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Complex oscillation of a second order linear differential equation with entire coefficients of (α,β)-order

Belaïdi Benharrat, Biswas Tanmay

Silesian Journal of Pure and Applied Mathematics

2021

Vol. 11, iss. 1

1--21

In this paper we study distribution of zeros and growth of solutions of second order linear equations depending on the coeﬃcients of the equation and their (α, β)-order. We obtain results in general form, which considerably extend some results from [21].

Admissible pair of spaces for non-correctly solvable linear differential equations

Chernyavskaya N. A., Dorel L. S., Shuster L. A.

Journal of Applied Analysis

2016

Vol. 22, nr 1

1--14

We consider the differential equation −y′(x)+q(x)y(x)=f(x),x∈R, where f∈Lp (R), p∈[1,∞), and 0≤q∈Lloc1 (R), ∫0−∞q(t)dt=∫∞0q(t)dt=∞, q0(a)=infx∈R∫x+ax−aq(t)dt=0 for any a∈(0,∞). Under these conditions, the above equation is not correctly solvable in Lp (R) for any p∈[1,∞). Let q∗(x) be the Otelbaev-type average of the function q(t), t∈R, at the point t=x; let θ(x) be a continuous positive function for x∈R, and Lp,θ (R)={f∈Llocp (R):∫ ∞−∞|θ(x)f(x)|pdx<∞},∥f∥p,θ:=∥f∥Lp,θ(R) =(∫ ∞−∞|θ(x)f(x)| pdx) 1/p. We show that if there exists a constant c∈[1,∞) such that the inequality c−1q∗(x)≤θ(x)≤cq∗(x) holds for all x∈R, then under some additional conditions for q the pair of spaces {Lp,θ (R);Lp (R)} is admissible for the considered equation.