We give some oscillation criteria for linear iterative functional equations. We compare obtained theorems with known results. We give applications to discrete equations too.

We obtain comparison theorems for the second-order half-linear dynamic equation ${\left[r\left(t\right)\Phi \left(y^{\Delta }\right)\right]}^{\Delta }+p\left(t\right)\Phi \left(y^{\sigma }\right)=0$, where $\Phi \left(x\right)={\left|x\right|}^{\alpha -1}\text{sgn}\hspace{0.17em}x$ with $\alpha >1$. In particular, it is shown that the nonoscillation of the previous dynamic equation is preserved if we multiply the coefficient $p\left(t\right)$ by a suitable function $q\left(t\right)$ and lower the exponent $\alpha$ in the nonlinearity $\Phi$, under certain assumptions. Moreover, we give a generalization of Hille-Wintner comparison theorem. In addition to the aspect of unification and extension, our theorems provide some new results even in the continuous and the discrete case.

We use a special space of integrable functions for studying the Cauchy problem for linear functional-differential equations with nonintegrable singularities. We use the ideas developed by Azbelev and his students (1995). We show that by choosing the function $\psi$ generating the space, one can guarantee resolubility and certain behavior of the solution near the point of singularity.

We deal with the nonlinear impulsive periodic boundary value problem $u^{″}=f\left(t,u,u^{\prime }\right)$, $u\left(t_i+\right)={\text{J}}_i\left(u\left(t_i\right)\right)$, $u^{\prime }\left(t_i+\right)={\text{M}}_i\left(u^{\prime }\left(t_i\right)\right)$, $i=1,2,\dots ,m$, $u\left(0\right)=u\left(T\right)$, $u^{\prime }\left(0\right)=u^{\prime }\left(T\right)$. We establish the existence results which rely on the presence of a well-ordered pair $\left({\sigma }_1,{\sigma }_2\right)$ of lower/upper functions $\left({\sigma }_1\le {\sigma }_2\hspace{0.17em}\text{on}\hspace{0.17em}\left[0,T\right]\right)$ associated with the problem. In contrast to previous papers investigating such problems, the monotonicity of the impulse functions ${\text{J}}_i$, ${\text{M}}_i$ is not required here.

The problem of nonuniqueness for a singular Cauchy-Nicoletti boundary value problem is studied. The general nonuniqueness theorem ensuring the existence of two different solutions is given such that the estimating expressions are nonlinear, in general, and depend on suitable Lyapunov functions. The applicability of results is illustrated by several examples.

The paper deals with the vector discrete dynamical system $x_{k+1}=A_k{x}_k+f_k\left(x_k\right)$. The well-known result by Perron states that this system is asymptotically stable if $A_k\equiv A=\text{const}$ is stable and $f_k\left(x\right)\equiv \tilde{f}\left(x\right)=o\left(\Vert x\Vert \right)$. Perron's result gives no information about the size of the region of asymptotic stability and norms of solutions. In this paper, accurate estimates for the norms of solutions are derived. They give us stability conditions for (1.1) and bounds for the region of attraction of the stationary solution. Our approach is based on the “freezing” method for difference equations and on recent estimates for the powers of a constant matrix. We also discuss applications of our main result to partial reaction-diffusion difference equations.

We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.

In the case of the wave equation, defined on a sufficiently smooth bounded domain of arbitrary dimension, and subject to Dirichlet boundary control, the operator $B^{*}L$ from boundary to boundary is bounded in the $L_2$-sense. The proof combines hyperbolic differential energy methods with a microlocal elliptic component.