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# Composite iterative schemes for maximal monotone operators in reflexive Banach spaces

- Prasit Cholamjiak
^{1}, - Yeol Je Cho
^{2}Email author and - Suthep Suantai
^{3}Email author

**2011**:7

https://doi.org/10.1186/1687-1812-2011-7

© Cholamjiak et al; licensee Springer. 2011

**Received:**15 February 2011**Accepted:**22 June 2011**Published:**22 June 2011

## Abstract

In this article, we introduce composite iterative schemes for finding a zero point of a finite family of maximal monotone operators in a reflexive Banach space. Then, we prove strong convergence theorems by using a shrinking projection method. Moreover, we also apply our results to a system of convex minimization problems in reflexive Banach spaces.

**AMS Subject Classification**: 47H09, 47H10

## Keywords

- Maximal monotone operator
- Shrinking projection method
- Proximal point algorithm
- Bregman projection
- Totally convex function
- Legendre function

## Introduction

Let *E* be a real Banach space and *C* a nonempty subset of *E*. Let *E** be the dual space of *E*. We denote the value of *x** ∈ *E** at *x 2 E* by 〈*x**, *x*〉. Let *A* : *E* → 2^{
E*
} be a set-valued mapping. We denote dom *A* by *domain* of *A*, that is, dom *A* = {*x* ∈ *E* : *Ax* ≠ ∅}and also denote *G*(*A*) by the *graph* of *A*, that is, *G*(*A*) = *f* (*x*, *x**) ∈ *E* × *E** : *x** ∈ *Ax*}. A set-valued mapping *A* is said to be *monotone* if 〈*x** - *y**, *x - y*〉 ≥ 0 whenever (*x*, *x**); (*y*, *y**) ∈ *G*(*A*). It is said to be *maximal monotone* if its graph is not contained in the graph of any other monotone operator on *E*. It is known that if *A* is maximal monotone, then the set *A*^{-1}(0*) = {*z* ∈ *E* : 0* ∈ *Az*} is closed and convex.

*λ*

_{ n }} ⊂ (0, ∞) and

*J*

_{ λ }is the resolvent of

*A*defined by

*J*

_{ λ }= (

*I*+

*λA*)

^{-1}for all

*λ >*0, and

*A*is a maximal monotone operator on

*E*. Such an algorithm is called the

*proximal point algorithm*. He proved that the sequence {

*x*

_{ n }} generated by (1.1) converges weakly to an element in

*A*

^{-1}(0) provided lim inf

_{n→∞}

*λ*

_{ n }> 0. Later, Kamimura and Takahashi [9] introduced the following iteration in a Hilbert space:

where {*α*_{
n
} } ⊂ [0, 1] and {*λ*_{
n
} } ⊂ (0, *∞*). The weak convergence theorems are also established in a Hilbert space under suitable conditions imposed on {*α*_{
n
} } and {*λ*_{
n
} }.

where {*α*_{
n
} }⊂ [0, 1], {*λ*_{
n
} } ⊂ (0, *∞*), *f* : *E* → ℝ is a Bregman function and *J*_{
λ
} = (∇*f* + *λA*) ^{-1} ∇*f* for all *λ* > 0. They also proved a weak convergence theorem of the proposed algorithm.

*A*

_{ i }:

*E*→ 2

^{ E* }(

*i*= 1, 2,...,

*N*) in a general reflexive Banach space

*E*as follows:

where
,
is an error sequence in *E* with
and
the Bregman projection with respect to *f* from *E* onto a closed and convex subset *K* of *E*. Those authors showed that the sequence {*x*_{
n
} } defined by (1.4) converges strongly to a common element in
under some mild conditions.

Motivated by the previous ones, we first introduce a composite iterative scheme which is different from (1.4) for finding a zero point of maximal monotone operators *A*_{
i
} : *E* → 2^{
E*
} (*i* = 1, 2,..., *N*) in reflexive Banach spaces. Using the shrinking projection technique, introduced by Takahashi et al. [12], we then prove that a sequence generated by the proposed algorithm converges strongly to an element in
under some appropriate control conditions. Finally, we also apply our result to a system of convex minimization problems.

## Preliminaries and lemmas

*E*be a real reflexive Banach space with a norm ||·|| and

*E** be the dual space of

*E*. Throughout this article,

*f*:

*E*→ (

*-∞*, +∞] is a proper, lower semi-continuous, and convex function, and the Fenchel conjugate of

*f*is the function

*f**:

*E** → (

*-∞*, +∞] defined by

*f*the domain of

*f*, that is, the set {

*x*∈

*E*:

*f*(

*x*)

*<*+∞). For any

*x*∈ int dom

*f*and

*y*∈

*E*, the

*right-hand derivative*of

*f*at

*x*in the direction

*y*is defined by

The function *f* is said to be *Gâteaux differentiable* at *x*
exists for any *y*. In this case, *f*^{o}(*x*, *y*) coincides with ∇*f* (*x*), the value of the *gradient* ∇*f* of *f* at *x*. The function *f* is said to be *Gâteaux differentiable* if it is Gâteaux differentiable for any *x* ∈ int dom *f*. The function *f* is said to be *Fréchet differentiable at x* if this limit is attained uniformly in ||*y*|| = 1. Finally, *f* is said to be *uniformly Fréchet differentiable* on a subset *C* of *E* if the limit is attained uniformly for *x* ∈ *C* and ||*y*|| = 1.

Let *E* be a reflexive Banach space. The Legendre function is defined from a general Banach space *E* into (-∞, +∞] (see [13]). According to [13], the function *f* is *Legendre* if and only if it satisfies the following conditions:

(L1) The interior of the domain of *f* (denoted by int dom *f* ) is nonempty, *f** is Gâteaux differentiable on int dom *f*, and dom ∇*f* = int dom *f* ;

(L2) The interior of the domain *f**(denoted by int dom *f**) is nonempty, *f** is Gâteaux differentiable on int dom *f**, and dom ∇*f** = int dom *f**.

*E*is reflexive, we always have (∂

*f*)

^{-1}= ∂

*f** (see [14]). This fact, when combined with the conditions (L1) and (L2), implies the following equalities [15]:

Also, the conditions (L1) and (L2), in conjunction with [13], imply that the functions *f* and *f** are strictly convex on the interior of their respective domains. Several interesting examples of the Legendre functions are presented in [13, 16]. Especially, the functions
with *s* ∈ (1, *∞*) are Legendre, where the Banach space *E* is smooth and strictly convex and, in particular, a Hilbert space. Throughout this article, we assume that the convex function *f* : *E* → (*∞*, +∞] is Legendre.

**Lemma 2.1**. [17]*If f* : *E* → ℝ *is uniformly Fréchet differentiable and bounded on bounded subsets of E, then* ∇ *f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E**.

*f*:

*E*→ (

*-∞*, +∞] be a convex and Gâteaux differentiable function. The function

*D*

_{ f }: dom

*f*× int dom

*f*→ [0, +∞) is defined as follows:

is called the *Bregman distance* with respect to *f*[18].

*Bregman projection*[19] of

*x*∈ int dom

*f*onto the nonempty, closed, and convex set

*C*⊂ dom

*f*is necessarily the unique vector satisfying

*f*:

*E*→ (

*-∞*, +∞] be a convex and Gâteaux differentiable function. The function

*f*is said to be

*totally convex*at

*x*∈ int dom

*f*if its modulus of total convexity at

*x*, that is, the function

*ν*

_{ f }: int dom

*f*× [0, +∞) → [0, +∞] defined by

*t >*0. The function

*f*is said to be

*totally convex*when it is totally convex at every point

*x*∈ int dom

*f*. In addition, the function

*f*is said to be

*totally convex on bounded sets*if

*ν*

_{ f }(

*B*,

*t*) is positive for any nonempty bounded subset

*B*of

*E*and

*t >*0, where the modulus of total convexity of the function

*f*on the set

*B*is the function

*ν*

_{ f }: int dom

*f*× [0, +∞) → [0, +∞] defined by

*C*be a nonempty, closed, and convex subset of

*E*. Let

*f*:

*E*→ ℝ be a Gâteaux differentiable and totally convex function and let

*x*∈

*E*. It is known from [20] that if and only if 〈∇

*f*(

*x*) - ∇

*f*(

*z*),

*y - z*〉 ≤ 0 for all

*y*∈

*C*. We also have

*f*is said to be

*sequentially consistent*[20] if, for any two sequences, {

*x*

_{ n }} and {

*y*

_{ n }}, in

*E*such that the first is bounded:

The following lemmas were proved by Reich and Sabach [11].

**Lemma 2.2**. [11]*Let f* : *E* → ℝ *be a Gâteaux differentiable and totally convex function. If x*_{0} ∈ *E and the sequence*
*is bounded, then the sequence*
*is also bounded*.

*f*is a Legendre function which is bounded, uniformly Fréchet differentiable on bounded, subsets of

*E*, then (see [22]). The

*Yosida approximation Aλ*:

*E*→

*E*,

*λ*> 0, is also defined by

for all *x* ∈ *E*. From Proposition 2.7 in [11], we know that
and 0* ∈ *Ax* if and only if 0* ∈ *A*_{
λ
}*x* for all *x* ∈ *E* and *λ* > 0.

*for all λ* > 0, *p* ∈ *A*^{-1}(0*) *and x* ∈ *E*.

## Strong convergence theorems

Now, in this section, we prove our main results of this article.

**Theorem 3.1**.

*Let E be a real reflexive Banach space and f*:

*E*→ ℝ

*a Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let A*

_{ i }:

*E*→ 2

^{ E* }(

*i*= 1, 2,...,

*N*)

*be maximal monotone operators such that*.

*Let*

*be such that*lim

_{n→ ∞}

*e*

_{ n }= 0.

*Define a sequence*

*in E as follows:*

*If*
*for each i* = 1, 2,..., *N, then the sequence* {*x*_{
n
}} *converges strongly to a point*

*Proof*. We divide our proof into six steps as follows:

**Step 1**. *F* ⊂ *C*_{
n
} for all *n* ≥ 1.

*i*= 1, 2,...,

*N*, we get that is a nonempty, closed and convex subset of

*E*. It is easy to see that

*C*

_{ n }is closed and convex for all

*n*≥ 1. Indeed, for each

*z*∈

*C*

_{ n }, it follows that

*D*

_{ f }(

*z*,

*y*

_{ n }) ≤

*D*

_{ f }(

*z*,

*x*

_{ n }+

*e*

_{ n }) is equivalent to

This shows that *C*_{
n
} is closed and convex for all *n* ≥ 1. It is obvious that *F* ⊂ *C* 1 = *E*.

This implies that *F* ⊂ *C*_{k+1}. By induction, we can conclude that *F* ⊂ *C*_{
n
}for all *n* ≥ 1.

**Step 2**. lim_{n→∞}*D*_{
f
}(*x*_{
n
}, *x*_{0}) exists.

Combining (3.3) and (3.4), we know that lim_{n→ ∞}*D*_{
f
}(*x*_{
n
}, *x*_{1}) exists.

**Step 3**. lim_{n→ ∞}||∇*f*(*y*_{
n
}) - ∇*f*(*x*_{
n
}+ *e*_{
n
})|| = 0

*m*,

*n*→

*∞*, we have

*D*

_{ f }(

*x*

_{ m },

*x*

_{ n }) → 0. Since

*f*is totally convex on bounded subsets of

*E*,

*f*is sequentially consistent by Butnariu and Resmerita [20]. It follows that

*||x*

_{ m }

*- x*

_{ n }

*||*→ 0 as

*m*,

*n*→

*∞*. Therefore, {

*x*

_{ n }} is a Cauchy sequence. By the completeness of the space

*E*, we can assume that

*x*

_{ n }→

*q*∈

*E*as

*n*→ ∞. In particular, we obtain

*f*is bounded on bounded subsets of

*E*, then ∇

*f*is also bounded on bounded subsets of

*E*. Moreover, if

*f*is uniformly Fréchet differentiable on bounded subsets of

*E*, then

*f*is uniformly continuous on bounded subsets of

*E*(see [24]). Using (3.5), we have

*f*is uniformly Fréchet differentiable on bounded subsets of

*E*, ∇

*f*is norm-to-norm uniformly continuous on bounded subsets of

*E*by Lemma 2.1. Hence, we have

*i*∈ {1, 2,...,

*N*} and for each

*n*≥ 1. We note that for each

*n*≥ 1. For any

*p*∈

*F*, by (3.2), it follows that

In a similar way, we can show that

for each *i* = 1,2,..., *N*.

*i*= 1, 2,...,

*N*. If (

*w*,

*w**) ∈

*G*(

*Ai*) for each

*i*= 1, 2,...,

*N*, then it follows from the monotonicity of

*A*

_{ i }that

*x*

_{ n }→

*q*and

*e*

_{ n }→ 0,

*x*

_{ n }+

*e*

_{ n }→

*q*. Therefore, for each

*i*= 1, 2,...,

*N*. Thus, from (3.14), we have

By the maximality of *A*_{
i
} , we have
for each *i* = 1, 2,..., *N*. Hence,
.

Hence, we have . This completes the proof.

As a direct consequence of Theorem 3.1, we also obtain the following result concerning a system of convex minimization problems in reflexive Banach spaces:

**Theorem 3.2**.

*Let E be a real reflexive Banach space and f*:

*E*→ ℝ

*a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let g*

_{ i }:

*E*→ (- ∞, ∞] (

*i*= 1, 2,...,

*N*)

*be proper lower semi-continuous convex functions such that*

*. Let*

*be a sequence in E such that*lim

_{n→ ∞}

*e*

_{ n }= 0.

*Define a sequence*

*in E as follows:*

*If*
*for each i* = 1, 2,..., *N, then the sequence* {*x*_{
n
} } *converges strongly to a point*
.

*Proof*. By Rockafellar's theorem [25, 26], ∂

*g*

_{ i }are maximal monotone operators for each

*i*= 1, 2,...,

*N*. Let

*λ*

^{ i }

*>*0 for each

*i*= 1, 2,...,

*N*. Then if and only if

Using Theorem 3.1, we can complete the proof.

**Remark 3.3**. By means of the composite iterative scheme together with the shrinking projection method, we can construct the proximal point algorithms for finding a common element in the set
. Moreover, our algorithm is different from that of Reich and Sabach [11] which is based on a finite intersection of sets.

**Remark 3.4**. Theorems 3.1 and 3.2 also hold in a uniformly convex and uniformly smooth Banach space with the generalized duality mapping.

## Declarations

### Acknowledgements

P. Cholamjiak would like to thank the Royal Golden Jubilee Project, the Thailand Research Fund; Y.J. Cho was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050); and S. Suantai was supported by the Thailand Research Fund, Thailand. Especially, P. Cholamjiak is grateful for the hospitality extended by Professor Sang Keun Lee, Chairman, and others in the Department of Mathematics Education, Gyeongsang National University, during his 3-month stay.

## Authors’ Affiliations

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