Bounded Mild Solutions for Semilinear Integro Differential Equations in Banach Spaces

In this paper we study the structure of various classes of spaces of vector-valued functions M(\mathbbR;X){\mathcal{M}(\mathbb{R};X)} ranging between periodic functions and bounded continuous functions. Some of these functions are introduced here for the first time. We propose a general operator theoretical approach to study a class of semilinear integro-differential equations. The results obtained are new and they recover, extend or improve variety of recent works.     

On Bounded Solutions of Some Classes of Two-Parameter Difference Equations in a Banach Space

We obtain criteria for the existence of bounded solutions of some classes of linear two-parameter difference equations with operator coefficients in a Banach space.     

Stability of Periodic Solutions of Ordinary Differential Equations in Banach Spaces

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Stability of Systems of Functional Differential Equations Containing a Small Parameter

New results on the stability of functional differential equations of the delayed type with a small parameter are reported. The right-hand side of the investigated system is assumed to satisfy Carathéodory-type conditions for the existence of a solution of the initial-value problem and to be globally continuous with respect to the parameter at the point . Under these assumptions the results can be used to investigate functional differential equations with rapidly oscillating coefficients. The concept of -stability is defined, and sufficient conditions for -stability are formulated in terms of Lyapunov-type functionals.     

Banach值随机微分方程解的存在唯一性

本文我们用逐次逼近的方法研究Banach值随机微分方程解的存在性和轨道唯一性.     

Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

Let A be a set, and let E be the Banach space of bounded functions : A R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,) : R × E E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to . Then there exists a solution u: [a,b] × [0,T] E of the problem ut(x,t) – uxx(x,t) = F(x,t,u(x,t)) ((x,t) R), u(x,t) = 0 ((x,t) R R).     

Computing Stability of Differential Equations with Bounded Distributed Delays

This paper deals with the stability analysis of scalar delay integro-differential equations (DIDEs). We propose a numerical scheme for computing the stability determining characteristic roots of DIDEs which involves a linear multistep method as time integration scheme and a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule. We investigate to which extent the proposed scheme preserves the stability properties of the original equation. We derive and prove a sufficient condition for (asymptotic) stability of a DIDE (with a constant kernel) which we call RHP-stability. Conditions are obtained under which the proposed scheme preserves RHP-stability. We compare the obtained results with corresponding ones using Newton–Cotes formulas. Results of numerical experiments on computing the stability of DIDEs with constant and nonconstant kernel functions are presented.     

Asymptotic Solution of the Cauchy Problem for a First-Order Differential Equation with a Small Parameter in a Banach Space

Mathematical Notes - The Cauchy problem for a first-order differential equation with a small parameter multiplying the derivative in a Banach space is considered. The right-hand side of the...     

Bounded Nonoscillatory Solutions of Nonlinear Functional Differential Equations

In this paper, we obtain some nonoscillatory theories of the functional differential equation （r（t）ψ（x（t））x （t）） ＋ f（t, x（t）, x（σ（t））） = 0, t ≥ t 0 , where r ∈ C 1 （[t 0 , ∞）; （0, ∞））, ψ∈ C 1 （R, R） and f ∈ C（[t 0 , ∞） × R × R, R）.     

Banach空间中非线性奇异微分方程边值问题的正解

本文通过构造一个特殊的锥,研究了Banach空间中一类奇异边值问题正解及多重正解的存在性.     

Banach空间中非线性脉冲积分微分方程的正解(英)

借助于锥理论，本文讨论Banach空间中非线性脉冲积分微分方程的解．给出一阶脉冲微分方程存在唯一正解的条件及混合型脉冲积分微分方程至少具有两解的条件．     

Asymptotic Expansion of Solutions of Parabolic Equations with a Small Parameter

The heat equation with a small parameter, $\left( {1 + \varepsilon ^{ - m} \chi \left( {\frac{x}{\varepsilon }} \right)} \right)ut = u_{xx} $ , is considered, where ε ∈ (0, 1), m < 1 and χ is a finite function. A complete asymptotic expansion of the solution in powers ε is constructed.     

Controllability of Nonlinear Differential Equations in Banach Space with Nonlocal Conditions

In this paper, we establish sufficient conditions for the controllability of semilinear evolution systems in Banach space with nonlocal conditions. We rely on a fixed-point theorem for compact maps due to Schaefer.     

Bounded Solutions of Some Second Order Nonlinear Differential Equations

We prove Landesman–Lazer type existence conditions forthe solutions bounded on the real line, together with theirfirst derivatives, for some second order nonlinear differentialequations of the form x'+cx'+f(t, x)=0. The proofs use upperand lower solutions techniques together with some limiting arguments.     

Stability of Stochastic Delay Differential Equation with a Small Parameter

Abstract

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution x ^?(·) of the underlying delay equation for ? > 0 small enough. Perturbed Liapunov function method is used in the analysis.     

Conditions for the Existence of Nonoscillating Solutions of Nonlinear Differential Equations with Delay and Pulse Influence in a Banach Space

For nonlinear differential second-order equations with delay and pulse influence in a Banach space, we establish necessary and sufficient conditions for the existence of their solutions nonoscillating with respect to a subspace.     

一类具有反射变元的泛函微分方程的有界解和概周期解

利用指数二分法及不动点定理,研究了一类具有反射变元的中立型泛函微分方程的有界解和概周期解的存在性.