Banach空间中Forward-Backward分裂法(简称FB分裂法)在信号处理、统计回归以及机器学习等领域有着广泛应用。本课题将对FB分裂法及其非精确算法的收敛性及收敛速度展开研究，且将减弱算法的强收敛性条件和加快算法的收敛速度。首先，用一致凸且q一致光滑(或一致光滑)的Banach空间相关不等式、次闭原理、Reich连续收敛定理以及实数列收敛定理等研究算法的收敛性；其次，用KM算法的性质、FB迭代算子的性质以及FB迭代算子的不动点方程等分析算法及其加速算法的收敛速度。在加速算法方面，我们将在迭代算法中引入松驰因子(该因子的取值范围为(0,2))；最后，将上述算法应用在信号处理和机器学习等实际问题中，通过数值实验检验算法的收敛性和收敛速度。本课题的研究结果将为FB分裂法收敛性和收敛速度的研究提供新思路。

The Forward-Backward splitting method (FB splitting method for short) in Banach space plays an important role in signal processing, statistical regression，machine learning and so on. In the subject, we will investigate the convergence and convergence rate of FB splitting method and its inexact algorithms. This subject will weaken the strong convergence condition of the algorithms and accelerate the convergence rate of the algorithms. Firstly, we will study the algorithms’ convergences according to some inequalities in the setting of uniformly convex and q-uniformly smooth (or uniformly smooth) Banach space, demiclosedness principle, Reich’s continuous convergence theorem, convergence theorem of real sequence and so on. Secondly, we will analyse the convergence rates of the algorithms and the accelerated algorithms by using the properties of KM iterative algorithm, the properties of FB iterative operator and the fixed point equation of FB iterative operator, and so on. In the aspect of accelerating algorithms, we will insert a relaxation factor (the value range of the factor is (0,2)) in the iterative algorithm. Finally, we will construct mathematical models for some application problems such as signal processing and machine learning, and check the convergence and the convergence rate of algorithms through numerical experiments. The results of this study may provide new insights for the researches on convergence and convergence rate of FB splitting method.