Eventually positive solutions for nonlinear impulsive differential equations with delays

Shao Yuan Huang¹ ; Sui Sun Cheng¹

¹Department of Mathematics Tsing Hua University Hsinchu, Taiwan 30043

Annales Polonici Mathematici, Tome 104 (2012) no. 1, pp. 43-70

Résumé

Several recent oscillation criteria are obtained for nonlinear delay impulsive differential equations by relating them to linear delay impulsive differential equations or inequalities, and then comparison and oscillation criteria for the latter are applied. However, not all nonlinear delay impulsive differential equations can be directly related to linear delay impulsive differential equations or inequalities. Moreover, standard oscillation criteria for linear equations cannot be applied directly since continuous coefficient functions and initial functions are required. Therefore we establish oscillation criteria for linear or nonlinear impulsive equations with piecewise continuous coefficients and initial functions. Our technique is based on transforming our problem into a fixed point problem in Banach spaces, and then establishing comparison theorems. Our results extend, improve and correct some well known results in the literature.

Détail

DOI : 10.4064/ap104-1-4

Keywords: several recent oscillation criteria obtained nonlinear delay impulsive differential equations relating linear delay impulsive differential equations inequalities comparison oscillation criteria latter applied however nonlinear delay impulsive differential equations directly related linear delay impulsive differential equations inequalities moreover standard oscillation criteria linear equations cannot applied directly since continuous coefficient functions initial functions required therefore establish oscillation criteria linear nonlinear impulsive equations piecewise continuous coefficients initial functions technique based transforming problem fixed point problem banach spaces establishing comparison theorems results extend improve correct known results literature