Morse theory and multiple periodic solutions of some quasilinear difference systems with periodic nonlinearities
(2017) Georgian Mathematical Journal — Vol. 24, n° 1, p. 103-112 (2017)
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- We consider the system of difference equations Δ ( Δ un - 1 1 - | Δ un - 1 | 2 ) = ∇ Vn ( un ) + h n , un = un + T ( n ∈ ℤ ) , with Δ un = un + 1 - un ∈ ℝ N, Vn = Vn ( x ) ∈ C 2 ( ℝ N , ℝ ), Vn + T = Vn, h n + T = h n for all n ∈ ℤ and some positive integer T, Vn ( x ) is ω i-periodic ( ω i > 0) with respect to each x i ( i = 1 , ... , N) and ∑ j = 1 T h j = 0. Applying a modification argument to the corresponding problem with a left-hand member of p-Laplacian type, and using Morse theory, we prove that if all its solutions are non-degenerate, then the difference system above has at least 2 N geometrically distinct T-periodic solutions.
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Jebelean, P., Mawhin, J., & Şerban, C. (2017). Morse theory and multiple periodic solutions of some quasilinear difference systems with periodic nonlinearities. Georgian Mathematical Journal, 24(1), 103-112. https://doi.org/10.1515/gmj-2016-0075 (Original work published 2017)