Geodesic
On one method of studying implicit singular differential inclusions
Vestnik rossijskih universitetov. Matematika, Tome 22 (2017) no. 6, pp. 1314-1320
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We propose a method of studying singular differential inclusions based on the representation of such an inclusion in the form of an operator inclusion in some space of measurable functions depending on the type of a given singularity. To the operator inclusion we apply the results on Lipschitz perturbations of multi-valued covering mappings. The article consists of three sections. In the first one we give the necessary definitions and formulate the theorem [A. Arutyunov, V.A. de Oliveira, F.L. Pereira, E. Zhukovskiy, S. Zhukovskiy // Applicable Analysis, 2015, 94, No 1] on the Lipschitz perturbations of multi-valued covering mappings. In the second section we introduce special metric spaces of integrable functions and obtain sufficient conditions of covering for the multi-valued Nemytskii operator in such spaces. Finally, using the mentioned results, we derive the existence conditions for the Cauchy problem for an implicit singular differential inclusion.
Keywords: implicit singular differential inclusion, Cauchy problem, covering multi-valued mapping, Lipschitz multi-valued mapping.
Mots-clés : existence of solution 
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E. A. Pluzhnikova; A. I. Shindyapin. On one method of studying implicit singular differential inclusions. Vestnik rossijskih universitetov. Matematika, Tome 22 (2017) no. 6, pp. 1314-1320. http://geodesic.mathdoc.fr/item/VTAMU_2017_22_6_a13/

[1] A. I. Shindyapin, “A boundary value problem for a singular equation”, Differ. Uravn., 20:3 (1984), 450–455 (In Russian) | MR | Zbl

[2] A. Shindiapin, “On linear singular functional-differential equations in one functional space”, Abstract and Applied Analysis, 7 (2004), 567–575 | DOI | MR | Zbl

[3] A. I. Shindiapin, E. S. Zhukovskiy, “Covering mappings in the theory of implicit singular differential equations”, Tambov University Reports. Series: Natural and Technical Sciences, 21:6 (2016), 2107–2112 (In Russian)

[4] E. R. Avakov, A. V. Arutyunov, E. S. Zhukovski\v{i}, “Covering mappings and their applications to differential equations not solved with respect to the derivative”, Differential Equations, 45:5 (2009), 627–649 | DOI | MR | Zbl

[5] “On the well-posedness of differential equations unsolved for the derivative”, Differential Equations, 47 (2011), 1541–1555 | DOI | MR | Zbl

[6] E. S. Zhukovskii, E. A. Pluzhnikova, “Covering mappings in a product of metric spaces and boundary value problems for differential equations unsolved for the derivative”, Differential Equations, 49:4 (2013), 420–436 | DOI | MR | Zbl

[7] A. Arutyunov, E. Avakov, B. Gel'man, A. Dmitruk, V. Obukhovskii, “Locally covering maps in metric spaces and coincidence points”, J. Fixed Points Theory and Applications, 5:1 (2009), 105–127 | DOI | MR | Zbl

[8] A. Arutyunov, V. A. de Oliveira, F. L. Pereira, E. Zhukovskiy, S. Zhukovskiy, “On the solvability of implicit differential inclusions”, Applicable Analysis, 94:1 (2015), 129–143 | DOI | MR | Zbl