On the method of finding periodic solutions of second-order neutral differential equations with piecewise constant arguments

• Main Author: Mukhiddin I Muminov
• Format: Article
• Language: English
• Published: SpringerOpen 2017-10-01
• Series: Advances in Difference Equations
• Subjects: differential equation; piecewise constant argument; periodic solution
• Online Access: http://link.springer.com/article/10.1186/s13662-017-1396-7

Abstract This paper provides a method of finding periodical solutions of the second-order neutral delay differential equations with piecewise constant arguments of the form x ″ ( t ) + p x ″ ( t − 1 ) = q x ( 2 [ t + 1 2 ] ) + f ( t ) $x''(t)+px''(t-1)=qx(2[\frac{t+1}{2}])+f(t)$ , where [ ⋅ ] $[\cdot]$ denotes the greatest integer function, p and q are nonzero constants, and f is a periodic function of t. This reduces the 2n-periodic solvable problem to a system of n + 1 $n+1$ linear equations. Furthermore, by applying the well-known properties of a linear system in the algebra, all existence conditions are described for 2n-periodical solutions that render explicit formula for these solutions.