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

Mukhiddin I Muminov

doi: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)+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 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.

Highlights

1 Introduction Certain functional differential equation of neutral delay type with piecewise constant arguments exists in the form of t+
This paper reports all conditions for the uniqueness, infiniteness and emptiness of n-periodic solutions of ( ) for f with n-periodicity
Some equations having a unique and infinite number of periodic solutions are emphasized as examples to authenticate the incorrectness of uniqueness results that were provided with other studies

Summary

Introduction

Where [·] denotes the greatest integer function, p and q are nonzero constants, and f (t) is a periodic function with positive integer period of n. Many useful methods such as Hale [ ], Fink [ ] and [ ] were developed to study the almost periodic differential equations Such equations have diversified application in the field of biology, neural networks, physics, chemistry, engineering, and so on [ – ]. Some equations having a unique and infinite number of periodic solutions are emphasized as examples to authenticate the incorrectness of uniqueness results that were provided with other studies. 3 2- and 4-periodic solutions we give the uniqueness conditions of periodic solutions of equation ( ) for the cases when f are - and -periodic functions. Let f be a continuous -periodic function and x be a -periodic solution of ( ) It follows from ( ) and -periodicity of x(t) that t+.

We set

Note that n