On the bivariate Shepard-Lidstone operators

We propose a new combination of the bivariate Shepard operators (Coman and Trîmbi?a?, 2001 [2]) by the three point Lidstone polynomials introduced in Costabile and Dell’Accio (2005) [7]. The new combination inherits both degree of exactness and Lidstone interpolation conditions at each node, which characterize the interpolation polynomial. These new operators find application to the scattered data interpolation problem when supplementary second order derivative data are given (Kraaijpoel and van Leeuwen, 2010 [13]). Numerical comparison with other well known combinations is presented.     

Inequality of Petrović and Giaccardi for Convex Function of Higher Order

In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generalization of the well-known Petrovis inequality.     

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTION FOR NONLINEAR SINGULAR 2mth-ORDER CONTINUOUS AND DISCRETE LIDSTONE BOUNDARY VALUE PROBLEMS

By fixed point theorem of a mixed monotone operators, we study Lidstone boundary value problems to nonlinear singular 2mth-order differential and difference equations, and provide sufficient conditions for the existence and uniqueness of positive solution to Lidstone boundary value problem for 2mth-order ordinary differential equations and 2mth-order difference equations. The nonlinear term in the differential and difference equation may be singular.     

On the Dirichlet and Lidstone Problems for a Higher-Order Linear Hyperbolic Equation

For a higher-order linear hyperbolic equation with nonsmooth coefficients, we consider the Dirichlet and Lidstone problems in a rectangle with nonclassical boundary conditions and prove that these problems are equivalent to the classical Dirichlet and Lidstone problems, respectively.     

Higher order boundary value problems on time scales

In this paper we study the existence of solutions for Lidstone boundary value problems on time scale. Firstly, by using Schauder fixed point theorem in a cone, we obtain the existence of solutions to a Lidstone boundary value problem (LBVP). Secondly, existence result for this problem is also given by the monotone method. Finally, by using Krasnosel'skii fixed point theorem, it is proved that the LBVP has a positive solution.     

Existence of multiple symmetric positive solutions of higher order Lidstone problems

This paper gives the sufficient conditions of the existence of at least three symmetric positive solutions for one type of higher order autonomous Lidstone problem by applying the five functionals fixed point theorem. The analogous result for higher order nonautonomous singular Lidstone problem is also proved here.     

Higher-order Lidstone boundary value problems for elliptic partial differential equations

The aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth-order elliptic Lidstone boundary value problems where the nonlinear functions depend on the higher-order derivatives. Sufficient conditions are given for the existence and uniqueness of a solution. It is also shown that there exist two sequences which converge monotonically from above and below, respectively, to the unique solution. The approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions. All the results are directly applicable to 2nth-order two-point Lidstone boundary value problems.     

Complementary Lidstone interpolation on scattered data sets

Recently we have introduced a new technique for combining classical bivariate Shepard operators with three point polynomial interpolation operators (Dell’Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished). This technique is based on the association, to each sample point, of a triangle with a vertex in it and other ones in its neighborhood to minimize the error of the three point interpolation polynomial. The combination inherits both degree of exactness and interpolation conditions of the interpolation polynomial at each sample point, so that in Caira et al. (J Comput Appl Math 236:1691–1707, 2012) we generalized the notion of Lidstone Interpolation (LI) to scattered data sets by combining Shepard operators with the three point Lidstone interpolation polynomial (Costabile and Dell’Accio, Appl Numer Math 52:339–361, 2005). Complementary Lidstone Interpolation (CLI), which naturally complements Lidstone interpolation, was recently introduced by Costabile et al. (J Comput Appl Math 176:77–90, 2005) and drawn on by Agarwal et al. (2009) and Agarwal and Wong (J Comput Appl Math 234:2543–2561, 2010). In this paper we generalize the notion of CLI to bivariate scattered data sets. Numerical results are provided.     

Bifurcation techniques for Lidstone boundary value problems

By employing bifurcation techniques, this paper investigates the existence of nontrivial solutions (single and multiple) for Lidstone boundary value problems depending on higher order derivatives. Our results improve on those in the literature.     

高阶差分方程Lidstone BVP的特征值问题

应用锥上的不动点定理和锥理论,我们在对非线性项的不同假设下分别得到了一类2m阶离散Lidstone BVP特征值的存在区间．     

Lidstone边值问题正解的单调迭代

考察了Lidstone边值问题正解的单调迭代. 这个迭代程序是从常值函数开始并且是可行的.     

Upper and lower estimates for positive solutions of the higher order Lidstone boundary value problem

We consider the higher order Lidstone boundary value problem. New upper and lower estimates for positive solutions of the problem are obtained. A discussion of the sharpness of the estimates is included.     

On 2nth-order Lidstone boundary value problems

This paper is concerned with the solutions of a class of 2nth-order Lidstone boundary value problems. Sufficient conditions for the existence and uniqueness of a solution are given. A monotone iteration is developed so that the iteration sequence converges monotonically to a maximal or a minimal solution. The approach to the problem is by the method of upper and lower solutions with a new maximum principle.     

Piecewise complementary Lidstone interpolation and error inequalities

The purpose of this paper is to develop piecewise complementary Lidstone interpolation in one and two variables and establish explicit error bounds for the derivatives in L_∞ and L₂ norms.     

2n阶Lidstone边值问题正解的逐次迭代

研究了2n阶Lidstone边值问题正解的逐次迭代,其中非线性项依赖于所有偶数阶导数.通过考察非线性项在某些有国介集合上的“高度”并利用单调迭代方法构造了一个逐次迭代程序.这个迭代程序从一个多项式开始并且是可行的.使用这个结论获得了m个正解的迭代方法,其中m是一个任意的自然数.     

Global behaviour of the components of nodal solutions for Lidstone boundary value problems

By employing bifurcation techniques, we investigate the global behaviour of the components of nodal solutions for Lidstone boundary value problems depending on higher order derivatives. Our results improve on those in the literature.     

Existence results for a fractional boundary value problem with fractional Lidstone conditions

Our aim in this work is to study the existence of solutions for a fractional Lidstone boundary value problems. We use some fixed point theorems to show the existence and uniqueness of solution under suitable conditions. Two examples are given to ilustrate the obtained results.     

Existence and uniqueness of positive solutions for a nonlinear fourth-order boundary value problem

This work presents sufficient conditions for the existence and uniqueness of a positive solution for a nonlinear fourth-order differential equation under Lidstone boundary conditions. Our analysis relies on a fixed point theorem in partially ordered sets.     

A discrete fourth-order Lidstone problem with parameters

Various existence, multiplicity, and nonexistence results for nontrivial solutions to a nonlinear discrete fourth-order Lidstone boundary value problem with dependence on two parameters are given, using a symmetric Green’s function approach. An existence result is also given for a related semipositone problem, thus relaxing the usual assumption of nonnegativity on the nonlinear term.     

On a new family related to truncated exponential and Sheffer polynomials

In this article, the truncated exponential and Sheffer polynomials are combined to introduce the 2-variable truncated-exponential based Sheffer polynomials (2VTESP) by using operational methods. Examples of certain special polynomials belonging to this family are considered. Operational correspondence between the 2VTESP and Sheffer polynomials is established, which is applied to derive the results for some members belonging to the 2VTESP family.