Some separation criteria and inequalities associated with linear second order differential operators.

*(English)*Zbl 0982.34032
Edmunds, David E. (ed.) et al., Function spaces and applications. Mainly the proceedings of the international conference on function spaces and applications to partial differential equations, New Delhi, India, December 15-19, 1997. New Delhi: Narosa Publishing House. 7-35 (2000).

The authors consider the symmetric second-order differential expression \(M(y)= -(py')+ qy\) defined on \(I= (a,\infty)\), \(a>-\infty\) and \(M_w(y)= W^{-1}M(y)\) in \(L^2\)-space with weight \(W\). \(M\) is assumed to satisfy the so-called minimal properties. The paper gives several new separation criteria (with left endpoint regular and right endpoint \(\infty\)). The first section gives the general idea of separation, its definition and the connection between separation and inequality. Further three significant results from this area which are representative of the type of theorems that have been obtained are stated. In the next section on “The separation criteria” a few theorems related to the main problem are enunciated along with relevant notes. These theorems are proved in the last section. Though the method of proof follows that of W. N. Everitt and M. Gierts [Proc. Lond. Math. Soc., III. Ser. 28, 352-372 (1974; Zbl 0278.34009)], there are some new criteria which are independent of the results of Everitt and Giertz.

For the entire collection see [Zbl 0958.00028].

For the entire collection see [Zbl 0958.00028].

Reviewer: N.D.Sengupta (Mumbai)

##### MSC:

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

34A40 | Differential inequalities involving functions of a single real variable |

##### Keywords:

differential operator; \(L_{\text{Loc}}(I)\) and \(L^2(w,I)\) spaces; separation inequality; limit-point
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\textit{R. C. Brown} et al., in: Function spaces and applications. Mainly the proceedings of the international conference on function spaces and applications to partial differential equations, New Delhi, India, December 15--19, 1997. New Delhi: Narosa Publishing House. 7--35 (2000; Zbl 0982.34032)