拟非扩张映射和平衡问题的弱收敛定理

A Weak Convergence Theorem for a Quasi-nonexpansive Mapping and an Equilibrium Problem

研究求解拟非扩张映射不动点和平衡问题的公共解问题.构造出了求解平衡问题和拟非扩张映射不动点的公共解的迭代算法,在较弱的条件下,证明了该迭代序列唯一弱收敛到所研究问题的某一公共解,并且该迭代序列在公共解集上的投影强收敛到该公共解.通过证明非扩张映射是满足定理条件(B)的拟非扩张映射,得到一个推论,即非扩张映射不动点与平衡问题的公共解的迭代算法及算法的弱收敛性结果.进一步,给出了例子说明存在满足本文条件(B)的拟非扩张映射,同时该映射不是一个非扩张映射.Tada和Takahashi(J.Optim.Theory Appl.,2007,133:359-370)论文中的一个主要结果(定理4.1)仅是本文定理的一种特殊情况.

This paper proposed an iterative method for finding a common solution of an equilibrium problem and a fixed point problem of a quasi-nonexpansive mapping. It has proved that the iterative sequences converge weakly to a common solution of the problems mentioned above,and that the projections of the iterative sequences onto the set of common solutions of the above problems converge strongly to the common solution. By proving that nonexpansive mappings are quasi-nonexpansive mappings satisfying condition （B） of the theorem, it is shown that iterative sequences converge weakly to a common solution of an equilibrium problem and a fixed point problem of a nonexpansive mapping. Furthermore, an example was given to show that there exists a qusi-nonexpansive mapping, which satisfies condition （B） but is not a nonexpansive mapping. The results take a main result （Theorem 4.1） of Tada and Takahashi （. J. Optim and Theory Appl. , 2007, 133. 359--370） as special cases.