Hilbert空间中一类新广义非线性变分不等式组问题
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摘要
变分不等式原理是当今数学技术中一个有力的研究工具,有重要的学术研究价值和意义。在运筹学、计算机科学、系统科学、工程技术、交通和经济与管理等方面有着广泛而重要的应用。利用η-次微分算子的预解算子技巧和辅助原理技术,研究Hilbert空间中的一类广义非线性变分不等式组问题,得出该问题的解。在此基础上,引入一个带有Lipschitz连续、强单调和松弛单调映射的辅助性问题,并且利用预解算子和集值压缩映像的不动点定理证明了解的存在性与唯一性,这一结果推广、改进和发展了相关文献的结果。
Variational inequality principle is a powerful research tool in the current mathematical technology,and has important academic research value and significance. It is widely and importantly applied in operations research,computer science,system science,engineering technology,transportation,economic and management and other aspects. In this essay,by using the resolvent operator technique and the auxiliary principle technique of η a differential operator,a class of generalized nonlinear variational inequalities problem in Hilbert space is researched,and the solution of the problem is obtained. On this basis,a auxiliary problem with Lipschitz continuous,strongly monotone and relaxed monotone mapping is introduced,and the existence and the uniqueness of the solution are proved by using the resolvent operator and the fixed point theorem of set-valued contractive mappings. This result has extended,improved and developed the results in relevant literatures.
Variational inequality principle is a powerful research tool in the current mathematical technology,and has important academic research value and significance. It is widely and importantly applied in operations research,computer science,system science,engineering technology,transportation,economic and management and other aspects. In this essay,by using the resolvent operator technique and the auxiliary principle technique of η a differential operator,a class of generalized nonlinear variational inequalities problem in Hilbert space is researched,and the solution of the problem is obtained. On this basis,a auxiliary problem with Lipschitz continuous,strongly monotone and relaxed monotone mapping is introduced,and the existence and the uniqueness of the solution are proved by using the resolvent operator and the fixed point theorem of set-valued contractive mappings. This result has extended,improved and developed the results in relevant literatures.
引文
[1]夏锦,苗放.解一类似变分不等式问题的预解式技术和辅助原理技术[J].四川师范大学学报:自然科学版,2002,25(5):85-486.
[2]陈细法.新的近似算法的广义非线性变分不等式系统[J].四川师范大学学报:自然科学版,2001,24(6):813-817.
[3]代宏霞.一类广义非线性变分不等式组解的存在唯一性[J].四川大学学报:自然科学版,2004,41(1):1-4.
[4]任晓.一类广义非线性变分不等式组解的存在性及迭代逼近[J].四川大学学报:自然科学版,2003,40(3):411-415.
[5]罗元松.Hilbert空间中一般形式的松弛余强制变分不等方程组解的迭代逼近问题[J].四川大学学报:自然科学版,2007,44(3):467-471.
[6]万波,邓磊.广义集值混合似变分不等式的迭代算法[J].西南师范大学学报:自然科学版,2008,33(3):1-4.
[7]张玉敏,张国春,刘英.Hilbert空间中关于松弛协强制映射的广义变分不等式组[J].河北大学学报:自然科学版,2012,32(5):453-457.
[8]彭再云,雷鸣,汪达成,等.希尔伯特空间中的广义余强制变分不等式体系及带误差的三步投影方法[J].四川师范大学学报:自然科学版,2009,32(2):168-171.
[9]丁协平.一类广义非线性隐拟变分包含[J].应用数学和力学,1999,20(10):1015-1024.
[10]邱洋青,何思宇.一类广义集值强非线性混合似变分不等式组的迭代算法及辅助原理[J].南昌工程学院学报,2013(1):18-24.
[11]张彦.一类新的变分包含系统[J].内江师范学院学报,2012(2):8-11.
[12]李观荣.一类新的广义变分包含系统及其迭代算法[J].湛江师范学院学报,2014(3):24-29.
[13]Ding X P,Luo C L.Perturbed proximal point algorithma for generalized quasi-variation-like inclusions[J].J Comput Appl Math,2000,210:153-165.
[2]陈细法.新的近似算法的广义非线性变分不等式系统[J].四川师范大学学报:自然科学版,2001,24(6):813-817.
[3]代宏霞.一类广义非线性变分不等式组解的存在唯一性[J].四川大学学报:自然科学版,2004,41(1):1-4.
[4]任晓.一类广义非线性变分不等式组解的存在性及迭代逼近[J].四川大学学报:自然科学版,2003,40(3):411-415.
[5]罗元松.Hilbert空间中一般形式的松弛余强制变分不等方程组解的迭代逼近问题[J].四川大学学报:自然科学版,2007,44(3):467-471.
[6]万波,邓磊.广义集值混合似变分不等式的迭代算法[J].西南师范大学学报:自然科学版,2008,33(3):1-4.
[7]张玉敏,张国春,刘英.Hilbert空间中关于松弛协强制映射的广义变分不等式组[J].河北大学学报:自然科学版,2012,32(5):453-457.
[8]彭再云,雷鸣,汪达成,等.希尔伯特空间中的广义余强制变分不等式体系及带误差的三步投影方法[J].四川师范大学学报:自然科学版,2009,32(2):168-171.
[9]丁协平.一类广义非线性隐拟变分包含[J].应用数学和力学,1999,20(10):1015-1024.
[10]邱洋青,何思宇.一类广义集值强非线性混合似变分不等式组的迭代算法及辅助原理[J].南昌工程学院学报,2013(1):18-24.
[11]张彦.一类新的变分包含系统[J].内江师范学院学报,2012(2):8-11.
[12]李观荣.一类新的广义变分包含系统及其迭代算法[J].湛江师范学院学报,2014(3):24-29.
[13]Ding X P,Luo C L.Perturbed proximal point algorithma for generalized quasi-variation-like inclusions[J].J Comput Appl Math,2000,210:153-165.