A subset E of a metric space (X, d) is totally bounded if and only if any sequence of points in E has a Cauchy subsequence. We call a sequence (xn) statistically quasiCauchy if st − limn→∞ d(xn+1, xn) = 0, and lacunary statistically quasi-Cauchy if Sθ − limn→∞ d(xn+1, xn) = 0.Weprove that a subset E of ametric space is totally bounded if and only if any sequence of points in E has a subsequence...

In this work, using the concept of natural density, we introduce $\left(
 p,q\right) $-analogue Stancu-beta operators rough $\lambda$%
 -statistically $\rho$-Cauchy convergence on triple sequence spaces. We define
 set Bernstein Stancu beta opeators statistical limit points of
 a spaces and obtain to $\lambda-$statistical convergence
 criteria associated with set. Also,...

For the Cauchy problem$$ {\displaystyle \begin{array}{c}{L}_1u\equiv Lu+\left(b,\nabla u\right)+ cu-{u}_t=0,\kern0.5em \left(x,t\right)\in D,\\ {}u\left(x,0\right)={u}_0(x),\kern1em x\in {\mathbb{R}}^N,\end{array}} $$for a nondivergent parabolic equation with growing lower-order term in half-space \( \overline{D}={\mathbb{R}}^N\times \left[0,\left.\infty \right)\right. \), N ≥ 3, we prove suffi...

The most basic concept is that of an infinite sequence (of real or complex numbers in these notes). For p ∈ Z, let Np = {k ∈ Z : k ≥ p}. An infinite sequence of (complex) numbers is a function a : Np → C. Usually, for n ∈ Np we write a(n) = an, and denote the sequence by a = {an}n=p. The sequence {an}n=p is said to converge to the limit A ∈ C provided that for each > 0 there is an N ∈ Z such th...

In this paper, we define several ideal versions of Cauchy sequences and completeness in quasimetric spaces. Some examples are constructed to clarify their relationships. We also show that: (1) if a quasi-metric space (X, ?) is I-sequentially complete, for each decreasing sequence {Fn} nonempty I-closed sets with diam{Fn} ? 0 as n ?, then ?n?N Fn single-point set; (2) let I be P-ideal, every pre...

let h be an infinite--dimensional hilbert space and k(h) be the set of all compact operators on h. we will adopt spectral theorem for compact self-adjoint operators, to investigate of higher derivation and higher jordan derivation on k(h) associated with the following cauchy-jencen type functional equation 2f(frac{t+s}{2}+r)=f(t)+f(s)+2f(r) for all t,s,rin k(h).

In this paper, we study the Cauchy problem for nonlinear Schrödinger equations with Coulomb potential $${\rm{i}}{\partial _t}u + \Delta u {K \over {\left| x \right|}}u = \lambda \right|^{p - 1}}u$$ $$1 < p \le 5\,\,{\rm{on}}\,\,{\mathbb{R}^3}$$ . Our results reveal influence of long range K∣x∣−1 on existence and scattering theories equations. particular, prove global when is attractive, i.e., K...

The abstract Cauchy problem for the non-stationary bulk queue M(t)|M[k, B]|1 Abstract We derived state probability equations describing the queue M (t)| M [k, B]|1 and formulated as an abstract Cauchy problem to investigate by means of the semi-group theory of bounded linear operators in functional analysis. With regard to the abstract Cauchy problem of this queue, we determined the eigenfuncti...

Definition 1.3 (Banach Spaces). It is easy to show that any convergent sequence in a normed linear space is a Cauchy sequence. However, it may or may not be true in an arbitrary normed linear space that all Cauchy sequences are convergent. A normed linear space X which does have the property that all Cauchy sequences are convergent is said to be complete. A complete normed linear space is calle...