We consider the nonlinear n-th order boundary value problem Lu=u(n)=f(t,u(t),u′(t),…,u(n−1)(t))=Nu given arbitrary bounded linear functional conditions Bi(u)=0, i=1,…,n and develop a method that allows us to study all such resonance problems of one, as well implementing more general constructive for deriving existence criteria in framework coincidence degree Mawhin. demonstrate applicability fo...

In this paper, we consider a semilinear parabolic equation ut = ∆u + u q ∫ t 0 u(x, s)ds, x ∈ Ω, t > 0 with nonlocal nonlinear boundary condition u|∂Ω×(0,+∞) = ∫ Ω φ(x, y)u (y, t)dy and nonnegative initial data, where p, q ≥ 0 and l > 0. The blow-up criteria and the blow-up rate are obtained.

By means of generalized Riccati transformation techniques and generalized exponential functions, we give some oscillation criteria for the nonlinear dynamic equation (p(t)x ∆ (t)) ∆ + q(t)(f • x σ) = 0, on time scales. We also apply our results to linear and nonlinear dynamic equations with damping and obtain some sufficient conditions for oscillation of all solutions.

By using averaging function and the approach developed by Philos and Kong, Kamenevtype and interval oscillation criteria are established for the even order differential equation with distributed deviating arguments, (r(t)|x(n−1)(t)|p−1x(n−1)(t))′ + β ∫ α F [t,ξ ,x(g(t,ξ ))]dσ(ξ ) = 0. The obtained results are extensions of existing ones for second order linear differential equations. Mathematic...

Nodular lymphocyte-predominant Hodgkin lymphoma (NLPHL) and T-cell/histiocyte-rich B-cell lymphoma (T/HRBCL) are distinct tumors and are treated differently. They are linked by a morphologic and probably a biologic continuum, which renders the differential diagnosis difficult. To develop criteria to distinguish the entities along the morphologic continuum, we correlated the lymph node architect...

We investigate the following nonlinear equations with p-Laplacian-like operators φ x′ t ′ f x t x′ t g x t e t : some criteria to guarantee the existence and uniqueness of periodic solutions of the above equation are given by using Mawhin’s continuation theorem. Our results are new and extend some recent results due to Liu B. Liu, Existence and uniqueness of periodic solutions for a kind of Lie...

Oscillation criteria for the higher order quasi-linear neutral delay differential equations of the form [r(t)ψ(u(t))|z(n−1)(t)|α−1z(n−1)(t)]′ + m ∑ i=1 qi(t) fi(|u(τi(t))|αi−1u(τi(t))) = 0, t t0, z(t) = u(t) + p(t)u(t −σ),α > 0,αi > 0 (i = 1,2,3, ...,m) , are established under the condition: ∫ ∞ t0 r− 1 α (s)ds = ∞ or ∫ ∞ t0 r− 1 α (s)ds < ∞ respectively, where n is even. The obtained results i...

We derive oscillation criteria for general-type neutral differential equations [x(t) +αx(t− τ) +βx(t+ τ)] = δ bax(t − s)dsq1(t,s) + δ ∫ d c x(t + s)dsq2(t,s) = 0, t ≥ t0, where t0 ≥ 0, δ = ±1, τ > 0, b > a ≥ 0, d > c ≥ 0, α and β are real numbers, the functions q1(t,s) : [t0,∞)× [a,b]→R and q2(t,s) : [t0,∞)× [c,d]→R are nondecreasing in s for each fixed t, and τ is periodic and continuous with ...

In this work we study the oscillatory behaviour of the differential equation of mixed type x′ (t) = ∫ 0 −1 x (t− r (θ)) dν (θ) + ∫ 0 −1 x (t+ τ (θ)) dη (θ) with delays, r (θ) , and advances, τ (θ) , both differentiable. Some analytical and numerical criteria are obtained in order to guarantee that all solutions are oscillatory.