Volume 24, Issue 1

Recently, the notion of positive linear operators by means of basic (or q -) Lagrange polynomials and 𝒜{\mathcal{A}}-statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means of q -Lagrange polynomials and A -statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911–6918]. In our present investigation, we introduce a certain deferred weighted 𝒜{\mathcal{A}}-statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1, t and t2{t^{2}} defined on a Banach space C⁢[0,1]{C[0,1]} for a sequence of (presumably new) positive linear operators based upon (p,q){(p,q)}-Lagrange polynomials. Furthermore, we investigate the deferred weighted 𝒜{\mathcal{A}}-statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.

Variational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. Some of these results apply to linear equations of a general form and of arbitrary orders or having a symmetry algebra of arbitrary dimension.

Under investigation in this paper is a two-dimensional Korteweg de Vries model, which is a spacial extension of the Korteweg de Vries model. An infinite number of nonlocal conservation laws are given which indicate the integrability of this model. Exact soliton solutions are then respectively derived by means of the multiple exp-function method.

This paper deals with a class of the stationary nonlinear Schrödinger Systems in ℝN{\mathbb{R}^{N}}. Our main tools are based on variational methods. More precisely, by using Ekeland’s Variational Principle, the Mountain Pass Theorem and the Nehari Manifold Method, an existence result is established.

Using the definitions of μ-th order lower and upper directional derivatives of vector-valued functions, introduced in Rahmo and Studniarski (J. Math. Anal. Appl. 393 (2012), 212–221), we provide some necessary and sufficient conditions for strict local Pareto minimizers of order μ for optimization problems where the partial order is introduced by a pointed polyhedral cone with non-empty interior.

This paper is devoted to study the global existence of solutions of the hyperbolic Dirichlet equation ut⁢t=L⁢u+f⁢(x,t) in ⁢ΩT=Ω×(0,T),u_{tt}=Lu+f(x,t)\quad\text{in }\Omega_{T}=\Omega\times(0,T), where L is a nonlinear operator and ϕ⁢(x,t,⋅){\phi(x,t,\cdot\,)}, f⁢(x,t){f(x,t)} and the exponents of the nonlinearities p⁢(x,t){p(x,t)} and μ⁢(x,t){\mu(x,t)} are given functions.

In this paper, we analyze the embedding cell method, an algorithm which has been developed for the numerical homogenization of metal-ceramic composite materials. We show the convergence of the iteration scheme of this algorithm and the coincidence of the material properties predicted by the limit with the effective material properties provided by the analytical homogenization theory in two situations, namely for a one-dimensional linear elasticity model and a simple one-dimensional plasticity model.

In this paper, we present a Hyers–Ulam stability result for the approximately linear recurrence in Banach spaces. An example is given to show the results in more tangible form.

In this paper a complex model describing thermo-elasto-plasticity, phase transitions (PT) and transformation-induced plasticity (TRIP) is studied. The main objective is the analysis of the corresponding initial and boundary value problem (IBVP) considering linearized thermo-elastic dissipation and a viscosity-like regularization.

In this paper, we investigate deficit distributions at ruin in a regime-switching Sparre Andersen model. A Markov chain is assumed to switch the amount and/or respective wait time distributions of claims while the insurer can adjust the premiums in response. Special attention is paid to an operator 𝐋{\mathbf{L}} generated by the risk process. We show that the deficit distributions at ruin during n periods, given the state of the Markov chain at time zero, form a vector of functions, which is the n -th iteration of 𝐋{\mathbf{L}} on the vector of functions being identically equal to zero. Moreover, in the case of infinite horizon, the deficit distributions at ruin are shown to be a fixed point of 𝐋{\mathbf{L}}. Upper bounds for the vector of deficit distributions at ruin are also proven.

We expand on a previous study by offering a generalized wave function associated with the parabolic cylinder function and a connection with a two-particle position-space wave function. We also provide an explicit formula for a wave function associated with a recent work by the present author and M. Wolf.