Existence of positive solutions to discrete second-order boundary value problems with indefinite weight

Abstract

Let T > 1 be an integer, T = {1, 2, ..., T}. This article is concerned with the global structure of the set of positive solutions to the discrete second-order boundary value problemswhere r ≠ 0 is a parameter, m : T → R changes its sign, m(t) ≠ 0 for t ∈ T and f : ℝ ℝ is continuous. Also, we obtain the existence of two principal eigenvalues of the corresponding linear eigenvalue problems. MSC (2010): 39A12; 34B18.

Global structure and one‐sign solutions for second‐order Sturm–Liouville difference equation with sign‐changing weight

“…uniformly for k ∈ [1, T ] Z and in every bounded interval of 𝜆. By Equation (15), it is easy to check that…”

“…When the weight function changes signs, the existence and multiplicity of nontrivial solutions for second-order differential problems were studied in. 4,[7][8][9][10] In the difference case, when m(k) ≥ 0, many authors have discussed the existence and multiplicity of solutions for discrete Sturm-Liouville problems (1), can be seen in other studies [11][12][13][14][15] and the references therein. But up to now, for the case that m(k) changes its sign, to the author's knowledge, there is no paper concerned with the unilateral global bifurcation of Equation (1).…”

Positive solutions of a discrete second-order boundary value problems with fully nonlinear term

“…Meanwhile, as a very important method, the bifurcation technique has also been introduced to discuss the discrete problem as (1.2). For example, by using the bifurcation technique, Gao et al [12] studied the continuum of the positive and negative solutions of the boundary value problem (1.2) and they also obtained the existence of positive solutions and negative solutions of (1.2). Meanwhile, Ma et al [27][28][29] and Gao [10] also used the same method to consider different discrete boundary value problems.…”

Existence of three solutions to the discrete fourth-order boundary value problem with four parameters

“…where ∆ denotes the forward difference operator defined by ∆u(k) = u(k + 1) − u(k), ∆ i+1 u(k) = ∆(∆ i u(k)), f, g : [2, T ] Z × R −→ R are two continuous functions, and α, β, λ, µ are real parameters and satisfy : λ > 0, µ > 0 and 1 + (T − 1).T α − + T.(T − 1) 3 .β − > 0, (1.2) where : α − = min(α, 0) and β − = min(β, 0).…”

“…In recent years, a great deal of work has been done in the study of the existence and multiplicity of solutions for discrete boundary value problem. For the background and recent results, we refer the reader to the monographs [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein. In this work , we will examine some applications of the variational methods to study the BVP (1).…”