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.
New △-convergence theorems of iterative sequences for asymptotically nonexpansive mappings in CAT(0) spaces are obtained. Consider an asymptotically nonexpansive self-mapping of a closed convex subset of a CAT(0) space . Consider the iteration process , where is arbitrary and or for , where . It is shown that under certain appropriate conditions on △-converges to a fixed point of .
The multiple-sets split equality problem (MSSEP) requires finding a pointx∈∩i=1NCi,y∈∩j=1MQjsuch thatAx=By, whereNandMare positive integers,{C1,C2,…,CN}and{Q1,Q2,…,QM}are closed convex subsets of Hilbert spacesH1,H2, respectively, andA:H1→H3,B:H2→H3are two bounded linear operators. WhenN=M=1, the MSSEP is called the split equality problem (SEP). If B=I, then the MSSEP and SEP reduce to the well-known multiple-sets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. One of the purposes of this paper is to introduce an iterative algorithm to solve the SEP and MSSEP in the framework of infinite-dimensional Hilbert spaces under some more mild conditions for the iterative coefficient.
LetH1, H2,andH3be real Hilbert spaces, letC⊆H1, Q⊆H2be two nonempty closed convex sets, and letA:H1→H3, B:H2→H3be two bounded linear operators. The split equality problem (SEP) is to findx∈C, y∈Qsuch thatAx=By. LetH=H1×H2; considerf:H→Ha contraction with coefficient0<α<1, a strongly positive linear bounded operatorT:H→Hwith coefficientγ̅>0, andM:H→His aβ-inverse strongly monotone mapping. Let0<γ<γ̅/α,S=C×QandG:H→H3be defined by restricting toH1isAand restricting toH2is-B, that is,Ghas the matrix formG=[A,-B]. It is proved that the sequence{wn}={(xn,yn)}⊆Hgenerated by the iterative methodwn+1=PS[αnγf(wn)+(I-αnT)PS(I-γnG*G)PS(wn-λnMwn)]converges strongly tow̃which solves the SEP and the following variational inequality:〈(T-λf)w̃,w-w̃〉≥0and〈Mw̃,w-w̃〉≥0for allw∈S. Moreover, if we takeM=G*G:H→H, γn=0, thenMis aβ-inverse strongly monotone mapping, and the sequence{wn}generated by the iterative methodwn+1=αnγf(wn)+(I-αnT)PS(wn-λnG*Gwn)converges strongly tow̃which solves the SEP and the following variational inequality:〈(T-λf)w̃,w-w̃〉≥0for allw∈S.
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