Oscillation Criteria For Second-order Neutral Equations With Distributed Deviating Argument

Along with the increasing development of science and technology,in the ?eld of natural science including physics, economies, biology, and medicines ?eld, many mathematical modes which are described by di?erential equations are proposed. Di?erential equations are powerful tools that describe the law of nature, but it is di?cult to ?nd their general solutions. Therefore, there has been an increasing interest in the study of the nature of solutions of di?erential equation in theory.The present paper employs a generalized Riccati transformation and Integral average technique to investigate the oscillation criteria for some class of neutral equations with distributed deviating argument, the results of which generalized and improved some known oscillation criteria.The thesis is divided into three sections according to contents.Chapter 1 we simply introduce the background and main contents of this paper.Chapter 2 We consider the Oscillation criteria for second-order neutral equations with distributed deviating argument which has the ?owing type:[r(t)(x(t) + c(t)x(t- τ))′]′+∫baq(t, ξ)x(g(t, ξ)dσ(ξ) = 0, t ≥ t0,In which τ > 0. The following conditions are assumed to hold without further mentioning:(A1) r(t), p(t) ∈ C(I, R), and 0≤ p(t) ≤ 1, r(t) > 0,∫∞t0(1/r(s))ds =∞, t ∈ I, I = [t0, ∞);(A2) q(t, ξ) ∈ C(I × [a, b], R+), and q(t, ξ) is not eventually zero on any half-linear [tu, ∞) × [a, b], tu≥ t0;(A3) g(t, ξ) ∈ C(I × [a, b], R+), g(t, ξ) ≤ t, ξ ∈ [a, b], g(t, ξ) has a continuous and positive partial derivative on I × [a, b] with respect to the ?rst variable t and nondecreasing to the second variable ξ, respectively, and lim inft→∞,ξ∈[a,b]g(t, ξ) = ∞;(A4) σ(ξ) ∈ C([a, b], ξ) is nondecreasing and the integral of Eq(1) is in the sense of Riemann-Stieltjes.we have try two ways to investigate the oscillation criteria for the Eq(1) and obtained several new oscillation criteria.Chapter 3 We consider the Oscillation criteria for second-order neutral equations with distributed deviating argument which has the ?owing type:[a(t)(ψ(x(t)))Z′(t)]′+∫bap(t, ξ)f [x(g(t, ξ)]dσ(ξ) = 0, t ≥ t0,In which Z(t) = x(t) + c(t)x(t- τ) andτ > 0. The following conditions are assumed to hold without further mentioning:(H1) a(t), c(t) ∈ C([t0, ∞), R+), andc(t) ≤ 1,∫∞t0(1/a(s))ds = ∞, R+=[0, ∞);(H2) p(t, ξ) ∈ C([t0, ∞) × [a, b], R+), and p(t, ξ)is not eventually zero on any half-linear [tu, ∞) × [a, b], tu≥ t0;(H3) g(t, ξ) ∈ C([t0, ∞) × [a, b], R+), g(t, ξ) ≤ t, ξ ∈ [a, b], g(t, ξ) has a continuous and positive partial derivative on I × [a, b] with respect to the?rst variable t and nondecreasing to the second variable ξ, respectively,and lim inft→∞,ξ∈[a,b]g(t, ξ) = ∞;(H4) σ(ξ) ∈ C([a, b], ξ) is nondecreasing and the integral of Eq(52) is in the sense of Riemann-Stieltjes.(H5) f(μυ) ≥ M f(μ)f(υ), f(x) ≥ x, in which M > 0.(H6) ψ ∈ C1(R, R), ψ(x) > 0, x ?= 0.we have try to investigate the oscillation criteria for the Eq(52) and obtained several new oscillation criteria.