Positive solutions of the periodic problems for quasilinear difference equation with sign-changing weight

• Published in: Advances in difference equations 2018-10, Vol.2018 (1), p.1-13, Article 393
• Main Authors: Xu, Man, Ma, Ruyun, He, Zhiqian
• Format: Article
• Language: English
• Subjects: Analysis; Bifurcation; Bifurcations; Continuity (mathematics); Difference and Functional Equations; Difference equations; Functional Analysis; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Periodic problem; Positive solution; Quasilinear difference equation; Sign-changing weight; Spectrum; Weight
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-018-1856-8

We show the existence of positive solutions of the periodic problem of the quasilinear difference equation { − ∇ [ ϕ ( Δ u k ) ] + q k u k = λ g k f ( u k ) , k ∈ T , u 0 = u T , u 1 = u T + 1 , where T = { 1 , 2 , … , T } with integer T ≥ 2 , ϕ ( s ) = s / 1 − s 2 , q = ( q 1 , … , q T ) ∈ R T , q k ≥ 0 for all k ∈ T and q k 0 > 0 for some k 0 ∈ T , g = ( g 1 , … , g T ) ∈ R T changes the sign on T , f is a continuous function, and λ ∈ R is a parameter. The proofs of the main results are based upon bifurcation techniques.