16.—Square Integrable Solutions of Perturbed Linear Differential Equations

Abstract

SynopsisThis paper is concerned with solutions of the ordinary differential equationwhere ℒ is a real formally self-adjoint, linear differential expression of order 2n, and the perturbed term f satisfiesfor some σ∈[0, 1]. Here λ(·) is locally integrable on [0,∞).In particular it is shown, under circumstances detailed in the text, that (*) possesses solutions in the Hilbert function space L2(0,∞).

On $$L_w^2 $$ -quasi-derivatives for solutions of perturbed general quasi-differential equations

“…Note that (4.1) and Theorem 2.1 imply that all solutions exist on the entire interval [a, b), see [2, Chapter 3], [10], [14] and [15].…”

“…, n and integrating (4.12) we obtain j (t, λ 0 ) ∈ L ∞ (a, b), this completes the proof. We refer to [10], [14] and [16] for more details.…”

“…These results were generalized by Ibrahim in [10] for general ordinary quasi-differential equations of nth order. Our objective in this paper is to extend the results of Ibrahim, Wong and Zettl in [10], [14], [15] and [16] to a general ordinary quasi-differential expression M of order n with complex coefficients and to prove, under suitable conditions on f , that the quasi-derivatives y…”

“…Also, in [14] Wong proved that all solutions of a perturbed linear differential equation belong to L 2 [0, ∞) assuming that all solutions of the unperturbed equation possess the same property. These results were generalized by Ibrahim in [10] for general ordinary quasi-differential equations of nth order.…”

On the Lw2‐boundedness of solutions for products of quasi‐integro differential equations

“…Introduction. In [8,11,15] where e 1 (t) and r 1 (t) are nonnegative continuous functions on [0,b). Our objective in this paper is to extend the results in [4,6,8,9,11,15] to more general class of quasi-integrodifferential equation in the form is the formal adjoint of τ j , j = 1, 2,...,n.…”