Decaying solutions for discrete boundary value problems on the half line

Abstract

Abstract. Some nonlocal boundary value problems, associated to a class of functional di¤erence equations on unbounded domains, are considered by means of a new approach. Their solvability is obtained by using properties of the recessive solution to suitable half-linear di¤erence equations, a halflinearization technique and a …xed point theorem in Fréchet spaces. The result is applied to derive the existence of nonoscillatory solutions with initial and …nal data. Examples and open problems complete the paper.

Decaying positive global solutions of second order difference equations with mean curvature operator

“…Here, we extend to second order difference equations with Euclidean mean curvature some results on globally positive decaying Kneser solutions stated in [13] for equations with p-Laplacian and b n < 0.…”

“…(3) Similar BVPs concerning the existence of Kneser solutions for difference equations with p-Laplacian operator are considered in [13] when b k < 0 for k ∈ Z + . It should be interesting to extend the solvability of the BVP (1)-(2) to the case in which the sequence b is negative and in the more general situation when the sequence b is of indefinite sign.…”

“…Corollary 3. Let (H i ), i=1,2,3, be satisfied and L c be defined by (13). The BVP (1)-( 2) has at least one solution if there exists λ > 0 such that for k ≥ 1…”

“…Other results in this direction are in [8,9], in which the multiplicity of solutions of certain BVPs involving the p-Laplacian is examined. Finally, in [13,15] for second order equations with p-Laplacian the existence of globally positive decaying Kneser solutions, that is solutions x such that x n > 0, ∆x n < 0 for n ≥ 1 and lim n→∞ x n = 0, is examined.…”

The Solvability of the Discrete Boundary Value Problem on the Half-Line

“…More information about properties of Fredholm operators can be found in [13,15,16] and the references therein. Many authors have considered a discrete version of (1) or its generalizations using different tools, see for example [2,[17][18][19][20][21] and the references therein. Lian et al,in [19], established the sufficient conditions for the existence of one and three solutions of the following problem:…”