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Zbl 1155.34012
Tian, Yu; Ge, Weigao
Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 69, No. 9, A, 2833-2842 (2008). ISSN 0362-546X

Summary: We investigate the following nonlinear first-order three-point boundary value problem on time scale $\Bbb T$: \align x^\Delta(t)+ p(t)x(\sigma(t))&= f(t,x(\sigma(t))), \quad t\in[0,T]_{\Bbb T},\\ x(0)- \alpha x(\xi)&= \beta x(\sigma(T)). \endalign By using several well-known fixed point theorems, the existence of positive solutions is obtained. Besides, the uniqueness results are obtained by imposing growth restrictions on $f$. In particular, Green's function for the above boundary value problem is established.
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B10 Multipoint boundary value problems
34B15 Nonlinear boundary value problems of ODE
39A10 Difference equations

Keywords: time scale; boundary value problem; positive solutions; fixed point

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