Periodic boundary value problems for first order difference equations

Abstract

In this paper, we describe a systematic development of the different types of perturbations methods in the theory of differential and integral equations. A special quadratic perturbation of the periodic boundary value problems of second order ordinary differential equations is studied in detail for different aspects of the solutions. An existence theorem is proved under mixed generalized Lipschitz and Carathéodory conditions and the existence results for extremal positive solutions are established for Carathéodory as well as discontinuity conditions. Our results include some known existence results for periodic boundary value problems of second order ordinary nonlinear differential equations as special cases. Mathematics subject classification (2010): 34K10.

Positive Periodic Solutions for First-Order Difference Equations with Impulses

“…ðÞ : Boundary value problems for difference equations and impulsive differential equation have been widely received attentions from many authors (see [1][2][3][4][5][6][7][8][9][10][11][12]) and reference therein. However, as far as we know, the theory of difference equation for boundary value problems (BVPs) with impulses is rather less, there are still lots of work and research that should be done.…”