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 \[ x^\Delta+ b(t)x= h(t,x),\quad t\in [a, c]_{\mathbb{T}}:= [a,c]\cap \mathbb{T} \] subject to the boundary conditions \[ G(x(a), x(\sigma(c)))= 0,\quad a,c\in\mathbb{T}, \] where \(h : [a,c]_{\mathbb{T}}\times \mathbb{R}^n\to \mathbb{R}^n\) is a continuous nonlinear function, \(t\) is from a so-called “time scale” \(\mathbb{T}\) (which is a nonempty closed subset of \(\mathbb{R}\)), \(x^\Delta\) is the generalized derivative of \(x\), the function \(b: [a,c]_{\mathbb{T}}\to \mathbb{R}\); \(a< c\) are given constants in \(\mathbb{T}\) 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.


39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
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