若干非线性算子的讨论及其在微分方程边值问题中的应用
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摘要
在本文中,我们主要讨论两类非线性方面的内容,其一为Banach空间中的非线性算子方程,其二为非线性算子理论在微分方程边值问题中的应用。所使用的方法为半序方法及迭代技巧等。
全文共分为三章。
在第一章(概论)中,我们对几类非线性算子(α(>1)-齐次算子,一类可迭代求不动点的算子,集值算子)的研究现状及我们在本文中将要做的工作进行了阐述;同时系统的介绍了用非线性知识处理微分方程边值问题的总体思路。
第二章,我们给出了几类非线性算子的不动点定理。
在§2.1中,我们在较弱的条件下给出了α(>1)-齐次算子具有唯一不动点的充分条件以及齐次算子的介值性定理,这类结论和方法在可查文献中尚不多见;
在§2.2中,我们给出了一类可迭代求不动点的算子的不动点存在性定理,并应用到一类积分方程之中;
在§2.3中,我们给出了集值映射的不动点定理,得到了类似于α-凹算子的不动点存在的充分必要条件,并应用于α-凸算子的不动点存在性的证明之中。这种用集值映射来讨论α-凸算子的思想在可查文献中尚未见到;
在§2.4中,我们利用不动点指数的性质给出了C[0,1]空间中全连续算子具有非零不动点新的存在性定理,其应用改进并概括了三点、两点及m-点微分方程边值问题。
在第三章中,我们利用非线性算子理论讨论了微分方程几种边值问题正解的存在性与多解性。一定程度上深化了研究微分方程边值问题的研究方法。
在§3.1中,利用锥拉伸锥压缩定理,§2.4中的结论及锥上的多解定理讨论了如下三点边值问题:
(Ⅰ) 半正三点边值问题
The purpose of this paper is to discuss two classes of nonlinear problems, one of which is nonlinear operator equations and the other is some applications of nonlinear operator theory to boundary value problems for differential equations.The methods employed are mainly partial ordering method and iterative techniques and so on.This paper includes three chapters.In chapter 1, we provide a research summary of several classes of nonlinear operators (α—homogeneous operators, a class of operators of which fixed points are obtained by iterative technique, set-valued operators) since the beginning when these operators' definitions were introduced, and the general thoughts in which we can deal with some integral equations,boundary value problems for differential equations by using nonlinear theory.In chapter 2, we present some fixed point theorems for several classes of nonlinear operators.In §2.1, sufficient conditions for the existence and uniqueness of fixed points to α—homogeneous operators and some mean value theorems of homogeneous operators are given under much weaker conditions, such results and methods have seldom been seen in literature available.In §2.2, some existence theorems of fixed pionts for a class of operators of which fixed points are obtained by iterative technique are given, and an application to a class of integral equations is considered.In §2.3, we obtain some new fixed point theorems for set-valued maps in ordered Banach spaces, and the sufficient and necessary conditions for existence of fixed points to certain operators similar to α—concave operators.In addition, we utilize the results to study the existence and uniqueness of positive fixed points for a class of α—convex operators. The idea of studying α—concave operators via set-valued maps has not been seen in literature available.In §2.4, a new nonzero fixed point theorem for completely continuous operators in C[0,1] is given by using the properties of fixed point index. Its applications improve and generalize previous results of three-point,two-point and m-point boundary value
problems for differential equations.In chapter 3,the existence and multiplicity of positive solutions for several classes of nonlinear boundary value problems for differential equations are discussed by using nonlinear operator theory. In some degree, we develop the methods of studying boundary value problems for differential equations.In §3.1, by using the fixed point theorem of cone expansion and compression, the results in §2.4 and multi- solution theorems in cones, we establish the existence and multiplicity of positive solutions to several classes of three-point boundary value problems as follows: (I) Semi-positone three-point BVPu"(t) + Xf(t,u(t)) = 0, i€(0,l),u(0) = 0, au(ri) = u(l).where 0 < 77 < 1, 0 < a < ^, f(t, u) > —M, here M is a positive constant. (II)Three-point BVP for differential equations with an advanced argumentu"(t) + a(t)f(u(h(t))) = 0, ie(0,l),u,(0) = 0, au(rj) = u(l).where 0 < r) < 1,0 < a < ±, h e C((0,1), (0,1]) satisfies t < h(t) < 1, t € (0,1). (III)Three-point BVP with the parameter in closed intervalu(0) = 0,au(T)) = m(1).where 0 Ao > 0.In §3.2. by using Leggett-Williams fixed point theorem, we establish the existence ofat least two positive solutions to the nonlinear semi-positone m-point boundary valueproblems t,u) = 0,t€(0,l),m-2 m-2u'(0) = ￡>?'(&), u(l) = J ?=1 t=l
全文共分为三章。
在第一章(概论)中,我们对几类非线性算子(α(>1)-齐次算子,一类可迭代求不动点的算子,集值算子)的研究现状及我们在本文中将要做的工作进行了阐述;同时系统的介绍了用非线性知识处理微分方程边值问题的总体思路。
第二章,我们给出了几类非线性算子的不动点定理。
在§2.1中,我们在较弱的条件下给出了α(>1)-齐次算子具有唯一不动点的充分条件以及齐次算子的介值性定理,这类结论和方法在可查文献中尚不多见;
在§2.2中,我们给出了一类可迭代求不动点的算子的不动点存在性定理,并应用到一类积分方程之中;
在§2.3中,我们给出了集值映射的不动点定理,得到了类似于α-凹算子的不动点存在的充分必要条件,并应用于α-凸算子的不动点存在性的证明之中。这种用集值映射来讨论α-凸算子的思想在可查文献中尚未见到;
在§2.4中,我们利用不动点指数的性质给出了C[0,1]空间中全连续算子具有非零不动点新的存在性定理,其应用改进并概括了三点、两点及m-点微分方程边值问题。
在第三章中,我们利用非线性算子理论讨论了微分方程几种边值问题正解的存在性与多解性。一定程度上深化了研究微分方程边值问题的研究方法。
在§3.1中,利用锥拉伸锥压缩定理,§2.4中的结论及锥上的多解定理讨论了如下三点边值问题:
(Ⅰ) 半正三点边值问题
The purpose of this paper is to discuss two classes of nonlinear problems, one of which is nonlinear operator equations and the other is some applications of nonlinear operator theory to boundary value problems for differential equations.The methods employed are mainly partial ordering method and iterative techniques and so on.This paper includes three chapters.In chapter 1, we provide a research summary of several classes of nonlinear operators (α—homogeneous operators, a class of operators of which fixed points are obtained by iterative technique, set-valued operators) since the beginning when these operators' definitions were introduced, and the general thoughts in which we can deal with some integral equations,boundary value problems for differential equations by using nonlinear theory.In chapter 2, we present some fixed point theorems for several classes of nonlinear operators.In §2.1, sufficient conditions for the existence and uniqueness of fixed points to α—homogeneous operators and some mean value theorems of homogeneous operators are given under much weaker conditions, such results and methods have seldom been seen in literature available.In §2.2, some existence theorems of fixed pionts for a class of operators of which fixed points are obtained by iterative technique are given, and an application to a class of integral equations is considered.In §2.3, we obtain some new fixed point theorems for set-valued maps in ordered Banach spaces, and the sufficient and necessary conditions for existence of fixed points to certain operators similar to α—concave operators.In addition, we utilize the results to study the existence and uniqueness of positive fixed points for a class of α—convex operators. The idea of studying α—concave operators via set-valued maps has not been seen in literature available.In §2.4, a new nonzero fixed point theorem for completely continuous operators in C[0,1] is given by using the properties of fixed point index. Its applications improve and generalize previous results of three-point,two-point and m-point boundary value
problems for differential equations.In chapter 3,the existence and multiplicity of positive solutions for several classes of nonlinear boundary value problems for differential equations are discussed by using nonlinear operator theory. In some degree, we develop the methods of studying boundary value problems for differential equations.In §3.1, by using the fixed point theorem of cone expansion and compression, the results in §2.4 and multi- solution theorems in cones, we establish the existence and multiplicity of positive solutions to several classes of three-point boundary value problems as follows: (I) Semi-positone three-point BVPu"(t) + Xf(t,u(t)) = 0, i€(0,l),u(0) = 0, au(ri) = u(l).where 0 < 77 < 1, 0 < a < ^, f(t, u) > —M, here M is a positive constant. (II)Three-point BVP for differential equations with an advanced argumentu"(t) + a(t)f(u(h(t))) = 0, ie(0,l),u,(0) = 0, au(rj) = u(l).where 0 < r) < 1,0 < a < ±, h e C((0,1), (0,1]) satisfies t < h(t) < 1, t € (0,1). (III)Three-point BVP with the parameter in closed intervalu(0) = 0,au(T)) = m(1).where 0
引文
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[2] 郭大钧,非线性泛函分析,济南:山东科学技术出版社,1985.
[3] M.A.Krasnoselskii, J.A.Lifshits, A.V.Sobolev, Positive linear systems-the motheod of positive operators, Sigma Series in Applied Mathematics, Vol.5, Heldermann Verlag, Berlin, 1989.
[4] V.Lakshmikantham and S.Leela, "Nonlinear Differential Equations in Banach spaces," Pergamon, Ehnsford,N.Y., 1981.
[5] M.A.Krasnoselskii, "Positive solutions of operators equations," Noordoff (Groningen), 1964.
[6] Klaus Deimling, Nonlinear Functional Analysis,Springer-Varlag, Berlin, 1985.
[7] 钟承奎,范先令,陈文塬,非线性泛函分析引论,兰州大学出版社,2004.
[8] 李福义,梁展东,φ-凹(凸)算子的不动点定理及其应用,系统科学与数学,1994,14(4):355-360.
[9] Bushell.P.J, Hilbert's metric and positive contraction mappings in a Banach space, Arch. Rational Mech. Anal, 1973, 52, 330-338
[10] A.J.B.Potter, Existence theorem for a nonlinear integral equation, J. London. Math. Soc, (2), 11(1975),7-10.
[11] P.J.Bushell, On a class of Volterra and Fredholm nonlinear integral equations, Math. Proc. Cambridge Philos. Soc, 79: 329-335(1976)
[12] P.J.Bushelt, The Cayley-Hilbert metric and positive operators, Linear Algebra and Its Applications, 84: 271-280(1986).
[13] 郭大钧,多项式型Hammerstein积分方程的正解极其应用,数学年刊,4(5),1983(A),645-656.
[14] 梁展东,王彩云,关于正α-齐次算子方程的一个定理及应用,数学学报,39(2),1996,204-208.
[15] 张庆政,正α-齐次超线性算子方程正解的存在唯一性及其应用,系统科学与数学,19(1),1999,29-33.
[16] 郭大钧,Hammerstein型非线性积分方程正解的个数,数学学报,1979,22(5),584-595.
[17] 郭大钧,一类凹与凸算子的不动点与固有元,科学通报,1985,30(15),1132-1135.
[18] Krasnoselskii, M.A. and Zabreiko, P.P. Geometrical Methods of Nonlinear Analysis[M], Moscow, 1975(in Russian).
[19] 郭大钧,非线性算子方程的正解及其对非线性积分方程的应用,数学进展,1984,13:294-310.
[20] 杜一宏,一类非紧算子的不动点定理及其应用,数学学报,1989,32(5),618-627.
[21] 王文霞,若干非线性算子与非线性方程的讨论,郑州大学博士学位论文,2003,5.
[22] 梁展东,王文霞,列压缩算子的不动点定理及其应用,数学学报,2004,47(1),173-180.
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