Stability of sets for impulsive functional differential equations via Razumikhin method

In this paper, we consider the functional differential equation with impulsive perturbations

Boundedness results for impulsive functional differential equations with infinite delays

In this paper, boundedness criteria are established for solutions of a class of impulsive functional differential equations with infinite delays of the form $$\begin{gathered} x'(t) = F(t,x( \cdot )), t > t^ * \hfill \\ \Delta x(t_k ) = I(t_k ,x(t_k^ - )), k = 1,2,... \hfill \\ \end{gathered} $$ By using Lyapunov functions and Razumikhin technique, some new Bazumikhin-type theorems on boundedness are obtained.     

具有脉冲的二阶三点边值问题存在性定理

In this paper, two existence theorems are given concerning the following 3-point boundary value problem of second order differential systems with impulses     

Existence and iteration of positive solution for a three-point boundary value problem with ap-Laplacian operator

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for BVPs $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u(0) - B(u\prime (\eta )) = 0, u\prime (1) = 0 \hfill \\ \end{gathered} \right.$$ and $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u\prime (0) = 0, u(1) + B(u\prime (\eta )) = 0 \hfill \\ \end{gathered} \right.$$ The main tool is the monotone iterative technique. Here, the coefficientq(t) may be singular att = 0,1.     

On the Volterra integrodifferential equations and applications

For the generator A of a C ₀-semigroup on a Banach space (X, ∥·∥), we apply the perturbation of Desch-Schappacher type to solve the Volterra integordifferential equation VE $$\left\{ \begin{gathered} \frac{{du\left( t \right)}}{{dt}} = A\left( {u\left( t \right) + \int_0^t {a\left( {t - s} \right)B_1 u\left( s \right)ds + B_2 u\left( t \right) + B_3 f\left( t \right)} } \right) \hfill \\ + \int_0^t {b\left( {t - s} \right)B_4 u\left( s \right)ds + B_5 u\left( t \right) + g\left( t \right),t \geqslant 0,} \hfill \\ u\left( 0 \right) = u_0 , \hfill \\ \end{gathered} \right.$$ > which can be applied to treat boundary value problems and inhomogeneous retarded differential equations.     

Global existence of small radially symmetric solutions to quadratic nonlinear wave equations in an exterior domain

We study the initial boundary value problem for the nonlinear wave equation: (*) $$\left\{ \begin{gathered} \partial _t^2 u - (\partial _r^2 + \frac{{n - 1}}{r}\partial _r )u = F(\partial _t u,\partial _t^2 u),t \in \mathbb{R}^ + ,R< r< \infty , \hfill \\ u(0,r) = \in _0 u_0 (r),\partial _t u(0,r) = \in _0 u_1 (r),R< r< \infty , \hfill \\ u(t,R) = 0,t \in \mathbb{R}^ + , \hfill \\ \end{gathered} \right.$$ wheren=4,5,u ₀,u ₁ are real valued functions and ∈₀ is a sufficiently small positive constant. In this paper we shall show small solutions to (*) exist globally in time under the condition that the nonlinear termF:?²→? is quadratic with respect to ? _t u and ? _t ² u.     

A survey and some new results on the existence of solutions of IPBVPS for first order functional differential equations

This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation

Some dichotomy results for the quenching problem

It is shown that any solution to the semilinear problem{ ut = uxx + δ(1-u)-p , (x, t) ∈ (-1 , 1) × (0 , T ), u( ±1 , t) = 0, t ∈ (0 , T ), u(x, 0) = u0(x) 1, x ∈ [ 1 , 1] either touches 1 in finite time or converges smoothly to a steady state as t →∞. Some extensions of this result to higher dimensions are also discussed.     

Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder

We study equations of the form $$\begin{gathered} u_{tt} + Lu + b(x,t)u_t = a(x,t)\left| u \right|^{\sigma - 1} u, \hfill \\ - u_t + Lu = a(x,t)\left| u \right|^{\sigma - 1} u \hfill \\ \end{gathered}$$ , whereL is a uniformly elliptic operator and 0<σ<1. In the half-cylinder II_0,∞={(x, t):x= (x ₁,...,x _n )∈ ω,t>0}, where ? ? ? _n is a bounded domain, we consider solutions satisfying the homogeneous Neumann condition forx∈?ω andt>0. We find conditions under which these solutions have compact support and prove statements of the following type: ifu(x, t)=o(t ^γ) ast→∞, then there exists aT such thatu(x, t)≡0 fort>T. In this case γ depends on the coefficients of the equation and on the exponent σ.     

The lattice point discrepancy of a torus in ℝ3

This article provides an asymptotic formula for the number of integer points in the three-dimensional body $$ \left( \begin{gathered} x \hfill \\ y \hfill \\ z \hfill \\ \end{gathered} \right) = t\left( \begin{gathered} (a + r\cos \alpha )\cos \beta \hfill \\ (a + r\cos \alpha )\sin \beta \hfill \\ r\sin \alpha \hfill \\ \end{gathered} \right),0 \leqq \alpha ,\beta < 2\pi ,0 \leqq r \leqq b, $$ for fixed a > b > 0 and large t.     

Oscillation of solutions of nonlinear partial differential equations of neutral type

In this paper, we deal with the oscillatory behavior of solutions of the neutral partial differential equation of the form $$\begin{gathered} \frac{\partial }{{\partial t}}\left[ {p\left( t \right)\frac{\partial }{{\partial t}}(u\left( {x,t} \right) + \sum\limits_{i = 1}^t {p_i \left( t \right)u\left( {x,t - \tau _i } \right)} )} \right] + q\left( {x,t} \right)f_j (u(x,\sigma _j (t))) \hfill \\ = a\left( t \right)\Delta u\left( {x,t} \right) + \sum\limits_{k = 1}^n {a_k \left( t \right)} \Delta u\left( {x,\rho _k \left( t \right)} \right), \left( {x,t} \right) \in \Omega \times R_ + \equiv G \hfill \\ \end{gathered} $$ where Δ is the Laplacian in EuclideanN-spaceR ^N, R₊=(0, ∞) and Ω is a bounded domain inR ^N with a piecewise smooth boundary δΩ.     

Three symmetric positive solutions for second-order nonlocal boundary value problems

Using the Leggett-Williams fixed point theorem,we will obtain at least three symmetric positive solutions to the second-order nonlocal boundary value problem of the form u(t)+g(t)f(t,u(t))=0,0相似文献   

Sharpness on the Lower Bound of the Lifespan of Solutions to Nonlinear Wave Equations

This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n = 2 and cubic nonlinearity (see the results of T. T. Li and Y. M. Chen in 1992). For this purpose, the authors consider the following Cauchy problem:

Forward dynamical problem for the Timoshenko beam

The initial boundary value problem

Numerical blow-up for a nonlinear heat equation

This paper concerns the study of the numerical approximation for the following initialboundary value problem

Low Regularity Solution of the Initial-Boundary-Value Problem for the ＂Good＂ Boussinesq Equation on the Half Line

we study an initial-boundary-value problem for the ＂good＂ Boussinesq equation on the half line
{δt^2u-δx^2u＋δx^4u＋δx^2u^2=0,t〉0,x〉0.
u（0,t）=h1（t）,δx^2u（0,t） =δth2（t）,
u（x,0）=f（x）,δtu（x,0）=δxh（x）.
The existence and uniqueness of low reguality solution to the initial-boundary-value problem is proved when the initial-boundary data （f, h, h1, h2） belong to the product space
H^5（R^＋）×H^s-1（R^＋）×H^s/2＋1/4（R^＋）×H^s/2＋1/4（R^＋）
1 The analyticity of the solution mapping between the initial-boundary-data and the with 0 ≤ s 〈 1/2. solution space is also considered.     

Nontrivial solution of a quasilinear elliptic equation with critical growth in ℝn

Suppose Δ_n u = div (¦ ?u ¦^n-2?u) denotes then-Laplacian. We prove the existence of a nontrivial solution for the problem $$\left\{ \begin{gathered} - \Delta _n u + \left| u \right|^{n - 2} u = \int {(x,u)u^{n - 2} in \mathbb{R}^n } \hfill \\ u \in W^{1,n} (\mathbb{R}^n ) \hfill \\ \end{gathered} \right.$$ wheref(x, t) =o(t) ast → 0 and ¦f(x, t)¦ ≤C exp(α_n¦t¦^n/(n-1)) for some constantC > 0 and for allx∈?;t∈? with α_n =nω _n ^1/(n-1) , ω_n = surface measure ofS ^n-1.     

Multivalued backward stochastic differential equations with time delayed generators

Our aim is to study the following new type of multivalued backward stochastic differential equation: $$\left\{ \begin{gathered} - dY\left( t \right) + \partial \phi \left( {Y\left( t \right)} \right)dt \ni F\left( {t,Y\left( t \right),Z\left( t \right),Y_t ,Z_t } \right)dt + Z\left( t \right)dW\left( t \right), 0 \leqslant t \leqslant T, \hfill \\ Y\left( T \right) = \xi , \hfill \\ \end{gathered} \right.$$ where ? _φ is the subdifferential of a convex function and (Y _t , Z _t ):= (Y(t + θ), Z(t + θ)) _θ∈[?T,0] represent the past values of the solution over the interval [0, t]. Our results are based on the existence theorem from Delong & Imkeller, Ann. Appl. Probab., 2010, concerning backward stochastic differential equations with time delayed generators.     

Some variations on Kac's formula for Brownian motion

We show that under suitable conditions $$\begin{gathered} E_x f\left\{ {a + \int_0^t \beta \left[ {b + \int_0^s {\alpha \left( {X_r } \right)dr, c + s, X_s } } \right]ds, b + \int_0^t {\alpha \left( {X_s } \right)ds, c + t, X_t } } \right\} \hfill \\ = e^{tG} f\left[ {a, b, c, x} \right] \hfill \\ \end{gathered} $$ whereX _t is a Brownian motion andG is the generator of a (C ₀) contraction semigroupe ^tG.     

Difference schemes for one class of variational inequalities and fourth-order mixed boundary-value problem

A difference scheme is constructed for the solution of the variational equation $$\begin{gathered} a\left( {u, v} \right)---u \geqslant \left( {f, v---u} \right)\forall v \varepsilon K,K \{ vv \varepsilon W_2^2 \left( \Omega \right) \cap \mathop {W_2^1 \left( \Omega \right)}\limits^0 ,\frac{{\partial v}}{{\partial u}} \geqslant 0 a.e. on \Gamma \} ; \hfill \\ \Omega = \{ x = (x_1 ,x_2 ):0 \leqslant x_\alpha< l_\alpha ,\alpha = 1, 2\} \Gamma = \bar \Omega - \Omega ,a(u, v) = \hfill \\ = \int\limits_\Omega {\Delta u\Delta } vdx \equiv (\Delta u,\Delta v, \hfill \\ \end{gathered} $$ The following bound is obtained for this scheme: $$\left\| {y - u} \right\|_{W_2 \left( \omega \right)}^2 = 0(h^{(2k - 5)/4} )u \in W_2^k \left( \Omega \right),\left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0(h^{\min (k - 2;1,5)/2} ),u \in W_\infty ^k \left( \Omega \right) \cap W_2^3 \left( \Omega \right)$$ The following bounds are obtained for the mixed boundary-value problem: $$\begin{gathered} \left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0\left( {h^{\min \left( {k - 2;1,5} \right)} } \right),u \in W_\infty ^k \left( \Omega \right),\left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0\left( {h^{k - 2,5} } \right), \hfill \\ u \in W_2^k \left( \Omega \right),k \in \left[ {3,4} \right] \hfill \\ \end{gathered} $$ .