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Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales. (English) Zbl 1155.34012
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:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B10 Nonlocal and multipoint boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 39A10 Additive difference equations
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##### References:
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