scholarly journals Nonlinear semigroups and the existence and stability of solutions of semilinear nonautonomous evolution equations

1996 ◽  
Vol 1 (4) ◽  
pp. 351-380 ◽  
Author(s):  
Bernd Aulbach ◽  
Nguyen Van Minh
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Nonlinear Operators ◽  
Nonlinear Semigroups ◽  
Evolution Semigroup ◽  
Existence And Stability ◽  
Functional Differential

This paper is concerned with the existence and stability of solutions of a class of semilinear nonautonomous evolution equations. A procedure is discussed which associates to each nonautonomous equation the so-called evolution semigroup of (possibly nonlinear) operators. Sufficient conditions for the existence and stability of solutions and the existence of periodic oscillations are given in terms of the accretiveness of the corresponding infinitesimal generator. Furthermore, through the existence of integral manifolds for abstract evolutionary processes we obtain a reduction principle for stability questions of mild solutions. The results are applied to a class of partial functional differential equations.

scholarly journals A Note on Impulsive Fractional Evolution Equations with Nondense Domain

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Zufeng Zhang ◽  
Bin Liu
Keyword(s):  
Differential Equations ◽  
Evolution Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Contraction Mapping Principle ◽  
Correct Formula ◽  
Probability Densities ◽  
Functional Differential ◽  
Fractional Functional Differential Equations ◽  
Integral Solutions

This paper is concerned with the existence of integral solutions for nondensely defined fractional functional differential equations with impulse effects. Some errors in the existing paper concerned with nondensely defined fractional differential equations are pointed out, and correct formula of integral solutions is established by using integrated semigroup and some probability densities. Sufficient conditions for the existence are obtained by applying the Banach contraction mapping principle. An example is also given to illustrate our results.

scholarly journals Approximate Controllability for Functional Equations with Riemann-Liouville Derivative by Iterative and Approximate Method

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Badawi Hamza Elbadawi Ibrahim ◽  
Zhenbin Fan ◽  
Gang Li
Keyword(s):  
Differential Equations ◽  
Approximate Method ◽  
Approximate Controllability ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Liouville Derivative ◽  
Liouville Fractional Derivative ◽  
Functional Control ◽  
Functional Differential

We discuss the functional control systems governed by differential equations with Riemann-Liouville fractional derivative in general Banach spaces in the present paper. First we derive the uniqueness and existence of mild solutions for functional differential equations by the approach of fixed point and fractional resolvent under more general settings. Then we present new sufficient conditions for approximate controllability of functional control system by means of the iterative and approximate method. Our results unify and generalize some previous works on this topic.

scholarly journals Almost periodic mild solutions of a class of partial functional differential equations

1998 ◽  
Vol 3 (3-4) ◽  
pp. 425-436 ◽  
Author(s):  
Bernd Aulbach ◽  
Nguyen Van Minh
Keyword(s):  
Differential Equations ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Partial Functional Differential Equations ◽  
Semilinear Equation ◽  
Functional Differential ◽  
Lipschitz Conditions

We study the existence of almost periodic mild solutions of a class of partial functional differential equations via semilinear almost periodic abstract functional differential equations of the form(*)                                                                       x′=f(t,x,xt).To this end, we first associate with every almost periodic semilinear equation(**)                                                                       x′=F(t,x).a nonlinear semigroup in the space of almost periodic functions. We then give sufficient conditions (in terms of the accretiveness of the generator of this semigroup) for the existence of almost periodic mild solutions of (**) as fixed points of the semigroup. Those results are then carried over to equation (*). The main results are stated under accretiveness conditions of the functionfin terms ofxand Lipschitz conditions with respect toxt.

scholarly journals Existence and Exponential Stability of Solutions to Stochastic Neutral Functional Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Ling Hu ◽  
Zheng Wu ◽  
Zhangzhi Wei ◽  
Lianglong Wang
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Exponential Stability ◽  
Functional Differential Equations ◽  
Banach Fixed Point Theorem ◽  
Stability Of Solutions ◽  
Neutral Functional Differential Equations ◽  
Semigroup Approach ◽  
Existence And Stability ◽  
Functional Differential

In this paper we consider the existence and stability of solutions to stochastic neutral functional differential equations with finite delays. Under suitable conditions, the existence and exponential stability of solutions were obtained by using the semigroup approach and Banach fixed point theorem.

Upper and Lower Solutions of Boundary Value Problems for Functional Differential Equations and Theorems on Functional Differential Inequalities

2000 ◽  
Vol 7 (3) ◽  
pp. 489-512 ◽  
Author(s):  
R. Hakl ◽  
I. Kiguradze ◽  
B. Půža
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Differential Inequalities ◽  
Bounded Operators ◽  
Nonlinear Operators ◽  
Functional Differential

Abstract Sufficient conditions are found for the existence of an upper and a lower solutions of the boundary value problem where and are linear bounded operators, and and are continuous, generally speaking nonlinear, operators. Kamke type theorems are proved on functional differential inequalities.

scholarly journals Existence and Stability Results for Second-Order Neutral Stochastic Differential Equations With Random Impulses and Poisson Jumps

2021 ◽  
Vol 1 (1) ◽  
pp. 1-18
Author(s):  
K. Ravikumar ◽  
K. Ramkumar ◽  
Dimplekumar Chalishajar
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Banach Contraction ◽  
Existence And Stability ◽  
Random Impulses ◽  
Functional Differential ◽  
The Stability ◽  
Stability Results

The objective of this paper is to investigate the existence and stability results of secondorder neutral stochastic functional differential equations (NSFDEs) in Hilbert space. Initially, we establish the existence results of mild solutions of the aforementioned system using the Banach contraction principle. The results are formulated using stochastic analysis techniques. In the later part, we investigate the stability results through the continuous dependence of solutions on initial conditions.

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