Volume 13, Issue 3

We consider the nonlinearly damped semilinear Petrovsky equation and prove the global existence of its solutions by means of the stable set method in combined with the Faedo–Galerkin procedure. Furthermore, we study the asymptotic behavior of solutions when the nonlinear dissipative term 𝑔 does not necessarily have a polynomial growth near the origin.

Let 𝑋 be a smooth and connected projective curve. Assume the existence of spanned 𝐿 ∈ Pic 𝑎 (𝑋), 𝑅 ∈ Pic 𝑏 (𝑋) such that ℎ 0 (𝑋, 𝐿) = ℎ 0 (𝑋, 𝑅) = 2 and the induced map ϕ 𝐿,𝑅 : 𝑋 → 𝐏 1 × 𝐏 1 is birational onto its image. Here we study the following question. What can be said about the morphisms β : 𝑋 → 𝐏 𝑅 induced by a complete linear system |𝐿 ⊗𝑢 ⊗𝑅 ⊗𝑣 | for some positive 𝑢, 𝑣? We study the homogeneous ideal and the minimal free resolution of the curve β (𝑋).

A functor responsible for second obstruction problems is defined and investigated on the category of topological spaces. In terms of this functor we formulate and prove a classification theorem for maps, which is the reformulation of all known classification theorems for the second obstruction.

We establish the existence of an entire solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow-up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply the Mountain Pass Lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz [Z. Angew. Math. Phys. 43: 270–291, 1992] on the existence of entire solutions of the nonlinear Schrödinger equation.

We prove that the maximal operator of the (𝐶, α )-means of quadratical partial sums of double Vilenkin–Fourier series is of weak type (1,1). Moreover, the (𝐶, α )-means of a function 𝑓 ∈ 𝐿 1 converge a.e. to 𝑓 as 𝑛 → ∞.

Sufficient conditions are found, under which for 𝑓 ∈ 𝐶([–π, π]) or 𝑓 ∈ 𝐿([–π, π]) the convergence of a sequence of trigonometric polynomials in the norms of these spaces implies the convergence of their conjugates in the same norms.

Uniqueness implies existence results are obtained for solutions of the fourth order ordinary differential equation, 𝑦 (4) = 𝑓(𝑥, 𝑦, 𝑦′, 𝑦″, 𝑦‴), satisfying 5-point, 4-point and 3-point nonlocal boundary conditions.

We give a new proof of the 𝑇(1) theorem for reflexive homogeneous Triebel–Lizorkin spaces , which uses neither maximal functions nor atomic decompositions. The substitute tool is an estimate on translations in the 𝐿 𝑝 (ℓ 𝑞 ) spaces.

We solve the old Kühnau's problem on the exact lower bound in the inverse inequality estimating the dilatation of a univalent function by its Grunsky norm and in the related Ahlfors inequality for Fredholm eigenvalues.

A resonance type theorem is proved, where conditions are given, which imply the non-compactness and a certain estimation of the measure of the non-compactness of operators in weighted ideal and symmetric function spaces. The application of the theorem to some concrete classes of operators is discussed.

We give upper bounds for the order of the elements in a characteristic set of a regular differential ideal or a radical of a finitely generated differential ideal with respect to some specific orderings. We then show how to compute characteristic sets of these ideals using algebraic methods.

Let 𝐾 be a nonempty closed convex subset of a real uniformly convex Banach space 𝐸 and 𝑆, 𝑇 : 𝐾 → 𝐾 two nonexpansive mappings such that 𝐹(𝑆) ∩ 𝐹(𝑇) := {𝑥 ∈ 𝐾 : 𝑆𝑥 = 𝑇𝑥 = 𝑥} ≠ ø. Suppose {𝑥 𝑛 } is generated iteratively by 𝑥 1 ∈ 𝐾, 𝑥 𝑛+1 = (1 – α 𝑛 )𝑥 𝑛 + α 𝑛 𝑆[(1 – β 𝑛 )𝑥 𝑛 + β 𝑛 𝑇𝑥 𝑛 ], 𝑛 ≥ 1, where { α 𝑛 }, { β 𝑛 } are real sequences in [0, 1]. In this paper, we discuss the weak and strong convergence of {𝑥 𝑛 } to some 𝑥* ∈ 𝐹(𝑆) ∩ 𝐹(𝑇).

The Cauchy problem for 𝑛-dimensional systems of linear functional differential equations is studied. Some new theorems on systems of functional differential inequalities are established.

The first boundary value problem with weight is investigated for a general-type second order elliptic type equation degenerating on the whole boundary.

Let 𝑓 be a 2π-periodic function in 𝐿 1 [–π, π] and be its lacunary Fourier series with small gaps. We have estimated Fourier coefficients of 𝑓 if it is of φ∧ 𝐵𝑉 locally. We have also obtained a precise interconnection between the lacunarity in such series and the localness of the hypothesis to be satisfied by the generic function which allows us to the interpolate the results concerning lacunary series and non-lacunary series.

A kind of higher order sub-and super-linear FDE with impulses is studied in this paper. Several criteria on the oscillations of solutions are given. In particular, in the case where the coefficients of equations are positive and continuous functions, we find some suitable impulse functions such that all solutions of the equation are oscillatory under the impulse control.