ordinary differential equation

Existence of periodic solutions for a semilinea

Analysis of the ranges of perturbed noninvertible operators

Available from STL Prague, CZ / NTK - National Technical LibrarySIGLECZCzech Republi

Generalized trigonometric functions in complex domain

summary:We study extension of $p$-trigonometric functions $\sin _p$ and $\cos _p$ to complex domain. For $p=4, 6, 8, \dots$, the function $\sin _p$ satisfies the initial value problem which is equivalent to (*) $$-(u')^{p-2}u''-u^{p-1} =0, \quad u(0)=0, \quad u'(0)=1$$in $\mathbb {R}$. In our recent paper, Girg, Kotrla (2014), we showed that $\sin _p(x)$ is a real analytic function for $p=4, 6, 8, \dots$ on $(-\pi _p/2, \pi _p/2)$, where $\pi _p/2 = \int _0^1(1-s^p)^{-1/p}$. This allows us to extend $\sin _p$ to complex domain by its Maclaurin series convergent on the disc $\{z\in \mathbb {C}\colon |z|2$. Finally, we provide some graphs of real and imaginary parts of $\sin _p(z)$ and suggest some new conjectures

p

We study extension of p-trigonometric functions sinp and cosp and of p-hyperbolic functions sinhp and coshp to complex domain. Our aim is to answer the question under what conditions on p these functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example, sin(z)=-i·sinh⁡i·z. In particular, we prove in the paper that for p=6,10,14,… the p-trigonometric and p-hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series for p-trigonometric and p-hyperbolic functions

A global bifurcation result for a class of semipositone elliptic systems

V článku se zabýváme jistou třídou soustav eliptických PDR závisejících na parametru s nelinearitou typu "semipoziton". K jejich studiu používáme teorii bifurkací. Podařilo se nám ukázat, že množina řešení obsahuje dvě disjunktní neomezené souvislé komponenty. V článku diskutujeme nodální vlastnosti řešení na těchto komponentách. Na závěr, jakožto důsledek těchto bifurkačních výsledků, dokážeme existenci a násobnost řešení v situaci, kdy se bifurkační parametr nalézá v blízkosti jednoduchého vlastního čísla asociovaného problému na vlastní čísla.We study a class of semipositone elliptic systems depending on a parameter using bifurcation theory. We show that there are two disjoint unbounded connected components of the solution set and discuss the nodal properties of solutions on these components. Finally, as a consequence of these results, we infer the existence and multiplicity of solutions for the bifurcation parameter in a neighborhood containing the simple eigenvalue of the associated eigenvalue problem

Prufer transformation for the p-Laplacian

Prufer transformation is a useful tool for study of second-order ordinary differential equations. There are many possible extensions of the original Prufer transformation. We focus on a transformation suitable for study of boundary value problems for the p-Laplacian in the resonant case. The purpose of this paper is to establish its basic properties in deep detail