Multivalued Solutions of an Iterative Equation with Variable Coefficients

In this paper, we consider a second order multivalued iterative equation with variable coefficients and the results on increasing solution and decreasing solution are obtained.

ITERATIVE SOLUTIONS FOR SYSTEMS OF NONLINEAR OPERATOR EQUATIONS IN BANACH SPACE

By using partial order method, the existence, uniqueness and iterative approximation of solutions for a class of systems of nonlinear operator equations in Banach space are discussed. The results obtained in this paper extend and improve recent results.

Iterative Solution for Systems of Nonlinear Two Binary Operator Equations

Using the cone and partial ordering theory and mixed monotone operator theory, the existence and uniqueness of solutions for some classes of systems of nonlinear two binary operator equations in a Banach space with a partial ordering are discussed. And the error estimates that the iterative sequences converge to solutions are also given. Some relevant results of solvability of two binary operator equations and systems of operator equations are improved and generalized.

Existence and Uniqueness of Analytic Solutions of Systems of Iterative Functional Equations

Let r be a given positive number. Denote by D=D r the closed disc in the complex plane C whose center is the origin and radius is r. Write A(D,D)={f: f is a continuous map from D into itself, and f|D ° is analytic}. Suppose G,H: D 2n+1 →C are continuous maps (n≥2), and G|(D 2n+1 ) °, H|(D 2n+1 ) ° are analytic. In this paper, we study the system of iterative functional equationsG(z,f(z),…,f n(z), g(z),…,g n(z))=0, H(z,f(z),…,f n(z), g(z),…,g n(z))=0, for any z∈D,and give some conditions for the system of equations to have a solution or a unique solution in A(D,D) ×A(D,D).

Spatiotemporal self-similar solutions for the nonautonomous (3+1)-dimensional cubic-quintic Gross-Pitaevskii equation

With the help of similarity transformation, we obtain analytical spatiotemporal self-similar solutions of the nonautonomous （3＋1）-dimensional cubic-quintic Gross-Pitaevskii equation with time-dependent diffraction, nonlinearity, harmonic potential and gain or loss when two constraints are satisfied. These constraints between the system parameters hint that self-similar solutions form and transmit stably without the distortion of shape based on the exact balance between the diffraction, nonlinearity and the gain/loss. Based on these analytical results, we investigate the dynamic behaviours in a periodic distributed amplification system.

Doubly Periodic Wave Solutions of Jaulent-Miodek Equations Using Variational Iteration Method Combined with Jacobian-function Method

One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The Jaulent-Miodek equations are used to illustrate effectiveness and convenience of this method, some new explicit exact travelling wave solutions have been obtained, which include bell-type soliton solution, kink-type soliton solutions, solitary wave solutions, and doubly periodic wave solutions.

Exact analytical solutions of three-dimensional Gross-Pitaevskii equation with time-space modulation

By the generalized sub-equation expansion method and symbolic computation, this paper investigates the （3＋1）dimensional Gross-Pitaevskii equation with time-and space-dependent potential, time-dependent nonlinearity, and gain or loss. As a result, rich exact analytical solutions are obtained, which include bright and dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic function solutions. With computer simulation, the main evolution features of some of these solutions are shown by some figures. Nonlinear dynamics of a soliton pulse is also investigated under the different regimes of soliton management.

DISCUSSION ON THE C^r－SOLUTIONS OF THE ITERATED EQUATION λ_1f（x）＋λ_2f^2（x）＝F（x）

In this paper, we consider the iterated equationλ1f(x) + λ2f2(x)=F(x)where f2(x)= f(f(x)), F (x) denotes known function and f(x) denotes the unknown function. There are given conditions for the existence, uniqueness and stability of C'-solutions ofthe iterated equation (*) and also there is a proved theorem for the continuous dependence of Cr-solutions of iterated equation (*) on the given function.

A Spectral Integral Equation Solution of the Gross-Pitaevskii Equation

The Gross-Pitaevskii equation (GPE), that describes the wave function of a number of coherent Bose particles contained in a trap, contains the cube of the normalized wave function, times a factor proportional to the number of coherent atoms. The square of the wave function, times the above mentioned factor, is defined as the Hartree potential. A method implemented here for the numerical solution of the GPE consists in obtaining the Hartree potential iteratively, starting with the Thomas Fermi approximation to this potential. The energy eigenvalues and the corresponding wave functions for each successive potential are obtained by a spectral method described previously. After approximately 35 iterations a stability of eight significant figures for the energy eigenvalues is obtained. This method has the advantage of being physically intuitive, and could be extended to the calculation of a shell-model potential in nuclear physics, once the Pauli exclusion principle is allowed for.

Approximation Solutions of Nonlinear Strongly Accretive Operator Equations by Ishikawa Iteration Procedure with Errors

<正>Let 1＜ρ≤2,E be a real ρ-uniformly smooth Banach space and T:E→E be a continuous and strongly accretive operator.The purpose of this paper is to investigate the problem of approximating solutions to the equation Tx=f by the Ishikawa iteration procedure with errors (?) where x_0 ∈ E,{u_n},{υ_n}are bounded sequences in E and{α_n},{b_n},{c_n},{a_n~'},{b_n~'},{c_n~'} are real sequences in[0,1].Under the assumption of the condition 0＜α≤b_n+c_n,An≥0, it is shown that the iterative sequence{x_n}converges strongly to the unique solution of the equation Tx=f.Furthermore,under no assumption of the condition(?)(b_n~'+c_n~')=0,it is also shown that{x_n}converges strongly to the unique solution of Tx=f.

Monotone Iterative Method for Nonlinear Discontinuous Differential Equations

A monotone iterative method for some discontinuous variational boundary problems is given, the convergence of iterative solutions is proved by the theory of partially ordered sets. It can be regarded as a generalization of the classical monotone iteration theory for continuous problems.

Analytical Solitary Wave Solutions to a (3+1)-Dimensional Gross—Pitaevskii Equation with Variable Coefficients

A(3+1)-dimensional Gross-Pitaevskii(GP)equation with time variable coefficients is considered,andis transformed into a standard nonlinear Schr(o|¨)dinger(NLS)equation.Exact solutions of the(3+1)D GP equationare constructed via those of the NLS equation.By applying specific time-modulated nonlinearities,dispersions,andpotentials,the dynamics of the solutions can be controlled.Solitary and periodic wave solutions with snaking andbreathing behavior are reported.

EXISTENCE THEOREMS OF EXTREMAL SOLUTIONS FOR A CLASS OF NONLINEAR INTEGRO－DIFFERENTIAL EQUATIONS

In this paper, the following initial value problem for nonlinear integro-differential equationis considered , whereUsing the method of upper and lower solutions and the monotone iterative technique .We obtain existence results of minimal and maximal solutions .

Exact Solutions of the Harry-Dym Equation

The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition, He＇s variational iteration, direct integration, and power series. We show that the two later methods are more successful than the two former to obtain more solutions of the equation.

Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation

This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.

Exact Solution of a Linear Difference Equation in a Finite Number of Steps

An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.

On the Iterative Solution to H_∞Control Problems

This paper addresses the problem for solving a Continuous-time Riccati equation with an indefinite sign of the quadratic term. Such an equation is closely related to the so called full information H∞ control of linear time-invariant system with external disturbance. Recently, a simultaneous policy update algorithm (SPUA) for solving H∞ control problems is proposed by Wu and Luo (Simultaneous policy update algorithms for learning the solution of linear continuous-time H∞ state feedback control, Information Sciences, 222, 472-485, 2013). However, the crucial point of their method is to find an initial point, which ensuring the convergence of the method. We will show one example where Wu and Luo’s method is not effective and it converges to an indefinite solution. Three effective methods for computing the stabilizing solution to the considered equation are investigated. Computer realizations of the presented methods are numerically compared on the computational platforms MATLAB and SCILAB.

Mixed Monotone Iterative Technique and Boundary Value Problem for a Class of Impulsive Integro-Differential Equations

In this paper,the existence and uniqueness of iterative solutions to the boundary value problems for a class of first order impulsive integro-differential equations were studied. Under a new concept of upper and lower solutions, a new monotone iterative technique on the boundary value problem of integro-differential equations was proposed. The existence and uniqueness of iterative solutions and the error estimation in certain interval were obtained.An example was also given to illustrate the results.

Existence of Solutions for Boundary Value Problems of Conformable Fractional Differential Equations

In this paper, we study a class of boundary value problems for conformable fractional differential equations under a new definition. Firstly, by using the monotone iterative technique and the method of coupled upper and lower solution, the sufficient condition for the existence of the boundary value problem is obtained, and the range of the solution is determined. Then the existence and uniqueness of the solution are proved by the proof by contradiction. Finally, a concrete example is given to illustrate the wide applicability of our main results.