POSITIVE SOLUTIONS FOR SOME 1-DIMENSIONAL BOUNDARY VALUE PROBLEMS OF LAPLACE-TYPE

This paper deals with the existence of triple positive solutions for the 1-dimensional equation of Laplace-type （φ（x′（t）））′＋q（t）f（t,x（t）,x′（t））=0,t∈（0,1）,subject to the following boundary condition：a1φ（x（0））-a2φ（x＇（0））=0,a3φ（x（1））＋a4φ（x＇（1））=0,where φ is an odd increasing homogeneous homeomorphism. By using a new fixed point theorem, sufficient conditions are obtained that guarantee the existence of at least three positive solu- tions. The emphasis here is that the nonlinear term f is involved with the first order derivative explicitly.

Solution of the Third Kind Boundary Value Problem of Laplace’s Equation Based on Conformal Mapping

In order to overcome the difficulty in solving the boundary value problem of electrostatic field with complex boundary and to give a new method for solving the third boundary value problem of Laplace’s equation, in this paper, the third boundary value problem of Laplace’s equation is studied by combining conformal mapping with theoretical analysis, the several analytical solutions of third boundary value problems of Laplace’s equation are gives, the correctness of its solution is verified through computer numerical simulation, and a new idea and method for solving the third boundary value problem of Laplace’s equation is obtained. In this paper, the boundary condition of the solving domain is changed by the appropriate conformal mapping, so that the boundary value problem on the transformed domain is easy to be solved or be known, and then the third kind boundary value of the Laplace’s equation can be solved easily;its electric potential distribution is known. Furthermore, the electric field line and equipotential line are plotted by using the MATLAB software.

Existence of Solutions for S-L Singular Boundary Value Problems with p-Laplacian Operators

The existence of solutions for singular nonlinear two point boundary value problems subject to Sturm Liouville boundary conditions with p Laplacian operators is studied by the method of upper and lower solutions. The proof is based on an application of Schauder’s fixed point theorem to a modified problem whose solutions are that of the original one. At the same time, Arzela Ascoli theorem is used to prove that the defined operator N is a compact map.

Existence of Multiple Positive Solutions for Higher-Order p-Laplacian Boundary Value Problems

The existence of multiple positive solutions for a class of higher order p Laplacian boundary value problem is studied. By means of the Leggett Williams fixed point theorem in cones, existence criteria which ensure the existence of at least three positive solutions of the boundary value problem are established.

一类带p-Laplace算子的非线性二阶m点共振边值问题解的存在性

基于拓扑横截方法连同障碍带技巧,得到一类带p-Laplace算子的非线性二阶m点共振边值问题解的新的存在性结果,并给出应用举例。

Optimal Shape Factor and Fictitious Radius in the MQ-RBF:Solving Ill-Posed Laplacian Problems

To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11).

具p-Laplace算子的分数阶脉冲微分方程奇异边值问题的解

用Banach压缩映像原理和Krsnoasel’skii不动点定理证明一类具有p-Laplace算子的分数阶脉冲微分方程奇异边值问题解的唯一性和存在性.

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.

Multiple positive solutions for a class of nonlinear four-point boundary value problem with p-Laplacian

This paper deals with the existence of multiple positive solutions for a class of nonlinear singular four-point boundary value problem with p-Laplacian：{（φ（u′））′＋a（t）f（u（t））=0, 0〈t〈1, αφ（u（0））-βφ（u′（ξ））=0,γφ（u（1））＋δφ（u′（η））0,where φ（x） = ｜x｜^p-2x,p 〉 1, a（t） may be singular at t = 0 and/or t = 1. By applying Leggett-Williams fixed point theorem and Schauder fixed point theorem, the sufficient conditions for the existence of multiple （at least three） positive solutions to the above four-point boundary value problem are provided. An example to illustrate the importance of the results obtained is also given.

EXISTENCE OF POSITIVE SOLUTIONS FOR p-LAPLACIAN SINGULAR BOUNDARY VALUE PROBLEMS OF FUNCTIONAL DIFFERENTIAL EQUATION

In this paper,the boundary value problems of p-Laplacian functional differential equation are studied.By using a fixed point theorem in cones,some criteria for the existence of positive solutions are given.

Picard Boundary Value Problems of Second Order p-Laplacian Differential Equations

Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.

Positive Solutions for Fourth-order Delay Differential Equation of Boundary Value Problem with p-Laplacian

In this work, we investigate the following fourth-order delay differential equation of boundary value problem with p-Laplacian（Φp（u000））0（t）＋a（t）f（t, u（t？τ）, u0（t））=0, 0〈t〈1;u000 （0）=u00 （0）=0, u0 （1）=αu0 （η）;u（t）=0, ？τ ≤t≤0. By using Schauder fixed-point theorem, some su？cient conditions are obtained which guar-antee the fourth-order delay differential equation of boundary value problem with p-Laplacian has at least one positive solution. Some corresponding examples are presented to illustrate the application of our main results.

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.

Existence Theorem about Triple Positive Solutions for a Boundary Value Problem with p-Laplacian

In this paper, by applying a fixed point theorem to verify the existence of at least three positive solutions to a three-point boundary value problem with p-Laplacian. The interesting point is the nonlinear term is involved with the first-order derivative explicitly.

Existence Results for m-Point Boundary Value Problem at Resonance with One-dimensional p-Laplacian

By using Mawhin＇s continuation theorem, the existence of a solution for a class of m-point boundary value problem at resonance with one-dimensional p-Laplacian is obtained. An example is given to demonstrate the main result of this paper.

Existence of Solutions for Boundary Value Problems of Vibration Equation with Fractional Derivative

In this paper, we investigate the solvability of boundary value problems for a class of vibration differential equation describing the fractional order damped system with signal stimulus. By presenting kernel function through the Laplace transform, and using the eigenvalue and the improved Leray-Schauder degree, the existence of solutions for boundary value problems is established.

A New Approach for Solving Boundary Value Problem in Partial Differential Equation Arising in Financial Market

In this paper, we present a new approach for solving boundary value problem in partial differential equation arising in financial market by means of the Laplace transform. The result shows that the Laplace transform for the price of the European call option which pays dividend yield reduces to the Black-Scholes-Merton model.

Positive Solutions for a Boundary Value Problem with a Derivative Argument

In this paper, several existence results of multiple positive solutions are obtained for a boundary value problem with p-Laplacian, by applying a fixed point theorem in cones. The interesting point is that the nonlinear term f is involved with the first-order derivative explicitly.

带p(t)-Laplace算子的分数阶Langevin方程参数型反周期边值问题解的存在性

利用Schaefer不动点定理,讨论一类带有p(t)-Laplace算子的分数阶Langevin方程参数型反周期边值问题,通过给出非线性项合理的假设条件得到了其解的存在性结果,并举例说明主要结果的应用.

带p-Laplace算子的离散混合边值问题负凸解的存在性

运用锥上的不动点定理讨论带p-Laplace算子的离散混合边值问题△[φ_(p)(△u(t-1))]=f(t,-u(t)),t∈[2,T-1]_(Z),△u(1)=0,u(T)=0负凸解的存在性,其中φ_(p)(s)=|s|^(p-2)s,p>1,f:[2,T-1]_(Z)×[0,+∞)→[0,+∞)连续.