In this paper, we study differential equations with piecewise constant argument of generalized (DEPCAGs) type, i.e., the argument is a general step function. They are hybrid equations combining properties of continuous and discrete equations. The play of the discrete part is always very important. The explicit solutions of the homogeneous and non-homogeneous linear DEPCAGs systems are obtained. Existence, uniqueness and stability of the solutions of the quasilinear DEPCAGs are under discussion. All previous results are improved. The importance of the advanced and delayed intervals will be clear. Cauchy and Green matrices type are deduced. The integral representation and Gronwall's inequality type obtained can be fruitfully applied to the investigation of stability, oscillations, controllability and many other problems of DEPCAGs.

We study the wavefront solutions of the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone reaction term $g: \mathbb{R}_{+} → \mathbb{R}_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. To this end, we describe the asymptotics of all wavefronts (including critical and non-critical fronts) at $-\infty$. We also prove the uniqueness of wavefronts (up to a translation). In addition, a new uniqueness result for a class of nonlocal lattice equations is presented.