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Xxxxxxxxxx, T.Β Xxxxxxxxx-Xxxxxxxxx, X.X., Xxxxx, O., Xxxxxxx, S., Xxxxxxx, A., Xxxxx, N., Xxxxxxx, G., Xxxxxxx, K., Xxxxxx, R., Xxxxxx, S. and Xxxxxx, G., 2017, May. BONSEYES: platform for open development of systems of artificial intelligence. In Proceedings of the computing frontiers conference (pp. 299-304).
Xxxxxxxxxx, T.Β H. Xxx, V.A. Shokarev 2023. This publication has been partially supported by the RUDN University named after Xxxxxxx Xxxxxxx, project no. 002092-0-000 and by the Ministry of Science and Higher Education of the Russian Federation (megagrant agreement No. 000-00-0000-0000). These issues are closely related to the inverse problem of the calculus of variations (IPCV) in the following statement: for a given equation, one needs to construct a functional such that its set of stationary points coincides with the set of solutions to this equation. There is a large number of works devoted to inverse problems of the calculus of variations: for ordinary differ- ential equations and partial differential equations [3], [4], [7], [9], [19], [20], [26], [27], operator equations [6], [21], [22], differential-difference equations [8], [17], [18], stochastic differential equations [23], [24], [25], fractional differential equations [1], [10], [14], [28]. In these works, nonlocal bilinear forms were mainly used to solve the IPCV. [16] for the representation of this equation in the form of Xxxxxxxx-Xxxxxxxxxxxx equations. The method of constructing functionals developed in [3] is extended to a fourth order ordinary differential equation. Local bilinear forms play a significant role in our study. In what follows we use the notation and terminology of [3], [20]. Let π , π be real linear normed spaces. The following definition and theorem will be employed in what follows.
1.1. ([20]) An operator π : π·(π ) β π β π is called potential on a set π·(π ) relative to a local bilinear form Ξ¦(π’; Β·, Β·) : π Γ π β R if there exists a G^ateaux differentiable functional πΉπ : π·(πΉπ ) = π·(π ) β R such that πΏπΉπ [π’, β] = Ξ¦(π’; π (π’), β) βπ’ β π·(π ), ββ β π·(ππ’β² ).
1.1. ([20]) Consider a G^ateaux differentiable operator π : π·(π ) β π β π and a local bilinear form Ξ¦(π’; Β·, Β·) : π Γπ β R such that for all fixed elements π’ β π·(π ), π, β β π·(ππ’β² ) the function π(π) = Ξ¦(π’ + πβ; π (π’ + πβ), π) belongs to the class πΆ [0, 1]. Then the operator π is potential on the convex set π·(π ) relative to Ξ¦ if and only if Ξ¦ (π’; ππ’β² β, π) + Ξ¦β²π’ (β; π (π’), π) = Ξ¦ (π’; ππ’β² π, β) + Ξ¦β²π’ (π; π (π’), β) (1.1) β«οΈ1 for all π’ β π· (π ), β, π β π· (ππ’β² ). Under this condition the potential πΉπ is given by πΉοΏ½οΏ½ [π’] = Ξ¦(π’0 + π(π’ β π’0); π (π’0 + π(π’ β π’0)), π’ β π’0) ππ + πΉπ [π’0], (1.2) where π’0 is a fixed element of π·(π ).
2. On indirect vari...
Xxxxxxxxxx, T.Β H. Xxx, V.A. Xxxxxxxx, On indirect representability of fourth order ordinary differential equation in form of Xxxxxxxx-Xxxxxxxxxxxx equations.
Xxxxxxxxxx, T.Β H. Xxx, V. A. Xxxxxxxx, On indirect representability of fourth order ordinary differential equation in form of XxxxxxxxβOstrogradsky equations, Ufa Mathematical Journal, 2023, Volume 15, Issue 3, 118β128 DOI: 10.13108/2023-15-3-118 Use of the all-Russian mathematical portal Xxxx-Xxx.Xx implies that you have read and agreed to these terms of use xxxx://xxx.xxxxxxx.xx/eng/agreement Download details: IP: 34.136.87.96 April 4, 2024, 17:28:43 XXXX 0000-0000 Ufa Mathematical Journal. Vol. 15. No 3 (2023). P. 118-128. doi:10.13108/2023-15-3-118
Xxxxxxxxxx, T.Β I. (2020), βSustainable development of domestic enterprises in modern economic conditionsβ, Vcheni zapysky TNU imeni V.I. Vernads'koho, vol. 31 (70), no. 2.