A discrete fourth-order Lidstone problem with parameters

Keywords: Difference equations Boundary value problems Symmetric Green's function Fixed points Fourth-order Discrete Beam Lidstone Semipositone a b s t r a c t Various existence, multiplicity, and nonexistence results for nontrivial solutions to a non-linear discrete fourth-order Lidstone boundary value problem with dependence on two parameters are given, using a symmetric Green's function approach. An existence result is also given for a related semipositone problem, thus relaxing the usual assumption of nonnegativity on the nonlinear term. 1. Introduction to the fourth-order discrete problem Recently there has been a large amount of attention paid to fourth-order differential equations that arise from various beam problems [4,6,11,16–19,21]. Similarly there has been a parallel interest in results for the analogous discrete fourth-order problem, for example [5,20], and in particular the discrete problem with Lidstone boundary conditions [1,12–15]. In what follows we seek to enrich the discussion found in the above cited literature by exploring two additional aspects of the discrete fourth-order Lidstone problem heretofore not considered, namely explicit dependence on two parameters and a semipositone result (relaxing the nonnegative assumption on the nonlinearity). With this goal in mind, we introduce the nonlinear discrete fourth-order Lidstone boundary value problem with explicit parameters b and k given by