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Core of a matrix in max algebra and in nonnegative algebra: A survey

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We consider functional-differential equation x ̇((g(t) )= f(t; x(h(t) ) ),t ∈ [0; 1], where function f satisfies the Caratheodory conditions, but not necessarily guarantee the boundedness of the respective superposition operator from the space of the essentially bounded functions into the space of integrable functions. As a result, we cannot apply the standard analysis methods (in particular the fixed point theorems) to the integral equivalent of the respective Cauchy problem. Instead, to study the solvability of such integral equation we use the approach based not on the fixed point theorems but on the results received in [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Coincidence points principle for mappings in partially ordered spaces // Topology and its Applications, 2015, v. 179, № 1, 13–33] on the coincidence points of mappings in partially ordered spaces. As a result, we receive the conditions on the existence and estimates of the solutions of the Cauchy problem for the corresponding functional-differential equation similar to the well-known Chaplygin theorem. The main assumptions in the proof of this result are the non-decreasing function f(t; •) and the existence of two absolutely continuous functions v,w, that for almost each t ∈ [0; 1] satisfy the inequalities v ̇(g(t) )≥f(t; v(h(t) ) ),w ̇(g(t) )≤f(t;w(h(t) ) ). The main result is illustrated by an example.

Keywords

coincidence point of mappings; partially ordered space; functional-differential equation; Cauchy problem; existence of solution; differential inequality theorem

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DOI

10.20310/2686-9667-2019-24-127-272-280

UDC

517.911, 517.929, 517.988.6

Pages

272-280

References

[1] N. Dunford, J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience publishers, New York, London, 1958. [2] N. V. Azbelev, V.P. Maksimov, L. F. Rakhmatullina, Introduction to the theory of functional differential equations, Nauka, Мoscow, 1957, 1991 (In Russian). [3] A. Shindiapin, “On linear singular functional-differential equations in one functional space”, Abstract and Applied Analysis, 179:1 (2015), 13–33. [4] E. A. Pluzhnikova, A. I. Shindyapin, “On one method of studying implicit singular differential inclusions”, Tambov University Reports. Series: Natural and Technical Sciences, 22:6-1 (2017), 1314–1320 (In Russian DOI: 10.20310/1810-0198-2017-22-6-1314-1320). [5] 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 DOI: 10.20310/1810-0198-2016-21-6-2107-2112. [6] T. V. Zhukovskaia, E. S. Zhukovskiy, “About antitone perturbations of covering mappings of ordered spaces”, Tambov University Reports. Series: Natural and Technical Sciences, 21:2 (2016), 371–374 (In Russian DOI: 10.20310/1810-0198-2016-21-2-371-374). [7] E. R. Avakov, A. V. Arutyunov, E. S. Zhukovskii, “Covering mappings and their applications to differential equations unsolved for the derivative”, Differential Equations, 45:5 (2009), 627–649. [8] A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy, “Coincidence points principle for mappings in partially ordered spaces”, Topology and its Applications, 7 (2004), 567–575. [9] A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy, “On coincidence points of mappings in partially ordered spaces”, Doklady Mathematics, 88:3 (2013), 710–713. [10] E. S. Zhukovskiˇı, “About orderly covering mappings and Chaplygin’s type integral inequalities”, Algebra i Analiz, 30:1 (2018), 96–127 (In Russian). [11] E. S. Zhukovskiy, “On ordered-covering mappings and implicit differential inequalities”, Differential Equations, 52:12 (2016), 1539–1556. [12] B. Z. Vulikh, A Short Course in the Theory of Functions of a Real Variable, Nauka, Мoscow, 1973.

Received

2019-05-23

Section of issue

Scientific articles

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