Discrete Second Order Boundary Value Problems

Ravi P. Agarwal, Donal O’Regan and Patricia J. Y. Wong
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Ravi P. Agarwal: National University of Singapore, Department of Mathematics
Donal O’Regan: National University of Ireland, Department of Mathematics
Patricia J. Y. Wong: Nanyang Technological University, Division of Mathematics

Chapter Chapter 17 in Positive Solutions of Differential, Difference and Integral Equations, 1999, pp 261-278 from Springer

Abstract: Abstract Let a, b (b > a) be nonnegative integers. We define the discrete interval [a, b] = {a, a + 1,..., b}. All other intervals will carry its standard meaning, e.g. [0, ∞) denotes the set of nonnegative real numbers. The symbol ∆ denotes the forward difference operator with step size 1, i.e. Δy(k) = y(k + 1) − y(k). Further for a positive m, Δ m is defined as Δ m y(k) = Δ m −1(Δy(k)). In this chapter we shall study positive solutions of the second order discrete boundary value problem 17.1 $$\begin{array}{*{20}{c}} {{{\Delta }^{2}}y(k - 1) + \mu f(k,y(k)) = 0,\quad k \in [1,T]} \\ {y(0) = 0 = y(T + 1)} \\ \end{array}$$ where μ > 0 is a constant and T > 0 is a positive integer. In fact, all the results we shall prove in this chapter are the discrete analogs of some of those established in Chapters 3, 4 and 7.

Keywords: Difference Equation; Discrete Analog; Nonnegative Solution; Nonnegative Real Number; Discrete Interval (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-94-015-9171-3_17

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