Carathéodory's method for a class of dynamic games

The method of equivalent variational methods, originally due to Carathéodory for free problems in the calculus of variations is extended to investigate feedback Nash equilibria for a class of n-person differential games. Both the finite-horizon and infinite-horizon cases are considered. Examples are given to illustrate the presented results.     

A higher order analogue of the Carathéodory-Julia theorem

A higher order analogue of the classical Carathéodory-Julia theorem on boundary derivatives is proved.     

An application of variational method to second-order impulsive differential equation on the half-line

In this paper, we study the multiplicity of solutions for second-order impulsive differential equation with a parameter on the half-line. By using a variational method and a three critical points theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. Also an example is given in this paper to illustrate the main results.     

A random Euler scheme for Carathéodory differential equations

We study a random Euler scheme for the approximation of Carathéodory differential equations and give a precise error analysis. In particular, we show that under weak assumptions, this approximation scheme obtains the same rate of convergence as the classical Monte–Carlo method for integration problems.     

A new algorithmic method for finding the Carathéodory extremal maps

In this paper we construct a new algorithmic method for finding Carathéodory extremal maps for some balanced domains including complex ellipsoids.     

Approximate solutions for a class of first order nonlinear difference equations

In this paper, a measure-theoretical approach to find the approximate solutions for a class of first order nonlinear difference equations is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a calculus of variations problem, some concepts in measure theory are used to approximate the solution. The procedure of constructing approximate solution in form of an algorithm is shown. Finally a numerical example is given.     

BOUNDARY VALUE PROBLEMS FOR SECOND ORDER MIXED-TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS

We use the topological degree method to deal with the generalized Sturm-Liouville boundary value problem (BVP) for second order mixed-type functional differential equation x(t)=f(t,xt,xt), 0≤t≤T. Existence principle and theorem for solutions of the BVP are obtained.     

Positive solutions for a class of fractional differential equations with integral boundary conditions

The existence of positive solutions for a class of fractional equations involving the Riemann–Liouville fractional derivative with integral boundary conditions is investigated. By means of the monotone iteration method and some inequalities associated with the Green function, we obtain the existence of a positive solution and establish the iterative sequence for approximating the solution.     

Boundary value problems for higher order linear impulsive differential equations

In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered.     

SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem.Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension,the existence of solutions of the above problem is proved.In this article,the complex analytic method is used,namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed,afterwards the above problem for the degenerate elliptic equations of second order is solved.     

A semi-analytical numerical method for solving evolution and elliptic partial differential equations

A new method for analyzing initial–boundary value problems for linear and integrable nonlinear partial differential equations (PDEs) has been introduced by one of the authors.     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

On bounded solutions of nonlinear first- and second-order equations with a Carathéodory function

In the paper we study the existence and uniqueness of bounded solutions for differential equations of the form: x^′−Ax=f(t,x), x^″−Ax=f(t,x), where A∈L(Rm), is a Carathéodory function and the homogeneous equations x^′−Ax=0, x^″−Ax=0 have nontrivial solutions bounded on R. Using a perturbation of the equations, the Leray-Schauder Topological Degree and Fixed Point Theory, we overcome the difficulty that the linear problems are non-Fredholm in any reasonable Banach space.     

Study of singular boundary value problems for second order impulsive differential equations

This paper studies the existence of extremal solutions for a class of singular boundary value problems of second order impulsive differential equations. By using the method of upper and lower solutions and the monotone iterative technique, criteria of the existence of extremal solutions are established.