On boundary value problems for second-order discrete inclusions
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We prove some existence theorems regarding solutions to boundary value problems for systems of second-order discrete inclusions. For a certain class of right-hand sides, we present some lemmas showing that all solutions to discrete second-order inclusions satisfy an a priori bound. Then we apply these a priori bounds, in conjunction with an appropriate fixed point theorem for inclusions, to obtain the existence of solutions. The theory is highlighted with several examples. 1. Introduction The theory of differential inclusions has received much attention due to its versatility and generality. For example, differential inclusions can accurately model discontinuous processes, such as systems with dry friction; the work of an electric oscillator; and autopilot (and other) control systems [8]. When considering these (or other) situations in discrete time, the modeling process gives rise to a discrete (or difference) inclusion, rather than a differential inclusion. In many cases, considering the model in discrete time gives a more precise or realistic description [1]. Let X and Y be two normed spaces. A set-valued map G : X → Y is a map that associates with any x ∈ X a set G(x) ⊂ Y . By CK(E), we denote the set of nonempty, convex, and closed subsets of a Banach space E. We say that G : Rn → CK(Rn ) is upper semicontinuous if for all sequences {ui } ⊆ Rn , {vi } ⊆ Rn , where i ∈ N, the conditions ui → u0 , vi → v0 , and vi ∈ G(ui ) imply that v0 ∈ G(u0 ). Since the upper semicontinuity plays an essential role in this paper, we illustrate this notion by the simple example [5, Example 4.1.1]. Example 1.1. The set-valued map f1 : R → R defined by {0} f1 (t) = [0,1]
for t = 0, for t ∈ R \ {0},
(1.1)
is not upper semicontinuous. On the other hand, the set-valued map f2 : R → R defined Copyright © 2005 Hindawi Publishing Corporation Boundary Value Problems 2005:2 (2005) 153–163 DOI: 10.1155/BVP.2005.153
154
On BVP for second-order discrete inclusions
by [0,1]
f2 (t) =
{0 }
for t = 0, for t ∈ R \ {0},
(1.2)
is upper semicontinuous. For more information about set-valued maps and differential inclusions, see Aubin and Cellina [3], Smirnov [8], or Erbe, Ma and Tisdell [6]. We are interested in the following boundary value problem (BVP) for second-order discrete inclusions:
∆2 y(k − 1) ∈ F k, y(k),∆y(k) ,
k = 1,...,T,
y(0) = A,
y(T + 1) = B, (1.3)
where A,B ∈ Rd are constants and F : {1,...,T } × Rd × Rd → CK(Rd ) is a set-valued (T+2)d to (1.3) is a vector y ¯ = { y(0),..., y(T + 1)} such map. A solution y¯ = { y(k)}T+1 k=0 ∈ R d that each element y(k) ∈ R satisfies the discrete inclusion for k = 1,...,T and the boundary conditions for k = 0 and k = T + 1. In Section 2, we show that under certain conditions on the right-hand side F, all solutions of (1.3) are bounded. The inequalities employed rely on growth conditions on F and on appropriate discrete maximum principles. Section 3 contains the appropriate operator formulations for (1.3) to be considered as a fixed point problem. In Section 4, we apply the
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