Research On Theories And Applications Of The Split Common Fixed Point Problem For Nonlinear Operators

The theory of the split common fixed point problem for nonlinear operators has been widely used in many practical problems,such as image restoration and signal processing.To solve many problems in real life and engineering technology can be modelled mathematically as to solve some split common fixed point problem.Therefore,this problem and its related theories have become a very active research topic of the fixed point theory for nonlinear operators.The extended research on the split common fixed point problem is mainly reflected in three aspects: space generalization,operator generalization and improved algorithm.Based on general Banach spaces or nonlinear operators,a suitable iterative algorithm is constructed,which makes the improved algorithm have the characteristics of better convergence,faster convergence speed and easier implementation,and obtains more meaningful and promising results.Firstly,in the framework of Hilbert spaces,on the one hand,the strong positive linear bounded operator is extended to a Lipschitz continuous and strong monotone nonlinear operator.Based on quasi-nonexpansive mappings,a generalized viscosity approximation method is proposed.Under appropriate parameter conditions,it is proved that the sequence generated by the algorithm strongly converges to a fixed point of the quasi-nonexpansive mapping.By using this main result,the split equality fixed point,the split equality zeros point,the split equality optimization,the split equality equilibrium and their mixture problems be considered,and some strong convergence results be obtained.On the other hand,based on demicontractive mappings and selection technology,an equivalent characterization for the split common fixed point problem is given,a viscosity approximation method with self-adaptive iteration step size is constructed,and it is proved that the algorithm strongly converges to the solution of the split common fixed point problem.By using the main result,the multiple sets split fixed point problem,the split variational inclusion problem and the split zeros problem are analyzed and studied,and some strong convergence theorems are obtained.Secondly,by using the uniform convexity and uniform smoothness of Banach spaces,combined with dual operator and its inverse operator,an equivalent characterization for solving the split common fixed point problem for strictly quasi-φ-pseudocontractive mappings is given.It is proved that solving the split common fixed point problem is equivalent to finding a fixed point of a nonlinear operator.Using the generalized projection technique and Lyapunov function,a shrinking projection algorithm is constructed.The step size of the algorithm does not involve the norm of the linear bounded operator.Under suitable conditions,it is proved that the algorithm strongly converges to a solution of the split common fixed point problem.Using this result,some related problems in Hilbert spaces and Banach spaces are considered,and some strong convergence theorems are obtained.Finally,in the framework of reflexive Banach spaces,the multiple sets split common fixed point problem for strictly Bregman quasi-pseudocontractive mappings is studied.In this chapter,we also give an equivalent characterization for solving the multiple sets split common fixed point problem,and prove that solving this problem is equivalent to finding a fixed point of a nonlinear operator.Using Bregman projection technique and Bregman distance function,combined with inertial algorithm,two iterative algorithms,hybrid projection and shrinking projection,are constructed.The iterative step size of these algorithms does not depend on the norm of the linear bounded operator.It is proved that both algorithms strongly converge to the solution of the split common fixed point problem.Using these main results,some related problems are studied,and some strong convergence theorems are obtained.Combined with the recent results,we generalize the space and operator of the split common fixed point problem,improve the existing iterative algorithms,and obtain some strong convergence theorems.Some related problems are discussed and analyzed by applying the obtained results.