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Systems of semipositone discrete fractional boundary value problems. (English) Zbl 1319.39002

Summary: We consider the existence of at least one positive solution to the discrete fractional system \[ \begin{cases} -\Delta^{\nu_{1}}y_1(t)=\lambda_1f_1(t+\nu_1-1, y_1(t+\nu_1-1), y_2(t+\nu_2-1)), \quad t\in [1,b+1]_{\mathbb N}, \\ -\Delta^{\nu_{2}}y_2(t)=\lambda_2f_2(t+\nu_2-1, y_1(t+\nu_1-1), y_2(t+\nu_2-1)), \quad t\in [1,b+1]_{\mathbb N}, \\ y_1(\nu_1-2)=y_1(\nu_1+b+1)=0, \\ y_2(\nu_2-2)=y_2(\nu_2+b+1)=0, \end{cases} \] where \(\nu_1, \nu_2\in (1,2]\). Due to the fact that \(f_1\) and \(f_2\) are allowed to be negative for some values of \(t, y_1\), and \(y_2\), we consider here the semipositone problem. In addition to discussing conditions under which this problem is guaranteed to have at least one positive solution for sufficiently small values of \(\lambda_1, \lambda_2>0\), we provide an example to illustrate the use of our results.

MSC:

39A10 Additive difference equations
26A33 Fractional derivatives and integrals
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