Positive periodic solutions for high-order differential equations with multiple delays in Banach spaces

Abstract

This paper deals with the existence of positive ω-periodic solutions for nth-order ordinary differential equation with delays in Banach space E of the form Lnu(t)=f(t,u(t−τ1),…,u(t−τm)),t∈R,\\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}$$L_{n}u(t)=f\\bigl(t,u(t-\\tau_{1}),\\ldots,u(t- \\tau_{m})\\bigr),\\quad t\\in\\mathbb{R}, $$\\end{document} where L_{n}u(t)=u^{(n)}(t)+sum_{i=0}^{n-1}a_{i} u^{(i)}(t) is the nth-order linear differential operator, a_{i}inmathbb {R} (i=0,1,ldots,n-1) are constants, f: mathbb{R}times E^{m}rightarrow E is a continuous function which is ω-periodic with respect to t, and tau_{i}>0 (i=1,2,ldots,m) are constants which denote the time delays. We first prove the existence of ω-periodic solutions of the corresponding linear problem. Then the strong positivity estimation is established. Finally, two existence theorems of positive ω-periodic solutions are proved. Our discussion is based on the theory of fixed point index in cones.