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Systems of discrete fractional boundary value problems with nonlinearities satisfying no growth conditions. (English) Zbl 1320.39001

Summary: We consider the coupled system of discrete fractional boundary value problems \[ \begin{aligned} & -\Delta^\nu_{\nu-2}x(t)=\lambda_1f(t+\nu-1,y(t+\mu-1)),\quad t\in[0,b+1]_{\mathbb N_0}\\ & -\Delta^\mu_{\mu-2}y(t)=\lambda_2g(t+\mu-1,x(t+\nu-1)),\quad t\in[0,b+1]_{\mathbb N_0}\end{aligned} \]
\[ \begin{aligned} & x(\nu-2)=H_1\left(\sum\limits^n_{i=1}a_iy(\xi_i)\right),\quad x(\nu+b+1)=0\\ & y(\mu-2)=H_2\left(\sum\limits^m_{j=1}b_jx(\zeta_j)\right),\quad y(\mu+b+1)=0, \end{aligned} \] where \(1<\nu\leq 2\) and \(1<\mu\leq 2\). In particular, we consider the case of nonlocal and possibly nonlinear boundary conditions, and demonstrate that this problem has at least one positive solution by imposing growth conditions only on the nonlocal terms \(H_1\) and \(H_2\). Of special note is that no growth conditions of any kind are imposed on the nonlinearities \(f\) and \(g\).

MSC:

39A05 General theory of difference equations
39A12 Discrete version of topics in analysis
39A99 Difference equations
26A33 Fractional derivatives and integrals
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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