Existence and comparison results for operator and differential equations in abstract spaces

In this paper we apply the chain methods developed in (Heikkilä and Lakshmikantham, 1994) to obtain new fixed point theorems and new existence and comparison results for operator equations in partially ordered sets. These results are then applied to discontinuous implicit functional differential equations in ordered Banach spaces.     

Existence results for operator equations in partially ordered sets and applications

In this paper, we prove existence results for operator equations in partially ordered sets, and apply the obtained results to operator equations in ordered Banach spaces and to semilinear functional parabolic and elliptic problems involving discontinuous nonlinearities.     

Iterative Einschliessung von Lösungen Nichtlinearer Operatorgleichungen und Bestimmung von Startnäherungen

In the present paper some Newton-like iteration methods are developed to enclose solutions of nonlinear operator equations of the kindF(x)=0. HereF maps a certain subset of a partially ordered vector space into another partially ordered vector space. The obtained results are proved without any special properties of the orderings by taking use of a new kind of a generalized divided difference operator, so that they even hold for nonconvex operators. Furthermore a method for constructing including starting points is presented and two examples are given.     

非混合单调二元算子方程(组)解的存在唯一性

利用锥理论及Banach压缩映射原理,在不要求上、下解条件及算子紧性与连续性的条件下,建立了一类满足更一般序关系条件的非混合单调二元算子方程组(?)解的存在唯一性定理,以及非单调二元算子方程T(x,x)=x和非单调一元算子方程Lx=x解的存在唯一性定理,推广了最近相关文献的研究结果.     

序Banach空间中算子方程的迭代求解及其应用

利用半序理论和混合单调算子技巧,研究序Ｂａｎａｃｈ空间中非线性算子方程解的存在唯一性,并给出了迭代序列收敛于解的误差估计．作为应用,讨论了序Ｂａｎａｃｈ空间中一类非线性积分方程的可解性,改进和推广了某些已知结果．     

Dhage iteration method for initial value problems for nonlinear first order hybrid integrodifferential equations

In this paper, we prove algorithms for the existence as well as the approximation of solutions to initial value problems (IVPs) for nonlinear first order ordinary integrodifferential equations using operator theoretic techniques in a partially ordered metric space. The main results rely on the Dhage iteration method embodied in the recent hybrid fixed point theorems of Dhage in a partially ordered normed linear space. The approximations to the solutions are obtained under weaker mixed partial continuity and Lipschitz conditions. The hypotheses and results are also illustrated by some examples.     

Recursion and group structures of soliton equations

The recursion operator method for nonlinear evolution equations integrable by the inverse spectral transform method is discussed. This method enables us to present the integrable equations in a compact and convenient form and to construct the infinite-dimensional groups of general Bäcklund transformations and the infinite-dimensional symmetry groups for these equations. Adjoint representation of the spectral problems plays a central role in the recursion operator method. Nonlinear integrable equations in 1+1 and 1+2 dimensions are considered.     

Positive fixed points for a class of nonlinear operators and applications

In this paper, we study the existence and uniqueness of positive solutions for a class of nonlinear operator equations on ordered Banach spaces. Various applications are also considered to illustrate our obtained results (existence of solutions to quadratic integral equations with a linear modification of the argument, positive solution of second-order Neumann boundary value problem, and positive definite solutions of a class of nonlinear matrix equations).     

半序概率度量空间中φ-μ-混合单调算子方程组的解(英文)

在半序概率度量空间中,引入φ-μ-混合单调算子的概念.同时,构建一些新的条件,并采用半序技巧和分布函数的性质获得了φ-μ-混合单调算子方程组的解.     

Fixed point theorems for multifunctions with applications to discontinuous operator and differential equations

In this paper we study existence of minimal and maximal fixed points for increasing multivalued functions in ordered topological vector spaces, and apply the obtained results to operator equations and quasilinear elliptic boundary value problems involving discontinuous nonlinearities.     

Existence and nonexistence of positive solutions for a class of nth-order three-point boundary value problems in Banach spaces

By using the fixed-point principle in cone and the fixed-point index theory for strict-set-contraction operator, this paper investigates the existence, nonexistence, and multiplicity of positive solutions for a class of nonlinear three-point boundary value problems of nth-order differential equations in ordered Banach spaces. In addition, an example is included to demonstrate the main results.     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

一类非线性算子方程解的存在唯一性及其应用

该文在更广泛的条件下,利用锥理论和Banach压缩映象原理证明了序Banach空间中一类非线性算子方程解的存在唯一性定理,并应用到Banach空间中二阶非线性混合型微分积分方程初值问题,改进并推广了已有的一些结果.     

Inexact Newton methods and nondifferentiable operator equations on Banach spaces with a convergence structure

In this study, we use inexact newton methods to find solutions of nonlinear, nondifferentiable operator equations on Banach spaces with a convergence structure. This technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way the metric properties of the examined problem can be analyzed more precisely. Moreover, this approach allows us to derive from the same theorem, on the one hand, semi-local results of Kantorovich-type, and on the other hand, global results based on monotonicity considerations. Furthermore, we show that special cases of our results reduce to the corresponding ones already in the literature. Finally, our results are used to solve integral equations that cannot be solved with existing methods.     

Linearizing equations with state-dependent delays

We investigate the differentiability of the composition operator and apply the result to equations with state-dependent delays.F. Colonius was supported by a Heisenberg grant from DFG at Universität Bremen.     

On the Newton-Kantorovich method inK-normed spaces

Nondifferentiable operator equations in spaces equipped with aK-norm — map taking values in the cone of an ordered Banach space — are studied. The majorant method is used to derive the convergence of a modified sequence of Newton-Kantorovich approximations.     

非线性算子方程迭代解的存在性定理及其应用

在Ｂａｎａｃｈ空间中,利用锥理论和单调迭代方法研究了一类非线性算子方程的解和最小最大耦合解的存在与迭代逼近定理,并应用到Ｂａｎａｃｈ空间中非线性Ｖｏｌｔｅｒｒａ型积分方程和常微分方程的初值问题．     

多粒度区间语言信息的C-OWH算子及其应用

研究了多粒度区间语言信息的集成问题,把连续的有序加权调和(C-OWH)平均算子拓展到多粒度区间语言环境中,提出了连续区间二元语义OWH(ITC-OWH)算子,并在此基础上提出了加权调和的ITC-OWH(WHITC-OWH)算子,有序加权调和的ITC-OWH(OWHITC-OWH)算子以及组合的ITC-OWH(CITC-OWH)算子,探讨了它们的一些性质。最后给出了基于这些算子的多粒度区间语言多属性群决策方法,实例分析结果表明该方法是可行的。     

Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces

This paper investigates the existence, nonexistence, and multiplicity of positive solutions for a class of nonlinear boundary-value problems of third order differential equations with integral boundary conditions in ordered Banach spaces by means of fixed-point principle in cone and the fixed-point index theory for strict set contraction operator. In addition, an example is worked out to demonstrate the main results.     

Positive solutions for nonlinear operator equations and several classes of applications

In this paper, we study a class of nonlinear operator equations x = Ax + x ₀ on ordered Banach spaces, where A is a monotone generalized concave operator. Using the properties of cones and monotone iterative technique, we establish the existence and uniqueness of solutions for such equations. In particular, we do not demand the existence of upper-lower solutions and compactness and continuity conditions. As applications, we study first-order initial value problems and two-point boundary value problems with the nonlinear term is required to be monotone in its second argument. In the end, applications to nonlinear systems of equations and to nonlinear matrix equations are also considered.