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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 solutions of the Lamé equation

Frits Beukers (2002)

Banach Center Publications

In this paper we give a summary of joint work with Alexa van der Waall concerning Lamé equations having finite monodromy. This research is the subject of van der Waall's Ph. D. thesis [W].

An almost-periodicity criterion for solutions of the oscillatory differential equation y ' ' = q ( t ) y and its applications

Staněk, Svatoslav (2005)

Archivum Mathematicum

The linear differential equation ( q ) : y ' ' = q ( t ) y with the uniformly almost-periodic function q is considered. Necessary and sufficient conditions which guarantee that all bounded (on ℝ ) solutions of ( q ) are uniformly almost-periodic functions are presented. The conditions are stated by a phase of ( q ) . Next, a class of equations of the type ( q ) whose all non-trivial solutions are bounded and not uniformly almost-periodic is given. Finally, uniformly almost-periodic solutions of the non-homogeneous differential equations...

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