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We consider a periodic pseudodifferential operator H = (- Δ)l + A (l > 0) in Rd which satisfies the following conditions: (i) the symbol of H is smooth in x, and (ii) the perturbation A has order smaller than 2l - 1. Under these assumptions, we prove that the spectrum of H contains a half-line. [ABSTRACT FROM AUTHOR]

This aritcle describes a simple alternative unified method of solving nonlinear regular perturbation problems. The procedure is based upon the manipulation of Taylor's approximation for the expansion of the nonlinear term in the perturbed equation. An essential feature of this technique is the relative simplicity used and the associated unified computational procedure that is employed. As such it should be of interest to teachers of applied mathematics courses, particularly those courses which include perturbation methods. One of the merits of this approach is that it leads on naturally to a scheme based on Taylor's expansions and, consequently, allows the regular perturbation method to be introduced into a course much earlier than is currently common. The method is illustrated by implementing it to four perturbation problems, including two algebraic equations and two initial-value problems. The new approach is compared and contrasted with the traditional perturbation scheme in order to demonstrate its relative merit. [ABSTRACT FROM AUTHOR]

In article [Abdel-Aziz, M.R., 2004, Relative perturbation theory for matrix eigenproblems in free vibration analysis (International J. of Computer Mathematics, Vol. 81, No. 4, pp. 505–514) 1], we have introduced bounds for eigenvalues of eigenproblems in free vibration analysis, when the perturbation happens in either the mass matrix or in the stiffness matrix. This article aims to bound the additive and multiplicative relative distances and ,respectively, where ? and µ are eigenvalues and one is the perturbation of the other. These bounds are investigated when the perturbation happens in both the mass and the stiffness matrices. Numerical experiments show that the investigated bounds are satisfied and their values are depend on the element-wise perturbation of the matrix. [ABSTRACT FROM AUTHOR]

A simple, straightforward and efficient method is proposed for the calculation of the period of anharmonic oscillators as an energy series. The approach is based on perturbation theory and the method of Lindstedt— Poincaré. [ABSTRACT FROM AUTHOR]