Christopher S. Goodrich
Summation equations with sign changing kernels and applications to discrete fractional boundary value problems

Comment.Math.Univ.Carolin. 57,2 (2016) 201-229.

Abstract:We consider the summation equation, for $t\in[\mu-2,\mu+b]_{\mathbb{N}_{\mu-2}}$, \begin{align*} 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)) \end{align*} 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.

Keywords: summation equation; sign-changing kernel; discrete fractional calculus; positive solution; nonlocal boundary condition

DOI: DOI 10.14712/1213-7243.2015.164
AMS Subject Classification: 39A05 39A12 39A99 26A33 47H07