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Volume 80 - 1978

Contents

Page 2 of 2

Research Article

  • Related oscillation criteria for higher order self-adjoint differential and integro-differential equations§

  • William T. Reid
  • Published online by Cambridge University Press:
    14 November 2011, pp. 333-345

  • Within recent years considerable attention has been devoted to extensions of the classical Sturmian theory of real linear homogeneous differential equations of the second order. In particular, such extensions have included not only self-adjoint systems of differential equations, but also higher order self-adjoint differential and integro-differential equations. For problems in these latter categories, however, only limited attention has been given to detailed application of the general oscillation and comparison criteria. The present paper is devoted to this area, and, in particular, it is shown how existing criteria may be exploited to obtain comparison theorems between equations of different orders. Although the presented results have ready extensions to vector differential and integro-differential equations of higher order, [see, for example, 5,6,7], for simplicity attention is restricted to scalar equations. Section 2 is devoted to the statement of known general criteria of oscillation for self-adjoint equations of higher order, with special applications of these criteria presented in Section 3. Finally, Section 4 sketches the framework of corresponding applications for self-adjoint higher order integro-differential equations.

Front matter

  • PRM volume 80 issue 1-2 Cover and Front matter

  • Published online by Cambridge University Press:
    14 November 2011, pp. f1-f2
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Back matter

  • PRM volume 80 issue 1-2 Cover and Back matter

  • Published online by Cambridge University Press:
    14 November 2011, pp. b1-b2
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Research Article

  • The interlacing of eigenvalues for periodic multi-parameter problems*

  • Patrick J. Browne
  • Published online by Cambridge University Press:
    14 November 2011, pp. 357-362

  • This paper studies a linked system of second order ordinary differential equations

    where xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.

Index

Front matter

  • PRM volume 80 issue 3-4 Cover and Front matter

  • Published online by Cambridge University Press:
    14 November 2011, pp. f1-f2
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Back matter

  • PRM volume 80 issue 3-4 Cover and Back matter

  • Published online by Cambridge University Press:
    14 November 2011, pp. b1-b7
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Page 2 of 2