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Admissible spaces for a first order differential equation with delayed argument

Nina A. Chernyavskaya, Lela S. Dorel, Leonid A. Shuster (2019)

Czechoslovak Mathematical Journal

We consider the equation - y ' ( x ) + q ( x ) y ( x - ϕ ( x ) ) = f ( x ) , x ∈ ℝ , where ϕ and q ( q ≥ 1 ) are positive continuous functions for all x ∈ ℝ and f ∈ C ( ℝ ) . By a solution of the equation we mean any function y , continuously differentiable everywhere in ℝ , which satisfies the equation for all x ∈ ℝ . We show that under certain additional conditions on the functions ϕ and q , the above equation has a unique solution y , satisfying the inequality ∥ y ' ∥ C ( ℝ ) + ∥ q y ∥ C ( ℝ ) ≤ c ∥ f ∥ C ( ℝ ) , where the constant c ∈ ( 0 , ∞ ) does not depend on the choice of f .

Algebraic methods for solving boundary value problems.

Lucas Jódar Sánchez (1986)

Stochastica

By means of the reduction of boundary value problems to algebraic ones, conditions for the existence of solutions and explicit expressions of them are obtained. These boundary value problems are related to the second order operator differential equation X(2) + A1X(1) + A0X = 0, and X(1) = A + BX + XC. For the finite-dimensional case, computable expressions of the solutions are given.

An algebraic approach for solving boundary value matrix problems: existence, uniqueness and closed form solutions.

Lucas A. Jódar Sanchez (1988)

Revista Matemática de la Universidad Complutense de Madrid

In this paper we show that in an analogous way to the scalar case, the general solution of a non homogeneous second order matrix differential equation may be expressed in terms of the exponential functions of certain matrices related to the corresponding characteristic algebraic matrix equation. We introduce the concept of co-solution of an algebraic equation of the type X^2 + A1.X + A0 = 0, that allows us to obtain a method of the variation of the parameters for the matrix case and further to find...

Asymptotic normality of eigenvalues of random ordinary differential operators

Martin Hála (1991)

Applications of Mathematics

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.

Boundary value problems for coupled systems of second order differential equations with a singularity of the first kind: explicit solutions

Lucas Jódar (1994)

Applications of Mathematics

In this paper we obtain existence conditions and an explicit closed form expression of the general solution of twopoint boundary value problems for coupled systems of second order differential equations with a singularity of the first kind. The approach is algebraic and is based on a matrix representation of the system as a second order Euler matrix differential equation that avoids the increase of the problem dimension derived from the standard reduction of the order method.

Boundary value problems for ODEs

Tadeusz Jankowski (2003)

Czechoslovak Mathematical Journal

We use the method of quasilinearization to boundary value problems of ordinary differential equations showing that the corresponding monotone iterations converge to the unique solution of our problem and this convergence is quadratic.

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