## 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.

### MSC:

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

### Keywords:

time scales; nonlinear boundary value problems
Full Text:

### References:

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