Curves of positive solutions of boundary value problems on time-scales. (English) Zbl 1077.34022

The authors prove an existence and uniqueness result for the nonlinear boundary value problem on a time scale \[ -[p(t)u^\Delta(t)]^\Delta + q(t) u^\sigma(t) = \lambda f(t,u^\sigma(t)) \] with Dirichlet conditions \(u(a) = u(b) = 0\) and certain sign and growth conditions on the nonlinearity \(f\). The symbols \(\Delta\) and \(\sigma\) are notions from time scales calculus. A time scale is an arbitrary closed subset of the reals. The result shows that there is a \(C^1\) curve \(\lambda \mapsto u\) of solutions, parameterized by \(\lambda \in [0,\lambda_{\max})\), \(\lambda_{\max}\) being the principal eigenvalue of an associated weighted eigenvalue problem. The main ingredients of the proof are a fixed-point theorem in a cone, maximum principle and generalizations of known techniques to the time scales case.


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