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First- and second-order dynamic equations with impulse. (English) Zbl 1100.39018
The authors present existence results for discontinuous first- and continuous second-order dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions. First, they consider first-order dynamic equation $$u^\nabla (t)=g(t,u(t)),\quad t\in J\backslash \{t_1\},$$ $$u(t_1^+)=I(u(t_1)),$$ $$B(u(0), u(T))=0,$$ where $J=[0,T]\cap \mathbb{T}$, $\mathbb{T}$ is a time scale with $0, t_1, T\in \mathbb{T}$ and $0<t_1<T$, $I: \mathbb{R}\to \mathbb{R}$ is continuous and nondecreasing, and $B: \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is continuous and for each $x\in [\alpha(0), \beta(0)], B(x, \cdot)$ is non-increasing, where $\alpha, \beta$ are lower and upper solutions of the above equation with $\alpha(t)\leq \beta(t)$ for all $t\in J$. For the discontinuous case with $g$($g$ is a Carathéodory-type function), they prove that the impulsive equation has a solution $u$ such that $u\in [\alpha, \beta]$ by using upper and lower solutions and Schauder’s fixed point theorem. Secondly, the authors are concerned with second-order dynamic equations $$y^{\Delta\Delta}(t)=f(t,y^\sigma(t)),\quad t\in \mathbb{T}^k\equiv [a,b]\setminus \{t_1\},$$ $$L_1(y(a),y^\Delta(a),y(\sigma^2(b)),y^\Delta(\sigma(b)))=0,$$ $$L_2(y(a),y(\sigma^2(b)))=0,$$ $$y(t_1^+)-y(t_1^-)=r_1,$$ $$y^\Delta (t_1^+)-y^\Delta (t_1^-)=I(y(t_1), y^\Delta (t_1^-)),$$ where $t_1\in \mathbb{T}$ with $a<t_1<b$ and $t_1$ right dense, $r_1\in \mathbb{R}$, $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is continuous. Assuming the existence of a lower solution $\alpha$ and an upper solution $\beta$ with $\alpha\leq \beta$ on $\mathbb{T}$, they prove that the above impulsive boundary value problem has a solution $u$ such that $u\in [\alpha, \beta]$ under $L_1, L_2$ and $I$ satisfying some appropriate monotone conditions.

##### MSC:
 39A12 Discrete version of topics in analysis
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