Abstract:
It is our purpose in this paper to prove two convergents of viscosity approximation scheme to a common fixed point of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, it is the unique solution in to a certain variational inequality, where ∶=∩∞=0() stands for the common fixed-point set of the family of multivalued nonexpansive mapping {}.

Abstract:
The convergence of three-step fixed point iterative processes for generalized multivalued nonexpansive mapping was considered in this paper. Under some different conditions, the sequences of three-step fixed point iterates strongly or weakly converge to a fixed point of the generalized multivalued nonexpansive mapping. Our results extend and improve some recent results.

Abstract:
Recently, we introduced a new coefficient as a generalization of the modulus of smoothness and Pythagorean modulus such as JX , p (t). In this paper, We can compute the constant JX , p (1) under the absolute normalized norms on 2 by means of their corresponding continuous convex functions on [0, 1]. Moreover, some sufficient conditions which imply uniform normal structure are presented. 2000 Mathematics Subject Classification: 46B20.

Abstract:
We define a mean nonexpansive mapping T on X in the sense that , . It is proved that mean nonexpansive mapping has approximate fixed-point sequence, and, under some suitable conditions, we get some existence and uniqueness theorems of fixed point. 1. Introduction Let be a Banach space, a nonempty bounded closed convex subset of , and ？:？ a nonexpansive mapping; that is, We say that has the fixed-point property if every nonexpansive mapping defined on a nonempty bounded closed convex subset of has a fixed point. In 1965, Kirk [1] proved that if is a reflexive Banach space with normal structure, then has the fixed-point property. Let be a nonempty subset of real Banach space and a mapping from to . is called mean nonexpansive if for each , In 1975, Zhang [2] introduced this definition and proved that has a fixed point in , where is a weakly compact closed convex subset and has normal structure. For more information about mean nonexpansive mapping, one can refer to [3–5]. 2. Main Results Lemma 1. Let be a mean nonexpansive mapping of the Banach space . If is continuous and , then T has a unique fixed point. Proof. The proof is similar to the proof of the Banach contractive theorem. If we let and as in Lemma 1, then the condition that is continuous may not be needed. Firstly, we recall the following two lemmas. Lemma 2. Let be a nonempty subset of Banach space and a mean nonexpansive self-mapping on with and . Let be a nonempty subset of ; one defines for any ; if the set is bounded, then is also bounded. Proof. Let and set as fixed; then for any , we have This implies that Hence, is bounded. The proof is complete. Lemma 3. Let be a nonempty subset of Banach space and a mean nonexpansive self-mapping on . If and , then for any , one has the following inequality: where and are two positive integers such that . Proof. By the definition of mean nonexpansive mapping, we have that this implies that where is an integer. When , the result is obvious. Suppose that (5) is true for ; that is, By the inequality (2) and (7), we have This implies from and that which follows that By induction, this completes the proof. Theorem 4. Let be a nonempty closed subset of Banach space and a mean nonexpansive self-mapping on . If and , then has a unique fixed point. Proof. For any , set ; by the definition of mean nonexpansive mapping, we have that Thus, the sequence is nonincreasing and bounded below, so exists. Suppose that ; then we have by Lemma 2 that the set is bounded, so there exists a positive number such that where and are two integers. Since and , for any , there

Abstract:
We introduce a new coefficient as a generalization of the modulus of smoothness and Pythagorean modulus of Banach space . Some basic properties of this new coefficient are investigated. Moreover, some sufficient conditions which imply normal structure are presented.

Abstract:
We present some sufficient conditions for which a Banach space X has normal structure in terms of the modulus of U-convexity, modulus of W -convexity, and the coefficient R(1,X), which generalized some well-known results. Furthermore the relationship between modulus of convexity, modulus of smoothness, and Gao's constant is considered, meanwhile the exact value of Milman modulus has been obtained for some Banach space.

Abstract:
We present some sufficient conditions for which a Banach space has normal structure in terms of the modulus of -convexity, modulus of -convexity, and the coefficient , which generalized some well-known results. Furthermore the relationship between modulus of convexity, modulus of smoothness, and Gao's constant is considered, meanwhile the exact value of Milman modulus has been obtained for some Banach space.

Abstract:
We introduce a new coefficient as a generalization of the modulus of smoothness and Pythagorean modulus of Banach space X. Some basic properties of this new coefficient are investigated. Moreover, some sufficient conditions which imply normal structure are presented.

Abstract:
We show some sufficient conditions on a Banach space concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zbag nu constant, the coefficient , the weakly convergent sequence coefficient WCS( ), and the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings. These fixed point theorems improve some previous results in the recent papers.

Abstract:
We show some sufficient conditions on a Banach space X concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zbag nu constant, the coefficient ε 0(X), the weakly convergent sequence coefficient WCS(X), and the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings. These fixed point theorems improve some previous results in the recent papers.