In this paper, we present the application of the homotopy analysis method for solving integrodifferential equations. In this method, a series is created, the successive elements of which are determined by calculating the appropriate integral of the previous element. In this elaboration, we prove that, if this series is convergent, then its sum is the solution of the objective equation. We formulate and prove the sufficient condition of this convergence, and we give also the estimation of error of an approximate solution obtained by taking the partial sum of the considered series. Moreover, we present in this paper the example of using the investigated method for determining the vibrations of the freely supported reinforced concrete beam as well as for solving the equation of movement of the electromagnet jumper mechanical system.

: Aim of this short paper is to construct in any infinite-dimensional Hilbert space a series with terms tending to zero such that some of its rearrangements possess the discrete set of limit points.

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Cz?stochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

In this paper, with reference to the previous work [WITUŁA, R.—SŁOTA, D.: δ-Fibonacci numbers, Appl. Anal. Discrete Math. 3 (2009), 310–329] concerning the, so called, δ-Fibonacci numbers, the concepts of δ-Lucas numbers, δ-Fibonacci and δ-Lucas polynomials are introduced. There are discussed the basic properties of such objects, as well as their applications, especially for description of certain polynomials and identities of algebraic and trigonometric type. Many from among these identities describe the binomial transformations of the respective integer sequences and polynomials. Similarly as for δ-Fibonacci numbers, also for δ-Lucas numbers some attractive identities–bridges are obtained, connecting these numbers in practice with every sequence of integer numbers.