Nonoscillatory half-linear difference equations and recessive solutions
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Recessive and dominant solutions for the nonoscillatory half-linear difference equation are investigated. By using a uniqueness result for the zero-convergent solutions satisfying a suitable final condition, we prove that recessive solutions are the “smallest solutions in a neighborhood of infinity,” like in the linear case. Other asymptotic properties of recessive and dominant solutions are treated too. 1. Introduction Consider the second-order half-linear difference equation
∆ an Φ ∆xn
+ bn Φ xn+1 = 0,
(1.1)
where ∆ is the forward difference operator ∆xn = xn+1 − xn , Φ(u) = |u| p−2 u with p > 1, and a = {an }, b = {bn } are positive real sequences for n ≥ 1. It is known that there is a surprising similarity between (1.1) and the linear difference equation
∆ an ∆xn + bn xn+1 = 0.
(1.2)
In particular, for (1.1), the Sturmian theory continues to hold (see, e.g., [15]), and also Kneser- or Hille-type oscillation and nonoscillation criteria can be formulated (see, e.g., [10]). Another concept recently extended to the half-linear case is the concept of a recessive solution (see [11]). We recall (see, e.g., [1, 8, 14]) that in the linear case, if (1.2) is nonoscillatory, then there exists a nontrivial solution u = {un }, uniquely determined up to a constant factor, such that lim n
un = 0, xn
(1.3)
where x = {xn } denotes an arbitrary nontrivial solution of (1.2), linearly independent of u. Solution u is called a recessive solution and x a dominant solution. Both solutions play Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:2 (2005) 193–204 DOI: 10.1155/ADE.2005.193
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Recessive solutions for half-linear equations
an important role in different contexts (see, e.g., [1, 4, 6] and references therein). In the linear case (see, e.g., [1, Chapter 6.3], [3, Theorem 6.8], [8, 14]), recessive solutions u and dominant solutions x can be equivalently characterized by the properties ∞
∞
1 = ∞, an un un+1 ∆un ∆xn < un xn
1 < ∞, an xn xn+1
for large n.
(1.4) (1.5)
As mentioned above, the concept of a recessive solution has been extended in [11] to the nonoscillatory half-linear equation (1.1) by the following way. Consider the generalized Riccati equation
∆wn − bn + 1 −
a wn = 0, Φ Φ∗ an + Φ∗ wn
n
(1.6)
where Φ∗ denotes the inverse function of Φ. If (1.1) is nonoscillatory, in [11] it is proved that there exists a solution w∞ = {wn∞ } of (1.6), such that an + wn∞ > 0 for large n, with the property that for any other solution w = {wn } of (1.6), with an + wn > 0 in some neighborhood of ∞, wn∞ < wn
for large n.
(1.7)
Such solution w∞ is called (eventually) a minimal solution of (1.6) and the solution u = {un } of (1.1), given by ∆un = Φ∗
wn∞ un , an
(1.8)
is called a recessive solution of (1.1). Since (1.1) is nonoscillatory, for any solution x = {xn } of (1.1), the sequence wx = {wnx }, where
wnx =
an Φ ∆xn , Φ xn
(1.9)
is, for large n, a solution of the generalized Riccati equation (1.6) and so property (1.7) coi
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