Hammerstein型积分方程非零解的存在性及应用

在不假定非线性项非负的情况下,利用半序理论讨论了Ｈａｍｍｅｒｓｔｅｉｎ型积分方程非零解的存在性,并将所得结果应用于常微分方程两点边值问题．     

Iterative Solutions of Nonlinear Equations of the Strongly Accretive Type

Let E be a real q-uniformly smooth Banach space. Suppose T is a strongly pseudo-contractive map with open domain D(T) in E. Suppose further that T has a fixed point in D(T). Under various continuity assumptions on T it is proved that each of the Mann iteration process or the Ishikawa iteration method converges strongly to the unique fixed point of T. Related results deal with iterative solutions of nonlinear operator equations involving strongly accretive maps. Explicit error estimates are also provided.     

Approximation of Solutions of Nonlinear Equations of Monotone and Hammerstein Type

Let X be a real uniformly smooth and uniformly convex Banach space with dual X ^*. Let A: X → X ^* be a bounded uniformly submonotone map. It is proved that a Mann-type approximation sequence converges strongly to Jx ^* where x ^* ∈ N(A). Furthermore, as an application of this result an iterative sequence which converges strongly to a solution of the Hammerstein equation u+KFu = 0 is constructed where, F:X→X^* and K:X^*→X are monotone-type mappings. No invertibility assumption is imposed on K. Moreover, neither K nor F need be compact. Finally, our method is of independent interest.     

一类Hammerstein积分方程的正分歧点

本文考虑有序Banach空间中形如x(t)=∫_Gk(t,s)[λAx(s) f(s,x(s),λ)]ds的Hammerstein型积分方程,给出了此类方程存在正分歧点的某些充分条件。     

Iterative solution of nonlinear equations of accretive and pseudocontractive types

Let E be a real uniformly smooth Banach space. Let A:D(A)=E→2E be an accretive operator that satisfies the range condition and A⁻¹(0)≠∅. Let {λ_n} and {θ_n} be two real sequences satisfying appropriate conditions, and for z∈E arbitrary, let the sequence {x_n} be generated from arbitrary x₀∈E by x_n+1=x_n−λ_n(u_n+θ_n(x_n−z)), u_n∈Ax_n, n?0. Assume that {u_n} is bounded. It is proved that {x_n} converges strongly to some x∗∈A−1(0). Furthermore, if K is a nonempty closed convex subset of E and T:K→K is a bounded continuous pseudocontractive map with F(T):={Tx=x}≠∅, it is proved that for arbitrary z∈K, the sequence {x_n} generated from x₀∈K by x_n+1=x_n−λ_n((I−T)x_n+θ_n(x_n−z)), n?0, where {λ_n} and {θ_n} are real sequences satisfying appropriate conditions, converges strongly to a fixed point of T.     

Banach空间中超线性Hammerstein型积分方程的解及其应用

本文利用不动点指数理论研究Ｂａｎａｃｈ空间中超线性Ｈａｍｍｅｒｓｔｅｉｎ型积分方程正解及非零解的存在性，并应用于Ｂａｎａｃｈ空间中超线性常微分方程的Ｓｔｕｒｍ－Ｌｉｏｕｖｉｌｌｅ问题，最后，本文给出了一个非线性常微分方程无穷组存在正解的例子．     

Iterative approximation of fixed points of Lipschitz pseudocontractive maps

Let be a -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ). Let be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset of and let be arbitrary. Then the iteration sequence defined by , converges strongly to a fixed point of , provided that and have certain properties. If is a Hilbert space, then converges strongly to the unique fixed point of closest to .

     

非线性发展方程解的渐近性的构造过程

本文借助Ｂａｎａｃｈ空间中的新不等式，通过迭代过程ｘｎ＋１＝ｘｎ－ａｎＡｘｎ（ｘ０∈Ｄ（Ａ），ｎ＝０，１，２，…）研究非线性发展方程ｄｕ／ｄｔ＝－Ａｕ解的渐近性，构造性地证明了，在一定条件下｛ｘｎ｝的极限点就是该发展方程的平衡点，推广了文献［１］的结论。     

Existence of Solutions of a Nonlinear Hammerstein Integral Equation

In this article, we are concerned with existence results for a nonlinear Hammerstein integral equation in L ¹ spaces. Making use of the De Blasi measure of weak noncompactness, we establish variants of some Krasnosel'skii type theorem involving the weak topology of Banach spaces.     

The Existence of Positive Almost Periodic Type Solutions for Some Nonlinear Delay Integral Equations

This paper presents the conditions of existence of positive almost periodic type solutions for some nonlinear delay integral equations, by using a fixed point theorem in the mixed monotone operators （Ma, Y.： On a class of mixed monotone operators and a kind of two-point bounded value problem. Indian J. Math., 41（2）, 211-220 （1999]]. Some known results are operators.     

一类Sylvester矩阵方程的一个迭代算法

考虑这样一类Sylvester矩阵方程:AX XB=C,A,B分别为n阶正半定、正定矩阵,C为n阶矩阵.给出了一个收敛的迭代算法.     

On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations

In this paper we present a characterization of those smooth piecewise polynomial collocation spaces that lead to divergent collocation solutions for Volterra integral equations of the second kind. The key to these results is an equivalence result between such collocation solutions and collocation solutions in slightly smoother spaces for initial-value problems for ordinary differential equations. For the latter problems Mülthei (1979/1980) established a complete divergence (and convergence) theory. Our analysis can be extended to furnish divergence results for smooth collocation solutions to Volterra integral equations of the first kind. AMS subject classification (2000) 65R20, 65L20, 65L60.Received May 2004. Accepted September 2004. Communicated by Tom Lyche.Hermann Brunner: This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).     

Banach空间非线性脉冲Hammerstein积分方程解的存在性

研究了Banach空间中定义在无穷区间R+上具有无穷多个脉冲点的Hammerstein积分方程解的存在性.利用MLnch不动点定理,建立了该类方程解的存在定理,并给出实例说明了该定理在无穷维脉冲积分方程组中的应用.     

Hammerstein非线性积分方程组的非平凡解及应用

该文利用拓扑方法和锥理论研究下列Hammerstein非线性积分方程组u(x)=∫_G k(x,y)f(y,u(y),v(y)) dy,v(x)=∫_Gk(x,y)g(y,u(y),v(y)) dy.在适当的条件下,证明了上述方程组非平解的存在性,并把所得结果应用于研究非线性二阶常微分方程组边值问题的非平凡解的存在性.     

非线性二元算子方程组的迭代求解方法

Using the cone and partial ordering theory and mixed monotone operator theory, the existence and uniqueness of solutions for some classes of systems of nonlinear two binary operator equations in a Banach space with a partial ordering are discussed. And the error estimates that the iterative sequences converge to solutions are also given. Some relevant results of solvability of two binary operator equations and systems of operator equations are improved and generalized.     

Iterative Positive Solutions to Nonlinear $q$-fractional Differential Equations with Integral Boundary Value Conditions

This paper is concerned with the existence of positive solutions to nonlinear $q$-fractional differential equations yielding to the integral boundary value conditions. Under sufficient conditions of the nonlinearity, by using some iterative techniques, we get that this problem has two positive solutions and a unique positive solution respectively. Our results improve some recent work.     

广义Volterra积分方程一类最大最小解的存在性

本文考虑Banach 空间中形如x(t)= u(t)+ ∫Gtf(t,s,x(s))ds的广义Volterra 积分方程,通过引入一类偏序证明了所述方程相应的最大最小解的存在性.     

Multilevel Iterative Algorithms for Accelerating Spectral Approximations of Integral Equations

In this paper, spectral collocation and Nyström methods for integral equations are discussed. Some multilevel correction schemes are presented for infinitely accelerating the covergence even if the exact solution is not so smooth.