Abstract

Charles proved the convergence of Picard-type iteration for generalized accretive nonself-mappings in a real uniformly smooth Banach space. Based on the theorems of the zeros of strongly quasi-accretive mappings and fixed points of strongly hemi-contractions, we extend the results to Noor iterative process and SP iterative process for generalized hemi-contractive mappings. Finally, we analyze the rate of convergence of four iterative schemes, namely, Noor iteration, iteration of Corollary 2, SP iteration, and iteration of Corollary 4.

1. Introduction and Preliminaries

In 2009, Charles [1] proved the convergence of Picard-type iteration for generalized accretive nonself-mappings in a real uniformly smooth Banach space. In this paper, we consider that the Noor iteration process and SP iteration process will be extended from the results of Charles [1].

In [2], Noor et al. proved the convergence of Noor iteration to fixed point of a pseudocontractive self-map defined in a real uniformly smooth Banach space. In [3], Qin and Yao showed the weak convergence of a Mann-like algorithm for nonexpansive and accretive operators. In [4], the author considers some generalized nonexpansive mappings on convex metric spaces and gives some sufficient and necessary conditions for a Noor-type iteration to approximate a common fixed point of an infinite family of uniformly quasi-supLipschitzian mappings and an infinite family of expansive mappings in convex metric spaces.

In 1953, the most general Mann iterative scheme now studied is the following: ,where is a real sequence satisfying appropriate conditions. In 1974, Ishikawa [5] introduced the Ishikawa iteration process as follows: for a convex subset of a Banach space and a mapping from into itself, for any given , the sequence in is defined bywhere and are two sequences in [0, 1] satisfying the conditions , for all and .

In 2000, Noor [6, 7] gave the following three-step iterative schemes for solving nonlinear operator equations in uniformly smooth Banach spaces.

Let be a nonempty convex subset of and let be a mapping. For a given in , compute the sequence by the iterative schemes:which is called the Noor iterative process, where are three real sequences in [0,1] satisfying some certain conditions.

In 2011, Phuengrattana and Suantai defined the SP iteration schemes [8] as follows.

Let be a nonempty convex subset of and let be a mapping. For a given in , define the sequence bywhere are three real sequences in [0,1] satisfying some certain conditions.

Definition 1. (see [1]). Given a gauge function , the mapping defined byis called the duality map with gauge function , where is any normed space. In the particular case, , the duality map is called the normalized duality map.

Proposition 1. (see [9, 10]). If a Banach space has a uniformly Gateaux differentiable norm, then is uniformly continuous on bounded subsets of from the strong topology of to the weak∗ topology of .

Definition 2. (see [11]). Let be an arbitrary real normed linear space. A mapping is called strongly hemi-contractive if , and there exists such that, for all ,holds for all . If , then is called hemi-contractive. Finally, is called generalized hemi-contractive with strictly increasing continuous function such that if, for all , there exists such thatIt follows from inequality (7) that is generalized hemi-contractive if and only if

Definition 3. (see [1, 12]). Let . The mapping is called generalized quasi-accretive with strictly increasing continuous function such that if, for all , there exists such that

Definition 4. (see [13, 14]). A mapping is called generalized Lipschitz if there exists a constant such that .

Proposition 2. (see [1]). If , the mapping is strongly hemi-contractive if and only if is strongly quasi-accretive, is strongly hemi-contractive if and only if is strongly quasi-accretive, and is generalized hemi-contractive if and only if is generalized quasi-accretive.

Proposition 3. (see [1]). Let be a uniformly smooth real Banach space, and let be a normalized duality mapping. Then,for all .

Proposition 4. (see [1]). Let and be sequences of nonnegative numbers and be a sequence of positive numbers satisfying the conditions and , as . Let the recursive inequalitybe given, where is strictly increasing continuous function such that it is positive on and . Then, , as .

2. Main Results

In this section, we will consider to extend the result of Charles [1] to the Noor iteration process and SP iteration process under the following assumptions.

First, we extend the result of Charles [1] to the Noor iteration process.

Theorem 1. Suppose is a nonempty closed convex subset of a real uniformly smooth Banach space . Suppose is a bounded generalized Lipschitz hemi-contractive mapping and . For arbitrary is a Noor iterative sequence defined by (3), where , , and . Then, there exists a constant such that if converges strongly to the unique fixed point of .

Proof. Since is a bounded generalized Lipschitz hemi-contractive mapping, there exists a strictly increasing continuous function with such thati.e.,for any , and exist a constant such thatLet be sufficiently large such that . Define . Then, since is bounded, we have that is bounded.
As is uniformly continuous on bounded subsets of , for , there exists a such that implies . Set .

Claim 1. is bounded.
Suffices to show that is in for all . The proof is by induction. By our assumption, . Suppose . We prove that . Assume for contradiction that . Then, since , we have that . We have the following estimates:Using (15), we obtainUsing (16), we obtainandUsing (17), we obtainUsing (19), we obtainThen,Therefore,Using Proposition 3 and the above formulas, we obtainSubstitute (23) into (24) and then substitute (24) into (25); since and , we havei.e., , a contradiction. Therefore, . Thus, by induction, is bounded. Then, are also bounded.