scholarly journals Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions

2020 ◽  
Vol 25 (6) ◽  
pp. 2271-2292 ◽  
Author(s):  
Zhihua Liu ◽  
◽  
Pierre Magal ◽  
◽  
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Infinite Delay ◽  
Continuous Functions ◽  
Functional Differential ◽  
Uniformly Continuous

On the approximation of continuous functions by analytic solutions of universal functional-differential equations

2004 ◽  
Vol 41 (1) ◽  
pp. 1-15 ◽  
Author(s):  
C. Elsner
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Analytic Solutions ◽  
Continuous Functions ◽  
Compact Interval ◽  
The Real ◽  
Real Analytic ◽  
Functional Differential ◽  
Real Analytic Solutions

The existence of an algebraic functional-differential equation P (y′(x), y′(x + log 2), …, y′(x + 5 log 2)) = 0 is proved such that the real-analytic solutions are dense in the space of continuous functions on every compact interval. A similar result holds for an algebraic functional-differential equation P(y′(x − 4πi), y′(x − 2πi), …, y′(x + 4πi)) = 0 (with i2 = −1), which is explicitly given: There are real-analytic solutions on the real line such that every continuous function defined on a compact interval can be approximated by these solutions with arbitrary accuracy.

scholarly journals Functional differential equations with infinite delay in Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 497-508 ◽  
Author(s):  
Jin Liang ◽  
Tijun Xiao
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Exponential Stability ◽  
Functional Differential Equation ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
Zero Solution ◽  
Fading Memory ◽  
Fundamental Operator ◽  
Functional Differential

In this paper, a definition of the fundamental operator for the linear autonomous functional differential equation with infinite delay in a Banach space is given, and some sufficient and necessary conditions of the fundamental operator being exponentially stable in abstract phase spaces which satisfy some suitable hypotheses are obtained. Moreover, we discuss the relation between the exponential asymptotic stability of the zero solution of nonlinear functional differential equation with infinite delay in a Banach space and the exponential stability of the solution semigroup of the corresponding linear equation, and find that the exponential stability problem of the zero solution for the nonlinear equation can be discussed only in the exponentially fading memory phase space.

scholarly journals On Delay-Independent Criteria for Oscillation of Higher-Order Functional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-6
Author(s):  
Yuangong Sun
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Continuous Functions ◽  
Higher Order ◽  
Oscillation Criteria ◽  
Functional Differential ◽  
Delay Dependent

We investigate the oscillation of the following higher-order functional differential equation:x(n)(t)+q(t)|x(t-τ)|λ-1x(t-τ)=e(t),whereq(t)ande(t)are continuous functions on[t0,∞),1>λ>0andτ≠0are constants. Unlike most of delay-dependent oscillation results in the literature, two delay-independent oscillation criteria for the equation are established in both the caseτ>0and the caseτ<0under the assumption that the potentialsq(t)ande(t)change signs on[t0,∞).

The functional-differential equation y′(t)=Ay(t) +By(qt) +Cy′(qt) +f(t)

1998 ◽  
Vol 9 (1) ◽  
pp. 81-91 ◽  
Author(s):  
HARALD LEHNINGER ◽  
YUNKANG LIU
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Fundamental Solutions ◽  
Continuous Functions ◽  
Variable Coefficients ◽  
Variation Of Constants Formula ◽  
Special Cases ◽  
Variation Of Constants ◽  
Functional Differential ◽  
Scalar Equations

An initial value problem for the functional differential equationy′(t) =Ay(t) +By(qt) +Cy′(qt) +f(t), t ≥ t0 > 0where A, B, C are complex matrices, q∈(0, 1), and f is a vector of continuous functions, is considered in this paper. Its solution is represented in terms of the fundamental solution via the variation-of-constants formula. For some special cases, the fundamental solutions are formulated as piecewise Dirichlet series. The variation-of-constants formula is used to analysis the asymptotic behaviour of the solutions of some scalar equations, including one with variable coefficients related to coherent states of the q-oscillator algebra in quantum mechanics.

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