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Positive solutions of nonlinear \(m\)-point boundary value problems on time scales. (English) Zbl 1185.34143

The authors study the following \(m\)–point boundary value problem \[ u^{\Delta\nabla} + f(t,u) = 0, \quad \beta u(0) - \gamma u^\Delta(0) = 0, \quad u(1) = \sum_{i=1}^{m-2} \alpha_i u(\epsilon_i), \]
where \(t,\epsilon_{i=1\dots m-2} \in [0,1] \cap \mathbb{T}\) (time scale). By applying Krasnoselskii and Legget–Williams type fixed point theorems in cones they show the existence of two resp.three positive solutions under various appropriate conditions. The results are illustrated by two examples.

MSC:

34N05 Dynamic equations on time scales or measure chains
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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