Abstract
We introduce the notion of abstract Volterra mapping acting in a partially ordered set. For an equation with such mapping, we define the notions of local, global, and maximally extended solutions and prove a theorem on its solvability. We apply this result to a discontinuous Uryson-type integral equation with respect to a spatiotemporal-dependent phase variable. In particular, such equations generalize a class of “switching” models of the electrical activity in the cerebral cortex.
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Acknowledgment
The work was supported by the Russian Foundation for Basic Research (projects no. 17-41-680975, 17-51-12064, 18-31-00227).
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Burlakov, E.O., Zhukovskiy, E.S. (2020). On Abstract Volterra Equations in Partially Ordered Spaces and Their Applications. In: Pinelas, S., Kim, A., Vlasov, V. (eds) Mathematical Analysis With Applications. CONCORD-90 2018. Springer Proceedings in Mathematics & Statistics, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-030-42176-2_1
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