Existence of periodic solutions for a class of ( ϕ 1 , ϕ 2 ) $(\phi _{1},\phi _{2})$ -Laplacian difference system with asymptotically ( p , q ) $(p,q)$ -linear conditions

Autor: Hai-yun Deng, Xiao-yan Lin, Yu-bo He
Jazyk: angličtina
Rok vydání: 2024
Předmět:
Zdroj: Boundary Value Problems, Vol 2024, Iss 1, Pp 1-12 (2024)
Druh dokumentu: article
ISSN: 1687-2770
DOI: 10.1186/s13661-024-01868-w
Popis: Abstract In this paper, we consider a ( ϕ 1 , ϕ 2 ) $(\phi _{1},\phi _{2})$ -Laplacian system as follows: { Δ ϕ 1 ( Δ u ( t − 1 ) ) + ∇ u F ( t , u ( t ) , v ( t ) ) = 0 , Δ ϕ 2 ( Δ v ( t − 1 ) ) + ∇ v F ( t , u ( t ) , v ( t ) ) = 0 , $$\begin{aligned} \textstyle\begin{cases} \Delta \phi _{1} (\Delta u(t-1) )+\nabla _{u} F(t,u(t),v(t))=0, \\ \Delta \phi _{2} (\Delta v(t-1) )+\nabla _{v} F(t,u(t),v(t))=0, \end{cases}\displaystyle \end{aligned}$$ where F ( t , u ( t ) , v ( t ) ) = − K ( t , u ( t ) , v ( t ) ) + W ( t , u ( t ) , v ( t ) ) $F(t,u(t),v(t))=-K(t,u(t),v(t))+W(t,u(t),v(t))$ is T-periodic in t. By using the mountain pass theorem, we obtain that the ( ϕ 1 , ϕ 2 ) $(\phi _{1},\phi _{2})$ -Laplacian system has at least one periodic solution if W is asymptotically ( p , q ) $(p,q)$ -linear at infinity. Our results improve and extend some known works.
Databáze: Directory of Open Access Journals
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