On generalized exponentially dichotomous systems of differential equations.

Abstract

In this paper linear homogeneous system of differential equations with continuous coefficients is considered on the semiaxis. The coefficient matrix is estimated at a rate of some positive continuous function, and its integral in the semiaxis diverges. This paper introduces the concept of a generalized exponential dichotomy of solutions of linear homogeneous systems of differential equations with continuous coefficients, which is more general than the concept of e - dichotomous system in the finite measure case. A generalized Lyapunov transformations-linear transformation is used. Installed partition of the space of solutions of a linear system of differential equations in the direct sum of the subspaces of solutions and the corresponding estimates in the subspaces solutions installed as well. Also we have known about below bounding of the mutual inclination of subspace. Generalized upper and lower central exponents of linear systems of differential equations are used to prove the existence of a homogeneous system of linear differential equations, which has the property of generalized exponential dichotomy on the semiaxis. We studied the linear non-homogeneous system of differential equations and using the concept of generalized exponential dichotomy of the solutions of the homogeneous system of linear differential equations with continuous coefficients we obtained the existence of at least one bounded solution of an inhomogeneous system of linear differential equations.