Existence and Multiplicities of Solutions for Asymptotically Linear Ordinary Differential Equations Satisfying Sturm-Liouville BVPs with Resonance

In this paper, we prove existence and multiplicities of solutions for asymptotically linear ordinary differential equations satisfying Sturm-Liouville boundary value conditions with resonance. Adding assumption H3 that is similar to (LL) in Theorem 1.1, by index theory and Morse theory, we obtain more nontrivial solutions.

UNIQUENESS OF SOLUTIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR THIRD QRDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].

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.

ASYMPTOTIC SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH TURNING POINTS

Boundary value problem; for third-order ordinary differential equations with turning points are studied as follows : epsilon gamma ' ' + f(x ; epsilon) gamma ' + g(x ; epsilon) gamma ' +h(x ; epsilon) gamma = 0 (- a < x < b, 0 epsilon 1), where f(x ; 0) has several multiple zero points in ( - n, b). the necessary conditions for exhibiting resonance is given, and the uniformly valid asymptotic solutions and the estimations of remainder terms are obtained.

ON DETERMINATION OF THE IMPULSIVE SOLUTIONS AND IMPULSIVE PROPERTIES TO LINEAR NON-HOMOGENEOUS MATRIX DIFFERENTIAL EQUATIONS

This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the pseudo state and the nonzero initial conditions of the input.Secondly,in order to characterise the dynamics of the LNHMDEs correctly,some important concepts such as the state,slow state(smooth state)and fast state(impulsive state)are generalized to the LNHMDE case and the solution of the LNHMDEs is separated into the smooth(slow)response and the fast(implusive)response.As a third result,a new characterization of the impulsive free initial conditions of the LNHMDEs is given.

Exact Travelling Wave Solutions of Two Nonlinear Schr&#246;dinger Equations by Using Two Methods

The special kind of (G’/G)-expansion method and the new mapping method are easy and significant mathematical methods. In this paper, exact travelling wave solutions of the higher order dispersive Cubic-quintic nonlinear Schr&#246;dinger equation and the generalized nonlinear Schr&#246;dinger equation are studied by using the two methods. Finally, the solitary wave solutions, singular soliton solutions, bright and dark soliton solutions and periodic solutions of the two nonlinear Schr&#246;dinger equations are obtained. The results show that this method is effective for solving exact solutions of nonlinear partial differential equations.

Existence of Periodic Solutions for Odd Order Ordinary Differential Equations via the Homotopy Method

This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form (1) where T is a fixed positive number and f satisfies some additional conditions which will be stated later.The periodicity problem has been one of main topics in the qualitative theory of ordinary

The Existence of Meromorphic Solutions to Non-Linear Delay Differential Equations

In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.

REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS

Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.

Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

ON THE ASYMPTOTIC SOLUTIONS FOR A CLASS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH SLOWLY VARYING COEFFICIENTS

In this paper we study the asymptotic expansions of the solutions for a class of second order ordinary differential equations with slowly varying coefficients. The defect of the known works on these problems is noted, and the results in [1 - 4] are improved and extended by means of the modified method of multiple scales.

Boundary Value Problems of Quasi-linear Ordinary Differential Equations on the Semi-infinite Interval

Existence of positive solution is established for boundary value problems of nonsingular for a class quasi-linear ordinary differential equation on the semi-infinite interval. The results are obtained by using the nonlinear alternative of Leray-Schauder method.

Construction of Periodic Solutions of One Class Nonautonomous Systems of Differential Equations

In this article we proposed a method for constructing approximations to periodic solutions of one class nonautonomous system of ordinary differential equations. It is based on successive approximation scheme using parallel symbolic calculations to obtain solutions in analytical form. We showed the convergence of the scheme of successive approximations on the period, and also considered an example of a second order system where the described scheme of calculations can be applied.

Roughness of Exponential Estimates for Linear Functional Differential Equations

Assume that the fundamental solution matrix U (t, s ) of x’(t)=L(t, x,) satisfies |U(t,s)|≤Ke-e(t-s) for t≥s.If|(t,φ)|≤δ|φ(0)|with δ【a/K, then the fundamental solution matrix of the perturbed equation x’(t)=L(t,x,)+(t ,x,) also possesses similar exponential estimate. For α=0, a similar result is given.

ON THE HOLOMORPHIC SOLUTION OF NON-LINEAR TOTALLY CHARACTERISTIC EQUATIONS WITH SEVERAL SPACE VARIABLES

In this paper the authors study a class of non-linear singular partial differential equation in complex domain C-t x C-x(n). Under certain assumptions, they prove the existence and uniqueness of holomorphic solution near origin of C-t x C-x(n).

New Exact Travelling Wave Solutions to Kundu Equation

Based on a first-order nonlinear ordinary differential equation with six-degree nonlinear term, we first present a new auxiliary equation expansion method and its algorithm. Being concise and straightforward, the method is applied to the Kundu equation. As a result, some new exact travelling wave solutions are obtained, which include bright and dark solitary wave solutions, triangular periodic wave solutions, and singular solutions. This algorithm can also be applied to other nonlinear evolution equations in mathematical physics.

Exponential Dichotomy and Eberlein-Weak Almost Periodic Solutions

We give sufficient conditions ensuring the existence and uniqueness of an Eberlein-weakly almost periodic solution to the following linear equation dx/dt(t) = A(t)x(t) + f(t) in a Banach space X, where (A(t)) t ∈□ is a family of infinitesimal generators such that for all t ∈□, A(t + T) = A(t) for some T > 0, for which the homogeneuous linear equation dx/dt(t) = A(t)x(t) is well posed, stable and has an exponential dichotomy, and f:□ →X is Eberlein-weakly amost periodic.

DIFFERENTIATOR SERIES SOLUTION OF LINEAR DIFFERENTIAL ORDINARY EQUATION

In this paper, the principle techinique of the differentiator method, and some examples using the method to obtain the general solution and special solution of the differential equation are introduced. The essential difference between this method and the others is that by this method special and general solutions can be obtained directly with the operations of the differentor in the differential equation and without the enlightenment of other scientific knowledge.

New Types of Travelling Wave Solutions From （2＋l）-Dimensional Davey-Stewartson Equation

In this paper, based on new auxiliary nonlinear ordinary differential equation with a sixtb-aegree nonnneal term, we study the （2＋l ）-dimensional Davey-Stewartson equation and new types of travelling wave solutions are obtained, which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations.

Inequalities for Sums and Products of Complex Zeros of Solutions to ODE with Polynomial Right Parts

The paper is devoted to non-homogeneous second-order differential equations with polynomial right parts and polynomial coefficients.We derive estimates for the partial sums and products of the zeros of solutions to the considered equations.These estimates give us bounds for the function counting the zeros of solutions and information about the zero-free domains.