Zdenek Smarda

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We show that the following system of difference equations x(n) = x(n-k)y(n-l)/(b(n)x(n-k) + a(n)y(n-l-k)); y(n) = y(n-k)x(n-l)/(d(n)y(n-k) + c(n)x(n-l-k)); n e No; where k, l e N, x(-i), y(-i) e R\{0}, i = 1; k+l, and sequences (a(n))neNo , (b(n))neNo , (c(n))neNo and (d(n))neNo are real, can be solved in closed form. For the case when the sequences a(n),(More)
In the paper, the existence of positive solutions is studied for the second-order delay differential equation with a damping term¨x(t) + a(t)˙ x(t) + b(t)x(h(t)) = 0 using a comparison with the integro-differential equation ˙ y(t) + t t 0 e − t s a(ξ)dξ b(s)y(h(s))ds = 0. Explicit non-oscillation criteria and comparison type results are derived.
  • Jaromı́r Baštinec, Josef Diblı́k, Zdeněk Šmarda, Donal O’Regan
  • 2009
A singular Cauchy-Nicoletti problem for a system of nonlinear ordinary differential equations is considered. With the aid of combination of Wa ˙ zewski's topological method and Schauder's principle, the theorem concerning the existence of a solution of this problem having the graph in a prescribed domain is proved.
Closed form formulas of the solutions to the following system of difference equations: x(n) = (y(n–1)y(n–2))/(x(n–1)(a(n) + b(n)y(n–1)y(n–2)), y(n) = (x(n–1)x(n–2))/(y(n–1)(α(n) + β(n)x(n–1)x(n–2)), n No, where a(n), b(n), α(n), β(n), n No, and initial values x(–i), y(–i), i {1, 2}, are real numbers, are found. The domain of undefinable solutions to the(More)