Approximate Controllability of Second-Order Neutral Stochastic Differential Equations with Infinite Delay and Poisson Jumps

The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations （jumps） resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.

INFINITE PERIODIC SOLUTIONS TO A CLASS OF SECOND-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

By means of variational structure and Z2 group index theory, we obtain infinite periodic solutions to a class of second-order neutral differential equations.

Oscillation of Second-Order Half-Linear Neutral Advanced Differential Equations

The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))')^(α))'+q(t)yα(σ(t))=0,t≥t_(0),when∫^(∞)r^(−1/α)(s)ds<∞.We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation.An example is provided to illustrate the results.

MULTI-PERIODIC SOLUTIONS TO A CLASS OF SECOND-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

By means of variational structure and Z 2 group index theory,we obtain infinite periodic solutions to a class of second-order neutral differential equations.

EXISTENCE OF SOLUTION AND APPROXIMATE CONTROLLABILITY OF A SECOND-ORDER NEUTRAL STOCHASTIC DIFFERENTIAL EQUATION WITH STATE DEPENDENT DELAY

This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam’s novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.

ON GLOBAL MEROMORPHIC SOLUTIONS OF SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS

The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.

Oscillation of a Second-order Impulsive Neutral Differential Equation

Oscillation theorems for a second-order impulsive neutral differential equation are established, which extend the main results developed by Li et alLi et al, Oscillation of second order self-coajugate differential equation with impuls[es. J Comput Appl Math 197（2006）： 78-88] to the considered equation. Two examples are also inserted to illustrate our main results.

Criteria for System of Three Second-Order Ordinary Differential Equations to Be Reduced to a Linear System via Restricted Class of Point Transformation

This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and . The necessary conditions for linearization by general point transformation and are found. The sufficient conditions for linearization by restricted class of point transformation and are obtained. Moreover, the procedure for obtaining the linearizing transformation is provided in explicit forms. Examples demonstrating the procedure of using the linearization theorems are presented.

New Interval Oscillation Criteria for Forced Second-order Half-linear Differential Equations

This paper discusses a class of forced second-order half-linear differential equations. By using the generalized Riccati technique and the averaging technique, some new interval oscillation criteria are obtained.

SINE TRANSFORM PRECONDITIONERS FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.

Solution of Second-Order Ordinary Differential Equations via Simulated Annealing

In this paper, we approach the problem of obtaining approximate solution of second-order initial value problems by converting it to an optimization problem. It is assumed that the solution can be approximated by a polynomial. The coefficients of the polynomial are then optimized using simulated annealing technique. Numerical examples with good results show the accuracy of the proposed approach compared with some existing methods.

OSCILLATION AND ASYMPTOTIC BEHAVIOR OF SOME SECOND-ORDER RETARDED DIFFERENTIAL EQUATIONS

In this paper, we consider the following second order retarded differential equations x″(t)+cx′(t)=qx(t-σ)-lx(t-δ) (1) x″(t)+p(t)x(t-τ)=0 (2) We give some sufficient conditions for the oscillation of all solutions of Eq. (1) in the case where q, ι, σ, δ are positive numbers and c is a real number. And also, we study the asymptotic behavior of the nonoscillatory solutions. If necessary, we give some examples to illustrate our results. At last, we study Eq. (2) with some conditions on p(t).

SECOND-ORDER ACCURATE DIFFERENCE METHOD FOR THE SINGULARLY PERTURBED PROBLEM OF FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.

OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

In this paper,the oscillation criteria for the solutions of the nonlinear differential equations of neutral type of the forms:[x(t)+p(t)x(σ(t))]″+q(t)f(x(τ(t)))g(x′(t))=0and[x(t)+p(t)x(σ(t))]″+q(t)f(x(t),x(τ(t)))g(x′(t))=0are obtained.

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(t)N′(t-τ(t))].

RAZUMIKHIN-TYPE THEOREM FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.

Oscillation in a Class of Neutral Differential Equations with Positive and Negative Coefficients

The oscillatory behavior of neutral differential equation with positive and negative coefficients is investigated by mathematics analysis technique and the fixed point principle. Some sufficient conditions for oscillation of neutral differential equation with positive and negative coefficients are obtained.

Oscillation for Solutions of Systems of High Order Partial Differential Equations of Neutral Type

In this paper, sane sufficient conditions are obtained for the oscillation for solutions of systems of high order partial differential equations of neutral type.

On the Existence of Positive Solutions for Impulsive Neutral Differential Equations

Aim To investigate the existence of positive solutions for impulsive neutral differential equations. Methods The Banach contraction principle was used to establish our results. Results and Conclusion The results of the existence of positive solutions for impulsive neutral differential equations are obtained.

Oscillation of Second Order Delay Differential Equations with Nonlinear Neutral Term

By using the averaging technique, we obtain new oscillation criteria for second order delay differential equation with nonlinear neutral term. These results generalize and improve some known results about neutral delay differential equation of second order.