Periodic solutions of nonlinear second-order difference equations
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We establish conditions for the existence of periodic solutions of nonlinear, second-order difference equations of the form y(t + 2) + by(t + 1) + cy(t) = f (y(t)), where c = 0 and f : R → R is continuous. In our main result we assume that f exhibits sublinear growth and that there is a constant β > 0 such that u f (u) > 0 whenever |u| ≥ β. For such an equation we prove that if N is an odd integer larger than one, then there exists at least one N-periodic solution unless all of the following conditions are simultaneously satisfied: c = 1, |b| < 2, and N arccos−1 (−b/2) is an even multiple of π. 1. Introduction In this paper, we study the existence of periodic solutions of nonlinear, second-order, discrete time equations of the form
y(t + 2) + by(t + 1) + cy(t) = f y(t) ,
t = 0,1,2,3,...,
(1.1)
where we assume that b and c are real constants, c is different from zero, and f is a realvalued, continuous function defined on R. In our main result we consider equations where the following hold. (i) There are constants a1 , a2 , and s, with 0 ≤ s < 1 such that f (u) ≤ a1 |u|s + a2
∀u in R.
(1.2)
(ii) There is a constant β > 0 such that u f (u) > 0 whenever |u| ≥ β.
(1.3)
We prove that if N is odd and larger than one, then the difference equation will have a N-periodic solution unless all of the following conditions are satisfied: c = 1, |b| < 2, and N arccos−1 (−b/2) is an even multiple of π. Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:2 (2005) 173–192 DOI: 10.1155/ADE.2005.173
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Periodic solutions of nonlinear second-order difference equations
As a consequence of this result we prove that there is a countable subset S of [−2,2] such that if b∈ / S, then
y(t + 2) + by(t + 1) + cy(t) = f y(t)
(1.4)
will have periodic solutions of every odd period larger than one. The results presented in this paper extend previous ones of Etheridge and Rodriguez [4] who studied the existence of periodic solutions of difference equations under significantly more restrictive conditions on the nonlinearities. 2. Preliminaries and linear theory We rewrite our problem in system form, letting x1 (t) = y(t),
(2.1)
x2 (t) = y(t + 1), where t is in Z+ ≡ {0,1,2,3,... }. Then (1.1) becomes
x1 (t + 1) 0 1 = −c −b x2 (t + 1)
x1 (t) 0 + x2 (t) f x1 (t)
(2.2)
for t in Z+ . For periodicity of period N > 1, we must require that
x1 (0) x1 (N) . = x2 (0) x2 (N)
(2.3)
We cast our problem (2.2) and (2.3) as an equation in a sequence space as follows. Let XN be the vector space consisting of all N-periodic sequences x : Z+ → R2 , where we use the Euclidean norm | · | on R2 . For such x, if x = supt∈Z+ |x(t)|, then (XN , · ) is a finite-dimensional Banach space. The “linear part” of (2.2) and (2.3) may be written as a linear operator L : XN → XN , where for each t ∈ Z+ ,
x1 (t + 1) x1 (t) Lx(t) = −A , x2 (t + 1) x2 (t)
(2.4)
the matrix A being
0 1 . −c −b
(2.5)
The “nonlinear part” of (2.2) and (2.3) may be written as a c
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