Continuous and discrete boundary value problems on the infinite interval: existence theory

Abstract

This paper presents existence criteria for continuous and discrete boundary value problems on the infinite interval, using the notion of upper and lower solution.

Non‐linear boundary value problems on the semi‐infinite interval: an upper and lower solution approach

“…where a = 1 -4• Po In terms of the new variable, the problem takes the form w"(z) + r^ vv' (z) = 0, 0 < z < oo, H<0)=1, w<oo) = 0, and note that 0 < a < 1. We remark here that the theory in [3] does not apply since /: [0, oo) x R 2 -> R (here /(?,>', u) = M/(1 -av)' /2 ) is not continuous (indeed/is not defined when y -I/a), and p 2 q (here /; = 1, q(l) = 2;) is not bounded on [0, oo). In [6], this problem was discussed using the shooting method technique.…”

Nonlinear Boundary Value Problems on Semi-Infinite Intervals using Weighted Spaces: An Upper and Lower Solution Approach

“…In the literature there are several existence results for (1/p)(py ) + Φ(t) f (t, y) = 0, t ∈ (a, +∞) with y(a) = 0, lim t→+∞ p(t)y (t) = 0; see [3,4,6,8,9,10,11,12,17,21,22,23,24,25] and the references therein. In [5,7] the authors discuss the existence of bounded solution for the equation (1/p)(py ) + Φ(t)f (t, y, py ) = 0, t ∈ (a, +∞) with some initial value condition. We refer the reader also to [1,2,13,14,15,16,18,17,18,19,20].…”

Unbounded solutions of second order discrete BVPs on infinite intervals

“…Later, similar methods were used for the existence of solutions to such discrete BVP and on the time scales, see [1,7]. In [25], Y. Tian, C. C. Tisdell and Weigao Ge established the existence of three (bounded) solutions of the discrete boundary value problem…”

“…Recently, the existence of linear and nonlinear discrete boundary value problems has been studied by many authors. We refer here to some works using the upper and lower solutions method, e.g., see [3,10,11,13,14,15,18,20,21,24,25,28,29] for finite interval problems, and [1,5,7,27] for infinite interval problems. Discrete infinite interval problems have also been studied by several other methods in [2,4,6,8,9,12,16,17,19,21,22,26].…”