Existence of Periodic and Subharmonic Solutions for Second-Order p -Laplacian Difference Equations
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Research Article Existence of Periodic and Subharmonic Solutions for Second-Order p-Laplacian Difference Equations Peng Chen and Hui Fang Received 26 December 2006; Accepted 13 February 2007 Recommended by Kanishka Perera
We obtain a sufficient condition for the existence of periodic and subharmonic solutions of second-order p-Laplacian difference equations using the critical point theory. Copyright © 2007 P. Chen and H. Fang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this paper, we denote by N, Z, R the set of all natural numbers, integers, and real numbers, respectively. For a,b ∈ Z, define Z(a) = {a,a + 1,... }, Z(a,b) = {a,a + 1,...,b} when a ≤ b. Consider the nonlinear second-order difference equation
Δ ϕ p Δxn−1
+ f n,xn+1 ,xn ,xn−1 = 0,
n ∈ Z,
(1.1)
where Δ is the forward difference operator Δxn = xn+1 − xn , Δ2 xn = Δ(Δxn ), ϕ p (s) is p-Laplacian operator ϕ p (s) = |s| p−2 s (1 < p < ∞), and f : Z × R3 → R is a continuous functional in the second, the third, and fourth variables and satisfies f (t + m,u,v,w) = f (t,u,v,w) for a given positive integer m. We may think of (1.1) as being a discrete analogue of the second-order functional differential equation
ϕ p (x ) + f t,x(t + 1),x(t),x(t − 1) = 0,
t∈R
(1.2)
which includes the following equation:
c2 y (x) = v y(x + 1) − y(x) − v y(x) − y(x − 1) .
(1.3)
2
Advances in Difference Equations
Equations similar in structure to (1.3) arise in the study of the existence of solitary waves of lattice differential equations, see [1] and the references cited therein. Some special cases of (1.1) have been studied by many researchers via variational methods, see [2–7]. However, to our best knowledge, no similar results are obtained in the literature for (1.1). Since f in (1.1) depends on xn+1 and xn−1 , the traditional ways of establishing the functional in [2–7] are inapplicable to our case. The main purpose of this paper is to give some sufficient conditions for the existence of periodic and subharmonic solutions of (1.1) using the critical point theory. 2. Some basic lemmas To apply critical point theory to study the existence of periodic solutions of (1.1), we will state some basic notations and lemmas (see [5, 8]), which will be used in the proofs of our main results. ∞ , that is, Let S be the set of sequences, x = (...,x−n ,...,x−1 ,x0 ,x1 ,...,xn ,...) = {xn }+−∞ S = {x = {xn } : xn ∈ R, n ∈ Z}. For a given positive integer q and m, Eqm is defined as a subspace of S by
Eqm = x = {xn } ∈ S | xn+qm = xn , n ∈ Z .
(2.1)
For any x, y ∈ S, a,b ∈ R, ax + by is defined by
ax + by = axn + byn
+∞
n=−∞ .
(2.2)
Then S is a vector space. Clearly, Eqm is isomorphic to Rqm , Eqm can be equipped with inner product x, y Eqm =
qm
xj yj,
∀x, y ∈ Eqm ,
(2.3)
∀x ∈ Eqm .
(2.4)
j =1
by which the norm · can be induced by
x
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