Multiple periodic solutions for resonant difference equations
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RESEARCH
Open Access
Multiple periodic solutions for resonant difference equations Jianming Zhang, Shuli Wang* , Jinsheng Liu and Yurong Cheng *
Correspondence: [email protected] College of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, P.R. China
Abstract In this paper, we study the existence of multiple periodic solutions for nonlinear second-order difference equations with resonance at origin. The approach is based on critical point theory, minimax methods, homological linking and Morse theory. Keywords: difference equations; periodic solutions; homological linking; critical group; Morse theory
1 Introduction In this paper, we consider the existence of multiple periodic solutions for the following nonlinear difference equations:
(P)
– u(k – ) = λm u(k) + f (k, u(k)), u() = u(N), u() = u(N + ),
k ∈ Z[, N],
where N > is a fixed integer, u(k) = u(k + ) – u(k), u(k) = (u(k)), f (k, ·) : R → R is a differential function satisfying f (k, ) = ,
k ∈ Z[, N],
(.)
and λm is the m + th eigenvalue of the linear periodic boundary value problem – u(k – ) = μu(k), k ∈ Z[, N], u() = u(N), u() = u(N + ).
(P )
Since (.) implies that (P) possesses a trivial periodic solution u ≡ , we are interested in finding nontrivial periodic solutions for (P). It follows from [] that all the eigenvalues of (P ) are μk = sin kπ , k ∈ Z[, N – ]. Thus μ = , μj = μN–j for j ∈ Z[, N ], where N N =
N– N
if N is odd, if N is even.
For the convenience of later use, we denote by = λ < λ < · · · < λN the distinct eigenvalues of (P ). Moreover, if N is odd then all eigenvalues of (P ) are multiplicity two except λ , © 2014 Zhang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zhang et al. Advances in Difference Equations 2014, 2014:236 http://www.advancesindifferenceequations.com/content/2014/1/236
Page 2 of 14
and let φ , φ , ψ , . . . , φ N , ψ N be the corresponding orthonormal eigenvectors; if N is even then all eigenvalues are multiplicity two except λ and λN , and let φ , φ , ψ , . . . , φN – , ψN – , φN be the corresponding orthonormal eigenvectors. Now we establish the variational framework associated with (P). Set T f(u) = f , u() , f , u() , . . . , f N, u(N) ,
T u = u(), u(), . . . , u(N) , and ⎛
⎜ ⎜– ⎜ ⎜. A = ⎜ .. ⎜ ⎜ ⎝ –
– .. .
– .. .
··· ···
.. . –
··· ···
.. . –
⎞ – ⎟ ⎟ ⎟ .. ⎟ . . ⎟ ⎟ ⎟ –⎠ N×N
Then we can rewrite (P) and (P ) as Au = λm u + f(u) and
Au = μu,
respectively. √ Let E = RN with inner product u, v = N u, u. Then k= u(k)v(k) and norm u = = λ u ≤ Au, u ≤ λN u , For p ≥ , define up = ( that
N
u ∈ E.
p /p k= |u(k)| ) ,
ap u ≤ up ≤ bp u,
(.)
then there exis
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