Abstract: In this chapter, complex vibrations of a flexible Euler-Bernoulli type beam driven by dynamical load and with various type of inputs on its edge are studied. The governing equations include damping terms with damping coefficients e1, e2 associated with deflection w and displacement u, respectively. Damping coefficients e1, e2 and the transversal load coefficients (q 0,E p) serve as control parameters.

Abstract: In the present paper sufficient conditions about the existence of almost periodic solutions of systems of impulsive differential-difference equations are obtained. The investigations are carried out by means piecewise continuous Lyapunov functions and Razumikhin techniques.

Abstract: Algorithms based on Sumudu transform are presented. The algorithms are straightforward to be implemented in computer algebra like Maple. A simple algorithm was implemented in Maple. It is shown with examples that the algorithms are powerful to calculate Maclaurin coefficients directly for a wide range of functions without human interaction.

Abstract: We establish stability inequalities for quasi-linear singularly perturbed twopoint boundary value problems. Our approach uses integral representations of the exact solution in terms of different approximate solutions, like the WKB asymptotic solutions. We prove our inequalities under conditions which are more general than those required for the similar results obtained by Lorenz in 1982 and by Kopteva in 2001. This is illustrated by several examples. Moreover, an example of the nonturning-point case shows that our inequalities are sharper.

Abstract: Department of Pure and Applied Mathematics, University of Fort Hare,Alice, 5700, RSA.E-mail: Gokecha@ufh.ac.zaAbstract We give sucient conditions for the existence of a stable (globally asymp-totically stable), bounded and uniform ultimate bounded solution to a certain fourthorder non-linear di erential equation using a single complete Lyapunov function with-out the use of a signum function or any stringent condition on the nonlinear terms.Theresults include and improve some existing results in literature.

Abstract: In this paper we prove that there exists only one family of classical Hamiltonian systems of two degrees of freedom with invariant plane $\Gamma=\{q_2=p_2=0\}$ whose normal variational equation around integral curves in $\Gamma$ is generically a Hill-Schr\"odinger equation with quartic polynomial potential. In particular, by means of the Morales-Ramis theory, these Hamiltonian systems are non-integrable through rational first integrals.

Abstract: where yk ≈ y(x, tk). Denote by S(t) the evolution operator associated with (1), that is, S(t)x = y(x, t), t > 0, x ∈ D, and let F(h) : D → D be the mapping F(h)(y) = F (h, y). Then the scheme (2) is dynamically consistent with (1) if F(h) and S(t) are topologically conjugate for every t > 0 and h > 0, that is, there exists a homeomorphism φ : D → D such that F(h) ◦ φ = φ ◦ S(t). Numerical schemes for particular systems in one or more dimensions constructed via the nonstandard finite method will be discussed.

Abstract: Multiplicity results are obtained for systems of second order dif- ferential equations with periodic or Sturm-Liouville boundary conditions. Re- sults rely on the notion of strict solution-tube. Different growth conditions of Wintner-Nagumo type are considered.

Abstract: This paper is aimed to formulate Campanato-type interior estimates for solutions of Rothe’s approximate equations to parabolic partial differential systems in non-divergence form, “independently of the approximation”. Combined with the estimates near the boundary in another paper, they will be applied to investigate the uniform Holder continuity of first derivatives of the solution.