The purpose of this paper is to present the notion of weak relatively nonexpansive multi-valued mapping and to prove the strong convergence theorems of fixed point for weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. In this paper, the notion of multivalued resolvent of maximal monotone operator has been also presented which is a weak relatively nonexpansive multivalued mapping and can be used to find the zero point of maximal monotone operator.

Iterative methods for approximating fixed points of multivalued mappings in Banach spaces have been studied by some authors, see for instance [

Let

Let

Let

As we all know that if

Next, we assume that

The generalized projection

If

Let

In [

The purpose of this paper is to present the notion of weak relatively nonexpansive multivalued mapping and to prove the strong convergence theorems for the weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. We first give the definition of weak relatively nonexpansive multivalued mapping as follows.

Let

We need the following Lemmas for the proof of our main results.

Let

Let

Let

Let

First, we show

Let

Next, we give an example which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping.

Let

We second claim that

Let

We first show that

Next, we show that

Since

Since

Finally, we prove that

When

Let

In this section, we apply the above results to prove some strong convergence theorem concerning maximal monotone operators in a Banach space

Let

Let

Let

Let

Since

By using Theorems

Let

This project is supported by the National Natural Science Foundation of China under Grant (11071279).