Abstract:
We prove a weak convergence theorem of the modified Mann iteration process for a uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space. We also introduce two kinds of new monotone hybrid methods and obtain strong convergence theorems for an infinitely countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings in a Hilbert space. The results improve and extend the corresponding ones announced by Kim and Xu (2006) and Nakajo and Takahashi (2003).

Abstract:
We introduce a new monotone hybrid iterative scheme for finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions of variational inequality problem, and the set of the solutions of the equilibrium problem in a Hilbert space. Strong convergence theorems of the purposed iteration are established. 1. Introduction Let be a nonempty convex subset of a Banach spaces . Let be a bifunction from to , where is the set of all real numbers. The equilibrium problem for is to find such that for all . The set of such solutions is denoted by . The set is called proximal if for each , there exists an element such that , where . Let , , and denote the families of nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of , respectively. The Hausdorff metric on is defined by for . A single-valued map is called nonexpansive if for all . A multivalued map is said to be nonexpansive if for all . An element is called a fixed point of (resp., ) if (resp., ). The set of fixed points of is denoted by . The mapping is called quasi-nonexpansive [1] if and for all and all . It is clear that every nonexpansive multivalued map with is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are not nonexpansive; see [2]. The mapping is called hemicompact if, for any sequence in such that as , there exists a subsequence of such that . We note that if is compact, then every multivalued mapping is hemicompact. A mapping is said to satisfy Condition (I) if there is a nondecreasing function with , for such that for all . In 1953, Mann [3] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping in a Hilbert space : where the initial point is taken in arbitrarily and is a sequence in . However, we note that Mann's iteration process (1.3) has only weak convergence, in general; for instance, see [4–6]. In 2003, Nakajo and Takahashi [7] introduced the method which is the so-called CQ method to modify the process (1.3) so that strong convergence is guaranteed. They also proved a strong convergence theorem for a nonexpansive mapping in a Hilbert space. Recently, Tada and Takahashi [8] proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space . In 2005, Sastry and Babu [9] proved that the Mann and Ishikawa iteration schemes for multivalued map with a fixed point converge to a fixed point of under certain

Abstract:
We consider the Ishikawa-type iteration process with errors for acontinuous real function on an arbitrary interval and prove the convergencetheorem. Furthermore, we give numerical examples to comparewith Mann and Ishikawa iteration processes with error sequences.

Abstract:
We introduce a new projection algorithm for solving the fixed pointproblem of relatively nonexpansive mappings in the framework of Banachspaces. We also prove the strong convergence theorem for suchmappings.

Abstract:
The purpose of this paper is to introduce a new hybrid projection algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of the variational inequality for an inverse-strongly monotone operator and the set of fixed points of relatively quasi-nonexpansive mappings in a Banach space. Then we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space.

Abstract:
The purpose of this paper is to introduce a new hybrid projection algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of the variational inequality for an inverse-strongly monotone operator and the set of fixed points of relatively quasi-nonexpansive mappings in a Banach space. Then we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space.

The present study was
conducted to investigate the effect of aqueous extracts of Ocimum sanctum L.
leaves on blood glucose, serum lipid profile and anti-oxidative activity to
protect various risk organs in DM rats. Ocimum sanctum L. leaves were
extracted using water, then the total phenolic content was determined. Three
groups of male Wistar rats were used including normal control rats, DM rats and
DM rats daily fed with aqueous extracts of Ocimum sanctum L. leaves
(AQOS) for three weeks. DM rats were induced by intraperitoneal injection of
streptozotocin (65 mg/kgbw). The results show that three weeks of diabetic
induction increased blood glucose, serum lipid profile and serum levels of
AST, ALT, ALP, LDH, CK-MB, creatinine and BUN. AQOS significantly decreased
blood glucose, serum lipid profile and serum levels of AST, ALT, ALP, LDH,
CK-MB, creatinine and BUN. The low level of serum insulin was also raised by
AQOS. AQOS suppressed high TBARS level and raised the activities of antioxidant
enzymes in the liver, kidney and cardiac tis- sues. Histopathological results
show that AQOS preserved the liver, kidney and myocardial tissues. It can be
concluded that AQOS had anti-hyperglycemic, anti-hyperlipidemic, and free
radical sca- venging effects providing organ protection from diabetes. The
phenolic compounds contained in AQOS might be responsible for these activities.

Abstract:
We introduce a new hybrid iterative scheme for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove the strong convergence theorem by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space. 1. Introduction Let be a real Banach space, and let be the dual of . Let be a closed and convex subset of . Let be bifunctions from to , where is the set of real numbers and is an arbitrary index set. The system of equilibrium problems is to find such that If is a singleton, then problem (1.1) reduces to find such that The set of solutions of the equilibrium problem (1.2) is denoted by . Combettes and Hirstoaga [1] introduced an iterative scheme for finding a common element in the solutions set of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. In 2004, Matsushita and Takahashi [2] introduced the following algorithm for a relatively nonexpansive mapping in a Banach space : for any initial point , define the sequence by where is the duality mapping on , is the generalized projection from onto , and is a sequence in . They proved that the sequence converges weakly to fixed point of under some suitable conditions on . In 2008, Takahashi and Zembayashi [3] introduced the following iterative scheme which is called the shrinking projection method for a relatively nonexpansive mapping and an equilibrium problem in a Banach space : They proved that the sequence converges strongly to under some appropriate conditions. 2. Preliminaries and Lemmas Let be a real Banach space, and let be the unit sphere of . A Banach space is said to be strictly convex if, for any , It is also said to be uniformly convex if, for each , there exists such that, for any , It is known that a uniformly convex Banach space is reflexive and strictly convex. The function which is called the modulus of convexity of is defined as follows: The space is uniformly convex if and only if for all . A Banach space is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for . The duality mapping is defined by for all . If is a Hilbert space, then , where is the identity operator. It is also known that, if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subset

Abstract:
We proposed in this paper a new iterative scheme for finding common elements of the set of fixed points of a finite family of quasi-nonexpansive mappings, the set of solutions of variational inclusion, and the set of solutions of generalized equilibrium problems. Some strong convergence results were derived by using the concept of W-mappings for a finite family of quasi-nonexpansive mappings. Strong convergence results are derived under suitable conditions in Hilbert spaces.

Abstract:
We proposed in this paper a new iterative scheme for finding common elements of the set of fixed points of a finite family of quasi-nonexpansive mappings, the set of solutions of variational inclusion, and the set of solutions of generalized equilibrium problems. Some strong convergence results were derived by using the concept of -mappings for a finite family of quasi-nonexpansive mappings. Strong convergence results are derived under suitable conditions in Hilbert spaces.