Abstract
In this paper, we establish the existence of nonoscillatory solutions to the neutral dynamic equation \t\t\t[x(t)−∫abp(t,η)x(g(t,η))Δη]Δ+∫cdω(t,ν)x(h(t,ν))Δν=0\\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}$$\\biggl[x(t)- \\int_{a}^{b}p(t,\\eta)x\\bigl(g(t,\\eta)\\bigr)\\Delta \\eta \\biggr]^{\\Delta }+ \\int_{c}^{d}\\omega(t,\\nu)x\\bigl(h(t,\\nu)\\bigr)\\Delta \\nu=0 $$\\end{document} on a time scale mathbb{T}. Some examples are given to illustrate the main results.
Highlights
1 Introduction In this paper, we will consider the existence of nonoscillatory solutions to the first-order neutral dynamic equation of the form b d x(t) – p(t, η)x g(t, η) η + ω(t, ν)x h(t, ν) ν = 0 (1.1)
Wang and Wu [5] established some sufficient conditions for the existence of positive solutions of the delay equation x(t) + p(t)x g(t) + q(t)x h(t) = 0
Ux + Sy ∈ X for any x, y ∈ X. (2) For x, y ∈ X and t ∈ T, we have b (Ux)(t) – (Uy)(t) ≤ p(t, η) η x – y ≤ α x – y, a which implies that U is a contraction mapping on X. (3) We show that S is completely continuous
Summary
We will consider the existence of nonoscillatory solutions to the first-order neutral dynamic equation of the form b d x(t) – p(t, η)x g(t, η) η + ω(t, ν)x h(t, ν) ν = 0. There has been much research activity concerning the nonoscillation of solutions of various equations on time scales, and we refer the reader to [1–4]. Wang and Wu [5] established some sufficient conditions for the existence of positive solutions of the delay equation x(t) + p(t)x g(t) + q(t)x h(t) = 0. Zhu and Wang [6] established the existence of nonoscillatory solutions to the neutral equation x(t) + p(t)x g(t) + f t, x h(t) = 0
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