Open Access

Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems

Fixed Point Theory and Applications20122012:90

https://doi.org/10.1186/1687-1812-2012-90

Received: 25 January 2012

Accepted: 24 May 2012

Published: 24 May 2012

Abstract

In this article, zero points of the sum of a maximal monotone operator and an inverse-strongly monotone mapping, solutions of a monotone variational inequality, and fixed points of a strict pseudocontraction are investigated. A hybrid projection iterative algorithm is considered for analyzing the convergence of the iterative sequences. Strong convergence theorems are established in the framework of real Hilbert spaces without any compact assumptions. Some applications of the main results are also provided.

AMS Classification: 47H05; 47H09; 47J25; 90C33.

Keywords

fixed point monotone operator strict pseudocontraction variational inequality zero point

1. Introduction

The theory of monotone operators has emerged as an effective and powerful tool for studying a wide class of unrelated problems arising in various branches of social, engineering, and pure sciences in unified and general framework. Two notions related to monotone operators have turned out to be very useful in the study of various problems involving such operators. The first one, which is inspired by the notion of subdifferential of a convex function, is the concept of enlargement of a given operator; see [13] and the references therein. It allows to make a quantitative analysis in different problems involving monotone operators, like for example variational inequalities, inclusions, etc. The second notion is the one of generalized sum of two monotone operators; see [4, 5] and the references therein. In recent years, much attention has been given to develop efficient numerical methods for treating zero point problems of monotone operators and fixed point problems of mappings which are Lipschitz continuous; see [628] and the references therein. The gradient-projection method is a powerful tool for solving constrained convex optimization problems and has extensively been studied; see [2931] and the references therein. It has recently applied to solve split feasibility problems which find applications in image reconstructions and the intensity modulated radiation theory; see [3235] and the reference therein.

In this article, zero points of the sums of a maximal monotone operator and an inverse-strongly monotone mapping, solutions of a monotone variational inequality, and fixed points of a strict pseudocontraction are investigated based on a hybrid iterative method.

The organization of this article is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a hybrid iterative method is proposed and analyzed. Strong convergence theorems for common elements in the zero point set of the sums of a maximal monotone operator and an inverse-strongly monotone mapping, the solution set of a monotone variational inequality, and the fixed point set of a strict pseudocontraction are established in the framework of real Hilbert spaces without any compact assumptions. In Section 4, applications of the main results are discussed.

2. Preliminaries

In what follows, we always assume that H is a real Hilbert space with inner product 〈· , ·〉 and norm || · ||. Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonlinear mapping. F(S) stands for the fixed point set of S; that is, F(S):= {x C : x = Tx}.

Recall that S is said to be nonexpansive iff
| | S x - S y | | | | x - y | | , x , y C .

If C is a bounded, closed, and convex subset of H, then F(S) is not empty, closed, and convex; see [36].

S is said to be κ-strictly pseudocontractive iff there exists a constant κ [0, 1) such that
| | S x - S y | | 2 | | x - y | | 2 + κ | | x - S x - y + S y | | 2 , x , y C .

It is clear that the class of κ-strictly pseudocontractive mappings includes the class of non-

expansive mappings.

Let A : CH be a mapping. A is said to be monotone iff
A x - A y , x - y 0 , x , y C .
A is said to be inverse-strongly monotone iff there exists a constant α > 0 such that
A x - A y , x - y α | | A x - A y | | 2 , x , y C .

For such a case, A is also said to be α-inverse-strongly monotone.

A is said to be Lipschitz continuous iff there exists a positive constant L such that
| | A x - A y | | L | | x - y | | , x , y C .
Recall that the classical variational inequality is to find an x C such that
A x , y - x 0 , y C .
(2.1)

It is known that x C is a solution to (2.1) if and only if x is a fixed point of the mapping Proj C (I - rA), where r > 0 is a constant, I stands for the identity mapping, and Proj C stands for the metric projection from H onto C. If A is α-inverse-strongly monotone and r (0, 2α], then the mapping Proj C (I - rA) is nonexpansive; see [37] for more details. It follows that V I(C, A), where V I(C, A) stands for the solution set of (2.1), is closed and convex.

A set-valued mapping R : H H is said to be monotone iff, for all x, y H, f Rx and g Ry imply 〈x - y, f - g> 0. A monotone mapping R : H H is maximal iff the graph G(R) of R is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping R is maximal if and only if, for any (x, f) H × H, 〈x - y, f - g〉 ≥ 0, for all (y, g) G(R) implies f Rx.

The class of monotone operators is one of the most important classes of operators. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of zero points for maximal monotone operators; see [3845] and the references therein. For a maximal monotone operator M on H and r > 0, we may define the single-valued resolvent J r : HD(M ), where D(M ) denotes the domain of M. It is known that J r is firmly nonexpansive and M -1(0) = F(J r ), where F (J r ):= {x D(M ): x = J r x}, and M -1(0): {x H : 0 Mx}.

In this article, zero points of the sums of a maximal monotone operator and an inverse-strongly monotone mapping, solutions of a monotone variational inequality, and fixed points of a strict pseudocontraction are investigated. A hybrid iterative algorithm is considered for analyzing the convergence of iterative sequences. Strong convergence theorems are established in the framework of real Hilbert spaces without any compact assumptions.

In order to prove our main results, we also need the following definitions and lemmas.

Lemma 2.1 [46]. Let C be a nonempty, closed, and convex subset of H, and S : CC a κ-strict pseudocontraction. Define a mapping S α x = βx + (1 - β)Sx for all x C. If β [κ, 1), then the mapping S β is a nonexpansive mapping such that F (S β ) = F (S).

Lemma 2.2 [47]. Let C be a nonempty, closed, and convex subset of H. Let S : CC be a nonexpansive mapping. Then the mapping I - S is demiclosed at zero, that is, if {x n } is a sequence in C such that x n x ̄ and x n - Sx n → 0, then x ̄ F ( S ) .

Lemma 2.3. Let C be a nonempty, closed, and convex subset of H, B : CH a mapping, and M : H H a maximal monotone operator. Then F(J r (I - sB)) = (B + M)-1(0).

Proof. Notice that
p F ( J r ( I - s B ) ) p = J r ( I - s B ) p p - s B p p + s M p 0 ( B + M ) - 1 ( 0 ) p ( B + M ) - 1 ( 0 ) .

This completes the proof.

Lemma 2.4 [48]. Let C be a nonempty, closed, and convex subset of H, A : CH a Lipschitz monotone mapping, and N C x the normal cone to C at x C; that is, N C x = {y H : 〈x - u, y〉, u C}. Define
W x = A x + N C x , x C , x C .

Then W is maximal monotone and 0 Wx if and only if x V I(C, A).

3. Main results

Now, we are in a position to give our main results.

Theorem 3.1. Let C be a nonempty, closed, and convex subset of H. Let S : CC be a κ-strict pseudocontraction with a nonempty fixed point set, A : CH an α-inverse-strongly monotone mapping, and B : CH a β-inverse-strongly monotone mapping. Let M : H H be a maximal monotone operator such that D(M) C. Assume that F : = F ( S ) ( B + M ) - 1 ( 0 ) V I ( C , A ) is not empty. Let{x n } be a sequence generated by the following iterative process:
x 1 C , C 1 = C , z n = Pro j C ( J s n ( x n - s n B x n ) - r n A J s n ( x n - s n B x n ) ) , y n = α n x n + ( 1 - α n ) ( β n z n + ( 1 - β n ) S z n ) , C n + 1 = { v C n : | | y n - v | | | | x n - v | | } , x n + 1 = Pro j C n + 1 x 1 , n 0 ,
(3.1)
where J s n = ( I + s n M ) - 1 , {r n } is a sequence in (0, 2α), {s n } is a sequence in (0, 2β), and {α n } and {β n } are sequences in (0, 1). Assume that the following restrictions are satisfied
  1. (a)

    0 ≤ α n a < 1, κβ n b < 1;

     
  2. (b)

    0 < rr n r' < 2α;

     
  3. (c)

    0 < ss n s' < 2β,

     

where a, b, r, r', s, and s' are real constants. Then the sequence {x n } converges strongly to Pro j F x 1 .

Proof. First, we show that C n is closed and convex for each n ≥ 1. From the assumption, we see that C1 = C is closed and convex. Suppose that C m is closed and convex for some m ≥ 1. We show that C m +1 is closed and convex for the same m. Let v1, v2 C m +1 and v = tv1 + (1 - t)v2, where t (0, 1). Notice that
| | y m - v | | | | x m - v | |
is equivalent to
| | y m | | 2 - | | x m | | 2 - 2 v , y m - x m 0 .
It is clearly to see that v C m +1. This shows that C n is closed and convex for each n ≥ 1. Put
v n = J s n ( x n - s n B x n ) ,
and
u n = S n z n ,
where S n is defined by
S n x = β n x + ( 1 - β n ) S x , x C .
We see from Lemma 2.1 that S n is nonexpansive with F (S n ) = F (S). Since A is α-inverse-strongly monotone, and B is β-inverse-strongly monotone, we see from the restriction (b) that
| | ( x - r m A x ) - ( y - r m A y ) | | 2 = | | x - y | | 2 - 2 r m x - y , A x - A y + r m 2 m | | A x - A y | | 2 | | x - y | | 2 - r m ( 2 α - r m ) | | A x - A y | | 2 | | x - y | | 2 , x , y C ,
(3.2)
and
| | ( I - s m B ) x - ( I - s m B ) y | | 2 = | | x - y | | 2 - 2 s m x - y , B x - B y + s m 2 | | B x - B y | | 2 | | x - y | | 2 - s m ( 2 β - s m ) | | B x - B y | | 2 | | x - y | | 2 , x , y C .
(3.3)
Now, we show that F C n for each n ≥ 1. Notice that F C = C 1 . Suppose that F C m for some m ≥ 1. For any p F C m , we see from (3.2), and (3.3) that
| | y m - p | | α m | | x m - p | | + ( 1 - α m ) | | u m - p | | α m | | x m - p | | + ( 1 - α m ) | | z m - p | | α m | | x m - p | | + ( 1 - α m ) | | ( v m - r m A v m ) - ( p - r m A p ) | | α m | | x m - p | | + ( 1 - α m ) | | ( x m - s m B x m ) - ( p - s m B p ) | | α m | | x m - p | | + ( 1 - α m ) | | x m - p | | = | | x m - p | | .
(3.4)
This shows that p C m +1. This proves that F C n . Note that x n = Pro j C n x 1 . For each p F C n , we have || x1 - x n || ≤ || x1 - p ||. Since B is inverse-strongly monotone, we see from Lemma 2.3 that (B + M)-1(0) is closed, and convex. Since A is Lipschitz continuous, we find that VI(C, A) is close, and convex. In view of Lemma 2.2, we obtain F(S) is closed, and convex. This proves that is closed and convex. It follows that
| | x 1 - x n | | | | x 1 - Pro j F x 1 | | .
(3.5)
This implies that {x n } is bounded. Since x n = Pro j C n x 1 and x n + 1 = Pro j C n + 1 x 1 C n + 1 C n , we have
0 x 1 - x n , x n - x n + 1 = x 1 - x n , x n - x 1 + x 1 - x n + 1 - | | x 1 - x n | | 2 + | | x 1 - x n | | | | x 1 - x n + 1 | | .
It follows that
| | x n - x 1 | | | | x n + 1 - x 1 | | .
This proves that lim n →∞ || x n - x1 || exists. Notice that
| | x n - x n + 1 | | 2 = | | x n - x 1 | | 2 + 2 x n - x 1 , x 1 - x n + 1 + | | x 1 - x n + 1 | | 2 = | | x n - x 1 | | 2 - 2 | | x n - x 1 | | 2 + 2 x n - x 1 , x n - x n + 1 + | | x 1 - x n + 1 | | 2 | | x 1 - x n + 1 | | 2 - | | x n - x 1 | | 2 .
It follows that
lim n | | x n - x n + 1 | | = 0 .
(3.6)
In view of x n + 1 = Pro j C n + 1 x 1 C n + 1 , we see that
| | y n - x n + 1 | | | | x n - x n + 1 | | .
This implies that
| | y n - x n | | | | y n - x n + 1 | | + | | x n - x n + 1 | | 2 | | x n - x n + 1 | | .
We, therefore, obtain from (3.6) that
lim n | | x n - y n | | = 0 .
(3.7)
On the other hand, we see from (3.3) that
| | y n - p | | 2 α n | | x n - p | | 2 + ( 1 - α n ) | | u n - p | | 2 α n | | x n - p | | 2 + ( 1 - α n ) | | z n - p | | 2 α n | | x n - p | | 2 + ( 1 - α n ) | | v n - p | | 2 = α n | | x n - p | | 2 + ( 1 - α n ) | | J s n ( x n - s n B x n ) - J s n ( p - s n B p ) | | 2 | | x n - p | | 2 - ( 1 - α n ) s n ( 2 β - s n ) | | B x n - B p | | 2 .
It follows that
( 1 - α n ) s n ( 2 β - s n ) | | B x n - B p | | 2 | | x n - p | | 2 - | | y n - p | | 2 | | x n - y n | | ( | | x n - p | | + | | y n - p | | ) .
In view of the restrictions (a), and (c), we find from (3.7) that
lim n | | B x n - B p | | = 0 .
(3.8)
Since J s n is firmly nonexpansive, we find that
| | v n - p | | 2 = | | J s n ( x n - s n B x n ) - J s n ( p - s n B p ) | | 2 v n - p , ( x n - s n B x n ) - ( p - s n B p ) = 1 2 ( | | v n - p | | 2 + | | ( x n - s n B x n ) - ( p - s n B p ) | | 2 - | | ( v n - p ) - ( ( x n - s n B x n ) - ( p - s n B p ) ) | | 2 ) 1 2 ( | | v n - p | | 2 + | | x n - p | | 2 - | | v n - x n + s n ( B x n - B p ) | | 2 ) = 1 2 ( | | v n - p | | 2 + | | x n - p | | 2 - | | v n - x n | | 2 - s n 2 | | B x n - B p | | 2 - 2 s n v n - x n , B x n - B p ) 1 2 ( | | v n - p | | 2 + | | x n - p | | 2 - | | v n - x n | | 2 + 2 s n | | v n - x n | | | | B x n - B p | | ) .
This finds that
| | v n - p | | 2 | | x n - p | | 2 - | | v n - x n | | 2 + 2 s n | | v n - x n | | | | B x n - B p | | .
(3.9)
It follows from (3.1) that
| | y n - p | | 2 α n | | x n - p | | 2 + ( 1 - α n ) | | u n - p | | 2 | | x n - p | | 2 - ( 1 - α n ) | | v n - x n | | 2 + 2 s n | | v n - x n | | | | B x n - B p | | ,
which in turn implies that
( 1 - α n ) | | v n - x n | | 2 | | x n - p | | 2 - | | y n - p | | 2 + 2 s n | | v n - x n | | | | B x n - B p | | | | x n - y n | | ( | | x n - p | | + | | y n - p | | ) + 2 s n | | v n - x n | | | | B x n - B p | | .
In view of the restriction (a), we see from (3.7), and (3.8) that
lim n | | v n - x n | | = 0 .
(3.10)
On the other hand, we see from (3.2) that
| | y n - p | | 2 α n | | x n - p | | 2 + ( 1 - α n ) | | u n - p | | 2 α n | | x n - p | | 2 + ( 1 - α n ) | | ( v n - r n A v n ) - ( p - r n A p ) | | 2 | | x n - p | | 2 - ( 1 - α n ) r n ( 2 α - r n ) | | A v n - A p | | 2 .
It follows that
( 1 - α n ) r n ( 2 α - r n ) | | A v n - A p | | 2 | | x n - p | | 2 - | | y n - p | | 2 | | x n - y n | | ( | | x n - p | | + | | y n - p | | ) .
In view of the restrictions (a), and (b), we find from (3.7) that
lim n | | A v n - A p | | = 0 .
(3.11)
Since Proj C is firmly nonexpansive, we arrive at
| | z n - p | | 2 = | | Pro j C ( v n - r n A v n ) - Pro j C ( p - r n A p ) | | 2 z n - p , ( v n - r n A v n ) - ( p - r n A p ) = 1 2 ( | | z n - p | | 2 + | | ( v n - r n A v n ) - ( p - r n A p ) | | 2 - | | ( z n - p ) - ( ( v n - r n A v n ) - ( p - r n A p ) ) | | 2 ) 1 2 ( | | z n - p | | 2 + | | x n - p | | 2 - | | z n - v n + r n ( A v n - A p ) | | 2 ) = 1 2 ( | | z n - p | | 2 + | | x n - p | | 2 - | | z n - v n | | 2 - r n 2 | | A v n - A p | | 2 - 2 r n z n - v n , A v n - r p ) 1 2 ( | | z n - p | | 2 + | | x n - p | | 2 - | | z n - v n | | 2 + 2 r n | | z n - v n | | | | A v n - A p | | ) ,
which finds that
| | z n - p | | 2 | | x n - p | | 2 - | | z n - v n | | 2 + 2 r n | | z n - v n | | | | A v n - A p | | .
(3.12)
This implies that
| | y n - p | | 2 α n | | x n - p | | 2 + ( 1 - α n ) | | u n - p | | 2 | | x n - p | | 2 - ( 1 - α n ) | | z n - v n | | 2 + 2 r n | | z n - v n | | | | A v n - A p | | ,
It follows that
( 1 - α n ) | | z n - v n | | 2 | | x n - p | | 2 - | | y n - p | | 2 + 2 r n | | z n - v n | | | | A x n - A p | | | | x n - y n | | ( | | x n - p | | + | | y n - p | | ) + 2 r n | | z n - v n | | | | A x n - A p | | .
In view of the restriction (a), we see from (3.7), and (3.11) that
lim n | | z n - v n | | = 0 .
(3.13)
On the other hand, we have
| | x n - y n | | = | | x n - α n x n - ( 1 - α n ) S n z n | | = ( 1 - α n ) | | x n - S n z n | | .
In view of (3.7), we see from the restriction (a) that
lim n | | x n - S n z n | | = 0 .
(3.14)
Note that
| | z n - x n | | | | z n - v n | | + | | v n - x n | | .
It follows from (3.10) and (3.13) that
lim n | | z n - x n | | = 0 .
(3.15)
In view of
| | x n - S n x n | | | | S n x n - S n z n | | + | | S n z n - x n | | | | x n - z n | | + | | S n z n - x n | | ,
we see from (3.14) and (3.15) that
lim n | | x n - S n x n | | = 0 .
(3.16)
Note that
| | S x n - x n | | | | S x n - S n x n | | + | | S n x n - x n | | β n | | S x n - x n | | + | | S n x n - x n | | ,
which yields that
( 1 - β n ) | | S x n - x n | | | | S n x n - x n | | .
In view of the restriction (b), we conclude from (3.16) that
lim n | | S x n - x n | | = 0 .
(3.17)

Since {x n } is bounded, there exists a subsequence { x n i } of {x n } such that x n i q . In view of Lemma 2.2, we obtain from (3.17) that q F(S). In view of (3.10), and (3.15), we see that u n i q , and z n i q , respectively. Now, we are in a position to show that q VI(C, A).

Define
W x = A x + N C x , x C , , x C .
Then W is maximal monotone. Let (x, y) G(W). Since y - Ax N C x and z n C, we have
x - z n , y - A x 0 .
On the other hand, we have from z n = Proj C (I - r n A1)v n that
x - z n , z n - ( I - r n A ) v n 0
and hence
x - z n , z n - v n r n + A v n 0 .
It follows that
x - z n i , y x - z n i , A x x - z n i , A x - x - z n i , z n i - v n i r n i + A v n i x - z n i , A x - A z n i + x - z n i , A z n i - A v n i - x - z n i , z n i - v n i r n i x - z n i , A z n i - A v n i - x - z n i , z n i - v n i r n i .

In view of the restriction (b), we obtain from (3.13) that 〈x - q, y〉 ≥ 0. We have q A - 10 and hence q VI(C, A).

Next, we prove that q (B + M) - 1(0). Notice that
x n - s n B x n v n + s n M v n ;
that is,
x n - v n s n - B x n M v n .
(3.18)
Let µ ν. Since M is monotone, we find from (3.18) that
x n - v n s n - B x n - μ , v n - ν 0 .
In view of the restriction (c), we see from (3.10) that
- B q - μ , q - ν 0 .

This implies that -Bq Mq, that is, q (B + M)-1(0). This completes q F . Assume that there exists another subsequence { x n i } of {x n } weak converges weakly to q F . We can easily conclude from Opial's condition (see [49]) that q = q'.

Finally, we show that q = Pro j F x 1 and {x n } converges strongly to q. This completes the proof of Theorem 3.1. In view of the weak lower semicontinuity of the norm, we obtain from (3.5) that
| | x 1 - Pro j F x 1 | | | | x 1 - q | | lim inf n | | x 1 - x n | | lim sup n | | x 1 - x n | | | | x 1 - >Pro j F x 1 | | ,

which yields that lim n | | x 1 - x n | | = | | x 1 - Pro j F x 1 | | = | | x 1 - q | | . It follows that {x n } converges strongly to Pro j F x 1 . This completes the proof.

We conclude from Theorem 3.1 the following results on nonexpansive mappings.

Corollary 3.2. Let C be a nonempty, closed, and convex subset of H. Let S : CC be a nonexpansive mapping with a nonempty fixed point set, A : CH be an α-inverse-strongly monotone mapping, and B : CH be a β-inverse-strongly monotone mapping. Let M : H H be a maximal monotone operator such that D(M) C. Assume that F : = F ( S ) ( B + M ) - 1 ( 0 ) V I ( C , A ) is not empty. Let {x n } be a sequence generated by the following iterative process:
x 1 C , C 1 = C , z n = Pro j C ( J s n x n - s n B x n ) - r n A J s n ( x n - s n B x n ) ) , y n = α n x n + ( 1 - α n ) S z n , C n + 1 = { v C n : | | y n - v | | | | x n - v | | } , x n + 1 = Pro j C n + 1 x 1 , n 0 ,
where J s n = ( I + s n M ) - 1 , {r n } is a sequence in (0, 2α), {s n } is a sequence in (0, 2β), and { α n } is a sequence in (0, 1). Assume that the following restrictions are satisfied
  1. (a)

    0 ≤ α n a < 1;

     
  2. (b)

    0 < rr n r' < 2α;

     
  3. (c)

    0 < ss n s' < 2β,

     

where a, r, r', s, and s' are real constants. Then the sequence {x n } converges strongly to Pro j F x 1 .

If A = 0, then Corollary 3.2 is reduced to the following.

Corollary 3.3. Let C be a nonempty, closed, and convex subset of H. Let S : CC be a nonexpansive mapping with a nonempty fixed point set, and B : CH be a β-inverse-strongly monotone mapping. Let M : H H be a maximal monotone operator such that D(M) C. Assume that F : = F ( S ) ( B + M ) - 1 ( 0 ) is not empty. Let {x n } be a sequence generated by the following iterative process:
x 1 C , C 1 = C , y n = α n x n + ( 1 - α n ) S J s n ( x n - s n B x n ) , C n + 1 = { v C n : | | y n - v | | | | x n - v | | } , x n + 1 = Pro j C n + 1 x 1 , n 0 ,

where J s n = ( I + s n M ) - 1 {s n } is a sequence in (0, 2β), and {α n } is a sequence in (0, 1).

Assume that the following restrictions are satisfied
  1. (a)

    0 ≤ α n a < 1;

     
  2. (b)

    0 < ss n s' < 2β,

     

where a, s, and s' are real constants. Then the sequence {x n } converges strongly to Pro j F x 1 .

If B = 0, then Corollary 3.2 is reduced to the following.

Corollary 3.4. Let C be a nonempty, closed, and convex subset of H. Let S : CC be a nonexpansive mapping with a nonempty fixed point set, A : CH a α-inverse-strongly monotone mapping. Let M : H H be a maximal monotone operator such that D(M) C. Assume that F : = F ( S ) M - 1 ( 0 ) V I ( C , A ) is not empty. Let {x n } be a sequence generated by the following iterative process:
x 1 C , C 1 = C , z n = Pro j C ( J s n x n - r n A J s n x n ) ) , y n = α n x n + ( 1 - α n ) S z n , C n + 1 = { v C n : | | y n - v | | | | x n - v | | } , x n + 1 = Pro j C n + 1 x 1 , n 0 ,
where J s n = ( I + s n M ) - 1 ,{r n } is a sequence in (0, 2α), {s n } is a sequence in (0, +∞), and {α n } is a sequence in (0, 1). Assume that the following restrictions are satisfied
  1. (a)

    0 ≤ α n a < 1;

     
  2. (b)

    0 < rr n r' < 2α;

     
  3. (c)

    0 < ss n < ∞,

     

where a, r, r', and s are real constants. Then the sequence {x n } converges strongly to Pro j F x 1 .

Let f : H → (-∞, +∞] be a proper convex lower semicontinuous function. Then the subdifferential of f is defined as follows
f ( x ) = { y H : f ( z ) f ( x ) + z - x , y , z H } , x H .

From Rockafellar [50], we know that ∂f is maximal monotone. It is not hard to verify that 0 f (x) if and only if f ( x ) = min y H f ( y ) .

Let I C be the indicator function of C, i.e.,
I C ( x ) = 0 , x C , + , x C .
Since I C is a proper lower semicontinuous convex function on H, we see that the subdifferential ∂I C of I C is a maximal monotone operator. It is clearly that J s x = Proj C x, x H. Notice that (B + ∂I C )- 1(0) = V I(C, B). Indeed,
x ( B + I C ) - 1 ( 0 ) 0 B x + I C x - B X I C x B x , y - x 0 x V I ( C , B ) .

In view of Theorem 3.1, we have the following.

Corollary 3.5. Let C be a nonempty, closed, and convex subset of H. Let S : CC be aα κ -strict pseudocontraction with a nonempty fixed point set, A : CH be an α-inverse-strongly monotone mapping, and B : CH be a β-inverse-strongly monotone mapping. Assume hat F : = F ( S ) V I ( C , B ) V I ( C , A ) is not empty. Let {x n } be a sequence generated by he following iterative process:
x 1 C , C 1 = C , z n = Pro j C ( Pro j C ( x n - s n B x n ) - r n A Pro j C ( x n - s n B x n ) ) , y n = α n x n + ( 1 - α n ) ( β n z n + ( 1 - β n ) S z n ) , C n + 1 = { v C n : y n - v x n - v } , x n + 1 = Pro j C n   + 1 x 1 , n 0 ,
where {r n } is a sequence in (0, 2α), {s n } is a sequence in (0, 2β), and {α n } and {β n } are sequences in (0, 1). Assume that the following restrictions are satisfied
  1. (a)

    0 ≤ α n a < 1, κβ n b < 1;

     
  2. (b)

    0 < rr n r' < 2α;

     
  3. (c)

    0 < ss n s' < 2β,

     

where a, b, r, r', s, and s' are real constants. Then the sequence {x n } converges strongly to Pr o j F x 1 .

4. Applications

Let F be a bifunction of C × C into , where denotes the set of real numbers. Recall the following equilibrium problem in the terminology of Blum and Oettli [51] (see also Fan [52]).
Find x C such that F ( x , y ) 0 , y C .
(4.1)

To study the equilibrium problem (4.1), we may assume that F satisfies the following conditions:

(A1) F(x, x) = 0 for all x C;

(A2) F is monotone, i.e., F(x, y) + F(y, x) = 0 for all x, y C;

(A3) for each x, y, z C,
lim >sup t 0 F ( t z + ( 1 - t ) x , y ) F ( x , y ) ;

(A4) for each xC,yF(x,y) is convex and lower semi-continuous.

Putting F(x, y) = 〈Ax, y - x〉 for every x, y C, we see that the equilibrium problem (4.1) is reduced to the variational inequality (2.1).

The following lemma can be found in [51, 53].

Lemma 4.1. Let C be a nonempty, closed, and convex subset of H and F:C xC→ a bifunction satisfying (A1)-(A4). Then, for any s > 0 and x H, there exists z C such that
F ( z , y ) + 1 s y - z , z - x 0 , y C .
Further, define
T s x = z C : F ( z , y ) + 1 s y - z , z - x 0 , y C
(4.2)
for all s > 0 and × H. Then, the following hold:
  1. (a)

    T s is single-valued;

     
  2. (b)
    T s is firmly nonexpansive; that is,
    T s x - T s y 2 T s x - T s y , x - y , x , y H ;
     
  3. (c)

    F(T s ) = EP (F );

     
  4. (d)

    EP(F) is closed and convex.

     
Lemma 4.2 [8]. Let C be a nonempty, closed, and convex subset of H, F a bifunction from C×C to which satisfies (A1)-(A4), and A F a multivalued mapping of H into itself defined by
A F X = { z H : F ( x , y ) y - x , z , y C } , x C , , x C .
(4.3)
Then A F is a maximal monotone operator with the domain D(A F ) C, E P ( F ) = A F - 1 ( 0 ) , where FP(F) stands for the solution set of (4.1), and
T s x = ( I + s A F ) - 1 x , x H , r > 0 ,

where T s is defined as in (4.2).

In this section, we consider the problem of approximating a solution of the equilibrium problem.

Theorem 4.3. Let C be a nonempty, closed, and convex subset of H. Let S : CC be a κ-strict pseudocontraction with a nonempty fixed point set, and F:C×C→ a bifunction satisfying (A1)-(A4). Assume that F : = F ( S ) E P ( F ) is not empty. Let{x n } be a sequence generated by the following iterative process:
x 1 C , C 1 = C , y n = α n x n + ( 1 - α n ) ( β n ( I + s n A F ) - 1 x n + ( 1 - β n ) S ( I + s n A F ) - 1 x n ) , C n + 1 = { v C n : y n - v x n - v } , x n + 1 = Pro j C n + 1 x 1 , n 0 ,
where A F is defined by (4.3), {s n } is a positive sequence, and {α n } and {β n } are sequences in (0, 1). Assume that the following restrictions are satisfied
  1. (a)

    0 ≤ α n a < 1, κβ n b < 1;

     
  2. (b)

    0 < ss n s' < ∞,

     

where a, b, s, and s' are real constants. Then the sequence {x n } converges strongly to Pro j F x 1 .

Proof. Putting A = B = 0, we immediately conclude from Lemmas 4.1 and 4.2 the desired conclusion.

Declarations

Authors’ Affiliations

(1)
School of Business and Administration, Henan University
(2)
Department of Mathematics, Huanghuai University

References

  1. Burachik RS, Iusem AN, Svaiter BF: Enlargements of maximal monotone operators with applications to variational inequalities. Set-valued Anal 1997, 5: 159–180. 10.1023/A:1008615624787MathSciNetView ArticleGoogle Scholar
  2. Revalski JP, Théra A: Enlargements and sums of monotone operators. Nonlinear Anal 2002, (48:):505–519.Google Scholar
  3. Svaiter BF, Burachik RS: ε -enlargements of maximal monotone operators in Banach spaces. Set-Valued Anal 1999, 7: 117–132. 10.1023/A:1008730230603MathSciNetView ArticleGoogle Scholar
  4. Moudafi A: On the regularization of the sum of two maximal monotone operators. Nonlinear Anal 2000, 42: 1203–1208. 10.1016/S0362-546X(99)00136-4MathSciNetView ArticleGoogle Scholar
  5. Moudafi A, Oliny M: Convergence of a splitting inertial proximal method for monotone operators. J Comput Appl Math 2003, 155: 447–454. 10.1016/S0377-0427(02)00906-8MathSciNetView ArticleGoogle Scholar
  6. Zhang SS, Lee JHW, Chan CK: Algorithms of common solutions for quasi variational inclusion and fixed point problems. Appl Math Mech 2008, 29: 571–581. 10.1007/s10483-008-0502-yMathSciNetView ArticleGoogle Scholar
  7. Qin X, Kang JI, Cho YJ: On quasi-variational inclusions and asymptotically strict pseudo-contractions. J Nonlinear Convex Anal 2010, 11: 441–453.MathSciNetGoogle Scholar
  8. Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J Optim Theory Appl 2010, 147: 27–41. 10.1007/s10957-010-9713-2MathSciNetView ArticleGoogle Scholar
  9. Qin X, Cho SY, Kang SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J Global Optim 2010, 48: 423–445. 10.1007/s10898-009-9498-8MathSciNetView ArticleGoogle Scholar
  10. Korpelevich GM: An extragradient method for finding saddle points and for other problems. Ekonomika i Matematicheskie Metody 1976, 12: 747–756.MathSciNetGoogle Scholar
  11. Yang S, Li W: Iterative solutions of a system of equilibrium problems in Hilbert spaces. Adv Fixed Point Theory 2011, 1: 15–26.Google Scholar
  12. Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl 2006, 128: 191–201. 10.1007/s10957-005-7564-zMathSciNetView ArticleGoogle Scholar
  13. Qin X, Su Y, Shang M: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math Comput Model 2008, 48: 1033–1046. 10.1016/j.mcm.2007.12.008MathSciNetView ArticleGoogle Scholar
  14. Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl 2011 2011., 10:Google Scholar
  15. Kim JK, Cho SY, Qin X: Some results on generalized equilibrium problems involving strictly pseudocontractive mappings. Acta Mathematica Scientia 2011, 31: 2041–2057. 10.1016/S0252-9602(11)60380-9MathSciNetView ArticleGoogle Scholar
  16. Kim JK, Cho SY, Qin X: Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings. J Inequal Appl 2010, 2010: 18. (Article ID 312602)MathSciNetGoogle Scholar
  17. Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J Math Comput Sci 2011, 1: 1–18.MathSciNetView ArticleGoogle Scholar
  18. Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J Comput Appl Math 2009, 225: 20–30. 10.1016/j.cam.2008.06.011MathSciNetView ArticleGoogle Scholar
  19. Lv S: Generalized systems of variational inclusions involving (A, η )-monotone mappings. Adv Fixed Point Theory 2011, 1: 1–14.Google Scholar
  20. Kang SM, Cho SY, Liu Z: Convergence iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J Inequal Appl 2010, 2010: 827082.MathSciNetGoogle Scholar
  21. Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi- ϕ -nonexpansive mappings and equilibrium problems. J Comput Appl Math 2010, 234: 750–760. 10.1016/j.cam.2010.01.015MathSciNetView ArticleGoogle Scholar
  22. Chang SS, Lee HWJ, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal 2009, 70: 3307–3319. 10.1016/j.na.2008.04.035MathSciNetView ArticleGoogle Scholar
  23. Chang SS, Lee HWJ, Chan CK: A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications. Nonlinear Anal 2010, 73: 2260–2270. 10.1016/j.na.2010.06.006MathSciNetView ArticleGoogle Scholar
  24. Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal Real World Appl 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017MathSciNetView ArticleGoogle Scholar
  25. Qin X, Cho SY, Kang SM: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings. J Comput Appl Math 2009, 233: 231–240. 10.1016/j.cam.2009.07.018MathSciNetView ArticleGoogle Scholar
  26. Shehu Y: A new iterative scheme for a countable family of relatively nonexpansive mappings and an equilibrium problem in Banach spaces. J Global Optim doi:10.1007/s10898–011–9775–1Google Scholar
  27. Saewan S, Kumam P: Modified hybrid block iterative algorithm for convex feasibility problems and generalized equilibrium problems for uniformly quasi- ϕ -asymptotically nonexpansive mappings. Abstr Appl Anal 2010., 22: (Article ID 357120)Google Scholar
  28. Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal 2010, 72: 99–112. 10.1016/j.na.2009.06.042MathSciNetView ArticleGoogle Scholar
  29. Polyak BT: Introduction to optimization, Optimization Software. Princeton Hall, New York 1987.Google Scholar
  30. Calamai PH, Moré JJ: Projected gradient methods for linearly constrained problems. Math Program 1987, 39: 93–116. 10.1007/BF02592073View ArticleGoogle Scholar
  31. Levitin ES, Polyak BT: Constrained minimization methods. Zh Vychisl Mat Mat Fiz 1966, 6: 787–823.Google Scholar
  32. Byrne B: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl 2008, 20: 103–120.MathSciNetView ArticleGoogle Scholar
  33. Censor Y, Elfving T, Kopf N, Bortfeld T: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl 2005, 21: 2071–2084. 10.1088/0266-5611/21/6/017MathSciNetView ArticleGoogle Scholar
  34. Censor Y, Bortfeld T, Martin B, Trofimov A: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys Med Biol 2006, 51: 2353–2365. 10.1088/0031-9155/51/10/001View ArticleGoogle Scholar
  35. Lopez G, Martin V, Xu HK: Perturbation techniques for nonexpansive mappings with applications. Nonlinear Anal 2009, 10: 2369–2383. 10.1016/j.nonrwa.2008.04.020MathSciNetView ArticleGoogle Scholar
  36. Browder FE: Nonexpansive nonlinear operators in a Banach space. Proc Nat Acad Sci USA 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041MathSciNetView ArticleGoogle Scholar
  37. Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal 2005, 61: 341–350. 10.1016/j.na.2003.07.023MathSciNetView ArticleGoogle Scholar
  38. Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. J Math Anal Appl 2007, 329: 415–424. 10.1016/j.jmaa.2006.06.067MathSciNetView ArticleGoogle Scholar
  39. Cho YJ, Qin X, Kang JI: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Anal 2009, 71: 4203–4214. 10.1016/j.na.2009.02.106MathSciNetView ArticleGoogle Scholar
  40. Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl 2003, 118: 417–428. 10.1023/A:1025407607560MathSciNetView ArticleGoogle Scholar
  41. Cho YJ, Kang SM, Zhou H: Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces. J Inequal Appl 2008, 2008: 598191.MathSciNetGoogle Scholar
  42. Eckstein J: Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming. Math Oper Res 1993, 18: 202–226. 10.1287/moor.18.1.202MathSciNetView ArticleGoogle Scholar
  43. Matsushita SY, Takahashi W: Existence theorems for set-valued operators in Banach spaces. Set-Valued Anal 2007, 15: 251–264. 10.1007/s11228-006-0030-8MathSciNetView ArticleGoogle Scholar
  44. Solodov MV, Svaiter FF: A hybrid projection-proximal point algorithm. J Convex Anal 1999, 6: 59–70.MathSciNetGoogle Scholar
  45. Qin X, Cho YJ, Kang SM: Approximating zeros of monotone operators by proximal point algorithms. J Glob Optim 2010, 46: 75–87. 10.1007/s10898-009-9410-6MathSciNetView ArticleGoogle Scholar
  46. Zhou H: Convergence theorems of fixed points for κ -strict pseudo-contractions in Hilbert spaces. Non-linear Anal 2008, 69: 456–462.View ArticleGoogle Scholar
  47. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc Symp Pure Math 1976, 18: 78–81.Google Scholar
  48. Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView ArticleGoogle Scholar
  49. Opial Z: Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull Am Math Soc 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleGoogle Scholar
  50. Rockafellar RT: Characterization of the subdifferentials of convex functions. Pac J Math 1996, 17: 497–510.MathSciNetView ArticleGoogle Scholar
  51. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math Stud 1994, 63: 123–145.MathSciNetGoogle Scholar
  52. Fan K: A minimax inequality and applications. In Inequality III. Edited by: Shisha. Academic Press, New york; 1972:103–113.Google Scholar
  53. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 2005, 6: 117–136.MathSciNetGoogle Scholar

Copyright

© Wu and Liu; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.