Geodesic
Abstract Volterra operators
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2008), pp. 3-17

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We propose a new general definition of Volterra operators. Several types of evolutionary operators, including Volterra ones in the sense of A. N. Tikhonov, satisfy this definition. For equations with generalized Volterra operators we introduce the notions of local, global, and maximally extended solutions. For solutions to nonlinear equations we formulate the existence, uniqueness, and extendability conditions. The theorems proved in this paper imply both known and new results on the solvability of concrete equations. We adduce an example of the application of obtained results to the study of the Cauchy problem for functional differential equations.
Keywords: abstract Volterra operators, Volterra invertibility of operators, extension of solutions, uniqueness of a solution, functional differential equations.
Mots-clés : local solutions 
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     author = {E. S. Zhukovskii and M. J. Alves},
     title = {Abstract {Volterra} operators},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--17},
     publisher = {mathdoc},
     number = {3},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2008_3_a0/}
}
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E. S. Zhukovskii; M. J. Alves. Abstract Volterra operators. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2008), pp. 3-17. http://geodesic.mathdoc.fr/item/IVM_2008_3_a0/