Green and generalized Green's functionals of linear local and nonlocal problems for ordinary integro-differential equations


Akhiev S. S.

ACTA APPLICANDAE MATHEMATICAE, vol.95, no.2, pp.73-93, 2007 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 95 Issue: 2
  • Publication Date: 2007
  • Doi Number: 10.1007/s10440-006-9056-z
  • Title of Journal : ACTA APPLICANDAE MATHEMATICAE
  • Page Numbers: pp.73-93

Abstract

A linear, completely nonhomogeneous, generally nonlocal, multipoint problem is investigated for a second-order ordinary integro-differential equation with generally nonsmooth coefficients, satisfying some general conditions like p-integrability and boundedness. A system of three integro-algebraic equations named the adjoint system is introduced for the solution. The solvability conditions are found by the solutions of the homogeneous adjoint system in an "alternative theorem". A version of a Green's functional is introduced as a special solution of the adjoint system. For the problem with a nontrivial kernel also a notion of a generalized Green's functional is introduced by a projection operator defined on the space of solutions. It is also shown that the classical Green and Cauchy type functions are special forms of the Green's functional.