Globally positive solutions for the third order differential equation with the damping term and delay, $$x^{\text{'}\text{'}\text{'}}+q\left(t\right)x^{\text{'}}\left(t\right)-r\left(t\right)f\left(x\left(\phi \left(t\right)\right)\right)=0,$$ are studied in the case where the corresponding second order differential equation $$y^{\text{'}\text{'}}+q\left(t\right)y=0$$ is oscillatory. Necessary and sufficient conditions for all nonoscillatory solutions of (*) to be unbounded are given. Furthermore, oscillation criteria ensuring that any solution is either oscillatory or unbounded together with its first and second derivatives are presented. The comparison of results with those...

We study a hybrid control system in which both discrete and continuous controls are involved. The discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set A or a controlled jump set C where controller can choose to jump or not. At each jump, trajectory can move to a different Euclidean space. We allow the cost functionals to be unbounded with certain growth and hence...

The paper concerns the solution of partial differential equations of evolution type by the finite difference method. The author discusses the general assumptions on the original equation as well as its discretization, which guarantee that the difference scheme is unconditionally stable, i.e. stable without any stability condition for the time-step. A new notion of the $A_n$-acceptability of the integration formula is introduced and examples of such formulas are given. The results can be applied to ordinary...

In this paper, we aim to study the global solvability of the following system of third order nonlinear neutral delay differential equations $$\frac{d}{dt}\left\{r_i\left(t\right)\frac{d}{dt}\left[{\lambda }_i\left(t\right)\frac{d}{dt}\left(x_i\left(t\right)-f_i(t,x_1(t-{\sigma }_{i1}),x_2(t-{\sigma }_{i2}),x_3(t-{\sigma }_{i3}))\right)\right]\right\}+\frac{d}{dt}\left[r_i\left(t\right)\frac{d}{dt}{g}_i(t,x_1\left(p_{i1}\left(t\right)\right),x_2\left(p_{i2}\left(t\right)\right),x_3\left(p_{i3}\left(t\right)\right))\right]+\frac{d}{dt}{h}_i(t,x_1\left(q_{i1}\left(t\right)\right),x_2\left(q_{i2}\left(t\right)\right),x_3\left(q_{i3}\left(t\right)\right))=l_i(t,x_1\left({\eta }_{i1}\left(t\right)\right),x_2\left({\eta }_{i2}\left(t\right)\right),x_3\left({\eta }_{i3}\left(t\right)\right)),t\ge t_0,i\in \{1,2,3\}$$ in the following bounded closed and convex set $$\Omega (a,b)=\left\{x\left(t\right)=(x_1\left(t\right),x_2\left(t\right),x_3\left(t\right))\in C([t_0,+\infty ),{ℝ}^3):a\left(t\right)\le x_i\left(t\right)\le b\left(t\right),\forall t\ge t_0,i\in \{1,2,3\}\right\},$$ where ${\sigma }_{ij}>0$, $r_i,{\lambda }_i,a,b\in C([t_0,+\infty ),{ℝ}^{+})$, $f_i,g_i,h_i,l_i\in C([t_0,+\infty )\times {ℝ}^3,ℝ)$, $p_{ij},q_{ij},{\eta }_{ij}\in C([t_0,+\infty ),ℝ)$ for $i,j\in \{1,2,3\}$. By applying the Krasnoselskii fixed point theorem, the Schauder fixed point theorem, the Sadovskii fixed point theorem and the Banach contraction principle, four existence results of uncountably many bounded positive solutions of the system are established.

Résumé. On présente une fonction continue f: c₀ → c₀ qui satisfait à une condition lipschitzienne par rapport à la mesure de non-compacité de Hausdorff (ou Kuratowski), mais telle que f n'est pas la somme d'une fonction dissipative et d'une fonction compacte. Cet exemple attache de l'importance au théorème d'existence de Sabina Schmidt (1989) pour des équations différentielles dans les espaces de Banach.