Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces

In this work, we study the existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces. We suppose that the undelayed part admits a resolvent operator in the sense given by Grimmer in [R. Grimmer, Resolvent operators for integral equations in a Banach space, Transaction of American Mathematical Society 273 (1982) 333–349]. The delayed part is assumed to be locally Lipschitz. Firstly, we show the existence of the mild solutions. Secondly, we give sufficient conditions ensuring the existence of the strict solutions.     

Asymptotic stability of impulsive stochastic partial integrodifferential equations with delays

In this paper, we study the existence and asymptotic stability in the p-th moment of mild solutions of nonlinear impulsive stochastic partial functional integrodifferential equations with delays. We suppose that the linear part possesses a resolvent operator in the sense given in Grimmer [R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc. 273(1) (1982), 333–349] and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is employed for achieving the required result. An example is provided to illustrate the results of this work.     

Exponential stability for some stochastic neutral partial functional integrodifferential equations with delays and Poisson jumps

In this work, we study the existence, uniqueness and exponential stability in mean square of mild solutions for some stochastic neutral partial functional integrodifferential equations with delays and Poisson jumps. Sufficient conditions for the existence and exponential stability in mean square of mild solutions are derived by means of the Banach fixed point principle. We suppose that the linear part has a resolvent operator in the sense given in Grimmer (Trans. Am. Math. Soc., 273(1):333–349, 1982). An example is provided to illustrate the results of this work.     

Existence and regularity of solutions for neutral partial functional integrodifferential equations with infinite delay

This paper is concerned with the existence and regularity of solutions for a class of neutral partial functional integrodifferential equations with infinite delay in Banach spaces. We use the theory of resolvent operator developed in R. Grimmer (1982) [29] to show the existence of mild solutions. We give sufficient conditions ensuring the existence of strict solutions. The phase space is axiomatically defined. Our results are applied to prove the existence and regularity of solutions to a Lotka–Volterra model with diffusion.     

Existence and Regularity of Solutions for Some Partial Integrodifferential Equations Involving the Nonlocal Conditions

The aim of this work is to prove some results about the existence and regularity of solutions for some partial integrodifferential equations with nonlocal conditions. We suppose that the linear part has a resolvent operator in the sens given by Grimmer. The non linear part is assumed to be continuous and Lipschitzian with respect to the second argument.     

Existence of Global Solutions for a Class of Abstract Second-Order Nonlocal Cauchy Problem with Impulsive Conditions in Banach Spaces

In this paper, we are concerned with a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. First, we study the existence of mild solutions for a class of second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces on an interval [0,a]. Later, we study a couple of cases where we can establish the existence of global solutions for a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. We apply our theory to study the existence of solutions for impulsive partial differential equations.     

Mild solutions of non‐Lipschitz stochastic integrodifferential evolution equations

In this work, we study the existence and uniqueness of mild solutions for stochastic partial integrodifferential equations under local non‐Lipschitz conditions on the coefficients. Our analysis makes use of the theory of resolvent operators as developed by R. Grimmer as well as a stopping time technique. Our results complement and improve several earlier related works. An example is provided to illustrate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion

This paper deals with the existence and uniqueness of mild solutions to neutral stochastic delay functional integro-differential equations perturbed by a fractional Brownian motion B ^H , with Hurst parameter H ∈ (1/2, 1). We use the theory of resolvent operators developed by R. Grimmer to show the existence of mild solutions. An example is provided to illustrate the results of this work.     

Behavior near hyperbolic stationary solutions for partial functional differential equations with infinite delay

The aim of this work is to investigate the asymptotic behavior of solutions near hyperbolic stationary solutions for partial functional differential equations with infinite delay. We suppose that the linear part satisfies the Hille–Yosida condition on a Banach space and it is not necessarily densely defined. Firstly, we establish a new variation of constants formula for the nonhomogeneous linear equations. Secondly, we use this formula and the spectral decomposition of the phase space to show the existence of stable and unstable manifolds. The estimations of solutions on these manifolds are obtained. For illustration, we propose to study the stability of stationary solutions for the Lotka–Volterra model with diffusion.     

Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion

This article focuses on controllability results of neutral stochastic delay partial functional integro-differential equations perturbed by fractional Brownian motion. Sufficient conditions are established using the theory of resolvent operators developed by Grimmer [Resolvent operators for integral equations in Banach spaces, Trans. Amer. Math. Soc., 273(1982):333–349] combined with a fixed point approach for achieving the required result. An example is provided to illustrate the theory.     

Well-posedness results for differential and integral equations containing Henstock–Kurzweil integrable vector-valued functions

In this paper, we prove existence, uniqueness and comparison results for solutions of differential and integral equations in Banach spaces containing Henstock–Kurzweil integrable functions from a compact real interval to an ordered Banach space.     

增算子不动点定理及其对含间断项非线性脉冲方程的应用

设E是Banach空间,本文在比C[I,E]更一般的Banach空间中得到了几个新的非连续增算子不动点定理.作为应用,我们研究了Banach空间中含间断项的非线性脉冲积分方程和微分方程的最大解和最小解的存在性.     

Strongly damped wave problems: Bootstrapping and regularity of solutions

The aim of the article is to present a unified approach to the existence, uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in Banach spaces. Our results are applied next to a number of examples appearing in literature, which fall into the class of strongly damped semilinear wave equations. The present work essentially extends the results on the existence and regularity of solutions to such problems. Previously, these problems have been considered mostly within the Hilbert space setting and with the main part operators being selfadjoint. In this article we present a more general approach, involving sectorial operators in reflexive Banach spaces.     

Trichotomies for abstract semilinear differential equations

In this paper we present some results concerning the existence and uniqueness of mild solutions to certain abstract semilinear differential equations and the asymptotic behavior of these solutions. The basic techniques used are the iterative method and the fixed point theory for differential equations in Banach space. However, the most pleasant here is that it can be applied to nonlinear equations without assuming the eigenvalues of the differential operator in the linear parts of the differential equation has non-zero real part.     

Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, by using the generalization of Darbo’s fixed point theorem, we establish the existence of global solutions of an initial value problem for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. Our results generalize and improve on the results of Guo et al. [F. Guo, L.S. Liu, Y.H. Wu, P. Siew, Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. 61 (2005) 1363–1382] in the sense that the conditions for existence of global solution in our theorem is simpler and less strict. To demonstrate the application of the theorem, we give the global solutions of two mixed boundary value problems for two classes of fourth order impulsive integro-differential equations.     

Existence for neutral impulsive functional differential equations with nonlocal conditions

In this paper, we study a class of neutral impulsive functional differential equations with nonlocal conditions. We suppose that the linear part satisfies the Hille-Yosida condition on a Banach space and it is not necessarily densely defined. We give some sufficient conditions ensuring the existence of integral solutions and strict solutions. To illustrate our abstract results, we conclude this work by an example.     

Local and global existence of mild solutions for impulsive fractional semilinear integro-differential equation

In this paper, we study the local and global existence of mild solutions for impulsive fractional semilinear integro-differential equations in an arbitrary Banach space associated with operators generating compact semigroup on the Banach space. Also, we review some applications of fractional differential equations.     

随机积分和微分方程在弱拓扑下解的存在性和比较结果*

在本文中,我们对非线性随机Volterra积分方程在Banach空间的弱拓扑下的随机解证明了几个存在定理.然后作为应用,我们得到了随机微分方程的弱随机解的存在定理.还得到了这些随机方程的极值随机解的存在性和随机比较定理.我们的定理改进和推广了[4,5,10,11,12]中的相应结果.     

Banach空间非线性脉冲积分方程解的存在性

利用Monch不动点定理和分段估计方法,本文研究Banach空间非线性脉冲积分方程解的存在性,但是我们不使用脉冲项的紧型条件和非紧型测度估计的限制性条件.作为一个应用,我们讨论Banach空间一阶非线性脉冲微分方程终值问题解的存在性.