| In this thesis,we study implicit midpoint iterative algorithm of nonlinear operator.Firstly,we mainly introduce development process and widespread use of fixed point theory.Meanwhile,we also introduce domestic and foreign research dynamic and achieved important results.Secondly,we introduce the generalized viscosity implicit rules of nonexpansive mappings in uniformly smooth Banach spaces by constructing the sequence {xn}:xn+1=?nxn+?nf?xn?+?nT(snxn+?1-sn?xn+1).Strong convergence theorems of the rules are proved under certain assumptions imposed on the parameters.As applications,we use our main results to solve fixed point problems of strict pseudocontractions in Banach spaces and variational inequality problems in Hilbert spaces.Thirdly,by constructing the sequence {xn}:xn+1=?nxn+?nf?xn?+?nTn(((xn+xn+1)/2),we introduce a modified viscosity implicit rules of one asymptotically nonexpansive mapping in Hilbert spaces.Some strong convergence theorems are given under certain assumptions imposed on the parameters.As an application,we apply our main results to solve finite variational inequality problems in Hilbert spaces.Finally,we introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in the intermediate sense in Hilbert spaces by constructing the sequence {xn}:xn+1=?nxn+?nf?xn?+?nTn(((xn+xn+1)/2).Meanwhile,we extend the sequence to {xn}:xn+1=?nxn+?nf?xn?+?nTn(((xn+xn+1)/2).We obtain some strong convergence theorems under certain assumptions imposed on the parameters. |
PDF Full Text Request