Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities

Multiplicity Results for Integral Boundary Value Problems of Fractional Order with Parametric Dependence

“…In some sense, the results given in this work follow similar steps to the ones obtained in [7] and [8] for second order differential and difference equations with periodic boundary value conditions. The main difference in our approach is that in this paper the Green function vanishes at t = 0 and s = 0, 1, which makes the main properties of the Green function used in those references not valid here.…”

Positive homoclinic solutions of n‐th order difference equations with sign‐changing Green's function

“…However, in most of the papers that do not use this theory, the results was mainly dependent of the constant sign of the associated Green's function to the periodic or Dirichlet problems 5,6 in bounded domains and, more recently, for homoclinic solutions of second-order equations. 7,8 In Cabada and Dimitrov 9 and Graef et al, 10 the authors studied the existence of positive solutions to the periodic boundary value problems. In the first one, the authors obtained existence, multiplicity, and nonexistence results for the following nonlinear fourth-order problem with parameter dependence { u(k + 4) + Mu(k) = g(k) (u(k)) + c(k), k ∈ {0, … , T − 1}, u(i) = u(T + i), i = 0, … , 3, in which the Green's function is nonnegative and attains the zero value at some points of its set of definition.…”

Existence of Solutions to Boundary Value Problems for a Fourth-Order Difference Equation

“…Difference equations are widely found in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamical systems, economics, biology, and so on. Many authors were interested in difference equations and obtained some significant results [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

“…By using variational methods and critical point theory, some new criteria are obtained for the existence of a nontrivial homoclinic solution. By Krasnoselskii's fixed point theorems in cones, Cabada and Dimitrov [13] obtained some existence, multiplicity, and nonexistence results for the nonlinear singular and nonsingular fourth-order equation depending on a real parameter…”