Volume 22, Issue 3

In the present paper, we consider the inverse problem for the perturbed Stark operator. We obtain necessary and sufficient conditions on the set of values, which could serve as scattering data for the considered operator, and we prove their sufficiency.

We prove the existence of a renormalized solution for the Dirichlet problem associated to the nonlinear elliptic equations div a ( x , u ,∇ u ) - div Φ( u ) + g ( x , u ,∇ u ) = f in Ω, in the general setting of Orlicz spaces, where Φ ∈ 𝒞 0 (ℝ,ℝ N ), the function g has a natural growth with respect to its third argument and satisfies the sign condition while the datum f belongs to W  -1 E M ¯ (Ω). No Δ 2 -condition is needed for the considered N -functions.

The classical Korovkin approximation theory deals with the convergence of a given sequence {Ln}${\lbrace L_{n}\rbrace }$ of positive linear operators on C [ a , b ]. When the sequence of positive linear operators does not converge to the identity operator it may be useful to use some summability methods. In this paper, we study some Korovkin type approximation theorems for the sequences of convolution operators via the Abel method, which is a sequence-to-function transformation. We also deal with the rate of Abel convergence.

In this work, we investigate the existence of solutions for a class of second order impulsive differential equations using either the implicit function theorem or bifurcation techniques by the mean of Krasnosel'ski theorem.

This paper gives the definition and analysis of Abel dynamic equations on a general time scale. As such, the results contain as special cases results for classical Abel differential equations and results for new Abel difference equations. By using appropriate transformations, expressions of Abel dynamic equations of second kind are derived on the general time scale. This also leads to a specific class of Abel dynamic equations of first kind. Finally, the canonical Abel dynamic equation is defined and examined.

For an arbitrary translation invariant basis B which does not differentiate an integral of some function, we construct a singular Lebesgue–Stieltjes measure having non-vanishing means with respect to B at almost every point.

In many applications it is important to be able to extend the (outer) unit normal vector field from a hypersurface to its neighborhood in such a way that the result is a unit gradient field. The aim of this paper is to provide an elementary proof of the existence and uniqueness of such an extension.

In this paper, we show that the category 𝖷𝖬𝗈𝖽/P${\mathsf {XMod}/P}$ of crossed modules over a fixed group P is an exact category. We give a complete proof of the embedding theorem of 𝖷𝖬𝗈𝖽/P${\mathsf {XMod}/P}$ into a set-valued functor category. As a consequence of this theorem, we show that the subcategory of finitely generated free crossed P -modules is dense in the category of crossed P -modules and each crossed P -module is represented as a coend.

This paper is concerned with the existence of positive solutions of an n th-order m -point impulsive boundary value problem. Existence results of at least three positive solutions are established via a fixed point theorem in a cone due to Avery–Peterson. Also, an example is given to illustrate the effectiveness of our result.

In this paper, we introduce a split general quasi-variational inequality problem which is a natural extension of a split variational inequality problem, quasivariational and variational inequality problems in Hilbert spaces. Using the projection method, we propose an iterative algorithm for a split general quasi-variational inequality problem and discuss some special cases. Further, we discuss the convergence criteria of these iterative algorithms. The results presented in this paper generalize, unify and improve many previously known results for quasi-variational and variational inequality problems.

For any three-coloring of the plane, it is shown that there are continuum many triangles of a prescribed type (acute-angled, right-angled, obtuse-angled, isosceles) such that the vertices of each of them carry all three colors.

In this paper, we consider the Schrödinger type operator H2=(-Δℍn)2+V2${H_2=(-\Delta _{\mathbb {H}^n})^2+V^2}$ on the Heisenberg group ℍ n , where Δℍn$\Delta _{\mathbb {H}^n}$ is the sub-Laplacian and the non-negative potential V belongs to the reverse Hölder class B q 1 for q1>Q2${q_1>\frac{Q}{2}}$ and Q > 5, where Q=2n+2${Q=2n+2}$ is the homogeneous dimension of ℍ n . The L p and weak type ( 1 , 1 ) estimates of the Riesz transform ∇ℍn2H2-1/2$\nabla _{\mathbb {H}^n}^2 H^{-1/2}_2$ related to the Schrödinger type operator H 2 are obtained, where ∇ℍn${\nabla _{\mathbb {H}^n}}$ is the gradient operator on ℍ n . Moreover, we show that ∇ℍn2H2-1/2$\nabla _{\mathbb {H}^n}^2 H^{-1/2}_2$ is a Calderón–Zygmund operator if V∈BQ/2${V\in B_{Q/2}}$ and there exists a positive constant C such that V(g)≤Cm(g,V)2${V(g)\le Cm(g,V)^2}$.

Conditions guaranteeing well-posedness of the problem u''=p0(t)u+q0(t)${u^{\prime \prime }=p_0(t)u+q_0(t)}$, u(a)=0${u(a)=0}$, u(b)=0${u(b)=0}$, are established. Here p0,q0:]a,b[→ℝ${p_0,q_0\colon ]a,b[\rightarrow \mathbb {R}}$ are locally Lebesgue integrable functions and may have singularities at t=a${t=a}$ and t=b${t=b}$.

In this paper, we study the spectrum of the operator D ( r ,0,0, s ) on a sequence space b v 0 , the space of all sequences x such that lim n →∞ x n = 0 and ∥ x ∥ = ∑ n =0,...,∞ | x n +1 - x n | < ∞.

Let u be a weak solution of the equation (-Δ) m u = f with Dirichlet boundary conditions in a smooth bounded domain Ω ⊂ ℝ n . In this paper, we obtain some estimates for the Green's function associated to this problem. Moreover, under appropriate conditions on p , we prove some weighted Sobolev a priori estimates for the solution u , where the weight is a power of the distance function. This result extends previous work by R. G. Durán and M. Sanmartino ([Indiana Univ. Math. J. 57 (2008), no. 7, 3463–3478] and [Anal. Theory Appl. 26 (2010), no. 4, 339–349]).

In this paper, we estimate the order of magnitude of the double Fourier coefficients of a function of the class (Λ1,Λ2)BV(p(n)↑∞,ϕ)${(\Lambda ^1,\Lambda ^2)\operatorname{BV}(p(n)\uparrow \infty ,\varphi )}$ over [0,2π]2${[0,2\pi ]^2}$.