# Abstract Volterra equation

A functional or functional-differential equation involving abstract Volterra operators (cf. Abstract Volterra operator). The equations are usually of the form $x(t)=(Vx)(t)$, on some interval $[t_0,T)$, or $\dot x(t)=(Vx)(t)$, in which case an initial value must be assigned: $x(t_0)=x_0$. A more natural initial condition would be: $x(t)=x_0(t)$ on $[t_0,t_1)$, $x(t_1)=x_1$, with $x_0(t)$ given. However, this type of initial condition can be reduced to the first one by suitably redefining the operator $V$. This procedure does not pose any problem, at least with regard to general existence theorems. For defining the stability concepts one must rely on the second type of boundary value problem.

Since L. Tonelli [a10] proved an existence theorem for the fundamental equation with an abstract Volterra operator, a good deal of research has been conducted in connection with these equations. A condition occurring in almost all existence results is that of compactness (or complete continuity, cf. also Completely-continuous operator) of the abstract Volterra operator. For a functional-differential equation this compactness condition is not necessary. A Lipschitz-type condition will assure the existence and uniqueness even for Banach-space-valued functions.

For an introduction to this subject see [a1], [a3], [a4], [a5], [a9].

The class of abstract Volterra equations contains as particular cases many classes of functional or functional-differential equations encountered in the literature, such as delay equations, integro-differential equations and integro-partial-differential equations. A general theory of dynamical processes with memory requires the further development of the theory of functional equations with abstract Volterra operators. Numerous applications of abstract Volterra equations can be found in [a2], [a3], [a6], [a7], [a8].

The term "abstract Volterra equation" may also refer to equations involving classical integral operators of Volterra type when the theory is framed in abstract spaces, see [a2], [a7], [a8]. For the case when the definition is based on the fact that the spectrum of the operator involved consists only of the zero point (which is related to the linear case) see [a15].

#### References

[a1] | N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, "Introduction to the theory of functional differential equations" , Nauka (1991) (In Russian) |

[a2] | V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Noordhoff (1976) |

[a3] | A.L. Bugheim, "Introduction to the theory of inverse problems" , Nauka (1988) (In Russian) |

[a4] | C. Corduneanu, "Integral equations and applications" , Cambridge Univ. Press (1991) |

[a5] | C. Corduneanu, "Equations with abstract Volterra operators and their control" , Ordinary Differential Equations and their Applications , Firenze-Bologna (1995) |

[a6] | L. Neustadt, "Optimization (a theory of necessary conditions)" , Princeton Univ. Press (1976) |

[a7] | J. Pruss, "Evolutionary integral equations" , Birkhäuser (1993) |

[a8] | M. Renardy, W.J. Hrusa, J.A. Nohel, "Mathematical problems in viscoelasticity" , Longman (1987) |

[a9] | G. Gripenberg, S.O. Londen, O. Staffans, "Volterra integral and functional equations" , Cambridge Univ. Press (1990) |

[a10] | L. Tonelli, "Sulle equazioni funzionali di Volterra" Bull. Calcutta Math. Soc. , 20 (1929) |

[a11] | V. Volterra, "Opere Matematiche" , 1–3 , Accad. Naz. Lincei (1954–1955) |

[a12] | I.W. Sandberg, "Expansions for nonlinear systems, and Volterra expansions for time-varying nonlinear systems" Bell System Techn. J. , 61 (1982) pp. 159–225 |

[a13] | M. Schetzen, "The Volterra and Wiener theories of nonlinear systems" , Wiley (1980) |

[a14] | A.N. Tychonoff, "Sur les équations fonctionnelles de Volterra et leurs applications à certains problèmes de la physique mathématique" Bull. Univ. Moscou Ser. Internat. , A1 : 8 (1938) |

[a15] | I.C. Gokhberg, M.G. Krein, "Theory of Volterra operators in Hilbert space and its applications" , Nauka (1967) (In Russian) |

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Abstract Volterra equation.

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