UNIFORM DECAY PROPERTIES OF LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

2008 ◽  
Vol 18 (01) ◽  
pp. 21-45 ◽  
Author(s):  
MONICA CONTI ◽  
STEFANIA GATTI ◽  
VITTORINO PATA
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Hyperbolic Type ◽  
Polynomial Decay ◽  
Convolution Kernel ◽  
Differential Inequalities ◽  
Uniform Decay ◽  
Decay Properties

We establish some new results concerning the exponential decay and the polynomial decay of the energy associated with a linear Volterra integro-differential equation of hyperbolic type in a Hilbert space, which is an abstract version of the equation [Formula: see text] describing the motion of linearly viscoelastic solids. We provide sufficient conditions for the decay to hold, without invoking differential inequalities involving the convolution kernel μ.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

scholarly journals Oscillation and non-oscillation of some neutral differential equations of odd order

1992 ◽  
Vol 15 (3) ◽  
pp. 509-515 ◽  
Author(s):  
B. S. Lalli ◽  
B. G. Zhang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solution ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
Neutral Term ◽  
Odd Order ◽  
Existence Criterion ◽  
Oscillation Of Solutions

An existence criterion for nonoscillatory solution for an odd order neutral differential equation is provided. Some sufficient conditions are also given for the oscillation of solutions of somenth order equations with nonlinearity in the neutral term.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

Oscillatory Criteria for Nonlinear nth-Order Differential Equations with Quasiderivatives

1996 ◽  
Vol 3 (4) ◽  
pp. 301-314
Author(s):  
Miroslav Bartušek
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Sign Condition ◽  
Proper Solutions

Abstract Sufficient conditions are given for the existence of oscillatory proper solutions of a differential equation with quasiderivatives Lny = f(t, L 0 y, . . . , L n–1 y) under the validity of the sign condition f(t, x 1, . . . , xn )x 1 ≤ 0, f(t, 0, x 2, . . . , xn ) = 0 on .

A note on meromorphic functions with finite order and of bounded l-index

2021 ◽  
Vol 18 (1) ◽  
pp. 1-11
Author(s):  
Andriy Bandura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Finite Order ◽  
General Problem ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Riccati Differential Equation

We present a generalization of concept of bounded $l$-index for meromorphic functions of finite order. Using known results for entire functions of bounded $l$-index we obtain similar propositions for meromorphic functions. There are presented analogs of Hayman's theorem and logarithmic criterion for this class. The propositions are widely used to investigate $l$-index boundedness of entire solutions of differential equations. Taking this into account we raise a general problem of generalization of some results from theory of entire functions of bounded $l$-index by meromorphic functions of finite order and their applications to meromorphic solutions of differential equations. There are deduced sufficient conditions providing $l$-index boundedness of meromoprhic solutions of finite order for the Riccati differential equation. Also we proved that the Weierstrass $\wp$-function has bounded $l$-index with $l(z)=|z|.$

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

Asymptotic Behavior for Nonoscillatory Solutions of Second Order Nonlinear Functional Differential Equation

2012 ◽  
Vol 616-618 ◽  
pp. 2137-2141
Author(s):  
Zhi Min Luo ◽  
Bei Fei Chen
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solutions ◽  
Nonlinear Functional ◽  
Nonlinear Functional Differential Equation ◽  
Functional Differential

This paper studied the asymptotic behavior of a class of nonlinear functional differential equations by using the Bellman-Bihari inequality. We obtain results which extend and complement those in references. The results indicate that all non-oscillatory continuable solutions of equation are asymptotic to at+b as under some sufficient conditions, where a,b are real constants. An example is provided to illustrate the application of the results.

scholarly journals Bifurcation of Bounded Solutions of Impulsive Differential Equations

2016 ◽  
Vol 26 (14) ◽  
pp. 1650242 ◽  
Author(s):  
Kevin E. M. Church ◽  
Xinzhi Liu
Keyword(s):  
Differential Equation ◽  
Integral Operator ◽  
Differential Equations ◽  
Impulsive Differential Equation ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Necessary Condition ◽  
Bounded Solutions ◽  
Pitchfork Bifurcations ◽  
Nonlinear Integral Operator

In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov–Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This leads to sufficient conditions for the presence of fold, transcritical and pitchfork bifurcations. Additionally, we provide a computable necessary condition for bifurcation in nonlinear scalar impulsive differential equations. Several examples are provided illustrating the results.

scholarly journals On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Nguyen Thanh Lan
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Necessary And Sufficient

For the differential equation , on a Hilbert space , we find the necessary and sufficient conditions that the above-mentioned equation has a unique almost periodic solution. Some applications are also given.

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