Wei Li, Agarwal Ravi P
1School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, China.
2Department of Mathematics, Texas A&M University-Kingsville, Kingsville, USA.
J Inequal Appl. 2018;2018(1):64. doi: 10.1186/s13660-018-1657-3. Epub 2018 Mar 27.
In a real uniformly convex and uniformly smooth Banach space, some new monotone projection iterative algorithms for countable maximal monotone mappings and countable weakly relatively non-expansive mappings are presented. Under mild assumptions, some strong convergence theorems are obtained. Compared to corresponding previous work, a new projection set involves projection instead of generalized projection, which needs calculating a Lyapunov functional. This may reduce the computational labor theoretically. Meanwhile, a new technique for finding the limit of the iterative sequence is employed by examining the relationship between the monotone projection sets and their projections. To check the effectiveness of the new iterative algorithms, a specific iterative formula for a special example is proved and its computational experiment is conducted by codes of Visual Basic Six. Finally, the application of the new algorithms to a minimization problem is exemplified.
在一个实一致凸且一致光滑的巴拿赫空间中，针对可数个极大单调映射和可数个弱相对非扩张映射，提出了一些新的单调投影迭代算法。在温和假设下，得到了一些强收敛定理。与之前相应的工作相比，一个新的投影集涉及投影而非广义投影，广义投影需要计算一个李雅普诺夫泛函。这在理论上可能会减少计算量。同时，通过考察单调投影集与其投影之间的关系，采用了一种寻找迭代序列极限的新技术。为检验新迭代算法的有效性，证明了一个特殊例子的具体迭代公式，并通过Visual Basic 6代码进行了计算实验。最后，举例说明了新算法在一个最小化问题中的应用。
