Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales. (English) Zbl 1155.34012

Tian, Yu; Ge, Weigao

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 9, 2833-2842 (2008).

Summary: We investigate the following nonlinear first-order three-point boundary value problem on time scale \(\mathbb T\):
\[ \begin{aligned} x^\Delta(t)+ p(t)x(\sigma(t))&= f(t,x(\sigma(t))), \quad t\in[0,T]_{\mathbb T},\\ x(0)- \alpha x(\xi)&= \beta x(\sigma(T)). \end{aligned} \]
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 to nonlinear boundary value problems for ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
39A10	Additive difference equations

Keywords:

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

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