Solutions of an advance-delay differential equation and their asymptotic behaviour
نویسندگان
چکیده
The paper considers a scalar differential equation of an advance-delay type \begin{equation*} \dot{y}(t)= -\left(a_0+\frac{a_1}{t}\right)y(t-\tau )+\left(b_0+\frac{b_1}{t}\right)y(t+\sigma )\,, \end{equation*} where constants $a_0$, $b_0$, $\tau $ and $\sigma are positive, $a_1$ $b_1$ arbitrary. behavior its solutions for $t\rightarrow \infty is analyzed provided that the transcendental \lambda = -a_0\mathrm{e}^{-\lambda \tau }+b_0\mathrm{e}^{\lambda \sigma } has positive real root. An exponential-type function approximating solution searched to be used in proving existence semi-global solution. Moreover, lower upper estimates given such
منابع مشابه
Asymptotic Behaviour of Solutions of Some Linear Delay Differential Equations
In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y′(x) = a(x)y(τ (x)) + b(x)y(x), x ∈ I = [x0,∞). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z′(x) = b(x)z(x), x ∈ I and a solution of the functional equation |a(x)|φ(τ (x)) = |b(x)|φ(x), x ∈ I.
متن کاملPeriodic solutions of fourth-order delay differential equation
In this paper the periodic solutions of fourth order delay differential equation of the form $ddddot{x}(t)+adddot{x}(t)+f(ddot{x}(t-tau(t)))+g(dot{x}(t-tau(t)))+h({x}(t-tau(t)))=p(t)$ is investigated. Some new positive periodic criteria are given.
متن کاملPositive Solutions of Neutral Delay Differential Equation
Neutral delay differential equations contain the derivative of the unknown function both with and without delays. Some new phenomena can appear, hence the theory of neutral delay differential equations is even more complicated than the theory of non-neutral delay equations. The oscillatory behavior of the solutions of neutral systems is of importance in both the theory and applications, such as...
متن کاملStability and asymptotic behaviour of solutions of the heat equation
where Ω is a bounded and smooth subset of Rn , n 1, m > 0 and p 1. Problem (1.1)–(1.3) (see Galaktionov, 1981; Samarskii et al., 1995) describes the propagation of thermal perturbations in a medium with a nonlinear heat conduction coefficient and a heat source depending on the temperature when u0 0. Local existence for the solutions of (1.1)–(1.3) has been proved when m > 1 (the so-called slow ...
متن کاملperiodic solutions of fourth-order delay differential equation
in this paper the periodic solutions of fourth order delay differential equation of the form $ddddot{x}(t)+adddot{x}(t)+f(ddot{x}(t-tau(t)))+g(dot{x}(t-tau(t)))+h({x}(t-tau(t)))=p(t)$ is investigated. some new positive periodic criteria are given.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Archivum mathematicum
سال: 2023
ISSN: ['0044-8753', '1212-5059']
DOI: https://doi.org/10.5817/am2023-1-141