Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sides

Sufficient conditions for uniform equi-asymptotic stability and uniform asymptotic stability of the zero solution of the retarded equation $$x'(t) = f(t, x_t), \qquad (x_t(s):= x(t+s),\ -h\le s\le 0)$$ are given. In the stability theory of non-autonomous differential equations a result is of Marachkov type if it contains some kind of boundedness or growth condition on the right-hand side of the equation with respect to $t$. Using Lyapunov's direct method and the annulus argument we prove theorems for equations whose right-hand sides may be unbounded with respect to $t$. The derivative of the Lyapunov function is not supposed to be negative definite, it may be negative semi-definite. The results are applied to the retarded scalar differential equation with distributed delay $$x'(t) = -a(t) x(t) + b(t) \int^t_{t-h} x(s) \,{\rm{d}}s,\qquad (a(t)>0),$$ where $a$ and $b$ may be unbounded on $[0,\infty)$. The growth conditions do not concern function $a$, they contain only function $b$. In addition, the function $t\mapsto a(t) - \int^{t+h}_t |b(u)| \,{\rm{d}}u$, measuring the dominance of the negative instantaneous feedback over the delayed feedback, is not supposed to remain above a positive constant, even it may vanish on long intervals

Existence of periodic solutions of pendulum-like ordinary and functional differential equations

The equation $$x''(t)=a(t,x(t))+b(t,x)+d(t,x)e(x'(t))$$ is considered, where $a:\mathbb{R}^2\to\mathbb{R}$, $b,d:\mathbb{R}\times C(\mathbb{R},\mathbb{R})\to\mathbb{R}$, $e:\mathbb{R}\to\mathbb{R}$ are continuous, and $a,b,d$ are $T$-periodic with respect to $t$. Using the Leray–Schauder degree theory we prove that a sign condition, in which $a$ dominates $b$, is sufficient for the existence of a $T$-periodic solution. The main theorem is applied to the equation of the forced damped pendulum

The complete degradation of acetanilide by a consortium of microbes isolated from River Maros

Chemical pollutants occurring in rivers may have severe effects on human health along with being harmful to the environment. Bioaugmentation is a potential tool for the removal of xenobiotics from soil and water therefore the objectives of this study were the isolation, identification and characterization of microbes with acetanilide- and aniline-degrading properties from the River Maros. Microbes isolated on minimal media containing acetanilide or aniline-HCl as a sole carbon and nitrogen source were considered as acetanilide- or aniline-degraders. The decomposition of acetanilide and aniline were followed by High Pressure Liquid Chromatography (HPLC). An acetanilide-degrading bacterium, identified as Rhodococcus erythropolis, was able to convert acetanilide to aniline, which was further decomposed by the fungal isolate Aspergillus ustus when the two microbes were co-cultivated in a minimal medium containing acetanilide as a sole carbon and nitrogen source. The strains isolated in this study might be used in approaches addressing the biodegradation of acetanilide and aniline in the environment