Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 635030, 13 pages
doi:10.1155/2011/635030
Research Article
Variational-Like Inclusions and Resolvent
Equations Involving Infinite Family of Set-Valued
Mappings
Rais Ahmad and Mohd Dilshad
Department of Mathematics, Aligarh Muslim University,
Aligarh 202002, India
Correspondence should be addressed to Rais Ahmad, raisain 123@rediffmail.com
Received 18 December 2010; Accepted 23 December 2010
Academic Editor: Qamrul Hasan Ansari
Copyright q 2011 R. Ahmad and M. Dilshad. 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 study variational-like inclusions involving infinite family of set-valued mappings and their
equivalence with resolvent equations. It is established that variational-like inclusions in real
Banach spaces are equivalent to fixed point problems. This equivalence is used to suggest an
iterative algorithm for solving resolvent equations. Some examples are constructed.
1. Introduction
The important generalization of variational inequalities, called variational inclusions, have
been extensively studied and generalized in different directions to study a wide class of
problems arising in mechanics, optimization, nonlinear programming, economics, finance
and applied sciences, and so forth; see, for example 1–7 and references theirin. The
resolvent operator technique for solving variational inequalities and variational inclusions
is interesting and important. The resolvent operator technique is used to establish an
equivalence between variational inequalities and resolvent equations. The resolvent equation
technique is used to develop powerful and efficient numerical techniques for solving various
classes of variational inequalities inclusions and related optimization problems.
In this paper, we established a relationship between variational-like inclusions and
resolvent equations. We propose an iterative algorithm for computing the approximate
solutions which converge to exact solution of considered resolvent equations. Some examples
are constructed.
2
Fixed Point Theory and Applications
2. Formulation and Preliminaries
Throughout the paper, unless otherwise specified, we assume that E is a real Banach space
with its norm · , E∗ is the topological dual of E, ·, · is the pairing between E and E∗ , d
is the metric induced by the norm · , 2E resp., CBE is the family of nonempty resp.,
nonempty closed and bounded subsets of E, and H·, · is the Housdorff metric on CBE
defined by
HP, Q max sup dx, Q, sup d P, y
x∈P
where dx, Q infy∈Q dx, y and dP, y
∗
mapping J : E → 2E is defined by
,
2.1
y∈Q
infx∈P dx, y. The normalized duality
Jx f ∈ E∗ : x, f x · f ,
f x, ∀x ∈ E .
2.2
Definition 2.1. Let E be a real Banach space. Let η : E × E → E; g, A : E → E be the singlevalued mapping, and let M : E → 2E be a set-valued mapping. Then,
i the mapping g is said to be accretive if
gx − g y , j x − y ≥ 0,
∀x, y ∈ E,
2.3
∀x, y ∈ E,
2.4
ii the mapping g is said to be strictly accretive if
gx − g y , j x − y ≥ 0,
and the equality hold if and only if x y,
iii the mapping g is said to be k-strongly accretive k ∈ 0, 1 if for any x, y ∈ E,
there exists jx − y ∈ Jx − y such that
gx − g y , j x − y ≥ k x − y
2
,
2.5
iv the mapping A is said to be r-strongly η-accretive, if there exists a constant r >
0 such that
Ax − A y , j η x, y
≥r x−y
2
,
∀x, y ∈ E,
2.6
v the mapping M is said to be m-relaxed η-accretive, if there exists a constant m >
0 such that
u − v, j η x, y
≥ −m x − y
2
,
∀x, y ∈ E, u ∈ Mx, v ∈ M y .
2.7
Fixed Point Theory and Applications
3
Definition 2.2. Let A : E → E, η : E × E → E be the single-valued mappings.
Then, a set-valued mapping M : E → 2E is called A, η-accretive if M is m-relaxed ηaccretive and A ρME E, for every ρ > 0.
∗
Proposition 2.3 see 8, 9. Let E be a real Banach space, and let J : E → 2E be the normalized
duality mapping. Then, for any x, y ∈ E
x
y
2
≤ x2
2 y, j x
y ,
∀j x
y ∈J x
y .
2.8
Definition 2.4. Let A : E → E, W : E × E → E, and let N : E∞ E × E × E · · · → E be the
mappings. Then,
i the mapping A is said to be Lipschitz continuous with constant λA if
Ax − A y ≤ λA x − y ,
∀x, y ∈ E,
2.9
ii the mapping W is said to be Lipschitz continuous in the first argument with
constant λW1 if
Wx1 , · − Wx2 , · ≤ λW1 x1 − x2 ,
∀x1 , x2 ∈ E.
2.10
Similarly, we can define Lipschitz continuity in the second argument.
iii the mapping N is said to be Lipschitz continuous in the ith argument with
constant βi if
N·, . . . , xi , . . . − N ·, . . . , yi , . . . ≤ βi xi − yi ,
∀xi , yi ∈ E.
2.11
Definition 2.5. Let A : E → E be a strictly η-accretive mapping, and let M : E → 2E be an
ρ,A
A, η-accretive mapping. Then, the resolvent operator Jη,M : E → E is defined by
ρ,A
Jη,M u A
ρM
−1
u,
∀u ∈ E.
2.12
Proposition 2.6 see 10. Let E be a real Banach space, and let η : E × E → E be τ-Lipschitz
continuous; let A : E → E be an r-strongly η-accretive mapping, and let M : E → 2E be an A, ηρ,A
accretive mapping. Then the resolvent operator Jη,M : E → E is τ/r − ρm-Lipschitz continuous,
that is,
ρ,A
ρ,A Jη,M x − Jη,M y ≤
where ρ ∈ 0, r/m is a constant.
τ
x−y ,
r − ρm
∀x, y ∈ E,
2.13
4
Fixed Point Theory and Applications
Example 2.7. Let E Ê, Ax Then, M is η-accretive.
√
√ √
√
x, My y, and ηx, y x − y for all x, y ≥ 0 ∈ E.
Example 2.8. Let M·, · : E × E → 2E be r-strongly η-accretive in the first argument.
Then, M is m-relaxed η-accretive for m ∈ 1, r r 2 , for r > 0.6180.
Let Ti : E → CBE, i 1, 2, . . . , ∞ be an infinite family of set-valued mappings, and
let N : E∞ E×E×E · · · → E be a nonlinear mapping. Let η, W : E×E → E; A, g, m : E → E
be single-valued mappings, let and B, C, D:E → CBE be set-valued mappings. Suppose
that M·, · : E × E → 2E is A, η-accretive mapping in the first argument. We consider the
following problem.
Find u ∈ E, wi ∈ Ti u, i 1, 2, . . . , ∞, a ∈ Bu, x ∈ Cu, and y ∈ Du such that
0 ∈ Nw1 , w2 , . . . − W x, y
ma
M gu − ma, u .
2.14
The problem 2.14 is called variational-like inclusions problem.
Special Cases
i If W 0, m 0, then problem 2.14 reduces to the problem of finding u ∈ E, wi ∈
Ti u, i 1, 2, . . . , ∞ such that
0 ∈ Nw1 , w2 , . . .
M gu, u .
2.15
Problem 2.15 is introduced and studied by Wang 11.
ii If W 0, m 0, N·, . . . N·, ·, then problem 2.14 reduces to a problem
considered by Chang, et al. 12, 13 that is, find u ∈ H, w1 ∈ T1 u, w2 ∈ T2 u
such that
M gu, u .
0 ∈ Nw1 , w2 2.16
It is now clear that for a suitable choice of maps involved in the formulation of
problem 2.14, we can drive many known variational inclusions considered and studied
in the literature.
In connection with problem 2.14, we consider the following resolvent equation
problem.
Find z, u ∈ E, wi ∈ Ti u, i 1, 2, . . . , ∞; a ∈ Bu, x ∈ Cu, y ∈ Du such that
Nw1 , w2 , . . . − W x, y
ρ,A
ma
ρ,A
ρ,A
ρ−1 Rη,M·,u z 0,
ρ,A
2.17
ρ,A
where ρ is a constant and Rη,M·,u I − AJη,M·,u , where AJη,M·,uz AJη,M·,u z
and I is the identity mapping. Equation 2.17 is called the resolvent equation problem.
Fixed Point Theory and Applications
5
In support of problem 2.17, we have the following example.
Example 2.9. Let us suppose that E Ê, Ti u −i, i, i 1, 2, . . . , ∞, Cu {π/2}, Bu 0, 1, and Du {1}.
We define for wi ∈ Ti u, i 1, 2, . . . , ∞, a ∈ Bu, x ∈ Cu and y ∈ Du.
i Nw1 , w2 , . . . min{−1, sin w1 , sin w2 , . . .},
ii ma sin−1 a
cos−1 a,
iii Wx, y xy,
iv Ax x − 1, for all x ∈ Ê,
v M·, x 1, for all x ∈ Ê,
Then, for ρ 1, it is easy to check that the resolvent equation problem 2.17 is satisfied.
3. An Iterative Algorithm and Convergence Result
We mention the following equivalence between the problem 2.14 and a fixed point problem
which can be easily proved by using the definition of resolvent operator.
Lemma 3.1. Let u, a, x, y, w1 , w2 , . . . where u ∈ E, wi ∈ Ti u, i 1, 2, . . . , ∞, a ∈ Bu, x ∈
Cu, and y ∈ Du, is a solution of 2.14 if and only if it is a solution of the following equation:
gu ma
ρ,A
Jη,M·,u A gu − ma − ρ Nw1 , w2 , . . . − W x, y
ma .
3.1
Now, we show that the problem 2.14 is equivalent to a resolvent equation problem.
Lemma 3.2. Let u ∈ E, wi ∈ Ti u, i 1, 2, . . . , ∞, a ∈ Bu, x ∈ Cu, y ∈ Du, then the
following are equivalent:
i u, a, x, y, w1 , w2 , . . . is a solution of variational inclusion problem 2.14,
ii z, u, a, x, y, w1 , w2 , . . . is a solution of the problem 2.17,
where
z A gu − ma − ρ Nw1 , w2 , . . . − W x, y
gu ma
ma ,
ρ,A
Jη,M·,u A gu − ma − ρ Nw1 , w2 , . . . − W x, y
3.2
ma .
6
Fixed Point Theory and Applications
Proof. Let u, a, x, y, w1 , w2 , . . . be a solution of the problem 2.14, then by Lemma 3.1, it is
a solution of the problem
ρ,A
Jη,M·,u A gu − ma − ρ Nw1 , w2 , . . . − W x, y
gu ma
3.3
ma ,
using the fact that
ρ,A
ρ,A
Rη,M·,u I − A Jη,M·,u ,
ρ,A
ρ,A
Rη,M·,u z Rη,M·,u A gu − ma − ρ Nw1 , w2 , . . . − W x, y
ρ,A
I − A Jη,M·,u
ma
A gu − ma − ρNw1 , w2 , . . . − W x, y
A gu − ma − ρ Nw1 , w2 , . . . − W x, y
ma
ma
ρ,A
− A Jη,M·,u A gu − ma − ρ Nw1 , w2 , . . . − W x, y
A gu − ma − ρ Nw1 , w2 , . . . − W x, y
− A gu − ma ,
ma
ma
3.4
which implies that
Nw1 , w2 , . . . − W x, y
ma
ρ−1 Rη,M·,u z 0,
ρ,A
3.5
with
z A gu − ma − ρ Nw1 , w2 , . . . − W x, y
3.6
ma ,
that is, z, u, a, x, y, w1 , w2 , . . . is a solution of problem 2.17.
Conversly, let z, u, a, x, y, w1 , w2 , . . . be a solution of problem 2.17, then
ρ Nw1 , w2 , . . . − W x, y
ρ Nw1 , w2 , . . . − W x, y
ρ,A
ma −Rη,M·,u z,
3.7
ρ,A
ma A Jη,M·,u z − z,
from 3.2 and 3.7, we have
ma
ρ Nw1 , w2 , . . . − W x, y
ρ,A
A Jη,M·,u A gu − ma − ρ Nw1 , w2 , . . . − W x, y
− A gu − ma − ρ Nw1 , w2 , . . . − W x, y
ma ,
ma
3.8
Fixed Point Theory and Applications
7
which implies that
gu ma
ρ,A
Jη,M·,u A gu − ma − ρ Nw1 , w2 , . . . − W x, y
ma ,
3.9
that is, u, a, x, y, w1 , w2 , . . . is a solution of 2.14.
We now invoke Lemmas 3.1 and 3.2 to suggest the following iterative algorithm for
solving resolvent equation problem 2.17.
Algorithm 3.3. For a given z0 , u0 ∈ E, wi0 ∈ Ti u0 , i 1, 2, . . . , ∞, a0 ∈ Bu0 , x0 ∈ Cu0 , and
y0 ∈ Du0 . Let
z1 A gu0 − ma0 − ρ N w10 , w20 , . . . − W x0 , y0
ma0 .
3.10
Take z1 , u1 ∈ E such that
gu1 ma1 ρ,A
3.11
Jη,M·,u1 z1 .
Since for each i, wi0 ∈ Ti u0 , a0 ∈ Bu0 , x0 ∈ Cu0 , and y0 ∈ Du0 by Nadler’s
theorem 14 there exist wi1 ∈ Ti u1 , a1 ∈ Bu1 , x1 ∈ Cu1 , and y1 ∈ Du1 such that
wi0 − wi1 ≤ HTi u0 , Ti u1 ,
a0 − a1 ≤ HBu0 , Bu1 ,
3.12
x0 − x1 ≤ HCu0 , Cu1 ,
y0 − y1 ≤ HDu0 , Du1 ,
where H is the Housdorff metric on CBE.
Let
z2 A gu1 − ma1 − ρ N w11 , w21 , . . . − W x1 , y1
ma1 ,
3.13
and take any u2 ∈ E such that
gu2 ma2 ρ,A
Jη,M·,u2 z2 .
Continuing the above process inductively, we obtain the following.
3.14
8
Fixed Point Theory and Applications
For any z0 , u0 ∈ E, wi0 ∈ Ti u0 , i 1, 2, . . . , ∞, a0 ∈ Bu0 , x0 ∈ Cu0 , and
y0 ∈ Du0 . Compute the sequences {zn }, {un }, {win }, i 1, 2, . . . , ∞, {a0 }, {x0 }, {y0 } by the
following iterative schemes:
ρ,A
i gun man ii
an ∈ Bun ,
an − an 1 ≤ HBun , Bun 1 ,
iii xn ∈ Cun ,
yn − yn
1
v for each i, win ∈ Ti un ,
1
3.16
xn − xn 1 ≤ HCun , Cun 1 ,
iv yn ∈ Dun ,
vi zn
3.15
Jη,M·,un zn ,
3.17
≤ HDun , Dun 1 ,
win − win
1
3.18
3.19
≤ HTi un , Ti un 1 ,
A gun − man − ρ N w1n , w2n , . . . − W xn , yn
man ,
3.20
where ρ > 0 is a constant and n 0, 1, 2, . . ..
Theorem 3.4. Let E be a real Banach space. Let Ti , B, C, D : E → CBE be H-Lipschitz
continuous mapping with constants δi , α, t, γ , respectively. Let N E∞ E × E × E · · · → E be
Lipschitz continuous with constant βi , let A, g, m : E → E be Lipschitz continuous with constants
λA , λg , λm , respectively, and let A be r-strongly η-accretive mapping. Suppose that η,W : E×E → E
are mappings such that η is Lipschitz continuous with constant τ and W is Lipschitz continuous in
both the argument with constant λW1 and λW2 , respectively. Let M : E × E → 2E be A, η-accretive
mapping in the first argument such that the following holds for μ > 0:
ρ,η
ρ,η
JM·,un zn − JM·,un−1 zn ≤ μun − un−1 .
3.21
Suppose there exists a ρ > 0 such that
λA λg
λm α λA
ρ
ρ
∞
βi δi
ρ λ W1 t
λ W2 γ
i1
r − ρm <
1 − λ2m α2
τ
μ2 − 2k ,
r
m < , λ2m α2 < 1
ρ
3.22
2k − μ .
2
Then, there exist z, u, ∈ E, a ∈ BE, and x ∈ CE, y ∈ DE, and wi ∈ Ti u that
satisfy resolvent equation problem 2.17. The iterative sequences {zn }, {un }, {an } {xn }, {yn },
and {win }, i 1, 2, . . . , ∞, n 0, 1, . . . generated by Algorithm 3.3 converge strongly to
z, u, a, x, y, wi , respectively.
Fixed Point Theory and Applications
9
Proof. From Algorithm 3.3, we have
zn
1
− zn A gun − man − ρ N w1n , w2n , . . . − W xn , yn
man − A gun−1 − man−1 − ρ N w1n−1 , w2n−1 , . . . − W xn−1 , yn−1
man−1 ≤ A gun − man − A gun−1 − man−1 3.23
ρ N w1n , w2n , . . . − N w1n−1 , w2n−1 , . . .
ρ W xn , yn − W xn−1 , yn−1
ρman − man−1 .
By using the Lipschitz continuty of A, g, and m with constants λA , λg , and λm , respectively,
and by Algorithm 3.3, we have
A gun − man − A gun−1 − man−1 ≤ λA gun − gun−1 λA man − man−1 ≤ λA λg un − un−1 λA λm an − an−1 ≤ λA λg un − un−1 λA λm HBun , Bun−1 3.24
≤ λA λg un − un−1 λA λm αun − un−1 λA λg λA λm α un − un−1 .
Since N is Lipschitz continuous in all the arguments with constant βi , i 1, 2, . . .,
respectively, and using H-Lipschitz continuity of Ti ’s with constant δi , we have
N w1n , w2n , . . . − N w1n−1 , w2n−1 , . . .
N w1n , w2n , . . . − N w1n−1 , w2n , . . .
N w1n−1 , w2n , . . .
≤ N w1n , w2n , . . . − N w1n−1 , w2n , . . .
N w1n−1 , w2n , . . . − N w1n−1 , w2n−1 , . . .
≤ β1 w1n − w1n−1
β2 w2n − w2n−1
···
···
···
10
Fixed Point Theory and Applications
≤
∞
βi win − win−1
i1
≤
∞
βi HTi un , Ti un−1 i1
≤
∞
βi δi un − un−1 ,
n 0, 1, 2, . . . .
i1
3.25
Since W is a Lipschitz continuous in both the arguments with constant λW1 , λW2
respectively, and C and D are H-Lipschitz continuous with constant t and γ , respectively,
we have
W xn , yn − W xn−1 , yn−1 ≤ λW2 yn − yn−1
λW1 xn − xn−1 ≤ λW2 γ un − un−1 λ W1 t
λW1 tun − un−1 3.26
λW2 γ un − un−1 .
Combining 3.24, 3.25, and 3.26 with 3.23, we have
zn
1
λA λm α un − un−1 − zn ≤ λA λg
ρ
∞
βi δi un − un−1 i1
ρ λ W1 t
λA λg
λW2 γ un − un−1 λm α λA
ρ
ρ
ρλm αun − un−1 ∞
βi δi
ρ λ W1 t
λ W2 γ
3.27
un − un−1 .
i1
By using Proposition 2.3 and k-strong accretiveness of g, we have
un − un−1 2 man ρ,η
JM·,un zn − man−1 ρ,η
− JM·,un−1 zn−1 − gun − un − gun−1 − un−1
≤ man − man−1 2
ρ,η
ρ,η
JM·,un zn − JM·,un−1 zn−1 2
2
− 2 gun − un − gun−1 − un−1 , jun − un−1 ≤ man − man−1 2
ρ,η
ρ,η
JM·,un zn − JM·,un−1 zn ρ,η
ρ,η
JM·,un−1 zn − JM·,un−1 zn−1 − 2 gun − un − gun−1 − un−1 , jun − un−1 ,
2
Fixed Point Theory and Applications
11
un − un−1 2 ≤ λ2m α2 un − un−1 2
ρ,η
ρ,η
ρ,η
JM·,un zn − JM·,un−1 zn ρ,η
JM·,un−1 zn − JM·,un−1 zn−1 2
2
− 2 gun − un − gun−1 − un−1 , jun − un−1 ≤ λ2m α2 un − un−1 2
2
un − un−1 ≤
τ
r − ρm
λ2m α2
2
μ2 un − un−1 2
zn − zn−1 2 − 2kun − un−1 2 ,
μ − 2k un − un−1 2
2
τ
r − ρm
2
zn − zn−1 2 ,
2
τ/ r − ρm
un − un−1 ≤
zn − zn−1 2 ,
2 2
2
1 − λm α μ − 2k
2
τ
un − un−1 ≤ 2 2
r − ρm
1 − λm α
μ2 − 2k
zn − zn−1 .
3.28
Using 3.28, 3.27 becomes
zn
1
− zn ≤
λA λg
λm α λA ρ
ρ ∞
ρ λ W1 t
i1 βi δi
r − ρm 1 − λ2m α2 μ2 − 2k
that is, zn
1
λ W2 γ
τ
zn − zn−1 ,
3.29
− zn ≤ θzn − zn−1 ,
where
θ
λA λg
λm α λA ρ
ρ ∞
ρ λ W1 t
i1 βi δi
r − ρm 1 − λ2m α2 μ2 − 2k
λ W2 γ
τ
.
3.30
From 3.22, we have θ < 1, and consequently {zn } is a Cauchy sequence in E. Since E
is a Banach space, there exists z ∈ E such that zn → z. From 3.28, we know that {un }
is also a Cauchy sequence in E. Therefore, there exists u ∈ E such that un → u. Since
the mappings Ti ’s, B, C and D are H-Lipschitz continuous, it follows from 3.16–3.19 of
Algorithm 3.3 that {an }, {xn }, {yn }, and {win } are also Cauchy sequences. We can assume that
win → wi , an → a, xn → x, and yn → y.
12
Fixed Point Theory and Applications
Now, we prove that wi ∈ Ti u. In fact, since win ∈ Ti ui and
dwi , Ti u ≤ wi − win
≤ wi −
win
d win , Ti u
max
sup d q2 , Ti u , sup d Ti u, q1
q2 ∈Ti un q1 ∈Ti u
wi − win
HTi un , Ti u
≤ wi − win
δi un − un−1 −→ 0
3.31
n −→ ∞,
which implies that dwi , Ti u 0. As Ti u ∈ CBE, we have wi ∈ Ti u, i 1, 2, . . . ∞.
Finally, by the continuity of A, g, m, N, and W and by Algorithm 3.3, it follows that
ma ,
A gun − man − ρ N w1n , w2n , . . . − W xn , yn
−→ z A gu − ma − ρ Nw1 , w2 , . . . − W x, y
ma
n −→ ∞,
zn
1
ρ,η
ρ,η
JM·,un zn gun − man −→ gu − ma JM·,u z
3.32
n −→ ∞.
From 3.32, and Lemma 3.2, it follows that
Nw1 , w2 , . . . − W x, y
ma
Nw1 , w2 , . . . − W x, y
ρ,η
ρ−1 z − A JM·,u z 0,
ma
ρ−1 RM·,u z 0,
ρ,η
3.33
that is, z, u, a, x, y, w1 , w2 , . . . is a solution of resolvent equation poblem 2.17.
Acknowledgment
This work is supported by Department of Science and Technology, Government of India,
under Grant no. SR/S4/MS: 577/09.
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