In this paper, we proposed a version of the Floquet theory for delay differential equations. We demonstrated that very natural assumptions for control in technical applications can lead us to a one-dimensional fundamental system. This approach allowed researchers to work with classical methods used in the case of ordinary differential equations. On this basis, new original unexpected results on the exponential stability were proposed. For example, in the equation $x^{\prime }(t)+a(t)x(t-\tau (t))=0$, $t\in [0,\mathrm{\infty })$, we avoided the assumption on the smallness of the product $\underset{t\in [0,\mathrm{\infty })}{sup}a(t)\underset{t\in [0,\mathrm{\infty })}{sup}\tau (t)<3/2$ for asymptotic stability. We obtained that in the case of $\omega$-periodic coefficient and delay, the fact that the period $\omega$ was situated in a corresponding interval can lead to exponential stability. We then applied our new tests of stability to the stabilization of a drone's flight, where smallness of the noted above product could not be achieved from a technical point of view. For an equation with periodic coefficient and delay, we got a formula of the solution's representation on the semiaxis.

In this study, we investigate the existence of at least one solution and the existence of an infinite number of solutions for a discrete fractional boundary value problem. Requiring an algebraic condition on the nonlinear term for small values of the parameter, and requiring an additional asymptotical behavior of the potential at zero, we investigate the existence of at least one nontrivial solution for the problem. Moreover, under suitable assumptions on the oscillatory behavior of the nonlinearity at infinity, for exact collections of the parameter, we discuss the existence of a sequence of solutions for the problem. We also present some examples that illustrate the applicability of the main results.

We introduce a novel definition of periodicity on arbitrary time scales, dependent on a strictly increasing and differentiable function. This removes the commonly used and restrictive assumption of a periodic time scale to define periodic functions. Our new definition furthermore allows for a wider class of functions to be studied using the theory of periodic systems. After providing crucial properties of these periodic functions, such as the translation invariance of integrals of periodic functions, we apply the concept of this new periodicity to linear dynamic equations. We provide necessary and sufficient conditions for a linear dynamic equation to have such a periodic solution and discuss its uniqueness.

In this paper, we introduce a novel formulation of dynamic Hardy-type inequalities on a time scale, motivated by a recently-established convexity approach in the Haar measure.The classical Hardy inequality is refined so that the classical Lebesgue-measure constant is replaced by the sharp constant $1$. We obtain time-scale analogues on finite intervals with best constants, and, for nonincreasing and nondecreasing functions, reversed inequalities with explicit weights described by incomplete $\beta$-functions. To establish our results, we employ two distinct time scales and apply the chain rule, together with the substitution rule, the derivative of inverse functions, and Fubini's theorem for delta integration. Our approach generalizes classical integral inequalities in the continuous setting, while yielding fundamentally new inequalities in the discrete setting. Furthermore, we explore the application of our results in the quantum case, demonstrating their broader relevance.