Hybrid methods for accretive variational inequalities involving pseudocontractions in Banach spaces

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Abstract

We use strongly pseudocontractions to regularize a class of accretive variational inequalities in Banach spaces, where the accretive operators are complements of pseudocontractions and the solutions are sought in the set of fixed points of another pseudocontraction. In this paper, we consider an implicit scheme that can be used to find a solution of a class of accretive variational inequalities.

Our results improve and generalize some recent results of Yao et al. (Fixed Point Theory Appl, doi:10.1155/2011/180534, 2011) and Lu et al. (Nonlinear Anal, 71(3-4), 1032-1041, 2009).

2000 Mathematics subject classification 47H05; 47H09; 65J15

Keywords

hybrid method accretive operator variational inequality pseudocontraction 

1. Introduction

Throughout this paper, we always assume that E is a real Banach space, 〈· , ·〉 is the dual pair between E and E*, and 2 E denotes the family of all the nonempty subsets of E. Let C be a nonempty closed convex subset of E and T : CE be a nonlinear mapping. Denote by Fix(T) the set of fixed points of T, that is, Fix(T) = {xC : Tx = x}. The generalized duality mapping J : E → 2E*is defined by
J ( x ) = { f * E * : x , f * = x , f * = x } , x E . Open image in new window

In the sequel, we shall denote the single-valued duality mapping by j. When {x n } is a sequence in E, x n x (x n x, x n x) will denote strong (respectively, weak and weak*) convergence of the sequence {x n } to x.

A mapping T with domain D(T) and range R(T) in E is called pseudocontractive if the inequality
x - y x - y + t ( ( I - T ) x - ( I - T ) y ) Open image in new window
(1.1)

holds for each x, yD(T) and for all t > 0. As a result of [1], it follows from (1.1) that T is pseudocontractive if and only if there exists j (x - y) ∈ J(x - y) such that 〈Tx - Ty, j(x - y)〉 ||x - y||2 for any x, yD(T). T is called strongly pseudocontractive if there exist j(x - y) ∈ J (x - y) and β ∈ (0, 1) such that 〈Tx - Ty, j(x - y)〉 ≤ β ||x - y||2 for any x, yD(T). T is called Lipschitzian if there exists L ≥ 0 such that ||Tx - Ty|| ≤ L||x - y||, ∀ x, yD(T). If L = 1, then T is called nonexpansive, and it is called contraction if L ∈ [0, 1).

Let E = H be a Hilbert space with inner product 〈· , ·〉. Recall that T : CH is called monotone if 〈Tx - Ty, x - y 0, ∀ x, yC. A variational inequality problem, denoted by VI(T, C), is to find a point x* with the property
x * C , such that T x * , x - x * 0 , x C . Open image in new window

If the mapping T is a monotone operator, then we say that VI(T, C) is monotone.

In [2], Lu et al. considered the following type of monotone variational inequality problem in Hilbert spaces(denoted by VI(1.2))
fi n d x * F i x ( T ) such that ( I - S ) x * , x - x * 0 , x F i x ( T ) , Open image in new window
(1.2)

where T, S : CC are nonexpansive mappings and Fix(T) ≠ ∅. Let W denote the set of solutions of the VI(1.2).

Very recently, Yao et al. [3] considered VI(1.2) in Hilbert spaces when T, S : CC are pseudocontractions.

In this paper, we consider the following variational inequality problem in Banach spaces (denoted by VI(1.3))
fi n d x * F i x ( T ) such that ( I - S ) x * , j ( x - x * ) 0 , x F i x ( T ) , Open image in new window
(1.3)

where T, S : CC are pseudocontractions. Let Ω denote the set of solutions of the VI(1.3) and assume that Ω is nonempty. Since I - S is accretive, then we say VI(1.3) is an accretive variational inequality.

For solving the VI(T, C), hybrid methods were studied by Yamada [4] where he assumed that T is Lipschitzian and strongly monotone. However, his methods do not apply to the VI(1.2) since the mapping I - S fails, in general, to be strongly monotone, though it is Lipschitzian. In fact the VI(1.2) is, in general, ill-posed, and thus regularization is needed. Let T, S : CC be nonexpansive and f : CC be contractive. In 2006, Moudafi and Mainge [5] studied the VI(1.2) by regularizing the mapping tS + (1 - t)T and defined {xs,t} as follows:
x s , t = s f ( x s , t ) + ( 1 - s ) [ t S x s , t + ( 1 - t ) T x s , t ] , s , t ( 0 , 1 ) . Open image in new window
(1.4)
Since Moudafi and Mainge's regularization depends on t, the convergence of the scheme (1.4) is more complicated. So Lu et al. [2] defined {xs,t} as follows by regularizing the mapping S:
x s , t = s [ t f ( x s , t ) + ( 1 - t ) S x s , t ] + ( 1 - s ) T x s , t , s , t ( 0 , 1 ) . Open image in new window
(1.5)

Note that Lu et al.'s regularization does no longer depend on t. And their result for the regularization (1.5) is under dramatically less restrictive conditions than Moudafi and Mainge's [5].

Very recently, Yao et al. [3] extended Lu et al.'s result to a general case, i.e., in the scheme (1.5), S, T are extended to Lipschitz pseudocontractive and f is extended to strongly pseudocontractive. But in [3], after careful discussion, we observe that a continuity condition on f is necessary. So, in this paper, we modify it.

Motivated and inspired by the above work, in this paper, we use strongly pseudocontrations to regularize the ill-posed accretive VI(1.3), and analyze the convergence of the scheme (1.5). The results we obtained improve and extend the corresponding results in [2, 3].

2 Preliminaries

If Banach space E admits sequentially continuous duality mapping J from weak topology from weak* topology, then by [6, Lemma 1], we get that duality mapping J is single-valued. In this case, duality mapping J is also said to be weakly sequentially continuous, i.e., for each {x n } ⊂ E with x n x, then J(x n ) ⇁ Jx[6, 7].

A Banach space E is said to be satisfying Opial's condition if for any sequence {x n } in E, x n x(n → ∞) implies that
lim sup n x n x < lim sup n x n y , y E , with  y x . Open image in new window

By [6, Lemma 1], we know that if E admits a weakly sequentially continuous duality mapping, then E satisfies Opial's condition.

Lemma 2.1([7]) Let C be a nonempty closed convex subset of a reflexive Banach space E, which satisfies Opial's condition, and suppose T : CE is nonexpansive. Then, the mapping I - T is demiclosed at zero, i.e.,
x n x , x n - T x n 0 implies x = T x . Open image in new window

Recall that S : CC is called accretive if I - S is pseudocontractive. We denote by J r the resolvent of S, i.e., J r = (I + rS)-1. It is well known that J r is nonexpansive, single-valued and Fix(J r ) = S-1(0) = {zD(S) : 0 = Sz} for all r > 0. For more details, see[8, 9, 10].

Let T : CC be a pseudocontractive mapping; then, I - T is accretive. We denote A = J1 = (2I - T)-1. Then, Fix(A) = Fix(T) and A : R(2I - T) → K is nonexpansive and single-valued. The following lemma can be found in [11].

Lemma 2.2([11]) Let C be a nonempty closed convex subset of a real Banach space E and T : CC be a continuous pseudocontractive map. We denote A = J1 = (2I - T)-1. Then,
  1. (i)
    [12, Theorem 6] The map A is a nonexpansive self-mapping on C, i.e., for all x, yC, there hold
    A x - A y x - y and A x C ; Open image in new window
     
  2. (ii)

    If limn→∞||x n - Tx n || = 0, then limn→∞||x n - Ax n || = 0.

     

We also need the following lemma.

Lemma 2.3 Let C be a nonempty closed convex subset of a real Banach space E. Assume that F : CE is accretive and weakly continuous along segments; that is F(x + ty) ⇀ F(x) as t → 0. Then, the variational inequality
x * C , F x * , j ( x - x * ) 0 , x C Open image in new window
(2.1)
is equivalent to the dual variational inequality
x * C , F x , j ( x - x * ) 0 , x C . Open image in new window
(2.2)
Proof (2.1) ⇒ (2.2) Since F is accretive, we have
F x - F x * , j ( x - x * ) 0 Open image in new window
and so
F x , j ( x - x * ) F x * , j ( x - x * ) 0 . Open image in new window
(2.2) ⇒ (2.1) For any xC, put w = tx + (1 - t)x*, ∀ t ∈ (0, 1). Then, wC. Taking x = w in (2.2), we have
F w , j ( w - x * ) = t F w , j ( x - x * ) 0 , Open image in new window
i.e.,
F w , j ( x - x * ) 0 . Open image in new window

Letting t → 0 in the above inequality, since F is weakly continuous along segments, it follows that (2.1) holds.

3. Main Results

Let C be a nonempty closed convex subset of a real Banach space E. Let f : CC be a Lipschitz strongly pseudocontraction and T, S : CC be two continuous pseudocontractions. For s, t ∈ (0, 1), we define the following mapping
x W s , t x : = s [ t f ( x ) + ( 1 - t ) S x ] + ( 1 - s ) T x . Open image in new window
It is easy to see that the mapping Ws,t: CC is a continuous strongly pseudocontractive mapping. So, by [13], Ws,thas a unique fixed point which is denoted xs,tC; that is
x s , t = W s , t x s , t = s [ t f ( x s , t ) + ( 1 - t ) S x s , t ] + ( 1 - s ) T x s , t , s , t ( 0 , 1 ) . Open image in new window
(3.1)
Theorem 3.1 Let E be a reflexive Banach space that admits a weakly sequentially continuous duality mapping from E to E*. Let C be a nonempty closed convex subset of E. Let f : CC be a Lipschitz strongly pseudocontraction, S : CC be a Lipschitz pseudocontraction, and T : CC be a continuous pseudocontraction with Fix(T) ≠ ∅. Suppose that the solution set Ω of the VI(1.3) is nonempty. Let for each (s, t) ∈ (0, 1)2, {xs,t} be defined by (3.1). Then, for each fixed t ∈ (0, 1), the net {xs,t} converges in norm, as s → 0, to a point x t Fix(T). Moreover, as t → 0, the net {x t } converges in norm to the unique solution x* of the following inequality variational(denoted by VI(3.2)):
x * Ω , ( I - f ) x * , J ( x - x * ) 0 , x Ω . Open image in new window
(3.2)

Hence, for each null sequence {t n } in (0,1), there exists another null sequence {s n } in (0,1), such that the sequence x s n , t n x * Open image in new window in norm as n → ∞.

Proof We divide our proofs into several steps as follows.

Step 1 For each fixed t ∈ (0, 1), the net {xs,t} is bounded.

For any zFix(T), for all s, t ∈ (0, 1), by (3.1), we have
x s , t z 2 = s [ t f ( x s , t ) + ( 1 t ) S x s , t ] + ( 1 s ) T x s , t z , J ( x s , t z ) = s t f ( x s , t ) f ( z ), J ( x s , t z ) + s ( 1 t ) S x s , t S z , J ( x s , t z ) + ( 1 s ) T x s , t T z , J ( x s , t z ) + s t f ( z ) z , J ( x s , t z ) + s ( 1 t ) S z z , J ( x s , t z ) s t β x s , t z 2 + s ( 1 t ) x s , t z 2 + ( 1 s ) x s , t z 2 + s t f ( z ) z x s , t z + s ( 1 t ) S z z x s , t z = ( 1 s t ( 1 β ) ) x s , t z 2 + s [ t f ( z ) z + ( 1 t ) S z z ] x s , t z , Open image in new window
which implies that
x s , t - z t f ( z ) - z t ( 1 - β ) + ( 1 - t ) S z - z t ( 1 - β ) 1 t ( 1 - β ) max { f ( z ) - z , S z - z } . Open image in new window
Hence, for each t ∈ (0, 1), {xs,t} is bounded. Furthermore, by the Lipschitz continuity of f and S, we obtain {f(xs,t)} and {Sxs,t}, which are both bounded for each t ∈ (0, 1). From (3.1), we have
T x s , t 1 1 - s x s , t + s 1 - s t f ( x s , t ) + ( 1 - t ) S x s , t . Open image in new window

So {Txs,t} is also bounded as s → 0 for each t ∈ (0, 1).

Step 2 xs,tx t Fix(T) as s → 0.

From (3.1), for each t ∈ (0, 1), we get
x s , t - T x s , t = s [ t f ( x s , t ) + ( 1 - t ) S x s , t - T x s , t ] 0 , a s s 0 . Open image in new window
(3.3)
It follows from (3.1) that
x s , t - z 2 = s t f ( x s , t ) - f ( z ) , J ( x s , t - z ) + s ( 1 - t ) S x s , t - S z , J ( x s , t - z ) + ( 1 - s ) T x s , t - T z , J ( x s , t - z ) + s t f ( z ) - z , J ( x s , t - z ) + s ( 1 - t ) S z - z , J ( x s , t - z ) ( 1 - s t ( 1 - β ) ) x s , t - z 2 + s t f ( z ) - z , J ( x s , t - z ) + s ( 1 - t ) S z - z , J ( x s , t - z ) . Open image in new window
It turns out that
x s , t - z 2 1 t ( 1 - β ) t f ( z ) + ( 1 - t ) S z - z , J ( x s , t - z ) , z F i x ( T ) . Open image in new window
Assume that {s n } ⊂ (0, 1) is such that s n → 0(n → ∞), by the above inequality we have
x s n , t - z 2 1 t ( 1 - β ) t f ( z ) + ( 1 - t ) S z - z , J ( x s n , t - z ) , z F i x ( T ) . Open image in new window
(3.4)
Since x s n , t Open image in new window is bounded, without loss of generality, we may assume that as s n → 0, x s n , t x t Open image in new window. Combining (3.3), Lemma 2.1 and 2.2, we obtain x t Fix(A) = Fix(T). Taking z = x t in (3.4), we get
x s n , t - x t 2 1 t ( 1 - β ) t f ( x t ) + ( 1 - t ) S x t - x t , J ( x s n , t - x t ) . Open image in new window
(3.5)

Since x s n , t x t Open image in new window and J is weakly sequentially continuous, by (3.5) as s n → 0, we obtain x s n , t x t Open image in new window. This has proved that the relative norm compactness of the net {xs,t} as s → 0.

Letting n → ∞ in (3.4), we obtain
x t - z 2 1 t ( 1 - β ) t f ( z ) + ( 1 - t ) S z - z , J ( x t - z ) , z F i x ( T ) . Open image in new window
(3.6)
So, x t is a solution of the following variational inequality:
x t F i x ( T ) , t f ( z ) + ( 1 - t ) S z - z , J ( x t - z ) 0 , z F i x ( T ) . Open image in new window
Letting C = Fix(T), F = t(I - f) + (1 - t)(I - S), by Lemma 2.3, we have the equivalent dual variational inequality:
x t F i x ( T ) , t f ( x t ) + ( 1 - t ) S x t - x t , J ( x t - z ) 0 , z F i x ( T ) . Open image in new window
(3.7)
Next, we prove that for each t ∈ (0, 1), as s → 0, {xs,t} converges in norm to x t Fix(T). Assume x s n , t x t Open image in new window as s n 0 Open image in new window. Similar to the above proof, we have x t F i x ( T ) Open image in new window, which solves the following variational inequality:
x t F i x ( T ) , t f ( x t ) + ( 1 - t ) S x t - x t , J ( x t - z ) 0 , z F i x ( T ) . Open image in new window
(3.8)
Taking z = x t Open image in new window in (3.7) and z = x t in (3.8), we have
t f ( x t ) + ( 1 - t ) S x t - x t , J ( x t - x t ) 0 , Open image in new window
(3.9)
t f ( x t ) + ( 1 - t ) S x t - x t , J ( x t - x t ) 0 . Open image in new window
(3.10)
Adding up (3.9) and (3.10), and since f is strongly pseudocontractive and S is pseudocontractive, we have
0 t ( I - f ) x t - ( I - f ) x t , J ( x t - x t ) + ( 1 - t ) ( I - S ) x t - ( I - S ) x t , J ( x t - x t ) - t ( 1 - β ) x t - x t 2 , Open image in new window

which implies that x t = x t Open image in new window. Hence, the net {xs,t} converges in norm to x t Fix(T) as s → 0.

Step 3 {x t } is bounded.

Since Ω ⊂ Fix(T), for any y ∈ Ω, taking z = y in (3.7) we obtain
t f ( x t ) + ( 1 - t ) S x t - x t , J ( x t - y ) 0 . Open image in new window
(3.11)
Since I - S is accretive, for any y ∈ Ω, we have
S x t - x t , J ( x t - y ) S y - y , J ( x t - y ) 0 . Open image in new window
(3.12)
Combining (3.11) and (3.12), we have
f ( x t ) - x t , J ( x t - y ) 0 , y Ω , Open image in new window
(3.13)
i.e.,
f ( x t ) - y + y - x t , J ( x t - y ) 0 , y Ω . Open image in new window
Hence,
x t - y 2 f ( x t ) - y , J ( x t - y ) = f ( x t ) - f ( y ) , J ( x t - y ) + f ( y ) - y , J ( x t - y ) β x t - y 2 + f ( y ) - y , J ( x t - y ) . Open image in new window
Hence,
x t - y 2 1 1 - β f ( y ) - y , J ( x t - y ) , Open image in new window
(3.14)
which implies that
x t - y 1 1 - β f ( y ) - y . Open image in new window

So {x t } is bounded.

Step 4 x t x* ∈ Ω which is a solution of variational inequality (3.2).

Since f is strongly pseudocontractive, it is easy to see that the solution of the variational inequality (3.2) is unique.

Next, we prove that ω w (x t ) ⊂ Ω; namely, if (t n ) is a null sequence in (0,1) such that x t n x Open image in new window as n → ∞, then x' ∈ Ω. Indeed, it follows from (3.7) that
( I - S ) x t , J ( z - x t ) t 1 - t ( I - f ) x t , J ( x t - z ) . Open image in new window
Since I - S is accretive, from the above inequality, we have
( I - S ) z , J ( z - x t ) t 1 - t ( I - f ) x t , J ( x t - z ) , z F i x ( T ) . Open image in new window
(3.15)
Letting t = t n → 0 in (3.15), we have
( I - S ) z , J ( z - x ) 0 , z F i x ( T ) , Open image in new window
which is equivalent to its dual variational inequality by Lemma 2.3
( I - S ) x , J ( z - x ) 0 , z F i x ( T ) . Open image in new window

Since Fix(T) is closed convex, then Fix(T) is weakly closed. Thus, x' ∈ Fix(T) by virtue of x t Fix(T). So, x' ∈ Ω.

Finally, we show that x' = x*, the unique solution of VI(3.2). In fact, taking t = t n and y = x' in (3.14), we obtain
x t n - x 2 1 1 - β f ( x ) - x , J ( x t n - x ) , Open image in new window
which together with x t n x Open image in new window implies that x t n x Open image in new window as t n → 0. Let t = t n → 0 in (3.13), we have
f ( x ) - x , J ( x - y ) 0 , y Ω . Open image in new window
(3.16)

It follows from (3.16) and x' ∈ Ω that x' is a solution of VI(3.2). By uniqueness, we have x' = x*. Therefore, x t x* as t → 0.

By Theorem 3.1, we have the following corollary directly.

Corollary 3.1([2, Theorem 3.3]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : CC be a contraction, S, T : CC be nonexpansive with Fix(T) ≠ ∅. Suppose that the solution set W of the VI(1.2) is nonempty. Let for each (s, t) ∈ (0, 1)2, {xs,t} be defined by (3.1). Then, for each fixed t ∈ (0, 1), the net {xs,t} converges in norm, as s → 0, to a point x t Fix(T). Moreover, as t → 0, the net {x t } converges in norm to the unique solution x* of the following inequality variational:
x * W , ( I - f ) x * , x - x * 0 , x W . Open image in new window

Hence, for each null sequence {t n } in (0,1), there exists another null sequence {s n } in (0,1), such that the sequence x s n , t n x * Open image in new window in norm as n → ∞.

Remark Theorem 3.1 improves and generalizes Theorem 3.1 of Yao et al.[3] in the following aspects:
  1. (i)

    Theorem 3.1 generalizes Theorem 3.1 in [3] from Hilbert spaces to more general Banach spaces;

     
  2. (ii)

    The mappings T in [3, Theorem 3.1] is weakened from Lipschitzian to continuous;

     
  3. (iii)

    We modify the condition of f, i.e., we suppose that f is Lipschitz strongly pseudocontractive.

     

The authors declare that they have no competing interests.

Notes

Acknowledgements

The first author was supported by the Research Project of Shaoxing University(No. 09LG1002), and the second author was supported partly by NSFC Grants(No.11071279).

Supplementary material

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© Wang and Chen; licensee Springer. 2011

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.

Authors and Affiliations

  1. 1.Department of MathematicsShaoxing UniversityShaoxingChina
  2. 2.Mathematical CollegeSichuan UniversityChengduChina
  3. 3.Department of MathematicsTianjin Polytechnic UniversityTianjinChina

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