A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.

SOLUTIONS TO DISCRETE MULTIPARAMETER PERIODIC BOUNDARY VALUE PROBLEMS INVOLVING THE p-LAPLACIAN VIA CRITICAL POINT THEORY

In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [C. Bonanno and P. Candito, Appl. Anal., 88（4） （2009）, pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and μ in some suitable intervals. The approaches we use are the critical point theory.

EXISTENCE THEOREMS FOR A SECOND ORDER THREE-POINT BOUNDARY VALUE PROBLEM WITH IMPULSES

In this paper,two existence theorems are given concerning the following 3-point boundary value problem of second order differential systems with impulses[HL(2:1,1Z;2,1Z]x″(t)=f(t,x(t),x′(t)),t∈(0,1),t≠t_k,k=1,2,...,m, Δx|_~t=t_k =I_k(x(t_k)),k=1,2,...,m, Δx′|_~t=t_k =J_k(x(t_k),x′(t_k)),k=1,2,...,m, x(0)=0,x(1)=αx(η).

The Existence of Solutions of Three-point Boundary Value Problems for Second Order Impulsive Differential Equation

We study the existence of solutions to the second order three-point boundary value problem：{x″（t）＋f（t,x（t）,x′（t））=0,t≠ti,△x（ti）=Ii（x（ti）,x′（ti））,i=1,2,…,k,△x′（ti）=Ji（x（ti）,x′（t））,i=1,2,…,k,x（0）=0=x（1）-αx（η）,where 0〈η〈1,α∈R,and f：[0,1]×R×R→R,Ii：R×R→R,Ji：R×R→R（i=1,2,…,k）are continuous. Our results is new and different from previous results. In particular, we obtain the Green function of the problem, which makes the problem simpler.

Existence of solutions of nonlinear two-point boundary value problems for 4nth-order nonlinear differential equation

Studies the existence of solutions of nonlinear two point boundary value problems for nonlinear 4n-th-order differential equationy (4n)=f(t,y,y′,y″,...,y (4n-1))(a)with the boundary conditions g 2i(y (2i)(a),y (2i+1)(a))=0,h 2i(y (2i)(c),y (2i+1)(c))=0,(i=0,1,...,2n-1)(b) where the functions f, g i and h i are continuous with certain monotone properties. For the boundary value problems of nonlinear nth order differential equationy (n)=f(t,y,y′,y″,...,y (n-1))many results have been given at the present time. But the existence of solutions of boundary value problem (a),(b) studied in this paper has not been covered by the above researches. Moreover, the corollary of the important theorem in this paper, i.e. existence of solutions of the boundary value problem.y (4n)=f(t,y,y′,y″,...,y (4n-1)) a 2iy (2i)(a)+a 2i+1y (2i+1)(a)=b 2i,c 2iy (2i)(c)+c 2i+1y (2i+1)(c)=d 2i,(i=0,1,...2n-1)has not been dealt with in previous works.

TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = 0 and αu（η） = u（1）, where η ∈ （0, 1）, α ∈ [0, 1）, and λ is a positive parameter. The nonlinear term f（t, u） is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ＊ ∈ （0, ＋∞], such that the above problem has at least two positive solutions for any λ ∈ （0, λ＊） under certain conditions on the nonlinear term f.

Existence of Multiple Positive Solutions for Second-order Three-point Boundary Value Problems on a Half-line

In this paper,we are concerned with the existence of multiple positive solutions to a second-order three-point boundary value problem on the half-line.The results are obtained by the Leggett-Williams fixed point theorem.

SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS

We mainly study the existence of positive solutions for the following third order singular multi-point boundary value problem{x^（3）（t） ＋ f（t, x（t）, x′（t）） = 0, 0 〈 t 〈 1,x（0）-∑i=1^m1 αi x（ξi） = 0, x′（0）-∑i=1^m2 βi x′（ηi） = 0, x′（1）=0,where 0 ≤ ai≤∑i=1^m1 αi 〈 1, i = 1, 2, ···, m1, 0 〈 ξ1〈 ξ2〈 ··· 〈 ξm1〈 1, 0 ≤βj≤∑i^m2=1βi〈1,J=1,2, ···, m2, 0 〈 η1〈 η2〈 ··· 〈 ηm2〈 1. And we obtain some necessa βi 〈=11, j = 1,ry and sufficient conditions for the existence of C^1[0, 1] and C^2[0, 1] positive solutions by constructing lower and upper solutions and by using the comparison theorem. Our nonlinearity f（t, x, y）may be singular at x, y, t = 0 and/or t = 1.

Precise integration method for a class of singular two-point boundary value problems

In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.

Positive solutions of three-point boundary value problems

In this paper, we consider existence of single or multiple positive solutions of three-point boundary value problems involving one-dimensional p-Laplacian. We then study existence of solutions when the problems are in resonance cases. The proposed approach is based on the Krasnoselskii＇s fixed point theorem and the coincidence degree.

Existence of Solutions for Second Order Three-Point Boundary Value Problems with Nonlinear Growth

The existence of solutions for second order three-point boundary value problems with nonlinear growth at resonance is studied by using Mawhin continuation theorem. The result shows that theorem 1 and 2 at least have one solution in c1[0,1]

Existence of Sign-changing Solution for Three-point Boundary Value Problems

In this paper, by using the fixed-point index theory, we study the existence of sign-changing solution of some three-point boundary value problems {y ''(t) + f(y) = 0, t ∈ [0, 1], y' (0) = 0, y(1) = αy(η), where 0 < α < 1, 0 < η < 1, f : R → R is continuous, strictly increasing and f(0) = 0.

Positive Solutions for Fractional Differential Equations with Multi-Point Boundary Value Problems

In this paper, a fractional multi-point boundary value problem is considered. By using the fixed point index theory and Krein-Rutman theorem, some results on existence are obtained.

Existence of Multiple Positive Solutions for Fourth Order Two-point Boundary Value Problems

The present paper tackles two-point boundary value problems for fourth-order differential equations as follows:Several existence theorems on multiple positive solutions to the problems are obtained, and some examples are given to show the validity of these results.

Existence of Two Solutions of Nonlinear m-Point Boundary Value Problems

Sufficient conditions for the existence of at least two positive solutions of a nonlinear m -points boundary value problems are established. The results are obtained by using a new fixed point theorem in cones. An example is provided to illustrate the theory.

Solvability of Third-order Three-point Boundary Value Problems with Caratheodory Nonlinearity

A class of third-order three-point boundary value problems is considered, where the nonlinear term is a Caratheodory function. By introducing a height function and considering the imtegration of this height function, an existence theorem of solution is proved when the limit growth function exists. The main tools are the Lebesgue dominated convergence theorem and the Schauder fixed point theorem.

Existence of Positive Solutions for Higher-Order Four-Point Boundary Value Problems with a p-Laplacian Operator

A class of higher-order four-point boundary value problems with a p-Laplacian operator is studied. By use of a fixed point theorem in cones, sufficient conditions for the existence of positive solutions for the boundary value problems are obtained.

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.

Multi-point Boundary Value Problems for Nonlinear Fourth-order Differential Equations with All Order Derivatives

By using fixed point theorem, multiple positive solutions for some fourth- order multi-point boundary value problems with nonlinearity depending on all order derivatives are obtained. The associated Green＇s functions are also given.

A class of boundary value problems for third-order differential equation with a turning point

A class of boundary value problems for a third-order differential equation with a turning point is considered. Using the method of multiple scales and others, the uniformly valid asymptotic expansion of solution for the boundary value problem is constructed.