Representation formula for solution of a functional equation with Volterra operator

Elena Litsyn

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

The following functional equation is under consideration,(0.1)L x = f with a linear continuous operator L, defined on the Banach space X00, Σ0, μ0 ; Y0) of functions x0 : Ω0 → Y0 and having values in the Banach space X22, Σ2, μ2 ; Y2) of functions x2 : Ω2 → Y2. The peculiarity of X0 is that the convergence of a sequence xn0 ∈ X0, n = 1, 2, ..., to the function x0 ∈ X0 in the norm of X0 implies the convergence xn0 (s) → x0 (s), s ∈ Ω0, μ0-almost everywhere. The assumption on the space X2 is that it is an ideal space. The suggested representation of solution to (0.1) is based on a notion of the Volterra property together with a special presentation of the equation using an isomorphism between X0 and the direct product X11, Σ1, μ1 ; Y1) × Y0 (here X11, Σ1, μ1 ; Y1) is the Banach space of measurable functions x1 : Ω1 → Y1). The representation X0 = X1 × Y0 leads to a decomposition of L : X0 → X2 for the pair of operators Q : X1 → X2 and A : Y0 → X2. A series of basic properties of (0.1) is implied by the properties of operator Q.

Original languageEnglish
Pages (from-to)1073-1089
Number of pages17
JournalJournal of Mathematical Analysis and Applications
Volume336
Issue number2
DOIs
StatePublished - 15 Dec 2007

Keywords

  • Functional equation
  • Representation formula
  • Volterra operator

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