Periodic boundary value problems for first-order impulsive dynamic equations on time scales. (English) Zbl 1165.34013
The authors consider the periodic boundary value problem for the first order dynamic equation on time scales
\[ \begin{aligned} &y^\triangle(t) = f(t,y(t)), \quad t \in [0,T]\cap{\mathbb T}, t \neq t_k, k = 1,\ldots,m,\\ &y(t_k^+) - y(t_k^-) = I_k(y(t_k^-)), \quad k = 1,\dots,m,\\ &y(0) = y(\sigma(T)), \end{aligned} \]
where \({\mathbb T}\) is a time scale, \(t_k \in {\mathbb T}\) for \(k = 1,\ldots,m\) are impulse points (\(0 < t_1 < \ldots < t_m < T\)), \(\sigma\) is the jump operator. New sufficient conditions for the existence of extremal solutions to this problem are obtained. The results are obtained by using the lower and upper solutions method and monotone iterative technique.
\[ \begin{aligned} &y^\triangle(t) = f(t,y(t)), \quad t \in [0,T]\cap{\mathbb T}, t \neq t_k, k = 1,\ldots,m,\\ &y(t_k^+) - y(t_k^-) = I_k(y(t_k^-)), \quad k = 1,\dots,m,\\ &y(0) = y(\sigma(T)), \end{aligned} \]
where \({\mathbb T}\) is a time scale, \(t_k \in {\mathbb T}\) for \(k = 1,\ldots,m\) are impulse points (\(0 < t_1 < \ldots < t_m < T\)), \(\sigma\) is the jump operator. New sufficient conditions for the existence of extremal solutions to this problem are obtained. The results are obtained by using the lower and upper solutions method and monotone iterative technique.
Reviewer: Jan Tomeček (Olomouc)
MSC:
| 34B37 | Boundary value problems with impulses for ordinary differential equations |
| 39A10 | Additive difference equations |
Keywords:
time scales; jump operators; impulsive dynamic equations; lower and upper solutions; periodic boundary value problemReferences:
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