Oscillation of a family of q -difference equations

• Published in: Applied mathematics letters Vol. 22; no. 6; pp. 871 - 875
• Main Authors: Baoguo, Jia, Erbe, Lynn, Peterson, Allan
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.06.2009; Elsevier
• Subjects: [formula omitted]-difference equation; Classification; Difference and functional equations, recurrence relations; Exact sciences and technology; Finite differences and functional equations; Mathematical analysis; Mathematics; Nonoscillation; Numerical analysis; Numerical analysis. Scientific computation; Oscillation; Sciences and techniques of general use; q -difference equation; Nonoscillation; Classification; Oscillation; Difference equation; Applied mathematics; q-difference equation
• ISSN: 0893-9659; 1873-5452
• DOI: 10.1016/j.aml.2008.07.014

We obtain the complete classification of oscillation and nonoscillation for the q -difference equation x Δ Δ ( t ) + b ( − 1 ) n t c x ( q t ) = 0 , b ≠ 0 , where t = q n ∈ T = q N 0 , q > 1 , c , b ∈ R . In particular we prove that this q -difference equation is nonoscillatory, if c > 2 and is oscillatory, if c < 2 . In the critical case c = 2 we show that it is oscillatory, if | b | > 1 q ( q − 1 ) , and is nonoscillatory, if | b | ≤ 1 q ( q − 1 ) .