Browsing by Subject "delay differential equation"

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Access status: Open Access ,
On existence and global attractivity of periodic solutions of nonlinear delay differential equations
(Wydawnictwa AGH, 2019) Qian, Chuanxi; Smith, Justin
Consider the delay differential equation with a forcing term $\tag{$\ast$} x'(t)=-f(t,x(t))+g(t,x(t-\tau ))+r(t), \quad t \geq 0$ $(∗)$ where $f(t,x): [0,\infty) \times [0,\infty) \to \mathbb{R}$, $g(t,x): [0,\infty) \times [0,\infty) \to [0,\infty)$ are continuous functions and ω-periodic in $t$, $r(t): [0,\infty) \to\mathbb{R}$ is a continuous function and $\tau \in (0,\infty)$ is a positive constant. We first obtain a sufficient condition for the existence of a unique nonnegative periodic solution $\tilde{x}(t)$ of the associated unforced differential equation of Eq. $(∗)$ $\tag{$\ast\ast$} x'(t)=-f(t,x(t))+g(t,x(t-\tau)), \quad t \geq 0.$ $(∗∗)$ Then we obtain a sufficient condition so that every nonnegative solution of the forced equation $(∗)$ converges to this nonnegative periodic solution $\tilde{x}(t)$ of the associated unforced equation $(∗∗)$. Applications from mathematical biology and numerical examples are also given.
Access status: Open Access ,
Oscillation criteria for third order nonlinear delay differential equations with damping
(2015) Grace, Said R.
This note is concerned with the oscillation of third order nonlinear delay differential equations of the form $\left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{$\ast$}$ $(*)$ In the papers [A.Tiryaki, M.F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M.F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear-functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation $(*)$ oscillates or converges to zero, provided that the second order equation $\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{$\ast\ast$}$ $(**)$ is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation $(*)$ oscillates if equation $(**)$ is nonoscillatory. We also establish results for the oscillation of equation $(*)$ when equation $(**)$ is oscillatory.
Access status: Open Access ,
Stochastic model of drug concentration level during IV-administration
(Wydawnictwa AGH, 2022) Dzhalladova, Irada; Růžičková, Miroslava
A stochastic model describing the concentration of the drug in the body during its IV-administration is discussed. The paper compares a deterministic model created with certain simplifications with the stochastic model. Fluctuating and irregular patterns of plasma concentrations of some drugs observed during intravenous infusion are explained. An illustrative example is given with certain values of drug infusion rate and drug elimination rate.