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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (3): 432-444.doi: 10.3969/j.issn.1005-3085.2015.03.012

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非阿基米德Menger概率2-度量空间中自映射的不动点问题研究

徐文清,   朱传喜,   吴照奇   

  1. 南昌大学数学系，南昌  330031
  • 收稿日期:2013-10-22 接受日期:2014-04-24 出版日期:2015-06-15 发布日期:2015-08-15
  • 基金资助:
    国家自然科学基金 (11361042; 11326099)；江西省自然科学基金 (20132BAB201001; 20142BAB211016)；江西省教育厅青年基金 (GJJ13012).

Study on Fixed Point Problems of Self-mappings in Non-Archimedean Menger Probabilistic 2-metric Spaces

XU Wen-qing,   ZHU Chuan-xi,   WU Zhao-qi   

  1. Department of Mathematics, Nanchang University, Nanchang 330031
  • Received:2013-10-22 Accepted:2014-04-24 Online:2015-06-15 Published:2015-08-15
  • Supported by:
    The National Natural Science Foundation of China (11361042; 11326099); the Natural Science Foundation of Jiangxi Province (20132BAB201001; 20142BAB211016); the Youth Research Foundation of the Education Department of Jiangxi Province (GJJ13012).

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摘要： 本文首先引入了Menger概率2-度量空间中的广义阿基米德三角范数以及非阿基米德Menger概率2-度量空间的定义，并给出了广义阿基米德三角范数的一个等价刻画．在此基础上，利用迭代方法，研究了非阿基米德Menger概率2-度量空间中满足不同类型压缩条件的自映射或自映射族的不动点或公共不动点的存在唯一性问题，得到了若干新的结果，改进和推广了之前文献中的相关结果．

关键词: Menger概率2-度量空间, 不动点, $g$-函数, 阿基米德$t$-范数

Abstract:

In this paper, we introduce the generalized Archimedean triangular norm in Menger probabilistic 2-metric spaces and Non-Archimedean probabilistic 2-metric spaces, and present a characterization of the generalized Archimedean triangular norm. Based on this, and utilizing the iterative method, we investigate the existence and uniqueness of the fixed points for self-mappings or common fixed points for a family of self-mappings under various contractive conditions in Non-Archimedean probabilistic 2-metric spaces. Many new results are obtained, which extend and generalize corresponding results in the literature.

Key words: Menger probabilistic 2-metric space, fixed point, $g$-function, Archimedean $t$-norm

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