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Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mappingTof a closed convex subsetCof a CAT(0) spaceX. Suppose that the set Fix(T)of fixed points ofTis nonempty. For a contractionfonCandt∈(0,1), letxt∈Cbe the unique fixed point of the contractionx↦tf(x)⊕(1-t)Tx. We will show that ifXis a CAT(0) space satisfying some property, then{xt}converge strongly to a fixed point ofTwhich solves some variational inequality. Consider also the iteration process{xn}, wherex0∈Cis arbitrary andxn+1=αnf(xn)⊕(1-αn)Txnforn≥1, where{αn}⊂(0,1). It is shown that under certain appropriate conditions onαn,{xn}converge strongly to a fixed point ofTwhich solves some variational inequality.

Let H 1 , H 2 , H 3 be real Hilbert spaces, C ⊆ H 1 , Q ⊆ H 2 be two nonempty closed convex sets, and let A : H 1 → H 3 , B : H 2 → H 3 be two bounded linear operators. The split equality problem (SEP) is finding x ∈ C, y ∈ Q such that Ax = By. Recently, Moudafi has presented the ACQA algorithm and the RACQA algorithm to solve SEP. However, the two algorithms are weakly convergent. It is therefore the aim of this paper to construct new algorithms for SEP so that strong convergence is guaranteed. Firstly, we define the concept of the minimal norm solution of SEP. Using Tychonov regularization, we introduce two methods to get such a minimal norm solution. And then, we introduce two algorithms which are viewed as modifications of Moudafi's ACQA, RACQA algorithms and KM-CQ algorithm, respectively, and converge strongly to a solution of SEP. More importantly, the modifications of Moudafi's ACQA, RACQA algorithms converge strongly to the minimal norm solution of SEP. At last, we introduce some other algorithms which converge strongly to a solution of SEP.

Abstract. In this paper the concepts of character contractibility, approximate character amenability (contractibility) and uniform approximate character amenability (contractibility) are introduced. We are concerned with the relations among the generalized concepts of character amenability for Banach algebra. We prove that approximate character amenability and approximate character contractibility are the same properties, as are uniform approximate character amenability and character amenability, as are uniform approximate character contractibility and character contractibility. For commutative Banach algebra, we prove that character contractibility and contractibility are the same properties. Moreover, general theory for those concepts is developed.

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