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 → 2E* 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.

The problem of finding zero points for maximal monotone operators plays an important role in optimizations. This is because it can be reduced to a convex minimization problem and a variational inequality problem. Many authors have studied the convergence of such problems in several settings, (see [1–6]). Initiated by Martinet [7], in a Hilbert space, Rockafellar [8] introduced the following iterative schemes:

(1.1)

where {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 infn→∞λ n > 0. Later, Kamimura and Takahashi [9] introduced the following iteration in a Hilbert space:

(1.2)

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 }.

In 2005, Kohsaka and Takahashi [10] studied the above iteration process in a more general setting, reflexive Banach spaces. In fact, those authors proposed the following algorithm:

(1.3)

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.

Very recently, in 2010, Reich and Sabach [11] proposed an algorithm for finding a zero point of maximal monotone operators A i : E → 2E* (i = 1, 2,..., N) in a general reflexive Banach space E as follows:

(1.4)

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 → 2E* (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

Let 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

We denote by dom 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, fo(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*.

Since 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*.

Let 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].

Recall that the Bregman projection[19] of x ∈ int dom f onto the nonempty, closed, and convex set C ⊂ dom f is necessarily the unique vector satisfying

Let 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

is positive, whenever 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

Let 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

(2.1)

Recall that the function 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 x0 ∈ E and the sequenceis bounded, then the sequenceis also bounded.

We know that the resolvent of A, denoted by , is defined as follows [21]:

It is known that , and is single-valued (see [21]). If 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.

Lemma 2.3. [11]Let A : E → 2E*be a maximal monotone operator such that A-1(0*) ≠ ∅. Then,

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 → 2E* (i = 1, 2,..., N) be maximal monotone operators such that. Letbe such that limn→ ∞e n = 0. Define a sequencein E as follows:

(3.1)

Iffor 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.

Since is closed and convex for each 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.

Now, suppose that F ⊂ C k for some . For any p ∈ F, by Lemma 2.3, we have

(3.2)

This implies that F ⊂ Ck+1. By induction, we can conclude that F ⊂ C n for all n ≥ 1.

Step 2. limn→∞D f (x n , x0) exists.

From and we have

(3.3)

By (2.1), for any p ∈ F ⊂ C n , we have

(3.4)

Combining (3.3) and (3.4), we know that limn→ ∞D f (x n , x1) exists.

Step 3. limn→ ∞||∇f(y n ) - ∇f(x n + e n )|| = 0

Since for m > n ≥ 1, by (2.1), it follows that

Letting 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

Since e n → 0, we also obtain

(3.5)

Since

We know from [23] that, if 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

Also, we have

and hence,

and, since e n → 0,

(3.6)

Since 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

(3.7)

Step 4. .

Denote for each 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

(3.8)

Since , by Lemma 2.3 and (3.8), it follows that

From (3.6) and (3.7), we get that . Since f is sequentially consistent,

(3.9)

Thus, from (3.6) and (3.9), it follows that

(3.10)

and hence,

(3.11)

Again, since , by Lemma 2.3 and (3.8), we know that

From (3.10) and (3.11), we have

Since f is sequentially consistent, it follows that

(3.12)

From (3.10) and (3.12), we have

and hence,

In a similar way, we can show that

, and

Hence, we can conclude that

(3.13)

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

Step 5.

For each i = 1, 2,..., N, we note that and so

From (3.13) and , we have

(3.14)

We note that for each 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

Since 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, .

Step 6. .

From , we have

Since F ⊂ C n , we also have

(3.15)

Letting n → ∞ in (3.15), we obtain

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 . Letbe a sequence in E such that limn→ ∞e n = 0. Define a sequencein E as follows:

(3.16)

Iffor 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 λi > 0 for each i = 1, 2,..., N. Then if and only if

which is equivalent to

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.