On a class of fourth-order nonlinear difference equations
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We consider a class of fourth-order nonlinear difference equations. The classification of nonoscillatory solutions is given. Next, we divide the set of solutions of these equations into two types: F+ - and F− -solutions. Relations between these types of solutions and their nonoscillatory behavior are obtained. Necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties. 1. Introduction Consider the difference equation
∆ an ∆ bn ∆ cn ∆yn
+ f n, yn = 0,
n ∈ N,
(1.1)
where N = {0,1,2,... }, ∆ is the forward difference operator defined by ∆yn = yn+1 − yn , and (an ), (bn ), and (cn ) are sequences of positive real numbers. Function f : N × R → R. By a solution of (1.1) we mean a sequence (yn ) which satisfies (1.1) for n sufficiently large. We consider only such solutions which are nontrivial for all large n. A solution of (1.1) is called nonoscillatory if it is eventually positive or eventually negative. Otherwise it is called oscillatory. In the last few years there has been an increasing interest in the study of oscillatory and asymptotic behavior of solutions of difference equations. Compared to second-order difference equations, the study of higher-order equations, and in particular fourth-order equations (see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]), has received considerably less attention. An important special case of fourth-order difference equations is the discrete version of the Schr¨odinger equation. The purpose of this paper is to establish some necessary and sufficient conditions for the existence of solutions of (1.1) with special asymptotic properties. Throughout the rest of our investigations, one or several of the following assumptions will be imposed: Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:1 (2004) 23–36 2000 Mathematics Subject Classification: 39A10 URL: http://dx.doi.org/10.1155/S1687183904308083
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On a class of fourth-order nonlinear difference equations
∞ ∞ (H1) ∞ i=1 (1/ai ) = i=1 (1/bi ) = i=1 (1/ci ) = ∞; (H2) y f (n, y) > 0 for all y = 0 and n ∈ N; (H3) the function f (n, y) is continuous on R for each fixed n ∈ N.
In [14] we can find the following existence theorem (some modification of Schauder’s theorem) which will be used in this paper. Lemma 1.1. Suppose Ω is a Banach space and K is a closed, bounded, and convex subset. Suppose T is a continuous mapping such that T(K) is contained in K, and suppose that T(K) is uniformly Cauchy. Then T has a fixed point in K. 2. Main results: existence of nonoscillatory solutions In this section, we obtain necessary and sufficient conditions for the existence of nonoscillatory solutions of (1.1) with certain asymptotic properties. We start with the following Lemma. Lemma 2.1. Assume that (H1) and (H2) hold. Let (yn ) be an eventually positive solution of (1.1). Then exactly one of the following statements holds for all sufficiently large n: (i) yn > 0, ∆yn > 0, ∆(cn ∆yn ) >
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