Non-resonant boundary value problems with singular $${\phi}$$ -Laplacian operators

Abstract

Abstract. In this paper, using Leray-Schauder degree arguments, critical point theory for lower semicontinuous functionals and the method of lower and upper solutions, we give existence results for periodic prob-r dt = 0. In particular we show that in this case we have non-resonance, that is periodic problemhas at least one solution for any continuous function e : [0, T ] → R. Then, we consider Brillouin and Mathieu-Duffing type equations for which r(t) ≡ b1 + b2 cos t and b1, b2 ∈ R.Mathematics Subject Classification (2010). 34B15, 34B16, 34C25, 35J20, 35J60.

Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity

“…In contrast with all the above-mentioned references on periodic solutions for indefinite singular equations in the classical case, there are no related results concerning the relativistic operator. Thus, our results represent a reasonable progress in this theory, complementing the literature on the periodic problem with singular φ−Laplacian operator (see for example [2,3,4,5,6,16,18,19,22]). Since no results are available for the relativistic equation, we considered previous works on the classical case in order to get an idea about what kind of conditions should be accurate for our problem.…”

“…In particular, (3) and (4) are equivalent when g behaves as (t − α) −k near α, but not if the singularity is too strong, for example g(t) ∼ e 1/(t−α) .…”

“…Theorem 1.1. Assume that the previous conditions (G1), (G2), (2) and (3) hold and that T < 2 min{t − α, β − t }. Then (1) has a T −periodic solution, provided that h :…”

Periodic Solutions of Second Order Nonlinear Difference Equations with Singularϕ-Laplacian Operator

“…The existence and multiplicity of solutions for (2) subjected to Dirichlet, Robin, periodic, or Neumann boundary conditions have been studied by various methods, such as the method of lower and upper solutions, topological degree theory, and critical point theory; see [4][5][6][7][8] and the references therein. An interesting question is which techniques and theorems regarding the continuous differential equations can be adapted for difference equations (see Kelly and Peterson [9], Agarwal [10], and Bereanu and Mawhin [11,12]).…”