两类非线性算子的不动点与固有值问题
|
摘要
本文首先研究了1-集压缩映象的若干不动点及固有值与固有元的存在性问题，然后讨论了具有凹凸性的一类1-型亚混合单调算子不动点的存在性问题。全文共分三章，每章可以看作是一个独立的部分。
第一章利用W．V．Petryshyn[5]中1-集压缩映象拓扑度的性质讨论了1-集压缩映象的若干固有值与固有元问题，得出了若干1-集压缩映象固有值与固有元定理，这些都是新结果。
在第二章，也是利用W．V．Petryshyn[5]中1-集压缩映象拓扑度的性质，并利用[9]中W．V．Petryshyn提出的一个要求映象满足(C)条件，得出了在不同的边界条件下1-集压缩映象的若干不动点存在性定理，是对[8]中Altman定理的重要推广。
第三章在不对算子的连续性及紧性作任何假设的条件下，在正规锥中，用单调迭代方法证明了一类具有凹凸性的非线性算子的不动点的存在性定理。
In this thesis, some existence problems of 1-set-contractive mapping's fixed points, eigenvalues and eigenvectors are discussed firstly. Then some problems of a class of 1-submixed monotone operators with convexity and concavity are investigated. This thesis consists of three chapters. And each chapter is an integrated paper.
In chapter l,some properties of 1-set-contractive mapping's degree in W.V.Petryshyn [5] are used to discuss some existence problems of 1-set-contractive mapping's eigenvalues and eigenvectors. Some existence theorems of 1-set-contractive mapping's eigenvalues and eigenvectors are obtained and these are all new ones.
In chapter 2,some properties of 1-set-contractive mapping's degree in W.V.Petryshyn [5] are also used, we have acquired some existence theorems of fixed points of 1-set-contractive mappings satisfying condition (c) in different edge conditions, which was mentioned in W.V.Petryshyn [9].These results are some important generalizations of Altman theorems in [8].
Chapter 3 is devoted to a class of nonlinear operators with convexity and concavity. Using monotone iterative method, some existence theorems of fixed points are concluded without any hypothesis of continuity and compactness.
第一章利用W．V．Petryshyn[5]中1-集压缩映象拓扑度的性质讨论了1-集压缩映象的若干固有值与固有元问题，得出了若干1-集压缩映象固有值与固有元定理，这些都是新结果。
在第二章，也是利用W．V．Petryshyn[5]中1-集压缩映象拓扑度的性质，并利用[9]中W．V．Petryshyn提出的一个要求映象满足(C)条件，得出了在不同的边界条件下1-集压缩映象的若干不动点存在性定理，是对[8]中Altman定理的重要推广。
第三章在不对算子的连续性及紧性作任何假设的条件下，在正规锥中，用单调迭代方法证明了一类具有凹凸性的非线性算子的不动点的存在性定理。
In this thesis, some existence problems of 1-set-contractive mapping's fixed points, eigenvalues and eigenvectors are discussed firstly. Then some problems of a class of 1-submixed monotone operators with convexity and concavity are investigated. This thesis consists of three chapters. And each chapter is an integrated paper.
In chapter l,some properties of 1-set-contractive mapping's degree in W.V.Petryshyn [5] are used to discuss some existence problems of 1-set-contractive mapping's eigenvalues and eigenvectors. Some existence theorems of 1-set-contractive mapping's eigenvalues and eigenvectors are obtained and these are all new ones.
In chapter 2,some properties of 1-set-contractive mapping's degree in W.V.Petryshyn [5] are also used, we have acquired some existence theorems of fixed points of 1-set-contractive mappings satisfying condition (c) in different edge conditions, which was mentioned in W.V.Petryshyn [9].These results are some important generalizations of Altman theorems in [8].
Chapter 3 is devoted to a class of nonlinear operators with convexity and concavity. Using monotone iterative method, some existence theorems of fixed points are concluded without any hypothesis of continuity and compactness.
引文
[1] H.Aman, Fixed point equations and Nonlinear eigenvalue problems in ordered Banach spaces, SIMA Review, 18(1976), 620-709.
[2] Jane Cronin, Eigenvalues of some Nonlinear operators, J.Math.Anal.Appl. 38(1972), 659-667.
[3] Da Jun Guo, Eigenvalues and Eigenvectors of Nonlinear operators, Chin.Ann.of Math. 2(Eng.Issue) 1981, 65-80.
[4] I.Massabo, C.A.stuart, Positive eigenvector of k-set-contractions, Nonliear.Anal.TMA, 3(1979), 35-44.
[5] W.V.Petryshyn, Remark on condensing and k-set-contractive mappings, J.Math.Anal.Appl. 39(1971), 717-741.
[6] Li Guo Zhen, The Fixed point Index and The Fixed Point Theorems of 1-set-contractive mappings, Proc.Amer.Math.Soc. Vol.104, No.4, 1988(1163-1170).
[7] 郭大钧，孙经先，抽象空间常微分方程，山东科学技术出版社．1989．
[8] 郭大钧，非线性泛函分析，山东科学技术出版社．2001．
[9] W.V. Petryshyn. Fixed point theorems for various classes of 1-set-contractive and 1-ball -contractive mapping in Banach spaces, Trans. Amer. Math. Soc. 182(1973), 323—352.
[10] W.V. Petryshyn, Strueture of the fixed points sets of k-set-contractions, Arch. Rational Mech. Anal. 40 (1970/71), 312—328. MR 42# 8358.
[11] R. D. Nussbaum, The fixed pint index and fixed point theorems for k-set-contractions, ph. D. Thesis, University of Chicago, Chicago, Ⅲ. 1969.
[12] 周恒，关于亚混合单调算子的不动点定理及其应用，山西大学学报，23(4)，309-311，2000．
[13] Zhang Ke-Mei, Xie Xue-Jun Solution and coupled minimal-maximal Quasi-solutions of Nonlinear Non-monotone operator Equations in Banach spaces, Journal of mathematical Research & Exposition. Vol. 23, No. 147-52, 2003.
[14] 张克梅，抽象空间中非单调算子方程解的存在唯一性，工程数学学报，Vol17．No．1．2000．
[15] 路慧芹，一类非线性算了方程的存在性及其应用，曲阜师范大学学报，Vol25，No．1，2001．
[16] 于立新，混合单调算子不动点的mann迭代序列的收敛性，曲阜师范大学学报，Vol27，No．1，2001．
[17] 孙钦福、李仁贵、栾世霞，T-混合单调算子不动点定理，曲阜师范大学学报，Vol25，No．4，1999．
[18] Zhang Zhi tao, New Fixed Point Theorems of Mixed Monotone Operators and Application Journal of Mathematical Analysis and Application 104. 307-319, 1996.
[2] Jane Cronin, Eigenvalues of some Nonlinear operators, J.Math.Anal.Appl. 38(1972), 659-667.
[3] Da Jun Guo, Eigenvalues and Eigenvectors of Nonlinear operators, Chin.Ann.of Math. 2(Eng.Issue) 1981, 65-80.
[4] I.Massabo, C.A.stuart, Positive eigenvector of k-set-contractions, Nonliear.Anal.TMA, 3(1979), 35-44.
[5] W.V.Petryshyn, Remark on condensing and k-set-contractive mappings, J.Math.Anal.Appl. 39(1971), 717-741.
[6] Li Guo Zhen, The Fixed point Index and The Fixed Point Theorems of 1-set-contractive mappings, Proc.Amer.Math.Soc. Vol.104, No.4, 1988(1163-1170).
[7] 郭大钧，孙经先，抽象空间常微分方程，山东科学技术出版社．1989．
[8] 郭大钧，非线性泛函分析，山东科学技术出版社．2001．
[9] W.V. Petryshyn. Fixed point theorems for various classes of 1-set-contractive and 1-ball -contractive mapping in Banach spaces, Trans. Amer. Math. Soc. 182(1973), 323—352.
[10] W.V. Petryshyn, Strueture of the fixed points sets of k-set-contractions, Arch. Rational Mech. Anal. 40 (1970/71), 312—328. MR 42# 8358.
[11] R. D. Nussbaum, The fixed pint index and fixed point theorems for k-set-contractions, ph. D. Thesis, University of Chicago, Chicago, Ⅲ. 1969.
[12] 周恒，关于亚混合单调算子的不动点定理及其应用，山西大学学报，23(4)，309-311，2000．
[13] Zhang Ke-Mei, Xie Xue-Jun Solution and coupled minimal-maximal Quasi-solutions of Nonlinear Non-monotone operator Equations in Banach spaces, Journal of mathematical Research & Exposition. Vol. 23, No. 147-52, 2003.
[14] 张克梅，抽象空间中非单调算子方程解的存在唯一性，工程数学学报，Vol17．No．1．2000．
[15] 路慧芹，一类非线性算了方程的存在性及其应用，曲阜师范大学学报，Vol25，No．1，2001．
[16] 于立新，混合单调算子不动点的mann迭代序列的收敛性，曲阜师范大学学报，Vol27，No．1，2001．
[17] 孙钦福、李仁贵、栾世霞，T-混合单调算子不动点定理，曲阜师范大学学报，Vol25，No．4，1999．
[18] Zhang Zhi tao, New Fixed Point Theorems of Mixed Monotone Operators and Application Journal of Mathematical Analysis and Application 104. 307-319, 1996.