Differential Equations with Impulse Effects: Multivalued by Nikolai A. Perestyuk

By Nikolai A. Perestyuk

This monograph is an creation to the speculation of standard differential equations with bounce stipulations at discrete moments of time. From the contents: Pulse differential equations and inclusions Linear platforms with multivalued trajectories approach to averaging in structures with pulse motion Averaging of differential inclusions Differential equations with discontinuous right-hand facet

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Extra info for Differential Equations with Impulse Effects: Multivalued Right-hand Sides with Discontinuities

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X0 y0 i ; i / and i D max. x0 y0 i ; i /. Proof. x/. x. x. x. x. x. x. iy0 ; y0 // t0 < ÄL x0 i

X/ and guarantee that the conditions of Lemma 6 are satisfied. t; y0 / of Eqs. 29). We take an arbitrary " > 0 and choose ı1 > 0 so small that max kx0 y0 kÄı1 for all i such that j x0 i y0 i j <" x0 i 2 Œt0 ; t0 C T . We choose ı as follows:   à 1 à L LT ı D min ı1 ; " 1 C e ; 1 NC where p is the number of points ix0 that belong to the interval Œt0 ; t0 C T . x0 /j > ". Theorem 5 is proved. 29) that is not inherent in the solutions of equations subjected to pulse action at fixed times. In Eqs.

17) is not true. Theorem 5 is proved. ij Remark 1. 13) belong to the domain D on the interval Œt0 ; T  together with their -neighborhoods. x/ tioned conditions are not equivalent. D are redundant. Note that the aforemen- Remark 1 is also valid for the subsequent theorems. Remark 2. x/W D ! x/W D ! 13) is weakly extendable. x/ are not continuously differentiable.

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