#### Abstract

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).

#### 1. Introduction

Let be a Banach space with norm , let denote the dual of and let denote the value of at . Let be an operator. The problem of finding satisfying is connected with the convex minimization problems. When is maximal monotone, a well-known method for solving the equation in Hilbert space is the proximal point algorithm (see [1]): and where and for all is the resolvent operator for T. Rockafellar [1] proved the weak convergence of the algorithm (1.1).

The modifications of the proximal point algorithm for different operators have been investigated by many authors. Recently, Kohsaka and Takahashi [2] considered the algorithm (1.2) in a smooth and uniformly convex Banach space and Kamimura et al. [3] considered the algorithm (1.3) in a uniformly smooth and uniformly convex Banach space ; and where is the duality mapping of . They showed that the algorithm (1.2) converges strongly to some element of and the algorithm (1.3) converges weakly to some element of provided that the sequences and of real numbers are chosen appropriately. These results extend the Kamimura and Takahashi [4] results in Hilbert spaces to those in Banach spaces.

In 2008, motivated by Kim and Xu [5], Li and Song [6] studied a combination of the schemes of (1.2) and (1.3); and for every where is the duality mapping of . They also proved strong and weak convergence theorems and give an estimate for the rate of convergence of the algorithm (1.4).

Very recently, Ibaraki and Takahashi [7] introduced the Mann iteration and Harpern iteration for new resovents of maximal monotone operator in a uniformly smooth and uniformly convex Banach space ; and where is the duality mapping of and is maximal monotone. They proved that Algorithm (1.5) converges strongly to some element of and Algorithm (1.6) converges weakly to some element of provided that the sequences and of real numbers are chosen appropriately.

Inspired and motivated by Li and Song [6] and Ibaraki and Takahashi [7], we study a combination of the schemes of (1.5) and (1.6); and for every where is the duality mapping of and is maximal monotone. When , Algorithm (1.7) reduces to (1.5) and, when , Algorithm (1.7) reduces to (1.6). Then, we prove strong and weak convergence theorems of the sequence and we also estimate the rate of the convergence of algorithm (1.7). Finally, by using our main result, we consider the problem of finding minimizes of convex functions defined on Banach spaces.

#### 2. Preliminaries

Let be a real Banach space with dual space When is a sequence in , we denote strong convergence of to by and weak convergence by , respectively. As usual, we denote the duality pairing of by , when and , and the closed unit ball by , and denote by and the set of all real numbers and the set of all positive integers, respectively. The set stands for and . An operator is said to be monotone if whenever . We denote the set by A monotone is said to be *maximal* if its graph is not properly contained in the graph of any other monotone operator. If is maximal monotone, then the solution set is closed and convex. If is reflexive and strictly convex, then a monotone operator is maximal if and only if for each (see [8, 9] for more details).

The normalized duality mapping from into is defined by We recall [10] that is reflexive if and only if is surjective; is smooth if and only if is single-valued.

Let be a smooth Banach space. Consider the following function: (see [11]) It is obvious from the definition of that for all . We also know that We recall [12] that the functional is called totally convex at if the function defined by is positive whenever . The functional is called totally convex on bounded sets if for each bounded nonempty subset of the function defined by is positive on .

It is well known that if a Banach space is uniformly convex, then is totally convex on any bounded nonempty set. It is known that (see [12]) if is totally convex on a bounded set , then for and and is strictly increasing on .

Lemma 2.1 (see [13]). *Let be a uniformly convex, smooth Banach space, and let and be sequences in . If or is bounded and , then .*

Let be a reflexive, strictly convex, smooth Banach space, and the duality mapping from into . Then is also single-valued, one-to-one, surjective, and it is the duality mapping from into . We make use of the following mapping studied in Alber [11]: for all and . In other words, for all and .

Lemma 2.2 (see [7]). *Let be a reflexive, strictly convex, smooth Banach space, and let be as in (2.5). Then
**
for all and .*

Let be a smooth Banach space and let be a nonempty closed convex subset of . A mapping is called generalized nonexpansive if and for each and , where is the set of fixed points of . Let be a nonempty closed subset of . A mapping is said to be sunny if A mapping is said to be a retraction if , for all . If is smooth and strictly convex, then a sunny generalized nonexpansive retraction of onto is uniquely decided if it exists (see [14]). We also know that if is reflexive, smooth, and strictly convex and is a nonempty closed subset of , then there exists a sunny generalized nonexpansive retraction of onto if and only if is closed and convex. In this case, is given by see [15]. Let be a nonempty closed subset of a Banach space . Then is said to be a sunny generalized nonexpansive retract (resp., a generalized nonexpansive retract) of if there exists a sunny generalized nonexpansive retraction (resp, a generalized nonexpansive retraction) of onto (see [14] for more detials). The set of fixed points of such a generalized nonexpansive retraction is . The following lemma was obtained in [14].

Lemma 2.3 (see [14]). *Let be a nonempty closed subset of a smooth and strictly convex Banach space . Let be a retraction of onto . Then is sunny and generalized nonexpansive if and only if
**
for each and , where is the duality mapping of .* Let be a reflexive, strictly convex, and smooth Banach space with its dual . If a monotone operator is maximal, then is closed and for all (see [14]). So, for each and , we can consider the set . From [14], consists of one point. We denote such a by . However is called a generalized resolvent of We also know that for each , where is the set of fixed points of and is generalized nonexpansive for each (see [14]). The Yosida approximtion of is defined by . We know that ; (see [14] for more detials). The following result was obtained in [14].

Theorem 2.4 (see [14]). * Let be a uniformly convex Banach space with a FrÃ©chet differentiable norm and let be a maximal monotone operator with . Then the following hold: *(1)*for each , exists and belongs to *(2)*if for each , then is a sunny generalized nonexpansive retraction of onto .*

Lemma 2.5 (see [7]). *Let be a reflexive, strictly convex, and smooth Banach space, let be a maximal monotone operator with , and for all . Then
**
for all , , and .*

Lemma 2.6 (see [16]). *Let be a sequence of nonnegative real numbers satisfying
**
where , and satisfy the conditions: , , and , . Then, .*

Lemma 2.7 (see [17]). *Let and be sequence of nonnegative real numbers satisfying
**
for all . If . Then has a limit in .*

#### 3. Convergence Theorems

In this section, we first prove a strong convergence theorem for the algorithm (1.7) which extends the previous result of Ibaraki and Takahashi [7] and we next prove a weak convergence theorem for algorithm (1.7) under different conditions on data, respectively.

Theorem 3.1. *Let be a uniformly convex Banach space whose norm is uniformly GÃ¢teaux differentiable. Let be a maximal monotone operator with and let for all . Let be a sequence generated by and
**
for every where , , satisfy , , and Then the sequence converges strongly to , where is a sunny generalized nonexpansive retraction of onto .*

*Proof. *Note that implies . In fact, if , we obtain and hence . So, we have . We denote a sunny generalized nonexpansive retraction of onto by . Let . We first prove that is bounded. From Lemma 2.5 and the convexity of , we have
for all . By (3.2), we have
for all . Hence, by induction, we have for all and, therefore, is bounded. This implies that is bounded. Since and for all , it follows that and are also bounded. We next prove that
Put for all . Since is bounded, without loss of generality, we have a subsequence of such that
and converges weakly to some . From the definition of , we have
for all . Since is bounded and as , it follows that
Moreover, we note that
By (3.7) and (3.8), we have
Since has a uniformly GÃ¢teaux differentiable norm, the duality mapping is norm to uniformly continuous on each bounded subset of . Therefore, we obtain from (3.9) that
This implies that as . On the other hand, from as , we have
If , then it holds from the monotonicity of that
for all . Letting , we get . Then, the maximal of implies . Put . Applying Lemma 2.3, we obtain
Finally, we prove that as . From Lemma 2.2, the convexity of and (3.2), we have
for all , where . It easily verified from the assumption and (3.4) that and . Hence, by Lemma 2.6, . Applying Lemma 2.1, we obtain Therefore, converges strongly to . Put in Theorem 3.1, then we obtain the following result.

Corollary 3.2 (see Ibaraki and Takahashi [7]). *Let be a uniformly convex and uniformly smooth Banach space and let be a maximal monotone operator with , let for all and let be a sequence generated by and
**
for every where , satisfy , and Then the sequence converges strongly to , where is the generalized projection of onto .*

Theorem 3.3. *Let be a uniformly convex and smooth Banach space whose duality mapping is weakly sequentially continuous. Let be a maximal monotone operator with and let for all . Let be a sequence generated by and
**
for every where , , satisfy , and . Then the sequence converges weakly to an element of .*

*Proof. *Let . Then, from (3.3), we have
for all By Lemma 2.7, exists. From and , we note that and are bounded. From (3.3) and (3.2), we have
for all and hence,
for all . Since and , . Applying Lemma 2.1, we obtain
Since is bounded, we have a subsequence of such that as . Then it follows from (3.20) that as On the other hand, from (3.20) and , we have
Let . Then, it holds from monotonicity of that
for all . Since is weakly sequentially continuous, letting , we get . Then, the maximality of implies . Thus, .

Let and be two subsequences of such that and . By similar argument as above, we obtain Put .

Note that From and , we have
respectively. Combining (3.23) and (3.24), we have
Since is strictly monotone, it follows that . Therefore, converges weakly to an element of .

Put in Theorem 3.3, then we obtain the following result.

Corollary 3.4 (see Ibaraki and Takahashi [7]). *Let be a uniformly convex and smooth Banach space whose duality mapping is weakly sequentially continuous. Let be a maximal monotone operator with , let for all and let be a sequence generated by and
**
for every where ,, satisfy , and Then the sequence converges weakly to an element of .*

#### 4. Rate of Convergence for the Algorithm

In this section, we study the rate of the convergence of the algorithm (1.7). We use the following notations in [6, 18]:

We recall [18] that, for a function satisfying , its pseudoconjugate , defined by is lower semicontinuous, convex and satisfies for all .

For a function , its greatest quasi-inverse , defined by is nondecreasing. It is known [18] that if .

For a function , its lower semicontinuous convex hull, denoted by , is defined by epi cl(co(epi. It is obvious that is lower semicontinuous convex and .

Proposition 4.1. *Let be uniformly convex and uniformly smooth. Then, for every , there exists such that, for all ,
*

*Proof. *Since is uniformly convex, is uniform convex on for all . Since the norm of is FrÃ©chet differentiable, its FrÃ©chet derivative . In [18, Proposition ?3.6.5] for and , where is an arbitrary positive real number, we get the function defined by
satisfies that if and only if , and is nondecreasing. Thus,
for all , and hence
for all , . It follows that
for all , . Since , we have
for all and .

Taking the supremum on both sides of (4.9) over , by [18, Lemma ?3.3.1(v)] (if , where , then , we get that
for all and . Since is nondecreasing and , we have . It follows from [20, Lemma ?3.3.1(iii)] that .

Interchanging and in (4.10) for , it also holds that
Thus, by taking , and adding side by side (4.10) and (4.11), we obtained

Theorem 4.2. *Let be a uniformly convex and uniformly smooth Banach space. Suppose that is maximal monotone with and is Lipschitz continuous at with modulus . Let be a sequence generated by and
**
for every where , , satisfying ?. If either and or , and , then converges strongly to and converges to . **Moreover, there exists an integer such that
**
where , , and . Also, one obtains
**
for all , where , and , is the greatest quasi-inverse of and is a positive number such that .*

*Proof. *Put . Since , we have . We separate the proof into two cases.*Case 1. *, and .

According to Theorem 3.3, we have exists, and are bounded, and hence for some . Since is Lipschitz continuous at with modulus , for some , we have whenever and . Since , we may assume for all . From Theorem 3.3, we have and as . Hence, there exists an integer such that for all . Since , we have
By for all . Since is uniformly continuous on each bounded set, (3.20), (3.21) and (4.16), we obtain By the uniform smoothness of , we have Since for all , we get
Hence, as It follows from Proposition 4.1 and (4.16) that there exists , which implies for all and , such that
for all . It follows from and (3.20) that
From (3.3), (3.2), and (4.18), we have
for all . Since and (4.19), we may assume and for all . By (4.20) and induction, we obtain
for all , where , , .

Next, we prove , and tend to . By and , we get
Thus is bounded. Since and , there exists some such that whenever . Then, we get
which implies
Meanwhile, we also have
On the other hand,
Since , , it follows from Lemma 2.6 that .

From , which implies that , and (3.20). It follows that
By (4.18), we have
for all , where

Since is uniformly convex, is uniformly totally convex on each bounded set of . Denote by the modulus of uniformly total convexity on the bounded set . Then and satisfies . From the definition of the greatest quasi-inverse of , we deduce that
From Lemma 2.5, we have
Since , it holds by (4.28), (4.29), and (4.30) that
for all . Let . Since and for , it follows in [18, Lemma ?3.3.1(i)] that and this implies that . By the first inequality of (4.20), (4.31) and the definition of , we have
*Case 2. * and .

From the proof of Theorem 3.1, we note that if then converges to , converges strongly to , and . By the same argument as in the proof of Case 1, we obtain for all , where , and are those of Case 1.

It remains to show that , and converge to . Since and , , it follows that
whenever large enough.

On the other hand,