Algebraic Systems With Lipschitz Perturbations

Abstract

By using variational methods, the existence of infinitely many solutions for a nonlinear algebraic system with a parameter is established in presence of a perturbed Lipschitz term. Our goal was achieved requiring an appropriate behavior of the nonlinear term f , either at zero or at infinity, without symmetry conditions.

Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix

“…f : R → R is continuous, and G = (g ij ) n×n is an n × n square matrix. The existence of positive solutions for system (1) has been extensively studied in the literature; see [2,3,5,6,10,11,13,15,17,20,21,24,[27][28][29] and the references therein. However, to the best of our knowledge, almost all obtained results require that the coefficient matrix G ≥ 0 or G > 0, where G ≥ 0 if g ij ≥ 0 and G > 0 if g ij > 0 for (i, j) ∈ [1, n] An n × n square matrix G is called a sign-changing coefficient matrix if its elements change the sign.…”

On a discrete elliptic problem with a weight

“…and they established the existence of one solution. Other works on the problem (1.3) can be found in [7,8] where the authors, using variational methods and maximum principle, proved the existence of infinitely many solutions and determined unbounded intervals of parameters such that (1.3) admits an unbounded sequence of solutions.…”