Global existence and nonexistence of solutions for second-order nonlinear differential equations

This paper deals with the global existence and nonexistence of solutions of the second-order nonlinear differential equation (φ(x^′))^′+λφ(x)=0${(\phi (x^{\prime }))}^{\prime }+\lambda \phi (x)=0$ satisfying x(0)=_x0$x(0)=x_0$ and x^′(0)=_x1$x^{\prime }(0)=x_1$, where λ   is a positive parameter and φ:(−ρ,ρ)→(−σ,σ)$\phi :(-\rho ,\rho )\to (-\sigma ,\sigma )$ with 0<ρ?∞$0<\rho ?\infty$ and 0<σ?∞$0<\sigma ?\infty$ is strictly increasing odd bijective and continuous on (−ρ,ρ)$(-\rho ,\rho )$. Necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic. Examples are given to illustrate our main result. Finally, a nonexistence result for the equation with a damping term is discussed as an application to our result.     

Global existence results for impulsive differential equations

This paper studies the global existence of solutions of the impulsive differential equation

     

An existence result for a class of second order evolution equations of mixed type

We give an existence and uniqueness result for a linear abstract evolution equation of second order with some coefficient in front of the second temporal derivative which may degenerate to zero and change sign.     

Oscillation criteria for second-order nonlinear differential equations with integrable coefficient

In this paper, we consider the second-order nonlinear differential equation

[a(t)|y′(t)|^σ−1y′(t)|′+q(t)f(y(t))=r(t)

where σ > 0 is a constant, a C(R, (0, ∞)), q C(R, R), f C(R, R), xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. Some new sufficient conditions for the oscillation of all solutions of (*) are obtained. Several examples which dwell upon the importance of our results are also included.     

Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations

in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions g_i(), , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.     

Nonoscillation and oscillation of second order half-linear differential equations

We study the oscillation problems for the second order half-linear differential equation ^′[p(t)Φ(x^′)]+q(t)Φ(x)=0, where Φ(u)=|u|^r−1u with r>0, 1/p and q are locally integrable on R₊; p>0, q?0 a.e. on R₊, and . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p≡1, our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712-723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180-188] on linear equations to the half-linear case for all r>0. These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39-47] on nonoscillation when 0<r<1 and on oscillation when r>1. The approach in this paper can also be used to fully extend Elbert's criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363-373].     

Global existence of solutions for stochastic impulsive differential equations

In this paper we obtain some results on the global existence of solution to Itô stochastic impulsive differential equations in M([0,∞),? ⁿ ) which denotes the family of ? ⁿ -valued stochastic processes x satisfying sup_t∈[0,∞) \(\mathbb{E}\)|x(t)|² < ∞ under non-Lipschitz coefficients. The Schaefer fixed point theorem is employed to achieve the desired result. An example is provided to illustrate the obtained results.     

Oscillation theorems for second-order half-linear differential equations

Oscillation criteria for the second-order half-linear differential equation

[r(t)|ξ′(t)|⁻¹ ξ′(t)]′ + p(t)|ξ(t)|⁻¹ξ(t)=0, t t₀

are established, where > 0 is a constant and exists for t [t₀, ∞). We apply these results to the following equation:

where , D = (D₁,…, D_N), Ω_a = x ^N : |x| ≥ a} is an exterior domain, and c C([a, ∞), ), n > 1 and N ≥ 2 are integers. Here, a > 0 is a given constant.     

Lyapunov type inequalities for second-order half-linear differential equations

In this paper, we establish some new Lyapunov type inequalities for second-order half-linear differential equations, which almost generalize and extend all related existing results in the literature.     

On oscillation of nonlinear second-order differential equations with damping term

The paper is concerned with oscillation of a novel class of nonlinear differential equations with a damping term. First it is demonstrated how known oscillation results for another intensively studied class of equations can be translated to the one in question, and vice versa. Advantages and drawbacks of such translation are carefully examined. Then an oscillation criterion for the new class of equations is established. The principal result of the paper is compared to those reported in the literature, and an illustrative example to which known oscillation criteria fail to apply is provided.     

New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping

Some new oscillation criteria are established for the nonlinear damped differential equation (r(t)y′)′+p(t)y′+q(t)f(y)=0 that are different from most known ones in the sense that they are based on a class of new functions Φ(t,s,r) defined in the sequel. Our results are sharper than some previous results which can be seen by the examples at the end of this paper.     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

On the stability and boundedness of solutions of a kind of third order delay differential equations

In this paper, we study Eq. (1.1) for asymptotic stability of the zero solution when and uniformly bounded and uniformly ultimate bounded of all solutions when      

Asymptotic stability and boundedness for functional differential equations

We study a system(D)x′=F(t,x _t) of functional differential equations, together with a scalar equation(S)x′=−a(t)f(x)+b(t)g(x(t−h)) as well as perturbed forms. A Liapunov functional is constructed which has a derivative of a nature that has been widely discussed in the literature. On the basis of this example we prove results for (D) on asymptotic stability and equi-boundedness. Supported in part by NSF of China, Key Project # 19331060     

Integral averaging technique for the interval oscillation criteria of certain second-order nonlinear differential equations

We present new interval oscillation criteria related to integral averaging technique for certain classes of second-order nonlinear differential equations which are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t₀,∞), rather than on the whole half-line. They generalize and improve some known results. Examples are also given to illustrate the importance of our results.     

New existence results of positive solution for a class of nonlinear fractional differential equations

In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Our main method to the problem is the method of upper and lower solutions and Schauder fixed point theorem. Finally, we give an example to illuminate our results.