Speciality:
01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
6.02.1940
Keywords: differential equations on networks,
differentiation and integration with respect to the measure,
Green function,
multipoint and non-standard boundary value problems,
oscillation spectral theorems,
Tchebyshev and Markov systems,
positive operators in the space with a cone,
focusing.
Subject:
Differential equations on networks. The concept of differential equation on network as a special type of differential equation (intermediate between ordinary and partial differential equations), the maximum principle established, the solvability of boundary value problems is investigated, Green function is constructed, the complete analugue of the theory of disconjugacy is developed, and for neworks of tree-type the oscillation structure of the spectrum is proved. Multipoint and non-standard boundary value problems. The estimates of Green function are obtained, the technique of normalising of Green function by the sign-changing weight is developed, oscillation theorems are obtained. Positive operators in the space with a cone. Discovered the property which called later the focusing one, the connection of this property with the estimate of the spectral gap is proved. Tchebyshev and Markov systems. The counterexample to the theorem of M. A. Rutman on integral representation of the Markov system is discovered. To repair the possibility of such representation the technique of the differentiation and integration with respect to the fractional measure is developed. Differentiation and integration with respect to the measure. The technique of the differentiation and integration with respect to the fractional measure applying to the analysis of multipoint discontinupus boundary problems and boundary problems on networks is developed. For equations of the second order with differentiation with respect to the measure the complete analogue of Sturm–Liouville theory is obtained.
Main publications:
Pokornyi Yu. V., Krasnoselskii M. A. Nenulevye resheniya uravnenii s silnymi nelineinostyami // Matem.zametki, 1969. T. 5, vyp. 2. S. 253–260.
Pokornyi Yu. V., Lazarev K. P. Nekotorye ostsillyatsionnye teoremy dlya mnogotochechnykh zadach // Differents. uravneniya. 1987. T. 23, # 4. S. 658–670.
Borovskikh A. V., Pokornyi Yu. V. Sistemy Chebysheva–Khaara v teorii razryvnykh yader Kelloga // Uspekhi matem. nauk. 1994. T. 49, # 3. S. 3–42.
Pokornyi Yu. V., Penkin O. M., Pryadiev V. L. O nelineinoi kraevoi zadache na grafe // Differents. uravneniya. 1998. T. 34, # 5. S. 629–637.
Pokornyi Yu. V. Integral Stiltesa i proizvodnye po mere v obyknovennykh differentsialnykh uravneniyakh // Dokl. AN. 1999. T. 364, # 2. S. 167–169.
