Approximation in eigenvalue problems for holomorphic fredholm operator functions I

The approximation of a holomorphic eigenvalue problem is considered. The main purpose is to present a construction by which the derivation of the asymptotic error estimations for the approximate eigenvalues of Fredholm operator functions can be reduced to the derivation of these estimations for the case of matrix functions. (Some estimations for the latter problem can be derived, in one's turn, from the error estimations for the zeros of the corresponding determinants.) The asymptotic error estimations are considered in part II of this paper, in [10]. By the presented construction also the stability of the algebraic multiplicity of eigenvalues by regular approximation is proved in Section 3

The presented construction, in essence, reproduces the constructions in [7] for the case of the compact approximation in subspaces and in [9] for the case of projection—like methods. It is simpler to use than similiar construction in [8], and allows unified consideration of the general case and the case of projection—like methods, what in [8, 9] was not achieved     

Existence of some neutral partial differential equations with infinite delay

This study intends to investigate a class of quasi-linear partial neutral functional differential equations with infinite delay. We assume that the linear part generates an analytic compact semigroup and the nonlinear part satisfies certain conditions. A sufficient condition is given to ensure the existence of mild and classical solutions. Finally, an example is given to illustrate our abstract results.     

On a new class of abstract neutral differential equations

In this work we introduce a new class of abstract neutral differential equations and study the existence of mild and strict solutions. Some concrete applications involving partial neutral differential equations are considered.     

一类脉冲分数阶半线性泛函微分方程的弱解的存在性

在本文中,我们通过利用线性算子的解析半群理论和不动点定理对一类脉冲分数阶半线性泛函微分方程的弱解的存在性进行讨论．     

On a solution to fractional order integrodifferential equations with analytic semigroups

In this paper we study a class of fractional order integrodifferential equations considered in an arbitrary Banach space. Using the theory of analytic semigroups we establish the existence, uniqueness and regularity of a mild solution to these fractional order integrodifferential equations.     

一类二阶迭代泛函微分方程的解析解

本文研究了一类二阶迭代泛函微分方程ｘ″（ｚ）＝ｍｊ＝０ｐｊｘｊ（ｚ）,ｚ∈Ｃ．其中ｍ为正整数,ｘｊ（ｚ）表示未知函数ｘ（ｚ）的ｊ次迭代,给出了这类方程满足初始条件解析解的几个存在性定理．     

Analytic solutions of an iterative functional differential equation

This paper is concerned with a functional differential equation x^′(z)=1/x(az+bx(z)), where a, b are two complex numbers. By constructing a convergent power series solution y(z) of a auxiliary equation of the form b²y^′(z)=(y(μ²z)−ay(μz))(μy^′(μz)−ay^′(z)), analytic solutions of the form for the original differential equation are obtained.     

On a Nonlinear Heat Equation Associated with Dirichlet–Robin Conditions

This article is devoted to the study a nonlinear heat equation associated with Dirichlet-Robin conditions. At first, we use the Faedo–Galerkin and the compactness method to prove existence and uniqueness results. Next, we consider the properties of solutions. We obtain that if the initial condition is bounded then so is the solution and we also get asymptotic behavior of solutions as t → + ∞. Finally, we give numerical results.     

Remark on invariant curves for planar mappings

Bounded solutions for a functional equation were investigated by Ng and Zhang [C. T. Ng, W. Zhang, Invariant curves for planar mapping, J. Difference Eqn. Appl. 3 (1997) 147-168]. In this paper, we consider its solutions which may be unbounded. We give not only existence and uniqueness of its solutions but also the convex and concave solutions. Moreover, some examples are given.     

Existence and stability in the α-norm for partial functional differential equations of neutral type

In this work, we study a general class of partial neutral functional differential equations. We assume that the linear part generates an analytic semigroup and the nonlinear part is Lipschitz continuous with respect to the é-norm associated to the linear part. We discuss the existence, uniqueness, regularity and stability of solutions. Our results are illustrated by an example. This work extends previous results on partial functional differential equations (Fitzgibbon and Parrot, Nonlinear Anal., TMA 16, 479–487 (1991), Hale, Rev. Roum. Math. Pures Appl. 39, 339–344 (1994), Hale, Resen. Inst. Mat. Estat. Univ. Sao Paulo 1, 441–457 (1994), Travis and Webb, Trans. Am. Math. Soc. 240 129–143 (1978), Wu and Xia, J. Differ. Equ. 124 247–278 (1996)). Mathematics Subject Classification (1991) 34K20, 34K30, 34K40, 47D06     

非线性分数阶积分微分发展方程适度解的存在唯一性（英文）

本文研究一类在Banach空间中分数阶积分微分发展方程的问题,利用分数阶幂算子和解析半群理论来证明所给方程适度解的存在唯一性.并进一步给出适度解的Holder连续性.     

Existence for neutral impulsive differential inclusions with nonlocal conditions

In this paper, by using a fixed point theorem for condensing multi-valued maps, we have investigated the existence of mild solutions for a class of semilinear impulsive neutral functional differential inclusions with nonlocal conditions. As an application, an example has been presented to illustrate the results obtained.     

Pseudo almost periodic solutions for a class of first order differential iterative equations

This paper is concerned with a class of first order differential iterative equations. Under proper conditions, we employ a novel argument to establish a criterion on the existence of pseudo almost periodic solutions. The obtained result complements with some existing ones.     

Periodic solutions of the third order functional differential equations

We obtain sufficient conditions for the existence of periodic solutions of the following third order nonlinear functional differential equations