Positive solution for system of nonlinear first-order PBVPs on time scales. (English) Zbl 1071.34017

Summary: We are concerned with the following system of nonlinear first-order periodic boundary value problems on time scale \(\mathbb{T}\) \[ \begin{gathered} x^\Delta_i(t)+ f_i(t, x_1(\sigma(t)), x_2(\sigma(t)),\dots, x_n(\sigma(t)))= 0,\quad t\in [0,T],\\ x_i(0)= x_i(\sigma(T)),\quad i= 1,2,\dots, n,\end{gathered} \] where \(f_i: [0, T]\times [0,+\infty)^n\to\mathbb{R}\) is continuous and there exists a constant \(M_i> 0\) such that \[ M_i x_i- f_i(t, x_1,x_2,\dots, x_n)\geq 0\quad\text{for }(x_1, x_2,\dots, x_n)\in [0,+\infty)^n,\quad t\in [0,T]. \] Some existence criteria for a positive solution are established by using a fixed-point theorem for operators on cone.


34B15 Nonlinear boundary value problems for ordinary differential equations
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