## 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.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations
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### References:

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