Asymptotic integration of higher order nonlinear delay differential equations via principal solutions

Article type: Research Article

Authors: Agarwal, Ravi P.^a | Ertem, Türker^{b; *} | Zafer, Ağacık^c

Affiliations: [a] Department of Mathematics, Texas A&M University–Kingsville, Kingsville, TX, USA. E-mail: agarwal@tamuk.edu | [b] Department of Industrial Engineering, Toros University, Mersin, Turkey. E-mail: turker.ertem@toros.edu.tr | [c] Department of Mathematics & Statistics, College of Engineering and Technology, American University of the Middle East, Kuwait. E-mail: agacik.zafer@aum.edu.kw

Correspondence: [*] Corresponding author. E-mail: turker.ertem@toros.edu.tr.

Abstract: We study the asymptotic integration problem for a general class of n-th order nonlinear delay differential equations of the form Lnx(t)=f(t,x(g(t))), where Lnx=x(n)+an−1(t)x(n−1)+⋯+a0(t)x. It is shown that if {u1,u2,…,un} is a set of principal solutions for Lnx=0, then the n-th order nonlinear delay differential equation above has solutions xk with the property that xk(t)=b1u1(t)+⋯+bkuk(t)+o(uk(t)), t→∞ for k=1,2,…,n.

Keywords: Higher order differential equation, delay differential equation, asymptotic integration, fixed point theory, principal solutions

DOI: 10.3233/ASY-161406

Journal: Asymptotic Analysis, vol. 101, no. 4, pp. 241-249, 2017