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Existence results for time scale boundary value problem. (English) Zbl 1107.34018

This paper deals with the existence of a positive solution for dynamic equations on a time scale \[ x^{\triangle \triangle}(t) + m(t) f(t, x(\sigma(t))) = 0, \quad t \in [a,b], \] and \[ x^{\triangle \triangle}(t) + m(t) f(t, x(\sigma(t)), x^{\triangle}(\sigma(t))) = 0, \quad t \in [a,b], \] satisfying the boundary conditions \[ \alpha x(a) - \beta x^{\triangle}(b) = 0, \quad \gamma x(\sigma(b)) + \delta x^{\triangle}(\sigma(b)) = 0, \] where \(f\) is continuous, \(m: (a,\sigma(b)) \to [0,\infty)\) is right-dense continuous and may be singular at both \(t=a\) and \(t=\sigma(b)\), \(\gamma \beta + \alpha \delta + \alpha \gamma (\sigma(b) - a) > 0\) and \(\delta = \gamma (\sigma^2(b)-a) \geq 0\). The existence of at least one positive solution is based on an application of the Krasnosel’skiĭ fixed-point theorem. Certain norm estimates needed to apply the cone-theoretic theorem of Krasnosel’skiĭ are obtained from the following assumption on \(f\), \[ f(t,x) \leq p(t) + q(t) x, \quad (t,x) \in [a,\sigma^2(b)] \times [0,\infty), \] where \(q: [a,\sigma(b)] \to [0,\infty)\) is continuous, in the case of the first dynamic equation. Similar condition on \(f\) are imposed for the second dynamic equation. The results in this case are based on in-depth analysis of the Green function.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
39A12 Discrete version of topics in analysis
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