Positive solutions to nonlinear first-order PBVPs with parameter on time scales. (English) Zbl 1161.34319

Sun, Jian-Ping; Li, Wan-Tong

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 3, 1133-1145 (2009).

Summary: In this paper we consider the following nonlinear first-order periodic boundary value problem with parameter on a time scale \(\mathbb T\)
\[ \begin{cases} x^\Delta(t)+p(t)x(\sigma(t))=\lambda f(x(t)),\quad t\in[0,T]_{\mathbb T},\\ x(0)=x(\sigma(T)),\end{cases} \]
where \(\lambda>0\) is a parameter. For suitable \(\lambda>0\), some existence, multiplicity and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index.

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
39A10	Additive difference equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

time scale; periodic boundary value problem; positive solution; fixed-point index

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