The topological structure of solution spaces of ordinary differential equations

Филиппов, В. В.

doi:10.1070/rm1993v048n01abeh000986

Cited by 21 publications

(30 citation statements)

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“…In 1992, Thieme [75] extended the Markus theorem to a more general case. Another analogue of the differentiable version of the Poincaré-Bendixson Theorem was proved by Filippov [28] in 1993. Using these results, Klebanov in 1997 [42] gave a precise description of orbits for some kind of planar equations and extended the results of Markus in a similar way to Thieme's theorems (the results of Klebanov and Thieme did not cover each other).…”

Section: Some Other Resultsmentioning

confidence: 99%

The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century

Ciesielski

2012

Open Mathematics

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Section: Some Other Resultsmentioning

confidence: 99%

The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century

Ciesielski

2012

Open Mathematics

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“…An apparatus for proving existence theorems for periodic solutions of equations with discontinuous right-hand side and differential inclusions is developed.KEY WORDS: differential equations and inclusions, topology of the solution space, existence theorems, Cauchy problem, periodic solutions.In this paper, we explain how to extend the applicability area of the approach [1] to the existence of periodic solutions for wider classes of ordinary differential equations and inclusions.We shall use the ideas developed in [2][3][4]. Part of the notation from [2] will be used without additional explanations (see also [3,4]).…”

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confidence: 99%

Existence of periodic solutions

Филиппов

1997

Math Notes

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ABSTRACT. An apparatus for proving existence theorems for periodic solutions of equations with discontinuous right-hand side and differential inclusions is developed.KEY WORDS: differential equations and inclusions, topology of the solution space, existence theorems, Cauchy problem, periodic solutions.In this paper, we explain how to extend the applicability area of the approach [1] to the existence of periodic solutions for wider classes of ordinary differential equations and inclusions.We shall use the ideas developed in [2][3][4]. Part of the notation from [2] will be used without additional explanations (see also [3,4]).Earlier, similar results were proved for equations with a continuous right-hand side or equations whose right-hand side satisfies the Carath~odory condition. As a consequence of the Cauchy problem theory that emerges in the context of our general approach (see [2][3][4][5]), our results turn out to be applicable to equations and inclusions with complicated discontinuities in their right-hand sides, including discontinuities in the space variables.w The length of the existence interval for solutions of the Cauchy problemWe assume that the right-hand side of the differential inclusion in question, Stated in this way, the uniqueness of the zero solution turns out to be similar to the following condition:any solution of inclusion (1) whose domain touches the segment [to, tl] can be continued to the entire segment It0, C1].Indeed, under natural restrictions on (1), the violation of this condition implies that inclusion (1) has a solution z(C), C0 _< t < C1, in the domain U0 such that [Iz(c)l[ --* cc as c -, tl or a solution z(t),Therefore, one can apply the considerations used in [6] to study the length of the existence interval for the solution. In order to simplify formulations, we shall not strive for the greatest possible generality.Recall some key notations from [2-4].The symbol Ri(U) denotes the set of all Z C Cs(U) that satisfy the condition if z E Z and a segment I lies in the domain ~r(z) of the function z, then zli E Z.

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“…Recently V.V. Filippov [4] and his students studied the space C v (X) of partial real-valued continuous functions with closed domains in X. (Let us mention that this space originates in [5,6].…”

Section: Introductionmentioning

confidence: 99%

On simultaneous extension of continuous partial functions

Künzi

Shapiro

1997

Proc. Amer. Math. Soc.

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Abstract. For a metric space X let Cvc(X) (that is, the set of all graphs of real-valued continuous functions with a compact domain in X) be equipped with the Hausdorff metric induced by the hyperspace of nonempty closed subsets of X ×R. It is shown that there exists a continuous mapping Φ : Cvc(X) → C b (X) satisfying the following conditions:is a linear positive operator such that Φ(1 K ) = 1 X .

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The topological structure of solution spaces of ordinary differential equations

Cited by 21 publications

References 6 publications

The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century

The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century

Existence of periodic solutions

On simultaneous extension of continuous partial functions

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