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Summation equations with sign changing kernels and applications to discrete fractional boundary value problems. (English) Zbl 1374.39001

Summary: We consider the summation equation, for \(t\in[\mu-2,\mu+b]_{\mathbb{N}_{\mu-2}}\), \[ y(t)=\gamma_1(t)H_1\left(\sum_{i=1}^{n}a_iy\left(\xi_i\right)\right)+\gamma_2(t)H_2\left(\sum_{i=1}^{m}b_iy\left(\zeta_i\right)\right)+\lambda\sum_{s=0}^{b}G(t,s)f(s+\mu-1,y(s+\mu-1)) \] in the case where the map \((t,s)\mapsto G(t,s)\) may change sign; here \(\mu\in(1,2]\) is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that \(G\) is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions \(H_1\) and \(H_2\). Finally, as an application of the abstract existence result, we demonstrate that by choosing the maps \(t\mapsto\gamma_1(t)\), \(\gamma_2(t)\) in particular ways, we can recover the existence of at least one positive solution to various discrete fractional- or integer-order boundary value problems possessing Green’s functions that change sign.

MSC:

39A05 General theory of difference equations
39A12 Discrete version of topics in analysis
26A33 Fractional derivatives and integrals
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