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Oscillatory and nonoscillatory solutions for first order impulsive dynamic inclusions on time scales. (English) Zbl 1164.34003
The authors deal with oscillatory properties of the first order dynamic inclusion on time scales $$ \aligned y^{\Delta }(t)&\in F\big (t,y(t)\big ),\ t\in J_{T}:=[t_{0},\infty )\cap \Bbb T, \quad t\not =t_{k},\quad k=1,\dots ,m,\dots , \\ y(t_{k}^{+})&=I_{k}\big (y(t_{k}^{-})\big ),\ y(t_{0})=y_{0} \endaligned \tag {1} $$ where $\Bbb T$ is a time scale which is supposed to be unbounded from above. Under the assumption that upper and lower solutions of (1) are well ordered (i.e. $\alpha \leq \beta $, where $\alpha $, $\beta $ are lower and upper solutions of (1), respectively), conditions on $F$ and $I$ in (1) are given which guarantee that this problem has at least one solution $y$ satisfying $\alpha (t) \leq y(t) \leq \beta (t)$. The proof of the main result of the paper is based on the nonlinear Leray-Schauder alternative.
34A60Differential inclusions
34A37Differential equations with impulses
39A10Additive difference equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
Full Text: EMIS EuDML