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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 Δ (t)] Δ +q(t)u σ (t)=λf(t,u σ (t))

with Dirichlet conditions u(a)=u(b)=0 and certain sign and growth conditions on the nonlinearity f. The symbols Δ and σ 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 λu of solutions, parameterized by λ[0,λ max ), λ 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:
34B15Nonlinear boundary value problems for ODE
39A12Discrete version of topics in analysis