Asymptotic formulas for the solutions of linear differential equations of the second order
We develop an elementary theory of Fourier and Laplace transformations for exponentially decreasing hyperfunctions. Since any hyperfunction can be extended to an exponentially decreasing hyperfunction, this provides simple notions of asymptotic Fourier and Laplace transformations for hyperfunctions, improving the existing models. This is used to prove criteria for the uniqueness and solvability of the abstract Cauchy problem in Fréchet spaces.
In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with Laplacian, where . We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as .
Boundary value problems for ordinary differential equations with random coefficients are dealt with. The coefficients are assumed to be Gaussian vectorial stationary processes multiplied by intensity functions and converging to the white noise process. A theorem on the limit distribution of the random eigenvalues is presented together with applications in mechanics and dynamics.
The asymptotic behaviour of the solutions is studied for a real unstable two-dimensional system , where is a constant delay. It is supposed that , and are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as are given. The method of investigation is based on the transformation of the real system considered to one...
The paper discusses the asymptotic properties of solutions of the scalar functional differential equation of the advanced type. We show that, given a specific asymptotic behaviour, there is a (unique) solution which behaves in this way.