The paper describes the general form of functional-differential equations of the first order with $m(m\ge 1)$ delays which allows nontrivial global transformations consisting of a change of the independent variable and of a nonvanishing factor. A functional equation $$f(t,uv,u_1{v}_1,...,u_m{v}_m)=f(x,v,v_1,...,v_m)g(t,x,u,u_1,...,u_m)u+h(t,x,u,u_1,...,u_m)v$$ for $u\ne 0$ is solved on $ℝ$ and a method of proof by J. Aczél is applied.

The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $$f(t,vy,wy+uvz)=f(x,y,z)u^{2}v+g(t,x,u,v,w)vz+h(t,x,u,v,w)y+2uwz$$ is solved on $ℝ$ for $y\ne 0$, $v\ne 0.$

Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number ${\varrho }_c\le 1$ such that |x|·|∇f| and ${|f\left(x\right)-c|}^{{\varrho }_c}$ are separated at infinity. If c is a regular value and ${\varrho }_c<1$, then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.

This paper is concerned with square integrable quasi-derivatives for any solution of a general quasi-differential equation of $n$th order with complex coefficients $M\left[y\right]-\lambda wy=wf(t,y^{\left[0\right]},...,y^{[n-1]})$, $t\in [a,b)$ provided that all $r$th quasi-derivatives of solutions of $M\left[y\right]-\lambda wy=0$ and all solutions of its normal adjoint $M^{+}\left[z\right]-\overline{\lambda }wz=0$ are in $L_w^2(a,b)$ and under suitable conditions on the function $f$.

We study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However, when the piecewise differential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we...