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Existence of solutions to first-order dynamic boundary value problems. (English) Zbl 1116.39009
The authors consider the existence of solutions to the first-order dynamic equation of the type $$x^\Delta+ b(t)x= h(t,x),\quad t\in [a, c]_{\bbfT}:= [a,c]\cap \bbfT$$ subject to the boundary conditions $$G(x(a), x(\sigma(c)))= 0,\quad a,c\in\bbfT,$$ where $h : [a,c]_{\bbfT}\times \bbfR^n\to \bbfR^n$ is a continuous nonlinear function, $t$ is from a so-called “time scale” $\bbfT$ (which is a nonempty closed subset of $\bbfR$), $x^\Delta$ is the generalized derivative of $x$, the function $b: [a,c]_{\bbfT}\to \bbfR$; $a< c$ are given constants in $\bbfT$ and $G$ is some known function describing a linear set of boundary conditions. The methods involve novel dynamic inequalities and fixed-point theory to yield new theorems guaranteeing the existence of at least one solution.

MSC:
39A12Discrete version of topics in analysis
34B15Nonlinear boundary value problems for ODE
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