We consider a quasilinear dynamic equation reducing to a half-linear equation, an Emden–Fowler equation or a Sturm–Liouville equation under some conditions. Any nontrivial solution of the quasilinear dynamic equation is eventually monotone. In other words, it can be either positive decreasing (negative increasing) or positive increasing (negative decreasing). In particular, we investigate the a...

Employing Riccati techniques and the integral averaging method, we establish interval oscillation criteria for the second-order Emden-Fowler neutral delay differential equation [|x′(t)|γ−1x′(t)]′ + q1(t)|y(t− σ)|α−1y(t− σ) + q2(t)|y(t− σ)|β−1y(t− σ) = 0, where t ≥ t0 and x(t) = y(t) + p(t)y(t − τ). The criteria obtained here are different from most known criteria in the sense that they are base...

Consider the Emden-Fowler sublinear dynamic equation (0.1) x(t) + p(t)x(σ(t)) = 0, where p ∈ C(T, R), where T is a time scale, 0 < α < 1, α is the quotient of odd positive integers. When p(t) is allowed to take on negative values, we obtain a Belohorec-type oscillation theorem for (0.1). As an application, we get that the sublinear difference equation (0.2) ∆x(n) + p(n)x(n+ 1) = 0, is oscillato...

When an unbounded domain is inside a slab, existence of a positive solution is proved for the Dirichlet problem of a class of semilinear elliptic equations that are similar either to the singular Emden-Fowler equation or a sublinear elliptic equation. The result obtained can be applied to equations with coefficients of the nonlinear term growing exponentially. The proof is based on the super an...