recently, mohiuddine and alghamdi introduced the notion of lacunary statistical convergence in a locally solid riesz space and established some results related to this concept. in this paper, some inclusion relations between the sets of statistically convergent and lacunary statistically convergent sequences are established and extensions of a decomposition theorem, a tauberian theorem to the l...

We obtain the maximum principles for the first-order neutral functional differential equation Mx t ≡ x′ t − Sx′ t − Ax t Bx t f t , t ∈ 0, ω , where A : C 0,ω → L∞ 0,ω , B : C 0,ω → L∞ 0,ω , and S : L∞ 0,ω → L∞ 0,ω are linear continuous operators, A and B are positive operators, C 0,ω is the space of continuous functions, and L∞ 0,ω is the space of essentially bounded functions defined on 0, ω ...

This is a sequel to the paper “Large time asymptotics for cubic nonlinear Schrödinger system in one space dimension” by same authors. We continue study Cauchy problem two-component of equations dimension. provide criteria large decay or non-decay $L^2$ small amplitude solutions terms Fourier transforms initial data.

we show that the category of convergence approach spaces is a simultaneously reective and coreective subcategory of the category of latticevalued limit spaces. further we study the preservation of diagonal conditions, which characterize approach spaces. it is shown that the category of preapproach spaces is a simultaneously reective and coreective subcategory of the category of lattice-valued p...

Let (X, d) be a metric space. The goal of these notes is to construct a complete metric space which contains X as a subspace and which is the “smallest” space with respect to these two properties. The resulting space will be denoted by X and will be called the completion of X with respect to d. The hard part is that we have nothing to work with except X itself, and somehow it seems we have to p...

The aim of this short note is to reconsider and to extend a former result of K. Mochizuki [Moc76], [MN96] on the existence of the scattering operator for wave equations with small dissipative terms. Contrary to the approach used by Mochizuki we construct the wave operator explicitly in terms of the parametrix construction obtained by a (simplified) diagonalization procedure, cf. [Yag97]. The me...

In this paper we consider the Cauchy problem on angular cutoff Boltzmann equation near global Maxwillians for soft potentials either in whole space or torus. We establish existence of unique mild solutions $L^p_vL^{\infty}_{T}L^{\infty}_{x}$ with polynomial velocity weights suitably large $p\leq \infty$, whenever initial perturbation weighted $L^p_vL^{\infty}_x$ norm can be arbitrarily but $L^1...

In this paper, we discuss the geometric properties of solution and lower bound estimate of ∆um−1 of the Cauchy problem for a degenerate parabolic equation with periodic source term ut =∆um+ upsint. Our objective is to show that: (1)with continuous variation of time t, the surface ϕ = [u(x,t)]mδq is a complete Riemannian manifold floating in space RN+1and is tangent to the space RN at ∂H0(t); (2...

It was recently proven by De Lellis, Kappeler, and Topalov in [17] that the periodic Cauchy problem for the Camassa-Holm equations is locally well-posed in the space Lip(T) endowed with the topology of H(T). We prove here that the Lagrangian flow of these solutions are analytic with respect to time and smooth with respect to the initial data. These results can be adapted to the higher-order Cam...

1 A Model Diffusion Problem . . . . . . . . . . . . . . . . . . . . 1 x.1 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 1 x.2 Continuous Function . . . . . . . . . . . . . . . . . . . 2 x.3 Normed Vector Space . . . . . . . . . . . . . . . . . . 2 x.4 Square Integrable Function . . . . . . . . . . . . . . . 3 x.5 Inner Product Space . . . . . . . . . . . . . . . . . . . 4 x.6 Cauch...