New existence results for nonlinear delayed differential systems at resonance

Ruipeng Chen,Xiaoya Li

doi:10.1186/s13660-018-1912-7

Abstract

This paper deals with the first-order delayed differential systems \t\t\t{u′+a(t)u=h(t)v+f(t,u(t−τ(t))),v′+b(t)v=g(t,u(t−τ(t))),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} u'+a(t)u=h(t)v+f(t,u(t-\\tau (t))), \\\\ v'+b(t)v=g(t,u(t-\\tau (t))), \\end{cases} $$\\end{document} where a, b, τ, h are continuous ω-periodic functions with int_{0}^{omega }a(t),dt=0 and int_{0}^{omega }b(t),dt>0; fin C(mathbb{R}times [0,infty ),mathbb{R}) and gin C( mathbb{R}times [0,infty ),[0,infty )) are ω-periodic with respect to t. By means of the fixed point theorem in cones, several new existence theorems on positive periodic solutions are established. Our main results enrich and complement those available in the literature.

Highlights

In the past few decades, there has been considerable interest in the existence of positive periodic solutions of the first-order delayed equation u + a(t)u = λb(t)f u t – τ (t), (1.1)where a, b ∈ C(R, [0, ∞)) are ω-periodic with ω a(t) dt > 0, ω b(t) dt > 0, and τ is a continuous ω-periodic function
By means of the fixed point theorem in cones, several new existence theorems on positive periodic solutions are established
Inspired by the existing literature, we study the existence of positive periodic solutions of the following first-order delayed differential system:

Summary

Introduction

By means of the fixed point theorem in cones, several new existence theorems on positive periodic solutions are established. 1 Introduction In the past few decades, there has been considerable interest in the existence of positive periodic solutions of the first-order delayed equation u + a(t)u = λb(t)f u t – τ (t) , (1.1) Inspired by the existing literature, we study the existence of positive periodic solutions of the following first-order delayed differential system:

Results

Conclusion