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高校应用数学学报
Applied Mathematics A Journal of Chinese Universities  2015, Vol. 30 Issue (2): 180-190    DOI:
    
Study on the existence of non-trivial solution of one kind $p$-Laplacian-like Neumann boundary value problems and iterative schemes
WEI Li, CHEN Rui
School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang 050061, China
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Abstract  In the paper, one kind $p$-Laplacian-like equations with Neumann boundaries are first converted to the form of operator equation involving the maximal monotone operators. After being proved that the operator equation exists solution, the $p$-Laplacian-like equations with Neumann boundaries are shown to have non-trivial solution. Secondly, some iterative schemes are constructed to approximate strongly to the zeros of maximal monotone operators through the relative resolvent of the maximal monotone operators. Finally, the relation between the solution of $p$-Laplacian-like equations with Neumann boundaries and the zeros of maximal monotone operators is being set up and then the iterative approximate solution of the $p$-Laplacian-like equations is obtained. The work done in this paper extends and complements some previous corresponding work.

Key words： $p$-Laplacian-like boundary value problems      relative resolvent      non-trivial solution      maximal monotone operator      iterative scheme     
Received: 18 November 2014      Published: 05 June 2018
CLC:  O177.91  
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WEI Li
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WEI Li, CHEN Rui. Study on the existence of non-trivial solution of one kind $p$-Laplacian-like Neumann boundary value problems and iterative schemes. Applied Mathematics A Journal of Chinese Universities, 2015, 30(2): 180-190.

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http://www.zjujournals.com/amjcua/     OR     http://www.zjujournals.com/amjcua/Y2015/V30/I2/180


一类$p$-Laplacian型Neumann边值问题非平凡解的存在性及迭代算法研究

首先将一类$p$-Laplacian型Neumann边值问题转化为含有极大单调算子的算子方程的形式, 得到算子方程解的存在性结论, 进而证明$p$-Laplacian型Neumann边值问题有非平凡解; 其次, 借助于极大单调算子的相对预解式构造出强收敛到极大单调算子零点的迭代序列; 最后, 建立$p$-Laplacian型Neumann边值问题的解与极大单调算子零点的关系, 得到解的迭代逼近序列. 推广和补充了以往的相关研究成果.

关键词： $p$-Laplacian型边值问题,  相对预解式,  非平凡解,  极大单调算子,  迭代算法 
[1] WEI Li, ZHANG Ya-nan. Implicit monotone projection scheme for infinite m-accretive mappings and inversely strongly accretive mappings and p-Laplacian systems[J]. Applied Mathematics A Journal of Chinese Universities, 2017, 32(2): 229-240.