Generalized solutions of ordinary linear differential equations in the Colombeau algebra
In this paper first order systems of linear of ODEs are considered. It is shown that these systems admit unique solutions in the Colombeau algebra .
In this paper first order systems of linear of ODEs are considered. It is shown that these systems admit unique solutions in the Colombeau algebra .
In this paper first order linear ordinary differential equations are considered. It is shown that the Cauchy problem for these systems has a unique solution in , where denotes the Colombeau algebra.
We study the existence of positive solutions of the integral equation in both and spaces, where and . Throughout this paper is nonnegative but the nonlinearity may take negative values. The Krasnosielski fixed point theorem on cone is used.
We study the existence of one-signed periodic solutions of the equations where , is continuous and 1-periodic, is a continuous and 1-periodic in the first variable and may take values of different signs. The Krasnosielski fixed point theorem on cone is used.
From the fact that the unique solution of a homogeneous linear algebraic system is the trivial one we can obtain the existence of a solution of the nonhomogeneous system. Coefficients of the systems considered are elements of the Colombeau algebra of generalized real numbers. It is worth mentioning that the algebra is not a field.
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