Oscillation Criteria Of Nonlinear Differential Equations With Damping

The theory of nonlinear differential equation with damping is an important areaof investigation since it represents a broad natural framework for modeling of manyreal world phenomena.In the paper, we mainly consider the oscillation criteria of nonlinear differentialequations with damping. This paper consists of three chapters.In Chapter 1, we study the second order nonlinear functional differential equation (r(t)ψ(x(t))φ(x'(t)))'+p(t)φ(x'(t))+f(t,x(t),x(T(t)),x'(t),x'(T(t)))=0,t≥t₀,where t₀＞0; r,τ∈C([t₀,∞),(0,∞)), lim_t→∞τ(t)=∞;ψ∈C(R,R),φ(μ)=|μ|^α-1μ,α＞0 is a constant; p∈C([t₀,∞),R); f∈C([t₀,∞)×R⁴, R). By us-ing general Riccati transformation, integral averaging technique, and the appropriatefunctionsμ∈U(a, b) and H∈H, we obtain some sufficient conditions for oscillationof all solutions of the equation, which extend the corresponding results of Cakmak[JMath Anal Appl, 2004, 300: 408-425], Li Wan-Tong[Applied Mathematics and Com-putation, 2004, 155: 451-468] and Tiryaki, Basci, Gulec[Computers and Mathematicswith Applications, 2005, 50: 1487-1498].In Chapter 2, we discuss the oscillation conditions of solutions of differentialequation (r(t)k₁(x,x'))'+p(t)k₂(x,x')x'+q(t)f(x)=0,t≥t₀,where t₀≥0, p,q∈C([t₀,∞),R), r∈C¹([t₀,∞),(0,∞)), f∈C(R,R), k₁∈C¹(R², R) and k₂∈C(R², R). The oscillation of equation was first studied by Ro-govchenko and Rogovchenko[J Math Anal Appl, 2003, 279: 121-134]. Recently, undersome assumptions, Tiryki and Zafer Mathematial and Computer Modelling, 2004,39:197-208] established some oscillation criteria. In their theorems, the condition" ((?)H)/((?)s)≤0 "on the auxiliary function H(t, s) was imposed. Meanwhile the conditionrequires thatμνand k₂(μ,ν) have the same sign which restrict the applications of thetheorems. In this chapter, by using general Riccati transformation and an appropriatefunction H(t, s), we establish some oscillation criteria of the equation as p(t) is nonneg-ative or of varying sign, respectively. Our results extend the corresponding results ofRogovchenko[Nonlinear Analysis, 2000, 41: 1005-1028], Tiryaki, A Zafer[Mathematialand Computer Modelling, 2004, 39: 197-208], Wong[J Math Anal Appl, 2001, 258:244-257] and Yang Xiaojing[Applied Mathematics and Computation, 2003, 136: 549-557]. In Chapter 3, the oscillation property and asymptotic behavior of solutions ofthird order nonlinear delay differential equation (r₂(t)(r₁(t)y'(t))')'+p(t)α(y(t))y'(t)+q(t)g(y(σ(t)))=0,t≥t₀ (*)are considered, where t₀≥0, r₁, r₂∈C¹ ([t₀,∞), (0,∈)), p, q∈C([t₀,∞), [0,∞)), andq(t)≠0 on any ray [T,∞] for some T≥t₀, g∈C(R,R)andg(μ)/μ≥M＞0,μ≠0;σ∈C¹([t₀,∞),R)and0＜σ(t)≤t,σ'(t)≥0, lim_{t→∞}σ(t)=∞,α∈C(R, (0;∞)),and there exist two constants C₁ and C such that 0＜C₁≤α(y)≤C≤∞,y∈R.Sufficient conditions are obtained for all non-oscillatory solutions of the equationstending to zero as t→∞, which extend the corresponding results of Saker [MathSlovaca, 2006, 56: 433-450], Tiryaki, Aktas[J Math Anal Appl, 2007, 325: 54-68].Meanwhile, by using an integral transformation, we prove that the oscillation of thethird order nonlinear delay differential equation with damping (r₂(t)(r₁(t)ψ(x(t))x'(t))')'+p(t)x'(t)+q(t)f(x(a(t)))=0,t≥t₀,is equivalent with that of equation (*), and obtain some oscillation criteria.