## Abstract

The following functional equation is under consideration,(0.1)L x = f with a linear continuous operator L, defined on the Banach space X_{0} (Ω_{0}, Σ_{0}, μ_{0} ; Y_{0}) of functions x^{0} : Ω_{0} → Y_{0} and having values in the Banach space X_{2} (Ω_{2}, Σ_{2}, μ_{2} ; Y_{2}) of functions x^{2} : Ω_{2} → Y_{2}. The peculiarity of X_{0} is that the convergence of a sequence x_{n}^{0} ∈ X_{0}, n = 1, 2, ..., to the function x^{0} ∈ X_{0} in the norm of X_{0} implies the convergence x_{n}^{0} (s) → x^{0} (s), s ∈ Ω_{0}, μ_{0}-almost everywhere. The assumption on the space X_{2} 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 X_{0} and the direct product X_{1} (Ω_{1}, Σ_{1}, μ_{1} ; Y_{1}) × Y_{0} (here X_{1} (Ω_{1}, Σ_{1}, μ_{1} ; Y_{1}) is the Banach space of measurable functions x^{1} : Ω_{1} → Y_{1}). The representation X_{0} = X_{1} × Y_{0} leads to a decomposition of L : X_{0} → X_{2} for the pair of operators Q : X_{1} → X_{2} and A : Y_{0} → X_{2}. A series of basic properties of (0.1) is implied by the properties of operator Q.

Original language | English |
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Pages (from-to) | 1073-1089 |

Number of pages | 17 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 336 |

Issue number | 2 |

DOIs | |

State | Published - 15 Dec 2007 |

## Keywords

- Functional equation
- Representation formula
- Volterra operator