Robert Mařík

Nonexistence of Positive Solutions of PDE's with p-Laplacian

We derive conditions on the function c : Rn → R which guarantee that the PDE with p-Laplacian div (❘∇ u❘p-2∇ u ) + c(x)❘u❘p-2u =0, p >1, has no positive solution either in Rn or in the domain Omega_r :={x ∈ Rn : ❘x❘ ≧r} with r arbitrarily large.

Conditional Oscillation of Half-Linear Differential Equations with Coefficients Having Mean Values

We prove that the existence of the mean values of coefficients is sufficient for second-order half-linear Euler-type differential equations to be conditionally oscillatory. We explicitly find an oscillation constant even for the considered equations whose coefficients can change sign. Our results cover known results concerning periodic and almost periodic positive coefficients and extend them to larger classes of equations. We give examples and corollaries which illustrate cases that our results solve. We also mention an application of the presented results in the theory of partial differential equations.

On Constants in Nonoscillation Criteria for Half-Linear Differential Equations

We study the half-linear differential equation ( 𝑟 ( 𝑡 ) Φ ( 𝑥  ) )  + 𝑐 ( 𝑡 ) Φ ( 𝑥 ) = 0 , where Φ ( 𝑥 ) = | 𝑥 | 𝑝 − 2 𝑥 , 𝑝 > 1 . Using the modified Riccati technique, we derive new nonoscillation criteria for this equation. The results are closely related to the classical Hille-Nehari criteria and allow to replace the fixed constants in known nonoscillation criteria by a certain one-parametric expression.

Oscillation of Half-Linear Differential Equations with Delay

We study the half-linear delay differential equation , , We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.

Oscillation criteria for neutral second-order half-linear differential equations with applications to Euler type equations

We study the second-order neutral delay half-linear differential equation [r(t)Φ(z′(t))]′+q(t)Φ(x(σ(t)))=0, where Φ(t)=|t|α−1t, α≥1 and z(t)=x(t)+p(t)x(τ(t)). We use the method of Riccati type substitution and derive oscillation criteria for this equation. By an example of the neutral Euler type equation we show that the obtained results are sharp and improve the results of previous authors. Among others, we improve the results of Sun et al. (Abstr. Appl. Anal. 2012:819342, 2012) and discuss also the case when σ∘τ≠τ∘σ.MSC:34K11, 34K40.

A half-linear differential equation and variational problem

In the paper new sufficient condition for nonnegativity ot the scalar p-degree fucntional are established in temrs of coupled points .

Coupled intervals for discrete p-degree functional

AbstractWe study the scalar p-degree functional J(x)=RN+1|xN+1|p+∑k=0N[Rk|Δxk|p−Ck|xk+1|p] over the class of sequences with zero boundary condition at the left endpoint and free right endpoint. We extend the linear concept of coupled intervals to give a necessary and sufficient condition for nonnegativity and positivity of this functional.MSC:39A12, 49K99.

A remark on connection between conjugacy of half-linear differential equation and equation with mixed nonlinearities

In this paper new criteria for conjugacy of half-linear ordinary differential equations are derived by using a Riccati transformation. These criteria are used to derive nonexistence and oscillation results for an equation with mixed nonlinearities, which is viewed as a perturbation of a half-linear equation.

On Eventually Positive Solutions of Quasilinear Second-Order Neutral Differential Equations

We study the second-order neutral delay differential equation , where , and . Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral . A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case.