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.     

Periodic solutions for perturbed differential equations

A variant of the Alekseev variation of constants integral equation is obtained relating the solutions of systems of the form $\text{x}\text{?}\text{= f(t, x, λ)}$ and $\text{y}\text{?}\text{= f(t, y, ψ(t, y)) + g(t, y)}$. For the case when f, g, and ψ have period P in t several theorems are given for the existence of periodic solutions extending known results when f is linear in x and does not depend on the parameter m-vector λ. Comparison with an older technique gives hypotheses where the method above is advantageous for establishing periodic solutions. An example is given for constructing limit cycles of autonomous second-order systems.     

Global existence of solutions for a class of nonlinear differential equations

By making use of a special Lyapunov-type function and applying the comparison method due to Conti, we prove global existence of solutions for a general class of nonlinear second-order differential equations that includes, in particular, van der Pol, Rayleigh, and Liénard equations, widely encountered in applications. Relevant examples are discussed.     

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

     

Quasi-periodic solutions for perturbed autonomous delay differential equations

This work discusses the persistence of quasi-periodic solutions for delay differential equations. We prove that the perturbed system possesses a quasi-periodic solution under appropriate hypotheses if an unperturbed linear system has quasi-periodic solutions. We extend some well-known results on ordinary differential equations to delay differential equations.     

Global existence results for impulsive functional differential equations

In this paper, we investigate the existence of solutions of impulsive delay differential equation

     

Existence of unique solutions for perturbed autonomous differential equations

We consider the existence of unique absolutely continuous solutionsfor x' = p(t)f(x) + p(t)h(t), t 0, x(0) = 0, where p, f, andh are positive almost everywhere, but none of them needs becontinuous or monotone. Moreover, p and f can be unbounded aroundzero. Our uniqueness results are not based on assumptions onthe differences f(x) – f(y), as it is usual in most uniquenessresults, and they are new even when p, f, and h are continuous.     

Simplified existence for solutions to stochastic differential equations

Nonstandard methods are used to give a simple construction of a solution to SDEs of the form , where are required only to be measurable, with, bounded. By working with an internal Brownian motion the proof avoids the complicated lifting and approximation arguments needed in previous existence proofs.     

Almost periodic solutions for forced perturbed impulsive differential equations

Sufficirnt condition for the existence of almost periodic solutions of forced perturbed systems of impulsive differential equations with impulsive effect at fixed Moments are considered.     

Convergence of solutions of perturbed nonlinear differential equations

Summary We use the nonlinear variation of parameters formula to investigate the convergence of the solutions of nonlinear perturbed systems of differential equations. This research was supported in part by the National Science Foundation under grant GP-11543. Entrata in Redazione il 9 ottobre 1971.     

On positive solutions of perturbed nonlinear differential equations

Some results are given concerning positive solutions of equations of the form x⁽ⁿ⁾ + P(t) G(x) = Q(t, x).Let class I (II) consist of all n-times differentiable functions x(t), such that x(t)>0 and x^{(n ? 1)}(t) ? 0 (x^{(n ? 1)}(t) ? 0) for all large t. Two theorems are given guaranteeing the nonexistence of solutions in class I and II, respectively, and three theorems ensure the convergence to zero of positive solutions. A recent result of Hammett concerning the second-order case is extended to the general case.     

Global solutions for abstract neutral differential equations

We study the existence of global solutions for a class of abstract neutral differential equation defined on the whole real axis. Some concrete applications related to ordinary and partial differential equations are considered.     

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.