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Zbl 1077.34022
Davidson, Fordyce A.; Rynne, Bryan P.
Curves of positive solutions of boundary value problems on time-scales. (English)
[J] J. Math. Anal. Appl. 300, No. 2, 491-504 (2004). ISSN 0022-247X

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.
[Stefan Hilger (Eichstätt)]

MSC 2000:: *34B15 Nonlinear boundary value problems of ODE
39A12 Discrete version of topics in analysis

Keywords: time scales; nonlinear boundary value problems

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