × K2 , compute
the sequences {xk } and {y k } such that
x
k+1
k
k
k
k
k
k
k
k k
= (1 − a − b )x + a Pk [x − ρS(x , y )] + b u
y k+1 = (1 − αk − β k )y k + αk QK [y k − ηT (xk , y k )] + β k v k ,
where PK is the projection of H1 onto K1 , QK is the projection of H2 onto K2 , ρ,
η > 0 are constants, S : K1 × K2 → H1 and T : K1 × K2 → H2 are any two
mappings, and uk and v k , respectively, are bounded sequences in K1 and K2 . The
sequences {ak }, {bk }, {αk }, and {β k } are in [0, 1] with (k ≥ 0)
0 ≤ ak + bk ≤ 1,
0 ≤ αk + β k ≤ 1.
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
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Algorithm 2. For an arbitrarily chosen initial point (x0 , y 0 ) ∈ K1 × K2 , compute
the sequences {xk } and {y k } such that
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xk+1 = (1 − ak − bk )xk + ak Pk [xk − ρS(xk , y k )] + bk uk
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y k+1 = (1 − ak − bk )y k + ak QK [y k − ηT (xk , y k )] + bk v k ,
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where PK is the projection of H1 onto K1 , QK is the projection of H2 onto K2 , ρ,
η > 0 are constants, S : K1 × K2 → H1 and T : K1 × K2 → H2 are any two
mappings, and uk and v k , respectively, are bounded sequences in K1 and K2 . The
sequences {ak } and {bk }, are in [0, 1] with (k ≥ 0)
Close
0 ≤ ak + bk ≤ 1.
Algorithm 3. For an arbitrarily chosen initial point (x0 , y 0 ) ∈ K1 × K2 , compute
the sequences {xk } and {y k } such that
xk+1 = (1 − ak )xk + ak Pk [xk − ρS(xk , y k )]
y k+1 = (1 − ak )y k + ak QK [y k − ηT (xk , y k )],
where PK is the projection of H1 onto K1 , QK is the projection of H2 onto K2 , ρ,
η > 0 are constants, S : K1 × K2 → H1 and T : K1 × K2 → H2 are any two
mappings. The sequence {ak } ∈ [0, 1] for k ≥ 0.
We consider, based on Algorithm 2, the approximation solvability of the SNVI
(1.1) − (1.2) problem involving strongly pseudomonotone and Lipschitz continuous
mappings in Hilbert space settings.
Theorem 2.1. Let H1 and H2 be two real Hilbert spaces and, K1 and K2 , respectively, be nonempty closed convex subsets of H1 and H2 . Let S : K1 × K2 → H1 be
strongly (r)− pseudomonotone and (µ)−Lipschitz continuous in the first variable
and let S be (ν)−Lipschitz continuous in the second variable. Let T : K1 × K2 →
H2 be strongly (s)−pseudomonotone and (β)−Lipschitz continuous in the second
variable and let T be (τ )−Lipschitz continuous in the first variable. Let k · k∗ denote
the norm on H1 × H2 defined by
k(x, y)k∗ = (kxk + kyk) ∀(x, y) ∈ H1 × H2 .
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
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In addition, let
s
1 − 2ρr + ρ +
θ=
s
σ=
1 − 2ηr + η +
ρµ2
2
ηβ 2
2
Close
+
ρ2 µ 2
+ ητ < 1
+ η 2 β 2 + ρν < 1,
let (x∗ , y ∗ ) ∈ K1 × K2 form a solution to the SNVI (1.1) − (1.2) problem, and let
sequences {xk }, and {y k } be generated by Algorithm 2. Furthermore, let
(i) hS(x∗ , y k ), xk − x∗ i ≥ 0;
(ii) hT (xk , y ∗ ), y k − y ∗ i ≥ 0;
(iii) 0 ≤ ak + bk ≤ 1;
P
P∞ k
k
(iv) ∞
k=0 a = ∞, and
k=0 b < ∞;
(v) 0 < ρ <
2r
µ2
and 0 < η <
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
2s
.
β2
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
Then the sequence {xk , y k } converges to (x∗ , y ∗ ).
Proof. Since (x∗ , y ∗ ) ∈ K1 ×K2 forms a solution to the SNVI (1.1)−(1.2) problem,
it follows that
x∗ = PK [x∗ − ρS(x∗ , y ∗ )] and
y ∗ = QK [x∗ − ηT (x∗ , y ∗ )].
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Applying Algorithm 2, we have
(2.1)
kxk+1 − x∗ k
k
k
k
k
k
k
k
∗
k
k
k
k k
= k(1 − a − b )x + a PK [x − ρS(x , y )] + b u
∗
∗
k
∗
k
k
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≤ (1 − a − b )kx − x k
k
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∗
− (1 − a − b )x − a PK [x − ρS(x , y )] − b x k
k
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k
k
∗
∗
∗
+ a kPK [x − ρS(x , y )] − PK [x − ρS(x , y )]k + M b
≤ (1 − a )kxk − x∗ k + ak kxk − x∗ − ρ[S(xk , y k ) − S(x∗ , y k )
k
+ S(x∗ , y k ) − S(x∗ , y ∗ )]k + M bk
≤ (1 − ak )kxk − x∗ k + ak kxk − x∗ − ρ[S(xk , y k ) − S(x∗ , y k )]k
+ ρk[S(x∗ , y k ) − S(x∗ , y ∗ )]k + M bk ,
k
Close
where
M = max{sup kuk − x∗ k, sup kv k − y ∗ k} < ∞.
Since S is strongly (r)− pseudomonotone and (µ)−Lipschitz continuous in the first
variable, and S is (ν)−Lipschitz continuous in the second variable, we have in light
of (i) that
(2.2)
kxk − x∗ − ρ[S(xk , y k ) − S(x∗ , y k )]k2
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
= kxk − x∗ k2 − 2ρhS(xk , y k ) − S(x∗ , y k ), xk − x∗ i
R. U. Verma
+ ρ2 kS(xk , y k ) − S(x∗ , y k )k2
vol. 8, iss. 1, art. 6, 2007
= kxk − x∗ k2 − 2ρhS(xk , y k ), xk − x∗ i + 2ρhS(x∗ , y k ), xk − x∗ i
+ ρ2 kS(xk , y k ) − S(x∗ , y k )k2
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≤ kxk − x∗ k2 − 2ρrkxk − x∗ k2 + ρ2 µ2 kxk − x∗ k2
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+ 2ρhS(x∗ , y k ), xk − x∗ i
≤ kxk − x∗ k2 − 2ρrkxk − x∗ k2 + ρ2 µ2 kxk − x∗ k2
+ 2ρhS(x∗ , y k ), xk − x∗ i
= [1 − 2ρr + ρ2 µ2 ]kxk − x∗ k2 + 2ρhS(x∗ , y k ), xk − x∗ i.
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On the other hand, we have
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∗
k
k
∗
∗
k
2
k
∗ 2
2ρhS(x , y ), x − x i ≤ ρ[kS(x , y )k + kx − x k ]
and
(2.3)
kS(x∗ , y k )k2
1
=
kS(x∗ , y k ) − S(xk , y k )k2 − 2[kS(xk , y k )k2
2
Close
1
− kS(xk , y k ) + S(x∗ , y k )k2
2
1
≤
kS(x∗ , y k ) − S(xk , y k )k2 − 2[kS(xk , y k )k2
2
1
k
k
∗ k
2
− | kS(x , y ) − S(x , y )k | ]
2
1
≤ kS(x∗ , y k ) − S(xk , y k )k2
2
µ2 k
≤ kx − x∗ k2 ,
2
where
kS(xk , y k )k2 −
1
| kS(xk , y k ) − S(x∗ , y k )k |2 > 0.
2
Therefore, we get
2 ρµ
∗ k
k
∗
2ρhS(x , y ), x − x i ≤ ρ +
kxk − x∗ k2 .
2
It follows that
2
ρµ
k
∗
k
k
∗ k
2
2
kx −x −ρ[S(x , y )−S(x , y )]k ≤ 1 − 2ρr + ρ +
+ (ρµ) kxk −x∗ k2 .
2
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
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As a result, we have
(2.4) kxk+1 − x∗ k ≤ (1 − ak )kxk − x∗ k + ak θkxk − x∗ k + ak ρνky k − y ∗ k + M bk ,
r
2
where θ = 1 − 2ρr + ρ + ρµ2 + ρ2 µ2 .
Similarly, we have
(2.5) y k+1 − y ∗ ≤ (1 − ak )ky k − y ∗ k + ak σky k − y ∗ k + ak ητ kxk − x∗ k + M bk ,
r
where σ =
1 − 2ηr + η +
ηβ 2
2
+ η2β 2.
It follows from (2.4) and (2.5) that
(2.6) kx
k+1
∗
− x k + ky
k+1
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
∗
−y k
≤ (1 − ak )kxk − x∗ k + ak θkxk − x∗ k + ak ητ kxk − x∗ k + M bk
+ (1 − ak )ky k − y ∗ k + ak σky k − y ∗ k + ak ρνky k − y ∗ k + M bk
= [1 − (1 − δ)ak ](kxk − x∗ k + ky k − y ∗ k) + 2M bk ,
where δ = max{θ + ητ , σ + ρν} and H1 × H2 is a Banach space under the norm
k · k∗ .
If we set
αk = kxk − x∗ k + ky k − y ∗ k,
β k = 2M bk
tk = (1 − δ)ak ,
for k = 0, 1, 2, ...,
in Lemma 1.2, and apply (iii) and (iv), we conclude that
kxk − x∗ k + ky k − y ∗ k → 0
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vol. 8, iss. 1, art. 6, 2007
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as k → ∞.
Hence,
kxk+1 − x∗ k + ky k+1 − y ∗ k → 0.
Consequently, the sequence {(xk , y k )} converges strongly to (x∗ , y ∗ ), a solution to
the SNVI (1.1) − (1.2) problem. This completes the proof.
Note that the proof of the following theorem follows rather directly without using
Lemma 1.2.
Theorem 2.2. Let H1 and H2 be two real Hilbert spaces and, K1 and K2 , respectively, be nonempty closed convex subsets of H1 and H2 . Let S : K1 × K2 → H1 be
strongly (r)− pseudomonotone and (µ)−Lipschitz continuous in the first variable
and let S be (ν)−Lipschitz continuous in the second variable. Let T : K1 × K2 →
H2 be strongly (s)−pseudomonotone and (β)−Lipschitz continuous in the second
variable and let T be (τ )−Lipschitz continuous in the first variable. Let k · k∗ denote
the norm on H1 × H2 defined by
k(x, y)k∗ = (kxk + kyk)
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
∀(x, y) ∈ H1 × H2 .
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In addition, let
s
θ=
1 − 2ρr + ρ +
s
σ=
1 − 2ηr + η +
ρµ2
2
ηβ
2
Contents
2
+
ρ2 µ 2
+ ητ < 1,
+ η 2 β 2 + ρν < 1,
let (x∗ , y ∗ ) ∈ K1 × K2 form a solution to the SNVI (1.1) − (1.2) problem, and let
sequences {xk }, and {y k } be generated by Algorithm 3. Furthermore, let
(i) hS(x∗ , y k ), xk − x∗ i ≥ 0
(ii) hT (xk , y ∗ ), y k − y ∗ i ≥ 0
(iii) 0 ≤ ak ≤ 1
P
k
(iv) ∞
k=0 a = ∞
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(v) 0 < ρ <
2r
µ2
and 0 < η <
2s
.
β2
Then the sequence {xk , y k )} converges strongly to (x∗ , y ∗ ).
Proof. Since (x∗ , y ∗ ) ∈ K1 ×K2 forms a solution to the SNVI (1.1)−(1.2) problem,
it follows that
x∗ = PK [x∗ − ρS(x∗ , y ∗ )]
and
y ∗ = QK [x∗ − ηT (x∗ , y ∗ )].
Applying Algorithm 3, we have
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
(2.7) kxk+1 − x∗ k
vol. 8, iss. 1, art. 6, 2007
= k(1 − ak )xk + ak PK [xk − ρS(xk , y k )] − (1 − ak )x∗
− ak PK [x∗ − ρS(x∗ , y ∗ )]k
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≤ (1 − ak )kxk − x∗ k + ak kPK [xk − ρS(xk , y k )] − PK [x∗ − ρS(x∗ , y ∗ )]k
≤ (1 − ak )kxk − x∗ k
+ ak kxk − x∗ − ρ[S(xk , y k ) − S(x∗ , y k ) + S(x∗ , y k ) − S(x∗ , y ∗ )]k
≤ (1 − ak )kxk − x∗ k + ak kxk − x∗ − ρ[S(xk , y k ) − S(x∗ , y k )]k
+ ρk[S(x∗ , y k ) − S(x∗ , y ∗ )]k.
Since S is strongly (r)− pseudomonotone and (µ)−Lipschitz continuous in the first
variable, and S is (ν)−Lipschitz continuous in the second variable, we have in light
of (i) that
(2.8)
kxk − x∗ − ρ[S(xk , y k ) − S(x∗ , y k )]k2
= kx − x∗ k2 − 2ρhS(xk , y k ) − S(x∗ , y k ), xk − x∗ i
+ ρ2 kS(xk , y k ) − S(x∗ , y k )k2
= kx − x∗ k2 − 2ρhS(xk , y k ), xk − x∗ i + 2ρhS(x∗ , y k ), xk − x∗ i
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+ ρ2 kS(xk , y k ) − S(x∗ , y k )k2
≤ kxk − x∗ k2 − 2ρrkxk − x∗ k2 + ρ2 µ2 kxk − x∗ k2
+ 2ρhS(x∗ , y k ), xk − x∗ i
≤ kxk − x∗ k2 − 2ρrkxk − x∗ k2 + ρ2 µ2 kxk − x∗ k2
+ 2ρhS(x∗ , y k ), xk − x∗ i
= [1 − 2ρr + ρ2 µ2 ]kxk − x∗ k2 + 2ρhS(x∗ , y k ), xk − x∗ i.
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
On the other hand, we have
2ρhS(x∗ , y k ), xk − x∗ i ≤ ρ[kS(x∗ , y k )k2 + kxk − x∗ k2 ]
and
(2.9)
vol. 8, iss. 1, art. 6, 2007
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kS(x∗ , y k )k2
(
1
=
kS(x∗ , y k ) − S(xk , y k )k2
2
)
1
− 2 kS(xk , y k )k2 − kS(xk , y k ) + S(x∗ , y k )k2
2
(
1
kS(x∗ , y k ) − S(xk , y k )k2
≤
2
)
1
− 2 kS(xk , y k )k2 − | kS(xk , y k ) − S(x∗ , y k )k |2
2
µ2
1
≤ kS(x∗ , y k ) − S(xk , y k )k2 ≤ kxk − x∗ k2 ,
2
2
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where
kS(xk , y k )k2 −
1 S(xk , y k ) − S(x∗ , y k )2 > 0.
2
Therefore, we get
2 ρµ
2ρhS(x , y ), x − x i ≤ ρ +
kxk − x∗ k2 .
2
∗
k
∗
k
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
It follows that
2
ρµ
2
kx −x −ρ[S(x , y )−S(x , y )]k ≤ 1 − 2ρr + ρ +
+ (ρµ) kxk −x∗ k2 .
2
k
∗
k
k
∗
k
2
As a result, we have
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
Title Page
kxk+1 − x∗ || ≤ (1 − ak )kxk − x∗ k + ak θkxk − x∗ k + ak ρνky k − y ∗ k,
r
2
where θ = 1 − 2ρr + ρ + ρµ2 + ρ2 µ2 .
(2.10)
Similarly, we have
k+1
∗
k
∗
k
k
∗
k
k
It follows from (2.9) and (2.10) that
(2.12)
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k
∗
ky
− y k ≤ (1 − a )ky − y k + a σky − y k + a ητ kx − x k,
r
2
where σ = 1 − 2ηr + η + ηβ2 + η 2 β 2 .
(2.11)
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kxk+1 − x∗ k + ky k+1 − y ∗ k
≤ (1 − ak )kxk − x∗ k + ak θkxk − x∗ k + ak ητ kxk − x∗ k
+ (1 − ak )ky k − y ∗ k + ak σky k − y ∗ k + ak ρνky k − y ∗ k
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= [1 − (1 − δ)ak ](kxk − x∗ k + ky k − y ∗ k)
k
Y
≤
[1 − (1 − δ)aj ](kx0 − x∗ k + ky 0 − y ∗ k),
j=0
where δ = max{θ + ητ , σ + ρν} and H1 × H2 is a Banach space under the norm
k · k∗ .
P
k
Since δ < 1 and ∞
k=0 a is divergent, it follows that
k
Y
lim
[1 − (1 − δ)aj ] = 0
k→∞
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
as k → ∞.
vol. 8, iss. 1, art. 6, 2007
j=0
Title Page
Therefore,
kx
k+1
∗
− x k + ky
k
k
k+1
∗
− y k → 0,
Contents
∗
∗
and consequently, the sequence {(x , y )} converges strongly to (x , y ), a solution
to the SN V I (1.1) − (1.2) problem. This completes the proof.
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References
[1] S.S. CHANG, Y.J. CHO, AND J.K. KIM, On the two-step projection methods
and applications to variational inequalities, Mathematical Inequalities and Applications, accepted.
[2] Z. LIU, J.S. UME, AND S.M. KANG, Generalized nonlinear variational-like
inequalities in reflexive Banach spaces, Journal of Optimization Theory and
Applications, 126(1) (2005), 157–174.
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
[3] L. S. LIU, Ishikawa and Mann iterative process with errors for nonlinear
strongly accretive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, 194 (1995), 114–127.
vol. 8, iss. 1, art. 6, 2007
Title Page
[4] H. NIE, Z. LIU, K. H. KIM AND S. M. KANG, A system of nonlinear variational inequalities involving strongly monotone and pseudocontractive mappings, Advances in Nonlinear Variational Inequalities, 6(2) (2003), 91–99.
[5] R.U. VERMA, Nonlinear variational and constrained hemivariational inequalities, ZAMM: Z. Angew. Math. Mech., 77(5) (1997), 387–391.
[6] R.U. VERMA, Generalized convergence analysis for two-step projection methods and applications to variational problems, Applied Mathematics Letters, 18
(2005), 1286–1292.
[7] R.U. VERMA, Projection methods, algorithms and a new system of nonlinear variational inequalities, Computers and Mathematics with Applications, 41
(2001), 1025–1031.
[8] R. WITTMANN, Approximation of fixed points of nonexpansive mappings,
Archiv der Mathematik, 58 (1992), 486–491.
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[9] E. ZEIDLER, Nonlinear Functional Analysis and its Applications I, SpringerVerlag, New York, New York, 1986.
[10] E. ZEIDLER, Nonlinear Functional Analysis and its Applications II/B,
Springer-Verlag, New York, New York, 1990.
Strongly Pseudomonotone
Nonlinear Variational
Inequalities
R. U. Verma
vol. 8, iss. 1, art. 6, 2007
Title Page
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